Proceedings of the 8th International Symposium on

F OF

IN THE LIGHT OF

NEW TECHNOLOGY

Edited by Sachio Ishioka • Kazuo Fujikawa

World Scientific

Proceedings of the 8th International Symposium on

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QUANTUM MECHANICS •

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Proceedings of the 8th International Symposium on

OF

IN THE LIGHT OF

NEW TECHNOLOGY BON—n

Advanced Research Laboratory

Hitachi, Ltd., Hatoyama, Saitama, Japan

22-25 August 2005

Edited by

Sachio Ishioka Advanced Research Laboratory Hitachi, Ltd., Japan

Kazuo Fujikawa Institute of Quantum Science Nihon University, Japan

\[p World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

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Proceedings of the 8th International Symposium (ISQM - Tokyo '05) FOUNDATIONS OF QUANTUM MECHANICS IN THE LIGHT OF NEW TECHNOLOGY

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The Eighth International Symposium on Foundations of Quantum Mechanics (ISQM-Tokyo '05)

Vll

PREFACE

The Eighth International Symposium on Foundations of Quantum Mechanics in the Light of New Technology (ISQM-TOKYO '05) was held on August 22-25, 2005 at the Advanced Research Laboratory, Hitachi, Ltd. in Hatoyama, Saitama, Japan. The symposium was organized by its own scientific committee under the auspices of the Physical Society of Japan, the Japan Society of Applied Physics, and the Advanced Research Laboratory, Hitachi, Ltd. Over 130 participants (23 from abroad) attended the symposium, and 30 invited oral papers, 7 contributed oral papers, and 37 poster papers were presented.

Just as in the previous seven symposia, the aim was to link the recent advances in technology with fundamental problems in quantum mechanics. It provided a unique interdisciplinary forum where scientists from different disciplines, who would otherwise never meet each other, convened to discuss basic problems of common interest in quantum science and technology from various aspects and "in the light of new technology."

Quantum Coherence, Decoherence, and Geometrical Phase was chosen as the main theme for the present symposium because of its importance in quantum science and technology. This topic was reexamined from all aspects, not only in terms of quantum computing, quantum information, and mesoscopic physics, but also in terms of the physics of precise measurement, spin related phenomena, and other fundamental problems in quantum physics. We were delighted that many active and eminent researchers in these fields accepted our invitation.

We are now very happy to offer the fruits of the symposium in the form of the proceedings to a wider audience. As shown in the table of contents, the proceedings include a special lecture on Einstein by Professor C.N. Yang, which was really well-timed with the World Year of Physics 2005, and 63 refereed papers in ten sections: quantum information and entanglement, quantum computing; quantum-dot systems, anomalous Hall effect and spin-Hall effect, spin related phenomena, superconductivity in nano-systems, novel properties of carbon nanotubes, novel properties of nano-systems, precise measurements, and fundamental problems in quantum physics. We will mention just some of the important key words here to give the flavor of the proceedings: quantum computation and communication, qubits, quantum dots, spintronics, mesoscopic spins, Berry phase, nanowires, Josephson junctions, and vortices in high-temperature superconductors. We hope that the proceedings will not only be the record of the symposium but also serve as a good reference book for experts on quantum coherence and decoherence and as an introductory book for newcomers in this field.

In conclusion, we thank the participants for their contribution to the symposium's success. Thanks are also due to all the authors who prepared manuscripts and to the referees who kindly reviewed the papers. We also thank the members of the Advisory Committee and Organizing Committee; without their invaluable assistance, the symposium would not have been a success. Finally, we would like to express our deepest gratitude to the Advanced Research Laboratory, Hitachi, Ltd. and its General Manager, Dr. Nobuyuki Osakabe, for providing us with financial support and an environment that was ideal for lively discussion. We also thank his staff members, in particular Mr. Yoshimasa Yamamoto and Ms. Maki Shinkai, for their efforts in making the symposium enjoyable as well as productive.

March 2006 Sachio Ishioka

Kazuo Fujikawa

IX

COMMITTEES

Chair:

H. Fukuyama, Tohoku University

Advisory Committee:

H. J. Kimble, California Institute of Technology S. Kobayashi, Tokyo University of Agriculture and Technology A. J. Leggett, University of Illinois at Urbana-Champaign J. E. Mooij, Delft University of Technology S. Nakajima, Superconductivity Research Laboratory,

International Superconductivity Technology Center N. Osakabe, Advanced Research Laboratory, Hitachi, Ltd. F. Wilczek, Massachusetts Institute of Technology C. N. Yang, Tsinghua University A. Zeilinger, Vienna University

Organizing Committee:

K. Fujikawa, Nihon University Y. lye, University of Tokyo N. Nagaosa, University of Tokyo Y. A. Ono, University of Tokyo F. Shimizu, University of Electro-Communications H. Takayanagi, NTT Basic Research Laboratories, NTT Corporation S. Ishioka, Advanced Research Laboratory, Hitachi, Ltd.

Sponsors:

The Physical Society of Japan The Japan Society of Applied Physics Advanced Research Laboratory, Hitachi, Ltd.

XI

CONTENTS

Preface vii

Committees ix

Opening Address H, Fukuyama 1

Welcoming Address N. Osakabe 3

Special Lecture

Albert Einstein: Opportunity and perception C.N. Yang

Quantum Information and Entanglement

Quantum optics with single atoms and photons H.J. Kimble 10

Quantum information system experiments using a single photon source Y. Yamamoto 16

Quantum communication and quantum computation with entangled photons A. Zeilinger 24

High-fidelity quantum teleportation and a quantum teleportation network for continuous variables

N. Takei, A. Furusawa 29

Long lived entangled states H. Hdffner, F. Schmidt-Kaler, W. Hansel, C. Roos, P.O. Schmidt, M. Riebe, M. Chwalla, D. Chek-al-Kar, J. Benhelm, U.D. Rapol, T. Korber, C. Becher, R. Blatt, 33

Quantum non-locality using tripartite entanglement with non-orthogonal states J. V. Corbett, D. Home 38

Quantum entanglement and wedge product H Heydari 42

xii

Analysis of the generation of photon pairs in periodically poled lithium niobate J. Soderholm, K. Hirano, S. Mori, S. Inoue, S. Kurimura 46

Generation of entangled photons in a semiconductor and violation of Bell's inequality G. Oohata, R. Shimizu, K. Edamatsu 50

Quantum Computing

Decoherence of a Josephson junction flux qubit Y. Nakamura, F. Yoshihara, K. Harrabi, J.S. Tsai 54

Spectroscopic analysis of a candidate two-qubit silicon quantum computer in the microwave regime

J. Gorman, D.G. Hasko, D.A. Williams 60

Berry phase detection in charge-coupled flux-qubits and the effect of decoherence H. Nakano, S. Saito, H. Takayanagi, R. Fazio 64

Locally observable conditions for the successful implementation of entangling multi-qubit quantum gates

H.F. Hofmann, R. Okamoto, S. Takeuchi 68

State control in flux qubit circuits: manipulating optical selection rules of microwave-assisted transitions in three-level artificial atoms

Y.-X. Liu, J.Q. You, L.F. Wei, C.P. Sun, F. Nori 72

The effect of local structure and non-uniformity on decoherence-free states of charge qubits

T. Tanamoto, S. Fujita 76

Entanglement-assisted estimation of quantum channels A. Fujiwara 80

Superconducting quantum bit with ferromagnetic 7t-Junction T. Yamashita, S. Takahashi, S. Maekawa 84

Generation of macroscopic Greenberger-Horne-Zeilinger states in Josephson systems T. Fujii, M. Nishida, N. Hatakenaka 88

Quantum-Dot Systems

Tunable tunnel and exchange couplings in double quantum dots S. Tarucha, T. Hatano, M. Stopa 92

Xlll

Coherent transport through quantum dots S. Katsumoto, M. Sato, H. Aikawa, Y. lye 99

Electrically pumped single-photon sources towards 1.3 um X. Xu, D.A. Williams, J.D. Mar, J.R.A. Cleaver 105

Aharonov-Bohm-type effects in antidot arrays and their decoherence M. Kato, H. Tanaka, A. Endo, S. Katsumoto, Y. lye 109

Nonequilibrium Kondo dot connected to ferromagnetic leads Y. Utsumi, J.Martinek, G. Schbn, S. Maekawa 113

Full counting-statistics in a single-electron transistor in the presence of strong quantum fluctuations

Y. Utsumi 117

Anomalous Hall Effect and Spin-Hall Effect

Geometry and the anomalous Hall effect in ferromagnets N.P. Ong, W.-L. Lee 121

Control of spin chirality, Berry phase, and anomalous Hall effect Y. Tokura, YTaguchi 127

Quantum geometry and Hall effect in ferromagnets and semiconductors N. Nagaosa 134

Spin-Hall effect in a semiconductor two-dimensional hole gas with strong spin-orbit coupling

J. Wunderlich, B. Kaestner, K. Nomura, A.H. MacDonald, J. Sinova, T. Jungwirth 140

Intrinsic spin Hall effect in semiconductors S. Murakami 146

Spin Related Phenomena

Theory of spin transfer phenomena in magnetic metals and semiconductors A.S. Nunez, A.H. MacDonald 150

Spin filters of semiconductor nanostructures T. Dietl, G. Grabecki, J. Wrobel 159

XIV

Experimental study on current-driven domain wall motion T. Ono, A. Yamaguchi, H. Tanigawa, K. Yano, S. Kasai 165

Magnetization reversal of ferromagnetic nano-dot by non local spin injection Y. Otani, T. Kimura 171

Theory of current-driven domain wall dynamics G. Tatara, H. Kohno, J. Shibata, E. Saitoh 177

Magnetic impurity states and ferromagnetic interaction in diluted magnetic semiconductors

M. Ichimura, K. Tanikawa, S. Takahashi, G. Baskaran, S. Maekawa 183

Geometrical effect on spin current in magnetic nano-structures M. Ichimura, S. Takahashi, S. Maekawa 187

Ferromagnetism in anatase Ti02 codoped with Co and Nb T. Hitosugi, T. Shimada, T. Hasegawa, G. Kinoda, K. Inaba, Y. Yamamoto, Y Furubayashi, Y. Hirose 191

Superconductivity in Nano-Systems

Nonlinear quantum effects in nanosuperconductors C. Carballeira, G Teniers, V.V. Moshchalkov, A. Ceulemans 194

Coalescence and rearrangement of vortices in mesoscopic superconductors A. Kanda, N. Shimizu, K. Tadano, Y. Ootuka, B.J. Baelus, F.M. Peeters, K. Kadowaki 200

Superconductivity in topologically nontrivial spaces M. Hayashi, T. Suzuki, H. Ebisawa, M. Kato, K. Kuboki 204

DC-SQUID ratchet using atomic point contact Y. Ootuka, H. Miyazaki, A. Kanda 208

Superconducting wire network under spatially modulated magnetic field H. Sano, A. Endo, S. Katsumoto, Y. lye 212

Simple and stable control of mechanical break junction for the study of superconducting atomic point contact

H. Miyazaki, A.Kanda, Y. Ootuka, T. Yamaguchi 216

Critical currents in quasiperiodic pinning arrays: One-dimensional chains and Penrose lattices

V. R. Misko, S. Savel'ev, F. Nori 220

XV

Macroscopic quantum tunneling in high-Tc superconductor Josephson junctions S. Kawabata 224

Novel Properties of Carbon Nanotubes

Carbon nanotubes and unique transport properties: Importance of symmetry and channel number

T. Ando 228

Optical processes in single-walled carbon nanotubes threaded by a magnetic flux J. Kono, S. Zaric, J. Shaver, X. Wei, S.A. Crooker, O. Portugall, G.L.J.A. Rikken, R.H. Hauge, R.E. Smalley 234

Non-equilibrium transport through a single-walled carbon nanotube with highly transparent coupling to reservoirs

P. Recher, N. Y. Kim, Y. Yamamoto 242

Novel properties of Nano-systems

Transport properties in low dimensional artificial lattice of gold nano-particles S. Saito, T. Arai, H. Fukuda, D. Hisamoto, S. Kimura, and T. Onai 246

First principles study of dihydride-chain structures on H-terminated Si(100) surface

Y. Suwa, M. Fujimori, S. Heike, Y. Terada, T. Hashizume 250

Electrical property of Ag nanowires fabricated on hydrogen-terminated Si(l 00) surface

M. Fujimori, S. Heike, T. Hashizume 254

Effect of environment on ionization of excited atoms embedded in a solid-state cavity M. Ando, C. Lee, S. Saito, Y. A. Ono 258

Development of universal virtual spectroscope for optoelectronics research: First principles software replacing dielectric constant measurements

T. Hamada, T. Yamamoto, H. Momida, T. Uda, T. Ohno, N. Tajima, S. Hasaka, M. Inoue, N. Kobayashi 262

Quantum Nerast effect H Nakamura, N. Hatano, R. Shirasaki 266

xvi

Precise Measurements

Quantum phenomena visualized using electron waves A. Tonomura 270

An Optical lattice clock: Ultrastable atomic clock with engineered perturbation H. Katori, M. Takamoto, R. Higashi, F.-L. Hong 276

Development of Mach-Zehnder interferometer and "coherent beam steering" technique for cold neutron

K. Taketani, H. Funahashi, Y. Seki, M. Hino, M. Kitaguchi, R. Maruyama, Y. Otake, H.M. Shimizu 282

Surface potential measurement by atomic force microscopy using a quartz resonator S. Heike, T. Hashizume 286

Fundamental Problems in Quantum Physics

Berry's phases and topological properties in the Born-Oppenheimer approximation K. Fujikawa 290

Self-trapping of Bose-Einstein condensates by oscillating interactions H. Saito, M. Ueda 294

Spinor solitons in Bose-Einstein condensates — Atomic spin transport J. Ieda 298

Spin decoherence in a gravitational field H. Terashima, M. Ueda 302

Berry's phase of atoms with different sign of the g-factor in a conical rotating magnetic field observed by a time-domain atom interferometer

A. Morinaga, H. Narui, A. Monma, T. Aoki 306

List of participants 309

Author Index 317

1

OPENING ADDRESS

HIDETOSHIFUKUYAMA Chair of the Organizing Committee. ISQM-TOKYO '05

Institute for Materials Research, Tohoku University Sendai-shi, Miyagi 980-8577, Japan

Good morning, ladies and gentlemen, and dear colleagues.

On behalf of the Organizing Committee of this conference, I would like to welcome all of you to this International Symposium on Quantum Mechanics in the Light of New Technology, ISQM-TOKYO'05.

This is the 8th of the series, which was started in 1983, when Professor Sadao Nakajima, then Director of the Institute for Solid State Physics, and Dr. Yasutsugu Takeda, then General Manager of Central Research Laboratory, Hitachi, Ltd., resonated in the idea to bridge between basic science and technology. Behind their decision, there was a beautiful experimental verification of the Aharonov-Bohm effect by Dr. Akira Tonomura, who, as far as I understand, received many inspiring suggestions and warm encouragements from Professor C.N. Yang to carry out this very difficult experiment.

Actually this experiment by Dr. Akira Tonomura was symbolic in demonstrating the fact that science and technology go hand in hand; namely, technological development can lead to further exploration of the mysteries of the nature and a new understanding of basic scientific facts leads to new technologies, or vice versa. Dr. Nakajima and Dr. Takeda had very good foresight when they started this activity. It is very fortunate for us to have Professor Nakajima here in this hall this morning, and we also expect to see Professor Yang later in the conference.

As I said before, this is the 8th symposium of the series, and it is true that science and technology have undergone tremendous

changes during these years. This is clearly seen from the program of the first conference. We can see the titles, such as Aharonov-Bohm Effect, Theory of Measurement, Quantum Tunneling, EPR paradox, Quantum Logic, Philosophy of Quantum Mechanics, Quantum Optics, Quantum Hall Effect, and so on; so many talks addressed to very basic but at the same time very abstract subjects. At that time people just talked about these kinds of things without worrying about the difficulties of realizing quantum coherence over macroscopic scales. Because of that, it was not easy for experimentalists to have creative interactions with theorists.

However, in this conference, many interesting experimental questions associated with the quantum coherence of not only electron waves but also of spins will be addressed. This trend will grow since scientific and technological interests are moving more and more toward materials on small scales, on the nano-scale, where obviously quantum coherence really manifests itself. It is my hope that this symposium will contribute to a deeper understanding of the properties of materials, which are governed by quantum mechanics, through stimulating discussion among colleagues who are interested in this subject but at the same time seldom meet each other because they are working in different fields. This conference, ISQM, is quite unique in that sense.

As I mentioned, this is the 8th of the series. When you look at the Chinese character eight, you can see its top part has a narrow space but at the bottom there is a wider space. The

2

bottom part is called "Sue" in Japanese, meaning "Future." The character "Eight" shows that the future is wide, bright, and holds many possibilities. Because of this particular structure, the number eight is considered to be a very good number. Then why not expect a very bright future for materials science where quantum coherence plays a crucial role? I hope all of you enjoy this conference.

Thank you very much for your attention.

3

WELCOMING ADDRESS

NOBUYUKIOSAKABE Advanced Research Laboratory, Hitachi, Ltd.

Hatoyama, Saitama 350-0395, Japan

Good morning, distinguished guests, ladies and gentlemen. On behalf of our laboratory, let me welcome you to Hatoyama and ISQM-TOKYO '05. It is a great pleasure for us to host this event for you all here today.

This symposium was initiated in 1983, as Professor Fukuyama explained. The motivation for Hitachi to host and sponsor the first ISQM came from Dr. Akira Tonomura's successful verification of the Aharonov-Bohm effect. In his study, new micro-fabrication technology developed in the semiconductor industry and an electron microscope technology developed at Hitachi were successfully combined to verify the important concept of the gauge field. In addition, through the strong thrust of Professor Chen Ning Yang, Hitachi decided to hold the first ISQM, i.e., the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology. The aim of the symposium has been to discuss the foundation of quantum mechanics achieved by using new technologies based on industrial innovation and to contribute to the scientific community in general.

Twenty two years have past since then. The site of the symposium has moved from the Central Research Laboratory in Tokyo to the Advanced Research Laboratory, Hitachi, Ltd., here at Hatoyama. Many new areas in physics have been discussed over the years. Today those fundamental quantum phenomena are now being used in the world of technology. Single electron charges can now controlled to make new semiconductor memories to breakthrough the barrier of modern device performance. Quantum entanglement will be used to secure future communication.

Macroscopic quantum tunneling and coherence will be the basis for quantum computing. The foundation of quantum mechanics will leverage industrial companies. Thus, because of all these possibilities, we found enormous value in hosting and sponsoring the symposium here again.

I hope all of you find the symposium rewarding. Before concluding, I would like to thank all the members of the Advisory and Organizing Committees for putting together such an exciting program. I also would like to thank all the invited speakers for coming to share their latest findings with the participants here. I very much look forward to hearing them.

Thank you very much for your attention.

4

ALBERT EINSTEIN: OPPORTUNITY AND PERCEPTION

CHEN NING YANG

Huang Ji-Bei & Lu Kai-Qun Professor Tsinghua University, Beijing 100084, China,

and Chinese University, Hong Kong

I

The year 1905 has been called Albert Einstein's

"Annus Mirabilis". It was during that year that

he caused revolutionary changes in man's

primordial concepts about the physical world:

space, time, energy, light and matter. How

could a 26-year-old clerk, previously unknown,

cause such profound conceptual changes, and

thereby open the door to the era of modern

scientific technological world? No one, of

course, can answer that question. But one can,

perhaps, analyze some factors that were

essential to his stepping into such a historic

role.

Fig. I Einstein as a Swiss patent clerk in 1905 when he revolutionized fundamental physics through the creation of the special theory of relativity.

First of all, Einstein was extraordinary lucky:

He was born at the right time, and was at the

peak of his creative powers when the world of

physics was shuddering from multiple crises.

In other words, there was the lucky opportunity

for him to change the course of physics, an

opportunity unmatched, perhaps, since the time

of Newton. Such lucky opportunities occur

very very infrequently. In E.T. Bell's Man of

Mathematics, Lagrange (1736-1813) was

quoted as having said:

Newton was assuredly the man of

genius par excellence, but we must

agree that he was also the luckiest: one

finds only once the system of the world

to be established.

Here Lagrange was referring to the words in

Newton's introduction to the third and final

volume of his great Principia Mathematica.

I now demonstrate the frame of the

system of the world.

While Lagrange was obviously envious of

Newton's lucky opportunity, we detect little

sentiment of a similar nature in what Einstein

had publicly said of Newton:

Fortunate Newton, happy childhood of

science... In one person he combined

the experimenter, the theorist, the

mechanic and, not least, the artist in

exposition. He stands before us strong,

certain, and alone...

5

Turning to Einstein's own times, he had the

opportunity to amend the system created by

Newton more than 200 years ago. This lucky

opportunity was of course open also to all

scientists of his time. Indeed electrodynamics

in a moving system had been a subject of

intense discussions since the Michaelson-

Morley experiment, first performed 1881,

repeated with greater precision in 1887.

Amazingly Einstein was already intensely

interested in this topic while still a student in

Zurich. He had written to his future wife

Milevain 1899:

/ returned the Helmholtz 's volume and

am now rereading Hertz's propagation

of electric force with great care

because I didn't understand

Helmholtz's treatise on the principle of

least action in electrodynamics. I'm

convinced more and more that the

electrodynamics of moving bodies as it

is presented today doesn't correspond

to reality, and that it will be possible to

present it in a simpler way.

[From Albert Einstein/Mileva Marie,

The love letters, Edited by Renn &

Schulmann, Translated by Smith.]

The search for this simpler way led, six years

later, to special relativity.

Many other scientists were also deeply

interested in the subject. Poincare (1854-1912),

one of the two towering mathematicians at the

time, was actively working on the same

problem. Indeed the name relativity was not

invented by Einstein. It was invented by

Poincare. One reads in his speech delivered

one year before 1905 (In Physics for a New

Century, AIP publication on History, volume 5,

1986):

The principle of relativity, according to

which the laws of physical phenomena

should be the same, whether for an

observer fixed, or for an observer

carried along in a uniform movement

of translation; so that we have not and

could not have any means of

discerning whether or not we are

carried along in such a motion.

This paragraph not only introduced the term

"relativity", but showed amazing insight which

is absolutely correct philosophically. However,

Poincare did not understand the full

implication in physics of this paragraph: Later

paragraphs in the same speech showed that he

failed to grasp the crucial and revolutionary

idea of the relativity of simultaneity.

Einstein was also not the first to write

down the great transformation formula:

x'=y(x-vt), y'=y, z'=z

1 Y"Vl-v2/c2

which had already been given by Lorentz

(1853-1928), after whom it was, and still is,

named. But Lorentz also failed to grasp the

revolutionary idea of the relativity of

simultaneity. He wrote later in 1915:

6

The chief cause of my failure was my

clinging to the idea that only the

variable t can be considered as the true

time and that my local time t' must be

regarded as no more than an auxiliary

mathematical quantity.

[Cf. Pais' biography of Einstein, p. 167]

I.e. Lorentz had the mathematics, but not the

physics, and Poincare had the philosophy, but

also not the physics. It was the 26-year-old

Einstein who dared to question mankind's

primordial concept about time, and insisted

that simultaneity is relative, thereby opening

the door to the new physics of the microscopic

world.

Almost all physicists today agree that it

was Einstein who had created special relativity.

Is that fair to Poincare and Lorentz? To discuss

this question let us quote from A. N.

Whitehead [in The Organization of Thought,

Greenwood Press, 1974, p. 127]:

To come very near to a true theory and

to grasp its precise application, are two

very different things, as the history of

science teaches us. Everything of

importance has been said before by

somebody who did not discover it.

Lorentz and Poincare indeed did seize the

lucky opportunity of the time, and had worked

very hard on one of the main problems, the

electrodynamics in a moving system. But they

both missed the crucial key point. They missed

because they had "clung" to old concepts, as

Lorentz himself later had said. Einstein did not

miss because he had a freer perception of the

meaning of space-time.

To have a free perception, one must

simultaneously be close to the subject under

investigation, and yet be able to examine it at a

distance. Indeed the often used term distant

perception shows the necessity of maintaining

a certain distance in any penetrating

discernment. But distant perception alone is

not enough. It must be matched by a detailed

close-up understanding of the problem at hand.

It is the ability to freely adjust, assess and

compare the close-up and distant views that

constitutes free perception. Pursuing this

metaphor, we might say that Lorentz had failed

because he had only a close-up view, while

Poincare had failed because he had only a

distant perception.

The great Chinese esthetician ^ % ?§

(1897-1986) had emphasized the importance of

"psychical distance" in artistic and literary

creativity. I think that idea is very much the

same as the distant perception discussed above,

but in another area of intellectual activity. In

the brilliant definitive scientific biography of

Einstein, Subtle is the Lord, by Pais, the author

chose one word to describe Einstein's character:

apartness, and quoted at the beginning of

Chapter 3:

Apart ...4. Away from others in action

or function; separately, independently,

individually.

f Oxford English Dictionary]

Indeed, apartness, distance, and free

perception are related concepts, referring to an

essential element in all human creativity, in

science, in art, and in literature.

Another historic achievement of

Einstein's in 1905 was his paper "0« a

7

heuristic point of view concerning the

generation and conversion of light" written in

March of that year. Historically this paper

launched the revolutionary idea of light as

quanta with discrete energy hv. The constant h

had already been introduced by Planck in 1900

in his bold theoretical study of black body

radiation. In subsequent years, however, Planck

got cold feet, and began to hedge. In stepped

Einstein in 1905, who not only did not hedge,

but pushed forward courageously with his

"heuristic point of view" of light quanta. That

this courageous push was not generally

appreciated can be gathered from the following

sentences in a document written by Planck,

Nernst, Rubens and Warburg, eight years later

in 1913, when they proposed Einstein for

membership in the prestigious Prussian

Academy:

In sum, one can say that there is hardly

one among the great problems in which

modern physics is so rich to which

Einstein has not made a remarkable

contribution. That he may sometimes

have missed the target in his

speculations, as, for example, in his

hypothesis of light-quanta, cannot

really be held too much against him,

for it is not possible to introduce really

new ideas even in the most exact

science without sometimes taking a

risk [From Pais, Subtle is the Lord,

1982,p.382]

Here the ridiculed "hypothesis of light-quanta"

referred to Einstein's bold proposal of 1905

mentioned above. Despite such general

derision, Einstein pushed further ahead, and in

papers of 1916-1917 established the value of

the momentum of the light quantum, leading

later to the epoch-making understanding of the

Compton effect in 1924.

The history of the birth of the

revolutionary idea of the light quanta can be

summarized as follows:

1905 Einstein's paper on E= hv

1916 Einstein's paper on P=E/c

1924 Compton effect

Throughout these years, before the Compton

effect was established in 1924, Einstein was

alone in his insightful perception, at a time

when entrenched conviction about waves was

sacred to the whole physics community.

Fig. 2 Einstein giving a lecture in 1922 at the College de France in Paris.

II

Between 1905 and 1924 Einstein's main

research interests were focused on the general

theory of relativity. As a scientific revolution

general relativity is unique in the history of

mankind. The grandeur of its conception, its

beauty, its sweeping scope, its spawning the

awesome science of cosmology, and the fact

that it was conceived and executed by one

8

single person, reminds me of the act of

creation in the old testament. (And I wonder

whether Einstein himself had thought of this

comparison.)

Fig. 3 Einstein in his study in his home on Haberlandstrafie in Berlin.

Of course, one would also naturally think

of other scientific revolutions, such as

Newton's Principia, special relativity, and

quantum mechanics. Some differences:

Newton's work had grandeur, had beauty, had

sweeping scope. Yes. But he had before him

the works of Galileo, of Kepler and of earlier

mathematicians and philosophers. He was not

alone at the time in searching for the law of

gravity. Special relativity and quantum

mechanics were both profound revolutions. But

they were hot topics worked on by many

people at the time. Neither was the creation of

a single person.

For general relativity, Einstein did not seize

any opportunity. He created the opportunity.

Alone, through deep perception, he conceived

the problem and after seven or eight years of

lonely struggle produced a new system of the

world of unimaginable beauty. It was an act of

pure creation.

Ill

General relativity represented the

geometrization of the gravitational field. It

quite naturally led to Einstein's push for the

geometrization of the electromagnetic field.

Thus was born his idea to formulate an overall

geometrization of all forces of nature, a unified

field theory, which gradually evolved into his

main research effort during the latter part of his

life. The last seminars that he gave, for

example, were in 1949-1950 at the Institute for

Advanced Study in Princeton, and the subject

was his latest attempt to incorporate the field

strengths F,,v of the electromagnetic field into

an unsymmetrical metric g,,v. This attempt, as

well as his earlier attempts in the same

direction, was also unsuccessful.

As a consequence of this lack of success,

and also because of the fact that, starting in the

late 1920s, he directed his attention almost

exclusively to this search, neglecting such

newly developing fields as solid state physics

and nuclear physics, he was often criticized,

even ridiculed. His devotion to the unified

field theory was called an obsession. An

example of this criticism is what I. I. Rabi

(1898-1988) had said in 1979 at the Einstein

centennial in Princeton:

When you think of Einstein's career

from 1903 or 1902 on to 1917, it was

an extraordinarily rich career, very

inventive, very close to physics, very

tremendous insights; and then, during

the period on which he had to learn

mathematics, particularly differential

geometry in various forms, he changed.

He changed his mind. That great

9

originality for physics was altered...

Was Rabi right? Did Einstein change?

The answer is, Einstein did change. Evidence

for this change can be found in his Herbert

Spencer lecture of 1933 bearing the title On the

Method of Theoretical Physics:

... the axiomatic basis of theoretical

physics cannot be extracted from

experience but must be freely

created...

Experience may suggest the

appropriate mathematical concepts,

but they most certainly cannot be

deduced from it...

But the creative principle resides in

mathematics. In a certain sense,

therefore, I hold it true that pure

thought can grasp reality, as the

ancients dreamed.

One may or may not agree with these very

concise statements, but one has to agree that

they powerfully and emphatically describe

Einstein's perception in 1933 about how to do

fundamental theoretical physics, a perception

that represents a profound change from his

earlier days.

So, Einstein did change. He himself was

keenly aware of this change. In the

Autobiographical Notes, published when he

was 70 years old, we read:

... and it was not clear to me as a

student that the approach to a more

profound knowledge of the basic

principles of physics is tied up with

the most intricate mathematical

methods. This dawned upon me only

gradually after years of independent

scientific work.

It is evident that in this passage the

independent scientific work was his long

struggle to formulate general relativity during

the period 1908-1915. That long struggle had

changed him. Did he change for the better?

Rabi would say: no, his changed perception

had become a futile obsession. We would say:

yes, his changed perception has altered the

future course of development of fundamental

physics:

Einstein's perception had permeated the

very soul of the research in fundamental

theoretical physics in the 50 years since

Einstein's death, serving as a lasting

testimonial to his courageous, independent,

obstinate and perceptive greatness.

Fig. 4 Einstein in Princeton

10

Q U A N T U M OPTICS WITH SINGLE ATOMS A N D PHOTONS

H. J. K I M B L E

Norman Bridge Laboratory of Physics MC 12-33

California Institute of Technology

Pasadena, CA 91125 USA

Across a broad front in physics, an important advance in recent years has been the increasing abil­ity to observe and manipulate the dynamical processes of individual quantum Systems. Within this context, an important physical system has been a single atom strongly coupled to the electromag­netic field of a high-Q cavity within the setting of cavity quantum electrodynamics (cavity QED). Another promising possibility is provided by the cooperative interaction of light with an atomic ensemble for writing and reading single quanta. Here, I present a brief overview of recent advances in the Quantum Optics Group at Caltech related to optical interactions with single photons and atoms and to applications in quantum information science.

Keywords: Quantum information; Quantum optics; Cavity QED.

1. Introduction

Because of several advantages, Quantum Op­tics continues to play an important role in the new science of Quantum Information, including for realizations of scalable quan­tum networks and for investigations of quan­tum dynamics of single quantum systems 1'2. Research in the Quantum Optics Group at Caltech addresses these themes by utilizing strong coupling in optical physics to control nonlinear optical interactions atom by atom and photon by photon. One application of this research is the implementation of a rudi­mentary quantum network, as illustrated in Figure 1 3 '4.

We are approaching the problem of con­trol of optical interactions on two different fronts. The first is within the setting of cav­ity quantum electrodynamics (cavity QED) with single atoms strongly coupled to the fields of high quality optical resonators. Al­though the prospects for quantum informa­tion processing via cavity QED are quite promising, several major scientific and tech­nical advances are required for the robust im­plementation of sophisticated quantum net­works. These challenges include control of

the external and internal degrees of free­dom for one atom strongly coupled to an optical cavity, the generation of entangle­ment among multiple cavities, and the labo­ratory realization of fault-tolerant protocols for quantum-state transformations. Much of our research within the arena of cavity QED is applicable to a broad spectrum of topics in Quantum Information Science, including fundamental primitives for general quantum-state synthesis and new protocols for the re­versible transfer of quantum states between different types of physical systems.

The second front for our research in­volves the collective interactions of ensem­bles of cold atoms with single photons. Here, we are developing the laboratory capabili­ties to implement the quantum network pro­tocols described in Ref. 5, including for entanglement-based quantum cryptography and teleportation of quantum states. We have also developed a theoretical protocol to realize a "heterogeneous" quantum network by entangling one atom in a cavity-QED sys­tem with a remotely located atomic ensem­ble.

Within this broad setting, recent labora-

11

Quantum Node

Quantum Channel iiQ

Fig. 1. The "quantum internet" composed of quantum nodes and quantum channels. The nodes are shown schematically as small quantum processing units based upon interactions in cavity QED, here illustrated with five atoms trapped inside a Fabry-Perot cavity (but which might incorporate atomic ensembles as well). Quantum channels distribute entanglement via photons in fibers, with the interface between light and matter made by way of quantum-state exchange protocols developed in Refs. 3 ' 4 .

tory advances include (1) the observation of the vacuum-Rabi spectrum for an individual atom trapped in an optical cavity 6, (2) the attainment of photon blockade for an optical cavity containing a single trapped atom 7, (3) a new experiment in cavity QED to achieve strong coupling of a single atom to the field of an ultrahigh-Q toroidal microresonator 8, (4) the generation of single photons from excita­tion stored in an atomic ensemble 9 ' 1 0 ' n and (5) the observation of measurement-induced entanglement between remote atomic ensem­bles 12. A brief summary of each of these activities follows.

2. Vacuum-Rabi spectrum for "one-and-the-same" atom

A cornerstone of optical physics is the in­teraction of a single two-level atom with the electromagnetic field of a high quality resonator. Of particular importance is the regime of strong coupling, for which the fre­quency scale g associated with reversible evo­lution for the atom-cavity system exceeds the rates (7, K) for irreversible decay of atom and cavity field, respectively. In the do­main of strong coupling, a photon emitted

by the atom into the cavity mode is likely to be repeatedly absorbed and re-emitted at the single-quantum Rabi frequency 2g be­fore being irreversibly lost into the environ­ment. This oscillatory exchange of excita­tion between atom and cavity field results from a normal mode splitting in the eigen­value spectrum of the atom-cavity system, which is manifest in emission and absorption spectra. The splitting has been dubbed the vacuum-Rabi splitting and serves as the hall­mark of strong coupling in cavity QED.

Strong coupling in cavity QED as ev­idenced by the vacuum-Rabi splitting pro­vides enabling capabilities for quantum in­formation science. Against this backdrop, experiments in cavity QED have made great strides over the past two decades to achieve strong coupling. However, without exception prior experiments related to the vacuum-Rabi splitting in cavity QED have required averaging over trials with many atoms (> 103

to 105 atoms) to obtain quantitative spectral information, even if individual trials involved only single atoms. By contrast, the imple­mentation of complex algorithms in quantum information science requires the capability

12

Fig. 2. A single atom is trapped inside an optical cavity in the regime of strong coupling by way of an intracavity FORT driven by the field £FORT 6- The transmission spectrum T\(wp) for the atom-cavity system is obtained by varying the frequency u>p of the probe beam £p and recording the output with single-photon detectors. Cooling of the radial atomic motion is accomplished with the transverse fields Q4, while axial cooling results from Raman transitions driven by the fields £FORT> ^Raman- An additional transverse field JJ3 acts as a repumper during probe intervals.

for repeated manipulation and measurement of an individual quantum system.

With this goal in mind, we have em­ployed the experimental arrangement shown in Fig. 2 and have succeeded in recording a complete probe spectrum for "one-and-the-same" atom 6. The vacuum-Rabi splitting has thereby been measured in a quantita­tive fashion for each atom in a sequence by way of a protocol that represents an impor­tant milestone in the development of cavity QED. Our experiment provides a critical first step towards more complex tasks in quan­tum information science, including, signifi­cantly, the realization of quantum networks. An essential component of this work is a new Raman scheme that we have developed for cooling atomic motion along the cavity axis, which leads to inferred atomic localization Az a i i a i ~ 33 nm.

3. Photon blockade

Sufficiently small metallic and semicon­ductor devices at low temperatures ex­hibit "Coulomb blockade," whereby charge transport through the device occurs on an electron-by-electron basis. In 1997, Imamoglu et al. proposed that an analogous effect might be possible for photon trans­port through an optical system by employ­ing photon-photon interactions in a nonlinear optical cavity 13>14. In this scheme, strong dispersive interactions cause the presence of a "first" photon within the cavity to block the transmission of a "second" photon, lead­ing to an ordered flow of photons in the trans­mitted field.

Although there is by now an extensive literature on photon blockade, this effect had not been previously observed. In Ref. 7

we reported the first observation of photon blockade by examining the light transmit­ted by an optical cavity containing one atom strongly coupled to the cavity field, as il-

13

Fig. 3. The upper panels illustrate the level structure used for implementation of the photon-blockade effect, while the lower panel offers a simple diagram of the experiment 7 . (a) Level diagram showing the lowest energy states for a two-state atom of transition frequency U>A coupled (with single-photon Rabi frequency go) to a mode of the electromagnetic field of frequency wc, with u ^ = u c = « o . Two-photon transmission is suppressed for a probe field uip (arrows) tuned to excite the transition |0) —> |1 , —), u>p = u>o — go. leading to sub-Poissonian statistics for the transmitted field, g '2 ' (0) < 1. (b) Eigenvalue structure for the {F = 4,mjj) «-» (F' = 5',m'F) transition in atomic Cesium coupled to two degenerate cavity modes ly,z- Although the eigenvalue structure is now much more complex than in (a), two-photon transmission is likewise blocked for excitation tuned to the lowest eigenstate (arrows), again leading to a more orderly flow of photons for the transmitted field (relative to the incident Poissonian photon stream), (c) Simple schematic of our experiment.

lustrated in Fig. 3. For coherent exci­tation at the cavity input, we investigated the photon statistics for the cavity output by way of measurements of the intensity correlation function g^(r), which demon­strate the manifestly nonclassical character of the transmitted field. Explicitly, we found 5(2)(0) = (0.13 ±0.11) < 1 with c/(2)(0) < #* 2 ) (T ) , SO that the output light is both sub-Poissonian and antibunched. Further, we ob­served that <?'2 '(T) rose to unity at a time T = TB — 45ns, which is consistent with the lifetime r_ = 48 ns for the first excited state

in the ladder of eigenstates associated with the blockade (see Fig. 3(a,b)). Over longer time scales, we found that the cavity trans­mission exhibited modulation arising from the oscillatory motion of an individual atom trapped within the cavity mode. We utilized this modulation to make an estimate of the energy distribution for the atomic center-of-mass motion and inferred a maximum energy E/kB ~ 250 ixK.

Our observations of photon blockade 7

advance optical physics beyond traditional nonlinear optics and laser physics into a new

14

Fig. -1. Scanning electron micrograph of a toroidal microcavity 8 . The principal and minor diameters are denoted by D and d, respectively. Our work at­tempts to t rap single Cesium atoms in the external evanescent field of the microcavity, as in the notional illustration in the figure.

regime with dynamical processes involving atoms and photons taken one by one.

4. Cavity Q E D wi th toroidal microcavities

In collaboration with the group of Professor K. Vahala at Caltech, we have undertaken an investigation of the suitability of toroidal mi­crocavities for strong-coupling cavity quan­tum electrodynamics, as illustrated in Figure 4. In Ref. 8 , we presented results from nu­merical modeling of the optical modes that demonstrate a significant reduction of the modal volume with respect to the whisper­ing gallery modes of dielectric spheres, while retaining the high-quality factors representa­tive of spherical cavities. Overall, we believe that the prospects for cavity QED in this setting are very promising. We have there­fore undertaken an experimental program to achieve strong coupling for individual Ce­sium atoms interacting with the evanescent fields of small toroidal resonators, including the construction of a new vacuum apparatus and associated optical components.

5. Scalable q u a n t u m networks wi th a tomic ensembles

As part of our efforts to realize quantum networks by way of the protocol of Duan, Lukin, Cirac, and Zoller 5, we have demon­strated the generation of single photons from an ensemble of cold Cesium atoms 9. Con­ditioned upon an initial detection from field 1 at 852 nm, a photon in field 2 at 894 nm is produced in a controlled fashion from ex­citation stored within the atomic ensemble. The single-quantum character of the field 2 is demonstrated by the violation of a Cauchy-Schwarz inequality, namely iu(l2,l2|li) = 0.24±0.05 £ 1, where w(l2 , l 2 | l i ) describes detection of two events (12,12) conditioned upon an initial detection l i , with w —* 0 for single photons and w —> 1 for coherent states.

We have also investigated the time de­pendence of nonclassical correlations for the two fields (1,2) generated by the atomic en­semble 10. The correlation function R(t 1^2) for the ratio of cross to auto-correlations for the (1,2) fields at times (£1, $2) is found to have a maximum value Rm&x = 292 ± 57, which significantly violates the Cauchy-Schwarz inequality R < 1 for classical fields. In our initial measurements, decoherence of quantum correlations was observed over r^ ~ 175 ns, as described by a model that we have developed. We have since implemented a new scheme to mitigate the effects of deco­herence, with coherence times now extended to Td ^ 10 /xs n . These are the first ob­servations of the temporal dependence of the joint probability pT(^i5^2) for the generation of correlated photon pairs from an atomic en­semble, which is critical for the protocol of Eef. 5. Our measurements of pT(t 1^2) are an initial attempt to determine the structure of the underlying two-photon wavepacket.

15

6. Entanglement of atomic ensembles

Quite recently, we have achieved entangle­ment between two atomic ensembles located on different tables in distinct apparatuses separated by 2.8 meters 12. Quantum inter­ference in the detection of a photon emit­ted by one of the samples projects the other­wise independent ensembles into an entan­gled state with one joint excitation stored remotely in 105 atoms at each site 5. Af­ter a delay of 1 fis, we confirm entanglement by mapping the state of the atoms to opti­cal fields, with measurements of these fields providing a quantitative lower bound for the entanglement of the ensembles. Our obser­vations provide a new capability for the dis­tribution and storage of entangled quantum states, including for scalable quantum com­munication networks 5.

7. Acknowledgements

This research is supported by the National Science Foundation, by the Advanced Re­search and Development Activity (ARDA), and by the Caltech MURI Center for Quan­tum Networks.

References

1. Quantum Computation and Quantum Infor­mation, M. A. Nielsen & I. L. Chuang (Cam­bridge University Press, Cambridge, United Kingdon, 2003).

2. The Physics of Quantum Information, D. Bouwmeester, A. Ekert, & A. Zeilinger (Springer-Verlag, Berlin, Germany, 2001).

3. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev. Lett. 78, 3221 (1997).

4. H.-J. Briegel, S. J. van Enk, J. I. Cirac, P. Zoller, in Ref. 2, pp. 192-197.

5. L.-M. Duan, M. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001).

6. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, Phys. Rev. Lett. 93, 233603 (2004).

7. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).

8. S. M. Spillane, T. J. Kippenberg, K. J. Va-hala, K. W. Goh, E. Wilcut, and H. J. Kim­ble, Phys. Rev. A 71, 013817 (2005).

9. C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble, Phys. Rev. Lett. 92, 213601 (2004).

10. S. V. Polyakov, C. W. Chou, D. Felinto and H. J. Kimble, Phys. Rev. Lett. 93, 263601 (2004).

11. D. Felinto, C.W. Chou, H. de Riedmat-ten, S.V. Polyakov, and H.J. Kimble, Phys. Rev. A 72, 053809 (2005); available as quant-ph/0507127.

12. C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, Nature (in press, 2005); available as quant-ph/0510055.

13. L. Tian and H. J. Carmichael, Phys. Rev. A 46, R6801 (1992).

14. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).

16

QUANTUM INFORMATION SYSTEM EXPERIMENTS USING A SINGLE PHOTON SOURCE

Y. YAMAMOTO

E. L. Ginzton Laboratory, Stanford University, Stanford, CA94305, USA

National Institute of Informatics, Tokyo, Japan

E-mail: yyamamoto@stanford.edu

This paper reviews the single photon sources based on semiconductor quantum dots and their applications to quantum information systems. By optically pumping a system consisting of a semiconductor single quantum dot confined in a monolithic microcavity, it is possible to produce a single photon pulse stream at the Fourier transform limit with a negligible jitter. This single photon source is not only useful for BB84 quantum key distribution (QKD), but also find applications in other quantum information systems such as Ekert91/BBM92 QKD and quantum teleportation gate for linear optics quantum computers.

Keywords: Single photon, Quantum information, Quantum indistinguishability

1. Introduction

In BB84 QKD [1], the security of systems using an ordinary Poisson light source can be threatened by an eavesdropper's photon splitting attack [2]. In the Ekert91 or BBM92 QKD using EPR-Bell photon pairs [3], additional bit errors occur due to the presence of two or more photon pairs per pulse [4]. Hopes are therefore pinned on the development of a light source that can generate single photons or single EPR-Bell photon pairs at a definite time.

Schemes have been proposed for implementing photonic quantum information processing techniques that are more advanced than QKD, such as quantum teleportation [5], quantum repeaters [6], and linear optics quantum computers [7]. For such applications, it is essential to generate a large number of single photons at definite time instances that satisfy the condition of "quantum indistinguishability".

Various methods are currently being investigated for the production of single photons at definite time instances, including those based on a single atom [8], atomic ensemble [9], single trapped ion [10], single molecules [11,12] or single solid-state lattice defects (diamond color centers) [13,14].

However, the results of these studies have yet to satisfy various conditions such as high efficiency, high speed and single-mode operation.

We have previously proposed and verified a method for producing single photons regularly by using the Coulomb blockade effect in a micro-pn junction [15,16]. An alternative way of generating single photons is the use of excitonic emission from a single semiconductor quantum dot excited by a pulse of light [17-20]. We will report in this paper BB84 QKD experiments using single photons generated in this way [21], conversion of indistinguishable single photons to entangled photon-pairs by post-selection [22] and quantum teleportation using indistinguishable single photons [23].

2. Generation of indistinguishable single photons

Figure 1(a) shows an AFM image of self-assembled InAs quantum dots in GaAs matrix produced by molecular beam epitaxy (MBE). Figure 1(b) shows an SEM photograph of a three-dimensional microcavity produced by using electron beam lithography and dry

17

etching. A single InAs quantum dot is embedded in the central optical cavity layer.

The lifetime of excitonic emissions from a quantum dot is normally 0.5-1 ns. On the other hand, the decoherence time of excitonic dipole caused by phonon scattering is about 1 ns at 4 K. That is, the natural width of spontaneous emissions is of about the same order as the homogeneous broadening caused by phonon scattering. If the lifetime of a single quantum dot can be reduced using the Purcell effect in a microcavity, then it should be possible that the temporal duration and

Fig. 1(a): AFM image of self-assembled InAs/GaAs quantum dots made by MBE deposition.

Fig. 1(b): GaAs/AlAs DBR micropost cavities enclosed in the center of InAs quantum dots produced by electron beam lithography.

spectral width of successively generated single photon pulses satisfy the Fourier transform limit. A single photon pulse stream of this sort should give rise to quantum interference phenomena as indistinguishable quantum particles.

Figure 2(a) shows the collision experiment for two single photons generated at 2 ns time intervals. The delay time of the Michelson interferometer was set to 2 ns, which is the same as the interval between the two single photon pulses. Figure 2(b) shows the joint counting probability with which the two detectors Tl and T2 detected the photons at 1=0 and t=% [20]. Peaks 1 and 5 in the central cluster centered on /=0 correspond to the case where the first photon takes a short path and the second photon takes a long path. Peaks 2 and 4 correspond to the case where the two photons both take a short path or both take a long path. The central peak 3 corresponds to the case where the first photon takes a long path and the second photon takes a short path, and only in this case are the two single photons incident on the output 50-50% beam splitter simultaneously. The fact that peak 3 is smaller than peaks 2 and 4 shows that quantum interference is actually occurring in the identical boson particle collisions. Figure 2(c) shows that this suppression' of the simultaneous counting probability reaches a maximum when the two pulses overlap completely, and disappears when the delay time exceeds the pulse width. The dip in coincidence counting probability at u=0 provides the quantum mechanical overlap V(T=0)~0.8 of the two single photon pulses [20].

3. Generation of EPE-Bell pairs with a single photon source

This experiment relies on two crucial features of our quantum dot single photon sources, namely its ability to suppress multi-photon

18

(a) Photon detector

-20 -10 0 tO 20

Delay time t«tj-t, (m)

200 0 200

Delay time (ps)

400

Fig. 2: Collision experiment involving two single photons generated with a 2 ns delay time difference. When the wave functions of the two single photons were overlapped at the output 50-50% beam splitter, the simultaneous counting probability was found to be lower due to two-photon interference.beam lithography.

pulses, and its ability to generate consecutively two photons that are quantum mechanically indistinguishable. The idea is to "collide" these photons with orthogonal polarizations at a non-polarizing beam splitter (NPBS). When the two optical modes corresponding to the output ports 'c' and 'd' of the NPBS have a simultaneous single occupation, their joint polarization state is expected to be the EPR-

Bell state:

HWP

SPCM V a, rQ-Hl-

fromQD

a H b a » _ | P \ \MEES1

Coincidence circuit = post-aeiection

Fig. 3(a): Experimental setup.

\v-)=-r(\H)c\v)d V2

\v)c\H)d)

The experimental setup is shown in Fig 3(a). A half wave plate was inserted in the longer arm of the interferometer to rotate the polarization to vertical. One time out of four, the first emitted photon takes the long path while the second photon takes the short path, in which case their wavefunctions overlap at the second non-polarizing beam-splitter (NPBS 2). Two single photon counter modules (SPCMs) in a start-stop configuration were used to record coincidence counts between the two output ports of NPBS 2, effectively implementing the post-selection. Single-mode fibers were used prior to detection to facilitate the spatial mode-matching requirements. They were preceded by quarter wave and polarizer plates to allow the analysis of all possible polarizations.

For given analyzer settings (a, /J), we

19

denote by C(a, p) the number of post-selected events normalized by the total number of coincidences in a time window. This normalization is independent of (oc, p) since the input of NPBS 2 are two modes with orthogonal polarizations. C(a, P) measures the average rate of coincidences throughout the time of integration.

A Bell's inequality test was performed for post-selected photon pairs. If we define the correlation function E(a, p) for analyser settings a and fi as:

E(a,p) =

C{a,p) +c(a ,/) -c(a ,/?) -c(a,{?)

C(a,/l)+c(a , / ) + c ( a , / ? ) + c ( « , / )

then local realistic assumptions lead to the inequality:

s =

\E (a, p)-B (a, fi)\ + \E (a, fi) + £ (a, fi')\ < 2

that can be violated by quantum mechanics. Sixteen measurements were performed for

all combination of polarizer settings among ae{0°, 45°, 90°, 135°} and Pe{22.5°, 67.5°, 112.5°, 157.5°}. The statistical error on S is quite large, due to the short integration time used to insure high stability of the QD device. Bell's inequality is still violated by two standard deviations, according to £-2.38+0.18. Hence, non-local correlations were created between two single independent photons by linear optics and photon number post-selection.

Complete information about the two-photon polarization state can be characterized by a reduced density matrix, where only the polarization degrees of freedom are kept. This density matrix can be reconstructed from a set of 16 measurements with different analyser settings, including circular [21]. We performed this analysis, know as quantum state tomography, on photon pairs. The reconstructed density matrix is shown in Fig.

3(b). It can be shown to be non separable, i.e. entangled, using the Peres criterion [22] (negativity -0.43, where a value of 1 means maximum entanglement).

We confirm that the observed degree of entanglement is consistent with the finite g2(0) value (-0.023) and V2(0) value (-0.8) [23].

4. Quantum teleportation with a single photon source

Following the idea of quantum teleportation gate [24], Knill, Laflamme and Milburn (KLM) [7] proposed a scheme for efficient linear-optics quantum computation (LOQC) based on the implementation of the controlled-sign gate (C-z gate) through teleportation. Since the C-z gate acts effectively on only one of the two modes composing the target qubit, a simplified procedure can be used where a single optical mode is teleported, instead of the two modes composing the qubit.

This procedure will be referred to as single mode teleportation to distinguish it from the usual teleportation scheme. In its basic version using one ancilla qubit, (i.e. two ancilla modes) this procedure succeeds half of the time. In its improved version using an arbitrarily high number of ancillas, it can succeed with a probability arbitrarily close to one [7].

We use quantum mechanically indistinguishable photons from a quantum dot single photon source. The single mode teleportation in its simplest form involves two qubits, a target and an ancilla, each defined by a single photon occupying two optical modes (see Fig. 4(a)). The target qubit can a priori be in an arbitrary state a\o)L +p\\)L where the logical | o) h and I \)L states correspond to the physical states | i\ I o>2 and I o\ \ i>2

respectively in a dual rail representation. The ancilla qubit is prepared with a beam-splitter (BS a) in the coherent superposition ^-(|o> I+|i>J = ^(|i>J|o>4+|o>J|i>4). One

20

Real component Imaginary component

Fig. 3(b): Reconstructed polarization density matrix for the post-selected photon pairs emitted by QD2.

rail of the target (mode 2) is mixed with one rail of the ancilla (mode 3) with a beam­splitter (BS 1), for subsequent detection in photon counters C and D. For a given realization of the procedure, if only one photon is detected at detector C, and none at detector D, then we can infer the resulting state for the output qubit composed of mode (1) and (4):

Vc = « |0>£ +/?|l), =a | l ) , |o) < + /?|o) l | l),

which is the initial target qubit state. Similarly, if D clicks and C does not, then the output state is inferred to be:

r» - *|o>4 -^ | i> , - a|i>.|o>4 -^)o>,| i)4

which again is the target state except for an

additional phase shift of 7t, which can be actively corrected for. Half of the time, either zero or two photons are present at counters C or D, and the teleportation procedure fails.

The success of the teleportation depends mainly on the transfer of coherence between the two modes of the target qubit. If the target qubit is initially in state I o)£ = 11), 10>2, then the ancilla photon cannot end up in mode (4) because of the postselection condition, so that the output state is always 11>, I o>4 as wanted.

target

ancilla BSa

Fig. 4(a): Schematic of single mode teleportation.

21

The same argument applies when the target qubit is in state 1i)j. However, when the target qubit is in a coherent superposition of |o>z

and | i) t, the output state might not retrieve the full initial coherence. We can test the transfer of coherence by preparing the target in a maximal superposition state:

where ^ is a phase that we can vary. If the initial coherence of the target qubit is not transferred to the output qubit, a change in ij> will not induce any measurable change in the output qubit. However, if changing </> induces some measurable change in the out qubit, then we can prove that the initial coherence was indeed transferred, at least to some extent.

The resulting coincidence counts were recorded for different phases tp of the target qubit. The result of the experiment is shown in Fig. 4(b) [25]. Complementary oscillations are clearly observed at counter A and at counter B, indicating that the initial coherence was indeed transferred to the output qubit. In other words, mode (2) of the target qubit was "replaced" by mode (4) of the ancilla without a major loss of coherence.

Detector A * Detector B

Voltage applied to piczo (V)

Fig. 4(b): Verification of single mode teleportation. Coincidence counts between detector A/C and B/C are plotted for different voltages applied to the piczo transducer, i.e. for different phase $ of the target qubit.

If the initial coherence was fully conserved during the single mode transfer, the state of the output qubit would truly be a|o), +pe"\i)L, and the single count rate at detector A (resp. B) would be proportional to cos2[-J (resp. sin2[-J ), giving a perfect contrast as the target phase <f> is varied. More realistically, part of the coherence can be lost in the transfer, resulting in a degradation of the contrast. Such a degradation is visible on Fig. 4(b). It arises mainly due to a residual dislinguishabilily between ancilla and target photons. Slight misalignments and imperfections in the optics also result in an imperfect mode matching at BS 1 and BS 2, reducing the contrast further. Finally, the residual presence of two-photon among pulses can reduce the contrast even more, although this effect is negligible here. The overlap v = j Vtarfanc between target and ancilla wave-packets, the two-photon pulses suppression factor g<2), as well as the non-ideal mode matching at BS 1 and BS 2 -characterized by the first-order interference visibilities l\, \\ - were all measured independently. The results are V ~ 0.75, g(2)(0) ~ 2%, r , ~ 0.92 and V2 ~ 0.91. The contrast C in counts at detector A or B when we vary the phase ^shouldbe:

This predicted value compares well with the experimental value of Ccxp ~ 0.6. The fidelity of teleportation is F = - ^ ~ 0.8 [25].

5. Quantum key distribution using single photons

Figure 5(a) shows a BB84 QKD test system using the single photon source [26], This system generates single photons at a repetition frequency of 76 MHz using a mode-locked Ti:Al203 laser. Alice uses random numbers generated in her computer to modulate the

'?:?.

polarization state to one of four states -horizontal (H), vertical (V), right-handed circular (R) or left-handed circular (L) - and sends it to Bob. Bob then splits the received photons into two paths with a 50-50% beam splitter and detects one path on an H-V basis and the other on an R-L basis. Since the photons are randomly divided between either of the output ports, this demodulation scheme is called passive demodulation. After this quantum transmission has taken place, Alice and Bob publicly compare the polarization bases they used, and store only the data for cases where they matched. Figure 5(b) shows a histogram of Alice and Bob's data [26]. As this figure shows, a strong correlation is formed when Alice and Bob use the same polarization base. After that, classical error correction is performed using a block coding technique. To increase the level of security so as to suppress the leakage of information to eavesdroppers to a level where it is ultimately negligible, classical privacy amplification is finally employed. Figure 5(c) shows how the repeater gain varies with the final key creation

rate per transmitted pulse in this system. In the present system, since the average number of photons sent out to the transmission path per pulse is 3x10 , it is inferior to schemes using an ordinary semiconductor laser in the region of small repeater gain (propagation loss), but in the high repeater gain region the small value of g(2|(0) makes it advantageous to use a single photon source. In the future it is expected that the theoretically predicted repeater gain of 60 dB will be assured by reducing optical losses inside a transmitter.

6. Future prospects

The generation of regulated single photons at the Fourier transform limit opened a door to various quantum information processing systems using photonic qubits. However, the present single photon source based on incoherent spontaneous emmision suffers from a non-ideal quantum mechanical overlap V(0)<1 as well as a finite multi-photon probability g(2)(0)it0. It is preferred that a future single photon source is based on

: Alice j

Lero pjohok

W»d*t«tOf; HaH-w.wetoi.gth plat*

Slit

JPbotoi

_ _Pump light

(a)

Channsl i Aftpbte "JSS^^Sm"

IS 20 25

Repeater gain (dB)

Fig. 5: An experimental BB84 quantum key distribution system using a single photon source, and the results obtained with this system

23

coherent deterministic process. When considering the future quantum

information technology, it will be necessary to encode information on the photonic qubits in quantum communication but store information on the nuclear spin qubits in quantum memory. Electron spins will no doubt play an important role as an interface between the two.

Acknowledgement

This work is partially supported by SORST on Quantum Entanglement (JST) and MURI on Photonic Quantum Information Systems (ARMY, DAAD19-03-1-0199).

References

1. C. H. Bennett and G. Brassard: Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 175 IEEE, New York, (1984)

2. N. Lutkenhaus: Phys. Rev. A 61, 2304 (2000)

3. A. K. Ekert: Phys. Rev. Lett. 67, 661 (1991); C. H. Bennett, G. Brassard and N. D. Mermin: Phys. Rev. Lett. 68, 557 (1992)

4. E. Waks, A. Zeevi and Y. Yamamoto: Phys. Rev. A 65, 052310 (2002)

5. P. G. Kwiat, K. Mattle, H. Weinfurther and A. Zeilinger: Phys. Rev. Lett. 75, 4337 (1995)

6. L. Duan, M. Lukin, J. Cirac and P. Zoller : Nature 414, 413 (2001)

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Santori, G. S. Solomon, and Y.Yamamoto: Phys. Rev. Lett. 92, 037903 (2004)

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24

QUANTUM COMMUNICATION AND QUANTUM COMPUTATION WITH ENTANGLED PHOTONS

ANTON ZEUJNGER

Institut fuer Experimentalphysik, Vienna University, Boltzmanngasse 5,A-1090 Vienna, Austria

IQOQI Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences,

Boltzmanngasse 3, A-1090 Vienna, Austria

E-mail: anton.zeilinger@univie.ac.at

Entangled photons provided the first evidence of a violation of Bell's inequality which implies a breakdown of the philosophical world-view of local realism. These experiments, having been extended over the years over ever longer distances, also provided the experimental foundation of the emerging field of quantum communication. There, quantum cryptography and quantum teleportation are key topics, which are already fairly advanced, being able to cover significant distances. Originally, it was thought that in a future quantum internet scheme, communication would be provided by photons while the local computers were to be realized with ions, atoms, atomic nuclei, solid state devices and the like. Interestingly, most recently the field of linear optics quantum computation has emerged with interesting possibilities and advantages.

Keywords: Quantum Communication; Quantum Computation; Entanglement.

1. Introduction

From a historical point of view, a most remarkable feature of entanglement is that it lay dormant for such a long time. There were very few reactions on the original Einstein-Podolsky-Rosen Paper1, most remarkably those of Niels Bohr2 and Erwin Schrodinger3. This may be seen for example by the number of citations which the EPR-Paper received (Fig. 1). The situation changed only in the 1960s and 1970s, after the discovery of Bell's theorem and after the first experiments4 which were made possible by the invention of the laser.

(0 c o S u

140

120

100

B0

60

4U

20

0£ »~w -~W.*- -••• , 1 1935 1945 1955 1965 1975 1985 1995 2005

year

Fig. 1. Number of citations of the EPR-Paper per year. In the beginning, the article was apparently ignored. It has gained wide recognition world-wide today.

From an information theoretic point of view, the most remarkable feature of entanglement is the fact that in a maximally entangled state, all information is encoded in joint properties of the individual systems while the individuals themselves carry no information whatsoever. This may for example be seen in the case of the antisymmetric polarization-entangled state of two photons.

Here and in the following, we assume the kets to be numbered from left to right. The state of equation 1 simply means that one has two photons, each one either horizontally (H) or vertically (V) polarized. They are orthogonal to each other in any basis, but before measurement, neither photon carries any information whatsoever.

2. Dense Coding and Cryptography

This property, that neither photon carries any polarization on its own, is essential in many applications, including entanglement-based quantum cryptography and quantum tele-

25

portation. The first application where this came to fruit is quantum dense coding5 as proposed by Bennett and Wiesner6. There, one is able to switch around between all four different Bell states, the maximal entangled states of two photons, Alice's and Bob's, by just manipulating Bob's photon alone. Alice, as the receiver, may then identify which of the four messages was sent by performing a suitable Bell state measurement. That way, she is able to identify which of the four Bell states was produced by Bob's manipulation, and thus receive the two bits of information sent by Bob.

In the first experiment where en­tanglement was put to work to transmit information in a way not possible with classical means, polarization-entangled photons were produced by type II down-conversion7. There, Bob can transfer between the four Bell states using XI2- and X14 -plates and Alice can then identify the Bell states by suitable polarization coincidence measurements. Since complete Bell state analyzers for two photons are hitherto unavailable, Alice was able to discriminate in the experiment between three possibilities only. Nevertheless, taking into account inefficiencies of the experiment, these resulted in a channel capacity of 1.6 bit per photon manipulated by Bob, which is significantly larger than the classical limit of 1 bit.

A conceptually rather interesting feature of this experiment is the property that it also constitutes a situation of quantum cryptography. The point is that by measuring Bob's photon alone, an eavesdropper has no possibility whatsoever to identify which

message was sent by Bob. This is because the density matrix of one photon alone represents a maximally mixed state. Therefore, as long as an eavesdropper has only access to Bob's channel, he has no possibility to identify the message sent.

3. Long-Distance Experiments

A most dramatic development in the time since the last ISQM is that quantum entanglement has left the laboratory. There exist now numerous different experiments utilizing quantum entanglement over large distances up to the order of 10 kilometers.

In terms of experimental stability, the most advantageous experiments are those where the entangled photons are distributed by glass fibers. This is because such fibers are usually put underground, and therefore such a communication is independent of external disturbances, in particular, the weather pattern. Distances of the order of 10 kilometers can easily be achieved as already demonstrated by the Geneva group8 in a test of Bell's inequality. In an experiment where the glass fiber ran under the river Danube9, quantum tele-portation over a distance of 600 meters could be demonstrated (Fig. 2). In that experiment, the maximally efficient quantum teleportation protocol possible with linear elements only was performed. At the receiver station, the unitary operation was applied to the arriving photon fast enough to form the necessary transformation, as suggested in the original protocol10.

26

Alice logic J

ad I h t\))\\\\\w\ ii I / / . Classical channel 1

f ""III) 600 m

Fig. 2. Long-distance quantum teleportation across the River Danube9. The quantum channel (fiber F) rests in a sewage-pipe tunnel below the river in Vienna, while the classical microwave channel passes above it. A pulsed laser (wavelength, 394 nm; rate, 76 MHz) is used to pump a barium borate (BBO) crystal that generates the entangled photon pair c and d and photons a and b (wavelength, 788 nm) by spontaneous parametric down-conversion. The state of photon b after passage through polarizer P is the teleportation input; a serves as the trigger. Photons b and c each arc guided into a single-mode optical-fiber beam splitter (BS) connected to polarizing beam splitters (PBS) for Bell-state measurement. Polarization rotation in the fibers is corrected by polarization controllers (PC) before each run of measurements. The logic electronics identify the Bell state as either y" (be) or y (be) and convey the result through the microwave channel (RF unit) to Bob's electro-optic modulator (EOM) to transform photon d into the input state of photon b.

Another recent application concerns the first bank transfer performed in the city of Vienna. There, all the information necessary to transmit during a banking transaction was encoded using a key generated by entangled photons. These photons were transmitted over underground optical fibers passing through the sewage system of Vienna.

An alternative for long distance quantum communication is sending the entangled photons through air via free-space links. The most promising avenue here is to finally exploit satellites. This is because then, one has to penetrate only a few kilometers of air, and the rest of the propagation is through vacuum. A distinct advantage of propagating through air is that there are no birefringent effects, that is, polarization is very well conserved during such communication. In a very recent experiment", it was actually possible to send

an individual entangled photon over a distance of 144 kilometers between two Canary Islands. Using such a long-distance link, it was feasible to actually encode some information in a quantum cryptography scheme.

The final goal will be to establish a source of entangled photons on a satellite or on the International Space Station ISS, and thus provide the possibility of exchanging quantum cryptographic keys between two arbitrary locations on Earth.

4. All-Optical Quantum Computation

It is well known that one can build a general quantum computer using a combination of different singlc-qubit-gates and one sufficiently complex two-qubit gale. Such a two-qubit-gate could, for example, be a CNOT-gate. The essential feature of all such two-qubit-gates is that one needs significant

27

interaction between two qubits. This means that the evolution of the second qubit must depend strongly on the quantum state of the first one.

While the quantum gates with interaction between two qubits are readily available in various atomic or solid state applications, this is not the case for photons. At present, no interaction exists between two photons which is strong enough, such that it can serve as the essential element in a two-qubit-gate. Nevertheless, it had been shown12 that with linear optical elements alone, one can build any quantum computer. The problem there remains that in such a linear optics scheme, the behavior of the algorithm is probabilistic. This means that only in a small fraction of cases, the optical quantum computer returns the calculation result desired.

This limitation has been shown to be overcome in a scheme proposed by Raussendorf and Briegel13. What they showed is that measurement of individual qubits together with active feed-forward can turn such a linear optics quantum computer into a deterministic one. The computer itself is based on cluster states, a specific class of multi-qubit entangled states.

In a recent experiment14, it was possible to demonstrate for the first time such cluster states for four qubits and implement some of the most basic schemes for one-way quantum computation. In that experiment, it was even possible to demonstrate successfully a Grover search algorithm15, where one tagged element out of four different ones was identified.

5. Outlook

It will be interesting to see to which extent long-distance quantum communication can be established using satellites in the near future. From a fundamental point of view, such ex­periments are interesting because they would eventually provide a test of quantum nonlocality over distances over the order of

v < | «*

Fig. 3. Experiment on all-optical one-way quantum computation 4. An ultraviolet laser pulse makes two passes through a nonlinear crystal (BBO), which is aligned to produce entangled photon pairs O" in the forward direction in modes a and b, and <D+ in the backward direction in modes c and d. Compensators (Comp), consisting of a half-wave plate (HWP) and an extra BBO crystal, are placed in each path to counter walk-off effects. Including the possibility of double-pair emission and the action of the polarizing beam-splitters (PBS), the four components of the cluster state can be prepared. The incorrect phase on the HHVV amplitude can easily be changed by using a HWP in mode a. The amplitudes can be equalized by adjusting the relative coupling efficiency of those photon pairs produced in the backward pass as compared to the forward pass. Polarization measurements were carried out in modes 1 to 4 using quarter-wave plates (QWPs) and linear polarizers (Pol) followed by single-mode fiber-coupled single-photon counting detectors behind 3-nm interference filters.

thousands of kilometers, a distance over which entanglement has not been tested at all thus far.

All-optical one-way computers in the next generation will include active feed-forward of the measurement result in such a way that further parameters in the computation are set, depending on earlier results. It is obvious that the main challenge for all-optical quantum computers is the development of sources of highly entangled multi-photon quantum states of detectors with high efficiency, which also would allow discriminating between different

28

numbers of photons. Nevertheless, all-optical quantum

computation provides the aspect of being extremely fast. Present technology would already allow one computational cycle, that is, one cycle of measurement and active feed­forward to be performed in less than 100 ns, and it is reasonable to expect that this can be pushed down into the 10 ns regime, which, if implemented on a grand scale, would imply a quantum computer with the order of 108

elementary calculations per second, a fascinating possibility.

Acknowledgments

This work was supported by the Austrian Science Fund (FWF) under project numbers SFB1520 and SFB1506, the European Space Agency under project QIPS, the European projects SECOQC and QAP, and by the ASAP program of the Austrian Space Agency (FFG).

References

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23, 807; 823; 844 (1935) [English translation by J. D. Trimmer, Proc. Am. Philos. Soc. 124, 323 (1980).]

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8. W. Tittel, J. Brendel, H. Zbinden and N. Gisin, Phys. Rev. Lett. 81, 3563 (1998).

9. R. Ursin, T. Jennewein, M. Aspelmeyer, R. Kaltenbaek, M. Lindenthal, P. Walther and A. Zeilinger, Nature 430, 849 (2004).

10. C. H. Bennett, G Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).

11. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Fiirst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter and A. Zeilinger, submitted for publication.

12. E. Knill, R. Laflamme, and GJ. Milburn, Nature 409, 46 (2001).

13. R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188(2001).

14. P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer and A. Zeilinger, Nature 434, 169 (2005).

15.L.K. Grover, Phys. Rev. Lett. 79, 325 (1997).

29

HIGH-FIDELITY Q U A N T U M TELEPORTATION A N D A Q U A N T U M TELEPORTATION NETWORK FOR CONTINUOUS VARIABLES

N O B U Y U K I TAKEI* a n d A K I R A FURUSAWA

Department of Applied Physics, School of Engineering,

The University of Tokyo, Tokyo 113-8656, Japan

CREST, Japan Science and Technology (JST) Agency, Tokyo 103-0028, Japan

*E-mail: takei@alice.tu-tokyo.ac.jp

We show the progress of continuous-variable quantum information processing, especially on re­alization of high-fidelity teleportation and entanglement swapping, and a quantum teleportation network.

Keywords: Quantum teleportation; Quantum entanglement; Entanglement swapping.

1. Introduction

Basic studies on quantum information sci­ence are being made with quantum optics intensively. Continuous-variable (CV) ap­proach attracts much interest because of rel­ative easiness to realize the experiments. Un­conditional quantum teleportation has been realized for the first time with this ap­proach 1, and various experimental successes of quantum teleportation and other protocols have been reported2,3 '4 '5 '6. In this paper, we show the progress of the CV approach, espe­cially on realization of high-fidelity telepor­tation and entanglement swapping7, a quan­tum teleportation network8'9.

2. High-fidelity quantum teleportation

We have demonstrated CV quantum telepor­tation beyond the no-cloning limit7. The no-cloning limit is defined with fidelity F = (V'inl/foutlV'in)11 and it is 2/3 for a coherent state input12. When the fidelity of teleporta­tion surpasses this limit, it warrants that the teleported state is the best remaining copy of the input state12. With this high-fidelity teleporter, one can teleport the nonclassical-ity of an input state. As an example, we have demonstrated teleportation of entanglement, namely, entanglement swapping7.

Fig. 1. Experimental setup for high-fidelity telepor­tation and entanglement swapping.

The experimental setup is shown in Fig. 1. Two pairs of bipartitely entangled EPR beams (EPR1,2) are created with four squeezed vacua from four subthreshold op­tical parametric oscillators (OPOs) and two beam splitters. One of the EPR1 beams is used for the input for the teleportation pro­cess and the other is used for the reference. The EPR2 beams are used for the resource of teleportation. A coherent state input is also used instead of the entangled beam in order to evaluate the fidelity. The fidelity 0.70±0.02 above the no-cloning limit of 2/3 is obtained7.

Figs. 2 and 3 show the experimen­tal results of teleportation of quantum entanglement7. Quantum entanglement be­tween the reference and the input and be-

30

-2.0

-4.0

(a) , (ii)

€mm*mxwm (Hi)

(b)

|Vv>^\Vy¥V^*v,y'u*l *>>irt»*>iif/*f#Mw*,Hl

tanglement between A and B as follows13'14,

(ii) (iii)

0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08 0.10

Time (s) Time (s)

Fig. 2. Correlation measurement for E P R l beams (a) The measurement result of the reference mode alone. Trace i shows the corresponding vacuum noise level ( ( A x ^ ) 2 ) = ( ( A p ^ ) 2 ) = 1/4. Traces ii and iii are the measurement results of ((Axref)

2) and {(Afiref)2), respectively, (b) The measurement result of the correlation between the input mode and the reference mode. Trace i shows the corre­sponding vacuum noise level {[A(x^J, — xin )]2) =

R(°> s(°)M2\ (IMPref + ? • „ ) ] > = V2- Traces ii and iii are the measurement results of ([A(xref — Xin)]

2) and ([A(pTef +pin)}

2), respectively.

0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.06 0.10

Time (s) Time (s)

Fig. 3. Correlation measurement results of the teleportation of quantum entanglement (a) The measurement result of the output mode alone. Trace i shows the corresponding vacuum noise level {(Ax<Z>t)

2) = <(Ap<°>t)2> = 1/4. Traces ii and

iii are the measurement results of {(Ax0ut)2) and ((Apout)2), respectively, (b) The measurement re­sult of the correlation between the output mode and the reference mode. Trace i shows the corre­sponding vacuum noise level ([A(xJ.e. — x0Jt)]

2) =

([A(p(°Jf + p^t)}2) = 1/2- Traces ii and iii are the measurement results of {[A(ccrey — x0ut )]2> and ([A(pref +pout)}2), respectively.

tween the reference and the output are checked with the sufficient condition for en-

( [ A ( X A - X B ) ] 2 ) + ( [ A ( P A + P B ) ] 2 ) < 1

(1)

where an electromagnetic field mode is rep­resented by an annihilation operator a whose real and imaginary parts (d = x + ip) corre­spond to quadrature-phase amplitude oper­ators with the canonical commutation rela­tion [x,p] = i/2 (units-free, with h = 1/2). The results satisfy the conditions as shown in Figs. 2 and 3 and it follows that telepor­tation of entanglement is succeeded.

3. A quantum teleportation network

classical information

^ ^ output

\\f>

Fig. 4. A quantum teleportation network.

We have also demonstrated a multipar­tite protocol, i.e., a quantum teleportation network. The heart of the protocol is mul­tipartite entanglement. In the experiment, we use three entangled light beams (modes A, B, and C) which can be verified with the inequalities listed below15,

I. ([A(xA - xB)]2>

+([A(PA + PB + 9cPc)}2) < 1 ,

<[A(xB -x c ) ] 2>

+([A{gApA+pB+pc)}2)<l,

( [ A ( x c - x A ) ] 2 )

+ ([A(pA + SBPB + pc)]2) < 1,

(2)

I I

III

31

where g&, g&, and 3c are some real num­

bers, and dj = £j + ipj is a field mode oper­

ator. For a quantum teleportation network

which is shown in Fig. 4, we set all g^s to

unity and the left hand sides of all inequal­

ities much smaller than one. In this case,

there is no bipart i te entanglement and we

need information from the third member for

success of teleportation between any combi­

nations of members.

Fig. 5 shows the experimental setup for

a quan tum teleportation network. Tripar-

titely entangled beams are created with three

squeezed vacua and two beam splitters. The

entangled beams satisfies all inequalities (2)

with all gj 's unity.

Fig. 6 shows the information gain depen­

dences of teleportation fidelities, where the

information gain corresponds to the amount

of information from the third member and

gi = 0 means no information and gi = 1

means all information. In the case of coher­

ent s tate inputs , the classical limit of the fi­

delity is F = 0.5 16 . In other words, the

fidelity of F > 0.5 means success of telepor­

tat ion. From the figure, we succeed in tele­

por ta t ion in three different combinations and

it shows success of the construction of a tele­

por ta t ion network. Note tha t teleportation

between any combinations fails without the

help of the third member, which corresponds

to the case of g* = 0 in this protocol. This

is because there is no bipart i te entanglement

between any combinations.

4 . S u m m a r y

We have demonstrated high-fidelity telepor­

ta t ion and entanglement swapping, a quan­

t u m teleportation network. By using the de­

veloped technology, we are now working on

realization of quantum error correction1 7 .

A c k n o w l e d g e m e n t s

We acknowledge T. Aoki and H. Yonezawa

for their contributions. This work was par t ly

coherent stale Tripartite entanglement source

Fig. 5. Experimental setup for a quantum telepor­tation network. This is the case for teleportation from Alice to Bob with the help of Claire.

supported by the M E X T and the M P H P T of

Japan .

R e f e r e n c e s

1. A. Furusawa et al., Science 282, 706 (1998). 2. W. P. Bowen et al., Phys. Rev. A 67, 032302

(2003). 3. T. C. Zhang et al., Phys. Rev. A 67, 033802

(2003). 4. X. Li et al., Phys. Rev. Lett. 88, 047904

(2002). 5. J. Jing et al., Phys. Rev. Lett. 90, 167903

(2003). 6. J. Mizuno et al., Phys. Rev. A 71 , 012304

(2005). 7. N. Takei et al., Phys. Rev. Lett. 94, 220502

(2005). 8. P. van Loock and S. L. Braunstein, Phys.

Rev. Lett. 84, 3482 (2000). 9. H. Yonezawa et al., Nature 431, 430 (2004).

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11. S. L. Braunstein et al., J. Mod. Opt. 47, 267 (2000).

12. F. Grosshans and P. Grangier, Phys. Rev. A 64, 010301(R) (2001).

13. L.-M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000).

14. R. Simon, Phys. Rev. Lett. 84, 2726 (2000). 15. P. van Loock and A. Furusawa, Phys. Rev.

A 67, 052315 (2003). 16. K. Hammerer et al., Phys. Rev. Lett. 94,

022321 (2005). 17. S. L. Braunstein, Nature 394, 47 (1998).

32

0.7

0.65

0.6

( 0.55

0.5

0.45

0.4

0.35

0.3

0.7

0.65

0.6

0.55

0.5

0.45

0.4

Classical Limit f i^iiH+^;

• •

0

b

0.2 0.4 0.6 gain gc

0.8 1

Classical Limit

0.2 0.4 0.6 0.8 gain g B

0.2 0.4 0.6 0.8 1 gain g A

Fig. 6. Information gain dependences of teleporta­tion fidelities. (a)The fidelity of teleportation from Alice to Bob with the help of Claire. (i)Teleportation without entanglement. (ii)Teleportation with entan­glement. The solid curves represent the theoreti­cal ones calculated from the experimental conditions. (b)The fidelity of teleportation from Alice to Claire with the help of Bob. (c)The fidelity of teleportation from Claire to Bob with the help of Alice.

33

LONG LIVED ENTANGLED STATES

H. HAFFNER1'2, F. SCHMIDT-KALER1'*, W. HANSEL1, C. ROOS1'2, RO. SCHMIDT1, M. RIEBE\M. CHWALLA1, D. CHEK-AL-KAR1 J. BENHELM2, U.D. RAPOL2,

T. KORBER1, C. BECHER1^, AND R. BLATT1-2

Institut fur Experimentalphysik, Innsbruck, Austria

Institut fur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften, Austria

Institut fur Theoretische Physik, Innsbruck, Austria

* Present address: Abteilung Quanten-Informationsverarbeitung, Universitat Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany

* Present address: Fachrichtung Technische Physik, D-66041 Saarbriicken, Germany

We investigate entangled states of two trapped 40Ca"1" ions. By encoding the quantum information in the Zeeman-manifold of the ground state, we observe entangled states lasting for more than 20 seconds.

Keywords: entanglement; quantum information; ion traps

1. Introduction

Entanglement is among the most counter­intuitive concepts in quantum mechanics. Still, it is believed to be the ingredient mak­ing a quantum computer1 much more power­ful than any classical machine. However, be­cause of the fragility of entanglement physi­cists widely assume that it is very hard —if not impossible— to construct such a quan­tum computer2. The reason is that any in­teraction with the environment which distin­guishes between the entangled sub-systems collapses the quantum state3. The notion of entanglement is also strongly linked with the so-called Schr6dinger-cat paradox: here, ab­surd consequences arise when a macroscopic object (a cat) is put in a superposition of two states (alive or dead). Such situations clearly do not comply with our notion of (macro­scopic) reality. Usually this observation is ex­plained with the collapse of the superposition states caused by the unavoidable entangle­

ment with the object's environment. There­fore the decoherence properties of entangled states play a central role in understanding the emergence of our classical world from quantum mechanics3. Consequently, there is a strong interest and need in generating entangled states and investigating their co­herence properties in well controlled physical systems.

2. Experimental Setup, Procedures and Results

For our experiments two Calcium-ions are suspended in free space inside a Paul trap with electromagnetic forces4. Superpositions of the two Zeeman levels of the S1/2 ground state represent the qubits. Each ion-qubit in the linear string is individually addressed by a series of tightly focused laser pulses on the S1/2 <—> D5/2 auxilary quadrupole tran­sition employing narrowband laser radiation near 729 nm. Doppler cooling and subse-

34

quent sideband cooling prepare the ion string in the ground state of the center-of-mass vi­brational mode. Optical pumping initializes the ions' electronic qubit states in the ( S ^ , mj = — l/2)-state. After preparing the en­tangled state with a series of laser pulses, the quantum state is read out by observing the laser induced flourescence with a photo mul­tiplier tube and/or with a CCD camera.

A sequence of three laser pulses address­ing the ions individually creates the entan­gled Bell state5

\*) = (\SD) + \DS))/V2 (1)

from the initialized ions. Here \S) and \D) re­fer to the internal states of each ion (see level scheme in Fig. 1). Via state tomography5 we

D& 2

_ e- ~~

Fig. 1. Qubit levels of 4 0 Ca+: While the two-level system ( S 1 / 2 , mj = - 1 / 2 <->D5/2, mj = - 1 / 2 ) decoheres due to spontaneous decay4, the electron spin in the S-state (Sj/2, mj = —1/2 <-> S j / 2 , mj = +1 /2) provides a stable quantum memory. This qubit is encoded via the S-D qubit and using a 7r-pulse to transfer the D-state population to the S-state manifold.

find an overlap of the experimentally gener­ated state with the ideal one (the fidelity) of up to 96%. For ^ we obtain coherence times of about 1 s, consistent with the fundamen­tal limit set by the spontaneous decay from the D5/2-level5. The long survival time of ^ is due to the fact that the constituents of the

superposition have the same energy and are thus insensitive to fluctuations common to both ions (e.g. laser frequency and magnetic field fluctuations). Similar results have been obtained with Bell states encoded in hyper-fine levels of Beryllium ions6'7.

As spontaneous emission ultimately lim­its the coherence time of entangled states en­coded in the {S,D}-state basis, one can en­hance the coherence time by transferring the populations into a basis spanned by the two Zeeman sub-levels of the Sx/2 ground state: |Si/2, mj = +1/2) = |0> and |S 1 / 2 ,m 7 = — 1/2) = |1). These states do not decay spon­taneously. In addition, we choose a superpo­sition with equal energies, e.g. [01) + |10), in the "decoherence free subspace"6. With this approach, we extend the coherence of the en­tangled state by more than one order of mag­nitude. In particular, we coherently transfer —just after the entangling operation— the population of the |D5/2 , m j = —1/2) state to the |0) state, while leaving the |1) population untouched (see Fig. 1). The fidelity of the re­sulting Bell state |*'> = (|01) + |10» /y/2 is 89 % where the loss of 7 % is due to imper­fect transfer pulses. To investigate decoher­ence, we insert a variable delay time before analyzing the state |\&'). After a delay of 1 s, full state tomography reveals that the fidelity of the entangled state is still 86 %. We plot the respective reconstructed density matrix in Fig. 2. Since the tomographic re­construction of the full density matrix re­quires many experimental cycles (« 1000), it is of advantage to employ a fidelity mea­sure that is based on a single density matrix element and thus is easier to access. Indeed, to determine a lower bound of the fidelity, it is sufficient8 to measure the density ma­trix element (01[yo|10) = poi.iOi as shown in the following: The fidelity of the entangled state |* ') = ( |01)+e^ |10)) /v /2 is given by F = {y'\p\*')=(poi,oi + Pio,io)/2 + Re (poi.ioe1^). In our implementation of the state 1 ') a magnetic field gradient leads to

35

0.5 >

0.4 -

0.3 - i-

IPlo.2-

0.1- ;

-0.1 - 1 • 00 " v .

Fig. 2. Tomographically reconstructed density ma­trix of the Bell state | * ' ) = (|01) + |10» /V2 after a waiting time of 1 s. The overlap with the ideal Bell state has dropped from 89% directly after prepara­tion to 86% after 1 s waiting time.

a deterministic evolution of the phase <j> in

time (see below). If we take this rotation

into account (i.e. determine the phase 0), the

fidelity becomes F = (poi.oi + Pw,io) /2 +

IPOI.IOI- Using the general property of the

density matrix y'poi.oi • Pio.io > boi.iol and

the triangular inequality, we obtain the lower

bound to the fidelity: F > Fm jn = 2 |/?oi,io|-

We plot the experimentally determined val­

ues for Fmin in Fig. 3. and find that the fi­

ll 1

0.8 4i—L_^_^ T T

0.2-

°0 5 10 15 20 25 30 Time (s)

Fig. 3. Minimum fidelity of the Bell state as a func­tion of the delay time as inferred from the density matrix element (01|p|10) (see text). A fidelity of more than 0.5 indicates the presence of entangle­ment. The line is a Gaussian fit to the data.

delity is larger than 0.5 for up to 20 s; thus at least up to this time the ions were still entangled8.

We considered the following reasons for the observed decay of entanglement: slow fluctuations of the magnetic field gradient which affect the ions differently, heating of the ion crystal, residual light scattering, and collisions. The latter two were excluded ex­perimentally: The scattering of residual light at 397 nm (S1/2 — P1/2 transition) which could induce a bit flip between the states |0) and |1) was measured by preparing the ions in |0) or |1). After waiting times ranging from 2 s to 18 s we probed the state with 7T-pulses from the |5i/2, m j = ±1/2) state to the \D5/2,mj = —1/2) state. We could not detect any bit flip transitions and conclude that on average less than 1 photon is scat­tered in 8 minutes. The pressure in our ion trap is less than 2 x 10~ n mbar and leads to an estimated rate of elastic collisions with the background gas of less than 3x 10~3 s~19. The heating rate of the ion crystal was mea­sured to be 1 phonon per second and vibra­tional mode. From the measured heating, we expect a degradation of our analysis pulses as the Rabi frequency depends on the motional state of the ion string. Thus we would ob­serve a decrease in the fidelity by 0.1 after 20 s. This effect, however, does not com­pletely explain our fidelity loss. A magnetic field gradient across the ion trap lifts the en­ergy degeneracy of the two parts of the su­perposition by AE = h x 30 Hz such that the relative phase <j> of the superposition evolves as <f>(t) = AEt/H . Thus relative fluctua­tions of the gradient by 10~3 within the mea­surement time of up to 90 minutes per data point could explain the observed decay rate of |(01|p|10)|. Generally, a slow dephasing mechanism (as compared to the experimen­tal cycle) such as fluctuations of the magnetic field gradient leads to a Gaussian decay of the coherence. A Gaussian fit to the data in Fig. 3 yields a time constant of 34(3) s for

36

the loss of coherence of the entangled state.

3. Discussion and Outlook

Previous experiments with single trapped Be+-ions have demonstrated that single par­ticle coherence can be kept for more than 10 minutes 10. Here we show that also entan­gled states can be preserved for many sec­onds: the two-ion Bell states in our investi­gations outlive the single particle coherence time of about 1 ms in our system11 by more than 4 orders of magnitudes. Even in the presence of an environment hostile for a sin­gle atom quantum memory, the coherence is preserved in a decoherence free subspace 6 . But more importantly, the coherence was kept non-locally, i.e. over distances which are many orders of magnitude larger than atomic or molecular dimensions.

With this at hand, we envision quan­tum computers using these long-lived entan­gled states for quantum memory and for ex­tended quantum information processing. For this one can encode one qubit in two ions. The challenge will be to process the infor­mation in this quantum memory without losing the protection from dephasing as it would happen with standard two-qubit gates (such as the Cirac-Zoller gate). A promis­ing approach is the use of off-resonant exci­tations. Single qubit operations on the log­ical qubits for instance can then be carried out by Raman transitions between the ba­sis states |0) = \SD) and |1) = \DS). As an example for such a single qubit operation in the logical basis, we show in Fig. 5 that starting from the |S'I?)-state (logical |0}), one can create |0 + 1) = \SD + DS) by apply­ing a single pulse to both ions simultaneously (see Ref. [12,13] for a more detailed explana­tion). In this experiment, the frequency of the pulse is detuned from the blue sideband by 10 kHz (see Fig. 4). In this way, we en­sure that the motional state is occupied only virtually but still transmits the coupling be­

tween the ions. The detuning also ensures that the total electronic excitation of the ions is conserved. The subsequent analysis with

~ ~ / » V ~ |DD,1> / \ |DD,0>

|SD,1> / \ - |DS,1> |SD,0> * * |DSt0>

|SS,1> |SS,0>

Fig. 4. Raman coupling between \SD) and \DS). The laser illuminates both ions simultaneously. The dashed line indicates the detuning of the laser from the blue sideband (the \DD, l)-level).

two individually addressed carrier-n/2 pulses shows (Fig. 5), that the operation indeed is coherent. In addition, we also observed Rabi-flopping between |0) and |1). Thus,

Phase (JI)

Fig. 5. Detection of a Bell state created with a sin­gle laser pulse addressed to both ions. For the anal­ysis, we applied a 7r/2-pulse to each ion and scanned the phase of one of the pulses. The contrast of the oscillation implies a minimal fidelity of 0.86 of the 7r/2-pulse.

only twice the resources (qubits + elemen­tary quantum gates in the decoherence free subspace) establish a basis with which, in principle, up to 4 orders of magnitude more operations as compared to the current setup can be realized before the quantum informa­tion is lost to the enviroment.

A c k n o w l e d g m e n t s

We gratefully acknowledge support by

the European Commission (CONQUEST

(MRTN-CT-2003-505089) and QGATES

(IST-2001-38875) networks), by the ARO

(No. DAAD19-03-1-0176), by the Austrian

Fonds zur Forderung der wissenschaftlichen

Forschung (FWF, SFB15), and by the Insti-

tu t fur Quanteninformation GmbH. T.K. ac­

knowledges funding by the Lise-Meitner pro­

gram of the F W F .

R e f e r e n c e s

1. M. A. Nielsen, I. L. Chuang: Quantum Com­putation and Quantum Information, (Cam­bridge University Press, Cambridge, 2000)

2. S. Haroche, J. M. Raimond: Physics Today 49, 51 (1996)

3. W.H. Zurek: Physics Today 44, 36 (1991) 4. F. Schmidt-Kaler, H. Haffner, S. Guide, M.

Riebe, G.P.T. Lancaster, T. Deuschle, C. Becher, W. Hansel, J. Eschner, C.F. Roos, R. Blatt: Appl. Phys. B 77, 789 (2003)

5. C. F. Roos, G. P. T. Lancaster, M. Riebe, H. Haffner, W. Hansel, S. Guide, C. Becher, J. Eschner, F. Schmidt-Kaler, R. Blatt: Phys. Rev. Lett. 92, 220402 (2004)

6. D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland: Science 291, 1013 (2001)

7. C. Langer, R. Ozeri, J. D. Jost, J. Chi-averini, B. DeMarco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B. Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosenband, T. Schaetz, P. O. Schmidt, D. J. Wineland: arXiv:quant-ph/0504076 (2005)

8. C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, C. Monroe: Nature 404, 256 (2000)

9. D.J. Wineland, C.R. Monroe, W.M. Itano, D. Leibfried, B.E. King, D.M. Meekhof: J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998)

10. J.J. Bollinger, D.J. Heinzen, W.M. Itano, S.L. Gilbert, D.J. Wineland: IEEE Trans. In-strum. Meas. 40, 126 (1991)

11. F. Schmidt-Kaler, S. Guide, M. Riebe, T. Deuschle, A. Kreuter, G. Lancaster, C. Becher, J. Eschner, H. Haffner, R. Blatt : J. Phys. B: At. Mol. Opt. Phys. 36, 623 (2003)

12. F. Schmidt-Kaler, H. Haffner, S. Guide, M.

37

Riebe, G. Lancaster, J. Eschner, C. Becher and R. Blatt, Europhysics Letters 65, 587 (2004).

13. A. S0rensen and K. M0lmer Phys. Rev. Lett. 82, 1971 (1999).

38

Q U A N T U M NON-LOCALITY USING TRIPARTITE ENTANGLEMENT WITH NON-ORTHOGONAL STATES

J .V. C O R B E T T

Mathematics Department, Macquarie University, Sydney, NSW 2019, Australia

E-mail: jvc@ics.mq.edu.au

D. H O M E

Department of Physics, Bose Institute, Kolkata 100 009, India

E-mail: dhom@bosemain. boseinst. ac. in

A system of three particles in an entangled state is used to display a non-Bell type of quantum non-locality. The measured values of particle 1 are changed by changing the state of particle 3 even though the preparation of 3 and the measurement of 1 occur at spatially separated locations and the particles never interact. The third particle, particle 2, has non-orthogonal states which do not change during the process. Therefore the effect cannot be explained in terms of a classical intervention mediated by particle 2.

Keywords: Entangled states; Einstein Locality; Non-orthogonal states.

1. Introduction

A system of three particles in an entangled state is used to display a quantum non-local effect, a direct violation of Einstein locality. The measured values of particular properties of particle 1 are affected by changes in the state of particle 3 even though the prepara­tion of 3 and the measurement of 1 occur at locations that are spatially separated and the particles never interact.

Particle 2, the third particle, has non-orthogonal states. Thus the tripartite en­tangled state is not Schmidt decomposable and not asymptotically equivalent to a GHZ state5.

The influence of particle 3 on particle 1 cannot be explained in terms of any classical intervention mediated by particle 2 because measurements on particle 2 do not detect any change in its state during the process.

To see how this describes a non-local ef­fect, we introduce three experimentalists Al­ice, Bob and Carol who perform experiments in regions that are spatially separated from each other. Particles 1 and 2 are first pre­

pared in the state \I/(1, 2).

* ( l , 2 ) = a ^ + ( l V + ( 2 ) + W _ ( l ) 0 _ ( 2 ) , (1)

where each of the single particle wave func­tions is normalised and (^+(1)|V'-(1)) = 0, but, <<M2)|<£_(2)>=72/0.

Particles 1 and 2 separate but continue to interact via a long-range residual interac­tion VR(1 , 2).

Particle 1 moves towards Bob and parti­cle 2 moves to Alice. Independently Carol prepares particle 3 and sends it to Alice. When particle 1 is distant from Alice but V R ( 1 , 2 ) is still non-negligible, Alice turns on an impulsive von Neumann interaction V;v(2,3)6 between 2 and 3. The states of particles 1, 2 and 3 become entangled in * / ( l , 2 , 3 ) given by

a ^ + ( l )0 + (2 )x+(3) + &V-( l )M2)x- (3) . (2)

The three particles separate and cease interacting as soon as the long-range residual interaction VR(1,2) becomes negligible.

Bob measures certain properties of par­ticle 1. The results of his measurements are seen to depend upon actions carried out by

39

Alice or Carol that affect the state of particle 3 but leave the state of particle 2 unchanged. An explicit dynamical model of the forma­tion of the entangled states is given in our paper1.

2. Non-locality

The non-locality can be observed at three dif­ferent levels. At each level the non-locality is discernible through statistical measurements in only one of the wings and does not use correlations between measurements in differ­ent wings, in contrast to the usual Bell-type arguments for quantum non-locality 3. Fur­thermore the non-locality is apparently inde­pendent of the way any hidden variables or realist models are formulated.

At the first, the results of Bob's measure­ments on particle 1 depend on whether the interaction Vjv(2,3) was turned on by Alice.

At the second, if the initial entangled state of 1 and 2 and the form of V/v(2,3) are fixed, the results of Bob's measurements depend upon the product of the strength and duration of the impulsive V/v(2,3) that Alice uses.

At the third level, if the form, strength and duration of VN(2, 3) are fixed, the results of Bob's measurements depend upon the ini­tial state of 3 prepared by Carol.

The first two levels depend upon the sep­aration of Bob from Alice, the third, on that between Carol and Bob, gives the sharpest example of Einstein non-locality.

2.1. The Sharpest Example

In the third scenario, Carol can prepare dif­ferent initial states of particle 3. For sim­plicity we suppose that there are only two possible states {XoiXo} *n a* Carol can pre­pare.

Then after Alice has turned on V/v(2,3), there are two possible final wave functions,

¥[(1,2,3) =

aV+( l)<M2)x;(3) + ty_(l)M2)XL(3). (3)

and ¥f (1,2,3) =

a<Ml)<M2)x;(3) + ty_(l)M2)x!(3) (4)

Each single particle wave function has norm 1. (^+(1)|^-(1)> = 0, <<M2)|<M2)} = 72 ^ 0 (x{(3) 1x1(3)) = 73 T^°> 3=r,s a n d 7 j ^ 7 | .

After the entangled state of particles 1 and 2 has been formed, particle 1 moves to­wards Bob and away from Alice, but particle 2 goes to Alice. Carol prepares particle 3 in one of the states Xo, f° r 3 — ri s> a n d sends 3 to Alice who turns on VAT (2,3) and forms the corresponding entangled state ¥{(1,2,3).

When all the interactions have ceased acting Bob measures a quantity A(l) of par­ticle 1 for which (ip+(l)\A(l)\ip-(l)) / 0, and finds that the observed value depends on which initial state of 3 was prepared by Carol. The the difference between the ex­pectation values of A(l) in the final states ¥[(1,2,3) and ¥f( l ,2 ,3) is given by

2SRe 06720(73 - 7f )• (5)

3. Conclusions

Thus a local change of parameter performed by Carol in a region spatially distant from Bob affects the statistical result of measure­ments performed by Bob. The key point is that although particle 3 send by Carol in­teracts impulsively with particle 2, which in turn interacts with particle 1, the informa­tion from Carol's initial state preparation sector is not simply transported in a clas­sical way by the interactions V R ( 1 , 2 ) and Vjv(2,3). This is shown by the fact that the state, the reduced density matrix, of 2 is un­changed by the changes made by Carol so that Alice cannot measure any change in the properties of 2.

40

That manipulations on particle 3 at lo­cations distant from particle 1, which does not directly interact with particle 3, can af­fect the outcomes of subsequent experiments on particle 1 clearly violates Einstein local­ity. Moreover, this non-locality does not use any realist model or require the use of hidden variables.

Einstein locality requires that the real factual situation of the system, particle 1, is independent of what is done with the sys­tem, particle 3, which is spatially separated from it. The systems are spatially separated when the distance between them is large compared to the range of the interactions be­tween them.

Einstein locality is necessary for the ex­istence of isolated systems and hence for the possibility of experimentally checking physi­cal laws.

In order that this violation of Einstein locality occurs the entangled states of par­ticle 2 must not be orthogonal and the long-range interaction V R ( 1 , 2 ) must be non-negligible compared to the kinetic energy when V/v(2,3) is turned on. The non-orthogonality of the states of 2 occur nat­urally in preparation and measurement pro­cedures that use von Neumann type inter­actions and therefore may be experimentally achieved. In general weak von Neumann type interactions produce non-orthogonal states for one of the interacting systems6.

All three particles are massive. The long-range interaction V R ( 1 , 2 ) could be the Coulomb interaction between charged parti­cles. For example, particle 1 could be an elec­tron and particle 2 could be a positive ion. In the entangled state of the system of ion and electron the states of the ion are assumed to be non-orthogonal. Particle 3 would then be an atom or molecule which interacts with the ion through an impulsive von Neumann in­teraction in a measurement process in which the atom or molecule acts as a pointer to the non-orthogonal states of the ion.

It is important that the interaction VR(1, 2) does not entangle the wave functions of particles 1 and 2. Therefore the underlying Hilbert spaces must be infinite dimensional7. The electron, particle 1, then behaves as a test particle with respect to the Coulomb force of the ionised atom, particle 2, on it.

4. Quantum Communication

Considered as a communications system, the set of possible messages from which the mes­sage to be transmitted is selected can be taken to be the pair of final tripartite states, {^i, \&f} which are not orthogonal,

= \a\2(Xr+,Xs

+) + \b\2{xr-,XS-)^0. (6)

Let pj = |\IJ{)(\I>j| and Pj,j = r, s be the probability of sending message j .

P = Prpr + PsPs is a mixed state. The von Neumann entropy S of p is less than its Shannon entropy H. The standard for­mal argument4 for the equality of S and H doesn't work here because the probabilities Pj are not obtained from an orthogonal de­composition of the state p.

For example, when pr = ps = , H = 1 and S is given by

i-*4E!>i*(i + in) _(i_-Jrl) lo&(1_ [r j ) (7)

The question of the relation between this re­sult and non-locality remains open. The role of non-orthogonality requires further study.

Acknowledgments

The authors acknowledge financial support for the visit of D.H. to Sydney from the Cen­tre for Time, Philosophy Department, Syd­ney University where J.V.C. is an Honorary Research Fellow. The research of D.H. is sup­ported by the Jawaharlal Nehru Fellowship.

References

1. J. V. Corbett and D. Home, Physics Letters A 333, 382-388 (2004).

2. J. V. Corbett and D. Home, Phys. Rev. A 62, 062103 (2000); J. V. Corbett and D. Home, Foundations of Quantum Mechanics in the Light of New Technology edited by Y.A.Ono and K. Fujikawa (World Scientific, Singapore, 2002), p. 68-71.

3. J.S.Bell, Physics (Long Island, NY)1, 195 (1964); D.M.Greenberger, M.A.Horne and A.Zeilinger Bell's Theorem, Quantum The­ory, and Conceptions of the Universe edited by M.Kafatos (Kluwer, Dordrecht, 1989), p. 69; N.D.Mermin, Phys.Rev.Lett. 65,3373 (1990); L.Hardy, Phys.Rev.Lett. 71,1665 (1993); A.Cabello, Phys.Rev.Lett.86, 1911 (2001),Phys.Rev.Lett.87, 010403-1 (2001).

4. A. Duwell, Studies in History and Philosophy of Modern Physics 34, 489 (2003).

5. A. V. Thapliyal. Phys. Rev. A 59, , (3336) 1999;

6. J. von Neumann, Mathematical Foundations of Quantum Mechanics, (Princeton Univer­sity, Princeton, 1955), pp. 443-445. See also P. Holland, Quantum Theory of Mo­tion (Cambridge University, England, 1993), pp. 339-341 and D. Home, Conceptual Foun­dations of Quantum Physics (Plenum, New York and London, 1997), pp. 68-70.

7. T.Durt, Z. Naturforsch 59a, 425-436 (2004); J. Gemmer and G. Mahler, Eur. Phys. J. D 17, 385 (2001).

42

QUANTUM ENTANGLEMENT AND WEDGE PRODUCT

H. HEYDAR1*

Institute of Quantum Science, Nihon University, Tokyo 101- 8308, Japan

* E-mail: hoshang@imit.kth.se

I derive a measure of entanglement for general pure multipartite states that coincides with the generalized concurrence for a general pure bipartite and three-partite states based on wedge prod­uct.

Keywords: Quantum entanglement; Concurrence; Wedge product.

1. Introduction duction to multilinear algebra.

Quantum entanglement is one of the most in­teresting properties of quantum mechanics. It has become an essential resource for quan­tum information (including quantum com­munication and quantum computing) devel­oped in recent years, with some potential applications such as quantum cryptography 1,2 and quantum teleportation 3. Quantifica­tion of a multipartite state entanglement 4 '5

is quite difficult and is directly linked to lin­ear algebra, geometry, and functional anal­ysis. The definition of separability and en­tanglement of a multipartite state was in­troduced in6, following the definition of bi­partite states, given by Werner 7. One of the widely used measures of entanglement of a pair of qubits is entanglement of forma­tion and the concurrence, that gives an ana­lytic formula for the entanglement of forma­tion 8>9'10. In resent years, there have been some proposals to generalize this measure on general pure bipartite states n.12*13-14

a n d on multipartite states 15>16'17. In this paper, I will construct a measure of entanglement that coincides with the generalized concur­rence for a general pure bipartite and three-partite state based on geometric algebra, us­ing wedge product, a useful tool from multi linear algebra, which is mostly used in dif­ferential geometry and topology in relation with differential forms. To make this note self-contained, we will give a preview intro-

2. Quantum entanglement

In this section we will define separable state and entangled state as well as give some ex­ample of entangled state. Let us denote a general pure composite quantum system Q = Qp

m(Ni,N2,...,Nm) = Q1Q2---Qm

with m subsystems, consisting of a state

I*) = (1)

i l = l » 2 = l »m = l

defined on a Hilbert space

nQ = HQl®HQ2®---®HQm (2)

= CN>®CN*®---®CN™,

where the dimension of j t h Hilbert space is given by Nj = dim(WgJ.). We are going to use this notation through this paper, i.e., we denote a pure pair of qubits by Qf(2, 2). Next,let PQ denote a density operator acting on the HQ. Then pg is said to be separable, which we will denote by PQP, with respect to the Hilbert space decomposition, if it can be written as

N m N

PQP = J2P><<8)PQJ> E ^ = 1 (3) fc=i j=i fc=i

for some positive integer N, where pk are positive real numbers and pj denote a den­sity operator on Hilbert space "HQJ. If /?g

43

represents a pure state, then the quantum system is separable if /OQ can be written as PQP = <8>Jli PQj. where pQ. is a density op­erator on HQ.. If a state is not separable, then it is called an entangled state. Some of the most important entangled states are called the Bell states or EPR states.

3. Multilinear algebra

In this section, we review the defini­tions and properties of multilinear algebra. Let us consider the complex vector spaces Vi, V 2 , . . . , Vm , where dim(V,) = Njt Vj = 1,2,. . . , m. Then, we define a tensor of type (m, n) on Vi, V2, • • •, V m as follows

T r T ( V 1 , V 2 , . . . , V m ) = (4)

£(v1,v2,...,vm;vr,v2*,...,v;) = Vi ® v2 ® • • • ® vm ® v? ® V5 <g> • • • <g> v;,

where V* = £(V,; C) Vj = 1,2,. . . , m is the space of linear applications Vj —• C and is called dual of Vj. For any basis e; of Vj and ei the dual basis of e defined by e^(e;) = <£?, we have the following linear representation

2 ® • • • ® eim (5) rp rpll,l2. •Jn l l

For example, we have 7^(Vi, V 2 , . . . , Vm) = Vj and r ^ V i , V 2 , . . . , Vm) = VJ. Let Sm

be a group of permutations (1,2, . . . , m ) . Then, we call tensor V\ ® v2 ® • • • ® vm e

• •, Vm) symmetric if

J«m = fff(l) ® • • • ® ^ ( m ) . (6)

Sm. The space of symmet-

V ( V i , V 2 , .

Vi (g> • • • C*

for all 7r € ric tensor is denoted by <S™(Vi, V 2 , . . . , Vm) . Moreover, we call the tensor v\ <S> v2 <8> • • • cS> vm € T^"(Vi, V 2 , . . . , Vm) skew-symmetric if

vi <g> • • • ® vm = e(7r)tv(i) ® • • • ® % ( m ) , (7)

for all 7r € S^, where e(7r) is the signature of permutation 7r. The space of symmetric ten­sor is denoted by A^(Vi, V 2 , . . . , V m ) . Fur­thermore, we have the following mapping

A l t m : T 0m ( V i , V 2 , . . . , V m ) (8)

— * A n V i , V 2 , . . . , V m )

where the alternating map is defined by Altm(ui <g> v2 <8> • • • ® vm) = Vi A v2 A • • • A

Vm = ^T Ex6Sm e W ^ d ) ® «*(2) ® • • • ® uff(m). For example, for m = 2 we have Alt2 : 7?(Vi, V2) —• Ag(Vi,V2) defined by t>i <2> v2 1—• «i A u2 = v\ <8> w2 — ^2 <8> t>i • The essential property of wedge product is that it is alternating on V, that is v A v = 0 for all vector v € V and L>I A v2 A • • • A vn = 0, whenever v\,v2,...,vn e V are linearly de­pendent.

4. Measure of entanglement for general bipartite state based on wedge product

From the wedge product, we can define a measure of entanglement for a general pure bipartite quantum system Q2(Ni,N2) as

C{Ql{NuN2)) = (N £ C^C^A , V i/i<f /

(9) where C^,„ is defined by C^^ = v^ A vv, v^ = (aM>i,ojM)2,...IaM,jv2), vv = (au,i,OLVt2,...,aVtN2)

a n d CM>I/ denote the complex conjugate of CM,„. That is, for a quantum system Q2(Ni,N2), if we write the coefficients ailii2, for all 1 < i\ < N\ and 1 < i2 < A 2 in form of a N\ x N2 matrix as below

/ "1,1 "2,1

"1,2 a2,2

ai,JV2

« 2 ,JV2

\

(10)

VofJVi.l OtNlt2 • • • &NUN2 )

then the vectors v^ and vv refer to different rows of this matrix. As an example let us look at the quantum system Q2(2,2) repre­senting a pair of qubits. Then the expression for a measure of entanglement for such state using the above equation (9) is given by

c(e?(2,2)) = (Mch2ch2y (11)

= (2A/"|ai,1a2,2 - a 2 , i a i , 2 1 2 ) 5

= 2|ailiQ!2,2 - 0(2,101,21,

44

where C\,2 is given by

Ci , 2 = « i A « 2 (12)

= ( a i , i , "1,2) ® ( a 2 , i , 012,2)

-("2,1,0:2,2) ® (0:1,1,0:1,2)

= (0 , a i ) i a 2 ) 2 -a 2 , iQ! i ,2 ,

ai,2«2,i - « 2 , 2 a i , i , 0 )

for A/" = 2. The measure of entanglement for

the general bipart i te s ta te denned in equa­

tion (9) coincides with the generalized con­

currence denned in 11>12>13.14 and in particu­

lar equation (11) tha t gives the concurrence

of a pair of qubits, first t ime denned in 9 ' 1 0 .

5. Genera l mul t ipar t i t e s t a t e a n d w e d g e p r o d u c t

Unfortunately, I couldn't directly generalize

this result, using equation (8), to a gen­

eral pure mult ipart i te state. For example,

for a general three-part i te quan tum system

Q2(AT1; N2, N3) with the coefficients ailti2,i3,

for all 1 < ii < Nu 1 < i2 < N2 and

1 < h < N3, we cannot identify the vectors

v^, vu and vT. To be able t o define an ex­

pression for degree of entanglement of a gen­

eral pure mult ipart i te s tate, I will construct a

wedge product between each subsystem in a

composite system as in the case of the bipar­

t i te state. So, let us consider the quantum

system Q^(Ni, N2, • • •, -/VTO) and define

Ck1iu...,kmim = otku...,kmaii,...,im- (13)

Then, a wedge product between each pair of

subsystem takes the following form

Ckil1,k2h,-,kjMj,...,kmlm = (14)

Q!fe 1 , f e2 , . . . , «5 : , - , . . . ,A ; m Q!i i , i 2 , - , ^ , . . . , im

~&ki,k2f-,lj,---,krn&li,l2,...,kj,...,lm-

Now, we can define a measure of entangle­

ment as follows

£(QUN1,...,Nm)) = (Mj2Yl

VK,L Vj

l ^ fc i (1 , . . . ,kj Alj,... ,kmlmkfci(1,• • • ,kj Alj,... ,kmlm )

= (N"52 ]Cla*l,fc2,...,fcmQ:*l,l2,...,lm

VK,L Vj

&ki,k2,...,kj-i,lj,kj + i,...,km

<Xll,l2,-,lj-l,kj,lj + 1 , - , l m \ ) ^ > (1 5)

where j = 1,2, . . . , m and multi-index

K = ( fc i , fc 2 , . . . , fem) ,L = (h,h,..-,lm)-

As an example, let us look at the gen­

eral pure three-part i te quan tum system

Q2(Ni,N2,N3). The above measure of en­

tanglement gives

£(QP3(Ni,N2,N3)) = (Af Yl E

ki,li;k2,h;k3,l3 j=l

l ^ f e i ' i i f e j A i j , k 3 l z ^ k i l l , k j Alj,k3l3 I j 2

= (M £ (16) ki,h;k2,l2',k3,l3

(|afci,fc2,fc3ah,i2,i3 ~ akuk2,l3ah,h,k3\

+ \ak1,k2Mah,l2,h ~ akul2M

ah,k2,h\ 1 i2\ 1

+ \ak1M,k3ah,h,l3 ~ aliM,k3

aki,l2,h\ P-This measure of entanglement coincides with

the generalized concurrence 1 5 and is equiva­

lent, but not equal, to our entanglement ten­

sor based on joint POVMs on phase space 18

A c k n o w l e d g m e n t s

The author acknowledges useful discussions

with Gunnar Bjork. The author also would

like to thank J a n Bogdanski. This work was

supported by the Wenner-Gren Foundations.

R e f e r e n c e s

1. C. H. Bennett and G. Brassard, Proc. IEEE Int. Conference on Computers, Systems and Signal Processing (IEEE, New York, 1984); C. H. Bennett, F. Bessette, G. Brassard, L.

45

Salvail, and J. Smolin, J. Cryptology 5, 3 (1992).

2. A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). 3. C. H. Bennett, G. Brassard, C. Crepeau, R.

Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).

4. M. Lewenstein, D. Brufi, J. I. Cirac, B. Kraus, M. Kus, J. Samsonowicz, A. San-pera, and R. Tarrach, J. Mod. Opt. 47, 2841 (2000).

5. W. Diir, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999).

6. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997).

7. R. F. Werner, Phys. Rev. A 40, 4277 (1989). 8. C. H. Bennett, D. P. DiVincenzo, J. Smolin,

and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).

9. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

10. W. K. Wootters, Quantum Information and Computaion, Vol. 1, No. 1 (2000) 27-44, Rin-ton Press.

11. A. Uhlmann Phys. Rev. A 62, 032307 (2000).

12. K. Audenaert, F. Verstraete, and B. De Moor, Phys. Rev. A 64 012316 (2001) .

13. P. Rungta, V. Buzek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev. A 64, 042315 (2001).

14. E. Gerjuoy, Phys. Rev. A 67, 052308 (2003). 15. S. Albeverio and S. M. Fei, J. Opt. B: Quan­

tum Semiclass. Opt. 3, 223 (2001). 16. D. D. Bhaktavatsala Rao and V. Ravis-

hankar, e-print quant-ph/0309047. 17. S. J. Akhtarshenas, e-print

quant-ph/0311166. 18. H. Heydari and G. Bjork, Quantum Infor­

mation and Computation 5, No. 2, 146-155 (2005)

19. H. Heydari and G. Bjork, J. Phys. A: Math. Gen. 38, 3203-3211 (2005).

46

ANALYSIS OF THE GENERATION OF PHOTON PAIRS IN PERIODICALLY POLED LITHIUM NIOBATE

J. S O D E R H O L M , K. H I R A N O , S. M O R I and S. INOUE*

Institute of Quantum Science, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan

* E-mail: inoQphys. est. nihon-u. ac.jp

S. K U R I M U R A

National Institute for Materials Science, 1-1 Namiki, Tsukuba-shi, Ibaraki 305-0044, Japan

The process of spontaneous parametric down-conversion (SPDC) in nonlinear crystals makes it fairly easy to generate entangled photon states. It has been known for some time that the conver­sion efficiency can be improved by employing quasi-phase-matching in periodically poled crystals. Using two single-photon detectors, we have analyzed the photon pairs generated by SPDC in a periodically poled lithium niobate crystal pumped by a femtosecond laser. Several parameters could be varied in our setup, allowing us to obtain data in close agreement with both thermal and Poissonian photon-pair distributions.

Keywords: Photon-pair generation; periodically poled lithium niobate; photon statistics.

1. Introduction

Through optical nonlinearities in a crys­tal, one pump photon can be converted into two photons with longer wavelengths. The fact that these down-converted pho­tons are always generated in pairs has made it possible to use them for many funda­mental quantum-mechanical experiments.1

These photon pairs are also suitable for fundamentally secure quantum-key distribu­tion, which is usually referred to as quantum cryptography.2 Recently, it was recognized that the efficiency of the photon-pair genera­tion can be improved by using periodically poled crystals.3 The ferroelectric polariza­tion of these crystals is periodically inverted, resulting in flips of the sign of the non­linear coupling x ^ - Through quasi-phase-matching one may then exploit larger non­linear coefficients than what is possible using birefringent phase matching.

In the present work, we have used single-photon detectors to analyze the down-converted photon pairs generated in a peri­

odically poled lithium niobate (PPLN) crys­tal under various conditions.

2. Exper imenta l se tup

The experimental setup is schematically de­picted in Fig. 1. Our 10-mm-long bulk PPLN crystal was doped with 5 mol% MgO and pumped by 200-fs-long pulses from a Tita-nium:sapphire laser with a repetition rate of 80 MHz. The poling periodicity of the crys­tal was 19.4 /mi, corresponding to degener­ate quasi-phase-matched parametric down-

PBS - \

j \ ^ H W P

J PPLN DM F L3 n l

50/50 FC D2

Fig. 1. Schematic drawing of our experimental setup used to analyze the photon pairs generated in a PPLN crystal. (See text for explanation.)

47

conversion at 1544 nm. However, in order to make the frequency of the degenerate photon pairs coincide with our interference filters' central frequency of 1550 nm, the wavelength of the laser was set to 775 nm. A half-wave plate (HWP) and a polarizing beam split­ter (PBS) were used to attenuate the laser power without distorting the laser mode and pulse. As the polarization direction of the laser beam depended on the rotation angle of the HWP, the intensity of the pump beam emerging from the PBS could be varied.

Three different pairs of lenses were used to focus the pump beam in the crystal, and collect the down-converted light. In the case with closest focus, we did not have two matching lenses at hand, so the focal length of LI was 8.2 mm, whereas that of L2 was 10 mm. In the other two cases, both LI and L2 had a focal length of 14.8 mm and 50 mm, respectively. Effective down-conversion was restricted to the focus, since the pump in­tensity was much higher there.

A dichroic mirror (DM) and a filter (F), with a full-width-at-half-maximum band­width of either 10 or 30 nm, were used to dis­criminate the down-converted light from the pump. The down-converted light was sub­sequently coupled into a fiber using a third lens (L3). After splitting the beam with a 50/50 fiber coupler (50/50 FC), both single and coincidence events at the single-photon detectors (Dl and D2) were recorded. The detectors were gated at a rate of 316 kHz and synchronized with the pump pulses.

3. Theory

Since our detectors cannot distinguish if one or more photons make them "click", the single-count rate at detector k (when neglect­ing dark counts) can be expressed as

m=0 n=0 ^ '

(1)

where R is the gate rate, p(m) denotes the probability for m photon pairs to be gener­ated, T is the transmittivity of the optical components, and r/k is the detector efficiency. Similarly, the coincidence-count rate, i.e., the rate with which both the detectors click dur­ing the same gate pulse, can be written as

x [ l - ( l - T r ? 2 ) 2 m - " ] . (2)

3.1. Photon-pair distributions

In the present setup, most photon pairs will be composed of photons with different wave­lengths. With no initial photons in the down-conversion modes, the nondegenerate SPDC process is known to generate the two-mode squeezed vacuum state. The photon-pair dis­tribution is then thermal4

where pi = sinh r is the average number of photon pairs, and the squeezing parameter r is proportional to the electric field of the pump.5 Assuming that the form of the pump pulse is independent of the pump power, we thus obtain fi = sinh \fKPave, where K is a constant and P a v e is the average pump power. We note that p « KPave as \/KPave <§; 1.

On the other hand, when the detected photons originate from many distinguishable down-conversion processes, the photon-pair distribution can be approximated6 by the Poissonian distribution

vme-u PPoi(m) = m] , (4)

where v is the average value of the total num­ber of photon pairs generated by a single pump pulse. As there are many distinguish­able processes, the average photon-pair num­ber in each of them is usually small (p -C 1), and therefore proportional to the average pump power. We then get v « fCPave, where

48

coun

t

0)

den

ity f

or c

oinc

s -a

0.004

0.003

0.002

0.001

• • '

Thermal photon-pair

.-* m -tr^*r\ .

distribution v *•

tf ^^~^ Poissonian •^-^"^ photon-pair distribution

£ 0.02 0.04 0.06 0.08

Probability for single count

Fig. 2. E x p e r i m e n t a l r e su l t s ( B ) o b t a i n e d w i t h a

focus l eng th of 8.2 m m a n d a 10-nm filter.

K. is a constant. In a recent experiment7 sim­ilar to the one reported here, but employing a semiconductor laser and a 3-cm-long PPLN waveguide, we have been able to fit both single and coincidence counts versus pump power to such a Poissonian photon-pair dis­tribution.

3.2. Dark counts

As there is a probability for the detectors to click even when no photon is incident, the ef­fects of these dark counts should be compen­sated for before comparing the experimen­tal data with the expressions in the previous section. Noting that the raw single-count rates S^ are due to either incident pho­tons, dark counts, or both, one finds that the photon-induced single-count rates are given by

: (ph) -,(raw)

- Sk

1 + SkR - l ' (5)

where 6k denotes the dark-count rate at detector k. Neglecting terms that in­clude the product 6162, the photon-induced coincidence-count rate can be expressed as

C-(ph) C ( r a w ) f l _ 5 ( p h ) a 2 _ a i 5 ( P h )

R - Si - 62 , (6)

where C(raw) is the raw coincidence-count rate.

CD

C 0 .3

o j.0.1 s (0 n o

Thermal photon-pair distribution \

\ Poissonian

photon-pair distribution 0.1 0.2 0.3 0.4 0.5 0.6 0.7

°" Probability for single count

Fig. 3 . E x p e r i m e n t a l d a t a o b t a i n e d w i t h a 10-nm

filter a n d pa i r s of lenses w i t h a focus l eng th of

14.8 m m ( • ) a n d 50 m m (A).

4. Experimental results

Using a laser emitting light with a wave­length of 1550 nm, the transmittivity of the optical components was measured. Also the detector efficiencies rji and r/2 were deter­mined in independent measurements. Plug­ging in these values into Eqs. (1) and (2), the single-count and coincidence-count rates for a thermal or Poissonian photon-pair distri­bution with a given average number of pho­ton pairs could be calculated. In Figs. 2-4, the resulting relations between the probabili­ties for a single and a coincidence count have been plotted as curves. Here, the single-count probability is the probability for one or two detectors to click, which is given by (Si + S2 - C)/R.

In Fig. 2, the data obtained with a fo­cal length of 8.2 mm and a 10-nm filter are presented. The different data points corre­spond to different pump powers, and the ef­fects of dark counts have been compensated for. These measurements are seen to be in agreement with a thermal photon-pair distri­bution. The results obtained after changing the focal length to 14.8 and 50 mm are shown in Fig. 3. We see that also with a focal length of 14.8 mm, the data are in agreement with the thermal distribution, whereas the longest focal length resulted in a distribution lying between the thermal and Poissonian photon-pair distributions. This can be understood

49

€ 1 0 t -I 3 'J o » a> 0-8 . 7 c Thermal photon-pair uj •§ 0 g distribution v /

1 ' X / 7 O 0.4 • /

£ • \ y \ .•&• 0.2 " ^ ^ T ' := .. ' _ ^ - ^ Poissonian 03 ....- i^——--'~~~*^ photon-pair distribution

•Q _ _ . . • " ' * ^ ^ . . . . . O 1^™" " —

£ 0.2 0.4 0.6 0.8 1.0 Probability for single count

Fig. 4. Experimental results (•) obtained with a focus length of 50 mm and a 30-nm filter.

considering the dispersion between the pump

and down-converted light in the crystal. For

a sufficiently large focus, i.e., for a sufficiently

long focal length of the lenses, photon pairs

generated in different locations within the fo­

cus will not emerge from the crystal a t the

same time. The contributing processes can

thus be distinguished, resulting in a photon-

pair distribution tha t is closer to Poissonian.

In Fig. 4, we present the da ta obtained

with a focal length of 50 m m and a 30-nm fil­

ter. In this case, our measurements are seen

to be close to the Poissonian curve. This can

be explained by the fact tha t a broader fil­

ter allows for a shorter coherence t ime of the

t ransmit ted down-converted light,6 , 7 which

makes it easier to distinguish the generated

photon pairs.

We note tha t a high number of photon

pairs were generated in some of our measure­

ments. Assuming that the down-converted

light generated with a focal width of 14.8 mm

indeed is described by a thermal photon-pair

distribution, and tha t the overall detection

efficiency equals the measured value of 2%,

the maximum average number of generated

photon pairs per pulse is found to be 60.

Similarly, the da ta presented in Fig. 4 cor­

responds to a maximum average number of

photon pairs of 100, if the generated pho­

ton pairs are assumed to have a Poissonian

distribution and the measured overall t rans-

mittivity of 2.8% is correct. In this case, the

power of the down-converted light after the

filter was measured to be 2 n W using a sim­

ple power meter.

Finally, we note tha t , in contrast t o our

experiment with a waveguide,7 we have not

been able to fit the measured single-count

and coincidence-count rates as functions of

pump power to the theoretical expressions

given in the previous section. We believe tha t

this is due to the high pump power and /o r

the very high peak intensity achieved with

the Titaniurmsapphire laser. This should in­

crease the tempera ture of the crystal, and

consequently change the properties of the

material and the phase-matching conditions.

5. C o n c l u s i o n s

We have shown tha t even with only two

single-photon detectors and a low overall de­

tection efficiency, insight into the photon

statistics can be gained. By changing the

experimental conditions, our experimental

da ta could be varied from the expected re­

sults for a thermal photon-pair distribution

to those for a Poissonian photon-pair distri­

bution. This was explained by the increasing

degree of distinguishability within the dis­

t r ibuted generation of photon pairs.

R e f e r e n c e s

1. L. Mandel, Rev. Mod. Phys. 71 , S274 (1999); A. Zeilinger, Rev. Mod. Phys. 71 , S288 (1999).

2. N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).

3. S. Tanzilli et al, Electron. Lett. 37, 26 (2001); K. Sanaka, K. Kawahara and T. Kuga, Phys. Rev. Lett. 86, 5620 (2001).

4. B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).

5. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cam­bridge, 1997), Chap. 16.

6. H. de Riedmatten et al, J. Mod. Opt. 51 , 1637 (2004).

7. S. Mori, J. Soderholm, N. Namekata and S. Inoue, quant-ph/0509186.

50

GENERATION OF ENTANGLED PHOTONS IN A SEMICONDUCTOR A N D VIOLATION OF BELL'S INEQUALITY

GORO OOHATAA'B, RYOSUKE SHIMIZUC, and KEIICHI EDAMATSUA'C* Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan ERATO Semiconductor Spintronics Project, Japan Science and Technology Agency, Japan

CREST, Japan Science and Technology Agency, Japan * E-mail: eda@riec.tohoku.ac.jp

We demonstrate the generation of entangled photon pairs via biexciton-resonant hyper parametric scattering in CuCl crystal. Quantum state tomography show that the generated photon pairs have a high degree of polarization entanglement. Also we show that the photon pairs violate Clauser-Horne-Shimony-Holt type Bell's inequality.

Keywords: Entanglement; Bell's inequality; Photon; Exciton; Quantum information.

1. Introduction

Entanglement is one of the essential re­sources of quantum info-communication technology. Atomic cascade emission1 and spontaneous parametric down-conversion2'3

have been the methods used so far to gen­erate highly entangled pairs of photons. However, for practical applications of quan­tum info-communication, entangled photon sources using semiconductors are in great de­mand. In practice, it has been demonstrated that a pair of single photons generated from a semiconductor quantum dot can be con­verted into an entangled photon pair4. Al­though it is in principle possible to generate a pair of entangled photons directly from a biexciton in a semiconductor5'6, yet the trials using biexciton emission from semiconduc­tor quantum dots have not been successful7. Recently, we reported the first experimen­tal evidence of the generation of entangled photons via biexcitons in a bulk semicon­ductor (CuCl) crystal8. However, in the previous report, insufficient elimination of accidental coincidences of photons originat­ing from independent biexcitons inhibited us from demonstrating nonlocal feature of en­tanglement such as Bell's inequality. There­after, by improving the experimental condi-

Biexciton

it ->

I'X*

;— —

Exciton 'X*'

Fig. 1. Schematic diagram of the resonant hyper parametric scattering via biexciton. The two pump (parent) photons (frequency: u>i) are converted to the two scattered (daughter) photons (LJS, UJSI).

tions, we succeeded to eliminate significantly the accidental coincidence that degrade the purity of entanglement. Here we show the experimental result that exhibits a higher de­gree of entanglement and violation of Bell's inequality.

2. Experiment

Figure 1 schematically shows the hyper para­metric scattering process in resonance to the biexciton state. The two pump photons (frequency:^) resonantly create the biexci­ton, and are converted into the two scat­tered photons (UJS , uv) satisfying the phase-matching condition that takes account of the exciton-polariton dispersion relation. Since the biexciton created in this process is in the

51

zero angular momentum (J=0) state, the po­larizations of the scattered photons are en­tangled so that their total angular momen­tum is also zero. Therefore the two-photon polarization state |\I/) is expressed as

| * } = i = ( | F ) 1 | J f f ) 2 + |F>1 |y)2)

=-^( | JL)1 | JR)2 + | JR)1 | i)2) (1)

where \H)t, \V)i, \L)i and \R)i represent the states for horizontal, vertical, left-circular and right-circular polarizations of each pho­ton (indicated by i=l,2), respectively. It is noteworthy that the mechanism of entan­glement generation in our scheme is simi­lar to that demonstrated in atomic cascade emission1.

The material we used in this study was copper chloride (CuCl) single crystal. Since CuCl has a large bandgap (~3.4 eV), it is suitable to generate photon pairs in the short wavelength region near ultraviolet. Further­more, the material has large binding ener­gies of the exciton (~200 meV) and biexciton (~30 meV). Because of these unique charac­teristics, CuCl is one of the best-investigated materials on the physics of excitons and biexcitons9. In particular, the "giant oscil­lator strength" in the two-photon resonance of the biexciton gives great advantage to our experiment9'10.

Figure 2 presents the schematic view of our experimental set-up. In the experiment, we used vapor-phase-grown thin single crys­tal of CuCl. The sample was put into a cold-finger cryostat and cooled down to ap­proximately 4 K. The pump source was the second harmonic light of a high repetition rate (1 GHz) femtosecond pulsed Tirsapphire laser. A low photon flux per pulse result­ing from the high repetition pump source is effective in suppressing the accidental co­incidence signals. The pump photon en­ergy was tuned by a zero-dispersion dou­ble monochromator to the two-photon exci-

Fig. 2. Schematic diagram of experimental set-up: (a) the light source and (b) the apparatus to measure the polarization correlation of photon pairs. ND: neutral density filter, PA: polarization analyzers, OF: optical fiber bundles, PMT: photomultipliers.

tation resonance (3.1861 eV) of the biexci­ton. Since a pair of photons indicated by HEP (high energy polariton) and LEP (low energy polariton) is emitted into different directions obeying the phase-matching con­dition, we put two optical fibers at appro­priate positions and led the pair of photons into two independent monochromators each of which selects photons corresponding to the HEP and LEP. The selected photons were detected by two photon-counting photomul­tipliers (PMTs) and the time delay (r) be­tween the photons was recorded by a time-interval analyzer.

3. Resul t s and Discussion

Figure 3 shows an example of the polariza­tion correlation measurements. Both linear (H and V) and circular (L and R) polariza­tion bases, we find clear polarization correla­tions for the coincidence signal at r = 0, in accordance with the theoretical prediction, Eq. (1), of the polarization-entangled state.

Figure 4 presents the density matrix p of the two-photon polarization state derived from quantum state tomographiy11'12 in­cluding various combinations of polarization correlation measurements, without artificial subtraction of the accidental coincidence sig­nals. The density matrix exhibits the fidelity F=(* |p |* )=0 .85 to the predicted maxi­mally entangled state, Eq. (1), beyond the

52

20

S 10 O 0 0

• 9 2 o

1 10

3 o g20

J 10 8 o

: VH

1 '

w ' I • I ' I I

" - - » " ' • • • -

• - • ' - • • - *

- HV

i i n r1-*- * • ' • - - - - ! n j

HH

1 '

LRR

LRL

iLR

1 LL

• •

'-

'Mi

*

uki

»* -50 0 50 -50 0 50

Delay Time x (ns)

Fig. 3. Photon correlation histogram for linear (left) and circular (right) polarization combinations. H, V, L and R represent the horizontal, vertical, left-circular and right-circular polarizations of each photon concerned, respectively. The central peaks at zero delay time (r = 0) are coincidence signals originating from the entangled photon pairs.

classical limit ( F = 0 . 5 ) . The values of en­

tanglement of formation (Ep), tangle (T)

and linear entropy (5x) obtained from the

density matr ix are Ep=0.65, T=0.56 , and

Sx=0 .31 , respectively. These values suggest

a small amount of degradation of our state

from the maximally entangled state; never­

theless it keeps a high degree of entanglement

and purity.

In connection to the entanglement, it

is interesting and important to examine the

nonlocality of the quantum s ta te 1 3 . We

evaluated Clauser-Horne-Shimony-Holt type

Bell's inequality1 4 of our photon pair s tate.

The obtained S-value is S = 2.34 ± 0 . 1 0 > 2;

our s tate undoubtedly violated the inequal­

ity. This is the first experimental demonstra­

tion of the violation of Bell's inequality by

use of photon pairs generated directly from a

semiconductor. The result manifests the use­

fulness of our approach for practical quantum

info-communication technology.

A c k n o w l e d g m e n t s

The authors are grateful to Professors T.

Itoh, H. Ishihara, and H. Ohno for valu­

able discussions. This work was supported

VH \ j / v v vv

Fig. 4. Real (Re) and Imaginary (Im) parts of the two-photon polarization density matrix recon­structed from the photon correlation measurements.

in pa r t by Strategic Information and Com­

munications R & D Promotion Programme

(SCOPE) of the Ministry of Internal Af­

fairs and and Communications, and by

Grant-in-Aid for Creative Scientific Research

(17GS1204) of the Ministry of Education,

Science, Sports and Culture.

R e f e r e n c e s

1. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47, 460 (1981); Phys. Rev. Lett. 49, 91 (1982)

2. P.G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995)

3. P.G. Kwiat et al., Phys. Rev. A 60, R773 (1999)

4. D. Fattal et al., Phys. Rev. Lett. 92, 037904 (20004)

5. S. Savasta, G. Martino, and R. Girlanda, Solid State Commun. I l l , 495 (1999)

6. O. Benson et al., Phys. Rev. Lett. 84, 2513 (2000)

7. C. Santori et al., Phys. Rev. B 66, 045308 (2002)

8. K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, Nature 431, 167 (2004)

9. Excitonic Processes in Solids, eds. M. Ueta et al. Springer-Verlag, Berlin, New York, Chap. 3 (1986) and references therein

10. T. Itoh and T. Suzuki, J. Phys. Soc. Jpn. 45, 1939 (1978)

11. D.F.V. James, P.G. Kwiat, W.J. Munro, and A.G. White, Phys. Rev. A 64, 052312 (2001)

12. A.G. White, D.F.V. James, W.J. Munro, and P.G. Kwiat, Phys. Rev. A 65, 012301 (2001)

53

13. W.J. Munro, K. Nemoto, and A.G. White, J. Mod. Opt. 48, 1239 (2001)

14. J.F. Clauser, M.A. Home, A. Shimony, and R.A. Holt, Phys. Rev. Lett. 23, 880 (1969)

54

DECOHERENCE OF A JOSEPHSON JUNCTION FLUX QUBIT

Y. NAKAMURA1'2'3*, F. YOSHIHARA2, K. HARRABI3, AND J.S. TSAI1'2'3

NEC Fundamental Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan

3CREST-JST, Kawaguchi, Saitama 332-0012, Japan * E-mail: yasunobu@ce.jp.nec.com

We have investigated decoherence mechanisms in a Josephson-junction flux qubit. Loss of the co­herence due to energy relaxation and dephasing was studied by measurements of the time evolution of qubit states subjected to simple control pulse sequences. As a function of qubit bias parameters, the characteristic time scales of the decoherence as well as the decay envelope functions varied, from which information about the fluctuation spectrum of the environment were deduced.

Keywords: Josephson junction; qubit; decoherence.

1. Introduction

Josephson-junction qubits are a unique quan­tum system which can be readily designed and built in a solid-state electric circuit1. Collective excitations of the superconduct­ing condensate, well isolated from quasipar-ticle excitations, are quantized according to the commutation relation between two con­jugate variables: the charge number and the phase. Moreover, non-linear inductance of the Josephson junction makes the energy level structure anharmonic. One can use the two lowest energy levels as an effective two-level system, i.e., a qubit2 '3.

Recent experimental works on Joseph-son junction qubits have been making much progress in terms of accurate co­herent qubit manipulations4, high-fidelity single-shot readout5, and coupling to an­other qubit6'7 or a quantum harmonic oscillator8'9,10. Also, coherence of the qubits has been improved largely since the first demonstration of Josephson junction qubit11. Decoherence times of the order of a microsecond have been achieved in several groups and with different types of qubit12-13 '14 '15. The ratio between the de-coherence time and the gate operation time can already be more than a hundred which

would be large enough for demonstrations of simple quantum logic circuits. Nevertheless, the ultimate limiting factor for qubit coher­ence has to be pursued for a further elon­gation of the decoherence time. In most of the early experiments, however, it was not easy to single out the dominant de-coherence source and understand the ob­served decoherence time quantitatively. One useful way to characterize the decoherence source is to change the operating point of the qubit by varying the bias parameters and ob­serve how the decoherence time depends on the parameters13 '15 '16 '17-18 '19 '20. The depen­dence is expected to be different for various origins of environmental fluctuations. Here we take such a strategy to investigate deco­herence mechanisms in a flux qubit.

2. Experimental Setup

Our flux qubit was fabricated by electron-beam lithography and shadow evaporation of Al films (Fig. 1(a)). It consists of a small superconducting loop intersected by four AI/AI2O3/AI Josephson junctions. One of the junctions is smaller than three others21 '22. The qubit loop shares a part and is inductively coupled with a bigger loop of a DC-SQUID which is used as a readout device

55

of the qubit state. For manipulating the qubit state, a set

of microwave current pulses is sent to a con­trol line inductively coupled to the qubit23. The resonant microwave pulses induce Rabi oscillations between the ground and excited states of the qubit. The amplitude and length of the pulse are adjusted so that a de­sired rotation of the qubit state is realized. For readout, we utilize a switching event of the SQUID from the supercurrent state to the voltage state under application of a bias current pulse23. The state-dependent persis­tent current along the qubit loop creates a small magnetic field which can be easily de­tected as the change of the critical current in the SQUID. In order to determine the pop­ulation in each eigenstate of the qubit after a certain control pulse sequence, we repeat

2\im

10 15 20

Time (ns)

Fig. 1. (a) Scanning-electron micrograph of the flux qubit and the readout SQUID. The small loop on the left hand side is the qubit loop, surrounded by the larger loop of the SQUID with two big junctions, (b) Eabi oscillations (dots) observed at the optimal point, together with a damped sinusoidal fit (solid line).

the control and readout procedure (and ini­tialization of the qubit by relaxation to the ground state as well) 104 times and count the switching probability Paw. The fidelity of the single-shot readout characterized by the visibility of Rabi oscillations was about 80% in the present sample (Fig. 1(b)). The measurements were performed at a temper­ature of 20 mK using a dilution refrigerator surrounded by a three-layer /z-metal shield at room temperature.

The effective Hamiltonian of the qubit can be written as H = — f<r2 — Y 0 " ^ where CT2, GX are Pauli matrices, and A is a tunnel splitting between two states with opposite di­rections of persistent current along the loop. e = 2J p#o(/ — /*) is energy bias between the two states. The energy difference between two eigenstates is given by EQI = y^e2 + A2 . Ip is a persistent current along the qubit loop. / = # e x / # o is a normalized magnetic flux in the qubit loop, $ e z is an externally applied flux through the qubit, and #Q is a flux quantum. f*(h) = 0.5 - # j b /#o is a normilized It-dependent flux; / = /* cor­responds to e = 0, i.e., degeneracy of the two persistent current states. The bias cur­rent of the SQUID, If,, affects the qubit only through the flux #jfa it induces in the qubit loop. Prom a result of spectroscopy measure­ment, the persistent current and the splitting energy were found to be Ip ~ 0.34 |iA and A/ft = 5.08 GHz, respectively.

There exist two externally controlled pa­rameters, / and /ft, for the Hamiltonian. Fluctuations of those parameters result in decoherence of the qubit. Dephasing is caused by variation of the qubit energy EQI . To minimize the sensitivity of EQI to the fluc­tuations of those parameters, one can choose bias points so that ^f- = 0 and 2$* = Q12, 13,22 fpkg £ o r m e r condition for an opti­mal flux bias is fulfilled when / = /*. As for lb, if the SQUID has a perfectly symmetric structure, small fluctuating current around lb = 0 always flows symmetrically through

56

two branches. However, for larger \Ib\ and in the presence of finite flux bias of the SQUID loop, the circulating current, and thus <&ib, depends on lb- Thus, only at lb = 0 the noise in If, is decoupled from the qubit, i.e., -gj- = 0 , and this is the optimal point for the bias current, i.e., ^f1 = 0. In gen­eral, if there is a slight asymmetry in the SQUID, the optimal condition is not neces­sary at Ii, = 0 but can be shifted a little to, say, I*h.

Relaxation of the qubit is induced by fluctuations of a transverse term in the Hamiltonian at the qubit frequency Eoi/h. The optimal point f = f* for dephasing due to flux noise in not necessarily the best point for a long relaxation time. However, as we see below, suppression of the dephasing time away from the optimal point is so strong that

80

60

40

20

i 1 1

- \ ,

1 1

1 1 1 1

T1 = 259 ns

I I I I

I (a)

I

-

-

-

-

Time (us)

Fig. 2. (a) TU (b) T2Ra and (c) T2echo mea­surements at the optimal point / = / * , /(, = /£ . Solid curves are exponential fits (exponential enve­lope for Ramsey signal) to the observed decay signals (circles).

it does not make sense to move away from that point.

We measured the energy relaxation time Ti by sending a sequence of two IT pulses. Population of the qubit ground state was measured as a function of the delay time between the two pulses. The experimental data was nicely fitted with an exponential decay giving T\ (Fig. 2(a)). Dephasing of the qubit was characterized by Ramsey inter­ference and spin-echo techniques. For Ram­sey interference two n/2 pulses with variable delay time in between were applied to the qubit. The carrier frequency of the pulses was detuned from the qubit frequency Eo\/h by 20 MHz so that the angle between rota­tion axes of each 7r/2 pulse rotates at the corresponding frequency and the fringes are visible. The switching probability vs. delay time was fitted with a 20 MHz sinusoidal function with an exponential decay envelope which gives T2Ramsf,y (Fig. 2(b)). The spin echo technique uses a sequence of two 7r/2 pulses with an intermediate w pulse located exactly in the middle of the delay time; the carrier frequency is not detuned. The echo technique is useful to reduce the effect of low frequency noise on the dephasing of the qubit. The experimental data was fitted with an exponential decay with a time constant of

T2echo (F ig . 2 ( c ) ) .

3. Results and Discussions

The experimental results indicate that: (i) At the optimal point ( / = /*, h = /£), echo-signal decay is dominated by energy re­laxation process (T2echo ~ 2Ti), indicating that there is no pure dephasing contribu­tion. On the other hand, Ramsey decay time is still shorter due to low frequency noises {T2Ramsey < T2echo) (Fig. 2). (ii) When the flux bias is shifted away from the opti­mal point ( / ^ /*, lb = I^), a characteris­tic Gaussian-type decay is observed in echo measurement (Fig. 3). In this case, it is not

57

T r -\ r (a) "

P,.d,o(l) = exp (-^tj exp (-r«/c„„t)

Rfogrf"^""0"0^ uoOnn°"ui, J I I I I l_

1 1 1 1 : 1 r (b) -

Pe(.j,„(t) = exp ( - y t ) exp (-rj„k„t2)

J i i i i i 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (us)

Fig. 3. Comparison of (a) exponential and (b) Gaussian fits (solid curves) to echo decay (circles) at / # /*> h = !£• The curve in (b) fits better. The corresponding / value is indicated in Fig. 4 by an arrow.

straightforward to define T^echo with an ex­ponential fit. Instead, we deduce the pure de-phasing rate Yyecho by fitting the decay with a product e x p ( - r i i / 2 ) e x p ( - r 2 e c ^ 2 ) . As

a function of / , Tvecho (YvRamsey obtained in a similar manner as well) scales linearly with QEZL (Fig. 4). These two facts can be explained by a presence of low-frequency flux 1 / / noise in the qubit20. A rough estimation

Fig. 4. r , / 2 ( . ) , / at if, = II. TlpRamsey and Tvecho are decay coeffi­cients of the Gaussian part of the envelope as seen in Fig. 3(b). Solid lines are 8E0i/df fits to TvRamsey

and Tyecho- The arrow indicates / value for the plots in Fig. 3.

0.6

0.5

0.4

0.3

0.2,

0.1'

I

- j[J$®

\ ,

^ 2 T ,

^ '2Ramsey

* ' 2echo

i

i

i 0.0 0.2 0.4

Fig. 5. 2Ti (o), T2Ramsey (o), and T2echo (•) vs. /(, at / = f*(Ib)- Scale of /;, is given as an output voltage of the source. The arrow indicates /;, value for the plots in Fig. 6.

(a)

I\,Ut) = exp (-^-tj exp (-VlZa)

J I I I I

- 1

" \

1 1 1 1 1 1 (b)

PccUt) = exp (-y t j exp (-[t.erJ,„(2) -

1 1 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time ( is)

Fig. 6. Comparison of (a) exponential and (b) Gaussian fits (solid curves) to echo decay (circles) at / = /*(/( ,) , h 7 !£• The curve in (a) fits better. The corresponding I/, value is indicated in Fig. 5 by an arrow.

from the fit in Fig. 4 gives us the noise spec­trum of S*(w) ~ ( l ( r 6 $o) 2 /w . (hi) When the bias current in the readout SQUID is shifted away from the optimal point ( / =

f*{h), h 7^ 4*) ; Ti, T2Ramsey, a n d T2echo

also decrease gradually (Fig. 5a). In this

aScale of Ii, is given as an output voltage of the source. In the set of measurements, a bias resistor whose resistance strongly depends on temperature

58

case, however, contrary to the case of a flux bias shift, the echo decay functions remain exponential, and no Gaussian component is seen (Fig. 6). Moreover, T2echo initially being close to 2Ti starts deviating from that and fi­nally approaches to T2Ramsey (Fig. 5). Con­sidering that the shift of /;, selectively intro­duces coupling of the qubit to the noise in the SQUID bias current, we can conclude from these observations that at large |/j, — /£ | the qubit dephasing is dominated by the current noise which has a relatively high-frequency component whose effect cannot be removed by the echo pulse sequence. It is also seen that ife-dependent part of T\ is proportional

to \-Qf) (data not shown). As Ti =

2f?(#02(i£)2^(¥) is e x P e c t e d at

/ = /* from a perturbation theory22 and as we know the derivatives from independent measurements, it is possible to estimate the current noise spectrum density at the qubit frequency Eoi/h.

Despite those observations and under­standings, there remain several things we have not understood in the decoherence mechanism of a flux qubit: e.g., (i) The ob­served behavior of Tj as a function of £oi m

a wide range (Fig. 7) is not understood in a simple model. Since the echo decay time is already limited by the energy relaxation time at the optimal bias point, it is very impor­tant to understand and improve the T-± time. Careful design of the electromagnetic envi­ronment of the qubit and caution for min­imizing coupling to microscopic degrees of freedom15'18 would be required. Although longer T\ times more than fis have been ob­served in similar samples (also in Ref. 11), the systematic tendency is not clear yet. (ii) The flux 1 / / noise suggested from r^cho

was used unintentionally. As a result, the absolute value and the linearity of I/, could not be determined. However, the qualitative argument above is not af­fected.

3 1 i- k aL I » , 11 m

0.1 r- ^ ~ ~ ^ ^ _ ^ -. 4 I I I > " — - : 0 5 10 15 20 25

Eo,/h (GHz)

Fig. 7. Ti vs. Eoi at /{, = JjJ\ E0i is varied by sweeping magnetic flux bias. The open circles are data points from f > f* and the filled circles are from / < /* . The solid curve depicts up to constant coefficient a dependence which is expected if the en­ergy relaxation is induced by frequency independent flux noise for the wide range of the qubit frequency Eoi/h.

measurements is consistent with what have been seen in the previous works24. However, the origin of the noise is still to be found.

4. Conclusion

We have investigated decoherence properties of a flux qubit. At the optimal bias condition with respect to the external magnetic field and the bias current of the readout SQUID, contribution from energy relaxation domi­nated the echo decay time. From the mea­surement of the decoherence times for var­ious bias conditions, the difference between the characteristics of the flux noise and cur­rent bias noise was revealed.

Acknowledgments

We thank O. Astafiev, A. Niskanen, P. Bertet, and A. Lupascu for useful discus­sions.

References

1. Yu. Makhlin, G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).

2. M. H. Devoret and J. M. Martinis, in Quantum Entanglement and Information Processing, eds. D. Esteve, J.-M. Rai-mond, and J. Dalibard (Elsevier, Amster­dam, 2004), p. 443.

59

3. J. M. Martinis, in Quantum Entanglement and Information Processing, eds. D. Esteve, J.-M. Raimond, and J. Dalibard (Elsevier, Amsterdam, 2004), p. 487.

4. E. Collin et al, Phys. Rev. Lett. 93, 157005 (2004).

5. I. Siddiqi et al, cond-mat/0507548. 6. T. Yamamoto et al, Nature425, 941 (2003). 7. R. McDermott et al, Science 307, 1299

(2005). 8. A. Wallraff et al, Nature 431, 162 (2004). 9. I. Chiorescu et al, Nature 431, 159 (2004).

10. J. Johansson et al, cond-mat/0510457. 11. Y. Nakamura, Yu. A. Pashkin, and J. S.

Tsai, Nature 398, 786 (1999). 12. D. Vion et al, Science 296, 886 (2002). 13. P. Bertet et al, Phys. Rev. Lett. 95, 257002

(2005). 14. A. Wallraff, et al, Phys. Rev. Lett. 95,

060501 (2005). 15. J. M. Martinis et al, Phys. Rev. Lett. 95,

210503 (2005). 16. A. Cottet, Ph.D. thesis, Universite Paris VI

(2002). 17. R. W. Simmonds et al, Phys. Rev. Lett. 93,

077003 (2004). 18. O. Astafiev et ai., Phys. Rev. Lett. 93,

267007 (2004). 19. T. Duty, D. Gunnarsson, K. Bladh, and P.

Delsing, Phys. Rev. 73 69, 140503(R) (2004). 20. G. Ithier et al, Phys. Rev. B 72, 134519

(2005). 21. J. E. Mooij et al, Science 285, 1036 (1999). 22. G. Burkard et al, Phys. Rev. B 71, 134504

(2005). 23. I. Chiorescu, Y. Nakamura, C. J. P. M. Har-

mans, and J. E. Mooij, Science 299, 1869 (2003).

24. F. C. Wellstood, C. Urbina, and J. Clarke, Appl Phys. Lett. 50, 772 (1987).

60

SPECTROSCOPIC ANALYSIS OF A CANDIDATE TWO-QUBIT SILICON QUANTUM COMPUTER IN THE MICROWAVE REGIME

J. GORMAN* and D. G. HASKO

Microelectronics Research Centre, University of Cambridge, Cambridge CB3 OHE, United Kingdom E-mail: john.gorman@cantab. net

D. A. WILLIAMS

Hitachi Cambridge Laboratory, Hitachi Europe Ltd, Cambridge CB3 OHE, United Kingdom

We report low temperature electron-transport measurements on two single-electron transistors operated simultaneously and coupled capacitively to two isolated double quantum-dots in silicon. The gate voltage dependencies and microwave induced responses are presented. A gate-voltage dependent anti-crossing type response with effective (3-value ~io5 and splitting energy 91 neV is observed. The sharp resonance peaks are attributed to population of charge states due to photon absorption from the incident microwave irradiation, and the peak splitting is attributed to the coupling between the isolated double quantum-dots.

Keywords: Quantum computer; quantum dot; silicon qubit.

1. Introduction r

A quantum computer will enable solutions to be found to classes of problems that are practically infeasible to solve by classical computation [1]. Solid-state systems are attractive candidates to build quantum computers because of their advantages over other systems, such as ease of fabrication, and well-developed addressing, manipulation and measurement techniques. One such 'top-down' solid-state approach is a patterned semiconductor quantum-dot system, where the electron confinement results in artificial atoms and molecules. These may be fabricated with sufficient decoupling from decoherence mechanisms, and offer a scalable qubit system. Single-electron transistors (SETs), as well as quantum-point contacts, are ideal measurement elements for potential charge qubit structures [2, 3]. Previous work has concentrated on the characterization of isolated double-quantum dots (IDQDs) measured by SET electrometers 1

through capacitive coupling [4, 5]. The f polarization energy of an IDQD was found to " be dependent on its dot diameter and the [. oxidation time during passivation [6]. More p

recently, charge-qubit operation of this system was demonstrated [7].

Here we show experimental results on the microwave-induced response of a circuit with two IDQDs and two SETs, towards the demonstration of entanglement and two-qubit operation in this scheme.

Fig. 1. (a) Scanning electron micrograph of the sample. The device consists of two IDQD and two SET elements, five in-plane DC gates, and an RF gate for microwave irradiation. The elements are coupled capacitively. (b) The material profile, showing the thicknesses of the individual layers, (c) Illustration of the IDQD after surface passivation, (d) The double-well confinement potential resulting in molecular states that embody the qubits.

61

2. Fabrication and method

Figure 1(a) shows a scanning-electron micrograph of the fabricated device, using high-resolution electron-beam lithography and reactive-ion etching to 'trench-isolate' the individual elements. Figure 1(b) shows the material profile in the perpendicular direction. The Phosphorous dopant concentration of the 35 nm-thick active region was 5.7xl019 cm"3. Passivation of the surface-charge trap states was achieved by thermal oxidation, which also reduced the dimensions of the confining silicon regions of the quantum-dots, illustrated in Figure 1(c). The resulting IDQD structures are made of silicon regions that are completely surrounded by amorphous oxide. The device was immersed in liquid helium at a temperature of T = 4.2 K. The RF gate forms part of a coplanar waveguide, which was matched the to room-temperature section.

3. Results

The single-electron transistors rely on the Coulomb-blockade effect to sense small changes in nearby charge through an induced modulation of their electrochemical potential energies. Therefore, a DC-characterization of the electrometers is necessary to establish the operating point for the SETs. Figure 2 shows two experiments where the gate voltages of the 'trimming-gates' VgX and Vg2 of the SETs were linearly swept for different source-drain potentials, and the currents II and 12 through the SETs were measured. Both devices show clear conductance oscillations due to Coulomb blockade. The periodicity of the oscillations is indicative of single-island behavior for both SETs. This is especially true for the response of SET2, where a gate capacitance of C , = 1.3 aF is observed. However, it is important to note that the gate capacitances will have dependencies on applied voltages and on the charge states of the IDQDs. The arrows in Figure 2 show the configured positions of the

SETs as highly-sensitive electrometers. The applied voltages were: V:= 3.06 V, V., = 2 mV, V, = 66.4 mV, and Vw, = 1 mV .

sd\ ' gl ' sd2

Having set-up the electrometers, a spectroscopic experiment was carried out by the application of a continuous-wave (CW) microwave signal at the RF gate, with output power P = -10 dBm at the sourcing unit. It was found that the cross-coupling between gate 4 and the electrometers was negligible, so that a large voltage (-Volts) applied to this gate resulted in a small (~mV) change in either electrometer. This allowed for the SET operating points to be maintained throughout the experiment. By contrast, experiments with similar voltages applied to Vg3 and Vg5 showed a large cross-coupling to their nearby SETs, and moved their operating points away from

Vg2 (V)

Fig. 2. Oscillations in the measured SET currents II and 12 as a function of gate voltages Vg\ and Vgj, for incrementing source-drain potentials V„d\ and Vsln around the linear-transport (Vsd ~ 0 V) regime. The arrows indicate the operating point of the SETs, where the gradient of the conductance function (cosh2) is large for change in applied gate voltage. The noise in 12 for Vg2 = 0.3-0.5 V is due to a two-state switching, which could be due to a random-telegraph signal (RTS) from a nearby trap state.

62

the charge-sensitive regime. This is shown in Figure 4(a), where the effect of incrementing Vg moves the operating point of SET2 and changes the visibility of the resonance observed at the microwave frequency / = 1.683469 GHz. Figure 3 shows the result where/was swept between 3.1 and 3.5 GHz for different Vg4. The upper graph shows measured II through SET1, and the lower graph 12 through SET2. Both responses show a broad centre-resonance, with an effective Q-value(/0/FWHM)«125, and centre frequencies /o

SET1 = 3.2667 GHz and f^" = 3.2625 GHz, respectively. It is unclear why there is a slight offset between the centre frequencies. A frequency shift in this broad resonance feature was not observed in this range of Vg4, suggesting that this may be a rectification-type process. However, a number of much sharper resonances were also observed. These can be seen in the range 3.15-3.2 GHz and 3.3-3.4 GHz for 11, and 3.2-3.25 GHz for 12. It was

II (nA)

Fig. 3. Response to CW microwave input observed in SET1 and SET2 in the frequency range 3.1-3.5 GHz. A broad central peak is observed, with numerous sharp resonance peaks on either side.

found that all of these sharper features could be shifted in frequency with gate voltages, with the strongest shift for changes in Vgi, VgA, and Vgs. Since these gates have a higher capacitive coupling to the IDQD structures than that of gates 1 and 2, the nature of these resonances is probably due to population of electron states with different polarizations in the IDQDs. Hence, an effective charge is induced on the SETs, which results in a change of their conductance and allows detection of the IDQD states. It is also evident that these sharp resonances do not occur at the same frequencies in both SETs, which implies that the IDQDs have different electron-state energies, and that SET1 mainly measures the electronic polarization of IDQD2, and SET2 that of IDQD1. Figure 4(a) shows a higher-resolution experiment performed to determine

1.6831 1.6832 1.6833 1.6834 1.6835 1.6836

/ ( G H z )

(b ) 11 (nA)

3.271 3.272 3,273 3.274

/ (GHz)

3.275 3.276

Fig. 4. (a) High 2-value resonance observed by SET2. The visibility and polarity of the resonance depends on the relative position of the SET on its Coulomb blockade oscillation, which can be tuned by Vgi. (b) Anti-crossing of two resonance peaks. Increasing Vg4 increases the splitting of the peaks, and modifies the occupation probability (left hand peak for Vg4 « 0, and right hand peak for Vg4 » 0).

63

the effective Q-\a\ue for one such resonance, which was measured to be ~2*105. This is consistent with measurements from an earlier device with only a single IDQD element [8]. It is important to note, however, that the detection is rather different than that of a 'traditional' spectroscopy experiment because of the strong non-linearity of the electrometers.

An in-depth analysis of the data in Figure 3 revealed that there were two main groups of resonances, which linearly shifted with different gradients for changes in Vg4. This meant that a few resonances inevitably crossed with each other for some values of Vg4. A frequency region close to the main peak f, showed an anti-crossing of two resonance peaks measured by SET1, shown in Figure 4(b). The two peaks around 3.2735 GHz are naturally split around Vg4 = 0, with an energy gap £ = 220 MHz = 91 neV, which increases as the magnitude of Vg4 is increased in either direction. The peak positions trace a nonlinear function similar to anti-crossing observed for interacting quantum states in quantum-dot systems [9, 10]. Since a single IDQD device results in only a single gradient shift in its spectrum for this type of experiment [8], it is concluded here that this novel observation is due to the interactions of the two IDQD system, mediated by the capacitive coupling. The two different gradients can be explained by the fact that gate 4 is situated in-line with the midpoint between the two IDQDs. Therefore, a change in Vg4 will influence the IDQD1 energy states in an opposite way to the influence on that of IDQD2, resulting in a net relative shift of the electron states of the two isolated structures. This explains the existence of the two gradients in the peak shift, and why Vg4 is so effective at modulating the energy-gap splitting ^.

4. Conclusion

In conclusion, we have successfully operated a novel device with two IDQDs and two SETs. Gate-voltage dependencies and microwave-

induced responses of the device were demonstrated. An anti-crossing response with tunable energy splitting was observed, indicating it may be possible to control the interaction between the qubit structures.

Acknowledgments

This work was supported by Hitachi Europe Ltd, and an UK EPSRC Basic Technology project 'Cryogenic Instrumentation for Quantum Electronics'.

References

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, (2000).

2. S. Gardelis et al, Phys. Rev. B 67, 73302 (2003).

3. M. H. Devoret and R. J. Schoelkopf, Nature 406, 1039(2000).

4. E. G. Emiroglu, D. G. Hasko, and D. A. Williams, Appl. Phys. Lett. 83, 3942 (2003).

5. E. G. Emiroglu, D. G. Hasko, and D. A. Williams, Microelectron. Eng. 73-74, 701 (2004).

6. M. G. Tanner, E. G. Emiroglu, D. G. Hasko, and D. A. Williams, Microelectron. Eng. 78-79, 195 (2005).

7. J. Gorman, D. G. Hasko, and D. A. Williams, Phys. Rev. Lett. 95, 090502 (2005).

8. E. G. Emiroglu, D. G. Hasko, and D. A. Williams, in 27 th International Conference on the Physics of Semiconductors (ICPS), edited by J. Menendez and C. G. Van der Walle (AIP Conf. Proc, 2004), Vol. 772, p. 1439.

9. T. Hayashi et al., Phys. Rev. Lett. 91, 226804/1 (2003).

10. W. G. van der Wiel et al., Rev. Mod. Phys. 75, 1 (2003).

64

BERRY PHASE DETECTION IN CHARGE-COUPLED FLUX-QUBITS A N D THE EFFECT OF DECOHERENCE

HAYATO NAKANO ( A ' B ' C \ SHIRO SAITO ( A ' C ) , HIDEAKI TAKAYANAGI (A 'C 'D) , AND

' 'NTT Basic Research Laboratories, Kanagawa 243-0198, Japan ( > Scuola Normale Superiore di Pisa, Toscana 56100, Italy

( ' CREST, Japan Science and Technology Agency, Saitama 332-0012, Japan * E-mail: nakano@will. brl. ntt. co.jp

ROSARIO FAZIO (B)

'• ' Scuola Normale Superiore di Pisa, Toscana 56100, Italy

We analyzed adiabatic geometric phase in a composite system; charge-coupled superconducting flux qubits. When we change two external magnetic fields along a contour, an finite off-diagonal Berry phase is accumulated between the ground and third excited states. The effect of decoherence on the Berry phase caused by magnetic field fluctuation is also examined numerically with a master equation. It seems that decoherence destroys energy-eigenstate superpositions much significantly than the values of Berry phase.

Keywords: superconducting flux qubits; composite system; Berry phase; decoherence.

1. In t roduc t ion

Superconducting nano-circuits are often used for experiments to demonstrate fundamen­tal quantum-mechanical phenomena in ar­tificial systems because the macroscopically coherent states are able to be easily con­trolled. Another important character of su­perconducting circuit with Josephson junc­tions is that they always have both charge and phase degree of freedoms at the same time. It makes the quantum behaviors in them very rich.

In the millennium year, Falci et al. pro­posed a Berry phase detection experiment in a superconducting charge-qubit with a SQUID geometry 1. In their proposal, the gate voltage for the Cooper-pair box and the external flux for the SQUID are adiabatically changed in order to encircle a finite Berry phase.

The dual analogy of their experiment us­ing a flux qubit with a small island 2 and a non-adiabatic two-qubit version 3 are able to be theoretically designed 2. It is a manifesta­

tion of the rich character of superconducting systems mentioned above.

Here, we propose and analyze a super­conducting circuit that is mainly composed of a charge-coupled flux-qubits[Fig. 1], in which we can detect a Berry phase by con­trolling only applied magnetic fluxes.

2. Sys tem analyzed

r-&

L-W

IT

0> M

•a - 1

qubit A "7T qubit B/7^7

Fig. 1. Effective circuit for a charge-coupled flux-qubit system.

The Lagrangian of the effective circuit in

65

Fig. 1 is given by

g A ( l + 2Q) 4 2e L = ~ ^ ^ '- =- 7 A - + - ^ 5 - 7A+

C A 2 2e

CB(l + 2a) 4 2e

+ ^~ 2~e ( 7 A

C*B ...2 2 2e

7B-

Cg \ - 2e T , , 2

+ 2-EJA cos[7A_/2] cos[7A+/2] -f- aEJA COS[7A-

-2TT/A] + 2BJB COS[TB-/2] COS[TB+/2]

+ aEJB COS[TB- + 2TT/B]

+ EJM cos[7A- - 7B- - 2TT(/A + / B + /M)].(1)

where

7D± = 7DI ± 7D2, (D = A, B) (2)

/ D E E $ D / $ O (D = A,B), $ 0 = V(2e). (3)

The Hamiltonian of this system can be approximated into

H = YIAOAZ + hsCTBz + A-A^Ax + &-B<*Bx

+QcAyCTBy + J(7AZ<TBZ + R&By, (4)

with Pauli operators for the qubit A and qubit B. h.A and hg are parameters that can be controlled by external magnetic fluxes ap­plied to the loops of the qubit A and the qubit B, respectively.

During the modification of the expres­sion we omitted terms which are constant for the change in fluxes. In Eq. 4, A D ( D = A , B )

is the tunneling matrix element of the qubit D, which is determined by -EJJD, -ECD = (2e)2/(2Co) and a. This and other parame­ters in Eq. (4) can be calculated numerically. Very roughly, they are estimated as follows.

hA « 4TT(/A - l/2)aE3A, (5)

hB « - 4 7 T ( / B - l/2)o£?jB , (6)

R « ABCgVg/(2e), (7)

J « -EJU COS[2TT/M], (8)

and Q is of the order of A A , A B or a little smaller.

3. Connection and Berry phase

The vector poten­tial (the connection) C(n) = (en |V^|e„) in the parameter space (HA, hB) of n-th energy-eigenstate can be calculated. Each connec­tion determines the Berry phase which is ac­cumulated in the the n-th energy-eigenstate after the one-cycle along a contour. However, the Berry phase of each eigenstate can not be observed. What can be measured is the Berry phase difference between two energy-eigenstates. For example, with the parame­ter; A = 1,Q = 0.8, R = 0.5, J = 0.1, we obtain approximately 0.1 radian of geomet­ric phase difference between the ground state and the third excited state after one adia-batic cycle, as described below.

Now we pay attention to the ground state and the third excited state. The differ­ence in Berry phases between the two states is determined by the difference SC = C(4) — C(l) in the connection. The calculated 5C are shown in Fig. 3(a). The contour;

HA

hB

\ = /O.8 + cos[0(t)]^ .52sin[0(t)] J '

with

0(0) = 0 — 0(100) = 2TT,

(9)

(10)

is used with the change of 0(t) from 0 to 2n. The line integration along the contour gives the Berry phase difference accumulated dur­ing the one cycle. Figure 3(b) shows the ac­cumulated Berry phase along the contour.

4. Results and Discussion

We actually have performed the calculation of the time evolution during the cycle nu­merically. In order to take into account the effect of decoherence, we calculated with a perturbative quantum master equation with the spectral density,

Je(uj) CJ

1+UJ2 (11)

66

The temperature was set k&T = 0.1. The two qubit system couples to the environment via OAZ + &Bz- This corresponds to the to­tal flux induced by the supercurrents in two qubits.

The initial state is the superposition (|eo) + \ez))/\/2. In order to eliminate the accumulation of dynamical phase, after one cycle (6(t) — 2it), TV pulse is applied and the state is reversed. The successive cycle moves on the contour in the opposite direc­tion. Therefore, the remaining accumulated phase should be just two times of the Berry phase difference acquired during one cycle.

Fig. 2. difference of the connection SC = C(4) — C(0) between the ground and 3rd excited states, (a) magnitude, (b) direction.

(a) eW | (b> eco

Fig. 3. (a) Berry phase acquired along the infinites­imal movement d on the contour, (b) accumulated Berry phase difference between the ground and 3rd excited states.

Figure 4 is an example of the time evo­lution. The fidelity for the initial state is plotted. Each of figures corresponds to the "starting", "in the vicinity of ir pulse applica­tion", or "end of two cycles". Horizontal axis

is "time xlOO". Here, T = 0.004, and the an­gular frequency of the Larmor oscillation is approximately 2TT. At "time xlOO = 10000", the 7r pulse was applied. If there were no Berry phase, the fidelity should place at a minimal at the end of the evolution ('time xlOO = 20000). However, the time of the minimal is shifted a little. This is the Berry phase accumulated during two cycles.

The effects of the decoherence are shown in Fig. 5. (a) shows the Berry phase accu­mulated during the two cycles vs. strength of the decoherence. (b) shows the decay of the amplitude of the Larmor oscillation. The obtained Berry phase quite agrees with the value shown in Fig. 3(b). (Note: Fig. 3(b) shows the Berry phase for one cycle, Fig. 5(a) shows the Berry phase for two cycles.) With a strong decoherence ( r > 0.02), the oscillation amplitudes are decayed by four times. This suggests that coherent super­position is destroyed and it becomes useless any longer. However, the Berry phase is only deviated by approximately 10%. This might show that Berry phase itself is robust against decoherence, as often discussed 4 '5.

G. De Chiara and G. M. Palma discussed decoherence effect on Berry phase with a slow fluctuation environment 4 . They did not find the Berry phase variation with the strength of decoherence. The weak change in accumulated Berry phase[Fig. 5(a)] with the strength of decoherence resembles the effect of the Lam shift on Berry phase, pointed out in the work by R. S. Whitney et al. 5 . How­ever, it should be examined more in detail to identify the origin because how our lowest order perturbation master equation reflects the "quadrupole" effect in their discussion is not so clear so far.

In actual experiments, the Berry phase shown in Fig. 5(a) or discussed in works above can be obtained by averaging a lot of results of measurement. Therefore, the distribution of each measurement result is also important. In our calculation, we ob-

67

tain a final density operator, which is a mixed state. A mixed state is a probabilistic super­position of pure states, each of which is an eigenstate of the density operator, and the eigenvalues correspond to probabilities. The pure states have different accumulated Berry phases. If these Berry phases are much dif­ferent from one another, can we claim that Berry phase is robust against decoherence ? It is another problem we should investigate. Our numerical calculation shows that deco­herence gives Berry phase difference between each pure states in the density operator, al­though the result is not shown in this paper. However, so far, we do not find a good mea­sure to estimate such kind of destruction in Berry phase.

Now we are preparing to carry out the non-Markov calculation which should include almost all important decoherence effects on geometric phase.

1 I I

1 t

(a)

.003 0 .004

r

(b)

I

Fig. 5. Effect of decoherence on (a) Berry phase (b) Oscillation amplitude of the superposition state.

1 V A / Y

V V 150

A / A

Fig. 4. Time evolution of the fidelity. This oscilla­tion is the Larmor oscillation for the superposition of the ground and the third excited states.

5. Conclusion

We analyzed Berry phase in charge-coupled superconducting flux qubits. When we change two external magnetic fields along a contour, an finite off-diagonal Berry phase is accumulated between the ground and third excited states. Decoherence on the Berry phase caused by magnetic field fluctuation seems to destroy energy-eigenstate superpo­sitions much significantly than the values of Berry phase.

References

1. G. Falci, R. Fazio, G.M. Palma, J. Siewert, and V. Vedral, Nature 407, 355 (2000).

2. H. Nakano, unpublished. 3. W. Xiang-bin and K. Matsumoto, Phys.

Rev. B 65, 172508(2002). 4. G. De Chiara and G.M. Palma, Phys. Rev.

Lett. 91, 090404 (2003). 5. R.S. Whitney, Y. Makhlin, A. Shnirman,

and Y. Gefen, Phys. Rev. Lett., 94, 070407 (2005).

68

LOCALLY OBSERVABLE CONDITIONS FOR THE SUCCESSFUL IMPLEMENTATION OF ENTANGLING

MULTI-QUBIT Q U A N T U M GATES

H O L G E R F . H O F M A N N

JST-CREST, Graduate School of Advanced Sciences of Matter, Hiroshima University, Kagamiyama 1-3-1, Higashi Hiroshima 739-8530, Japan

RYO O K A M O T O A N D SHIGEKI T A K E U C H I 2

Research Institute for Electronic Science, Hokkaido University, Kita-12, Nishi-6, Kita-ku, Sapporo 060-0812, Japan, 2 JST-CREST

The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.

Keywords: quantum computation, process fidelity, entanglement capability

1. Introduction

Quantum gates are designed to perform tasks that are far more complex than those achieved by classical gates. To test ex­perimental realizations of quantum gates, it is therefore necessary to identify the char­acteristic non-classical features that define the quantum nature of the gate. In recent reports of experimental gates, these non-classical features were either characterized by the capability of generating entanglement 1,2,3,4 o r ky- fmj quantum process tomogra­phy 5>6'7>8. However, the complexity of such tests will increase rapidly as the number of qubits increases. A more intuitive and sim­ple method for testing quantum gate perfor­mances is therefore desirable.

In the following we show that the quan­tum coherent properties of a gate operation can be tested efficiently by using a pair of operations on two complementary sets of or­thogonal input states 9>1 0 'n . The classi­cally defined fidelities of these two operations then provide upper and lower bounds for the

quantum process fidelity. This estimate of the process fidelity can then be used to pre­dict the performance of the gate for other in­put states. In particular, the fidelity of gen­erating maximally entangled outputs from product state inputs can be estimated us­ing only the fidelities of local non-entangling operations. The entanglement capability of multi-qubit gates can thus be evaluated with­out actually generating any entanglement.

2. Noisy gate operations and process fidelity

Ideal quantum gates are described by a uni­tary operation UQO, representing the effects on both the amplitude and the phase of an arbitrary quantum state input. Using the computational basis {| nz)} for the input states, this ideal operation can be defined by

#oo|nz>=|/n>- (1)

However, any experimental implementation will include noise effects due to decoherence. To classify these noise effects, it is useful to

69

expand them into a set of orthogonal unitary operations.

For TV qubits, this is easily accomplished by applying all possible combinations of phase and amplitude errors to the input states. Specifically, a phase error is repre­sented by the Pauli matrix Z and an ampli­tude error is represented by the Pauli ma­trix X. The set of local operations applied to the input are then given by the single qubit-operations J, X, Z, and ZX = iY. The multi-qubit error can therefore be repre­sented by a product of phase and amplitude errors, 11 - = $iAj, where $i is a product operator of N local I and Z operators de­scribing the distribution of phase errors and Aj is an product operator of N local / and X operators describing the distribution of am­plitude errors. The indices i and j can be given by iV-bit binary numbers, so that the location of the errors are given by the dig­its with a value of 1. With this definition of error operators, a set of 4N orthogonal oper­ations representing the possible errors of t/oo can be constructed by

Uij = C/ooIIjj,

with TriQUki} = 2N6ikSjl. (2)

Any operator can now be expressed as a lin­ear combination of the unitary operations

Uij. In general, a noisy experimental process

is described by a superoperator E(p) that transforms an arbitrary input density matrix /5jn into the corresponding output density matrix pont = -E(/5m). This superoperator can be written as a process matrix with ma­trix elements XijM by expanding it in terms of the orthogonal unitary operations Uij,

E(P) = E XimUiiPUl- (3) i,j,k,l

The process fidelity is then defined as the overlap of the process matrix with the ideal process Uoo, given by the matrix element Xoo.oo = ^process, and the diagonal elements

XijAj °f the process matrix can be interpreted as the probabilities of the phase and ampli­tude errors ij.

3. Observable effects of quantum errors

The classical operation performed by a quan­tum gate is tested by preparing input states in the computational basis states | nz) and measuring the fidelities of the predicted out­put states | / „} . The classical fidelity of this operation is equal to the average probability of obtaining the predicted results for all 2N

possible inputs,

n

= ^NT,<^\E(\nz)(nz\)\fn). (4) n

Using the definition of the output state given in eq. (1), the definition of the errors in eq. (2), and the definition of the process matrix in eq. (3), it can be shown that the classical fidelity is given by

Fz = X^Xio.io- (5)

The classical fidelity Fz can therefore be identified with the sum of the 2N diagonal elements of the process matrix correspond­ing to phase errors only.

Eq.(5) is a quantitative expression of the fact that the classical fidelity Fz is not sen­sitive to phase errors in the Z-basis inputs. In order to obtain information about these phase errors, it is therefore necessary to use a different input basis. As the operator rep­resentation of the errors indicate, it is con­venient to use the eigenstates of the local X operators for this purpose. This set of states {| nx)} forms an orthogonal basis set which is complementary to the computational ba­sis in the sense that the overlap between any two states taken from the different basis sets

70

is

Kn* I »',>|2 = Q ) • (6)

The effect of the unitary operation C/oo o n

the complementary basis states is given by

Uoo\nx)=\gn). (7)

Since the input states | nx) are eigenstates of the amplitude errors Aj this operation is not sensitive to the amplitude errors. The classical fidelity Fx of the complementary op­eration is therefore given by the sum of the 2N diagonal elements of the process matrix corresponding to amplitude errors only,

F* = ^N ] C P(9n I nx) = J2 Xojfij • (8) ™ 3

By combining the two complementary classi­cal fidelities, it is therefore possible to obtain a fidelity measure that is sensitive to all error syndromes,

Fx + Fz-1 = xoo.oo - 5 Z XVMJ- (9)

Since the only positive contribution to this fi­delity measure is the process fidelity xoo,oo = -Fprocess J this measure is necessarily equal to or lower than the process fidelity. On the other hand, each of the classical fidelities Fz

and Fx is necessarily equal to or greater than the process fidelity. Therefore, it is possible to obtain lower and upper bounds of the pro­cess fidelity with 9

Fz + Fx-1 < Fprocess < Min{Fz,Fx}. (10)

This estimate of the process fidelity provides lower limits for all other properties of the quantum gate. It is thus possible to predict the wide range of possible quantum opera­tions from the operations on only 2N+1 local input states.

4. Application to entangling operations

One of the essential features of quantum computation is the generation of multi qubit

entanglement. The most simple case of N-qubit entanglement generation is a series of controlled-NOT operations, where the Z-state of the first qubit decides whether the other qubits are flipped or not,

#oo =|o z)(o 2 | a / " - 1 * | i 2 )<i 2 | ®xN~\ ( i i )

This operation generates a maximally entan­gled ./V-qubit Greenberger-Horne-Zeilinger (GHZ) state if the first qubit input is in an X eigenstate and the remaining N — 1 qubit inputs are in Z eigenstates,

£>oo|0T ,02 ,02 , . . .) = - J=( |0 2 , 0 2 , 0 2 , . . . )

+ |1 2 , 1 2 , 1 2 , . . . ) )

Uoo\lx,0x,0z,...) = - W | 0 2 , 0 2 , 0 2 , . . . )

- | l z , l „ l „ . . . »

'• (12)

However, if all input qubits are in the Z or X basis, the operation Uoo produces only lo­cal Z or X basis output states. The process fidelity -Fprocess c a n therefore be estimated from the local classical fidelities Fx and Fz. Since the fidelity Fen^ of the entanglement generating operation (12) is necessarily equal to or greater than -Fprocess, the minimal fi­delity of ./V-qubit entanglement is equal to Fz+Fx- 1.

To obtain a measure of the entanglement capability, it is useful to consider that any state with a GHZ state component of more than 50% ((GHZ | p \ GHZ) > 1/2) is an ./V-qubit entangled state. The minimal en­tanglement capability C can thus be defined as 2F e n t — 1, and the estimate obtained from the local classical fidelities Fz and Fx reads

C > 2FZ + 2FX - 3. (13)

The multi-qubit gate is therefore capable of generating entanglement if the average of the local classical fidelities Fz and Fx is greater than 3/4. A pair of local non-entangling op-

71

erations is thus sufficient to demonstrate the entanglement capability of the ./V-qubit gate.

5. Conditions for the violation of local realism

Since the local classical fidelities Fz and Fx define a minimal amount of multi-qubit entanglement capability, it is possible to know whether the gate operation violates the expectations associated with local real­ism without ever generating actual entangle­ment.

In the case of a 3 qubit operation, the GHZ paradox is based on the correlation 12

KGwl = X®X®X-X®Y®Y

-Y®X®Y -Y®Y®X. (14)

For any combination of classical variables X,Y = ± 1 , this correlation is equal to ±2. However, the quantum mechanical expecta­tion values of the operator - ^QJJZ a r e ^ " twice as high as the result of any local hidden variable theory. Quantum states can there­fore violate the GHZ inequality I ^ Q J J Z I — ^ by as much as a factor of two.

If the entangling operation given by eq. (12) works perfectly, it generates output states with the extremal value of I-KQJJZI =

4, clearly violating local realism. If the pro­cess fidelity is smaller than one, the minimal value of the above correlation is obtained by assuming that all errors result in the opposite eigenvalue of KQjj£,

l-^GHZI - ^process - 4. (15)

Local realism is therefore violated for ^process > 3/4. This condition is necessarily fulfilled if the average of the classical fideli­ties Fz and Fx exceeds 7/8,

±(Fz+Fx)>7/8. (16)

An average classical fidelity greater than 7/8 in the two local operations on X and Z basis input qubits therefore ensures that the oper­ation of the three qubit gate can generate a

violation of local realism from non-entangled inputs. Similar conditions can be derived for any number of qubits.

6. Conclusions

The performance of a quantum gate can be estimated efficiently by testing only two sets of complementary input states, e.g. the com­putational Z-basis and the X basis generated by local Hadamard gates on Z-basis inputs. By obtaining estimates of the process fidelity, it is possible to predict the performance of the gate for any other set of input states. As a result, the performance of entangling gates can be estimated from the non-entangling lo­cal operations observed in the Z and X basis. The quantum features of such multi-qubit gates can thus be predicted from the perfor­mance of entirely classical local operations. In particular, it is possible to show that a gate operation violates local realism without actually generating any entangled state.

References

1. F. Schmidt-Kaler et al., Nature (London) 422, 408 (2003);

2. J.L. O'Brien et al, Nature (London) 426, 264 (2003);

3. S. Gasparoni et al, Phys. Rev. Lett. 93, 020504 (2004);

4. Z. Zhao et al., Phys. Rev. Lett. 94, 030501 (2005).

5. J.L. O'Brien et al., Phys. Rev. Lett. 93, 080502 (2004);

6. Y.-F. Huang et al., Phys. Rev. Lett. 93, 240501 (2004).

7. N.K. Lang et al., e-print quant-ph/0506262 (2005).

8. N. Kiesel et al., e-print quant-ph/0506269 (2005).

9. H.F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005).

10. R. Okamoto, H.F. Hofmann, S. Takeuchi, and K. Sasaki, e-print quant-ph/0506263 (2005).

11. H.F. Hofmann, e-print quant-ph/0407165, to be published in Phys. Rev. A.

12. N.D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).

72

STATE CONTROL IN FLUX QUBIT CIRCUITS: MANIPULATING OPTICAL SELECTION RULES OF MICROWAVE-ASSISTED

TRANSITIONS IN THREE-LEVEL ARTIFICIAL ATOMS

YU-XI LIU ( A ) , J. Q. YOU ( i 4 ) , L. F. WEI ( j 4 ) , C. P. S U N ^ ' B \ AND FRANCO NORI ( j 4 'C )

(A) Frontier Research System, RIKEN, Wako-shi 351-0198, Saitama, Japan (B) Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China (C) Center for Theoretical Physics, Physics Department, Center for the Study of Complex

Systems, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

We analyze optical selection rules of microwave assisted transitions in a superconducting flux qubit circuit (SQC). We show that the parities of the states relevant to the superconducting phases in the SQC are well-defined at a special value <3>o/2 of the external magnetic flux $e. When <Se *o/2, the symmetry of the potential of the SQC is broken, and a so-called A-type cyclic three-level artificial "atom" is formed, where one- and two-photon processes can coexist.

Keywords: Flux qubit; Adiabatic state control.

1. In t roduc t ion

Analogous to natural atoms, superconduct­ing quantum circuits (SQCs) can possess dis­crete levels. Micro-chip electric circuits 1 '2

show that quantum optical effects can also appear in artificial atoms, allowing quantum information processing in such circuits by using cavity quantum electrodynamics (e.g., Refs. 1 , s '4 and a similar proposal 5 ) .

A superconducting loop with three junc­tions threaded by an externally applied mag­netic flux (AMF) can be considered as a flux qubit 6 '7 '8. Because for an appropriate AMF, this device can have two stable states, which are well isolated from other upper states and correspond to dc currents flowing in opposite directions. The computational basis states of the qubit are defined by these two states of opposite currents.

Quantum optical technology, developed for atomic systems, can be applied to manip­ulate quantum states of artificial atoms. For example, the selective population transfer, based on the stimulated Raman adiabatic passage (STIRAP) with A-type atoms 9, has been applied to superconducting flux qubits 10. How to probe the decoherence

of flux qubit states using electromagnetically induced transparency has also been inves­tigated n by setting the AMF away from the optimal point. In these proposals, e.g., in Ref. n , an auxiliary state is required to form a three-level system with A-type tran­sitions, where the transitions between two lowest states are forbidden. But, the A-type transitions cannot be really formed in the flux qubit circuits. Because the transition between two lowest states (qubit states) is also possible when the AMF is away from the optimal point. Then the classical fields applied for A-type transitions in flux qubit circuits might result in a non-adiabatic evo­lution. Recently, the effect of pulses on the adiabatic operation of the flux qubit studied by us 12 after considering all possible transi­tions.

Here, we mainly discuss how the phases of pulses affect the adiabatic operation of the qubit states by virtue of the third auxiliary state. This paper includes three parts: we discuss the interaction between the flux qubit and external magnetic fields, the Hamilto-nian will be given in Sec. 2; then the adia­batic evolution is studied in Sec. 3; and fi­nally, conclusions are given.

73

Fig. 1. (a) Superconducting flux qubit circuit, each junction is marked by an X, $ ( t ) and <£e de­note time-dependent and time-independent magnetic fluxes, respectively. Here, we choose a = 0.8. C J I = Cj2 = Cj and Cj3 = a C j . (b) Transition diagram of three energy levels versus reduced flux / . The right and left figures correspond to / = 0.5 and / ^ 0.5 in (b), respectively. The dotted-line with the cross sign in (b) means that the transition between these two levels is forbidden.

2. In terac t ion b e t w e e n flux q u b i t a n d e x t e r n a l m a g n e t i c field

We consider a flux qubit circuit composed of

a superconducting loop with three Joseph-

son junctions (e.g., Refs. 6 ' 7 ) , as shown in

Fig. 1(a). The two larger junctions have

equal Josephson energies JBJI = Ej2 = Ej

and capacitances C J I = Cj2 = C j , while

for the third junction: £ j 3 = aEj and

Cj3 = aCj, with a < 1. The Hamiltonian

of this circuit is

Hn 2MV 2M„

+ U(<Pp,<Pm), (1)

where the momenta can be writ ten as Pp =

—ihd/difp and Pm = —ihd/dipm. The ef­

fective masses are Mp = 2Cj($o/27r) 2 and

Mm = Mp(l + 2a). The effective potential

U(tPp,<Pm) IS

U(ipp,<Pm) = 2-Ej(l - COS tpp COS tpm)

+ aE3[\-caa(2irf + 2tpm)],

where (pp = (tpi + <f2)/2 and <pm = {tpi -

<fi2)/2 are defined by the phase drops <pi

and <f2 across the two larger junctions; / =

3>e/<]>o is the reduced magnetic flux, applied

through the qubit loop. We found 12 t ha t

the three lowest energy levels are well sepa­

rated from other higher levels near the point

/ = 1/2 for given parameters , e.g. a = 0.8

and Ej = 40f?c with the charge energy Ec =

e 2 / 2 C j . Then, the two lowest levels form a

two-level artificial atom, called qubit . W i t h

an auxiliary th i rd level, a three-level artificial

a tom is formed. Usually, the third level can

assist to perform the qubit operations by ap­

plying the appropriate classical or quantized

optical fields.

We consider quantum transit ions be­

tween different energy levels, induced by a

t ime-dependent magnetic flux $(£) through

the superconducting loop. For a small $(£),

the <fm— dependent per turbat ion Hamilto­

nian reads

Hi{<pm,t) = / * (* ) = I*™ cosfajt), (2)

where

/ = -(2TraEj/$0)sm(2Trf + 2<pm) (3)

is the circulating supercurrent when the ap­

plied magnetic flux $(£) = 0. The ampli tude

$ i is now assumed to be t ime-independent.

The transitions are determined by the ma­

tr ix elements Uj = (i\I$a(t)\j) for the ith

and j t h eigenstates \i) and \j), with eigen­

values Ei and Sj, respectively.

When the reduced magnetic flux / =

1/2, the effective potential U(ipp,<pm) in

Eq. (2) has a center of inversion symme­

try, tha t is, U(<pp,(pm) = U(-<pp,-<pm)-

And the interaction Hamiltonian in Eq. (2)

is an odd function of the variables ipp and

ipm. Then, the microwave-assisted transi­

tions have the same selection rule as for

the electric-dipole interaction in usual a toms.

Thus , a S-type (or ladder) t ransi t ion can be

formed for the lowest three levels, as shown

in the right par t of Fig. 1(b). However, if

the reduced magnetic flux / deviates from

this special point, then the symmetry of the

potential U(<pp, ipm) is broken and the selec­

tion rule does not exist. In this case, the

transit ions between any two of the lowest

three levels are possible, a A- type t ransi­

tion is formed, as shown in the left pa r t of

Fig. 1(b).

74

3. Adiabatic evolution

Now we consider the interaction between the qubit and the time-dependent three mi­crowave electromagnetic field

2

*a(t)= £ [*m„(i)e- iuW + h.c], m > 7 i = 0

which includes three different pulses, and the ®mn (t) vary slowly on the time-scale of the pulses, where u;mn are the pulse carrier fre­quencies. If u>mn is resonant or near-resonant with the transitions among the non-adiabatic (i.e., diabatic) states \m) (m = 0,1,2), the total Hamiltonian in the interaction picture can be written under the rotating wave ap­proximation as

2

# i n t = J2 (nmn(t)e^™t\m)(n\+h.C.), m>n=0

where the complex Rabi frequencies Qmn(t) = (m\I$mn(t)\n) and the detuning A m n =wm-wn- tjmn with wm = sm/h.

The instantaneous adiabatic eigenvalues of Hint are given by

Ek = 2|»(t)|

v/3 cos > 2w(k - 1)]

corresponding to the adiabatic states \Ek) with k = 1, 2, 3. The parameter 6 can be determined by

3\/3 cos 9 = -Re[n01(t)fl12(t)fl2o(t)e' i u / t i

\m\3' with \n(t)\2 = \n12(t)\

2+\n2o(t)\2+\^oi(t)\2

and w' = Aoi + A12 — Ao2- The eigenvalues are sensitive to both the total phase <j> of the product fioi(i)fii2(i)fi2o(i) = fl'(i) and de-tunning LO' . It can be found that 6 = 0 or •K when |fi0i(*)| = |012(*)| = |£22o(*)| and /? = w't + (p = nir. In such case, there are energy level crossings, the adiabatic description of the time evolution is no longer correct. Com­paring with the typical A-type atom 9, the time-evolved zero eigenvalue £3 = 0 (corre­sponding to £1 = \Q(t)\, E2 = -|ft(*)|) can also be found for Hmi when (3 = (2p+ l)7r/2,

for integer p. In this case, the evolution is adiabatic. Contrary to the A-type atoms, the eigenvalues only depend on the amplitudes of the pulses 9.

Fig. 2. The phase-dependent eigenvalues Ek at t = 3r (solid curves) and t = 3 . 5 T (dotted curves), with time delays T\ = T2/2 = T3/2 = 2r. Here eigenvalues are given in order E\, E3, E2 from the top curve to the bottom curve for the same color and /3 = u>'t + <p.

Let us now consider Rabi frequencies Qmn(t) as Gaussian envelops with the width Ti (i = 1, 2, 3). For example, we assume that the amplitudes, phases, and central times of f*2i(*) (fiio(t), and fi2o(*)) are fi0 (0.9 fi0, and 0.85 fl0), 4>i {4>2 and <j>$), and T\ (T2

and T3) with fio > 0. Fig. 2 plots a few snapshots of the eigenvalues Ek (t) versus the phase /3 of fi'(t), for the given pulse central times T\ = T2/2 = T3/2 = IT. It shows that two of the eigenvalues Ek are closer to each other for three points (<fi = —7r, 0, 7r) in the range \cj>\ < n, when t = 3r and u>' = 0. If the pulses related to ilmn do not have a signifi­cant overlap (e.g., when t = 3.5r), then the three energy levels are well separated. Here, we should emphasize that different central times Ti of pulses correspond to the different time evolution for given Rabi frequencies f2;, so Ti should be well chosen such that at least |fti(*)| = |ft2(*)l = \^3(t)\ can be avoided.

The control of the quantum state can be realized by choosing the classical pulses such that the diabatic components in the adia-

75

batic basis states \Ek) can be changed during

the adiabatic evolution. The adiabatic evolu­

tion requires that the nonadiabatic coupling

(Ek\-jftEi) and adiabatic energy differences \Ek — Ei\ satisfy 1 3 the condition

FM

(Ek\&E,)

(Ek-Et) < 1 . (4)

For given pulses, it implies tha t the adia­

batic condition depends not only on the total

detuning w' and individual detunings A m „ ,

but also on the total phase <f>. For exam­

ple, the time evolutions of Fki are illustrated

in Fig. 3 by showing only one representa­

tive top curve for the central t imes (e.g.,

T\ = 2 r 2 / 3 = r 3 / 2 = 2 T ) in the resonant

case (e.g., A m „ = 0), for two special phases

<j> = •K and <f> = 7r/2. For fixed envelopes of

the Rabi frequencies, Fig. 3 shows tha t the

adiabatic condition F « < 1 is valid when

4> = 7r/2, but it is invalid when <f> = n.

In conclusion, we analyze the adiabatic

evolution of quantum s ta tes of a flux qubit. We show tha t its quantum states can be con­

trolled via the amplitudes and phases of the

three applied pulses.

R e f e r e n c e s

J.Q. You and F. Nori, Phys. Rev. B 68, 064509 (2003); Physica E (Amsterdam) 18, 33 (2003); J.Q. You, J.S. Tsai, and F. Nori, Phys. Rev. B 68, 024510 (2003). A. Wallraff et al, et al., Nature 431, 162 (2004); I. Chiorescu ibid. 431, 159 (2004). Yu-xi. Liu, L.F. Wei, and F. Nori, Euro­phys. Lett. 67, 941 (2004); Phys. Rev. A 71 , 063820 (2005); ibid. 72, 033818 (2005). A.M. Zagoskin et al., Phys. Rev. A 70, 060301 (R) (2004). L.F. Wei, Yu-xi Liu, and F. Nori, Europhys. Lett. 67, 1004 (2004); Phys. Rev. B 71 , 134506 (2005). I. Chiorescu et al., Science 299, 1869 (2003). T.P. Orlando et al., Phys. Rev. B 60, 15398 (1999). J.Q. You, Y. Nakamura, and F. Nori, Phys. Rev. B , 71 , 024532 (2005). M.O. Scully and M.S. Zubairy, Quantum Op­tics (Cambridge, Cambridge, England, 1997). Z. Zhou, S.I. Chu, and S. Han, Phys. Rev.

B 66, 054527 (2002). K.V.R.M. Murali et ai, Phys. Rev. Lett. 93,

087003 (2004) 12. Yu-xi Liu, J.Q. You, L.F. Wei, C.P. Sun,

and F. Nori Phys. Rev. Lett. 75, 087001 (2005).

13. C.P. Sun, Phys. Rev. D 41, 1318 (1990).

10.

11.

t/T 4

Fig. 3. The representative top curves of F^^t) axe plotted for two phases <j> = ir (dashed curve) and 4> = 7r/2 (solid curve) with the pulses given in Fig. 2 for central times T\ = 2T2/3 = T3/2 = 2T; here, Am.n = 0. The dash-dotted line denotes F^i(t) = 1.

76

THE EFFECT OF LOCAL STRUCTURE A N D NON-UNIFORMITY ON DECOHERENCE-FREE STATES OF CHARGE QUBITS

T E T S U F U M I T A N A M O T O * A N D SHINOBU F U J I T A

Advanced LSI Technology Laboratory, Toshiba Corporation, Kawasaki 212-8582, Japan

* E-mail: tetsufumi. tanamoto @toshiba. co.jp

We analyze robustness of decoherence-free (DF) subspace in charge qubits when there are a local structure and non-uniformity that violate collective decoherence measurement condition. We solve master equations of up to four charge qubits and a detector as two serially coupled quantum point contacts (QPC) with an island structure. We show that robustness of DF states is strongly affected by local structure as well as by non-uniformities of qubits.

Keywords: charge qubitjquantum dot;quantum point contact;decoherence

1. Introduction

Decoherence-free(DF) states * are effectively designed for collective decoherence environ­ment, even if there is a small symmetry breaking perturbation parameterized by rj in the order of 0(r?) (rj -C l ) 2 . Nowadays, ex­periments for DF states have been successful up to four qubits in photon system3 and in nuclear magnetic resonance (NMR)4. How­ever, in solid-state qubits, it seems to be more difficult to realize DF states for the col­lective decoherence environment than in the case of optical or NMR qubits. This is be­cause we could not prepare plenty of qubits with mathematically exact size, because the sizes of Cooper-pair box5'6 or quantum dots (QD)7 '8 '9 are less than hundreds of nm, and moreover, the solid-state qubit system is ar­ranged compactly resulting in being often af­fected by local structures such as electrodes or electrical wires.

Here we theoretically describe the effect of non-uniform environment on DF states of charge qubits composed of coupled QDs, con­sidering the measurement process, by using time-dependent density matrix (DM) equa­tions. The charge distribution of the qubits changes the QPC current capacitively, result­ing in detection and corruption of charged state (backaction)10. Thus, measurement is a basic and important decoherence process that induces shot noise environment here and

should be investigated in detail. In particu­lar, we discuss the effect of a local structure, an island in the QPC detector, on the DF states combining with the non-uniformity of qubits based on the setup shown in Fig. 1(a), and compare them with two-qubit states shown in Fig. 1(b). We treat four-qubit DF states written as:

|4ri4]>(1234) = | [ | 0 1 ) - | 1 0 ) ] ( 1 2 ) ® [ | 0 1 } - | 1 0 ) J ( 3 4 ) ,

l*241>(1234) = ^ [ 2 I ° 0 1 1 > ~ I 0 1 0 1 ) - l ° 1 1 0 )

- |1001> - | 1 0 1 0 ) + 2 | 1100 ) ] ( 1 2 3 4 )

l44l>(1234) = l*l11)(1432) (1)

where |1001)(1234) = |1)1|0>2|0)3|1>4 and so on. These DF states are compared with two qubit Bell states: \a) = (| | | ) + | TT»/\/2, \b) = (I II) -I TT»/>/2, |c> = (| |T) +| Tl »A/2,|d>=(UT>-IU»A/2.

Fig. 1. Qubits that use double dot charged states are capacitively coupled to a QPC detector.

77

2. Formulation

We show the formulation for Fig. 1(a). The Hamiltonian for the combined qubits and the QPC for Fig. 1(a) is written as H = Hqb+Hqpc+Hini. Hqb describes the interact­ing four qubits: ifqb = J2t=i {^iaix+U<yiz) + Hi=iJi,i+-L<?iz<7i+iz, where fi, and e* are the inter-QD tunnel coupling and energy differ­ence (gate bias) within each qubit. Here, spin operators are used instead of annihilation op­erators of an electron in each qubit. Ji^+\ is a coupling constant between two nearest qubits, originating from capacitive couplings in the QD system n . | | ) and | | ) refer to the two single-qubit states in which the ex­cess charge is localized in the upper and lower dot, respectively.

The two serially coupled QPCs are de­scribed by

^c=^^EiJiaSciaS+ViaS[c\aads+d\ciaS^

(2)

• • i • i

a = L,Ri =T,1

+ J2 Ed4ds + Ud^d^ .

Here, CiLS(ciRS) is the annihilation opera­tor of an electron in the i^th (iflth) level (*L(*A) = 1, ».,n) of the left (right) elec­trode, ds is the electron annihilation operator of the island between the QPCs, EiLS(EiRS) is the energy level of electrons in the left (right) electrode, and Ed is that of the is­land. Here, we assume only one energy level on the island between the two QPCs, with spin degeneracy. ViLS (ViRS) is the tunneling strength of electrons between the left (right) electrode state i^s (IRS) and the island state. U is the on-site Coulomb energy of double occupancy in the island.

Hint is the capacitive interaction be­tween the qubits and the QPC, that induces dephasing between different eigenstates of OiZ 1. Most importantly, it takes into account the fact that localized charge near the QPC increases the energy of the system electro­statically, thus affecting the tunnel coupling

l

0.8

©•0.6

0.4 •

0.2 -.

:.=r3$> weak ffl*> weak

• |b> strong |c> strong

I^Wjetran^i fftHbstrang

20 40 60 80

Fig. 2. Time-dependent fidelity of four-qubit states (I*!,41) and |*jj41)) and two-qubit non-DF states (|6) and |c)). The 'weak' means a weak measurement case of C = 0.2 and the 'strong' means a strong mea­surement case of C = 0.6. Q = 2ro, Jij = 0 ej = 0.

between the left and right electrodes:

2

-Hint -22

+£

lL,S U = l

4

(ciLSd<>+dsCiLs)

y^VjH^sOj.

,i=3

(ct ds+diaRS) (3)

where 6ViaiS (a = L, R) is an effective change of the tunneling strength between the elec­trodes and QPC island. Hereafter we neglect the spin dependence of Via and 5Viati. We assume that the tunneling strength of elec­trons weakly depends on the energy Viai = Via(Eia) and electrodes are degenerate up to the Fermi surface \ia. Then qubit states in­fluence the QPC tunneling rate TL and FR

by T^1 = r r 1 +T^ and T^1 = T^1+T? through i f 0 = 27rpa(/xQ)|y/Q

±)(^)|2 and r f Y =2vpa{iia+U)\v£\iia+U)\2, depend­ing on the qubit state aiz = ±1 (V/a (/xQ) = Via(na) ± SVia(fj,a) and pa{Ha) is the den­sity of states of the electrodes (a = L,R)). The values of r\ s are determined by the geometrical structure of the system. The strength of measurement is parameterized by Ar* as T^±) = r i 0 ± Ar*. The measurement strength C is related to the tunneling rates as Ti = r 0 ( l ± C) ( r 0 is an unit)12. We call | | | ) , | IT), | Tl), and | TT) \A) ~ \D) respectively, and four-qubit states are writ-

78

ten as \AA), \AB), ..,\DD). For uniform two qubits, rA = r0(i - 0/2, rB = r c

= r 0 ( l - C 2 ) / 2 and TD = r 0 ( l + C)/2 with C = Ar / r 0 .

The DM equations of four qubits and de­tector at zero temperature of Fig. 1(a) are derived as in Ref.12 by

4

-Z2inj IPgj (z1)z2~Paz19j {z2) ]

dt

-\/r%r%lpbzlz2+PbzU,

rg+rf+rg+rg •">Z2Zi ob'

Pziz2

L^i J^gr(z\)Z2 Hz1gr(z2)'

+ v / r L r I K , 2 + v / r f l r ^ d^-[iJz2z,-[ri+Yi]]P%lZ2 dt

~ I Z * n i ^ r ( * i ) * a ^ , * i 9 r ( * a ) ]

-\fi?r%\Pzlza+P,U), (4)

where 21,22 = AA,AB,...,DD and, p" lZ2 , ph

z\z2 and p^ i02 are density matrix elements when no electron, one electron and two elec­trons exist in the QPC island, respectively.

Jz2zi = Jz2 — Jzi-i JAA = z J i e i + "A.2 + "^23j

JAB = J2i ei ~ e4 + J12 — J23, —, JDD =

- T2i ei+Ji2+J23- gi(zi) and gr{zi) are intro­duced for the sake of notational convenience and determined by the relative positions be­tween qubit states as in Ref.12. We have 768 equations for four-qubits.

To see the decoherence effect ex­plicitly, we study time-dependent fidelity, F{t) = Tr[p(0)p'(t)] on the rotating co-ordinate as p'(t) =e i£n*<7" tp(*)e"*£ :n*<T" t

(ft'i = y/Qf+ef/4) to eliminate the bonding-antibonding coherent oscillations of free qubits (trace is carried out over qubit states).

3. Numerical Resul t s

Figure 2 shows the effect of the local struc­ture, that is, an island in the QPC detector, on the fidelity of DF states. It is seen that the local structure greatly degrades the fidelity of qubit states. In particular, the degradation is large when the strength of measurement increases. Figure 2 also shows that non-DF two-qubit states are better than four qubit DF states when the strength of measurement is large. This result indicates that DF states using plenty of qubits will not be preferable when there are local structures. In Ref12, we showed that \b) state is a candidate of logical state in the weak measurement case, and \c) is a candidate in the strong measure­ment case, by exactly solving the DM equa­tions analytically. Figure 2 also suggests that \c) state is better in the strong measurement case.

Fig. 3. Time-dependent fidelity of four-qubit DF

states (|*i41), |*2*') a n d l*34')) under various fluc­

tuations : (i)H3 = (1 — f])Q, €3 = Tjro and Tg =

(1 - »?)r(±). (ii)n2 = rc3 = (1 - »/)n, e2 = es = ro and r2

±) = r^±) = (i-77)r(±). (hi)n4 = (i-»?)n, e4 = r ^o and T^± ) = ( l - r ? ) r < ± ) . (a) 77=0.01 and C=0.6 (strong measurement), (b) ?7=0.05 and £=0.2 (weak measurement). Q = 2ro, Jij = 0 tj = 0. r" = r

Figure 3 shows the combined effect of local QPC island structure and the non-uniformities of the qubits. In case (i), only 3rd qubit fluctuates as Q3—> ^3(1 —77), £3—• £3(1 — 7/) and T3 —• r 3 ( l —77). In case (ii), the 2nd and 3rd qubits fluctuate. In case (iii) only 4th qubit fluctuates. All of these

79

non-uniformities are introduced in a manner similar to that described in Ref. 12.

Figure 3 (a) shows that non-uniformity of qubit does not change the fidelity of l ^ ) and |*3 ) in the strong measurement case, when compared with Fig. 2. This is because the local structure has already greatly de­graded the fidelity without non-uniformity (Fig. 2). On the other hand, |#[41), which is a product state of two singlet states, degrades due to the non-uniformity. Thus, the combi­nation of local structure and non-uniformity degrades all the four-qubit states even when the non-uniformity is small (1%).

Figure 3 (b) shows the robustness of four-qubit DF states depends on the dis­tribution of non-uniformities in the qubits. Similar to the results in Ref. 12, the fi­delities of case (iii) in |*j,4]> and |#3

41) are

smaller than those of other distributions of non-uniformities. For | * i ) , the distribution of case (ii) is the largest, because both the two singlets 2- 1 ( |01)- |10)) ( I 2 ) and (|01>— 110}) (34) are affected in the configuration of non-uniformity.

Figures 4 (a) and (b) show the fideli­ties for four-qubit DF states as a function of non-uniformity rj at r 0 t = 50. Basically, as r) increases, the fidelity decreases and the degradation strongly depends on the distri­bution of non-uniformity. Moreover, finite bias e (pure dephasing region) reduces the fi­delity more than in the zero bias cases. Fig­ure 4 (b) can be compared with the Fig. 3 in Ref.12 where QPC detector has no island structure: fidelity of the present island QPC detector is less than that of the structureless QPC in Ref.12. In particular, even at 77 = 0, fidelity in the present case degrades because the island in the QPC detector violates the collective decoherence environment.

m

^ 0 . 4

"*- 0.2

0.8: :*£•»»•#•***££* (a)

O n ^ J.

^24t>e=0 " 8 . > E = 0 . 1 B i>s=0 «*.

0% 5% 10% 15% 5% 10% 15% 7

Fig. 4. Fidelities of four-qubit DF states at t = 50r^" as a function of non-uniformity 77. (a) Non-uniformity for 2nd and 3rd qubits (case (ii)). (b) Non-uniformity for 4th qubit (case (iii)). f2 = 2ro, Jij = 0 and C=0.2.

cuss the robustness of DF states when there are a local structure and non-uniformities. We found that local structure is an ob­stacle to using DF states other than non-uniformities. We also showed that two-qubit non-DF states are candidates for the logical qubits.

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4. Conclusion

We have solved master equations of four and two qubits with QPC detector, and dis-

80

ENTANGLEMENT-ASSISTED ESTIMATION OF Q U A N T U M CHANNELS

AKIO FUJIWARA Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mail: fujiwara@math. wani. osaka-u. ac.jp

This paper reviews some recent developments in asymptotic estimation theory for a parametric family {Tgjg of quantum channels, putting emphasis on an active interplay between noncommu-tative statistics and information geometry.

Keywords: quantum channel; statistical estimation; unitary channel; Pauli channel; entanglement; representation theory; asymptotics.

1. Introduction

Let 7i be a Hilbert space that represents the physical system of interest and let S(H) be the set of density operators on Ji. It is well known that a dynamical change Y : S(H) —> S(H) of the physical system, called a quan­tum channel, is represented by a trace pre­serving completely positive map. Neverthe­less, it is a different matter how one can identify the quantum channel that one has in a laboratory. Since almost every quan­tum protocol assumes a priori knowledge of the behavior of the quantum channel under consideration, there is no doubt that identi­fying the channel is of fundamental impor­tance in quantum information theory. It is, however, not very long since the quantum channel identification problem1 was directed proper attention, and the theory of finding the optimal estimation scheme has not been fully developed.

In this paper, we review some re­cent developments in asymptotic estimation theory for a parametric family {Tg ; 0 = (91,...,6d) 6 O} of quantum channels. In particular, we focus on the extension2 (id <g>

r0)®n : s{{H ® w)®n) -• S((H ® n)9n), where n is an arbitrary positive integer, and study the role of quantum entangle­ment and the degree n of extension, putting emphasis on an active interplay between noncommutative statistics3 and information

geometry4. All the proofs omitted here are found elsewhere5'6.

2. SU{2) channel

We first study the estimation problem of a unitary channel5. Suppose an unknown op­eration r acting on S(C2) is noiseless, in that there is a unitary operator U 6 SU(2) such that r : p i-> UpU*. Our problem is to es­timate the unknown U. Let us introduce a local coordinate system 6 of SU(2) around the identity / by the exponential map:

Ug = £V,02i(?3) = Exp I i ^2 9kak 1 ,

where <7i, 02,03 are the standard Pauli ma­trices. By a suitable rearrangement of the constituent Hilbert spaces H = C2, we iden­tify (id ® r„)®" with id®" <g> r f n . Once an input state ip € H®n®n®n is fixed, we have a quantum statistical model:

p»:=(id®n®r®n)(W<Vl),

and the problem of estimating the unknown unitary operation Ug e 517(2) is reduced to estimating the parameter 0 of the model pg.

Let us decompose H®n into irreducible subspaces under the 517(2) action as follows:

A V STab (A) /

81

where A runs over all possible Young frames and STab (A) stands for the set of standard tableaux on A. Then

A \ S T a b (A)

Given an input state ip e H®n ® H®n, we decompose it as

f = E E «W*[AI. A STab(A)

where tp^ is a unit vector on the invariant subspace H®n®H^, and the coefficients oW satisfy the normalization

E E i«[AIia = i-A STab(A)

Accordingly, the symmetric logarithmic derivative (SLD) Fisher metric g of the out­put state model pg is decomposed as

s = E E ia[A1i2s[A1 a) A STab(A)

where g^ is the SLD metric of the model with respect to the input state ip^ e H®n ® W'A1. Thus evaluating g is reduced to evalu­ating each component g^.

Theorem 2 .1 . Let V>[A] e H®n <g> H[A! 6e decomposed into the Schmidt form

p

where p = dim W'A1. Then

trgW<tr\gW] l J a=ME

where [ - ] Q =ME indicates that it is evaluated for a maximally entangled ip^.

Theorem 2.1 asserts that there is no in­put state that satisfies g^ > [g^] = M E and M * [9 W]

l a = M E simultaneously. In this [A] sense, a maximally entangled input VME *S

optimal on each invariant subspace H®n <g> 7Y'A'. In what follows, we restrict ourselves to input states tf) that are superpositions of

YMF,-

Let us mention the achievability of the SLD Cramer-Rao inequality:

Ve[M^]>(j^)~\ (2)

where Vg [M^] is the covariance matrix of the locally unbiased estimator (POVM) M(") for the parameter 0, and J^n) := g{6) is the SLD Fisher information matrix with respect to the coordinate system 6.

Theorem 2.2. When the input ip is a super­position of maximally entangled states ^ g , there is a locally unbiased estimator M^ that attains the lower bound in (2).

According to Theorem 2.2, the problem amounts to finding the optimal input state ip that maximizes the SLD metric g. By a di­rect computation, the metric [g^] _ M E that corresponds to the following Young frame

/• \

I I I I I I I i ^ •

is evaluated at the identity (6 = 0) as

$L where p = dimW^l = n — 2k + 1. As a consequence,

and the maximum is taken at A = Asym, where

^sym

Since g is a convex combination of con­stituent g^ as in (1), we conclude that the optimal input is V'ME •

In summary, the optimal strategy for es­timating an unknown unitary U$ € SU(2) exhibits

3 / - N - 1

m i n V0 M(n)

2n « ) • (3)

82

ai=o 1/2 1

Fig. 1. The structure of output manifold for n = 1. When a i = 0 or 1, it collapses to 2-dimensional sphere S2 of unit radius; when a i = 1/2, it is iso­metric to 3-dimensional real projective space R P 3

of radius 2; otherwise it is diffeomorphic, but is not isometric, to MP3 of any radius.

The implication of this result is profound7. In the standard (classical) statistics, it is commonly believed that the estimation error approaches zero in the rate of O (1/n). By contrast, for estimating an unknown 51/(2) channel, the estimation error approaches zero asymptotically in the rate of O ( l / n 2 ) , as Eq. (3) asserts.

Let us interpret these results in terms of differential geometry. Given an input i/> e H®n ® H®n, the set M{ij>) := {pe = (id®" ®rfn)(|V>(V'l)} of output states is re­garded as a differentiable manifold equipped with a local coordinate system 6, and the SLD metric is the infinitesimal version of the statistical distance between points on M(tjj). The simplest case n = 1 (hence p = 2) is illustrated in Fig. 1, where the global struc­ture of the output manifold M(tp) is shown for each degree a = (a i , 1 — c*i) of entan­glement of ip. As a i increases toward 1/2, the manifold inflates and hence points on the manifold are getting separated from each other. This is the geometrical mechanism be­hind the optimality of a maximally entan­gled input. In fact, the larger the SLD dis­tance of two nearby quantum states becomes, the easier one can distinguish these states, as the quantum Cramer-Rao inequality asserts. The situation is much the same for general n. When the input is a maximally entangled

state V'ME' ' ^ e manifold of output states maximally inflates, and becomes isometric to SU(2)/{±I} ^ SO(3) ^ R P 3 of radius r„ = V4 (n2 + 2n)/3.

In conclusion, the degree a of entangle­ment is related to the "shape" of the man­ifold of output states, and the degree n of extension to its "radius."

3. Pauli channel

We next investigate the estimation problem of a Pauli channel6. In view of applications, Pauli channels form a reasonably large class of quantum channels, including many impor­tant submodels such as the bit flip channel, the phase flip (=phase damping) channel, and the depolarizing channel.

A Pauli channel T : <S(C2) - • S(C2) is denned by

3

r ( T ) = X)Pfc(CTfcrcrfe)' fc=0

where p = (po,Pi,P2,P3,) is a proba­bility vector, <To = / , and o~i,o~2,o~3 are the Pauli matrices. Obviously there is a one-to-one correspondence between the set of Pauli channels and the three dimensional probability simplex Vz :=

{p = (Pi)i=o ; yPi ^ °> E i = o ^ = l} • Let­ting 8 = (9X) i<A<3 be a global coordinate system of the probability simplex V3, one can specify a Pauli channel T by the correspond­ing coordinate 9, whereby we can denote r = Te as rV(r) = E L o PkV) (*k ra*k).

Theorem 3.1. The SLD Fisher information Jg takes the maximum if the input is the iid extension of maximally entangled states. The lower bound of the corresponding Cramer-Rao inequality

^K>]>(J ( ,( B ,)"1 = ^(^ (1))"1

is uniformly achievable if and only if 9 is Vm-affine in V3.

83

Theorem 3.1 implies that the optimal es­timation strategy is of iid nature: one cannot acquire more information than JQ ' per ex­tension even if one invokes other entangled inputs and/or collective measurements that straddle possibly larger Hilbert spaces.

In summary, the optimal strategy for es­timating an unknown Pauli channel exhibits

- l min Vg MM

M(n) km (4)

This asymptotic behavior is the same as in the classical case, and is in remarkable con­trast to that for SU(2) case. However, the geometrical mechanism behind Theorem 3.1 is similar. Intuitively speaking, the "shape" of the output manifold can be transformed into the classical parameter space V3 by ad­justing the degree a of entanglement, and the "radius" of the manifold of output states is scaled as r„ oc y/n.

That the output manifold can be identi­fied with V3 in the sense of information ge­ometry further allows us to treat submod­els of generalized Pauli channels in a unified manner. For example, each Ve-autoparallel submanifold M. of V3 forms an exponential family and has an efficient estimator for the Vm-affine coordinate system of M. There­fore, each Ve-autoparallel submanifold of V3

corresponds to a statistically tractable sub­model of generalized Pauli channels. We might as well call such a model a V e-autoparallel submodel, see Fig. 2. Any other (smooth) submodel of Pauli channels can be regarded as a curved exponential family4.

4. Concluding remarks

We have shown that there are at least two classes of quantum channels that exhibit es­sentially different asymptotic behaviors: the estimation error is of O ( l / n 2 ) for SU(2) channels, while it is of O (1/n) for Pauli channels. It can be shown that the same asymptotic properties hold for arbitrary fi­nite dimensional H, that is, for SU(N)

( 1 , 0 , 0 , 0 )

( 0 , 0 , 0 , 1 )

( 0 , 1 , 0 ,

( 0 , 0 , 1 , 0 )

Fig. 2. The set V3 of Pauli channels and some standard one-dimensional Ve-autoparallel submod­els: (a) depolarizing channel, (b) bit flip channel, (c) bit-phase flip channel, (d) phase flip channel.

channels5 and generalized Pauli channels6. It is an open question whether there are classes of quantum channels that exhibit dif­ferent asymptotic rates O ( l / n s ) with s ^ 1,2. Nevertheless, the present study clearly demonstrates the usefulness of differential ge­ometrical methods in quantum channel esti­mation theory.

References

1. For a comprehensible review, see: A. Fuji-wara, Quant. Inform. Comput., 6&7, 479-488 (2004).

2. For a more general extension (A ® r^)®n

with A a known channel, see: A. Fujiwara, Phys. Rev. A 63, 042304 (2001).

3. A. S. Holevo, Probabilistic and Statisti­cal Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).

4. S. Amari and H. Nagaoka, Methods of In­formation Geometry, Transl. Math. Mono­graphs, vol. 191 (AMS, Providence, 2000).

5. A. Fujiwara, Phys. Rev. A 65, 012316 (2002); A. Fujiwara, in preparation.

6. A. Fujiwara and H. Imai, J. Phys. A: Math. Gen. 36, 8093-8103 (2003).

7. Also, in a different setting of covariant Bayesian estimation, an analogous asymp­totic property has been obtained: G. Chiri-bella et al., Phys. Rev. Lett. 93, 180503 (2004); E. Bagan et al., Phys. Rev. A 70, 030301(R) (2004); M. Hayashi, quant-ph/0407053.

84

SUPERCONDUCTING Q U A N T U M BIT WITH FERROMAGNETIC T T - J U N C T I O N

T. YAMASHITA 1 *, S. T A K A H A S H I 1 , A N D S. M A E K A W A 1 ' 2

Institute for Materials Research, Tohoku University, Sendai, Miyagi, 980-8577, Japan 2 CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama, 332-0012, Japan

* E-mail: taro@imr.tohoku.ac.jp

We propose a new qubit consisting of a superconducting ring with an ordinary Josephson junction and a ferromagnetic 7r-junction. In the system, two degenerate stable states appear in the phase space without an external magnetic field due to a competition between the 0- and 7r-states. Because of quantum tunneling between the two degenerate states, bonding and antibonding states, which are used as a bit, are formed. For manipulating the states of the qubit, only small external magnetic field around zero is required. This feature leads to a large-scale integration and a construction of the qubit with a smaller size which is robust to decoherence by external noises.

Keywords: quantum computing; Josephson effect; ferromagnet/superconductor hybrid structures.

Quantum computing has attracted a great deal of interest in the recent years 1. As a candidate for an elemental unit of a quan­tum computer (qubit), many proposals have been done. Among the proposals, solid-state devices have great advantages in large-scale integration and flexibility of layout, but still face the problem of reducing the decoher­ence effect due to their coupling to the en­vironment. Recently, there are many works on superconducting qubits in which the free­dom of superconducting phase is utilized as a bit (flux qubit) 2 '3. Three-junction type of the flux qubit proposed by Mooij et al. con­sists of a superconducting loop with three Josephson junctions, and the bonding and antibonding states formed by applying an ex­ternal magnetic field are used as a bit 2.

Furthermore, due to recent advances in microfabricating techniques, extensive works on spin-electronics have been done 4. In a superconducting spin-electronic de­vice, ferromagnetic 7r-j unctions have been studied extensively 5 '6. In a superconduc-tor/ferromagnet (SC/FM) junction, the su­perconducting order parameter penetrates into FM and oscillates due to the exchange field in FM 5. As a result, in a SC/FM/SC

Superconductor

- Ferromagnet

Insulator

Fig. 1. Schematic diagram of a superconduct­ing ring with an insulator and a ferromagnet. The superconductor/insulator/superconductor junc­tion is an ordinary 0-junction, and the superconduc­tor/ferromagnet/superconductor junction is a metal­lic 7r-j unction.

junction the system is stable at the phase difference equal to 7r when the order param­eters in two SC's take different sign, and this state is called u7r-state". Experimentally, cusp structures in the temperature and FM's thickness dependences of the critical current have been observed as an evidence for the transition between the 7r-state and 0-state 6.

In this paper, we propose a new qubit consisting of a superconducting ring with an insulator and a ferromagnet (Fig. 1). In this system, the supercon­ductor/insulator/superconductor (SC/I/SC) junction is an ordinary Josephson junction with Josephson energy UQ = —EQ cos 6.

85

-0.6 0.0 0.5 1.0 1.5 2.0

Fig. 2. Free energy F (solid line) and Josephson current / (dashed line) as a function of 6 in a metallic SC/FM/SC junction for hex = 0.31fiF. Here A 0 is the superconducting gap at zero temperature.

Here, EQ is the coupling constant in the

S C / I / S C junction and 9 is the phase dif­

ference between the SC's. On the other

hand, the S C / F M / S C junct ion is a metallic

7r-junction. We show tha t the qubit does not

require an external magnetic field for forming

the coherent two states, and only small ex­

ternal field is required for distinguishing the

states. This feature leads to a construction of

the smaller-size qubit , and thus this qubit is

advantageous to large-scale integration and

expected to have a long decoherence t ime.

First we derive the Josephson energy

U-n of the metallic 7r-junction, by using the

Bogoliubov-de Gennes (BdG) equation 7 .

Solving the BdG equation, we obtain the An-

dreev bound state energy Ea{9) for spin a 8 .

The phase dependent par t of the free energy

in the S C / F M / S C junction is then expressed

in terms of E„ as

F{6) = - f c B T ^ l n T,E,

2 cosh 2kRT (1)

where T is the tempera ture and kn is

the Boltzmann constant. At low temper­

atures, the free energy (Eq. (1)) reduces

to F(9) = -(l/2)J2atEaEa(9). Figure 2

shows the 9 dependence of the free energy

F (solid line) and of the Josephson cur­

rent J = (2e/h)(dF/dO) (dashed line) in

the S C / F M / S C junction. Here we assume

tha t the interfaces between SC and F M are

a metallic contact: Z = mV/h2kF = 0,

where V is the strength of the ^-function

type of potential at the interfaces (m is the

electron mass, kp is the Fermi wave num­

ber) 8 . In obtaining these results, we have

assumed tha t the exchange field is hex =

0.31/XF (HF is the Fermi energy), the thick­

ness of the F M is L = 10/fcjr, and the co­

herence length at zero tempera ture is £o =

lOOO/fcp- As shown in the solid line in

Fig. 2, F has a minimum at 9 = n (n-

junction), and is approximately expressed as

F « -En |COS((0 + TT) /2 ) | , where En is the

Josephson coupling constant . This leads to

a current with the nonsinusoidal form, which

is approximately I « —(7^/2) sin ((6 + n)/2)

for 0 < 6 < 2n with the critical current

In = 2eE7!/h. This form shows small devi­

ations from the dashed line in Fig. 2 when

6 is near 0 or 2ir. However, these devia­

tions do not affect the following discussion,

and the approximate relations given here are

valid for the cases of 0.28/J.F ;$ hex < 0.34/ijr

and 0 .79/UF < hex < 0 .87 /XF- In addition,

it is shown tha t these relations are valid for

more realistic cases with small but finite Z 9 . Therefore, if we choose the appropriate

values for hex/np a n d kpL, this form for the

Josephson energy of the metallic 7r-j unction

is a good approximation, and in the follow­

ing it is used for the Josephson energy of the

metallic 7r-junction in the ring.

We now discuss the superconducting

qubit which is shown in Fig. 1. In the

ring, the S C / I / S C (0-junction) and the

S C / F M / S C (7r-junction) junctions have the

Josephson energies UQ = —EQ cos #o and

UK = — E„ \cos((9n + 7r)/2)|, respectively.

Here, -EO(TT) is the coupling constant in the

0- (7T-) junction. The superconducting phase

differences, 9Q and 9^, are for the 0- and n-

junctions, respectively. In this case, the to­

tal flux in the ring $ satisfies the relation

6„ - 0 O = 2 T T $ / $ 0 - 27TZ, where $ 0 = h/2e

is the unit flux and I is an integer. The total

Hamiltonian of the ring is expressed as H =

K + UO + U^ + UL, where K = -±Ec{d2/39l)

86

is the electrostatic energy, Ec = e2/2C is the Coulomb energy for a single charge, C is the capacitance of the 0-junction, and UL is the magnetic energy stored in the ring. The magnetic energy UL is given by UL = ($ - $ext)2/2I<s, where La is the self-inductance of the ring and $ e x t is the exter­nal magnetic flux. The total Hamiltonian H is analogous to that describing the motion of a particle with kinetic energy K and in a potential Utot = UO + U^ + UL- Using the re­lation between the phase and the total flux, the potential Utot becomes a function of 6n

and $: Utot = Utot (&*, $)• In order to obtain the state of the ring, we seek the solution at which Utot is minimum. First, we minimize Utot with respect to <fr, i.e., dUtot/d$ — 0, which yields $ (0O) = /3($O/2TT) sin0O + $ext, where (3 = 2irI0Ls/(&0 and I0 = 2eE0/h is the critical current in the 0-junction. Sub­stituting this equation in the expression for Utot, w e obtain

Utot/Eo = - c o s 0O

—a cos 90+TT , 0

sin 0Q

- s in 0O

$ 0 -(2)

where a = E„/E0. Throughout this paper, we use a = 3 and (3 = 3.0 x 10 - 3 . The value of a can be adjusted by changing the thickness and the area of the insulating bar­rier or of the ferromagnet, and that of /? is reasonable for the micrometer-size ring and the Josephson junction with several hundred nano-ampere of the critical current.

Figure 3 shows the 0o dependence of the normalized Utot- As shown in the solid line in Fig. 3, Utot has double minima located at 0O « 7r/2 and 37r/2 under no external mag­netic field. For ||> (0O « ir/2) and | | ) (0O w 37r/2) states, supercurrents / of magnitude < Io flow in the clockwise and counterclockwise directions, respectively. Because of the quan­tum tunneling between |f) and | | ) , the bond­ing |0) oc |T) + II) and the antibonding |1) ex |t) — | |) states are formed, hence yielding a

R

>.0 0.5 1.0 1.5 2.0

9jn

Fig. 3. Normalized Utot as a function of 0o for &ext/$o = 0 (solid line), 0.01 (dashed line), and -0.01 (dotted line).

two-level quantum system. From numerical calculations for the ratio E0/Ec = 30, we es­timate that the energy gap AE between the ground state |0) and the first exited state |1) is A.E ss 1.6 x 10_ 2 3J, which corresponds to the frequency of v = AE/h « 24 GHz 2. In order to confirm this two-level quan­tum state, microwave absorption measure­ments (Rabi oscillation) can be used. The dashed and dotted lines in Fig. 3 indicate the normalized Utot a s a function of 0o in an external magnetic field ($ext/$o = ±0.01). By applying a small external magnetic field to the ring, the degeneracy of the states | | ) and | | ) is lifted. For $ext = + ( - ) 0 . 0 1 $ 0 , the component of |T) (||))is larger than that of | | ) (|t)) in the ground state |0), and vice versa in the first excited state |1). There­fore, finite supercurrents flow in opposite di­rections for |0) and |1) states in a finite ex­ternal magnetic field. As a consequence, the |0) and |1) states are distinguished by mea­suring the current flowing in the ring with a superconducting quantum interference de­vice (SQUID) placed around the ring.

Our qubit has coherent states which re­quire no external magnetic field, thus only a small external magnetic field is enough to manipulate and detect the states, as com­pared to the half unit flux $o/2 required in the proposal of Ref. 2. For instance, even if the size of the qubit is several 100 nm's, only

87

magnetic fields of the order of a millitesla are needed for manipulation of our qubit. This feature enables one to make qubits with smaller size which is advantageous in large-scale integration. This type of qubit is also resistant to external noise and has longer de-coherence time 2. Here we discuss the dis­sipation in the metallic n junction. In the small metallic 7r junction, the discrete An-dreev bound states are formed. For L -C £0, only one Andreev bound state exists. The Andreev bound state energy is Ea « 0 (gap-less) for 8n = 0 and 2n. In the ring, the quan­tum tunneling occurs between 90 ~ 0„ ~ 7r/2 and ss 37r/2 (Fig. 3). In this phase re­gion, the Andreev bound state energy is in the region of 0.75A < Ea < A, where A is the superconducting gap. This indicates that the metallic IT junction is well gapped in the phase region where the quantum tun­neling occurs. Therefore, the quasiparticle tunneling which can cause the dissipation is strongly suppressed at low temperatures and low voltages.

In order to realize the universal quan­tum logic gates, a controlled-NOT (CNOT) gate is needed as well as single qubit rota­tions by using the Rabi oscillation discussed above. In our qubit, the CNOT gate is real­ized in the following way: When two qubits A and B have an inductive coupling, the en­ergy gap in qubit A depends on the state of qubit B. In other words, the energy gap for qubit A is AJSAO(AI) when qubit B is in state |0) (|1)). If a microwave pulse with the fre­quency AEAi/h is applied to qubit A, then the state of qubit A is changed only if qubit B is in state |1).

The qubit proposed here has the follow­ing advantages: (i) the geometry is simple, i.e., only two Josephson junctions are used, (ii) the qubit is constructed without an ex­ternal magnetic field, and only a small ex­ternal magnetic field is needed for measuring the state. Furthermore, owing to these ad­vantages, (iii) the size of the qubit can be

reduced. This makes large-scale integration possible, and leads to a reduction of the de-coherence due to the coupling to the envi­ronment. The qubits offer a new route for spin-electronics to quantum computing.

T.Y. was supported by JSPS Research Fellowships for Young Scientists. This work was supported by NAREGI Nanoscience Project, Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and by a Grant-in-Aid from MEXT, CREST-JST and NEDO of Japan.

References

1. M.A. Nielsen and I.L. Chuang, Quan­tum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).

2. J.E. Mooij et al. Science 285, 1036 (1999); I. Chiorescu et al., Science 299, 1869 (2003); I. Chiorescu et al., Nature (London) 431, 159 (2004).

3. T. Yamashita et al, Phys. Rev. Lett. 95, 097001 (2005); T. Yamashita et al, cond-mat/0507199.

4. Spin Dependent Transport in Magnetic Nanostructures, edited by S. Maekawa and T. Shinjo (Taylor and Francis, London and New York, 2002).

5. A.I. Buzdin et al., JETP Lett. 35, 178 (1982); Z. Radovic et al., Phys. Rev. B68, 014501 (2003); E.A. Dernier et al, Phys. Rev. B55, 15174 (1997).

6. V.V. Ryazanov et al., Phys. Rev. Lett. 86, 2427 (2001); T. Kontos et al., Phys. Rev. Lett. 89, 137007 (2002); S.M. Frolov et al., Phys. Rev. B70, 144505 (2004).

7. P.G. de Gennes, Superconductivity of Met­als and Alloys (W. A. Benjamin, New York, 1966), chap. 5.

8. A. Furusaki, in Physics and Application of Mesoscopic Josephson Junctions, edited by H. Ohta and C. Ishii (Phys. Soc. Jpn., Tokyo, 1999), pp. 79-96.

9. The value of Z was estimated to be between 0 and 0.31 from measurements of the conduc­tance in FM/SC point contacts (see, for ex­ample, Ref. 10). It follows from our numerical calculations that the form of the free energy F given in the text is valid when Z < 0.7.

10. G.J. Strijkers et al, Phys. Rev. B63, 104510 (2001).

GENERATION OF MACROSCOPIC GREENBERGER-HORNE-ZEILINGER STATES

IN JOSEPHSON SYSTEMS

T O S H I Y U K I F U J I I * , M U N E H I R O NISHIDA* and N O R I Y U K I HATAKENAKA*'1"

* AdSM, Hiroshima University, Higashi-Hiroshima, 7S9-8530, Japan 'Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521,

Japan

We theoretically investigate generation of the entangled three macroscopic objects, i.e., macro­scopic Greenberger-Horne-Zeilinger states, in Josephson systems by using the Cirac-Zoller scheme. We propose a quantum breather in a long Josephson junction as a macroscopic artifical atom and derive the Jaynes-Cummings-type interaction Hamiltonian required in the scheme. Macroscopic Greenberger-Horne-Zeilinger states play a crucial role in fundamental tests of quantum mechanics versus local realism at a macroscopic level.

Keywords: GHZ; Entanglement; Josephson effect.

1. In t roduct ion

Quantum mechanics successfully explains physical phenomena at microscopic scales but still involves conceptual problems unre­solved. One such problem is an applicabil­ity of quantum mechanics at a macroscopic scale. Superconducting nanodevices appear, at present, to be unique among the can­didate systems for demonstrating quantum-mechanical aspects at a macroscopic scale because the collective (flux) coodinate rep­resenting the motion of 109 Cooper pairs acting in the system. In fact, macroscopic quantum tunneling 1 and macroscopic super­positions 2 '3 have been observed in such su­perconducting nanodevice families.

Entanglement is, on the other hand, at the heart of quantum mysteries addressed by Einstein, Podolsky and Rosen (EPR) 4, and has been predominantly studied in micro­scopic systems like photons and atoms based on the Bell inequalties in the connection to local realism. In 1994, an innovative progress has been made by Greenberger, Home and Zeilinger (GHZ) 5. They developed a proof of the Bell theorem without inequalities. En­tanglement of more than two particles leads

to a strong conflict with local realism for non-statistical predictions of quantum mechanics. The GHZ state for spin-1/2 particles has a simple form

I * G W = ^ ( I TI)I T2)l h}-\ li)l l2)l j 3 ) ) ,

(1) where | | i) and | U) are spin-up and spin-down states along some axis for the ith par­ticle. In such three-particle correlations, only a single set of observations is required in or­der to demolish the local-reality assumption. In the other word, this entangled state pro­vides an "always" versus "never" test of local realism.

Polarization entaglement of three spa­tially separated photons as a microscopic system has been already observed 6 . Re­cently, one proposal for preparation of GHZ states has been presented using multiple su­perconducting quantum-interference device qubits in cavity QED 7. However, in the existing superconducting nanocircuits with strong constant interqubit coupling, two-qubit logic gates could be easily imple­mented, while single-qubit gates cannot be strictly achieved. It seems to be difficult to manipulate only one qubit and leave the

89

other one unaffected. The generation scheme of the GHZ state

for two-level atoms has been established by Cirac and Zoller 8 based on the exchange of a single photon between two-level atoms suc­cessively passing through a resonant cavity. Since only a single atom interacts with a sin­gle photon in the cavity, no technical prob­lems mentioned above are expected in this scheme. The Cirac-Zoller scheme requires the Jaynes-Cummings Hamiltonian in atom-EM field interaction. In this paper, we pro­pose a quantum breather as a macroscopic artificial atom and study breather-field inter­actions, resulting in the Jaynes-Cummings Hamiltonian.

2. Q u a n t u m Brea the r s

The dynamics of the phase difference across the Josephson junction obeys the sine-Gordon equation,

_ _ _ _ = o n ^ (2)

where the spatial coordinate x is normal­ized to the Josephson penetration depth Xj, the time t to the inverse Josephson plasma frequency w j 1 . It is well known that the sine-Gordon equation involves fluxon (kink) and antifiuxon (antikink) solutions. In ad­dition, there also exists a breather solution expressed as

^ , t ) = 4 t a n - ^ 7 S i n ^ M , (3) \UJBrCOSh'yxJ

where u>Br and 7 are the breather frequency and the Lorentz factor, respectively. As shown in Fig. 1, the breather is an elemen­tary topological excitation in a long Joseph-son junction, which is regarded as a bound state of a fluxon and an antifiuxon.

If the junction area becomes smaller and smaller, the quantum nature of the Josephson phase is enhanced. Finally, we have to treat the Josephson phase differ­ence quantum-mechanically. According to

Fig. 1. Schematic of a moving breather

the Bohr-Sommerferd quantization rule,

fTdtrdxn(x,t)^^=2nh(n+~S\ ,

(4) where the generalized momentam is defined as n(x, t) = d(j)/dt, the breather frequency UJBT and its energy are quantized as 9

w*= w j(^fe(n +0)' (5)

and

E» = 16Ej^(J^(n+iy), (6)

where Ej is the Josephson coupling energy and n is an integer. The energy levels are not equidistant so that the lowest two lev­els serve as an macroscopic artifical atom. In addition, breathers can move along a Joseph-son transmission line. Therefore, qunatum breathers are quite similar to atoms in the Cirac-Zoller scheme.

3. Interaction Hamiltonian

Now let us consider the interaction between a breather and an electromagnetic (EM) field. Our model is shown in Fig. 2. Here we assume that the breather is an oscillating fluxon and antifiuxon coupled together. The fluxon and antifiuxon are considered to be created by the oppositely circulating current / each other, and are interacting with the

90

magnetic component of an electromagnetic field in the cavity in the form of I$/2 with $ being the magnetic flux in the area enclosed by the circulating current;

E=\l* + \(-l)*

= >*J B[x~ -B X + \

(7)

(8)

(9)

The first and second term in the right hand side of Eq. (7) are contributions to the in­teraction energy from the fluxon and the an­tinuxon, respectively. Here we also assume that the current flows in the ring of radius Aj. The circulating current is evaluated by combining the following two relations: The magnetic flux density in the center of the ring produced by the current / is expressed as

Bl ~ 2A7" (10)

where HO is the vacuum permeability. On the other hand, the spatial derivative of the phase difference in a long Josephson junction is related to the magnetic flux density by the expression

2eddx

— — (2sechz)|x=o

h

edXj

(11)

(12)

(13)

where d is the length of the insulating layer. Here we introduced the fluxon solution (j>(x,t) = 4 t an _ 1 ( e _ x ) in sine-Gordon equa­tion into Eq. (11). Comparing the above two expressions Eqs. (10) and (13), the circulat­ing current yields

Hoed'

Therefore, the interaction Hamiltonian is ex­pressed as

Hj = hn\2j dB Hoed dx'

(15)

Note that the breather amplitude is propor­tional to the distance 5 between a fluxon and an antifluxon that is the observable of the system. In this expression, the variables 6 and B then carry the infromation of the breather and the EM field, respectively. In contrast to fluxons, a magnetic-field gradient is required for the breather-EM field interac­tion.

In the second quantization formalism, the magnetic component of a single-mode EM field of the frequency u in the cavity is given by

fh \1/2

B(x, t) = i( ° J coskx(ak(t) - al(t)),

(16) where <vk(t) and ak{t) are a creation and an annihiration operators for photons in the cavity with the volume V. The cosine term is nothing but the standing wave in the cav­ity. On the other hand, the breather variable 8 is also reexpressed into second quantiza­tion formalism by introducing the eigenstate of Hs with an eigenvalue En. Based on the completeness relation Y2n \n)(n\ = I> the m " teraction Hamiltonian is then represented by

Hi = — iak sin kx

ln=0 J lm=0 J

x(ak(t) - afk(t)) (17)

with

hn\2j (hw\io a = — (

Hoed \ V

1/2 (18)

In the interaction picture, the interaction Hamiltonian is expressed as

Hi = — iak sin kx

x(tto|l><0|e™*«+fl,i|0)<l|e- fc '* t)

x(ake-iut - a{eiut), (19)

where g^ = (i\S\j) and UJB = {E\ - E0)/K. Here we ignore the diagonal terms that reno-malize the energy levels. In the rotating-

91

wave approximation, the final effective inter­action Hamiltonian is then given by

Hi = — iaksmkx\g\o\ (o+afc — o_aM ,

(20) where a+ = |1)(0| and cr_ = |0)(1|. This is equivalent to the Jaynes-Cummings Hamilto­nian that describes an atom-field interaction in quantum optics.

Three quantum breathers, therefore, could be entangled to form macroscopic GHZ states by the exchange of a single photon in a high Q resonant cavity based on the Cirac-Zoller scheme.

4. Summary and discussions

We have discussed generation of a macro­scopic Greenberger-Horne-Zeilinger state in a long Josephson junction by using the Cirac-Zoller scheme developed in cavity QED. We have proposed a quantum breather as a can­didate of a macroscopic object and have de­rived the Jaynes-Cummings-type Hamilto­

nian required for the scheme. By applying the scheme, three quantum breathers are en­tangled in the form of the macroscopic GHZ states. Our proposed breather is an advan­tage for a study in a time-domain experiment in which one measures a set of two time cor­relations of the macroscopic object 10. This will be an important goal in the next gen­eration of experiments what can definitively rule out the alternative hypothesis of macro-realism.

Acknowledgments

We would like to thank Professors A. Usti­nov and S. Tanda for their fruitful discussion and comments. This work was supported in part by a research grant from The Mazda Foundation and a Grant-in-Aid for COE Re­search (No. 13CE2002) and the 21st Cen­tury COE program on "Topological Science and Technology" from the Ministry of Ed­ucation, Sports, Science and Technology of Japan.

References

Fig. 2. A model of the breather-EM field interac­tion: (a) a profile of the breather, (b) a profile of the magnetic field due to the breather, (c) a model of the breather-EM field interaction

2.

3.

4.

R. F. Voss and R. A. Webb, Phys. Rev. Lett. 47, 265 (1981). C. H. van der Wal et al, Science 290, 773 (2000). J. R. Friedman et al., Nature(London) 406, 43 (2000). A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). D. M. Greenberger, M. A. Home, and A. Zeilinger, in Bell's theorem, Quantum The­ory, and Conceptions of the Universe, ed. M. Kafatos (Kluwer Academics, Dordrecht, The Netherlands, 1989), pp. 73-76. D. Bouwmeester et al., Phys. Rev. Lett. 82, 1345 (1999). C. Yang and S. Han, Phys. Rev. A70, 062323 (2004). J. I. Cirac and P. Zoller, Phys. Rev. A50, R2799 (1994). ' R. F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D l l , 3424 (1975).

10. A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 587 (1985).

8

9.

92

TUNABLE TUNNEL AND EXCHANGE COUPLINGS IN DOUBLE QUANTUM DOTS

S. TARUCHA Department of Applied Physics, University of Tokyo and ICORP- Japan Science & Technology Agency,

Bnnkyo-ku, Tokyo 113-8656, Japan E-mail: tarucha@ap. t. n-tokyo.ac.jp

T. HATANO Quantum Spin Information Project, ICORP, Japan Science & Technology Agency, Atsugi-shi, Kanagawa,

E-mail: hatano@tarucha.jst.go.jp

M. STOPA Department of Engineering and Applied Sciences, Harvard University, St. Cambridge, MA 02138, USA

E-mail: stopa@deas.harvard. edu

We prepare a novel double dot device in a hybrid vertical-lateral geometry to study the electronic configurations of the tunnel-coupled and exchange-coupled states. In this device two two-dimensional quantum dots coupled side by side, and current flows vertically. From transport measurements we derive important parameters such as the inter- and intra-dot Coulomb energies, the tunnel coupling energy and the potential detuning between the two dots to influence the electronic configurations in a double dot. Thus we confirm the formation of the Heitler-London two-electron state in the weak tunnel coupling regime, and also find that the tunnel and exchange couplings are tunable with coupling gate voltage and magnetic field.

Keywords, exchange coupling; quantum dot; electron spin.

1. Introduction

Electron spin is a natural two-level system and

can be used for making quantum bits (spin

qubits) with quantum dots (QDs) [1]. Universal sets of quantum gates can then be prepared by manipulating individual spins and also spin-spin (i.e. exchange) interaction between electrons localized on the two dots. The two-electron state relevant for manipulating the exchange interaction can be represented by a Heitler-London (HL) model [2]. The tunnel coupling between the dots of a double dot is one of the important parameters in determining the exchange interaction, and has been studied, for example, using the microwave spectroscopy [3,4] and the integrated quantum point contacts [5]. Two-electron filling of the tunnel-coupled states forms either a spin singlet state or a triplet state, where one electron is localized predominantly in each dol (o form a HI. two-

electron state. The splitting between the singlet and triplet states is equivalent to the exchange energy, and so depends on the inter-dot tunnel coupling [2]. The electronic properties of tunnel-coupled states have been studied for various double dot devices, such as laterally coupled double dots [6], vertically

Fig. 1. (a) Scanning electron micrograph of the device. The double quantum dot is located inside the circular mesa, (b) Schematic of hybrid vertical-lateral double dot device. The two dots are vertically confined in the quantum well, and are laterally confined by a harmonic potential which is made of the depletion layer.

93

coupled double dots [7] and self-assembled double dots [8]. However, only a few experimental studies reported to date for exchange-coupled states in double QDs.

In this work we have prepared a novel double dot device in a hybrid vertical-lateral geometry, which allows us to evaluate the way of electron filling in the double dot. In this device two two-dimensional (2D) QDs coupled side by side, and current flows vertically (See Fig. 1(b)). From measurements of stability diagrams in the plane of two side gate voltages and nonlinear conductance characteristics we derive important parameters such as the inter-and intra-dot Coulomb energies and the tunnel coupling energy to influence the electronic configurations in the double dot structure. Thus we confirm the filling of the HL two-electron state with one electron localized in each dot. Further, we use coupling gate voltage and magnetic field to modulate the inter-dot tunnel coupling and exchange coupling.

2. Hybrid Vertical-lateral Double Dot Device

Our device is made from a double-barrier heterostructure consisting of an undoped 12 nm InGaAs well and two undoped AlGaAs barriers of thicknesses 8.5 nm and 7.0 nm [9, 10]. Fig. 1(a) is the scanning electron micrograph of the device. Each circular-shaped mesa contains a 2D QD coupled to the side of the other. Current flows vertically from a conductive substrate, through the parallel double dot, to a common metal contact on the top surface (Fig.l (b)). There are four split gates, two of which (side gates) tune the number of electrons in each dot independently. The other two gates (center gates) pinch the barrier between the two dots, and modulating the voltage, VcQ, applied to them changes principally the inter-dot barrier height. In Fig. 1(a), four line mesas emerge from the double dot mesa which serve to split the Schottky gate metal allowing the four, aforementioned gates

to be independently biased. The mesas are sufficiently narrow (100 nm) that current only flows through the metal on the top, and not through the semiconductor. The measurements described here are all carried out in a dilution refrigerator with a base temperature below 60 mK.

3. Stability Diagram

In the small source-drain bias, Vsd, limit, a series of conductance oscillations or Coulomb oscillations are observed in the plane of two side gate voltages, FsL and VsR. The typical experimental data of Coulomb oscillations (peak positions) is shown in Fig. 2(a) for the center gate voltage VcQ = -2.4 V In this figure we find that the anti-crossing gap size fluctuates substantially from one anti-crossing to another. The charging energies of the two dots, as determined by non-linear transport measurements, are about 1 meV Using this to establish the gate-dot capacitances [6], we find that the anti-crossing gap ranges from 0.4 meV down to 0.1 meV. This variation of the gap size reflects the difference in the strength of tunnel coupling [10]. For increased FcG

(smaller center barrier) the anti-crossing gaps widen until, at FcG= -2.0 V in Fig. 2 (b), some of the anti-crossings are barely discemable. In this limiting case the dots have merged and the charging energy has dropped to 0.4 meV, nearly half that of the individual dots (See Fig. 3). Note that this behavior is not homogeneous throughout the plane in Fig. 2(b) but rather

V6L(V) VsL(V)

Fig. 2. Coulomb peak positions in the plane of two side gate voltages VsL and V&. for center gate voltage Vca = -2.4 V(a)and-2.0V(b).

94

depends on side gate voltages. This cross capacitive effect results in an increase of barrier height as the side gates become less negative, suggesting that the laterally placed gates tend to pull the dots apart even as they induce more electrons.

Now we zoom in part of Fig. 2(a), where the anti-crossings are well defined, and discuss the way of filling. Figure 3 (a) shows the magnified plot of Fig. 2(a) in the side gate voltage ranges of rsL = -1.35 to -1.0 V and l'sR

= -1.5 to -1.13 V. Crossing Coulomb oscillations in gate voltage sweeps along the thin solid (dashed) line adds an electron to dot L (R) (Figs. 3 (b) and (c)). The combined Coulomb and quantum mechanical tunnel coupling between the dots inhibits simultaneous addition of electrons to both dots, thereby opening up diagonal gaps, or "anti-crossings," at the Coulomb oscillation vertices, resulting in the characteristic hexagonal or "honeycomb" stability diagram [6]. Each honeycomb cell represents a region where a given pair of electron numbers of the two dots

Fig. 3. (a) Magnified plot of Fig. 2(a) in the gate voltage ranges of V*. = -1.35 to -1.0 V and VA - -1.5 to -1.13 V. (b)-(d) Way of filling between two dots, depending on the path for sweeping gate voltages. Dot L, and R are predominantly filled along the thin solid line (b), and thin dashed line (c), respectively. The two dots are equivalently filled at the Coulomb oscillation vertices (along the bold solid line) (d). (e) Center gate voltage dependencies of mean value of Coulomb peak spacings at various Coulomb oscillation vertices.

are stable, e.g. (A'L, NR). At cell boundaries the total energies, E(NL, NR), of neighboring charge states are degenerate. With the knowledge about the honeycomb diagram we are able to evaluate various characteristic energies, including intra-dot Coulomb energy, I intraL 0'm^R) for the left (right) dot, inter-dot Coulomb energy, Vmta, and tunnel coupling energy, It. Vmtra

L (VmtiaR) is derived from the

Coulomb peak spacing along the thin solid (dashed) arrow. The energy of l'm,c,+2t is evaluated from the diagonal gap size along the bold solid line (later discussed in detail). Fig. 3(e) shows the change of mean diagonal gap size averaged over various anti-crossings with VCQ. As I'CQ is made more negative, the mean gap size decreases due to the reduced tunnel coupling.

4. Linear and Nonlinear Responses

The HL picture is only valid when the inter-dot tunnel coupling is small. Here, we check

Fig. 4. (a) Charging diagram in the plane of two side gate voltages, K,L and V& measured for V& = 8 liV. The center gate voltage is fixed at Vco

= -2.4 V. (b) Differential conductance d/.ydKd by sweeping both the gate voltage and the source-drain bias The gale voltages I',,. and I 'K are swept along the path c-d. The observed Coulomb diamond in the center is very symmetrical, like that of a single dot. On the other hand, for sweeps along the path a-b and e-f, the diamonds for the double dot are clearly asymmetric (See Fig.5). This implies that the highest filled state is only delocalized across the two dots along the path c-d. (c) Charging energies, EC(HL) and E^NI) for the HL (solid line) and NI (dashed line) addition of two electrons, using V^ = 1-8 meV, V^,* = 1.4 meV, VMa -=0.175 meV and I = 0.08 meV. The spacing (zfV) between measured Coulomb oscillations, weighted by the average lever arms, is shown as open triangles.

95

the relevance of this assumption in our double dot device [10]. Fig. 4(a) shows the magnified intensity plot of Coulomb oscillations in the * SL-J SR plane. Note the device is different from that used for the experiment of Figs. 2 and 3. We estimate the electron numbers in each dot to be less than 10 (and level spacings are much greater than kBT). The "diagonal vertex lines" (DVL) mark the boundaries a single electron tunnels from one dot to the other, i.e.E(NL+l, MO = E (NL, NR+l). Along this DVL the single-particle states with energies, £j, and £R, of dot L and R are always aligned, and the peak spacing, AF* measures the sum of V^ta and 2( in the HL model. AV for sweeping gate voltages parallel to the DVL increases away from the DVL, because the states are energetically detuned between the two dots and It is replaced by (4r+/J2)1/2, where A is the detuning energy A = %- %. For sweeping gate voltages along the path with A >, =, or < 0, two electrons are added (predominantly) to dot R and then dot L (path a-b in Fig. 4(a)), to dot L and then dot R (path e-f), or half an electron to each dot two times (path c-d). They are only added to the two dots equivalently along the DVL. From the change in A V with A we can learn about the way of electron filling. In the non-linear transport regime. Coulomb diamonds [11] are obtained showing conductance as a function of both gate voltage and source-drain bias. In the double dot case, the gate voltage is swept along any chosen path in the I SL-I'SR plane.

Fig. 4(b) shows the Coulomb diamond for sweeping the gate voltages along the DVL. The shape of the Coulomb diamond is symmetric, reflecting that the first and second electrons are added equivalently to the two dots. In our analysis below, we suppose that the region, in Fig. 4(a), marked (NL, MO corresponds to a filled "core," indicated by two empty states, and we concentrate on the filling of these two states at the Fermi surface. The slopes of the Coulomb diamonds reveal the

"lever arms", a, = CJC„, where i=L or R, C,* and Cu are the side gate capacitance and self-capacitance, respectively, of dot i [6]. Because of possible difference between «L and aR, the Coulomb diamond appears symmetric only when the two dots are equivalently filled [12]. Then we can confirm that the lateral width, ATsd, of the Coulomb diamond is equivalent to 2(2/+I 'mter)/e. In general for A £ 0, the slopes of the Coulomb diamonds become composite of the two lever arms aL and «R, and can be described as [12],

few)-' =" 1 (1)

where TJ is the fraction of an electron added to dot L. We use this equation to analyze the asymmetric Coulomb diamonds, as observed for A # 0 (Insets to Fig. 5)

Contributions from It and Vmtl.r to the symmetric Coulomb diamond width A\ "sd (Fig. 4(c)) are distinguished by measuring the magnetic field, B, dependence of the anti-crossing gap size, eAVsd/2, (converted to energy with the dot capacitances), (Fig. 6). As B increases, the tunnel coupling is expected to become negligibly small due to contraction of

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 A (meV)

Fig. 5. A dependence of the fractions rj and 1-17 for an electron added to dot L and dot R. The black squares indicate the average of the values rj calculated from both data of Coulomb diamonds measured for various A values using Eq. (1). The solid and dashed lines are calculations of 7 = sin20 and l-tj= cos2# with a parameter r-0.082(=(0.079+0.084)/2) mcV. Insets: Coulomb diamonds plotted in the same way as shown in Fig. 4(b) but along the path a-b (Left) and e-f (Right), respectively. The diamond shapes are asymmetric for Af-0.

96

the wave functions localized in the two dots. However, the inter-dot Coulomb interaction should be practically independent of the wave function extent and therefore, roughly independent of B. The stability diagram at B = 2 T is shown in the upper inset of Fig. 6. The anti-crossing gap size (circle) is much smaller than that at B = 0 T (Fig. 4(a)) and has a sharp form like the classical honeycomb diagram (i.e. / = 0) [6,12]. We conclude that at high B, the saturation value of eAFsd/2 is given simply by Winter = 0.175 meV, and therefore the variation of the anti-crossing gap size between B = 0 and the saturation limit is a measure of It. The value of the tunneling coefficient extracted in this way is t = 0.075 meV.

0.35

0.30-

! 0.25-

0.20-

0.15

Fig. 6. Magnetic field, B, dependence of the anti-crossing gap size (/fV) converted to energy with the dot capacitances. The variation of /N between 3 - 0 and the saturation limit is a measure of 2r = 0.15 meV. Upper inset: Stability diagram at B = 2 T.

5. Formation of Heitler-London State

The manner in which the two electrons fill the double dot can be described by two limiting cases, which we term the HL case and the non-interacting (NI) case. In both cases the first electron enters the lowest single particle state (remember we are treating the other electrons as an inert core). This state is a symmetric combination of states, fa and fa, localized on the left and on the right, & = fasin20 + facos20 , where 0 = tan'[(zV20 +

{\+(AI2tf}xn]. Exactly on the DVL (Fig. 4(a)), A = 0 and 0 = n/4. As 0 -» 0 (n/2), fa - » £ (fa). Thus, ?] = sin20 for the first electron.

In the non-interacting limit the second electron occupies the opposite spin state of the same spatial state, and the two particle state (suppressing the spin part) is simply fa(f])fa(r2). The correlated HL limit, on the other hand, takes the (spatially symmetrized) two electron state as (fa(r\)fa(r2) + fa(r\)fa(r2)), which has identically one electron in each dot, independent of A Then, the final state is always (NL+1JJR+1). This implies that for the second (primed) electron, rf =1- r] = cos2 ft For the moment we will assume that the second electron forces the creation of two localized states and pushes the first electron into one and occupies the other, i.e. the HL limit. From the fit of Eq. (1) with / as a parameter to the data set of the Coulomb diamond slopes (several additional sweeps parallel to the path c-d in Fig. 4(a) have been performed), we obtain t = 0.079, and 0.084 meV for the first, and second electron, respectively (Fig. 5). The / values agree well with that estimated from the B dependence of the anti-crossing size (Fig.6).

We also calculate rj and 1- 77 for various A values. Thus obtained functions are well reproduced by calculations of sin2# and cos2 0, which indicate that as A becomes large, an electron initially localized in dot R becomes delocalized in the symmetric state, and then localized again but in dot L.

For adding the second electron, the interaction energy should be taken into account. First we estimate the energy necessary to add two electrons to the same spatial state (NI model). If we ignore exchange and other higher order interaction terms, then the NI addition energy to go from 0 to 2 electrons, proportional to the length of the diagonal lines such as a-b, c-d and e-f in Fig. 4(a) or, alternatively, to the spacing between Coulomb oscillations, is EC(NI) = 7j2VmU3

L+ (1-

97

7)2frintraR+277(l-7)frmter. The HL state, by contrast, has a constant Coulomb energy of Pinter, so that its addition energy is EC(HL) = Winter + EA - Es, where EAXS) = (at + %)/2 ± [r + (A/2)2]1/2. Note that the A dependence of the HL energy is contained entirely in the single particle energies, whereas for the NI model, tj depends on A and so the Coulomb energy varies as we move away from the DVL.

The expressions for EC{NI) and EC(HL) depend on the intra-dot Coulomb matrix elements, which are given approximately by the charging energies of the two dots (I r

m&aL =

1.8 meV and FmtraR= 1.4 meV). The value of

Winter = 0.175meV is estimated from the reduction of the anti-crossing with increasing B (Fig. 6). Using these values of I'mtra, Fmter

and t, we calculate EC(NI) and EC{HL) versus A, and compare with the experimental data AFpp measured for various DVLs in Fig. 4(c). In this figure we have plotted AI'PP weighted by the average lever arms (open triangles). The first feature to note is that for small |A| the HL addition energy is far below the NI energy, justifying our assumption of its validity in this regime. For sufficiently large |A| the gap between Es and EA exceeds the Coulomb cost of double occupancy and the NI state becomes lower in energy. The experimental data of the spacing between measured Coulomb oscillations agrees well with EC(HL). Thus our two electron state is a HL state up to around the borders of the honeycomb cell. EC(HL) we have calculated in Fig. 4(c) does not include the term to favour a spin singlet state, i.e. 4^/(1 "intra- Winter) (=/). Actually we are able to estimate this energy J = 0.021 meV, using t = 0.08 meV, I'mtia=1.4meV and t"mtei. = 0.175meV. The J is much smaller than It, J-"inter and Fmtra, indicating that the HL model is a good approximation.

Note when the inter-dot tunnel coupling is large, the NI state is more favoured than the HL state, because EC(NI) does not explicitly include t. We see a signature of the

NI states in Fig. 2 (b) in the side gate voltage ranges of VsL = -1.8 to -1.4 V and I "sR = -1.8 to -1.3 V. In this region, the anti-crossing gaps are very wide. The anti-crossing itself is not well identified, and the stability diagram resembles that for a single dot [6].

Finally, we discuss the relevance of our coupled dot device for implementing quantum gate and quantum entanglement. In our device the exchange coupling J between two electron spins in two dots is tunable with magnetic field and central gate voltage. The temporal control of J between J = 0 and 4- 0 enables swapping of two spins [2], by applying a pulsed voltage to the center gate between the dots. The operation time rswap is, for example,

200 ps for J = 0.02 meV. This J can be too small to perform reliable quantum gate operations, and the rSWaP is not short enough as compared to the decoherence time. We have recently obtained J > 0.1 meV in a similar but smaller double dot device. Thus estimated rSwap ~ 30 ps meets our criteria as described before.

Conclusions

We have prepared a novel double dot device in a hybrid vertical-lateral geometry, which allows us to precisely evaluate the way of filling the one-electron and two-electron states. From measurements of stability diagrams in the plane of two side gate voltages and nonlinear conductance characteristics we derive important parameters such as the inter-and intra-dot Coulomb energies and the tunnel coupling energy to influence the electronic configurations in the double dot structure. We confirm the filling of the HL two-electron state with one electron localized in each dot in the weak tunnel coupling regime. We also demonstrate that the tunnel coupling (and thereby the exchange coupling) is tunable with center gate voltage and magnetic field.

98

Acknowledgments

We thank S. Teraoka, A. Shibatomi, S. Amaha, and D.G. Austing for experimental help and useful discussions. Part of this work is financially supported by the Grant-in-Aid for Scientific Research A (No. 40302799), by SORST-JST, by the Japan Society for the Promotion of Science, by Focused Research and Development Project for the Realization of the World's Most Advanced IT Nation, IT Program, MEXT, and by the DARPA-QUIST program (DAAD19-01-1-0659).

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10. T. Hatano, M. Stopa, T. Yamaguchi, T. Ota, K. Yamada, and S. Tarucha, Phys. Rev. Lett. 93, 066806 (2004).

11. S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, LP. Kouvenhoven, Phys. Rev. Lett. 77, 3613 (1996); S. Tarucha, D.G. Austing, Y. Tokura, W.G. van der Wiel, and L.P. Kowenhoven, Phys. Rev. Lett. 84, 2485 (2000).

12. T. Hatano, M. Stopa, and S. Tarucha, Science, 309, 268 (2005).

99

COHERENT TRANSPORT THROUGH QUANTUM DOTS

*S. KATSUMOTO, M. SATO, H. AIKAWA, and Y. IYE Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

* E-mail: kats@issp.u-tokyo.ac.jp

We report experiments on the transport through interference circuits, in which semiconductor quantum dots are embedded. The interference picks up coherent transport through the dots, which provides information on electronic states including many body effects in them. Here we focus on the role of spin scattering in a quantum dot to the coherent transport through it. We present evidence of quantum decoherence due to the spin-flip scattering in a dot through the Aharonov-Bohm (AB) amplitude when the Kondo effect does not emerge. Around a resonance peak of the dot, the interference effect appears as the Fano effect, which also gives information on the phase shift through the dot. We have observed the combined effect of the Fano effect and the Kondo effect — the Fano-Kondo effect in a side-coupled quantum dot system, which manifests the quantum coherence of the Kondo singlet state and at the same time the phase shift locking to IT/2. In an AB ring system, the Fano-Kondo effect has been also observed and the parameter of interference is tuned by the AB flux.

Keywords: Spin scattering; Fano effect; Kondo effect.

1. Introduction

Process of quantum decoherence can be viewed as a dispersion of quantum informa­tion into an "environment", which has an in­finite number of degree of freedom. The pro­cess is triggered by generation of an entan­glement between the system under consider­ation and a freedom of the environment. In early days of mesoscopic physics, saturation of phase coherence length at low tempera­tures was attributed to spin scattering, which leads to the entanglement of the spins of con­duction electrons and the localized moments, hence to that of the spins and the orbitals. Therefore it mediates indirect interaction be­tween electrons. In the end of 1990s, there was a debate on the existence of "intrinsic de-coherence", which persists down to absolute zero1,3. Here, spin scattering also plays a key role and the question even extends to the ex­istence of decoherence triggered by spin scat­tering.

The triggering of the decoherence means the beginning of entanglement between the system and the environment. However if

there is some mechanism which localizes the entanglement in a finite area, the quantum coherence should survive. In the case of spin flip scattering, such a mechanism is the Kondo effect2, which localizes the entangle­ment in the size of Kondo cloud. On the other hand, the Kondo cloud is made of an entangled state with a huge number of free­dom, and may be fragile for decoherence by external perturbations4.

We hence pose a question whether the in­terference of a Kondo cloud can be observed in a mesoscopic interference circuit. Because a Kondo cloud is always in resonance with the Fermi energy, such an interference should appear in a combination with a resonance, i.e., as the Fano effect. Such an effect — the Fano-Kondo effect is manifestation of quan­tum coherence of a Kondo cloud in an in­terference circuit and at the same time, can bring about information on the phase shift of an electron through the Kondo cloud.

In this article, we report experimental observation of spin-flip triggered quantum decoherence and observation of the Fano-Kondo effect, in which the quantum coher-

100

ence is recovered. The phase shift locking to TT/2 reflecting the resonant feature is deduced from the lineshape of the Fano-Kondo peaks.

2. Exper imen t s

In the pioneering works performed in the Weizmann institute5, they took care to mini­mize the effect of multiple reflection or encir­cling so as to obtain the information of the phase-shift directly from the AB oscillation.

In the present work, we have adopted the two-wire setup, from which the direct in­formation is not available though high ratio of the coherent component in conductance is achievable and phenomena characteristic to coherent transport are expected.

We used two methods to define quan­tum wires and dots in two-dimensional elec­tron gas (2DEG) formed at GaAs/AlGaAs hetero-structure (sheet carrier density 3.8 x 1015 m - 2 , mobility 80 m2/Vs).

vc

v t Vz V* ~~

(a) (b)

Fig. 1. (a) Scanning electron micrograph of an AB ring structre with wrap gate quantum dots, (b) Gate configuration for side-coupled quantum dot struc­ture.

In the first method, we adopt so called wrap gate structure to define quantum dots and wires. Quantum wire circuits are de­fined, by using electron beam lithography and wet etching. Fine patterns of metal­lic gates fabricated on the circuits form tun­neling barriers, which define quantum dots and control the parameters. A representa­tive structure, which contains two QDs in a

ring is shown in Fig. 1(a). In the second, a dot and a wire are de­

fined in the conventional split gate method. This is used only for the side coupled geom­etry. Note that this is not applicable to the ring geometry other than by using compli­cated air bridge method. Micrograph of a representative structure is shown in Fig. 1(b).

The samples were placed in a shielded box thermally attached to a mixing chamber of a dilution refrigerator and cooled down to 30mK. Metallic resistors, which act as high frequency noise filters are inserted in the leads just at the boundary of the shielded box. Magnetic field up to 7T was applied by a superconducting solenoid.

3. Resul ts and Discussion

3.1. Decoherence due to Spin Scattering

The AB amplitude is often considered as a measure of quantum coherence. When the number of electrons AT in a QD is odd, it works as a magnetic impurity and the spin-flip scattering causes decoherence and the amplitude diminishes. When N is even, the electrons pass through the dot via the top-

ni U 1 Li 1 -0.19 -0.185 -0.18 -0.175 -0.17

Gate Voltage (V)

Fig. 2. Amplitude of the AB oscillation over ten periods as a function of the dot gate voltage. The positions of Coulomb peaks corresponding to a spin pair state are indicated by broken lines. The AB amplitude is asymmetric with respect to the center of the peak. The reduction of the amplitude at the center of the peak is due to the n jump of the AB phase.

101

most empty level where no spin-flip can oc­cur, hence keeping the AB amplitude. As a result the AB amplitude should change alter­natively with N6, which constitutes evidence of decoherence due to spin-flip scattering7.

We prepared an AB ring plus a QD sys­tem. In a real system, the situation is not so simple, e.g., correlation effect in a QD may induce a high spin state, which also can scat­ter an electron with spin-flip and largely vio­lates the simple rule of alternative change of the AB amplitude.

In order to avoid such effects, we should look for a "spin-pair" state, below which the energy levels form a closed shell. Further­more, the spin-pair state itself should form the next closed shell. Selection of such a state solves almost all of the above problems. The high-spin problem is automatically solved. Because the orbitals are the same for the states with Krammer's degeneracy, difference in the interference amplitude due to that in the orbitals can be ignored. And under such conditions, the electrostatic potential effect modifies the AB amplitude, if it exists, in the same way for the two Coulomb peaks, which gives inverse effect to that by the spin scattering.

We have picked up a pair of Coulomb peaks from the addition energy spectum, which have characteristics of a spin-pair from every aspect. Figure 2 shows the AB ampli­tude for these peaks integrated over 10 pe­riods, in which they have no level-crossing. Just as predicted in the theories, the AB am­plitude clearly is suppressed for N = odd. The sharp dips at the Coulomb peaks are due to 7r jump of the AB phase, which comes from the two-terminal nature. The results hence constitute evidence of the decoherence triggered by spin-flip scattering.

3.2. The Fano-Kondo effect

A single spin-flip scattering results in an en­tangled state between the conduction spin

and the localized spin as

-L(M>|<u>-|*r)|dT», (i)

here s and d denote conduction and local­ized electrons respectively and j and J. indi­cate spin states. (1) is written as a vector in the Hilbert space and cannot represent deco­herence. Even the suppression of the inter­ference is not rigorously equal to quantum decoherence as clearly shown by a newtron interference experiment8.

Instead, we should think (1) as a trigger of decoherence. After the scattering, the con­duction electron and the localized electron are spatially far apart and interact with other degrees of freedom independently, which re­sults in real quantum decoherence, i.e., an increase in the quantum information entropy.

In the neutron experiment, the interfer­ence pattern was recovered with a further ro-taion of neutron spin to in. Then a natu­ral question is whether such an effect that prevents the dispersing of quantum informa­tion exists or not in solids. The Kondo effect is a candidate to have such property form­ing a Kondo singlet state (Kondo cloud), in which the conduction electrons are multiply scattered and the quantum information does not disperse into the environment though the state itself is composed of numbers of spin-flip scatterings.

The quantum coherence of a Kondo sin­glet can be examined through the interfer­ence of the singlet itself. Because a Kondo singlet is always in resonance with the Fermi surface, the interference is associated with resonance, and hence can be viewed as the Fano effect. It appears when a resonance and an interference coexist, as a character­istic distortion in the system conductance G as

G(e)<x(e + q)2/{e2 + l), (2)

where e is the energy difference from the res­onance position normalized with the width

102

Fig. 3. Conductance of the wire measured at several coupling strength tuned by Vm- The base tempera­ture is 30mK. Vm varies stepwise by 6 mV and the data are offset by 0.3e2//i for clarity.

of the resonance, and q is Fano's asymmetric parameter 9.

Then the Kondo effect in an interference circuit should appear as a mixed effect of the Kondo resonance and the Fano resonance, which is called the Fano-Kondo effect10. An experimental observation of the Fano-Kondo efect evidences the quantum coherence of the Kondo singlet by itself, and at the same time, gives information on the phase shift through the Kondo singlet.

We adopted so called side-coupled quan­tum dot configuration, which is shown in Fig. 1(b) and schematically in the upper of Fig.311. The mutual coupling strength be­tween the dot and the wire is tuned by gate voltage Vm and the electrostatic potential is by Vg. The lower of Fig.3 shows the wire conductance as a function of Vg and from top to bottom the coupling strength is in­creased. When the coupling is weak we observe ordinary Fano lineshapes in the Vg

dependence. With increasing the coupling

strength, the conductance of two regions be­tween the Fano resonances decrease and the neighboring Fano lineshapes are bound into broad dip structures.

These are clear sign of the Fano-Kondo effect12. This claim can be examined from various aspects and here we show the tem­perature dependence. The upper panel of Fig. 4(a) shows detailed temperature depen­dence of G(Vg) at Kondo valley A. We ob­tained the Kondo temperature I K from the form,

G(T) = Go - Gi (Tg/(T2 + Tg))s, (3)

where Glt TK = TK/V21/3 - 1 and s are fit­ting parameters. Go was fixed to 1.8e2//i, the wire conductance far from the anti-resonance. Examples of the fitting are shown in Fig. 4(b). The data below 120 mK are not taken into account in the fitting because of the electron temperature saturation. From

Fig. 4. (a) Upper: Conductance as a function of gate voltage at temperatures from 750 mK to 50 mK with the temperature step of 50 mK. Lower: Kondo temperatures TK obtained from the temperature de­pendence, (b) Examples of the fitting to obtain T K . The gate voltages adopted here are indicated by ar­rows in (a), (c) Phase shift of the QD obtained for the data at 50mK and 750mK based on the model shown in (d). (d) Schematic diagram of a simple quantum circuit model of the system. The factor 2 of A6 in a,2 is due to the reflection.

103

the fitting, s = 0.25 ± 0.04 was obtained, which is in accordance with the prediction for spin 1/2 impurities. The obtained 7k are plotted against Vs in the lower panel of Fig. 4(a). 7k depends quadratic on Vg

with the bottom around the mid-point of the Kondo valley. This dependence agrees well with the Kondo temperature, 7k = v/n7exp(7re(e + U)/TU)/2, where r is the dot-wire coupling and e is the single elec­tron level measured from the Fermi level. We obtained U = 0.39 ± 0.03 meV and r = 0.30 ± 0.02 meV by fitting the above function to TK, which are in good agreement with those obtained from the Coulomb dia­monds.

The nearly symmetric lineshape in Fig.4(a) means that the Fano parameter q is around zero. Hence we assume q = 0 and discuss the phase shift of Kondo state, which can be obtained from the analysis along the scattering formalism for conductance. A sim­ple model of the system : a three branch S-matrix and a reflector (the QD), is shown in Fig. 4(d). Assuming the absence of reflection from the electrodes we put a3 = 0. And tak­ing the QD as a transmitter plus a perfect re­flector, we put ai = &2 exp(2iA0) and a\ = 1 for unitary input. Figure 4(d) shows thus calculated A9(Vg). The phase shift variation approaches to the stationary line of 7r/2 at the Kondo valley with temperature lowering.

The above observation manifests firstly the quantum coherence of the Kondo sin­glet and secondly its resonant feature with the Fermi surface of the electrodes. In other words, the symmetry of the present Kondo cloud is SU(2), which is consistent with the fact that the Kondo dips appear alternatively in the Coulomb oscillation. This is in con­trast to the existing result 13, where only locking to 7T at pre-Kondo region was ob­served. The difference may come from the width of the energy levels14, or from that in the geometry of the interferometer though we have no conclusive opinion at present.

3.3. Tuning of the Fano-Kondo effect in an AB ring

The Fano-Kondo resonance is also expected in an AB ring-QD hybrid10, which is more suitable to see the degree of quantum coher­ence and to control the sign of interference through magnetic flux, which does not affect the channel number or other transport pa­rameters other than the phase difference.

Figure 5 shows the appearance of the Kondo effect in a QD embedded in an AB ring though the reference arm is pinched off. In order to observe the clear Kondo reso­nance, we needed to apply a perpendicular magnetic field of 0.42T, which is common phenomenon in experiments though the real reason is not known yet.

We observed clear AB oscillation with opening the reference arm of the AB ring, which enables us to extract "coherent" com-

-0.84 -0.82 -0.80 -0.78 2 -0.76

Fig. 5. Upper: Two Kondo peaks of a QD embed­ded in an AB ring (the reference arm is pinched off) at various temperatures under magnetic field of 4.2T. Lower: Gray scale plot of the conductance at 50mK. The abscissa is common for the upper and the lower panels. Clear Kondo resonances are observed in the middle of Coulomb diamonds.

104

Gate Voltage (V)

Fig. 6. Upper: Total conductance of the AB ring-QD hybrid with the open reference arm. Lower: "Co­herent" component of the conductance for four rep­resentative AB flux shift from the magnetic field of 0.4795T (i.e. the field in the upper panel). <po is the flux quantum h/e. The abscissa (the gate voltage of the QD) is common.

ponent in the conductance by subtracting

slowly varying background. The conduc­

tance lineshape and the coherent component

for a period of the AB oscillation (<fi0 = h/e)

are shown in Fig.6. Surprisingly the coher­

ent component can be almost perfectly anti­

symmetric (Acfi = 0, +<^o/2) or symmetric

(±fo/4). The results at the first sight show a

strange outlook which breaks the Onsager

symmetry. T h a t is, the two-terminal na­

ture of the device should erase the pri­

mary oscillation at resonances and result

in "frequency double" at the Fano-Kondo

resonance1 0 . However the present device has

a finite width in the arms and this residual

two-dimensionality results in multiple pa ths

in the ring, and causes smooth shift of the

AB phase instead of 7r-jump. If we inter­

pret the results along this line, they are in

agreement with the theory. The decrease

in the V^-derivative of the conductance for

Acp = 0, + 0 o / 2 in the middle of the Fig.6

again indicates the tendency of the phase-

shift locking to 7r/2.

In summary, we have examined coherent

t ransport through quan tum wire - quantum

dot hybrid systems. It has been shown tha t

a spin flip scattering triggers quantum deco-

herence. On the other hand, the Kondo sin­

glet s tate tunred out to be quantum mechan­

ically coherent. The analysis of the Fano-

Kondo lineshape indicates the locking of the

phase shift to 7r/2.

We thank K. Kobayashi for the collabo­

ration in an early stage of experiments.

References

1. P. Mohanty and R. A. Webb, Phys. Rev. B 55, R13452 (1997).

2. J. Kondo, in Solid State Physics Vol. 23, p. 13 (Academic Press, New York, 1969).

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5. R. Schuster, E. Buks, M. Heiblum, D. Ma-halu, V. Umansky, and H. Shtrikman, Nature 385, 417 (1997).

6. J. Konig and Yuval Gefen, Phys. Rev. B 65, 045316 (2002).

7. H. Aikawa, K. Kobayashi, S. Katsumoto, and Y. lye, Phys. Rev. Lett 92, 176802 (2004).

8. H. Rauch, A. Zeilinger, G. Badurek, A. Will­ing, W. Bauspiess, and U. Bonse, Phys. Lett. 54A, 425 (1975).

9. K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. lye, Phys. Rev. Lett. 88, 256806 (2002).

10. W. Hofstetter, J. Konig, and H. Scholler, Phys. Rev. Lett. 87, 156803 (2001).

11. M. Sato, H. Aikawa, K. Kobayashi, S. Kat­sumoto, and Y. lye, Phys. Rev. Lett. 92, 066801 (2005).

12. K. Kang, S. Y. Cho, J.-J. Kim, and S.-C. Shin, Phys. Rev. B 63, 113304 (2001).

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14. U. Gerland, J. von Delft, T. A. Costi, and Y. Oreg, Phys. Rev. Lett. 84, 3710 (2000); Phys. Rev. Lett. 95, 066801 (2005).

105

ELECTRICALLY P U M P E D SINGLE-PHOTON SOURCES TOWARDS 1.3 Aim

XIULAI XU*, DAVID A. W I L L I A M S and J O N A T H A N D. M A R

Hitachi Cambridge Laboratory, Cavendish Laboratory, Madingley Road, Cambridge CBS OHE,

United Kingdom

*E-mail: xx757@cam.ac.uk

J O H N R. A. CLEAVER

Microelectronics Research Centre, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, United Kingdom

We demonstrate electrically pumped single-photon emission from single quantum dots in a lateral p-i-n junction without metal apertures. In such a structure, strong antibunching effects of exciton and biexciton emission are observed and imply that single-photon emission has been achieved. The devices can be operated with electric pulses with repetition rates of more than 100 MHz. New wafers with emission wavelength around 1.3 /Jtn are investigated for future development.

Keywords: Single-photon sources; InAs quantum dots; Lateral p-i-n junction.

1. Introduction

To implement quantum communications, single-photon sources are in demand to avoid the simple beam-splitter attack1; and sin­gle photons also have potential applications in physical realizations of entanglement for linear optical quantum computation2. For this purpose, several single-quantized sys­tems have been investigated such as single atoms3, single ions4, single molecules5, single quantum dots6 '7 and single nitrogen-vacancy centres in diamond8. Among them, semi­conductor InAs quantum dots are promising as they have several advantages: good sta­bility, high repetition rate, electrolumines­cence, and compatibility with semiconduc­tor processing techniques. To meet practi­cal needs, it is desirable to be able to excite a specific single quantum dot electrically for single-photon emission at 1.3 fim.

Single-photon sources pumped electri­cally were first realized using quantum dots formed by etched mesoscopic heteroj unctions with electric injection9. However, low tem­peratures of the order of 50 mK were re­

quired. Recently, Yuan et al.10 have put InAs quantum dots into an intrinsic region of a conventional p-i-n structure. In order to obtain emission from one single dot, metal apertures were made on the surface to se­lect the emission site, but the low coinci­dence between quantum dots and metal aper­tures produced a low yield of functional sites. In this work, we demonstrate electrically-pumped single-dot emission in a lateral p-i-n junction without metal apertures, a device structure which potentially is well suited to integration into complete systems.

2. Device structure and measurements

The structure of our devices is shown in Fig. 1(a). The lateral p-i-n junction, without quantum dots, was reported previously11. In the work reported here, it has been modified to include quantum dots. The structure is designed so that the lower n-type channel is fully depleted if the upper p-type region is in place. When the upper layer is removed, the n-type channel becomes conducting. A

Fig. 1. (a) Schematic vertical section of lateral p-i-n structure, (b) SEM image of the device, the white bar is 2 (im. (c) Electroluminescence spectra from a single dot with a range of injection currents. Each spectrum is shifted for clarity.

p-n junction then forms at the interface be­neath the edge. Kaestner et al.11 successfully obtained electroluminescence from the active region at the etched edge and showed by sim­ulation the current flow geometry within this region. In this work, a very low density of InAs quantum dots was put into the intrin­sic region of this lateral junction. A heavily doped p-type layer on the surface is used to make a non-annealed ohmic contact on the surface. After the fabrication processes, a wire-shaped device is formed (as shown in Fig. 1(b)), the dots at the etched edge in the active channel are excited at low voltages whilst when there is no dot near the edge those far away from the edge can be excited by increasing the voltage. By controlling the voltages, single-dot emission can be obtained from most devices because of the long active channel. The structure is then self-selecting for emission from a single dot, with no need for metal apertures.

The devices were mounted in a He-flow cryostat and cooled to 5K. The emitted light was collected by a long working distance objective (NA 0.5), and dispersed through a 0.46 in spectrometer; the spectrum was detected with a cooled charge coupled de­

vice camera. The spectrum of quantum dots at 1.3 //m was measured with NIR photomultiplier tubes. Correlation mea­surement of electroluminescence was per­formed using a Hanbury Brown and Twiss (HBT) setup12 with a 50/50 beamsplitter and two single-photon-counting avalanche photodiodes (APDs). The two APDs were connected to start and stop inputs of a time-to-amplitude converter, whose output was stored in a time-correlated photon-counting card. The resulting histograms show the large number of photon pairs with arrival time separations of r = tstart~tatop- The his­tograms are equivalent to the second-order correlation function g^(r) when r is much shorter than the average time between de­tected photons. The time resolution of the correlation card is around 190 ps.

3. Resul ts and discussion

Electroluminescence spectra were collected from one device at 5 K with different currents (shown in Fig. 1(c)). At low current, only one peak can be observed, at 1.2995 eV. With increasing applied voltage, another peak ap­pears at around 3 meV less than the exci-ton emission. The dependences of the high and low energy peak intensities on current density are linear and quadratic respectively for small currents. We associate these two peaks with exciton (X) and biexciton (XX) emission.

With the HBT system, we measured the second order-correlation function:

}«(r) = » ? t (1) V ' (I(t))2

where I(t) is the intensity at time t, and (• • •) indicates averaging over t. For an ideal single-photon source, 5 ^ ( r ) should be equal to 0 at zero time delay ( T = 0 ) . Correlation measurements of exciton and biexciton elec­troluminescence are shown in Fig. 2. Fig­ure 2(i) plots the correlation result of the X

107

line (width 400 /feV) for low current injec­tion, at 90 fiA. Clear antibunching at r = 0 can be observed, which shows that the simul­taneous emission of two photons is largely suppressed. gy

b (0) is 0.54 ± 0.05 with the presence of background light, and does not reach the theoretical minimum, zero. If we consider the dark counts, the corrected value is 0.48 ± 0.06. However, the s(2)(0) with­out dark counts still does not go to the ideal value, zero, which may be due to the fi­nite time resolution and background stray light. We fitted the histogram with g^2\r) = 1 — aexp (—r/6), where 1-a and b are g^2\0) and the recombination lifetime, respectively. The thick line in the figure shows the fitting results. The exciton emission lifetime is mea­sured to be 1.07 ns, which is similar to other results10'13-14.

-20 -10 0 10

Delay time (ns)

Fig. 2. Second-order correlation function g ' 2 H T ) measured (thin lines) and fitted (thick line) of elec­troluminescence of: (i) exciton peak; (ii) biexciton peak; (iii) central position between exciton and biex­citon peaks. Inset: Spectrum and correlation posi­tion for (iii).

Generation of cascade photons15 and en­tangled photon pairs16 has been proposed by using exciton/biexciton in one quantum dot. Figure 2(ii) shows the correlation measure­ment of a XX line at a large injection cur­rent, where the biexciton emission was dom­inant in the spectrum. The </2^ (0) equates to 0.43 ± 0.03, indicating the antibunching na­ture of biexcitons. The lifetime of biexcitons

is around 350 ps, which is much shorter than that of exciton emission. The faster decay of a biexciton relative to an exciton agrees with the results of time-resolved experiments on quantum dots17. Theoretically if two in­dependently excited single-photon emitters (such as exciton and biexciton emission) are present and can be fed into the same corre­lation measurement simultaneously, the sec­ond correlation function at zero time delay 5(2)(0) should be equal to 0.514. To mea­sure this, we adjusted the injection current to make the intensities of exciton and biexci­ton emission equal, and moved the gratings of the spectrometer to the central position where the exciton and biexciton spectra over­lap. The spectrum and correlation position of this measurement are shown in the inset of Fig. 2. We can consider emission at this position as being due to both exciton and biexciton, two single emitters. The correla­tion result shows a shallower dip than that of exciton/biexciton (Fig. 2(iii)). The fitted parameter a around 0.22 ± 0.04 is rather less than half of that in the exciton and biexciton histograms alone, as expected.

Delay (ns)

Fig. 3. Second-order correlation function g(2> (T) for electroluminescence of exciton peak 100 MHz pulse electrical injection, with height of 5 V and a width 800ps superimposed on 15 V DC voltage.

One advantage in using InAs quantum dots as single-photon sources is that they can be operated at high repetition rates, limited by the recombination time for ex-

108

citons (around 1 ns). Therefore pulse in­

jection at high repetition rate has been

investigated6 , 7 '1 0 for obtaining high-speed

regulated single-photon sources. Figure 3

shows a correlation histogram of exciton

emission, driven with 100 MHz pulses with

height of 5 V and a width 800ps superim­

posed on 15 V DC voltage. Antibunching can

be observed at zero t ime delay, which implies

the possibility of using single-photon sources

at high repetit ion rates. Again, </2 '(0) is

0.45 ± 0.06, and not zero. For this, an addi­

tional reason may be tha t the R F pulse signal

was broadened between the pulse generator

and the device.

900 1000 1100 1200 1300 Wavelength (nm)

Fig. 4. Photoluminescence spectra as a function of excitation power from quantum dots around 1.3 jj-ra.

4 . C o n c l u s i o n a n d future works

We have demonstrated electroluminescence

from excitons and biexcitons at one single

quan tum dot, introduced into the intrinsic

layer of a lateral p-i-n junction; in this case,

a metal mask is not needed to select the

emission light from a single quantum dot.

Antibunching of exciton and biexciton emis­

sion a t zero t ime delay implies single-photon

emission, which can be operated at high rep­

eti t ion rates. For communications, it is de­

sirable to have electrically pumped single-

photon sources at 1.3 or 1.55 fim. Figure 4

shows the excitation power dependent photo-

luminescence spectra of quantum dots which

are emit t ing around 1.3 /jm. Similar lateral

structures with low density dots emitt ing at

1.3 /xm are being fabricated for single-photon

emission.

A c k n o w l e d g m e n t s

We wish to acknowledge Dr M. Hopkinson

and Dr Huiyun Liu from the University of

Sheffield for growth of the QD wafer. The

project is supported by the Foresight Link

Award Nano-electronics at the quantum edge

from the UK Depar tment of Trade and In­

dust ry and Hitachi Europe Ltd.

R e f e r e n c e s

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and H. Weinfurter, Phys. Rev. Lett. 85, 290 (2000).

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Williams, Jpn. J. Appl. Phys. 41, 2513 (2002).

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109

AHARONOV-BOHM-TYPE EFFECTS IN ANTIDOT ARRAYS AND THEIR DECOHERENCE

MASANORI KATO, HIROYASU TANAKA, AKIRA ENDO, SHINGO KATSUMOTO

AND YASUHIRO IYE

Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha Kashiwa, Chiba 277-8581 Japan

We have investigated the Aharonov-Bohm(AB)-type quantum oscillations in triangular and square arrays of antidots fabricated from two-dimensional electron systems in GaAs/AlGaAs heterostructure.

Keywords: Aharonov-Bohm oscillation, two-dimensional electron system, antidot array; vortex glass transition, GaAs/AlGaAs

1. Introduction

Since its first experimental observations1'2, the Aharonov-Bohm (AB) effect in electron transport has been extensively studied in a wide range of mesoscopic conductors not only in the diffusive metallic transport regime but also in the ballistic (or semi-ballistic) transport regime realized in high mobility two-dimensional electron system (2DES) in semiconductor heterostructures.

The AB effect is a manifestation of quantum interference effect and shows up as oscillatory resistance of a ring-shaped sample at low temperatures as a function of external magnetic field. It is customary to distinguish two types of the AB effect: The Altshuler-Aronov-Spivak (AAS) effect {hlle oscillation) is attributed to interference of a time-reversed pair of electron waves. As long as the spin-orbit effect is small, the AAS interference is always constructive at zero magnetic field. By contrast, the phase of the AB interference (in its narrower sense), which is responsible for hie oscillation, is sample specific. This fundamental difference is highlighted in an assembly of identical rings. While the AB interference effect tends to be smeared by the random phase averaging in such a system, the AAS effect is immune to the ensemble averaging, as demonstrated experimentally3.

Antidot lattices fabricated from 2DES constitute another interesting class of systems to study the AB effect. An antidot lattice can be viewed as a network consisting of many rings. The first observation of the AB oscillation in antidot lattice was made by Nihey et al? and by Weiss et al5 They detected S-periodic small oscillations superposed on the commensurability peak of magnetoresistance in square antidot lattices. Later, the AAS oscillation was observed in triangular antidot lattice6.

When one was reminded that dephasing of the AB interference should occur in an array of rings due to ensemble averaging, the very fact that AB oscillation could be observed in antidot lattice of macroscopic overall size, was rather surprising. Subsequent theoretical studies have attributed the effect to oscillatory fine structure of the density of state spectra, as calculated, for instance, by periodic orbit theory. For this reason, it is customary to call the effect AB-"type" oscillation so as to distinguish it from the ordinary AB effect in the single ring case. There is an intermediate case of finite antidot lattices7, in which a small number of antidots constitute a mesoscopic size system that becomes phase coherent as a whole at low enough temperatures.

Despite a great deal of the theoretical efforts8, full understanding of the physical

110

origin of the AB-type oscillation in macroscopic antidot lattice is yet to be worked out.

We have investigated the AAS and AB-type oscillations in triangular and square antidot lattices both near zero magnetic field and in high fields. Due to the page limitation, we focus on the low field behavior in this paper. Various aspects of the AB-type oscillation effect in the quantum Hall regime are discussed elsewhere9'10.

2. Experiment

The samples used in the present study were fabricated from a GaAs/AlGaAs single heterostructure wafer with electron density «= 3.8xl015 m"2 and mobility p= 60 m2/Vs. Using electron beam lithography and shallow wet etching, triangular and square arrays of antidots as shown in the inset of Fig. 1 were fabricated on the active area of Hall bar samples with AuGe/Ni Ohmic contact pads. The lattice period was a= 1 um and the antidot diameter was d= 600nm. Because of the depletion region of width -lOOnm around each dot, the effective antidot diameter becomes c?*~8Q0nm, leaving the effective width of the narrowest part of the channel between the antidot ~100nm. The 2DES density was controlled by a Au-Ti Schottky front gate, which also changed the effective channel width. For the "large" array samples the total number of antidots was ~104, while for the "small" array samples it was -50.

Transport measurements were made by a standard ac lock-in technique at 13 Hz. The sample was directly immersed in the 3He-4He mixture of the mixing chamber of a dilution refrigerator at a base temperature 30mK. Magnetic fields up to 15 T were applied perpendicular to the 2DES plane.

_

_

/ '

'. • ' '

i i I 1

1 T=30mK Vo=-60mV

t

II 1 1 , , 1

0 c 1 -n r "n i c 1 1 ^u_ -2 0 2 4 S 8 10 12 14

Magnetic Field (T)

Figure 1. Inset. The left figure is a schematic potential landscape for electrons in an antidot lattice. The right panel shows an SEM image of the triangular lattice of antidots. The lattice constant is a= 1 \xm and the antidot diameter is d= 600nm. Main Panel: Magnetoresistance at 30mK.

3. Results and Discussions

The main panel of Fig. 1 shows the magnetoresistance trace of the large array triangular lattice sample. The effect of the antidot array is most prominent in the enhancement of zero-field resistance and the large negative magnetoresistance in the low field range. By contrast,, the overall shape of the magnetoresistance trace in the high field regime, i.e. the Shubnikov-de Haas oscillation and the quantum Hall effect, is not so different from those of unpatterned 2DES samples. However, when the fine details of the trace exhibits different kinds of oscillatory behavior.

The inset of Fig. 2 is an expanded view of the peak at zero field of the trace in Fig.l, which clearly shows oscillatory behavior. In order to highlight the oscillatory part, the low field magnetoresistance with the smooth background subtracted, are shown in the main panel. Oscillations with two distinct frequencies are readily recognized. The rapid oscillation at around zero magnetic field has a period hl2eS=2.6mT (S = (V3/2)?2 = 0.80//m2

being the unit cell area of the triangular lattice) and has a maximum at zero magnetic field, so it is identified as the AAS oscillation.

I l l

The smaller amplitude oscillation at somewhat higher field has a period h/eS=5.2mT, and is the AB-type oscillation.

-0.04 -0.02 0,00

Magnetic Field (T)

Figure 2. Inset: Magnetorestance traces of the triangular lattice sample at low magnetic field. Main Panel; The oscillatory part of the magnetoresistance highlighted by subtracting the smooth background. The AAS (hlle) and the AB-type (hie) oscillations are clearly identified.

Similar phenomena were also observed in the square lattice samples. It is worth noting that the earlier studies4"6 have reported that the low field magnetotransport behavior is quite different between the square and the triangular antidot lattices. For instance, the AAS oscillation was observed only in the latter, and the former showed a slightly positive magnetoresistance with a "matching" peak while a large negative magnetoresistance characterized the latter. These differences have been attributed to the fact that the ballistic electron trajectories tend to be more strongly back-scattered in the triangular lattice than the square lattice. While such differences in behavior is manifest among samples with small to intermediate values of aspect ratio, the difference tends to diminish for systems with larger aspect ratios (i.e. narrower channel) such as those studied here.

In order to study the decoherence mechanisms in these systems, we measured the temperature dependence of the amplitude of the AAS and AB oscillations. Figure 3 shows the temperature dependence of the

AAS oscillation observed in the square lattice sample. The data can be fitted to the following functional form reflecting the temperature dependence of the inelastic scattering time T OC T~P .

4AAsxexp{-aT") (1)

with p~l.6. Our earlier study on triangular lattices have yielded a similar exponent p~ 1.5 and 1.1.

(b) V^-t

: •

•

K f l 0

^ V .

Magnetic Fielc

1

<T)

~

-

200 400 600

Temperature (mK)

Figure 3. (a) Magnetorestance traces of the square antidot lattice sample at different temperatures. The inset shows the AAS (h/2e) oscillation around zero magnetic field. (b) Temperature dependence of the AAS oscillation. The solid curve is a fit to the functional form exp(-a7''')

The AB-type oscillation in antidot arrays is attributed to oscillatory fine structure in the density of states of the system8, and its temperature dependence is expected to arise from temperature smearing of the density-of-state feature. Figure 4 shows the AB-type oscillation in the square lattice sample at low magnetic fields. The temperature dependence of the amplitude in this case is fitted to the functional form

112

aT 2K kn /^\ Aw oc , a = §-» (A>

s in^ar) AE

which is shown by the solid curve. In our earlier study on triangular lattice samples, we found that the experimental results show some deviation from the above functional form, i.e. saturation of AB-type oscillation amplitude for T -> 0 was found to be slower than eq.(2). We attributed it to the temperature dependence of the phase coherent area9. The present result on the square lattice is somewhat at variance with the previous result, and appears to be explained rather simply by thermal smearing. The origin of the different experimental outcome is unknown at the moment.

. \ "

•S

-

150

100

\ Sf50

\ ° *\

MllA JilL i AyiAfo.fWrfUhif

. . . 500 mK

.05 0.10 0.15 Magnetic Field (T)

"\« ^ N . •

2IHI 400 601)

Temperature (mK)

Figure 4. Inset: The AB-type oscillation in the square antidot lattice sample. Main panel: The temperature dependence of the amplitude, the solid curve represents a fit to a functional form aT/s'mh(aT).

4. Conclusion

We have studied quantum coherent oscillatory phenomena in antidot arrays with triangular and square symmetries. The difference in the lattice symmetry becomes less important for antidot arrays with larger aspect ratios. The temperature dependence of the AAS oscillation reflects that of the inelastic scattering time. On the other hand, the

temperature dependence of the AB-type oscillation is understood in terms of thermal smearing of oscillatory fine structure in the density of states.

Acknowledgements The work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sport, Science and Technology (MEXT), Japan.

References

1. D.Yu. Sharvin and Yu.V. Sharvin: Pis'ma Zh. Eksp. Theor. Fiz. 34 (1981) 285. [JETP Lett. 34 (1981) 272.]; B.L. Altshuler, A.G. Aronov and B.Z. Spivak: Pis'ma Zh. Eksp. Theor. Fiz. 33 (1981) 101. [JETP Lett. 33(1981)94.]

2. R.A. Webb, S. Washburn, C.P. Umbach and R.B. Raibowitz: Phys. Rev. Lett. 54 (1985)2696.

3. C.P. Umbach, C. van Haesendonck, R.B. Laibowitz, S. Washburn and R.A. Webb: Phys. Rev. Lett. 56 (1986) 386.

4. F. Nihey and K. Nakamura: Physica B 184 (1993)398.

5. D. Weiss, K. Richter, A. Menshing, R. Bergmann, H. Schewzer, K. von Klitzing and G. Weimann: Phys. Rev. Lett. 70 (1993)4118.

6. F. Nihey, S.W. Hwang and K. Nakamura: Phys. Rev. B 51 (1995)4649.

7. R. Schuster, K. Ensslin, D. Wharam, S. Kuehn, J.P. Kotthaus, G. Boehm, W. Klein, G. Traenkle and G. Weimann: Phys. Rev. B 49 (1994) 8510.

8. S. Ishizaka, F. Nihey, K. Nakamura, J. Sone and T. Ando: Phys. Rev. B 51 (1995) 9881; S. Uryu and T. Ando: Phys. Rev. B 53 (1996) 13613.

9. Y. lye, M. Ueki, A. Endo and S. Katsumoto, J. Phys. Soc. Jpn. 73 (2004) 3370.

lO.M.Kato, H.Tanaka, A.Endo, S.Katsumoto and Y.Iye, to appear in the Proceedings of EP2DS-16; M. Kato, H. Tanaka, A. Endo, S. Katsumoto and Y. lye, to appear in the Proceedings of LT24.

113

NONEQUILIBRIUM KONDO DOT CONNECTED TO FERROMAGNETIC LEADS

Y. U T S U M I 1 ' 2 , J. M A R T I N E K 2 ' 3 , G. S C H O N 2 and S. M A E K A W A 4

Condensed Matter Theory Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan ' Institut fur Theoretische Festkorperphysik, Universitdt Karlsruhe, 76128 Karlsruhe, Germany

Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan, Poland Institute for Materials Research, Tohoku University, Sendai 980 8577, Japan

We studied theoretically the Kondo effect in a quantum dot coupled to ferromagnetic leads. Spin-dependent charge fluctuations induce a kinetic exchange interaction between a spin inside the quantum dot and the lead magnetizations, resulting in a splitting of a zero bias anomaly. A left-right asymmetry in tunnel barriers significantly affects the nonlinear differential conductance. Our results agree qualitatively well with recent experiments. We also show that the decoherence of the Kondo resonance is significant in this system.

Keywords: Kondo effect; Kinetic exchange interaction; Decoherence; Spin-dependent transport

1. Introduction

The Kondo effect in a quantum dot (QD) is the perhaps simplest and best controllable example of a mesoscopic strongly correlated system. Although Coulomb blockade sup­presses real charge fluctuations, the QD and leads can exchange spins by cotunneling pro­cesses. It leads at low temperatures to the Kondo effect, the formation of a resonant state in the QD near the Fermi energy.

Spintronics devices provide possibili­ties to manipulate electron spin degree of freedom. A well-known example is the tunnel magnetoresistance (TMR) in a ferromagnetic tunnel junction, a ferro­magnetic/insulator/ferromagnetic junction, which relies on switching the magnetizations of the electrodes between parallel (P) and antiparallel (AP) orientations by an applied magnetic field. It is also known that in magnetic multilayers the exchange coupling, i.e. the Ruderman-Kittel-Kasuya-Yosida in­teraction, is induced between layers.

Therefore, it is intriguing to investigate the Kondo effect in the QD coupled to fer­romagnetic leads. In this article, we will show that in our system an exchange cou­

pling is induced between the QD spin and the magnetizations of ferromagnetic leads. The exchange interaction competes with the Kondo effect and eventually splits the zero-bias anomaly. Our results agree qualita­tively well with recent experiments with C6o single molecule QDs attached to ferromag­netic leads1. We also discuss the decoher­ence effects associated with the nonequilib-rium Kondo effect in the presence of spin splitting.

2. Kinetic Exchange Interaction

We introduce the single-site Anderson model coupled to spin-polarized leads [Fig. 1 (a-1) and (a-2)] (h=kB = l);

+ ^ £rka alk<rarka+^2 Vr dlarka + h . C . , (1) rka rko

where, da is the annihilation operator of elec­trons with spin-<r (o-=^,.\.) in the QD. The energy of the single QD level eo < 0 (mea­sured relative to the Fermi level of the lead electrons) may be tuned by a gate voltage, and U is the onsite Coulomb interaction. The operator arka annihilates an electron in

114

the lead r = L,R with wave number A; and

spin a. The ferromagnetism in the leads is

accounted for by the spin-dependent disper­

sion erka and, hence, spin-dependent density

of s tates vra{ui) = Ylk^i^ ~ £rka)- We as­

sumed tha t the tunneling ampli tude Vr is in­

dependent of the spin and wave number.

(a-D (a-2)

If f t t I I

Fig. 1. A single level quantum dot coupled to two ferromagnetic leads: (a-1) parallel and (a-2) antipar-allel alignments. We allow for a left-right asymmetry in tunnel barriers, (b) The electron fluctuations in­ducing the kinetic exchange interaction.

The origin of the exchange interaction is

spin-dependent charge fluctuations between

the QD and the lead r. This effect was dis­

cussed in the frame of a two-stage scaling

theory2 and confirmed by numerical renor-

malization group studies3 , 4 . Here, we derive

the exchange interaction using perturbation theory in Vr. When a spin a is inside the

QD, the energy gain caused by second order

processes [Fig. 1 (b)] reads

Aer„ = 2TT " D - € O

• 1 1 1 -U + €0

2TT '" D + U + eo'

where a is the opposite spin of a and D is the

conduction band width. The spin-dependent

coupling strength, Tra = 2n\Vr\2i/a(uj) is ap­

proximated as a constant. The first (sec­

ond) term is the energy gain by electron

(hole) fluctuations, which stabilize the ma­

jori ty (minority) spin in the QD. Then the

'spin spli t t ing' Ae r = Ae rf — Aer.j., i.e. the

exchange energy, becomes

A e « - P r L ] n ^ + P,*]»#±*.(2) 7T - C o T U + e0

Here D > -£n and

pr = ( r r t - r r 4 ) / ( r r t + r r 4) , (3)

is the spin polarization factor, while T r = (rr-j- + r r,i ) / 2 . Equation (2) shows that the electron (hole) fluctuations induce the fer­romagnetic (antiferromagnetic) exchange in­teraction. The spin splitting quenches spin-flip scattering processes for low energies be­low |Ac r | . It suppresses the Kondo correla­tion and leads to a splitting of the Kondo resonance2 .

The total spin splitting is the sum of con­tributions from both leads, Af = Ae^ + AeR.

For P alignment [Fig. 1 (a-1)], the spin split­ting reads,

AtP = aP(TL + rR), P = PL = PR, (4)

and for A P alignment [Fig. 1 (a-2)],

AeAP = a P(TL - TR), P = PL = -PR,

(5) where a is a constant. In the following we will consider U —• oo, where the exchange interaction is ferromagnetic.

3 . N o n l i n e a r C o n d u c t a n c e

The splitting of the Kondo resonance can

be observed by measuring the nonlinear

differential conductance. To see what it

looks like, we will present, numerical results.

Unfortunately, s tandard methods, such as

the slave-boson mean field approximation' '

and the non-crossing approximation out of

equilibrium6 , do not work since they can­

not treat correctly the spin-dependent charge

fluctuations and the Kondo correlations si­

multaneously. Therefore, we used a real-time

diagrammatic technique8 , 1 0 , and developed

an approximation free from such shortcom­

ings 9 . The approximation is an extension

of the resonant tunneling approximation1 0

which produces qualitatively reasonable re­

sults in the presence of a magnetic field above

the Kondo temperature .

Figure 2 shows the nonlinear differential

conductance for various temperatures for P

alignment and symmetric coupling, T/, = TR

(The horizontal axis is normalized by T =

115

T t + TR). At low temperatures a splitting of

the zero-bias anomaly appears. Two grow­

ing peaks with decreasing temperature in­

dicate the Kondo correlations. The magni­

tude of the split t ing almost coincides with

the spin splitting Ae [Eq. (2)]. We also ob­

serve the valley to become more pronounced

as the temperature is lowered. It indicates

tha t the suppression of the zero-bias anomaly

is a t t r ibuted to the quenching of the spin-flip

scattering.

p 1 1 1 1 1 1 1 r T7P-O.0O5

J I I I 1 1 I I L -0.4 -0.2 I) 0.2 0.4

eV/T

Fig. 2. (a) The nonlinear differential conductance for the P alignment (PL = PR = 0.1 and T t = rR). Solid, dashed, dotted and dot-dashed lines are results for various temperatures T/V = 0.005,0.01,0.025, and 0.05, respectively. Parameters are to/T = —2, and D / r = 50.

Figure 3 shows the nonlinear differen­

tial conductance for (a) P and (b) AP align­

ments. Solid lines are for symmetric cou­

pling. Dotted and dashed lines are for asym­

metric coupling, TR/TL = 0.5 and 0.25. For P

alignment, we observe the splitting and slight

asymmetry for asymmetric coupling. For AP

alignment the splitting shrinks or even van­

ishes. The shrinking of the splitting can be

understood from Eq. (5). For A P alignment,

contributions from left and right leads chan­

cel each other . Especially, for the symmetric

case, they perfectly compensate each other

and the zero-bias anomaly appears again.

We also observe a large asymmetry in the

splitting of the two peaks in the A P align­

ment . Intuitively, it can be explained by the

following: In the regime \eV\ < |Ae,4p|, a

spin-up occupies the QD level. Above a pos­

itive bias voltage eV> |Af,4p| , spin-flip tun­

neling events from a left lead spin-down state

to a right lead spin-up s tate contribute to the

current. In this case, because both initial and

final states are minority-spin states, a small

peak is expected. On the other hand, at neg­

ative bias voltages eV < —|A£^p| , the spin-

flip tunneling processes are between majority

spin states. Thus the peak in the differential

conductance is strong.

The presented results agree qualitatively

well with recent experiments1 . Figure 2 cap­

tures the features of Fig. 3 (A) in Ref. 1,

while the conductances for symmetric and

asymmetric samples in Fig. 3, corresponds

to Fig. 2 (C) and Fig. 2 (A) of Ref. 1, re­

spectively.

•0.4 -U.2 0 0.2 0.4 -0.4 41.2 0 0.2 0.4

cv/r ev/r

Fig. 3. (a) The differential conductance for the P alignment (P^ = P/i = 0.2). Solid lines, dashed lines and dotted lines are results for TR/Ti, = 1,0.5, and 0.25. (b) The differential conductance with the AP alignment. Parameters: T/V = 0.005. The other parameters are the same as for Fig. 2.

4. D e c o h e r e n c e Effect

One impor tant feature of the Kondo effect in QD is the strong decoherence in the presence of the magnetic field and/or out of equilib­rium even a t zero tempera ture 6 , 1 1 . T h e deco­herence effect is important since it provides the cutoff energy of Kondo correlations.

The transverse spin relaxation t ime, T2, is related to the imaginary part of the off-diagonal component of the self-energy in spin space9 . In the presence of the Zeeman splitting Ez, within the Markov approxima­tion and second order per turbat ion theory, it

116

reads (T=0)

h/T2 * Y^{Taa\eV\ + %s\eV + <TEZ\)I%

a

(6)

where Too1 = ^La^Ra1 h\- The rate is pro­

portional to the sum of the spin preserving

(first term) and and the spin flip (second

term) co-tunneling rates.

The higher order contribution enhances

Taa> by the Kondo correlations. Moreover

it renormalizes the Zeeman energy as Ez -+

Ez + Ae. Therefore, the decoherence is also

enhanced. In order to see the effect, we plot

the spin resolved DOS for P alignment within

the non-crossing approximation for the off-

diagonal component of the reduced propaga­

tor in spin space9 (solid line in Fig. 4) but in

equilibrium. By comparing the DOS with­

out the decoherence effect (dotted line), we

can see the decoherence effect strongly sup­

presses the split Kondo resonance.

— i 1 1 1 r—

0.2-

A t

Fig. 4. The spin resolved DOS for parallel align­ment (PL = PR = 0.2) with (solid line) and with­out (dotted line) the decoherence effect. Parameters: T / r = 0.005 and the other parameters are the same as for Fig. 2.

5. S u m m a r y

In summary, we have demonstrated the spin

splitting of the Kondo effect in a QD cou­

pled to ferromagnetic leads and have ana­

lyzed the nonlinear t ransport . The split­

ting of the zero-bias anomaly itself is robust

against temperature , though the hight of the

split peaks are suppressed with increasing

temperature . An asymmetry in the coupling

strengths does not affect much the splitting

for the parallel alignment. For the antipar-

allel alignment, the spin splitting is reduced,

and it even vanishes for the symmetric set­

ups. The reason is tha t for antiparallel align­

ment, the exchange interactions with the left

and the right lead perfectly compensate each

other for the symmetric case. Our results

agree qualitatively well with recent experi­

ment. We also discussed decoherence effects.

Second order perturbat ion theory shows that

the t ransport voltage and the magnetic field

induce decoherence. Higher order processes

enhance decoherence effect by the Kondo cor­

relations and also by the induced spin split­

ting.

A c k n o w l e d g m e n t s

Y.U. was supported by the Special Postdoc­

toral Research Program of RIKEN and the

DFG-Forschungszentrum CFN.

R e f e r e n c e s

1. A. N. Pasupathy et al., Science 306, 86 (2004).

2. J. Martinek, Y. Utsumi, H. Imamura, J. Barnas, S. Maekawa, J. Konig, and G. Schon, Phys. Rev. Lett. 91 , 127203 (2003).

3. J. Martinek, M. Sindel, L. Borda, J. Barnas, J. Konig, G. Schon, and J. von Delft, Phys. Rev. Lett. 91 , 247202 (2003).

4. M.-S. Choi et al., Phys. Rev. Lett. 92, 056601 (2004).

5. P. Coleman, Phys. Rev. B 35, 5072 (1987). 6. Y. Meir et al., Phys. Rev. Lett. 70, 2601

(1993). 7. Y. Kuramoto, Z. Phys. B 53, 37 (1983). 8. H. Schoeller and G. Schon, Phys. Rev. B 50,

18436 (1994). 9. Y. Utsumi, J. Martinek, H. Imamura, G.

Schon, and S. Maekawa, Phys. Rev. B 71, 245116 (2005).

10. J. Konig, J. Schmid, H. Schoeller, and G. Schon, Phys. Rev. B 54, 16820 (1996).

11. J. Paaske, A. Rosch, J. Kroha, and P. W61fle, Phys. Rev. B 70, 155301 (2004).

117

FULL COUNTING-STATISTICS IN A SINGLE-ELECTRON TRANSISTOR IN THE PRESENCE OF STRONG QUANTUM FLUCTUATIONS

Y. U T S U M I 1 ' 2

Condensed Matter Theory Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Institut fur Theoretische Festkoperphysik, Universitdt Karlsruhe, 76128 Karlsruhe, Germany

Using the Majorana representation and the Schwinger-Keldysh approach, we evaluate the distribu­tion of current through a single-electron transistor for the large conductance. The strong quantum fluctuations induce the lifetime broadening of charge-state levels out of equilibrium, as well as the renormalization effect. We find that the lifetime broadening suppresses higher cumulants of current fluctuations.

Keywords: Quantum noise; Quantum dot; Coulomb blockade

1. In t roduc t ion

The nonequilibrium fluctuating properties of carrier at zero temperature, T = 0, is beyond the description of the 'fluctuation-dissipation theorem'. The measurement of the nonequi­librium 'noise' has been the highlight of the mesoscopic quantum physics. Recently, 'full counting statistics' (FCS) has pervaded as the powerful concept for the description of nonequilibrium current fluctuations1.

The concept is the following: Suppose the measurement of the number of trans­mit electrons q through a quantum conduc­tor during the time to. After many measure­ments, one can plot the distribution P as a function of q as shown in Fig. 1. The FCS theory enables one to obtain the cumulant generating function,

oo

W(A)= ! > ( < 7 ) e i 9 A

q= — oo

oo -

= E^n»(? 'A)n- (!) 71=0

Previous studies have been mainly devoted to the mean (q) = {{Sq)), the average current (/) = e(q)/to, and the variance (Sq2) = ((Sq2)), the zero-frequency current noise Su = 2e2((Sq2))/to. The mean cor­responds to the peak position of the distri­bution and the variance corresponds to the

width of the distribution [Fig. 1]. Higher cu­mulants contain also the important informa­tion. The skewness {{Sq3)) and the kurtosis {{Sq4)) characterize the asymmetry and the sharpness of the distribution [Fig. 1]. The FCS can provide the full information and the vivid picture of the nonequilibrium current fluctuations.

A K)M

Fig. 1. The distribution of the number of transmit electrons q.

The FCS is attractive concept especially because it will promote our understanding on the nonequilibrium strongly-correlated meso­scopic systems. But only a few attempts have been done, for now: An open quantum dot coupled to electrodes by single-channel point contacts2 and the nonequilibrium two-channel Kondo model for a quantum dot in the Kondo regime3. Both of them are lim­ited to the Toulouse point, the exactly solu­ble case, and capture one aspect, not whole picture, of the electron correlations.

118

In this paper, we evaluate the FCS of the 'single-electron transistor' (SET), which is the most basic example of the strongly cor­related mesoscopic systems, for the large con­ductance regime where the quantum fluctu­ations of charge are enhanced.

2. Model and Calculations

The SET consists of a metallic island cou­pled to left, right and gate electrodes with small total capacitance Cs = CL + C R + CG [Fig. 2(a)]. It also exchanges electrons with left (drain), right (source) electrodes through tunnel junctions with resistance R^ and i?R. But the single-electron charging en­ergy Ec = e2/(2Cs) can exceed the source-drain bias voltage V (eV > 0) and insulate the island [Coulomb blockade]. One can re­duce the Coulomb barrier, the energy for im­pinging an electron into a charge-neutral is­land, Ao = Ec(l — 2Qc/e) by gate-induced charge QQ. Then an electron starts to tun­nel through the island one by one [sequen­tial tunneling], when the voltage drop in the junction r, jir = KreV [KL/R = ± C R / L ( C L +

C R ) - 1 ] , reaches the condition fiR< A O < / ^ L -

(a) T L e i v i . e " ^ I I m m IR

CL _

V/2 v o~ 1, I1

7 — u r ^ ±° cR ~ -V/2

,1 '1

-00 C+ +00

-oo-ihp

Fig. 2. (a) The equivalent circuit of a SET transis­tor, (b) The closed time-path going from - c o to 00 (C+), going back to —00 (C_) , connecting the imag­inary time path Cr and closing at t = — 00 — ihj3.

The physics characteristic for strong electron-correlations emerges when the par­allel resistance i?x = ^L-RR/(-RL + -RR) re­

duces to the order of the resistance quan­tum RK = h/e2 and quantum fluctuations are promoted4. The main consequence is the renormalization of the charging energy and the conductance. In the regime, where the dimensionless conductance g = RH/RT

is smaller than unity, g < 1, the perturba-tive renormalization group theory predicted that the renormalization factor logarithmi­cally depends on the cutoff energy A, which is T or Ao {h = ATR = 1), as ZQ = 1/{1 + 5ln(£'c/A)/(27r2)}5. Based on the real­time diagrammatic technique, the theory of the average current accounting for such the logarithmic renormalization was presented before6. However, the technique appears not to be tractable for the extention to the FCS. In the following, we evaluate the theory of FCS for SET in such a regime using the Ma­jorana representation7 and the Schwinger-Keldysh approach8.

Since we are interested in the effect of the strong Coulomb interaction on FCS, we will consider the case when the inverse of the RC time T = RTCZ is bigger than the charging energy EcTocg< 1. In this case, the charge-state levels are well resolved and thus two charge states, a state with charge neutrality and a state with one excess electron, are rele­vant. The effective Keldysh action describing SET then consists of parts for the charging energy and the tunneling S = Sch+St. In the Majorana representation, they read9,10

5 c h = / dt{c*{idt - A 0 ) c + l-<f>dt<t>},

St « -[dtdt'c*(t)<t>(t)a(t,t')<j>(t')c{t'), Jc

where c and <j> are Dirac and Majorana fermions. C is the closed time-path shown in Fig. 2(b). Here a = aL + OR is a particle-hole Green function describing the tunneling of an electron from an electrode into the is­land. In the Keldysh space, the particle-hole

119

Green function is expressed by a 2x2 matrix,

ar(W) = - ^ ( W - ^ r ) ( j c o t - y , (2) where, aT

0 = RK/(i/K2RT) is the dimensionless junction conductance. To obtain CGF, one need to perform the rotation in the Keldysh space as,

d£(w) =e ! 'K r A T l / 25 r(w)e- i K r A T l / 2 , (3)

where T\ is the x component of the Pauli matrix. Then the CGF follows form the in­tegration over fermions;

W ( A ) = l n / D[c*,c,^]eis^\ (4)

where 5(A) is the effective action contains the 'rotated' particle-hole Green function, Eq. (3).

We compute Eq. (4) by the perturba-tive expansion in OIQ. By accounting for the leading logarithms, we obtain9

W(A) as ~ fdu ln[l

+T F (u , ) / L ( W ){ l - / R ( W )} (e i A - l )

+ T » / R ( W ) { 1 - / L ( w ) } ( e - i A - l ) ] , (5)

where fj(ui) = f(tu—fj,r). Equation (5) looks similar to the Levitov-Lesovik formula11. But the effective transmission probability

TF(u,)= - Q L H " R ( " ) ( 6 )

where S^ is the Hilbert transformation of a ,10, accounts for strong quantum charge-fluctuations.

3. Resul ts and Discussions

3.1. Small Conductance

In the limit of ao—>-0, Eq. (5) reproduces the 'orthodox' theory12 in the sequential tunnel­ing regime and the cotunneling theory13 in the Coulomb blockade regime. Figure 3(a) shows the current (I = eq/to) distribution for various Ao for the symmetric SET at zero temperature. The distribution widens

when we sweep Ao, i.e. the gate charge, from the center of the sequential tunneling regime Ao/eK = 0 to the center of the Coulomb blockade regime Ao/eV = 1. At Ao = 0, the CGF is W(A) M 2 g ( e i A / 2 - l ) , where eq/t0 = G0V/2 and l/G0 = Rh+RR. The fac­tor e1*/2 results in the sub-Poissonian value of the Fano factor 5/ / / (2e(/))?al /2 indicat­ing that tunneling processes are correlated. The meaning of the 'correlated process' can be understood from the distribution function

P(<i) = £^ , g R =o p p(?L)fp(gR)<W + g B ) /2-The distributions of transmitted electrons through junctions L and R, q±, and <7R, fol­low the same Poissonian distribution func­tion Pp(q) = qqe~v/q\. Then the Kronecker delta implies that qi, and gpt, are not indepen­dent variables and thus correlated.

As Ao increases and approaches the threshold Ao = eV/2, the left junction be­comes the bottleneck, W(A) sa JOILI (esA —

1) and the Poissonian value is approached 5///(2e(/)) s» 1. In the Coulomb block­ade regime the CGF is also the Poissonian, W ( 1 )(A)«*o7+ ( e i A - l ) , where 7+ is the co-tunneling rete. Since elementary tunneling processes are different in two regimes, near the threshold, there is the regime where nei­ther the 'orthodox' theory nor the 'cotunnel­ing theory' works.

3.2. Renormalization and Lifetime Broadening

As the conductance increases, quantum fluc­tuations are promoted. However, for ZQT <^ eV [A = max(|z0Ao|,27rT, |eV|/2)], the 'or­thodox' theory with renormalized parame­ters 2o<*o and ZQAQ is valid. It is consis­tent with the RG result5. But the loga­rithmic renormalization fails in the regime TK = £ c e _ 1 / ( 2 a o ) / ( 2 7 r ) » A where subhead­ing logarithms are important.

In contrast, when the renormalization ef­fect is negligible, eV 3>Tk, we can approx­imate S f (w) RS — inaoeV, i.e. the lifetime

120

a ao=10l—-=»/

- / J y^~ i i • i

' " \ eV/Ec=0 .1 "

^ 1 (b-1) "

_^_ - / | \ \ -

1 1 I 1 1

-0.5 0 0.5 A0/eV

a" to.

CO.

1 1 1

(b-2)

a0=10 •—J f /^ \ 10'2 1 / " 4 5xl0"2—-1 /

i i r i I I

-1 -0.5 0 0.5 1 A0/eV

Fig. 3. (a) The zero-temperature current distribu­tion at eV = 0.2EQ for various Ao ( J?L = ^R and CL = C R ) . Panels (b-1) and (b-2) show the skew­ness and that normalized by the average value at eV = 0.\Ec and for various conductance.

broadening effect is dominant . Then the

C G F at A 0 = 0 reads

W(\)K2q{{e'x'2-l)-2a0(elA-l)

2„2(J3\/2_J\/2>. + 7 rao(e + 0(ag)}.(7)

The higher cumulants are suppressed as «o

increases and the lifetime broadening is en­

hanced: {{Sqn))/((Sq)) = 2 1 - " { l - 4 a 0 ( 2 n - 1 -

l) + 0(al)}. Figure 3(b-l) shows the skewness ({Sq3))

as a function of Ao- A double-peak structure

growing with increasing the conductance is

observed. The structure is absent in the

'Fano factor' {(Sq3))/{(Sq)) [Fig. 3(b-2)]. Such

a characteristic behavior would be measur­

able for a present-day experiment1 4 .

4. S u m m a r y

We evaluated the FCS of SET for large con­

ductance. Out of equilibrium, the quan tum

fluctuations induce the lifetime broadening

of charge-state levels, as well as the renor-

malization effect. Higher cumulants of cur­

rent fluctuation are suppressed by the life­

t ime broadening effect. We found the char­

acteristic structure in the skewness.

A c k n o w l e d g m e n t s

Y.U. was supported by the Special Postdoc­

toral Research Program of RIKEN and the

DFG-Forschungszentrum CFN.

References

1. Quantum Noise in Mesoscopic Physics, Vol. 97 of NATO Science Series II: Mathemat­ics, Physics and Chemistry edited by Yu. V. Nazarov (Kluwer Academic Publishers, Dordrecht/Boston/London, 2003).

2. M. Kindermann and B. Trauzettel, Phys. Rev. Lett. 94, 166803 (2005).

3. A. Komnik and A.O. Gogolin, Phys. Rev. Lett. 94, 216601 (2005).

4. D. S. Golubev, J. Konig, H. Schoeller, and G. Schon, Phys. Rev. B 56, 15782 (1997) and references therein.

5. K. A. Matveev, Sov. Phys. JETP. 72, 892 (1991).

6. H. Schoeller, and G. Schon, Phys. Rev. B 50, 18436 (1994).

7. H. J. Spencer, and S. Doniach, Phys. Rev. Lett. 18, 23 (1967).

8. K.-C. Chou et al., Phys. Rep. 118, 1 (1985); A. Kamenev, cond-mat/0412296.

9. Y. Utsumi, cond-mat/0503653. 10. Y. Utsumi et al., Phys. Rev. B 67, 035317

(2003). 11. L. S. Levitov, and G. B. Lesovik, JETP lett.

58, 230 (1993); cond-mat/9401004. 12. D. A. Bagrets, and Yu. V. Nazarov, Phys.

Rev. B 67, 085316 (2003). 13. L. S. Levitov, and M. Reznikov, Phys. Rev.

B 70, 115305 (2004). 14. B. Reulet, J. Senzier and D. E. Prober,

Phys. Rev. Lett. 91 , 196601 (2003).

121

GEOMETRY AND THE ANOMALOUS HALL EFFECT IN FERROMAGNETS

N. P. ONG and WEI-LI LEEt Department of Physics, Princeton University, New Jersey 08544, U.S.A.

The geometric ideas underlying the Berry phase and the modern viewpoint of Karplus and Lut-tinger's theory of the anomalous Hall effect are discussed in an elementary way. We briefly review recent Hall and Nernst experiments which support the dominant role of the KL velocity term in ferromagnets.

I. INTRODUCTION

Geometry crops up in physics in unex­pected ways. The example in this talk traces an idea proposed 50 years ago. Mired in controversy from the start, it simmered for a long time as an unresolved problem, but has now re-emerged as a topic with modern appeal. In 1954, Karplus and Lut-tinger (KL) [1] discovered the earliest in­stance of the Berry phase [3] in solids when they calculated the charge current of elec­trons moving in a Bravais lattice with bro­ken time-reversal symmetry (a ferromag-net). They uncovered a quantity 12(k) that acts like a "magnetic field" in recip­rocal space to produce a transverse veloc­ity eE x fi. The velocity leads to a dis-sipationless Hall current which, according to KL, accounts for the anomalous Hall ef­fect (AHE) seen in all ferromagnets [1]. Be­cause Berry-phase physics lay far into the future, the KL theory encountered stiff re­sistance. Prom the 60's to late 70's, Smit's rival skew-scattering theory [2] gained as­cendancy, with apparent support from ex­periments on dilute Kondo systems, e.g. CuMn (in hindsight, these experiments are irrelevant because the host lattice retains time-reversal invariance).

In the past 4 years, interest in topologi­cal currents derived from the quantum Hall effects has led to the re-examination of the KL theory from the modern viewpoint [4-9]. The new treatments show that KL's early perturbative approach has a much broader generality [4, 5]. Experiments [10-13] show that the Berry phase plays a key role in the Hall effect of magnetic materials. The evi­dence that the KL theory provides the cor­

rect explanation of the AHE and Nernst ef­fect in ferromagnets is reviewed here.

In a Hall experiment on a ferromagnet, the observed Hall resistivity pxy is the sum of the AHE resistivity p' and the usual Lorentz term RQH (RQ is the ordinary Hall coefficient). We have

pxy(T,H) = R0(T)H + p'xy(T,H). (1)

At a fixed T, the strong H dependence of the AHE term simply reflects the rotation of all domains into alignment. This is expressed as p'xy(T,H) = RS{T)M(T,H) where the anomalous Hall coefficient Rs (T) is the scale factor relating the magnetization M to p'xy. Our interest is on the value of p'xy in the impurity scattering regime.

In this talk, we introduce the geometric ideas underlying the KL theory and review recent experiments. [Readers familiar with holonomy may skip to Sec. IV.]

II. GEOMETRY

Imagine that a beetle moves a unit vector v along a closed path C on a curved sur­face (e.g. a latitude on the globe) with the constraint that v(t) must always lie in the local tangent plane, i.e. v • e3 = 0 where e3 is the local normal vector (Fig. la). Along the path, we impose the further restriction that the direction of v is perturbed mini­mally (Fig. lb) . Clearly, v cannot remain absolutely parallel to itself; the next best thing is to ensure that v never rotates about e3, i.e. any change dv must be parallel to ± e 3 (Fig. lc), viz.

e3 x dv = 0. (2)

122

(a) e3 (b)

• av

(c)

FIG. 1: (a) Transport of v along the curve C on a sphere. If C is a latitude, the tangent planes are given by the cone with apex A. (b) Flatten­ing the cone on a table shows that v remains parallel to its initial direction if it is prevented from rotating around the local normal vector e3 (parallel transport). However, measured rel­ative to the latitude, v makes an angle a(C) on returning to the starting point (holonomy). (c) In general, v lies in the tangent plane (white rectangle). In parallel transport, the change dv must not have any component in the tangent plane, i.e. ±es | |dv.

The process satisfying Eq. 2 is known as "parallel transport" (or Levi-Civita trans­port) [14]. If C is a circle on the sphere, the tangent planes are part of a cone (Fig. la). By flattening the cone (Fig. lb) we.see why the no-rotation condition is termed paral­lel transport. In the "flattened" space, v is always fixed in direction. However, on the curved surface, the beetle finds that v does not return to its initial direction when the path is completed. Instead, it makes a "ge­ometric" angle a(C) that is path dependent (holonomy, Fig. lb).

To measure a, we need a local coordinate frame [&i(t), e2(t)], where t parametrizes C (on the sphere, these are usually the unit vectors along the latitude and longitude). At each point R(t), we expand

v(t) = cosa(t) ei(t) + sina(t) e2(t)- (3)

As we move along C, the triad [v, w,e3] (where w = e3 x v) rotates about the nor­mal by a(t), relative to the local frame [ei,e2,e3]. We note that Eq. 2 implies

e3 x rfw = 0. (4)

Using Eq. 3 in Eq. 2 with the normaliza­tion conditions ei • de\ = 0 etc, we find the elegant result

da — e i • de2 = ^21- (5)

On completing the trip, the total angle a(C) is the integral of the "connection form" W21 which encodes the overlap between de% and ei [14].

Holonomy seems less mysterious if we re­gard v as the quasi-fixed direction relative to which the local frame rotates as we tra­verse C. This is evident for the flattened cone in Fig. lb.

These results may be written more com­pactly using complex vectors with the in­ner product (a, b) = a* • b . We write n = [ei + ie-^/y/2 and V> = [v + iw]/%/2. Equa­tion 3 becomes t/>(4) = n(i)e~ , aW, with the angle a(t) now appearing as a phase. The parallel transport conditions Eqs. 2, 4 and 5 now have the compact forms

n* • di> = 0, da = —n* • idh. (6)

III. BERRY PHASE

In the Berry-phase problem [3] a param­eter (the coordinate Q of the nucleus) is slowly taken around a closed curve C, while the bound electron is assumed to remain in the same eigenstate |n, Q) parametrized by Q (Fig. 2a). This assumption is a con­straint analogous to that on v in Sec. II. When the circuit is completed, the electron ket \ip) does not return to its starting state \n, Qo). Instead, it acquires a phase x(C) called the Berry phase. We now compare this system with the previous example.

The path C traced by Q(t) lies on a sur­face in parameter space Q. At each Q, the eigenstate \n,Q(t)) may be regarded as the local coordinate frame defined in Hilbert

123

space (in place of the tangent plane), i.e. |n,Q(£)) is analogous to h(t) in Eq. 6. As C is traversed, the ket of the electron \tp) is parallel transported, and hence ac­quires a phase angle relative to \n, Q), viz. M = |n,Q>e-'*(Fig. 2b).

(a) (b)

FIG. 2: (a) Electron constrained to same eigen-state \n, Q) as nuclear coordinate Q is taken around a closed path C. (b) Parallel transport of ket |?/>) in Hilbert space. As Q changes, \ip) acquires a phase angle relative to \n, Q).

Imposing the parallel-transport condition (n,Q\6\il>) = 0 (Eq. 6), we find 8X = — (n, Q\iS\n, Q). On completing the path C, the total Berry phase x(C) is the line integral

x(C) = - ^ d Q . ( n , Q | t V Q | n , Q > . (7)

Physically, as Q changes, the electronic ket \ip) stays "parallel" to its initial direction (in the sense of Levi-Civita), while the lo­cal reference ket \n, Q) rotates relative to it. The Berry phase is the phase angle between them.

The form of Eq. 7 suggests that it is fruitful to view the integrand as a (Berry) vector potential A(Q) = (n,Q|iVq|n,Q). We may then regard x(C) as an Aharanov-Bohm (AB) phase caused by an effective magnetic field B(Q) = V Q X A(Q) that lives in parameter space (B(Q) is called the Berry curvature).

IV. ANOMALOUS VELOCITY

For an electron in a periodic potential, the eigenstates are the Bloch states

tfnk(r) = - ~ L e i k r u n k ( r ) = (r|nk). (8)

If we ignore the spin, the band index n and wave-vector k are sufficient to fully charac­terize its state.

Let us perturb the electron by adding a static potential V(r). In Bloch representa­tion, the Hamiltonian is

H = V(R) + en(k), (9)

where R = ?Vk is the Wannier coordinate indexing the lattice sites and e„(k) is the unperturbed band energy. The potential V causes the wave vector k to drift along a path C in reciprocal space at the rate

dV hk=-i[k,H] = - M . (10)

In principle, transitions to other bands n' ^ n will occur with finite amplitude. How­ever, for weak V, it is customary to assume that the electron always remains in the same band n. This assumption is a constraint analogous to those in the preceding exam­ples. Hence we should expect the electron's ket vector \ip) to acquire a Berry phase, i.e. along C, (v\ij)) = u„k(r)e~*x(k) where

x (k) = - / dk' • X(k') , (11) Jc

with the Berry vector potential X(k) given by

X(k) = f d3r < k ( r ) i V k « n k ( r ) (12) J cell

(X has dimensions of length). With the sub­stitution k —* Q, Fig. 2b equally well de­picts the parallel transport of |i/>) as k(t) changes in reciprocal space.

As in Eq. 7, we may regard x as the AB phase caused by a magnetic field

ft(k) = V k x X(k) (13)

124

that lives in k space. The Berry curvature tt alters the re­

sponse of the electron to V in an essential way. To see this, we perform a gauge trans­formation to remove the AB phase at the cost of adding the vector potential X to iV k

in the argument of V in Eq. 9. The trans­formed Hamiltonian is then

H' = V(iVk + X) + en(k). (14)

There are 2 ways to view H'. The role of X(k) as a vector gauge potential is now manifest. Moreover, in the argument of V, the position operator is now given by

x = R + X(k) (15)

instead of just R. We may view X(k) as an intracell coordinate that locates the electron within the unit cell. Equation 15 implies that x fails to commute with itself. Instead we have

[xu Xj] = ieijkClk. (16)

With the Hamiltonian H', tik is un­changed from Eq. 10, but the group velocity is (using Eq. 16)

ftv = - i[x,ff ' ] = Vke„(k) + (?£\ xfl(k).

(17) The extra term involving il is called the (Luttinger) anomalous velocity.

If V = —eE • x (e is the elemental charge and E an electric field), the semiclassical equations of motion become (we now in­clude a weak, physical magnetic field B for completeness)

ftk = eE + ev x B (18)

hv = Ven - e E x f i . (19)

V. ANOMALOUS HALL CURRENT

In the Boltzmann-equation approach, the charge current density in the absence of B is

J = e £ > k L / k + S k ] , (20) k,s

where /£ is the equilibrium distribution

function and gk = ev k r • E(— -£-) the cor­rection caused by E| |x (r is the transport relaxation time). Normally, the term in /£ sums to zero by symmetry, as it must, leav­ing the term in gk to give the usual con­ductivity a (the Hall conductivity axy = 0). Now, with v k given by Eq. 19, the term in /£ yields a sizeable current transverse to E, viz.

J" = y E x E"( k ) /k (21) k,s

Because it derives from /£ , J # has the re­markable feature that it is independent of r , as found by KL [1]. We may write the AHE conductivity as

a'xy=n^(£l), (22)

where (Q.) = n - 1 X) kf i*( k ) /k is the

weighted average of the Berry curvature. Under the time-reversal operation,

fi(k) —> —fi(—k), while under inversion, fi(k) —» il(—k). Hence, if both symmetries are present, t l(k) = 0 for all k. Further, to obtain a finite value in the sum in Eq. 21, we must break time-reversal symmetry (breaking inversion symmetry alone is insufficient). In a ferromagnet, the uniform magnetization M breaks time-reversal symmetry for the spins. This symmetry-breaking is communicated to the charge currents via spin-orbit coupling. In the simple case of a ferromagnetic semiconduc­tor, Nozieres and Lewiner (NL) [15] have calculated X(k) = — Ak x S, where A is the spin-orbit coupling parameter and S ~ M. Equation 21 then gives for the NL model an AHE current J # = 2ne2AE x S that is linear in the carrier density n and M, but independent of r .

The quantity that is measured is the Hall resistivity p'xy = axyp

2. Since p ~ (m - ) - 1 , Eq. 22 immediately implies that [12]

p'xy = Anp2 (23)

with p the resistivity and A = e2(Q)/h. In an experiment in which both n and r can

125

be varied, the KL theory predicts that p'xy/n scales like pa with a = 2. By contrast, skew-scattering theories predict axy ~ nr so that p' is linear in p.

VI. SPINEL FERROMAGNET

These predictions have to be tested in the impurity-scattering regime because, if r is dominated by phonon/magnon scattering, both theories predict a = 2. Moreover, it would be desirable to change n greatly (in addition to r ) without destroying the mag­netization. These conditions are met in the spinel ferromagnet CuCr2Se4_xBrx which is a metal with a Curie temperature Tc ~ 400 K (for x = 0) [12]. Interaction between the local moments on adjacent Cr ions is ferro­magnetic because the Cr-Se-Cr bond is 90°. The holes which reside on the Se bands con­tribute negligibly to the exchange. Hence tuning the hole density n (by changing the Br content x) hardly affects the magnetiza­tion. As x increases from 0 to 1, n decreases from 7 x 1021 to ~ 1020 cm - 3 , while T de­creases by a factor of 70 to give a change in resistivity p exceeding 3 decades (measured at 5 K) [12]. However, M a t 5 K remains nearly unchanged (Tc decreases to 250 K). Typical profiles of pxy vs. H are shown in Fig. 3.

In Fig. 4, \p'xy\/n measured in 12 crystals at 5 K is plotted against p. As p varies over 3 decades, p'xy/n increases by 6 decades, fol­lowing Eq. 23 with a = 1.95. This implies that a'xy is linear in n, but independent of T. However, the sharp sign change near x = 0.4 is not understood at present. [To com­pare with the skew-scattering prediction, we have also plotted p'xy (without dividing by n) against p. The data yield an exponent of 1.4 instead of 1. The disagreement with skew scattering lies well outside the uncer­tainties of the data.]

The experiment provides an estimate of the average Berry curvature (fi). Using Eq. 22, we find that the parameter A = 2.24 x 10"25 (SI units) gives (Q)i = 0.30 A.

K0H(T)

FIG. 3: Traces of the observed pxy vs. H in CuCr2Se4-xBrx for x = 0.85 (top panel) and 1.0 (bottom) [12]. At each T, pxy varies like M vs. H (Ro is negligible). In the x= 1.0 sample, p'xy at 5 K is possibly the largest AHE resistivity observed to date in a ferromagnet.

VII. ANOMALOUS NERNST EFFECT

The AHE corresponds to a charge cur­rent that flows transverse to E. In addition to the charge, the electron current carries heat. This may be observed by the anoma­lous Nernst effect [13]. In the Nernst ex­periment, an applied temperature gradient — VT||x produces a transverse charge cur­rent which is antisymmetric in H||z, viz. Jy = ayx(—VT), where a^ is the Peltier conductivity tensor. With the Luttinger ve­locity, we have

k,s ^

(24) where fee is Boltzmann's constant, p the chemical potential, and hv° = Ve(k). In deriving this, we have made the substitution hk/r —> eE in Eq. 19. Defining the quan­tity (fi)£ as the angular average of fi(k) over

126

24 to

10"2

a

? 10"4

<«

"s- io"5

Q.

10"6

r

a =

MM, I

1.95

0 (

y° 0.10

• I I I

0.60

/ • 0.50

0.25

I '

rm 0.85

1.0 '.

-

1

_

1

_

•

10"r 10"" 10"5 10"* 10"3

Resistivity p at 5 K ( Sim )

FIG. 4: Plot of the saturation value of p'xy/n versus p (both measured at 5 K) in log-log scale for 12 crystals of CuCr2Se4_xBrx [12]. Open (solid) circles indicate negative (positive) sign for p'xy at 5 K (p'xy for the x = 0 sample is not resolved). The solid line is a fit to \p'xy\/n = Apa with a = 1.95 and A = 2.24 x 10 - 2 5 (SI units).

the Fermi Surface (FS), we may simplify Eq.

ax 7T2 ek%T 2 fd(U)eeM

3 h 3 V de X- (25)

With the density of states Af ~ y/e, the last factor reduces to £ IA/F if the e-dependence of ft is negligible.

In CuCr2Se4_ xBr x , axy behaves at low T as [13]

a. xy n (26)

with J\fp the free-electron density of states, and A ~ 3 4 A 2 . The agreement with Eq. 25 is encouraging although the numerical co­efficient seems too large to be simply iden­tified with Q. We note, however, t ha t the free-electron jV° used in Eq. 26 implies tha t A is enhanced by the t rue effective mass. T h e 2 experiments probe different ways of averaging £l(k) over k.

We acknowledge support from the Na­tional Science Foundation (grant DMR 0213706), and helpful comments from F. D. M. Haldane.

^Present address of WLL: Johns Hopkins Univ, Dept. Phys. and Astron., Baltimore, MD 21218, USA

[1] R. Karplus, J. M. Luttinger, Phys. Rev. 95, 1154 (1954); J. M. Luttinger, Phys. Rev. 112, 739 (1958).

[2] J. Smit, Physica (Amsterdam) 21, 877 (1955); ibid. Phys. Rev. B 8, 2349 (1973).

[3] Geometric Phases in Physics, ed. Alfred Shapere and Frank Wilczek (World Scien­tific, 1989).

[4] Ganesh Sundaram and Qian Niu, Phys. Rev. B 59, 14915 (1999).

[5] M. Onoda, N. Nagaosa, J. Phys. Soc. Jpn. 71, 19 (2002).

[6] S. Murakami, N. Nagaosa, S. C. Zhang, Science 301, 1348 (2003).

[7] T. Jungwirth, Qian Niu, A. H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002).

[8] Yugui Yao et al., Phys. Rev. Lett. 92,

037204 (2004). [9] F. D. M. Haldane, Phys. Rev. Lett. 93,

206602 (2004). [10] P. Matl et al., Phys. Rev. B 57, 10248

(1998). [11] Y. Taguchi et al.., Science 291, 2573

(2001). [12] Wei-Li Lee, Satoshi Watauchi, V. L. Miller,

R. J. Cava and N. P. Ong, Science 303, 1647 (2004).

[13] Wei-Li Lee et al., Phys. Rev. Lett. 93, 226601 (2004).

[14] Differential Geometry, J. J. Stoker (Wiley, New York 1969).

[15] P. Nozieres, C. Lewiner, J. Phys. (France) 34, 901 (1973).

127

CONTROL OF SPIN CHIRALITY, BERRY PHASE, A N D ANOMALOUS HALL EFFECT

Y. T O K U R A *

Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan, Spin Superstructure Project, ERATO, JST, Tsukuba 305-8562, Japan

Correlated Electron Research Center (CERC), AIST, Tsukuba 305-8562, Japan

* E-mail: tokura@ap.tu-tokyo.ac.jp

Y. T A G U C H I

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan,

E-mail: taguchi@imr. tohoku. ac.jp

Nontrivial spin texture on some topological lattice can give rise to the quantal phase in the conduc­tion electron hopping term, and hence affect the transverse transport as gigantic fictitious magnetic field. In Nd2Mo207 (NMO) and related R2M02O7 (R being a rare-earth ion) compounds, the Ising anisotropy of Nd (or R) moments on the pyrochlore lattice interact ferromagnetically with each other and anitiferomagentically with Mo spins, giving rise to the non-coplanar configuration with spin chirality. The Hall resistivity in NMO changes sign with increasing magnetic field applied along [111] direction, while it monotonously approaches zero with the field applied along [100] or [110] direction. This is in accord with the prediction by the Berry phase theory and is interpreted in terms of field-induced reversal of spin chirality on the pyrochlore lattice, i.e. the transformation from the two-in two-out to three-in one-out configuration of the Nd (and hence Mo) moments.

Keywords: Berry phase; spin chirality; anomalous Hall effect.

To test the idea of Berry phase of con­duction electrons in a solid, the pyrochlore type compound ( ^ ^ O ? ) provides an im­portant arena because of its geometrical frus­tration nature. The pyrochlore lattice is composed of two sublattices of A-site and .B-site, which are structurally identical but are displaced by half a lattice constant from each other. In each sublattice, the ions locate at the vertex of corner-sharing tetrahedra, as schematically shown in Fig. 1(a). Its pro­jection onto the (111) plane is the Kagome-lattice, and one B ion is located at the center of a hexagon formed by six A ions (and vice versa).

Among many pyrochlore-type materials, molybdate system is of particular interest be­cause it shows a transition from spin glass insulator to ferromagnetic metal with the change of >l-site ion l. Figure 1(b) shows the electronic phase diagram 2 plotted against

-

~«&R E'sSl.

1

Gd T

1

Nd,

S m , ' '

* ,'

FMM

. . . 1

w •

-

•

•

pi ' 1 1 1 1 1.05 1.1

Ionic Radius (A)

Fig. 1. (a)Schematic figure of the A-site (or B-site) sublattice of the pyrochlore structure, (b) Elec­tronic phase diagram of pyrochlore-type molybdate where spin freezing temperatures (closed squares) and Curie temperatures (closed triangles) are plot­ted against ionic radius of A-site ion. FMM and SGI denote ferromagnetic metal and spin glass insulator, respectively.

the ionic radius of the i4-site species. Similar phase diagrams have also been obtained by several groups 1 , s . It has been indicated by an optical study 2 that the electron correla­tion effect is of substantial importance in the

128

metallic phase. In such a correlated metal­lic state, a spin state strongly affects the charge dynamics. The most dramatic mani­festation of such interplay between spin state and charge dynamics is the so-called colos­sal magnetoresistance phenomena in man-ganites 4, in which the longitudinal con­ductivity strongly depends on the Mn i2g core spin configuration (together with ad­ditional charge/orbital ordering instabilities and/or Jahn-Teller interaction). This ef­fect is viewed as a consequence of the mod­ified magnitude of the transfer integral by the spin configuration 5. The phase of the transfer integral should also be modified by the non-trivial spin texture and could pro­duce the gauge flux 6. Such a gauge flux acts as a fictitious magnetic field and affects the charge dynamics in the same way as a real magnetic field does. Recently, anoma­lous Hall effect or transverse conductivity has been discussed in terms of this phase (termed Berry phase) in several ferromag­netic transition-metal oxides, for example, in perovskite-type manganites 7 '8 '9, molybdates with pyrochlore structure 10>n,12,13,14^ a n d rutile-type Cr02 15. Weak coupling regime has also been examined very recently in con­nection with the canonical spin glass sys­tem 16. According to the Berry phase the­ories, the conduction electron feels a ficti­tious magnetic field produced by the non-coplanar spin configuration or spin chirality, and the transverse conductivity appears. On the other hand, there are some criticism 17 '18

against the interpretation of the anomalous Hall effect in Nd2Mo2C>7 in terms of the Berry phase theory. Here, we present re­cent experimental evidence that the anoma­lous Hall effect in the pyrochlore ferromagnet does arise from the spin chirality.

In Fig.2(a), we show the temperature variation of the magnetization at H = 0.5 T and neutron scattering intensity of (200) peak for the Nd2Mo207 crystal, whose Curie temperature was determined from ac-

susceptibility measurement to be 89 K. The Nd moments align antiparallel to the Mo Ad spins at the Curie temperature and begin to grow rapidly around a crossover temperature (« 40 K), resulting in a reduction of the to­tal magnetization at low temperatures. At around this crossover temperature, the neu­tron scattering intensity of (200) reflection begins to grow. The Nd moments and the Mo spins form "umbrella structure" n due to the strong (111) Ising anisotropy of the Nd mo­ment and the antiferromagnetic interaction between the Nd moment and the Mo spin. The (200) reflection represents the ordering of the transverse component of the umbrella structure. The Mo spin moment at low tem­peratures was deduced to be approximately 1.4/iS/Mo from the analysis of magnetiza­tion data and neutron diffraction data n .

Temperature (K)

Fig. 2. (a) Temperature dependence of magnetiza­tion at H = 0.5 T and background-subtracted and properly normalized neutron scattering intensity / of the (200) reflection, which represents the ordering of the transverse component, (b) Temperature vari­ation of the Hall conductivity for Nd2Mo207 and Gd2Mo2C>7. The data for Gd compound are multi­plied by 3.

We show in Fig.2(b) the Hall conduc-

129

tivity axy = pn/(p2xx + p | ) at H = 0.5 T

for Nd2Mo207 and Gd2Mo207 . The most remarkable feature is that the Hall conduc­tivity of Nd2Mo2C>7 for each direction con­tinuously increases down to 2 K, and finally saturates only below 2 K (not shown here). Such a behavior is totally different from the conventional behaviors of the anomalous Hall effect 19 '20. By contrast, the Gd compound shows a conventional temperature depen­dence as the ferromagnetic metal: The abso­lute value of the Hall conductivity takes max­imum near the Curie temperature (= 45 K), and decreases toward low temperatures. Fur­thermore, the absolute value itself is much smaller than that of the Nd compound. It seems that the conventional theories can hardly explain this remarkable difference be­tween two materials with very similar lon­gitudinal transport properties since they are based upon the spin-orbit interaction of the conduction electrons, which would be least affected by the rare-earth species. On the other hand, the Berry phase theory can nat­urally account for the difference. The Nd3 +

(4/3) is of Ising-like anisotropy with (111) axis while Gd3 + (4/7) without orbital angu­lar momentum is of Heisenberg-like one 21. Therefore, the spin chirality is produced in the Mo spin system via the f-d interaction only in the case of the Nd compound. This is the reason why the unconventional be­havior of Hall conductivity is observed for Nd2Mo207 and also for Sm2Mo207 with the Ising anisotropy of the Sm moment 12, but not for Gd2Mo207 .

The Berry phase theory has an impor­tant prediction that the sign of the Hall resis­tivity changes when a strong magnetic field is applied along [111] direction, whereas it does not when the field is applied along [100] or [110] direction. To make a more precise statement, we define the fictitious magnetic field that penetrates a single Mo-tetrahedron

b Mo as a vector sum:

b Mo = / _, (Si • Sj x Sk)~njjk,

where n^-j, is a normal vector (with unit length) of a triangle formed by site i, j , and k. A relation that b Mo • H < 0 always holds when the applied field is along [100] or [110] direction. On the other hand, b Mo • H changes its sign when strong field is applied along [111] direction. Thus, the Hall resistiv­ity that arises from the fictitious field changes sign when the field is applied along [111] di­rection.

In Figs.3(a) and (b), we display schematic configurations of the Nd moments and Mo spins, where the magnetic field is ap­plied along [100] direction. In the both panel, the upper tetrahedron represents the Nd-tetrahedron and the lower one corresponds to the Mo-tetrahedron. Figure 3(a) shows a case where weak field (e.g. H = 0.5 T) is applied along [100] direction. The Mo spins are almost ferromagnetically aligned along the field direction. The Nd moments feels a downward effective field due to the anti-ferromagnetic interaction with the Mo spins. Thus, the Nd moments, which are subject to strong single-ion anisotropy, form the "2 in, 2 out" configuration. Then, in turn, Mo spins are tilted from the direction of net mag­netization by the antiferromagnetic interac­tion with Nd moments. In the pyrochlore structure, Mo site 1 locates at the center of a hexagon formed by two Nd 2', two Nd 3' and two Nd 4'. Taking into account the an­tiferromagnetic interaction between Mo and Nd, we can infer that the transverse com­ponent of the Mo spin 1 points parallel to that of the Nd moment 1'. Thus, Mo spin system acquires the same "tilting habit" as the Nd moment system. In this particular spin configuration, b Mo • H is negative. Fig­ure 3(b) depicts the situation where all the Nd moments are reversed toward the field di­rection by a strong enough field (e.g. H =

130

(a) ff|| [100]

Ndl '

H

A

Mol

(b) # | | [100]

H II [110]

Nd2Mo207

T= 1.7 K 70 m K ^ || [100])

50mK(W || [110])

H < —

Nd2Mo207

T- 1.6K

-W || [110]

[100]

[110]

10 20 30 Magnetic Field (T)

Fig. 3. (a), (b)Schematic figures of spin configurations for Nd- and Mo-tetrahedra when the field is applied along [100] direction. In each panel, upper and lower tetrahedra correspond to Nd- and Mo-tetrahedra, respectively. The relationship among the Mo spin 1, 2, and 3 is schematically shown at the right hand side of each Mo-tetrahedron. Open arrows depict the directions of the applied magnetic field. See text for details. (c)Magnetization curves for the applied field along [100] and [110] direction. Horizontal dot-dashed line represents the contribution from Mo spin moment. Thick horizontal bars indicate the expected values of saturation moment for each field direction under the assumption of strong Ising anisotropy of Nd moments. See text for details. (d)pn is plotted as a function of magnetic field which is applied along [100] or [110] direction.

5 T). Similarly to the case of Fig. 3(a), the anisotropy of the Nd moment system is trans­mitted to the Mo spin system. The Mo spin configuration in Fig. 3(b) is obtained from the configuration in Fig.3(a) by operating a rotation around the field direction by 180°. Importantly, the spin chirality is invariant under the global spin rotation, thus, the fic­titious magnetic field that penetrates the Mo tetrahedron points to the same direction as in Fig.3(a). Therefore, the Hall resistivity will not change sign in terms of the Berry phase model, even when all the Nd moments are reversed by the field applied along [100] direction. The situation is analogous when the field is applied along [110] direction; "2 in, 2 out" tilting habit for the transverse Mo moments is similar to the case of H || [100], and hence there will be no change in sign of the Mo spin chirality and hence in sign of Hall effect as well.

In Fig.3(c), we show the field depen­dence of magnetization for fields applied along [100] and [110] direction at 1.7 K and 50 (or 70) mK. The estimated value of the Mo id moment in this temperature range is ~ lAfiB n> a n d the averaged Nd mo­ment should vanish at the field of Ho « 3T. This consideration is consistent with recent results of neutron diffraction in a magnetic field 18-22. For the respective field directions, the magnetization curves are almost identi­cal at 1.7 K and 50 (or 70) mK. Above 10 T, the magnetization nearly saturates, and these saturation moments are roughly in ac­cord with the estimated values (indicated by horizontal thick bars) under the following as­sumption: We assumed that the longitudi­nal component of the Mo spin moment is field-independent (because of the small tilt­ing angle) and that the Nd moment is of strong Ising-anisotropy with its effective mo-

131

(a) # | | [111] j * (b) # | | [111]

/Nd2 '

10 20 Magnetic Field (T)

Fig. 4. (a), (b)Schematic figures of spin configurations for Nd- and Mo-tetrahedra when the field is applied along [111] direction. In each panel, upper and lower tetrahedra correspond to Nd- and Mo-tetrahedra, respectively. The relationship among the Mo spin 1, 2, and 3 is schematically shown at the right hand side of each Mo-tetrahedron. Open arrows depict the directions of the applied magnetic field. See text for details. (c)Magnetization curves for the applied field along [111] direction. Horizontal dot-dashed line represents the contribution from Mo spin moment, thick horizontal bars indicate the expected values of magnetization when the magnetic configurations of all the Nd tetrahedra are "3 in, 1 out" and "2 in, 2 out", respectively. See text for details. (d)pn is plotted as a function of magnetic field which is applied along [111] direction.

ment and Ising axis being gesJ w 2.3 and the (111) direction, respectively. Smooth nature of magnetization curves below 10 T even at 70 (or 50) mK suggests that the Nd moments flip incoherently from site to site, not in a co­operative and metamagnetic manner. There­fore, the local spin structure, and hence the local spin chirality, is different from the aver­aged spin structure during the magnetization process.

In Fig.3(d), the Hall resistivity is plotted as a function of magnetic field, which is ap­plied along [100] or [110] direction. For the both field directions, the Hall resistivity de­creases monotonously, and approaches zero. At the magnetic field of Ho ( « 3 T), both of the longitudinal and transverse component of the averagedNd moment vanishes. Then, the transverse component of averaged Mo mo­ment also vanishes. However, as noted above, each Nd moment does not vanish at all even at H0, and hence each Mo moment does have

a transverse component, namely, local spin chirality. Therefore, finite Hall resistivity ap­pears at HQ- What should be compared with the experimental result is the averaged ficti­tious field generated from the local Mo spin chirality(= (Si • 52 x S3)), but not the ficti­tious field calculated from such an averaged spin strucuture (= ( S i ) • (S2) x ( S3)) 18 as determined from the neutron diffraction ex­periment. At high magnetic fields, the Mo spins are aligned along the field direction. Therefore, the tilting angle of the Mo spin becomes smaller and smaller as the field is increased. This reduction of the tilting angle results in the decrease of Hall resistivity.

Figure 4(a) represents a case where the field is applied along [111] direction. Here, the field is already strong enough to reverse the averaged Nd moment, but not yet so strong (e.g. H — 5 T) as to make all the spins satisfied in terms of Zeeman energy. This

132

state also corresponds to the "2 in, 2 out" structure. Figure 4(b) shows a spin configu­ration where a magnetic field (e.g. H = 12 T) applied along [111] direction forces all the Nd spins to align so as to satisfy the Zeeman energy. In this case, Nd tetrahedron consists of the "3 in, 1 out" (or "1 in, 3 out") con­figuration. Then, the Mo spin system again acquires the same "tilting habit" as that in Nd moment system, namely, the "3 in, 1 out" structure, where b Mo • H is positive. This means that the fictitious field operating on conduction electrons changes its sign, and hence the sign of the Hall resistivity will be reversed.

In Fig.4(c), we plot the magnetization curve for the field applied along [111] direc­tion at 1.7 K and 65 mK. Surprisingly, the magnetization process is temperature inde­pendent also for this field direction at these low temperatures. The saturation moment at high fields almost coincides with the ex­pected value for the "3 in, 1 out" structure under the same assumption as above, but ap­parently larger than the expected value for the "2 in, 2 out" configuration. The magne­tization process below about 10 T is gradual also for this field direction. Therefore, the spin configuration of the Mo tetrahedra is random mixture of the "2 in, 2 out" and the "3 in, 1 out" structure with the ratio chang­ing as the field is increased 23, and finally becomes the "3 in, 1 out" configuration for all the tetrahedra above 13-15 T.

The Hall resistivity for the applied field along [111] is displayed in Fig.4(d) . Clearly, it changes sign at 7.5 T. With increasing magnetic field from zero, the number of tetrahedron with the "3 in, 1 out" struc­ture increases. In the high field limit, all the tetrahedra become the "3 in, 1 out" config­uration as evidenced by the magnetization curve. The local fictitious magnetic field is antiparallel to the local magnetization in the case of "2 in, 2 out" while parallel in the case of "3 in, 1 out", as mentioned above.

Therefore, in the course of increasing field from zero to strong enough field to make all the tetrahedra "3 in, 1 out" configuration, the internal fictitious field cancels out to be zero in the bulk limit. This field corresponds to 7.5 T, where the Hall resistivity vanishes. With further increasing field, the number of the tetrahedra with the "3 in, 1 out" config­uration becomes dominant, and the fictitious field for the conduction electron changes sign. This results in the sign reversal of the Hall resistivity, as observed.

In summary, we have clarified unconven­tional behaviors of anomalous Hall effect in a ferromagnet Nd2Mo2C>7 with pyrochlore structure. The temperature dependence of the Hall conductivity of Nd2Mo2C*7 is to­tally different from the prediction of exist­ing theories, but well understood in terms of the Berry phase theory. The magnetization curve shows little temperature dependence below 2 K and the values of saturation mo­ments are in accord with the assumption that the Nd moment is of Ising-anisotropy with the magnitude and Ising axis being « 2.3/^B

and (111) direction, respectively. The Hall resistivity changes sign when the field is ap­plied along [111] direction, but does not when the field is applied along [100] or [110] direc­tion. This fact is consistent with the predic­tion by the Berry phase theory, evidencing the spin-chirality mechanism of the anoma­lous Hall effect for Nd2Mo207 and related compounds.

Acknowledgments

We would like to thank Y.Oohara, H. Yoshizawa, N. Nagaosa, T. Sasaki, S. Awaji, Y. Iwasa, T. Tayama, T. Sakakibara, S. Iguchi, T. Ito, N. Hanasaki, and I. Kezsmaxki for their colloboration throughout this work. This work was in part supported by a Grant-In-Aid for Scientific Research Priority Area from the MEXT,Japan.

133

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134

Q U A N T U M GEOMETRY A N D HALL EFFECT IN FERROMAGNETS A N D SEMICONDUCTORS

N A O T O NAGAOSA

CREST, Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

* E-mail: nagaosa@appi. t u-tokyo. ac.jp Correlated Electron Research Center (CERC),

National Institute of Advanced Industrial Science and Technology (AIST),

Tsukuba Central 4, Tsukuba 305-8562, Japan

Geometrical phase for the Bloch states in solids leads to the motion of electrons transverse to the applied electric field. This leads to the intrinsic Hall effect, respresentative examples of which are the anomalous Hall effect in ferromagnets and spin Hall effect in semiconductors. This intrinsic Hall current is essentially dissipationless, but the disorder effect is crucial to understand the realistic situation. We review here the conventional view and recent new perspectives developed for these phenomena.

Keywords: Berry, anomalous Hall effect, spin Hall effect

1. Introduction

Quantum transport phenomena in solids are the manifestation of the quantum mechanics of electronic waves. The interference associ­ated with the phases of the waves is a cen­tral issue, and Aharonov-Bohm (AB) effect 1

is one of the representative examples where the external vector potential modifies inter­ference pattern. In this paper, we focus on the Hall effect not by the external magnetic field but by the nature of the Bloch wave-functions in periodic lattice 2. Since there is no magnetic field in real space, there is no AB effect in the usual sense. Instead, there appears another type of AB effect, i.e., AB effect in momentum space.

From the symmetry argument, time-reversal symmetry breaking is required for the non-zero axy — ayx. Therefore, we are lead to consider magnets for the nonzero Hall effect even without the extremal mag­netic field. The phenomenon of our interest here is the anomalous Hall effect (AHE) in ferromagnets. This phenomenon was found nearly a century ago2'3, but its origin has been controversial up to now. Empirically,

the Hall resistivity pn is often expressed as

PH = -PxV = RoH + 4TTRSM, (1)

where RQ is the usual Hall coefficient while Rs is called the anomalous Hall coefficient. In this expression, it is assumed that the anomalous contribution is proportional to the magnetization M, which is often ob­served experimentally. However, this rela­tion is not trivially true, and actually recent experiments have shown the nonmonotonous dependence of the anomalous contribution on M even including the sign change, which gives an important clue to the origin of the AHE as will be discussed later.

Theories of AHE were initiated by Karplus-Luttinger (KL) 4. They considered the band structure in the ferromagnetic state taking into account the relativistic spin-orbit interaction perturbatively. Note that the lat­tice periodicity is preserved even with the spin-orbit interaction. They pointed out that the conventional Boltzmann transport the­ory can not explain the AHE because the inter-band matrix elements of the current op­erator are crucial. The velocity due to this

135

inter-band effect is called "anomalous veloc­ity" and is interpreted as the time deriva­tive of the position operator, Wannier cen­ter, measured from the center of unit cell 5 . This is purely the band structure effect and nonequilibrium states with the steady cur­rent flow were not considered. Therefore, soon after its publication, KL theory was criticized by Smit 6, claiming that the impu­rity scatterings are essential for the steady state analysis, in particular for AHE. He attributed the mechanism of AHE due to the impurity scattering itself 6. Let Wk,k' be the transition probability due to the im­purity potential from the initial state k to the final state k', then it could happen that Wk,k' / Wk'tk when one considers higher or­der Born approximation including spin-orbit interaction. This is called the skew scattering and has been considered up to now the major mechanism of AHE. Another mechanism of AHE due to impurity, i.e., a side jump model, has also been proposed 7. These impurity-induced mechanisms are often called "extrin­sic", while that from the band structure, "in­trinsic" . If the extrinsic mechanism is domi­nant, it is hopeless to develop a generic the­ory for AHE since it depends on the details of the scattering potentials and other details of the model.

On the other hand, Kondo 8 applied this idea of skew scattering to analyze AHE in the s-d model. Because conduction electrons are scattered by fluctuating local spins, the spin-orbit interaction leads to the skewness, and gives AHE. Treating the s-d exchange in­teraction perturbatively, this theory predicts the relation pH oc< ( M - < M > ) 3 >, which explained many experimental data in the high temperature region, i.e., near the tran­sition temperature. Recently Ye et al. 9 re­visited this problem in the context of AHE in manganites. In this case, the exchange inter­action between conduction electron's spins and localized spins is very strong10, and in this limit the idea of "spin chirality" u ' 1 2

plays a crucial role. This spin chirality has been focused on gauge theories of resonat­ing valence bond (RVB) state related to high temperature superconductors 13. This spin chirality acts as a gauge field, and is expected to produce a Hall effect when its uniform component is finite. Ye et al. 9 argued that a term AM • b is generated by the spin-orbit in­teraction where M is the magnetization and b is the fictitious magnetic field. In the fer­romagnetic state, there appears uniform M and hence b, which leads to an anomalous Hall effect analogous to the usual Hall ef­fect caused by the external magnetic field. In their theory, the microscopic mechanism for b is thermally activated Skyrmion exci­tations, and hence the AHE is maximized near the ferromagnetic Tc and is quenched at low temperatures. Lyanda-Geller and Sala-mon 14 considered the same problem from the incoherent limit where hopping is treated perturbatively. This approach is much solid compared with the previous one, since it does not introduce an uncontrolled continuum ap­proximation. These theories can be regarded as a re-interpretation of Kondo's theory 8 , since the model itself is the same and the temperature dependence of AHE is similar.

The new perspectives on AHE recently attained are the following: First, the rather complicated formula given by KL can be ex­pressed in terms of the Berry phase gauge connection 15 of Bloch wavefunctions in the momentum space 16>17. This made it pos­sible to have a clear geometrical picture for the AHE in the momentum space, and also clarified the non-perturbative topological na­ture of this effect 16. Second, accurate first-principles band structure calculations have been done for SrRuOs 18 and Fe 19, and good quantitative agreements with experi­ments were obtained. The importance of band crossing as the source (sink) of gauge field recognized by Onoda and Nagaosa16

urged finer mesh of the k-points in the Bril-louin zone and careful analysis of the con-

136

vergence on the mesh-size. This clarified rich structure of the gauge field distribution in the momentum space, and the AHE was identified as the fingerprint of this distri­bution. Third, there appeared several low temperature experiments on AHE were re­ported, showing that the behaviors are con­sistent with the gauge theory , but inconsis­tent with any impurity induced mechanisms 17,18,20,21 Lastly this idea of the momentum space gauge field has motivated many new ideas, such as the intrinsic spin Hall effect22

and optical Hall effect23. We will give a re­view on these recent developments below.

2. Topological p ic tu re of AHE

Here we consider non-interacting electrons and assume there is no disorder for the mo­ment. Then each single-particle state is spec­ified by the crystal momentum k and the band index n. Then be expressed b y 24,25,16

<Txy = e2J2f(£n(k))bnz(k), (2) nk

where bnz(k) is the ^-component of bn(k) = Vfe x an(k) for each Bloch state specified by (n, k), and the vector potential anix{k) is de­fined by

anti,(k) = — i < nkld/dk^nk >, (3)

which corresponds to the overlap integral of two Bloch wavefunctions at neighboring points in the fc-space, Therefore the trans­verse conductivity axy represents the gauge field distribution in the momentum space. In particular, when the chemical potential is lo­cated in the gap, the fc-integral in eq.(2) is performed over the whole first Brillouin zone, leading to (e2/h) x integer in two-dimensions. This integer is the topological number called Chern number 24>25. This nontrivial topolog­ical structure of the gauge field anil{k) is as­sociated with the existence of the Dirac mag­netic monopoles 26 in the momentum space.

Namely V*, • bn(k) is not necessarily zero and has the delta-functional singularities. This is because the degeneracy points in the momen­tum space, i.e., the band crossings, act as the Dirac monopoles when the effective Hamil-tonian describing the region near the cross­ing contains all the three components of the Pauli matrices. Note that eq.(2) is obtained only when the infinitesimal imaginary part i<5 is neglected, which is justified when the sys­tem is insulating with a gap 24-25

j but is not justified a priori when the system is metallic with the dissipative processes. Therefore, we should go back to the Kubo formula when the impurity effect should be analyzed seriously.

Onoda and Nagaosa 16 studied the dis­tribution of the gauge field bn(k) for a sim­ple two dimensional 3 band model of t2g

orbitals coupled with spin-orbit interaction. We found that the uniform magnetization produces the nonzero Chern number for each band. This is because the spin-orbit in­teraction lifts the degeneracies of the band crossing to produce the shifts of the Chern numbers. This is the reason why the spin-orbit interaction can not be treated pertur-batively in sharp contrast to the previous theoretical treatments. Actually the distri­bution of the gauge field strength in momen­tum space is found to be characterized by the sharp peak structures corresponding to the near degeneracy of the bands. This sug­gests that the Hall effect is a kind of "finger­print" of the band structure and very sensi­tive to the distribution of the singular struc­ture of the gauge field. The axy as a func­tion of the magnetization mz exhibits the non-monotonous behavior in contrast to the simple relation axy oc mz. This nontrivial behavior has been found both in the first principles band calculation and in the ex­periment in SrRuOa, a metallic ferromagnet with Tc of 130K 18. In this material, the non-monotonous temperature dependence includ­ing even the sign change observed experimen­tally is reproduced by the calculated magne-

137

tization dependence of the Hall conductiv­ity. Similar conclusion was also obtained for the AHE in Fe 19, where the sharp spiky structure of the Berry phase distribution was found.

As for the impurity effects, Onoda et al.27considered the self-energy effect in a model of AHE, and found the crossover from the intrinsic region, where axy is nearly con­stant, to the extrinsic region, where pxy is nearly constant with the increasing scatter­ing strengh. However, it is known that the skew scattering is dominant in the limit of weak potential scattering 28, which is repre­sented by the vertex corrections. Recently Onoda et al.29 have studied the role of this vertex correction due to impurity scatterings when the intrinsic Hall effect is enhanced resonantly by the band crossing/degeneracy 29. We have found that a crossover from the extrinsic to the intrinsic regions occurs at much weaker disorder strength than the crossover mentioned above; This means that the monopole structure also play an impor­tant role for the intrinsic mechanism to be dominant over the extrinsic one.

3. Spin chirality

Up to now, we have discussed that the rela-tivistic spin-orbit interaction plays the essen­tial role in AHE. However, it is now estab­lished that the spin-orbit interaction in the conduction electron is not necessarily needed for AHE 30 '21. For example, let us consider a double exchange model

H = - J2 tij(4acja + h.c.) ij,a

- JH^2Si- c\aaa0ci0, (4) i

where JH is the ferromagnetic Hund's cou­pling and a is the vector of Pauli spin matri­ces. In the strong coupling limit JH —* oo, the conduction electron spin is forced to align parallel to that of the localized spin at each

site, and the effective transfer integral t^f be­tween the different sites i and j is given by

ttf = tij < XilXj >= Uidaii ^(jf), (5)

where \\% > is the two-component spin wave-function corresponding to the direction of the spin field at site i, 6ij is the angle between the two spin fields at i and j . Here, the phase fac­tor emij has the following geometrical mean­ing. Let Si, Sj, Sk be the spins at sites i, j , and k, respectively. Consider a conduction electron moving in the background of these spins as i —* j —> k —> i. Then the phase factor e

l(aa+ajk+aki) picked up by this elec­tron is given by elil/2 with O being the solid angle subtended by the three spins on the unit sphere. The Q. is called the (sclar) spin chirality, and is proportional to Si • Si x Sk for slowly varying spins. Therefore, the phase factor of ifj^ acts as the fictitious mag­netic field for the conduction electrons. Oh-gushi et al.30 considered the ground state of the non-coplanar spin configuration on the Kagome lattice, and found that the gauge field distribution in momentum space is pro­duced similar to the discussion in section 2. Two experiments21'31 confirmed that the pyrochlore-lattice structured Nd2Mo207 has the spin-chirality induced AHE. Here this lattice structure is a three dimensional gen­eralization of the Kagome lattice.

4. Spin Hall effect

The concept of the gauge field and anoma­lous velocity is not restricted to the ferromag­netic case, but is more general. In particular, the generalization to the nonmagnetic state is quite interesting. However the Hall con­ductivity axy — <jyX does not appear for the T-symmetric state as discussed in the intro­duction. This is due to the cancellation of the up and down spin contributions, even though the anomalous velocity is finite for both spin species. Therefore, we expect a spin current perpendicular to the electric field. This is

138

called spin Hall effect, and has been of great recent interests both theoretically and exper­imentally. The extrinsic spin Hall effect was first proposed by Dyakonov and Perel 1971 32, extended later by Hirsch33 and Zhang34, but recent focus has been on the intrinsic effect driven by the gauge field 22>35. Mu­rakami et al.22 showed theoretically that in p-hole doped GaAs and Ge, the spin current is induced perpendicular to the applied elec­tric field. In these semiconductors, the 4-bands near the top of the valence bands are described by the Luttinger Hamiltonian as

with

H° = T,cliH^K* (6)

with

H^k) = _ (<n + f)*2 272(fc • S)'

The degeneracy plays an essential role for the gauge field, and the F-point corresponds to this case; the two doubly degenerate bands touches there. Since there is Kramers de­generacy at each fc-point, the Berry connec­tion is not the simple phase factor (U(l)), but is given by the 2 x 2 matrix (SU(2)). Namely, one should consider the non-Abelian SU(2) gauge connection/field for the quan­tum topology of the nonmagnetic electronic systems. The gauge structure around this monopole can be grasped as follows. At each fc-point, there are two pseudo-spin states, i.e., parallel and anti-parallel to k = k/\k\. This is called helicity. The gauge field de­pends on the helicity of the state, i.e., it has the opposite sign for different helicity states. In other words, the anomalous veloc­ity depends on the spin state of the electron, and the external electric field can induce the transverse spin current even though there is no net transverse charge current. This idea was applied to the doped p-holes of GaAs/Ge and the spin current j'j was predicted to be 22,36

T" ~ 67T2" ^ F ~ k p ^ (8)

J- <jHe ijk

Ek (7)

where kp and kF are the Fermi wave num­bers for the heavy and ligh hole bands, re­spectively., This means that when an electric field is applied, the spin current transverse to it is generated with the spin polarization perpendicular to both of these directions.

There are two recent experimental re­ports on the spin Hall effect focusing on the spin accumulation near the edge of the sam­ple of n-type 37 and p-type 38 GaAs. For more details, the readers are referred to the article by Murakami in this Proceedings.

5. Conclusions

The interference of the electronic waves in solids is the central issue of condensed matter physics, which inevitably leads to the gauge theoretical description. The quantum trans­port phenomena are most sensitive to this interference, and the Hall effect we discussed here is a fingerprint of this gauge structure in the momentum space. This idea can be generalized to many directions, of which the electric polarization is a most interesting is­sue 39 '40. Combined magnetic, dielectric, and transport properties in the strongly corre­lated electronic systems driven by the gauge structure will be the next major field in the near future.

6. Acknowledgements

The author acknowledges Z. Fang, S. Mu­rakami, K. Ohgushi, M. Onoda, S. Onoda, R. Shindou, N. Sugimoto, K. Terakura, Y. Taguchi, Y. Tokura, H. Yoshizawa, and S.C. Zhang, for collaborations on this issue and K. Fujikawa, H. Fukuyama, F.D.M. Haldane, A.H. MacDonald, S. Maekawa, Q. Niu, N.P. Ong, M. Salamon, and M. Sato for fruit­ful discussions. This work is supported by NAREGI and Priority Areas Grants and

139

Grant-in-Aid for COE research and Kiban S

grant from the Ministry of Education, Cul­

ture, Science and Technology, Japan.

R e f e r e n c e s

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28. J. M. Luttinger, Phys. Rev. 112 (1958) 739. 29. S. Onoda, N. Sugimoto, and N. Nagaosa,

preprint (2005). 30. K. Ohgushi, S. Murakami, and N. Nagaosa,

Phys. Rev. B 62 (2000) R6065. 31. Y. Taguchi, T. Sasaki, S. Awaji, Y. Iwasa,

T. Tayama, T. Sakakibara, S. Iguchi, T. Ito, and Y. Tokura Phys. Rev. Lett. 90 (2003) 257202.

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T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92 (2004) 126603.

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140

SPIN HALL EFFECT IN A SEMICONDUCTOR TWO-DIMENSIONAL HOLE GAS WITH STRONG SPIN-ORBIT COUPLING

J. WUNDERLICH*

Hitachi Cambridge Laboratory, Cambridge CB3 OHE, United Kingdom E-mail: wunderlich@phy. cam. ac. uk

B. KAESTNER

Hitachi Cambridge Laboratory, Cambridge CB3 OHE, United Kingdom

K. NOMURA and A.H. MACDONALD

Department of Physics, University of Texas at Austin, Austin TX 78712-1081, USA

J. SINOVA

Deparment of Physics, Texas A&M University, College Station, TX 77843-4242, USA

T. JUNGWIRTH

Institute of Physics ASCR, Cukrovarnicka' 10, 162 53 Praha 6, Czech Republic School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Semiconductor spintronics is based on the controlled generation of localized spin densities. Finite spin densities in semiconductors have traditionally [1] been generated by external magnetic fields, by circularly polarized light sources, or by spin injection from spin-aligning materials, such as ferromagnets. Recently there has been considerable interest [2] in an alternate strategy in which edge spin densities are generated electrically via the spin Hall effect (SHE) [3,4], i.e., in a planar device by the current of spins oriented perpendicular to the plane that is generated by and flows perpendicular to an electric field. The SHE has traditionally been thought of as a consequence of spin-dependent chirality in impurity scattering that occurs in systems with spin-orbit (SO) coupling [5, 6]. Recently it has been recognized that the SHE also has an intrinsic contribution due to SO coupling in a perfect crystal [7, 8]. In this work, we study SHE induced edge spin accumulation in a two-dimensional hole gas (2DHG) with strong SO interactions. The 2D hole layer is a part of a p-n junction light-emitting diode with a specially designed coplanar geometry which allows an angle-resolved polarization detection at opposite edges of the 2D hole system. In equilibrium the angular momenta of the spin-orbit split heavy-hole states lie in the plane of the 2D layer. When an electric field is applied across the hole channel, a nonzero out-of-plane component of the angular momentum is detected whose sign depends on the sign of the electric field and is opposite for the two edges. Microscopic quantum transport calculations show only a weak effect of disorder, suggesting that the clean limit spin-Hall conductance description (intrinsic spin-Hall effect) might apply to our system.

1. Introduction

The Hall effects are among the most

recognized families of phenomena in basic

physics and applied microelectronics. The

ordinary and quantum Hall effects, which, e.g.,

proved the existence of positively charged

carriers (holes) in semiconductors and led to

the discovery of fractionally charged

quasiparticles [9], occur due to the Lorentz

force that deflects like-charge carriers towards

one edge of the sample creating a voltage

transverse to the current. In the anomalous Hall

effect [10], the spin-orbit interaction plays the

role of the force that deflects like-spin carriers

to one edge and opposite spins to the other

edge of the sample. In a ferromagnetic material,

this leads to a net charge imbalance between

the two sides, which allows one to detect

magnetization in the conductor by simple

electrical means. Here we report the

experimental observation of a new member of

141

the Hall family - the spin-Hall effect (SHE). As an analogue of the anomalous Hall effect but realized in nonmagnetic systems, the SHE opens new avenues for inducing and controlling spin currents in semiconductors without applying magnetic fields or introducing ferromagnetic elements. Predictions of the SHE were reported, within different physical contexts, in several seminal studies [11,6-8], and currently its microscopic origins are the subject of an intense theoretical debate [12-19]. Experimentally, the SHE has been elusive because in nonmagnetic systems the transverse spin currents do not lead to net charge imbalance across the sample, precluding the simple electrical measurement.

2. Experiment

To demonstrate the SHE, we have developed a novel p-n junction light-emitting diode (LED) microdevice, in a similar spirit to the one proposed in Ref. [7] but distinct in that it couples two-dimensional hole and electron doped systems. Its coplanar geometry and the strong spin-orbit (SO) coupling in the embedded two-dimensional hole gas (2DHG), whose ultrasmall thickness diminishes current induced self-field effects, are well suited for inducing and detecting the SHE. When an electric field is applied across the hole layer, a nonzero out-of-plane component of the spin is optically detected whose sign depends on the sign of the field and is opposite for the two edges, consistent with theory predictions.

The LEDs were fabricated in (AI, Ga)As/GaAs heterostructures grown by molecular-beam epitaxy and using modulation donor (Si) and acceptor (Be) doping in (Al,Ga)As barrier materials. The planar device features were prepared by optical and electron-beam lithography. A schematic of the wafer and numerical simulations of conduction and valence band profiles are shown in Figs. 1(a) and 1(b) [20].

Fig. 1. Basic characteristics of the LED. (a) The schematic cross section of the coplanar p-n junction LED device. At forward hias of order of the GaAs band gap, electrons move from the 2DEG to the 2DHG where they recombine. The highest intensity of the emitted light is in the p region near the junction step edge, (b) Numerical simulations of the conduction and valence band profiles along the (001) growth direction (z axis) in the unetched part of the wafer near the step edge. Black lines correspond to an unbiased p-n junction and red lines to a forward bias of 1.5 V across the junction. In the detailed image of the upper (AI,Ga)As/GaAs interface (with the conduction band shifted down in energy for clarity), we indicated possible sub-GaAs-gap radiative

recombinations with the 2DHG involving either 3D electrons in the conduction band (red arrow) or impurity states (black arrow). The black line corresponds to unbiased wafer 1 and the blue line to wafer 2. (c),(d) Photoluminescence spectra (black lines) and electroluminescence spectra (red lines) at low (lower curves) and high (higher curves) bias measured in wafers 1 and 2 at 4.2 K. Impurity (/), excitonic (A"), and 3D electron to 2D hole transitions (A, B, C) are identified. We focus on peak B, which has small overlap with the / and X lines.

Two wafers were investigated that differ in the 3nm undoped Al,Ga,.^As spacer at the upper interface, with wafer 1 having x = 0.5 and wafer 2: x = 0.3. The heterostructures are p-type; the band bending leads to the formation of an empty triangular quantum well in the conduction band at the lower interface and an occupied triangular quantum well near the upper interface, forming a 2DHG. The coplanar p-n junction is created by removing acceptors from a part of the wafer by etching, leading to population of the previously depleted conduction band well and depletion of the DHG in that region.

142

The diode has a rectifying l-V characteristic, and the onset of the current is accompanied by electroluminescence (EL) from the p-region near the junction step edge [20]. The current, /LED « lOOuA, is dominated by electrons moving from the n to the p region; the opposite hole current is negligibly small due to lower mobility of holes [20].

The interpretation of EL peaks shown in Figs. 1(c) and 1(d) starts from a comparison with measured photoluminescence (PL). Following previous PL studies [21,22] of (Al, Ga)As/GaAs single heterojunctions, we assign the sharp high energy PL peaks (X) to well known GaAs exciton lines, and the lower energy peaks (J) to twodimensional electron gas (2DEG) to acceptor or donor to 2DHG transitions. The other PL recombination process identified previously in literature [21,22], which involves 3D electron to 2D hole transition in the p region (or the analogous process in the ^-region), is unlikely to contribute to our PL spectra due to the large built-in electric field in the unbiased structure. In the EL spectra, the excitonic peaks are clearly visible and, consistently, their position is independent of wafer and applied p-n junction bias. A nonzero EL spectral weight at peaks / suggests that recombination processes with impurity states may contribute in the low energy part of EL. The peaks / are expected to shift to lower energies under forward bias because of the weaker confinement at the interfaces. Therefore, EL in the spectral range between the PL / and X peaks originates from another mechanism, i.e., from the 3D electron to 2D hole recombination. The broad peak B is the most striking example of such a process. Additional related peaks, A and C, can be identified in wafer 1 at high bias or in wafer 2 at low bias, respectively. These EL peaks shift to higher energies with increasing /LED> which is reminiscent of the behavior of 3D electron to 2D hole transition peaks as a function of the excitation power, reported in the PL

measurements in single AlGaAs/GaAs heterojunctions [21]. Figure 2(a) presents our microscopic calculations of the 2DHG energy structure [23].

•02 O0 02 - 3 - 2 - 1 0 1 2 3 1.500 1.505

k, (nm'l ii | i | E[cV|

Fig. 2. Spin-polarization in the absence of the SHE driving current, (a) Theoretical energy spectrum of the 2DHG along Ihe ky component of the wave vector (parallel to /LED) of the heavy-hole (HH) and the light-hole (LH) subband, spin-split due to the SO coupling. Unbiased wafer 1 was assumed in the calculations. A small shift of the subband energies towards the top of the valence band is expected for wafer 2 and under bias. Schematic color plot demonstrates circularly polarized light emission induced by transitions from the asymmetrically occupied conduction band to the HH+ subband. (b) Theoretical spinpolarization components along ky in the HH+ and HH- subbands.

(An infinitesimal magnetic field along: direction was added to break the degeneracy at k = 0). Except for a small region near k, = 0, spins are oriented in plane and perpendicular to the k vector, (c) Circular polarization, defined as the relative difference between intensities of left and right circularly polarized light, plotted as a function of the in-plane magnetic field along the x axis, measured in wafer 1. The observation angle is a - 85°, which is the maximum detection angle allowed in our experimental setup. The experiment measures, to high accuracy, the x component of the light polarization vector which is proportional to (sx)- (d) Spectral plot of the circular polarization near peak B for field ±3 T. (e) Same as (c) for magnetic and light detection axis along the normal to the 2DHG plane, (f) Same as (d) for field 0 and ±3 T. At zero magnetic field the z component of the polarization vanishes. All measurements are done at 4.2 K.

The confining potential of the unbiased wafer 1 was assumed in these numerical simulations. Only one heavy-hole (HH) and one light-hole (LH) bound state forms in the quantum well. The calculated spin orientation as a function of the wave vector component ky parallel to the /LED is plotted in Fig. 2(b). Near k = 0, the HH

143

subband states with total angular momentum jz

= ±3/2 are split from the LHs withy'z= ±1/2 due to different HH and LH effective masses, which forces the HH spins to align with the z axis. (Note that in the four-band representation of the valence band states [23] the total angular momentum and spin operators of holes are related as j = 3s.) For larger wave vectors, the mixing with LH states together with the SO interaction in the asymmetric quantum well [23] result in the splitting of the HH subband and in a vanishingly small angular momentum component (sz). The nonzero in-plane component of spin is oriented perpendicular to the 2D k vector. For the measured hole density, /?2D = 2 x 1012 cm"2, only the HH 2D subband is occupied by holes. Assuming that the optical recombination is dominated by states close the hole subbands Fermi wave vectors [22] and realizing that the 3D electron bands occupation under forward bias is strongly asymmetric along ky, we can now assign peaks A and B to hole transitions to the split HH subbands. Peak C is assigned to the LHs which may become populated in biased wafer 2 due to a weaker 2DHG confinement, compared to the one considered in Fig. 2. For simplicity, we now focus on peak B where the role of impurity and excitonic recombinations can be safely neglected. The detection of spin-polarization phenomena is done by measuring circular polarization (CP) of the light. Because of optical selection rules, a finite CP along a given direction of the propagating light indicates a finite spinpolarization in this direction of carriers involved in the recombination. In Figs. 2(c) and 2(d) we plot the CP of peak B for light detection axis close to the 2DHG plane and perpendicular to 7LED. The nonzero CP detected in the absence of an external magnetic field, consistent with the results in Fig. 2(b), is an experimental demonstration of a spin-polarization effect

induced by asymmetric band occupation [24]. Data taken at finite magnetic field applied along the x direction show the expected [23] cooperative or competing effects of the field and the SO coupling, depending on the sign of the field. A comparison between these data and data in Figs. 2(e) and 2(f), showing CP measurement at magnetic fields and light detection axis along the z direction, confirms the strongly anisotropic Lande' g factor typical of the SO coupled 2DHG [23]. Figures 2(e) and 2(f) confirms that the CP is purely in plane at zero external magnetic field, which is a crucial piece of evidence in the demonstration of the SHE, which we now discuss. In the SHE, nonzero (sz) occurs as a response to external electric field and the carries with opposite spins are deflected to opposite edges of the sample parallel to the SHE driving electrical current. A microdevice that allows us to induce and detect such a response is shown in Fig. 3(c). A /^-channel current is applied along the x direction, and we measure CP of the light propagating along z axis while biasing one of the LEDs at either side of the p-channel. The occurrence of the SHE upon applying Ip is demonstrated in Fig. 3(d,e). When biasing, e.g., LED 1, peak B polarization of the light emitted from the region near the corresponding p-n junction step edge is nonzero and its sign, i.e., the sign of (sz) of the holes, flips upon reversing Ip. The detected polarization of about 1% represents a lower bound of the SHE induced spin accumulation at the edge because the emitted light intensity decreases only gradually when moving from the biased junction towards the opposite side of the 2DHG channel [20]. Consistently to the unique SHE phenomenology, the polarization changes sign if detected from opposite edges of the sample when either LED 1 or LED 2 was activated. The opposite sign of these two signals confirms opposite (sz) at the two edges, establishing the SHE origin.

144

1.500 1.505 1510 1.315 1.520 Enwgy[eV]

Fig. 3. Schematic configuration of lateral p-n junction to detect spin accumulation (a): 2DHG channel bordered by an n-type region which forms LED. Driving a current /LED through the LED results in electroluminescence indicated by the yellow line. Driving a current Ip along the 2DHG channel induces the SHE spin accumulations at the edges of the 2DHG. Light emission from the p-n junction recorded by aCCD camera (b). Electron microscope image of the microdevicc with symmetrically placed LEDs at both edges of the 2DHG channel (c). Emitted light polarization of recombined light in the p-n junction, (d) and (c), for the current flow indicated in (c).

In Fig. 3(a) we show schematically the experimental setup with the LED current, /LP:D , driving the electroluminescence at the edge of the hole strip, and the 2DHG channel current, Ip, inducing the SHE edge spin accumulation. Digital camera image of the light emitted from the p-n junction is shown in Fig. 3(b). Fig. 3(c) is a scanning electron microscope image of the whole microdevice with symmetrically placed LEDs at both edges of the 10 urn channel. Circular polarization of the spectral peak corresponding to recombination of electrons with the spinpolarized 2DHG states at the edges of the channel are shown in Figs. 3(d) and (e). For more details on our co-planar LED devices and on the spectroscopical analysis of the electroluminescence data, see Refs. [4] and citations therein. The magnitude of the /,, current in the 1 Oum channel was adjusted so that the corresponding longitudinal electric field, Ex « O.I5Vum"', was comparable to the field used in the experiment in the 1.5 urn channel [4]. Spin polarizations of order ~ 1% observed in both devices confirm the expected independence of the SHE signal on the width of the diffusive channel [27].

We now compare experimental and calculated [27] magnitudes of the edge spin polarization. For more details to the theoretical analysis, see Refs. [27]. Numerical data in Fig. 4 imply an approximate general form of the dimensionless edge polarization,

tin ' 4itvh-hn

where gs is a numerical factor. In the measured system the 2DHG density n = 2 x 10l2cm'2, VfHOWs, Aso/Ep* 0.4 and (V/r)/£,, w 0.1. The system is therefore in the strong SO coupling regime and the corresponding numerical factor gs = 7.4.

Fig. 4. Spatial profile of the r-component of accumulated spin density as a function of spin-orbit coupling for (v7ij/£p*0.1 and L = 44k,.:' (a), the disorder strength for Aso/Ey 0.4 and L = 44A;."' (b), and of the system size for (V/r)/£F M 0.3 and ASO/EF * 0.4 (c). Schematic cartoon of SHE induced spin accumulation in the strong SO coupling regime (d).

From Eq. (1) we obtain that the expected polarization for the measured sample reaches - 8 % and the width of the accumulation area is of order Lso ~10nm. The theoretical - 8 % polarization is consistent with the measured value o f - 1 % assuming an effective LED recombination width of the order ~100nm. This number cannot be precisely determined experimentally because of the resolution limit set up by the wavelength of the emitted light (~800nm). However, comparisons between experiments in the 1.5 urn and 10 urn 2DHG

145

channels, analysis of the digital images of the active p-n junction area, and simulations of device I-V characteristics [4] confirm a sub-micron width of the recombination region near the p-n junction. In summary we have studied the edge spin accumulation due to the SHE. The theoretical spin-density is independent of the disorder scattering rate in magnitude and is localized within a spin-precession length of the edge. The magnitude increases with the strength of the SO coupling. For the parameters used in the experimental study, we predict an 8% spin-polarization at the edge. These predictions are consistent with the experimental finding of - 1 % optical polarization averaged over a distance ~10Z-so-The experiments also confirmed that opposite spin accumulations at the edges can be be separated over large distances in diffusive conduction channels and that their magnitude is not a affected by the channel width. We conclude by noting that, beside the demonstrated ability to induce and detect spin polarization in the absence of external magnetic fields, our novel planar design allows the fabrication of more complex microdevices that integrate on a single semiconductor chip these two functionalities with transmission of the spin information between different parts of a device. Lithographically defined gate electrodes may provide additional control of the SHE, and therefore of the overall device performance, by varying 2DHG density or SO coupling.

Acknowledgments

We thank Shoucheng Zhang for many useful discussions. The work was supported by the Grant Agency and Academy of Sciences of the Czech Republic through Grant No. 202/02/0912 and Institutional Support No. AV0Z10100521 and by the Japan Society for the Promotion of Science Grant Agency and Academy of Sciences of the Czech Republic,

EPSRC, Welch Foundation, and Department of Energy.

References

1. I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).

2. C. Day, Physics Today February, 17 (2005). 3. Y. K. Kato, et al., Science 306, 1910 (2004). 4. J. Wunderlich, et al., Phys. Rev. Lett. 94,

047204 (2005). 5. M. I. Dyakonov and V. I. Perel, Phys. Lett. A p.

459(1971). 6. J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 7. S. Murakami et al., Science 301, 1348 (2003). 8. J. Sinova, et al, Phys. Rev. Lett. 92, 126603

(2004). 9. R. E. Prange and G M. Girvin, "The Quantum

Hall Effect" (Springer-Verlag, New York, 1990).

10. "The Hall Effect and Its Applications", edited by C. L. Chien and C. R. Westgate (Plenum, New York, 1980).

11. M. I. Dyakonov and V. I. Perel, Phys. Lett. 35A, 459(1971).

12. J. Hu, B. A. Bernevig, and C. Wu, Int. J. Mod. Phys. B 17, 5991 (2003).

13. E. I. Rashba, Phys. Rev. B 68, 241315 (2003). 14. J. Schliemann and D. Loss, Phys. Rev. B 69,

165315(2004). 15. S.-Q. Shen, Phys. Rev. B 70, 081311(R) (2004). 16. J. Inoue, et al., Phys. Rev. B 70, 041303 (2004). 17. N. A. Sinitsyn et al, Phys. Rev. B 70, 081312

(2004). 18. L. Sheng, D. S. Sheng, and C. S. Ting, Phys.

Rev. Lett. 94, 016602 (2005). 19. K. Nomura, et al., Phys. Rev. B 71, 041304

(2005). 20. B. Kaestner, et al., cond-mat/0411130. 21. W. Ossau, E. Bangert, and G Weimann, Solid

State Commun. 64, 711 (1987). 22. A Y Silov et al., J. Appl. Phys. 73, 7775 (1993). 23. R. Winkler, "Spin-Orbit Coupling Effects in

Two-Dimensional Electron Hole Systems", Springer Tracts in Modern Physics Vol. 191 (Springer-Verlag, Berlin, 2003).

24. A. G Mal'shukov and K. A. Chao, Phys. Rev. B 65, 241308 (2002).

25. E. M. Hankiewicz, et al. Phys. Rev. B 70, 241301 (2004).

26. A. Bernevig and S.-C. Zhang (private communication).

27. K. Nomura et al., accepted for publication in Phys. Rev. B

146

INTRINSIC SPIN HALL EFFECT IN SEMICONDUCTORS

SHUICHI MURAKAMI

Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

* E-mail: murakamWappi.t. u-tokyo.ac.jp

We theoretically propose that the spin Hall effect occurs in p-type semiconductors even without an aid of impurity scattering. An effect of disorder is also investigated. We also discuss a criterion for nonzero spin Hall effect. With this criterion, we predict that some band insulators have nonzero spin Hall effect.

Keywords: Spin Hall effect; semiconductor; Berry phase

1. Introduction

The spin-orbit (SO) coupling gives rise to various phenomena of spin-orbit-coupled transport, which give us new insights into the emerging field of spintronics. Among them is the spin Hall effect (SHE), where an ex­ternal electric field induces a transverse spin current. Such effect was predicted to oc­cur extrinsically, i.e. due to impurities, more than three decades ago.1 On the other hand, two groups 2'3 independently predicted that such effect can occur intrinsically, based on a band structure of the system; therefore it can appear even in relatively clean samples. Murakami et al. predicted the effect for p-type semiconductors, 2 and Sinova et al. pre­dicted the effect in n-type semiconductors in heterostructure.3

In this paper we explain this intrinsic SHE and its impurity effect. Recently the effect of impurities has been debated, since in some systems a least amount of impuri­ties gives an exact cancellation between the extrinsic SHE and the intrinsic SHE. We dis­cuss the reason for this cancellation. For the Rashba model, representing 2D n-type semi­conductors in heterostructures, we show that the cancellation occurs under rather gen­eral conditions. Meanwhile, for general sys­tems the total SHE is nonzero. Furthermore, in some band insulators such as PbTe, the

charge conductivity vanishes, and the spin Hall current flows without dissipation; we can call it a spin Hall insulator.4 We will also briefly explain two recent experiments on the SHE in semiconductors.5,6

2. Spin Hall effect in p-type Semiconductors

We begin with semiclassical description of the SHE by introducing a "Berry phase in momentum space", and apply it to the p-type semiconductors.2 When the wavevec-tor k of a Bloch state gradually changes under an external electric field, the quan­tum state acquires a phase change called the Berry phase. The Berry phase in momentum space, namely the field strength of a U(l) gauge field in momentum space, is written as Bn(k) = Vr x An(k), where Ani(k) =

—i (nk\ ^ 7 nk\ is a gauge connection, and

\nk) is the Bloch wavefunction in the n-th band. The Berry phase gives an additional term to the conventional semiclassical equa­tion of motion as,7

i r = 4 ^ P + £x £„(£), (i) « dk

hk = -e(E + £x B{x)). (2)

The second term in the RHS of (1) represents the effect of the Berry phase, and is called

147

an anomalous velocity. This term gives rise to the Hall effect. By summing over all the occupied state we can calculate the intrinsic Hall conductivity.

(a) bulk (b) heterostructure

UJ 1.5

CB\^

fy

kt k? '

\ HH

LH\ •

- . so

\

1.7

> T 16

V Ul 15

; ' V \ J^ . i l l

Wavenumber k(nm' )

Wavenumberk(nm')

Fig. 1. Band structure for GaAs. CB, HH, LH, SO represent the conduction, heavy-hole, light-hole and split-off bands, respectively, (a) represents a bulk crystal, while (b) expresses a band structure in heterostructure, where the Rashba coupling lifts the double degeneracy of the conduction band.

In semiconductors with diamond struc­ture (e.g. Si) or zincblende structure (e.g. GaAs), the valence bands are composed of two doubly degenerate bands: heavy-hole (HH) and light-hole (LH) bands. They are degenerate at k = 0 (see Fig. 1). The va­lence bands near k = 0 are well described by the Luttinger Hamiltonian

H = 2m 7i + 2^2 fe2-272(A;-5)2

(3) where S is the spin-3/2 matrices represent­ing the total angular momentum. By diago-nalizing this Hamiltonian, we get B\(k) = A(2A2 - l)k/k3 where X = k • S denotes the helicity. Hence, the anomalous velocity is parallel to Ex k, giving rise to a deflec­

tion from the (conventional) velocity v = ^t. The direction for this deflection is opposite for different signs of the helicity A, refer­ring to whether the spin S and the wavevec-tor k are parallel or antiparallel. This shift amounts a spin current perpendicular to the electric field, resulting in the SHE. If the elec­tric field is along /-axis, the spin current with Sl spin flowing toward the j-direction is 2

3) = I ^ ( 3 ^ ~ UfiUiiEi, (4)

where fc" and k\ are the Fermi wavenumber for the HH and LH bands, respectively. Spin Hall conductivity (SHC) for p-GaAs is eval­uated as a$ ~ 80fi and o-g ~ 7fi disorder effect is also calculated. Within the first-order Born approximation, the correc­tion due to disorder approaches zero in the clean limit. It is not so trivial, since it is not the case in the Rashba model, as we discuss in the next section. Thus the disorder is ir­relevant in a relatively clean p-type sample.

y

'cm x for n = 1019cm 3

"'cm x for n 1016cm-3.2 The

y

E p-GaAs

L

Fig. 2. Schematic of the spin Hall effect in p-type semiconductors. By applying an electric field along +z direction, +x spins will flow toward +y direction, and —x spins will flow toward —y direction.

We note that the above result has been derived within the semiclassical approach. If we attempt to incorporate the quantum na­ture, for example by using the Kubo formula,

148

we have to discuss what is the "spin current operator". As the spin is not conserved in the presence of the SO coupling, the defini­tion of the spin current is not unique.18

3. Spin Hall Effect in n-type Semiconductors in Heterostructure

In n-type semiconductors with diamond or zincblende structure, the SO coupling is small. On the other hand, however, if they are incorporated into two-dimensional het­erostructure, the inversion symmetry is bro­ken, and the SO coupling becomes relevant. The Hamiltonian is approximated as

H=^-+\(ax k)z, (5)

where <r, is the Pauli matrix. The second term is called the Rashba term representing the SO coupling, giving rise to a splitting of the conduction band (see Fig. 1 (b)). The Rashba coupling A can be experimentally de­termined, and can be controlled by the gate voltage. Sinova et al. applied the Kubo for­mula to this Rashba Hamiltonian.3 For this procedure, they denned the spin current Jz

to be a symmetrized product of the spin Sz

and the velocity vy = ^-. By assuming no disorder, the resulting SHC is e/(87r);3 this is independent of the Rashba coupling A, re­flecting an interband nature of the SHE.

With regards to an impurity effect, Inoue et al. 9 found an important result. They as­sumed random impurities with a J-function potential. In a clean limit, within lad­der approximation, they obtained a vertex-correction contribution —•£- to the SHC, which precisely cancels the intrinsic value j ^ - . This result holds under more gen­eral conditions.10 '11 '12 '13 '14 '15 As noted in Refs. 12, 14, this cancellation may come from the fact that the "conventional" spin cur­rent operator Jz = | {vy, Sz} is proportional to ^ 5 " , which is discussed to vanish in the clean limit.

To exploit this problem the Keldysh formalism is useful. We can show that this cancellation occurs exactly even for longer ranged impurities, within higher or­der Born approximation and for finite relax­ation time.16 This cancellation stems from the above-mentioned relationship Jz <x^Sy. This kind of relationship does not hold for general systems, and the SHE does not van­ish in general.17 As the spin current is not uniquely defined, one can alternatively define the spin current by an associated spin accu­mulation. An idea along this line by Zhang et al.8 leads to a "conserved" spin current operator Js = j[(ySz). By considering this conserved spin current, the resulting SHC be­comes nonzero.16

4. Spin Hall Insulator

Let us discuss a criterion for nonzero SHE. As the SHE is caused by the SO coupling, it should be commonly found in various kinds of materials. If the bands split by the SO coupling are all filled, the SO coupling is not effective for the SHE. Thus the bands in a multiplet split by the SO coupling should have different fillings. For the valence band of semiconductors, the SO coupling splits the bands into the HH and LH bands, thereby inducing nonzero SHC for p-type semicon­ductors. On the other hand, the conduc­tion bands are degenerate for a bulk crys­tal. Therefore even electron-doping does not give different fillings between the conduction bands. Thus in the bulk n-type semiconduc­tors the SHE vanishes. Only when the con­duction bands are split e.g. in heterostruc­ture (Fig. 1(b)), electron-doping gives rise to nonzero SHE as shown by Sinova et al.3

This criterion does not exclude insula­tors. Indeed some band insulators, where the SO-split bands have different fillings, can have nonzero SHC. We call them "spin Hall insulators". 4 One of them is zero-gap semi­conductors such as HgTe, where the conduc-

149

tion and the valence bands touch at the T-point. This degeneracy can be lifted by a uniaxial strain, opening a finite gap at the T-point. Another candidate is narrow-gap semiconductors such as PbTe, which have a direct gap at the four L points. By using simplified models for these systems, we find the SHC to be as ~ e/a where a is the lat­tice constant. In contrast with the quantum Hall systems, the SHC is not quantized even when the system is insulating.

(a) conventional (b) zero-gap (c) narrow-gap

ck

t .&.

\

Fig. 3. Band structures for (a) conventional semi­conductors (GaAs), (b) zero-gap semiconductors (HgTe) and (c) narrow-gap semiconductors (PbTe).

5. Concluing Remarks

This SHE has been experimentally observed by two groups. 5 ,e Kato et al.5 used a 3D n-type semiconductor, and measured a spatial profile of spin accumulation of the sample, due to the spin current by the SHE. Wunder-lich et al.6 fabricated a light-emitting diode (LED) and measured a circular polarization of light from the 2D p-type semiconductor in the LED. Whether the results are explained within intrinsic or extrinsic SHE (or both) is still under intensive research.

This spin current is even under time-reversal, and thus the SHE can be induced without breaking time-reversal symmetry. Therefore, it does not require any magnetic field, which is ideal for spintronics applica­tions. This effect is potentially important

for efficient spin injection into semiconduc­tors without the need for metallic ferromag-nets, opening up a new possibility for spin-tronic devices with low power consumption.

Acknowledgments

This work is partially based on collaborations with N. Nagaosa, S. Onoda, N. Sugimoto, and S.-C. Zhang. The author is supported by Grant-in-Aid (No. 16740167) from the Min­istry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

References

1. M. I. D'yakonov and V. I. Perel', ZhETF Pis. Red. 13, 657 (1971) [JETP Lett. 13, 467 (1971)].

2. S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).

3. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald Phys. Rev. Lett. 92, 126603 (2004).

4. S. Murakami, N. Nagaosa, and S.-C. Zhang, Phys. Rev. Lett. 93, 156804 (2004)

5. Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, Science 306, 1910 (2004).

6. J. Wunderlich. B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 1047204 (2005).

7. G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999).

8. P. Zhang, J. Shi, D. Xiao, Q. Niu, cond-mat/0503505.

9. J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 67, 033104 (2003); ibid. B70, 041303 (2004).

10. E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004).

11. R. Raimondi, P. Schwab, Phys. Rev. B71, 033311 (2005).

12. O. V. Dimitrova, cond-mat/0405339. 13. A. Khaetskii, cond-mat/0408136. 14. O. Chalaev, D. Loss, cond-mat/0407342. 15. S. Y. Liu, X. L. Lei, cond-mat/0411629. 16. N. Sugimoto, S. Onoda, S. Murakami and

N. Nagaosa, preprint (2005). 17. S. Murakami, Phys. Rev. B69, 241202

(2004). 18. S. Murakami, N. Nagaosa, and S.-C. Zhang,

Phys. Rev. B69,235206 (2004).

150

THEORY OF SPIN TRANSFER PHENOMENA IN MAGNETIC METALS AND SEMICONDUCTORS

ALVARO S. N U N E Z and A.H. M A C D O N A L D

Physics Department, University of Texas at Austin, Austin TX 78712

Spin-transfer refers to the transfer of spin angular momentum between transport quasiparticles and the magnetic condensate of a ferromagnetic metal. It is ususally regarded as a consequence of total spin conservation. We discuss a theory that views spin-transfer from a more microscopic point of view and sees it as an example of a more general class of phenomena that occurs in any magnetic metal or semiconductor, and indeed in any system in which interactions contribute importantly to the quasiparticle self-energy. Our theory of spin transfer in ferromagnets does not rest on an appeal to conservation of total spin, can assess whether or not current-induced magnetization precession and switching in a particular geometry will occur coherently, and can estimate the efficacy of spin-transfer when spin-orbit interactions are present. We illustrate our theory by applying it to a toy-model two-dimensional-electron-gas ferromagnet with Rashba spin-orbit interactions and mention examples of related phenomena in other systems, most of which have not yet been realized experimentally.

Keywords: Nanomagnetism; Spin Transfer; Current induced dynamics.

1. In t roduc t ion

In recent years fundamental aspects of mag­netism that are obscured in bulk materi­als have been cleanly identified and sys­tematically studied in magnetic nanostruc-tures. These new phenomena, includ­ing giant-magneto resistance1, inter-layer coupling2 and spin transfer, have collectively weaved a rich phenomenological tapestry that has already enabled several new tech­nological applications3 and promises more in the future. The transfer4'5 of magnetiza­tion from quasiparticles to collective degrees of freedom in transition metal ferromagnets has received attention recently because of experimental6and theoretical7progress that has motivated basic science interest in this many-electron phenomenon, and because of the possibility that the effect might prove to be a useful way to write magnetic infor­mation. The key theoretical ideas that un-derly this effect were proposed some time ago4'5 and rest heavily on bookkeeping which follows the flow of spin-angular momentum through the system. Recent advances in

nanomagnetism have made it possible to compare these ideas with experimental ob­servations and explore them more fully.

The non-equilibrium state of a current-carrying many-electron system can be quite complicated. It is remarkable then, that in the case of a nanomagnet with a spin-valve geometry, the main effect of a trans­port bias voltage is to introduce a non-equilibrium torque that acts on the mag­netic condensate and has an extremely sim­ple form. The influence of a bias voltage on order parameter dynamics is an example of a type of non-equilibrium physics that ap­pears likely to arise with greater frequently as nanoscale transport is explored in more and more contexts. For example, it has re­cently been argued8 on the basis of mean-field-theory considerations, that the mag­netic transition temperatures and other ther­modynamic magnetic properties of a mag­netically isolated film can also be altered by transport bias voltages. These two examples motivate a more general and formal examina­tion of the equilibrium statistical mechanics of collective variables in conductors with a

151

bias voltage than is attempted here. In this article we focus our attention on

a theory of spin transfer that does not rest on an appeal to conservation of total spin, fo­cusing instead on its origin in the change in the exchange-field experienced by quasipar-ticles in the presence of non-zero transport currents. Our discussion of spin-transfer sees the effect of a specific example of a larger class of phenomena that occur in any inter­acting electron system that can be described by a time-dependent mean-field theory. Our approach can assess whether or not the current-driven magnetization dynamics in a particular geometry will be coherent, and can predict the efficacy of spin-transfer when spin-orbit interactions are present. Although we can employ any time-dependent mean field theory of a metallic ferromagnet we ex­pect that for magnetic metals most applica­tions would be in the framework of ab ini­tio spin-density-functional theory9 (SDFT) which is accurate for many of these systems.

Our approach makes it clear that closely related phenomena occur in any physical system with interactions between quasipar-ticles and collective coordinates, even sys­tems without broken symmetries. We briefly discuss applications to10 semiconduc­tor electron-electron bilayers and antiferro-magnetic metal nanoparticle circuits.

2. Microscopic Theory of Spin Transfer

Our picture of spin-transfer is summarized schematically in Fig.(l). In SDFT, order in a metallic ferromagnet is characterized by ex­cess occupation of majority-spin orbitals, at a band energy cost smaller than the exchange-correlation energy gain. (Adopting the com­mon terminology of magnetism, we refer to the spin-indenpendent and spin-dependent parts of the exchange-correlation fields of SDFT below as scalar and exchange poten­tials.) In the ordered state, majority and

minority spin quasiparticles are brought into equilibrium by an exchange field that is ap­proximately proportional to the magnetiza­tion magnitude and points in the majority-spin direction. The spin-orientation of the singly occupied majority-spin orbitals is the collective-coordinate, the magnetization ori­entation, that plays the lead role in most magnetic phenomena. The non-equilibrium current-carrying state of a ferromagnetic metal thin film can then be described us­ing a scattering or non-equilibrium Greens function formulation of transport theory11. The current is due to electrons in a narrow transport window with width eV centered on the Fermi energy, and can be evaluated by solving the quasiparticle Schroedinger equa­tion for electrons incident from the high-potential-energy side of the film. The spin-transfer effect occurs when the spin-polarization of these transport electrons is not parallel to the magnetization, producing a transport induced exchange field around which the magnetization precesses. We ex­pand on this picture below and illustrate its utility by applying it to a toy-model two-dimensional ferromagnet with Rashba12

spin-orbit interactions.

2.1. Quasiparticle Spin Dynamics

We start by considering single-particle Hamiltonians of the form

n=^ + V(r)-lA(f).r, (1)

where V{r) and A(r) are arbitrary scalar and exchange potentials and f is the Pauli spin-matrix vector. In the local-spin-density approximation9 (LSDA) of SDFT, A(r) = Ao(n(f),m(f))m(r) where m is a unit vec­tor, m = mm is the total spin-density at r obtained in equilibrium by summing over all occupied orbitals, and the mag­nitude of the exchange field (A0(n,m))

152

Equi l ibr ium Ferromagnet

O O

oo O O Q Q

I

o o o o o

Singly o c c u p i e d Orbitals

Spin Transfer Torque

t o o O O Q O ^» ^ Transport | | V ^ Orbitals Ij,' eV

Eqp

Fig. 1. Left panel: Ground state of a metallic ferromagnet. The low-energy collective degree of freedom is the spin-orientation of singly occupied orbitals. Right panel: Quasiparticles experience a strong exchange field A that brings majority and minority spins into equilibrium. Because this field is parallel to the magnetization it does not produce a torque. In an inhomogeneous ferromagnet, the spin orientation of the transport orbitals in a window of width eV at the Fermi energy can differ from the magnetization orientation. The spin-transfer torque is produced by the transport-orbital contribution to the exchange field.

is the quasiparticle spin-splitting of a po­larized uniform electron gas. The spin-density contribution from a single orbital * a is sa(r) = *^(f) f * a ( r ) / 2 . The time-dependent quasiparticle Schroedinger equa­tion therefore implies that

where the spin current tensor for orbital a is defined by,

^ ( f ) = 2^Im (•i(f^V**-W) • ^ This equation exhibits the separate contribu­tions to individual quasiparticle spin dynam­ics from convective spin flow, the source of the conservative term, and precession around the exchange field A. Both sides of Eq.(2) vanish when the quasiparticle spinor solves a time-independent Schroedinger equation.

2.2. Collective Magnetization Dynamics:

The time-dependence of the total magnetiza­tion is obtained by summing Eq.(2) over all occupied orbitals.

^ = £v,/i,(r) 1 h

A x fh(r) J

(4)

where Jlaj is the contribution to the spin-

current from orbital a. The main point we wish to make here is that (in the LSDA) A is proportional to m at each point in space-time so that (at least in the absence of transport currents) the second term on the right vanishes. The collective magne­tization dynamics13 is driven not by the large effective fields seen by the quasiparti­cles, but by external and demagnetization fields and spin-orbit coupling effects that have been neglected to this point in the dis­cussion, and by the divergence of the collec­tive spin-current14 in the first term. A com­plete description of magnetization dynam­ics would require that the neglected terms be included, and that damping due to mag-netophonon and other couplings be recog­nized. In practice, thin film magnetiza­tion dynamics can usually be successfully de­scribed using a partially phenomenological micromagnetic theory approach15 in which the long-wavelength limit of the microscopic physics is represented by a small number of material parameters that specify magnetic anisotropy, stiffness, and damping. We adopt that pragmatic approach here, replacing the microscopic Eq.(4) by the phenomenological

153

Landau-Liftshitz equation

dm „ ,-. „ dm . „ , — = m x Heff + a m x — , (5)

where a is the damping parameter,

Heff(r) = JL (6) om[r)

is the effective field that drives the long-wavelength collective dynamics of an electri­cally isolated sample, and EMM[m] is the mi-cromagnetic energy functional.

2.3. Spin- Transfer

When current flows through a ferromagnet, the transport orbitals are few in number and make a negligibly small contribution to the magnitude of the magnetization. In an inho-mogeneous magnetic system, however, they can make an important contribution to the exchange field A as we now explain. The slow dynamics of the collective magnetization can be ignored in the transport theory, appeal­ing to an adiabatic approximation. Our ap­proach to spin-transfer is based on a scat­tering theory formulation11 in which proper­ties of interest can be expressed in terms of scattering solutions of the time-independent Schroedinger equation defined by the instan­taneous value of A. Transport electrons will in general make a contribution to the spin-density that is small but perpendicular to the magnetization a . We define this trans­port contribution to the spin-density as m t r . Because it is perpendicular to the magneti­zation, its contribution to the exchange-field experienced by all quasiparticles

m t r

<5HST = A0(n,m) (7) m

produces a spin-torque that can be compa­rable to that produced by Heff. It follows a The transport orbitals will also, in general, con­tribute to the total spin-density component in the direction of the magnetization. This effect alters the exchange field along the magnetization direction and does not produce a spin torque.

that the influence of a transport current on magnetization dynamics is captured by re­placing Heff in Eq.(5) by Heff 4- <5HST- This proposal is the central idea of our paper. We note here that the above doesn't correspond, by means of Eq.(6), to just a correction in EMM!™]- The net correction <5HST to the ef­fective dynamics under non-equilibrium con­figurations can be separated into two con­tributions. One "conservative" part which can be written as a corresponding correction to EMM[™], and one "non-conservative" part that pump (or drain) energy to (from) the system. A remarkable feature of the spin-valve geometry is that this non-conservative part is the dominant part (an example of this is given in Fig. (3a)). In this way the main effect of the spin-transfer torques in the dy­namics is to create an "effective damping"b. Although the behavior of this term makes it compete, in the dynamical equations of the magnet, with the damping it is important to note that its origins are not in a disorganized reservoir but in a coherent precession of the electrons in the transport window.

The separation we have made here be­tween transport orbitals and condensate or­bitals is reminiscent of the separation be­tween conduction electrons and local mo­ments that is often made in models of magnetic systems. In diluted-magnetic-semiconcutor ferromagnets, for example, these models often have a quantitative16 va­lidity. In transition metal ferromagnets so-called s — d models of this type has some qualitatively validity, but since the transi­tion metal bands cross the Fermi level, can­not be justified systematically. In the s — d model description of spin-transfer, the mag­netic condensate is associated entirely with the local moments and transport with the

bNote that , this "effective damping", can compen­sate the intrinsic damping4 '5 , signaling the instabil­ity that precedes the switching, in current induced switching experiments.

154

conduction electrons. The mean-field ex­change interaction between local moments and transport electrons that carry current through a region with a non-collinear magne­tization produces a torque through the mech­anism described above. In our formulation of spin-transfer torque theory, the separation between transport orbitals and the conden­sate order parameter is based only on the existence of a transport energy window near the Fermi energy.

Our proposal can be related to the com­mon approach in which spin-transfer is com­puted from spin current fluxes. In the ab­sence of spin-orbit coupling, summing over all transport orbitals and applying Eq.(2) implies a relationship between the transport magnetization and the transport spin cur­rents:

A(r) x mtv(r) hViJ"'\?) (8)

where J*'1 is the spin-current tensor summed over all transport orbitals. Note that the net spin current flux through any small volume is always perpendicular to the magnetization. It follows from Eq.(8) that

SHsrif) = V ^ ' ^ X r f l . (9) m

When Eq.(9) is inserted in Eq.(5) it implies a contribution to the local rate of spin-density change in any small volume proportional to the net flux of spin current into that volume; in other words it implies that the bookkeeping theory of spin-transfer applies locally, a prop­erty that can be traced in this instance to the local spin-density-approximation (LSDA) of SDFT. The local approximation for ex­change interactions has its greatest validity when the magnetization varies slowly on an atomic length scale, in long-wavelength spin waves or in typical domain walls for example. This observation helps explain why a simple spin-transfer argument17 is able to account for the influence of a current on spin-waves in a homogeneous ferromagnet18 and on the

propagation of a domain wall19. When spin-orbit interactions are present, Eq.(9) is no longer valid.

Eqs.(5) and (7) provide explicit expres­sions for the effective magnetic fields that drive magnetization precession at each point in space and time. Using these equations it is possible to explore the consequences of spatial variation in spin-transfer torque magnitude and direction, and of spin-orbit interactions. These have a dominant im­portance in ferromagnetic semiconductors20, where spin transfer effects have been success­fully demonstrated21.

3. Toy-Model Calculation

We illustrate our theory by evaluating mtT(r) for a toy model containing a ferromag­netic two-dimensional electron system with Rashba spin-orbit interactions. The model system, illustrated in Fig. (2), is intended to capture key features of the spin-transfer ef­fect. We take the width of the pinned magnet to infinity, neglect the paramagnetic spacer that is required in practice to eliminate ex­change coupling between the two magnets, and assume for simplicity that there is no band offset between the two ferromagnets and that the two exchange fields are equal in magnitude. Current flows from the pinned magnet, through the free magnet, into a paramagnetic metal that functions as a load. The spin-orbit interaction is assumed to be confined to the free magnet region. c The above simplifications allow us to write the Hamiltonian of the system as:

v2

H = £ A(x) • s + {\(x),px z} • s, 2m

where A correspond to the local exchange vector, and A is non-zero only on the re­gion with spin-orbit coupling. For this

cTo ensure Hermiticity we write Hso = ({^>p} x %)' s, where the symbol {•, •} denotes the operator anti-conmutator .

155

C « ^ f t 6 .

* * N > * *'» < * * * . , ,

Fig. 2. Toy model described in the text, a 2DBG with ferromagnetic regions. In our calculations we apply periodic boundary conditions in the transverse y direction. A spin-transfer torque is present when the two magnetization directions are not aligned. The inset shows the Fermi surfaces of the two ferromagnets in which are identical in the absence of spin-orbit coupling and and indicates the transverse channel ky range over which one of the two Schroedinger equation solutions is an evanescent spinor. The Schroedinger equation solutions for electrons incident from x —* — oo can be solved by elementary but tedious calculations in which the spinors and their derivatives are required satisfy appropriate continuity conditions at the interfaces.

model we evaluated mtr(r) in a current-carrying system using the Landauer-Buttiker approach11. In the linear response regime, this requires that the Schroedinger equation be solved at the Fermi energy for all trans­verse channels for electrons incident from the left.

It is helpful at this point to make contact with the usual description of spin-transfer. In its simplest version, spin-transfer theory assumes complete transfer, i.e. that the in­coming current is spin-aligned in the fixed magnet direction and the outgoing current is spin-aligned in the free magnet direction. To the extent that the complete transfer as­sumption is valid, the torque is in the plane defined by the two magnetization orienta­tions, which we refer to as the transfer plane. Microscopically7 the component of the out­going current perpendicular to the transfer plane is expected to be very small because of interference between precessing magneti­zations in different channels.

It follows from Eq.(7) that the spatially

averaged spin orientation of the transport electrons should be approximately perpen­dicular to the transfer plane. It can be veri­fied that this is indeed true by directly eval­uating m^ffi. This simple intuitive argu­ment is not exact, however. In particular, the incoming spin current is not necessarily po­larized along the pinned magnet magnetiza­tion, because of interference between incident and reflected quasiparticle waves that com­plicates the spin-transfer torques and also be­cause it fails to account for electrons that are described by spinors with evanescent compo­nents. (See the inset of Fig. (2)). In any mi­croscopic calculation these effects and others conspire to produce a relatively small com­ponent of the torque that is perpendicular to the transfer plane, and correspondingly to a component of rhir{r) that is in the transfer plane.

In Fig.(3(a)) we plot values of mtr(r) per unit current averaged over the free mag­net space as a function of the angle between the two magnetization orientations, in the

156

§ on

— — W-IOX, W-3»,

— W-2J,

X/A

Fig. 3. (a) Transport spin density per unit current in the case without spin-orbit interaction, m^ is the component perpendicular to the transfer plane ("non-conservative")while mj.r is the smaller component in the transfer plane ("conservative") that is contributed by evanescent spinors. Both components are normalized to the maximum m£ which occurs for 9 = n/2. (b)Non-equilibrium spin density per unit current perpendicular to the transfer plane for different spin-orbit interaction strengths. It follows that from these results that the spin-transfer torque is reduced in efficiently and altered in angle dependence by spin-orbit interactions. (c)Spin transfer efficiency, g, normalized by the ST efficiency in the absence of spin-orbit coupling, as a function of the spin-orbit strength, for several widths of the free magnet. The spin-transfer effect becomes weak when the spin-orbit splitting is comparable with the exchange splitting.

case without spin-orbit interaction. We have taken the free magnet orientation be the y direction and the pinned magnet to be in the x — y plane with polar angle 6.

When spin-orbit interactions are in­cluded, the strength of the spin-transfer torque must be evaluated using the trans­port spin densities. The bookkeeping argu­ment, based on total spin conservation, is no longer valid. The quasiparticle spins not only are no longer conserved due to momentum-dependent effective magnetic fields that rep­resent spin-orbit coupling. As we see in Fig.(3(b)), the spin-transfer effect is not only reduced in magnitude but its dependence on 6 no longer approximates the simple com­plete transfer expression. A measure of how the effect is destroyed by the spin-orbit inter­action is given by the magnitude of the spin

transfer efficiency g, defined as the value of the in plane torque per unit current at the op­timum geometry, 6 = n/2. In Fig.(3(c)] we show the efficiency as a function of the spin orbit interaction strength. We see that when the spin-orbit interaction strength is compa­rable to the exchange spin splitting the effect is strongly reduced except for the case of ex­tremely thin layers.

4. Discussion

We have presented a formalism that allow us to evaluate the interplay between trans­port currents and magnetization dynamics in very general circumstances. This for­malism can address open issues in magneto-transport theory including the possible im­portance of incoherent nanomagnet magneti­zation dynamics in metal spin-transfer phe-

157

nomena, and the influence of the spin-orbit interactions on spin-transfer in diluted mag­netic semiconductor ferromagnets. Our the­ory of spin-transfer is formulated in terms of the change in the effective Hamiltonian that describes all quasiparticles, even ones well away from the Fermi energy, when a conduc­tor is placed in a non-equilibrium state by connecting it to two reservoirs with different chemical potentials. From this point of view, related phenomena occur in nearly any elec­tronic systems, although they will not always lead to experimental effects that are as inter­esting and experimentally robust as the ones that occur in ferromagnetic metals.

The approach to electron-electron inter­action related non-linear transport effects explained in this article has recently been applied10 to quantum Hall bilayers and to circuits that contain antiferromagnetic met­als. In the case of quantum Hall bilay­ers, the collective coordinates of interest are the interlayer phase and population dif­ferences, which play the same role as az-imuthal and polar angles of the magneti­zation in a ferromagnet, pseudospins rather than spins. Bilayer quantum Hall systems have spontaneous interlayer phase coher­ence (pseudospin ferromagnetism) and pseu-dospin transfer torques have been invoked to explain the sudden drop in interlayer conduc­tance with bias voltage seen in experiment22. We anticipate that similar transport effects can occur even in systems that do not have interlayer phase coherence, notably in bilayer electron systems in the absence of an exter­nal magnetic field. In antiferromagnetic met­als circuits, it has recently been predicted10

that large spin-transfer torques appear be­cause of quasiparticle scattering properties related to combined spatial and spin sym­metries. In this case the spin-torques can­not be related to conservation of total spin. Other examples include quantum wells with tilted magnetic fields, in which the Hartree-potential that defines the two-dimensional

transport channel is itelf altered by a bias voltage. In all these effects, the quasiparti­cle band structure can no longer can be re­garded as fixed for a given system. Instead changes in quasiparticle band structure, and non-equilibrium changes in the quasiparticle Hamiltonian density matrix appear as inter­dependent responses to circuit bias voltages.

Acknowledgments

This work was supported by the Welch Foun­dation and by the National Science Foun­dation under grant DMR-0115947. We ac­knowledge helpful interactions with Ger-ritt Bauer, Jack Bass, Joaquin Fernandez-Rossier, Rembert Duine, Paul Haney, and Maxim Tsoi.

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11. R. Landauer, Z. Phys. B 68, 217 (1987); M. Buttiker, IBM J. Res. Dev. 32, 317 (1988); H.U. Baranger and A.D. Stone, Phys. Rev. B 40, 5169 (1989); Supriyo Datta,Electronic Transport in Mesoscopic Systems, (Cam­bridge University Press, 1995). For examples of successful applications of scattering formu­lations to investigate ferromagnetic transport effects see G.E.W. Bauer et al., J. Appl. Phys. 75 6704 (1994) and J. Mathon, J. Phys. D 35, 2437 (2002).

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14. The association between magnetization dy­namics and spatial structure and spin-currents has been discussed previously in a number of contexts. See for example Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403 (2002); Rev. B 67, 140404(R) (2003); Jan Heurich, Jiirgen Konig, and A. H. MacDonald, Phys. Rev. B 68, 064406 (2003) and work cited therein.

15. See for example, W.F. Brown, Micromag-netics, New York, Interscience Publishers, 1963; A. Aharony Introduction to the The­ory of Ferromagnetism (Clarendon Press,

Oxford, 1996); R. Skomski and J.M.D. Coey Permanent Magnetism, (IOP Publish­ing, Bristol, 1999).

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19. Gen Tatara and Hiroshi Kohno, Phys. Rev. Lett. 92, 086601 (2004).

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21. D. Chiba, et al. Phys. Rev. Lett. 93, 216602 (2004) ; M. Yamanouchi, et al. Nature 428, 539 (2004)

22. LB. Spielman et al. Phys. Rev. Lett. 84, 5808 (2000); Phys. Rev. Lett. 87, 036803 (2001).

159

SPIN FILTERS OF SEMICONDUCTOR NANOSTRUCTURES

T. D I E T L *

Institute of Physics, Polish Academy of Sciences and ERATO Semiconductor Spintronics Project, Japan Science and Technology Agency, al. Lotnikow 32/46, PL-02668 Warszawa, Poland; also,

Institute of Theoretical Physics, Warsaw University, Poland

* E-mail: dietl@ifpan.edu.pl

G. G R A B E C K I and J. W R O B E L

Institute of Physics, Polish Academy of Sciences, al. Lotnikow 32/46, PL-02668 Warszawa,

Poland

Our recent works on spin niters built of semiconductor nanostructures are reviewed. It is shown that in appropriately design ballistic wires spin filtering can occur in the tens' millitesla range of the external field. In the case of high mobility electrons at the GaAs/AlGaAs interface, spin-related effects are revealed by applying an in-plane local magnetic field produced by micromagnets of ferrmagnetic metals. The observed effects are attributed to switching between Zeeman and Stern-Gerlach modes - the former dominating at low electron densities. Due to a large spin-orbit coupling in PbTe, which results in \g*\ f» 66, the spin-splitting can become larger than ID quantization energy in rather weak magnetic fields. Evidences for generation of spin polarized current carried by many ID subbands in PbTe nanoconstrictions are described.

Keywords: Spintronics; Nanostructures

1. Introduction

Long spin coherence times in semiconductors have triggered considerable efforts towards developing devices, in which functionalities would involve spin degrees of freedom1. An important building block of such devices is a spin-filter, which could serve for either generating or detecting spin-polarized cur­rents and, indeed, spin filtering capabilities of quantum point contacts2 and quantum dots3 '4 '5 have recently been demonstrated. In those devices, a spin dependent bar­rier occurs as a result of the Zeeman spin-splitting generated by a strong uniform ex­ternal magnetic field. Also the Stern-Gerlach (S-G) effect has been theoretically consid­ered as a possible spin-filter in spin-logic processors6. There are, however, fundamen­tal arguments against the occurrence of the S-G effect for beams of electrons9, a prob­lem that attracts persistent attention10 '11 '12. At the same time, the progress in fabri­

cation of hybrid ferromagnet-semiconductor microstructures7'8 has made it possible to ad­dress various aspects of electron transport in the presence of an inhomogeneous magnetic field13'14'15'16'17'18. For example, the present authors have proposed a way to achieve spin separation by using a Stern-Gerlach appara­tus for the conduction electrons residing in a quantum well and exposed to a gradient of the in-plane magnetic field16.

In this paper, we summarize the outcome of our resent works aiming in developing spin filters of semiconductor nanostructures. In particular, we discuss the effect of a local in-plane magnetic field on ballistic currents in a quantum wire patterned of GaAs/AlGaAs heterostructure19. The results are obtained for a ferromagnet-semiconductor hybrid de­vice which is highly optimized in order to toggle between Zeeman-like (uniform field) and Stern-Gerlach-like (field gradient) inter­nal spin barriers. By comparing our findings to results of conductance computations by

160

the recursive Green-function method, we find out that the Zeeman effect dominates, partic­ularly at low carrier densities. Furthermore, we discuss how rather unique characteristics of PbTe for the spin ballistic transport20, such as a huge value of the static dielec­tric constant, es m 1000, and a large magni­tude of the Lande factor of electrons in PbTe, \g*\ « 66, can be exploited to develop a de­vice generating spin polarized current carried by many ID subbands21. Owing to spin fil­tering and detecting capabilities that occur in a weak external magnetic field, the struc­tures discussed here emerge as a perspective component of spintronic devices.

2. GaAs/AlGaAs-based hybrid structures

Figure 1(a) presents a micrograph of a GaAs/AlGaAs-based device, whose design results from an elaborated optimization process16,19. A two-dimensional electron gas (2DEG) resides 95 nm below the top sur­face of a quantum wire of the geometri­cal width smoothly increasing from 0.7 to 1.4 fira, chemically etched from a modula­tion Si-doped (001) GaAs/(Al,Ga)As het-erostructure, which was grown in the Drude Institute in Berlin. The wire is patterned along the [110] crystal axis, for which the di­rection of a fictitious magnetic field brought about by spin-orbit effects will be parallel to the field gradient and, therefore, will weakly affect spin dynamics22. The electron mobil­ity prior to nanofabrication is fx = 1.76 x 106 cm2/Vs in the dark. Hence, with the electron density n « 2.3 x 1011 cm"2 , the mean free path is of the order of the chan­nel length. A local magnetic field is pro­duced by NiFe (permalloy, Py) and cobalt (Co) films. The micromagnets of dimensions 40 x 7 x 0.1 fim3 reside in 0.15 ± 0.05 /am deep groves on the two sides of the wire, so that the 2DEG is approximately at the cen­ter of the field. To prevent stripe oxidation

and to avoid accumulation of electrostatic charges, both micromagnets are covered with a thin (20 nm) protecting AuPd layer and connected to contact pads. Additional nar­row groove patterned on the wire exit define two regions acting as electron counters. This separating trench is filled with AuPd and also connected electrically. Annealed films of AuGe constitute ohmic contacts between the 2DEG and current leads. The micromagnets are also used as side gates which, together with illumination by infrared radiation, serve for controlling the number of occupied ID subbands.

Fig. 1. (a) Scanning electron micrograph of the spin-filter device. Fixed AC voltage VQ is applied be­tween emitter (E) and "counters" (1), (2); VQ is the DC gate voltage. The external in-plane magnetiz­ing field (B||) is oriented as shown, (b) The in-plane magnetic field By (wider part of the channel is in front) calculated for half-plane, 0.1 fim thick mag­netic films separated by a position dependent gap W(x) and magnetized in the same directions (sat­uration magnetization as for Co). (c)By calculated for antiparallel directions of micromagnet magnetiza­tions, (d) Counter currents I\ and I2 as a function of the gate voltage at Vo = 100 /zV and By = 0; upper curve (shown in gray) was collected during a different thermal cycle and after longer infra-red illumination. After Ref. 19.

Hall magnetometry was applied in or­der to visualize directly the magnetizing process of the two micromagnets in ques­tion. Hall microbridges, patterned of GaAs/(Al,Ga)As:Si heterostructures grown in the Weizmann Institute in Rehovot, con-

161

tain a 2DEG at 47.5 nm below the top sur­face on which Py and Co micromagnets, analogous to those in the spin-filter device, are deposited. It has been found that the Co and Py micromagnets have differing co­ercive fields but similar saturation magneti­zations M s

1 9 . Thus, in the spin filter device, we can compare the electric currents through the counters in the presence of the virtually uniform magnetic field (parallel magnetiza­tion directions) to the case when a strong field gradient is present. The spatial distrib­ution of the magnetic field, evaluated under assumption that the values of Ms correspond to that of Co, [i0Ms = 0.179 T, are presented in Figs. 1(b) and 1(c). We see that depending on the relative magnetization directions the electrons will experience the local magnetic field B up to 0.3 T [Fig. 1(b)] or the local field gradient up to 106 T/m [Fig. 1(c)].

Electron transport measurements for the spin filter device have been carried out in a dilution refrigerator at 100 mK employ­ing a standard low-frequency lock-in tech­nique. According to results presented in Fig. 1(d), conductance plateaux are clearly resolved. Their heights imply that the to­tal transmission coefficient is about 0.7, a value consistent with the presence of the reflecting barrier separating the two coun­ters. Since during these measurements mi­cromagnets are not magnetized, a visible dif­ference in counter currents provides informa­tion about the degree of structure symme­try. What should we expect when the spin-dependent potential barriers are switched on? Classically, the presence of the S-G effect should manifest itself by a gradient-induced symmetric enhancement of the cur­rent through both counters at given emitter-counter and gate voltages.

As shown in Fig. 2, we detect a cur­rent increase in counters when a field gradi­ent is produced by an appropriate cycle of the external magnetic field. The range of magnetic fields where the enhancement is ob­

served corresponds to the the presence of the field gradient according to the Hall magne-tometry, which also shows that Py magneti­zation diminishes almost twofold prior to a change in the direction of the external mag­netic field. This effect, associated with the formation of closure domains in soft magnets, explains why the current changes appear be­fore the field reversal. The relative change A7 of counter current depends on VG, AI/I increases from 0.5% at zero gate voltage to 50% close to the threshold. Furthermore, for VG about —0.8 V AI is negative. It has been checked that results presented in Fig. 2 are unaltered by increasing the temperature up to 200 mK and independent of the magnetic field sweep rate.

< c

~"? 7 T ~ « — ™ ~ flejfj fawn

Va = -0.8V

' i

i •

-0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1

B,| (T) B„ (T)

Fig. 2. The counter current I\ as a, function of the in-plane magnetic field for various gate voltages. Af­ter Ref. 19.

The expected magnitude of AI/I is eval­uated within the model of quantum ballistic transport, which was developed previously22

by employing the recursive Green function method. We note that the key feature of our experimental configuration is a dramatic re­duction of the influence of the Lorentz force by electron confinement. In particular, the effect of the in-plane magnetic field BXtV is much reduced by the interfacial electric field and the corresponding quantization of elec­tron motion in the z direction. Furthermore, since a residual field Bz brought about by misalignment of the magnet centers tends to vanish in the branching region, its influ-

162

ence on electron dynamics will be small23, in agreement with a weak asymmetry of data in Fig. 2. Under these assumptions, elec­tron dynamics is governed by the potential V(x,y) determined by the device geometry, taken in the form corresponding to Fig. 1(a) as well as by the Pauli term, g*sB, where B = [0,By(x,y),0] with By(x,y) displayed in Figs. l(b,c) for both magnetization con­figurations. Because of low density of elec­trons in the quantum wire, we expect a con­siderable enhancement of the electron Lande factor. While the role of many body effects in confined systems is under an active de­bate presently, we take their existence into account by allowing for an enhancement of the Lande factor to the value |<?*| = 2.0.

The computed conductance Go of our model device is shown in Fig. 3(a) as a function of the electrostatic potential bar­rier height. Black and grey lines correspond to the Zeeman-like and Stern-Gerlach-like spin dependent barriers, respectively. As expected, when both micromagnets are po­larized in the same direction an additional spin-resolved narrow plateaux shows up. Re­versing the magnetization of Py pole while leaving the Co pole unaffected corresponds to the transition from the black to the grey curve. As a result, at the transition region between plateaux AGo is either positive or negative. At the quantized plateaux, the con­ductance does not depend on the type of the spin barrier, and the spatial distribution of total current density is only slightly modified by the presence of the field gradient. Under these conditions, however, the spatial sepa­ration of the potential wells for the particu­lar spin directions makes the electric current at opposite edges of the device to be strongly "left" or "right" spin polarized, up to 50% for Go = 1. This indicates that the S-G effect is present under our experimental conditions though it contributes weakly to the current enhancement visible in Figs. 2 and 3.

Vp„ (mV) Vp,,, (mV)

Fig. 3. (a) Go and G calculated for parallel (black) and antiparallel (grey) magnetizations. G is com­puted taking into account the presence of non-zero bias Vb = 100 ^V; G+~ corresponds to the Stern-Gerlach configuration. The hard wall model poten­tial V(x, y) and minimal channel width WQ = 0.5 fim is used. An additional electrostatic potential Vpot is adiabatically introduced to simulate the gate volt­age, separating groove is not modelled and scatter­ing is absent. Conductances are shown as a function of VPot for electron energy E = 2 meV. (b) Rel­ative changes of the conductance AG. Solid line: A G = ( G + - - G++) / (G) , where G++ corresponds to the spin-filter configuration; (G) is the mean value. Dashed line: AG = [G(B(|0) - G(B||0.3 T)]/(G) for the non-magnetic device. After Ref. 19.

It is clear that Go should be averaged over a non-zero energy window correspond­ing to the applied emitter/counter voltage. We defined the observed conductance as G = f1*2 Go(E)dE, where E is the electron energy and /i2 — Mi = e^o- The averaging procedure narrows the plateaux but quite remarkably, the non-zero bias (Vb = 100 //V), which is ~ 3 times larger than the expected spin split­ting (30 /j,eV for g* = 2 ) , does not smear out the changes of the conductance associated with the presence of the Zeeman barrier. Ac­tually, it extends the regions of conductance changes towards the quantized plateaux of Go- We see that the computed magnitude of the effect compares favorably with the exper­imental findings. Except for the immediate vicinity of the pinch-off region, the model de­scribes the magnitude of the current changes, and explains why the sign of the effect can be negative for some values of Va­

in conclusion, the experimental and the­oretical studies discussed here demonstrate that semiconductor nanostructures of the

163

kind shown in Fig. 1 can serve to generate and detect spin polarized currents in the ab­sence of an external magnetic field. More­over, the degree and direction of spin polar­ization at low electron densities can easily be manipulated by gate voltage or a weak external magnetic field. While the results of our computations suggest that the spin sep­aration and thus Stern-Gerlach effect occurs under experimental conditions in question, its direct experimental observation would re­quire incorporation of spatially resolved spin detection.

3. PbTe-based nanoconstr ict ions

The multilayers used for fabrication of nanoconstrictions by electron beam lithog­raphy and wet etching consisted of PbTe quantum wells embedded by Bi-doped Pbo.92Euo.08Te barriers grown by molecular beam epitaxy onto [111] BaF2 at Kepler Uni­versity in Linz24. For this particular Eu com­position, the barrier is as high as ~ 240 meV. Furthermore, the fourfold L-valley degener­acy of the conduction band in PbTe is lifted, so that the relevant ground-state 2D subband is formed of a single valley with the long axis parallel to the [111] growth direction. As discussed recently20, the paraelectric charac­ter of PbTe results in efficient screening of Coulomb scattering potentials, so that sig­natures of ballistic transport can be observed despite of significant amount of charged de­fects in the vicinity of the channel. At the same time, the electron density can be tuned over a wide range by biasing a p-n junc­tion that is formed between the p + interfacial layer and the n-type quantum well20.

As shown in Fig. 5, a precise conduc­tance quantization is observed for nanocon­strictions patterned of the multilayers with various width of the quantum well. In the case of the widest quantum well, the channel cross section assumes a quasi-circular cross-section, so that ID excited subbands are

orbitally degenerate25 and an untypical se­quence of the steps is visible, G = 2ie2/h, where i = 1,3,6,.... Furthermore, a rather large magnitude of electron spin splitting for the magnetic field along the growth direction, corresponding to the Lande factor \g * | as 66, can be exploited to develop an efficient spin aligner. According to results displayed in Fig. 6, spin-degeneracy of the steps starts to be removed well below 1 T, and it has become possible to generate spin current carried by a number of ID subbands21.

RELATIVE GATE VOLTAGE V -V,h [V]

Fig. 4. Conductance quantization in PbTe nano­constrictions patterned of PbTe quantum wells of various thickness. Missing conductance steps indi­cate orbital degeneracy of ID states for circular shape of the wire. Inset shows AFM image of sample con­taining several nanoconstrictions of different widths. After Ref. 20.

In conclusion, from the results presented in this section it appears clearly that hy­brid structures containing ballistic wire of PbTe and ferromagnetic micromagnets may show strong filtering capabilities in the ab­sence of an external magnetic field. Further­more, according to our preliminary studies of PbTe/In system, this heterostructure ex­hibits excellent characteristics of Andreev re­flection indicating that is can be possible to integrate both spin filter and spin detector in one structure.

164

6.

1 2 3 4

MAGNETIC FIELD [T]

Fig. 5. Transconductance dG/dVg (gray scale) showing dependence of ID subbands on the mag­netic field and gate voltage in PbTe nanocon­striction as a function of magnetic field and gate voltage for PbTe nanoconstriction of a wide (Pb,Eu)Te/PbTe/(Pb,Eu)Te quantum well. After Ref. 21.

A c k n o w l e d g m e n t s

We thank K. Prone, E. Kaminska, A.

Lusakowski, and A. Piotrowska in Warsaw,

G. Bauer and G. Springholz in Linz, R. Hey

and K. H. Ploog in Berlin as well H. Shtrik-

man in Rehovot for long staying and fruitful

collaboration. This work was supported in

par t by KBN grant (PBZ-044/P03/2001).

R e f e r e n c e s

1. S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger, Science 294, 1488 (2001); T. Dietl, Acta Phys. Polon. A 100 (suppl.), 139 (2001); H. Ohno, F. Matsukura and Y. Ohno, JSAP International 5, 4 (2002); I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).

2. R.M. Potok et al, Phys. Rev. Lett. 89, 266602 (2002).

10

11

12

13

14

15

16 17

18 19

20.

21. 22.

23.

24.

25.

J.A. Folk et al, Science 299, 679 (2003). M. Ciorga et al, Appl. Phys. Lett. 80, 2177 (2002). R. Hanson et al., cond-mat/0311414, un­published. C.H.W. Barnes, J.M. Shilton, and A.M. Robinson, Phys. Rev. B 62, 8410 (2000). M. Johnson et al., Appl. Phys. Lett. 71 , 974 (1997). T. Matsuyama et al., Phys. Rev. B 65, 155322 (2002). N. F. Mott, Proc. Roy. Soc. London A 124, 425 (1929). H. Batelaan, T. J. Gay, and J. J. Schwendi-mann, Phys. Rev. Lett. 79, 4517 (1997). M. Garraway and S. Stenholm, Phys. Rev. A 60, 63 (1999). G. A. Gallup, H. Batelaanm, and T. J. Gay, Phys. Rev. Lett. 86, 4508 (2001). A. Nogaret, S. J. Bending, and M. Henini, Phys. Rev. Lett. 84, 2231 (2000). D.N. Lawton et al., Phys. Rev. B 64, 033312 (2001). S. M. Badalyan and F.M. Peeters, Phys. Rev. B 64, 155303 (2001). J. Wrobel et al., Phys. E 10, 91 (2001). J. Fabian and S. Das Sarma, Phys. Rev. B 66, 024436 (2001). G. Grabecki et al, Physica E 21, 451 (2004). J. Wrobel et al, Phys. Rev. Lett. 93, 246601 (2004). G. Grabecki et al, Phys. Rev. B 72, 125332 (2005); G. Grabecki et al, Physica E, in print. G. Grabecki et al, Physica E 13, 649 (2002). A. Lusakowski, J. Wrobel, and T. Dietl, Phys. Rev B 68, 081201(R) (2003). C.W.J. Beenakker and H. van Houten, Phys. Rev. Lett. 63, 1857 (1989). A.Y. Ueta, G. Springholz, G. Schinagl, G. Marschner, and G. Bauer, Thin Sol. Films 306, 320 (1997). A.G. Scherbakov, E.N. Bogachek, and U. Landman, Phys. Rev. B 53, 4054 (1996).

165

EXPERIMENTAL STUDY ON CURRENT-DRIVEN DOMAIN WALL MOTION

T. ONO\ A. YAMAGUCHI, H. TANIGAWA, K. YANO and S. KASAI

Institute for Chemical Research, Kyoto University, Uji, 611-0011, Japan E-mail: ono@scl. kyoto-u. ac.jp

Current-driven domain wall (DW) motion for a well-defined single DW in a micro-fabricated magnetic wire with submicron width was investigated by real-space observation with magnetic force microscopy. Magnetic force microscopy visualizes that a single DW introduced in a wire is displaced back and forth by positive and negative pulsed-current, respectively. Effect of the Joule heating, reduction of the threshold current density by shape control, and magnetic ratchet effect are also presented.

Keywords'- current-driven domain wall motion," spin-transfer; spintronics, magnetic force microscopy.

1. Introduction

Manipulation of the magnetization by an electric current is a key technology for future spintronics. Underlying physics is that a spin-polarized current can apply a torque on the magnetic moment through direct transfer of spin angular momenta of conduction electrons to localized spins when the spin direction of the conduction electrons has a relative angle to the local magnetic moment.

In general, ferromagnets are composed of magnetic domains, within each of which magnetic moments align. The directions of magnetization of neighboring domains are not parallel. As a result, there is a magnetic domain wall (DW) between neighboring domains. The direction of moments gradually changes in a DW as illustrated in Fig. 1(a). Here, arrows show the direction of magnetic moments. What will happen if an electric current flows through a DW? Suppose a conduction electron passes though the DW from left to right. During this travel, the spin of conduction electron follows the direction of local magnetic moments because of the s-d interaction (Fig. 1(b)). As a reaction, the local magnetic moments rotate reversely (Fig. 1(c)), and, in consequence, the electric current can displace the DW.1'2

This current-driven DW motion provides a new strategy to manipulate a magnetic configuration without the assistance of a

magnetic field, and will improve drastically the performance and functions of recently proposed spintronic devices, whose operation is based on the motion of a magnetic DW.3,4

Reports on this subject have been increasing in recent years from both theoretical5"11 and experimental12"23 points of view because of its scientific and technological importance. In the following, the result of direct observation of the current-driven DW motion in a micro-fabricated magnetic wire is presented.16,22'23

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DW

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(b)

(c)

Fig. 1 Schematic illustration of the current-driven domain wall motion; (a) a domain wall between two domains in a magnetic wire. The arrows show the direction of magnetic moments, (b) The spin of conduction electron follows the direction of local magnetic moments because of the s-d interaction, (c) As a reaction, the local magnetic moments rotate reversely, and in consequence, the electric current displaces the domain wall.

166

Magnetic force microscopy (MFM) is used to show that a single DW can be displaced back and forth by positive and negative pulsed-currents.

2. Experiments

Samples were fabricated onto thermally oxidized Si substrates by means of an e-beam lithography and a lift-off method. MFM observations were performed at room temperature. CoPtCr low moment probes were used in order to minimize the influence of the stray field from the probe on the DW in the wire.

3. Results and discussion

3.1. Real-space observation of the current-driven DW motion by MFM

An L-shaped magnetic wire with a round comer was used for the experiments so that a single DW can be introduced into the wire. One end of the L-shaped magnetic wire was connected to a diamond-shaped pad, which acted as a DW injector24' and the other end was sharply pointed to prevent a nucleation of a DW from this end.25 The width and the height of the wire were 240 run and 10 nm, respectively.

Figures 2(a)-2(k) are successive MFM images with one pulsed-current applied between each consecutive image. The current density and the pulse duration were 6.7 * 10uA/m2 and 0.5 us, respectively. The DW is imaged as a dark contrast, which corresponds to the stray field from negative magnetic charge. Thus, in this case, a tail-to-tail DW is realized. Each pulse displaces the DW opposite to the current direction. The average displacement per one pulse did not depend on the polarity of the pulsed-current. The average DW displacement was proportional to the pulse duration, which indicates that the DW had a constant velocity during the application of the pulsed-current. The average DW velocity of 3 m/s was

obtained for the current density of 6.7 x 10 A/m2. It was found that the DW velocity increased with the current density. It was confirmed that the head-to-head DW are also displaced opposite to the current direction. The fact that both head-to-head and tail-to-tail DWs were displaced opposite to the current direction

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Fig. 2 Successive MFM images with one pulse applied between each consecutive image. The current density and the pulse duration were 6.7 * 10n A/m2 and 0.5 Us, respectively. Note that a tail-to-tail domain wall is introduced, which is imaged as a dark contrast.

167

indicates clearly that the DW motion was not caused by the magnetic field generated by the pulsed-current.

3.2. Effect of the Joule heating

The current-driven DW motion makes it possible to switch the magnetic configuration without an external magnetic field, and is of great importance for future spintronic devices. However, the current density required is relatively high, of the order of 10u A/m2, and it is necessary to investigate the effect of the Joule heating for practical applications.

We estimated the sample temperature by

4000

3000

2000

1000

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Fig. 3 (a) Resistance of the sample during the application of the pulsed-current as a function of the current density, (b) Standard temperaturedependence of the sample resistance with constant current 1 uA.

comparing me sample resistance during the

application of a pulsed-current with the independently measured temperature dependence of the sample resistance. The sample investigated in the above paragraph was used in the experiments.

Figure 3(a) shows the sample resistance during the application of a pulsed-current, Rpu\se, as a function of current density. Rpuise

increased with me current density, and Rpuise for the threshold current density was almost twice the resistance at room temperature. This result suggested the high current density caused considerable Joule heating of the sample.

The temperature dependence of the resistance measured with constant current of 1 uA is shown in Fig. 3(b). Above 300 K, the resistance was almost proportional to the temperature. We crudely extrapolated the linear temperature dependence of the resistance between 300 K and 400 K into higher temperatures. The dashed line is the extrapolated temperature dependence of the resistance.

By comparing the result of Fig. 3(a) with that of Fig. 3(b), we found that the sample temperature was about 750 K for the threshold current density of 6.7 x lfj11 A/m2. The temperature was raised to 830 K for the current density of 7.5 x 1011 A/m2, which is very close to the Curie temperature of bulk Ni8iFei9 (850 K). When the current density exceeded 7.5 * 1011 A/m2, an appearance of the multi-domain structure in the wire was observed. These experimental results suggested the sample temperature exceeded the Curie temperature of Ni8]Fej9 when the current density exceeded about 7.5 x 10HA/m2.

3.3. Reduction of the threshold current density by shape control

As described in the above paragraph, the high threshold current density, Jc, inevitably results in the severe sample heating. Reduction of Jc is required from both fundamental study and practical applications.

168

It has been suggested theoretically that Jc is proportional to the product of the hard-axis magnetic anisotropy, K±, and the DW width, k, in the case of a thick DW and a weak pinning.5

Here, the thick DW means that the thickness of the DW is much larger than the Fermi wavelength of conduction electrons which is satisfied in usual ferromagnetic metals, and the weak pinning means that a pinning potential for a DW is much smaller than KJa, where a is the Gilbert damping factor. Thus, the theory predicts that Jc can be decreased by reducing K±. For samples of Ni8iFei9, K± can be controlled by the sample shape, because the crystal magnetic anisotropy of Ni8iFei9 is negligibly small and the magnetic anisotropy is dominated by the magnetostatic energy.

We fabricated samples of Ni8iFe19 with different cross-sectional shape by varying the width, w, and the height, h, of the wires; (w, h) = (110 nm, 64 ran), (110 nm, 73 ran), (110 nm, 78 nm), (110 nm, 113 nm), (240 nm, 10 nm), and (240 nm, 25 nm). Jc was determined by the MFM direct observation.

As shown in Fig. 4, Jc decreased with an increase of the aspect ratio, hlw. This is

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qualitatively consistent with the theoretical prediction, since the magnetic anisotropy of the

sample decreases with increasing the aspect ratio. We succeeded in decreasing Jc by a factor of 2 by the shape control.

3.4. Magnetic ratchet effect

In a previous report26, we have shown that the depinning field of the DW from the asymmetric notch artificially introduced into a submicron magnetic wire depends on the propagation direction of the DW, and that the asymmetric notch works as an asymmetric potential barrier against the propagation of a magnetic DW. This phenomenon could be called a "magnetic ratchet effect". A similar effect was found in the experiment on submicron magnetic wires with a triangular structure using a magneto-optical Kerr effect.27 We examined whether the magnetic ratchet effect can be produced not only by a magnetic field but also by an electric current through the wire.

An arched Ni8iFe19 wire (25 nm in thickness) with asymmetric notches was fabricated as shown in Fig. 5(a).

After the introduction of a DW in the wire (Fig. 5(b)), a pulsed-current with the duration of 2 us was applied through the wire from left to right. The MFM observation was carried out after applying each pulse with increasing the intensity of the pulsed current. It was confirmed that the threshold current density for the current-driven DW motion was 7.2 x 1011

A/m2 for the leftward propagation. Figures 5(c)-5(e) are successive MFM images with one pulsed-current of 7.2 * 1011 A/m2 between each consecutive image. Each pulsed-current displaced the DW notch by notch to the leftward direction.

We tried to determine the threshold current density for the rightward propagation by switching the current polarity. However, the DW did not move rightward up to the current density of 8.9 * 1011 A/m2. Above this current density, an appearance of the multi-domain structure was observed (Fig. 5(f)), indicating that the sample temperature

169

exceeded the Curie temperature of Ni8iFei9 by the Joule heating because of the high current density. Therefore, we found a clear difference of the threshold current density between the leftward propagation and the rightward propagation. The DW moved more easily in the direction along which the slope of the asymmetric notch was less inclined. This situation is the same as in the case of the DW motion in magnetic wires with asymmetric notches by an external

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motion but also for the current-driven DW motion.

4. Summary

Results of experimental study of the current-driven DW motion by means of the magnetic force microscopy were presented. It was demonstrated that the DW position in the wire can be controlled by tuning the intensity, the duration and the polarity of the pulsed-current. The relatively high threshold current density inevitably results in the severe sample heating because of the Joule heating. It was found that the threshold current density can be reduced by shape control. The magnetic ratchet effect in the current-driven DW motion as demonstrated for the magnetic wire with asymmetric notches.

Acknowledgments

The present work was partly supported by MEXT Grants-in-Aid for Scientific Research in Priority Areas, JSPS Grants-in-Aid for Scientific Research, and Industrial Technology Research Grant Program in '05 from NEDO of Japan.

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Fig. 5 (a) AFM image of the arched wire with asymmetric notches, (b) MFM image of the initial magnetization configuration, A tail-to-tail DW was observed, which was imaged as a dark contrast. (c)-(e) Successive MFM images with one pulsed current applied between each consecutive image. The pulsed current with the duration of 2 us flowed through the wire from left to right. The applied current density was 7.2 x 10" A/m2. Each pulsed current displaced the DW notch by notch, (f) MFM images after the application of a pulsed current with the current density of 8.9 x 10" A/m2 in the opposite direction. Multidomain structure appeared in the wire.

magnetic field.26 Thus, it is concluded that the asymmetric notch works as an asymmetric potential not only for the field-driven DW

References

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Phys. Rev. Lett., 87,026601 (2001).. 5. G Tatara and H. Kohno, Phys. Rev. Lett. 92,

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Suzuki, Europhys. Lett. 69, 990 (2005). U .S . E. Barnes and S. Maekawa, Phys. Rev.

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12. J. Grollier, P. Boulenc, V. Cros, A. Hamzic, A. Vaures, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509 (2003).

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14. M. Klaui, C. A. F. Vaz, J. A. C. Bland, W. Wemsdorfer, G Fani, E. Cambril ,and L. J. Heyderman, Appl. Phys. Lett., 83, 105 (2003).

15. T. Kimura, Y. Otani, K. Tsukagoshi ,and Y. Aoyagi, J. Appl. Phys., 94, 07947 (2003).

16. A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett, 92, 077205 (2004).

17. C. K. Lim, T. Devolder, C. Chappert, J. Grollier, V. Cros, A. Vaures, A. Fert, and G Faini, Appl. Phys. Lett., 84, 2820 (2004).

18. N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke ,and R. P. Cowbura, Europhys. Lett, 65, 526 (2004).

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20. E. Saitoh, H. Miyajima, T. Yamaoka ,and G Tatara, Nature, 432, 203 (2004).

21. M. Klaui, C. A. F. Vaz, J. A. C. Bland, W. Wemsdorfer, G Fani, E. Cambril, L. J. Heyderman, F. Nolting, and U. Rudiger, Phys. Rev. Lett., 94, 106601 (2005).

22. A.Yamaguchi, S. Nasu, H. Tanigawa, T. Ono, K. Miyake, K. Mibu, and T. Shinjo, Appl. Phys. Lett., 86, 012511 (2005).

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171

MAGNETIZATION REVERSAL OF FERROMAGNETIC NANO-DOT BY NON LOCAL SPIN INJECTION

Y. Otani* and T. Kimura Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8581, Japan

Frontier Research System {FRS) RIKEN.2-1 Hirosawa, Wako Saitama 351-0198, Japan E-mail: votani&issp.u-tokvo.ac.ip

We here demonstrate non-local spin injection into a nano-scale ferromagnetic particle configured in a lateral spin valve structure to switch its magnetization without charge current. The non-local spin injection is found to align the magnetization of the nano-magnctic particle along the magnetization of the spin injector. The magnitude of the spin current for reversal is determined to be about 158 uA, which is reasonable compared with the estimated value of the spin current required for switching free layer in conventional nano-pillar structures.

Keywords: magnetization reversal; spin injection; spin accumulation.

1. Introduction

The current induced magnetic switching is a key phenomenon for future spin electronic devices including a next generation magnetic random access memory. The switching mechanism proposed by Slonc/cwski shows that the torque exerted on the magnetization is proportional to the injected spin current [1]. This means that the spin current is essential for the magnetization switching by the spin injection. Most of the spin-transfer devices so far investigated consist of vertical multilayered nanopillars in which typically two magnetic layers are separated by a nonmagnetic metal layer [2-4]. In such vertical structures, the charge current always flows together with the spin current, thereby undesirable Joule heat is generated.

Our recent experiments have demonstrated that the spin currents are effectively absorbed into an additionally connected metallic wire with a small spin resistance [5, 6J. This implies that the spin current without a charge flow can be selectively injected into a ferromagnetic particle with a small spin resistance such as a Permalloy particle, replaced with the wire. This spin current may contribute to the magnetization reversal. To test this idea, a nano-scale ferromagnetic

particle is configured for a lateral non-local spin injection device as in Figs. 1(a) and (b).

(a)F2 AA /si i/c

( b ) F 2

-4 F1

^

A

Is; ! " Ic

^

Fig. 1 Schematic illustrations of (a) local spin injection and (b) non-local spin injection, (c) SEM image of the fabricated lateral spin injection device and (d) magnified image, around the Py

172

2. Experimental Procedures

Multi-terminal spin injection device used in this study was fabricated by means of conventional e-beam lithography followed by lift-off techniques. Figures 1 (c) and (d) show the scanning electron microscope (SEM) images of a fabricated device. The device consists of a large Permalloy (Py) pad 30 nm in thickness, a Cu cross 100 nm in width 80 nm in thickness, and a Py nano-scale particle 50 nm in width, 180 nm in length and 6 nm in thickness. A gold wire 100 nm in width 40 nm in thickness is connected lo the Py parliclc to minimize the effective spin resistance to obtain high spin current absorption into the Py particle [5].

The magnetic field is applied along the easy axis of the Py particle. Note that the dimensions of Py pad and Cu wires arc set large enough so that the charge current up to 15 mA can flow through them. Py layer is grown by using an electron beam evaporator with a base pressure of 2 x 109 Torr. The Cu and Au wires are evaporated by a resistance heating evaporator with a base pressure of 3 x 108 Torr. The interface between the Py and Cu and that between the Py and Au are well cleaned by Ar-ion milling prior to each deposition. Very low resistance of the interface assures a good ohmic contact. The distance between the Py pad and the particle is 400 nm. The resistivities of Py and Cu wires are respectively 10.2 uiicm and 1.14 uTicm at 77 K. All the measurements are performed at 77 K by means of conventional lock-in technique.

3. Results and Discussion

Before showing the experimental results, we discuss spin-current absorption into the electrically floating additional wire. We have demonstrated that the spin current distribution can be calculated by the model base on the spin-resistance circuit and that the spin current favors to flow in the subsection which has small spin resistance [5, 6]. The spin resistance is given by 2p lA1/(l-«1

2jS', , where pt, X,, al and S, are respectively the

resistivity, spin diffusion length, spin polarization and the cross section of the wire. For example, the spin resistance Rs for the Cu wire with the cross section S, of 100 nm x 80 nm can be calculated as 2.85 ft We took the value of 1 //m for the spin diffusion length of Cu wire obtained in our previous experiments [7]. The effective cross section for the spin resistance of the Py particle in Fig. 1(d) is given by the junction area between the Py particle and the Cu wire because the spin diffusion length of the Py is very short. Therefore, we obtain the spin resistance Rs

y

for the Py particle as 0.08 ft. Here, we use the spin polarization aPy =0.2 and spin diffusion length XPy = 2 nm obtained in our experiments [7]. Important is that Ry is about 2 orders of magnitude smaller than Rg" . Thus, the spin current favors to be absorbed into the Py particle although the particle is electrically floating. Such absorption can be employed as a method to inject the spin current non-locally.

To begin with, the non-local spin valve (NLSV) measurements are performed to

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Fig. 2 (a) Non-local spin valve signal together with the schematic probe configuration. The solid and dotted lines correspond to positive and negative field sweeps, respectively, (b) The NLSV signal after the pulsed current injection as a function of the current amplitude with the corresponding magnetization configurations.

173

understand the spin accumulation behaviors in our devices. Figure 2(a) shows the NLSV signal with the inset of the measurement probe configuration. We can see a clear spin signal of 0.18 mfi. The resistance changes at low and high fields correspond to the relative magnetic switching of the Py pad and particle, from parallel (P) to anti-parallel (AP) states and vice versa. These results prove that the spin current from the Py pad is absorbed into the Py particle. It should be noted that the measured spin signal is smaller than the value expected from our previous experiments [6-8]. As will be mentioned later, this is because the Py particle used in the present study has smaller spin resistance than the values estimated from our lateral spin-valve experiments.

Then, we examine the effect of the non-local spin injection into the Py particle with using the same probe configuration. Before performing the non-local spin injection, the magnetization configuration is set in the AP configuration by controlling the external magnetic field. The non-local spin injection is performed by applying large pulsed currents up to 15 mA with the same current probes for the NLSV measurement in the absence of magnetic field. Note that the current pulse is triangular shape with the duration of Is. After the non-local spin injection, the NLSV signal is successively measured to determine the magnetic state of the Py particle.

In this way, as shown in Fig. 2(b), the NLSV signal after the non-local spin injection as a function of the amplitude of the pulsed current is obtained. When the magnitude of the pulsed current is increased positively in the AP state, no signal change is observed up to 15 mA. On the other hand, for the negative scan, the abrupt signal change is observed at -14 mA. The change in resistance at -14 mA is 0.18 imQ, corresponding to that of the transition from AP to P states. The magnetization direction of the Py particle is

confirmed to be parallel to the Py pad by sweeping the magnetic field with measuring the NLSV signal. After the transition from the AP to P states, the current is positively increased in the P state. However, we observe no signal change even though the amplitude of the pulsed current was increased up to 15 mA.

In this device, there are 2 equivalent AP states where the magnetization of the Py particle directs towards either left or right in Fig. 2 (b). Both AP states are transformed into the P state in the same manner. We can exclude as follows a possibility that the magnetization of the Py particle is switched by the current-induced Oersted field. In the probe configuration for the non-local spin injection, the charge current passes through the Cu cross and induces the Oersted field. However, the induced field is normal to the substrate and thus does not switch the magnetization of the Py particle since the demagnetizing field of nearly 1 T is far bigger than the Oersted field.

The similar measurement is performed with a different probe configuration to study an influence of spin current distribution. We also obtain a clear spin signal of 0.19 mQ, slightly larger than that in the previous configuration in Fig. 2 because of inhomogeneous spin current distribution. Then, the similar spin injection measurements are performed. Similarly the clear transition from the AP to P states is observed whereas the reverse P to AP transition is not observed. For this experiment, the switching occurs at -13.3 mA slightly smaller than that of the previous configuration (Fig. 2 (a)). This is because the larger spin accumulation at the interface induces the larger spin current than in the previous configuration. In this probe configuration, the distribution of the Oersted field is different from the previous experiment. No remarkable difference is observed between the two experiments. This supports that the observed AP to P transition is not originated by the Oersted field. Note here that for both cases the

174

Oersted field exerted normal to the substrate causes a deviation from the collinear magnetic configuration between Py pad and particle [11], and this may assist the magnetization switching of the Py particle due to the spin torque, resulting in the reduction of the switching current.

We estimate the magnitude of injected spin current into the Py particle in the AP state When the electrons are injected from the Py pad into the Cu wire (negative current), the Cu wire is magnetized in parallel to the Py pad due to the spin accumulation. The continuity of the chemical potential at the interface also brings about the spin splitting in the Py particle, as shown in Figs. 3(a), (b) and (c). The spin-dependent chemical potential of the Py particle in anti-prallel to the Py pad is given by [10, 12]

M\ = M,

Mi=Mt

aPy 1 + aPy Xp;

aPy * ~ aPy X?

(1)

(2)

Defining the spin current as Is = / t -I± =-(SPycr<t/e)(d/Lbt/dx) +(SPyai/e)(djui_/dx) yields the injected (absorbed) total spin current into the Py particle:

Is = {{-apyjjpyS

2eA, Mt =

•Py

Mi

eRP/ (3)

This means that in the AP states the spin current along the Py pad is injected into the Py particle. This discussion also stands for the P configuration as in Fig. 3(c). Therefore, the spin current induced by non-local spin injection with the negative current injection aligns the magnetization of the Py particle along the Py pad. The magnitude of the injected spin current can be deduced from the

intensity of the spin signal by using Eq. (3). The relation between the induced spin splitting in the chemical potential //, at the interface and the obtained spin signal Rmsv of the NLSV measurement is given by Mi =eicRMjSV/aPy . Here, ic is the applied charge current for the measurement. Therefore, when we inject a pulsed current with amplitude of /^p, the injected spin current I'gJ into the Py particle can be calculated as

J'nj _ Mi RNLSV'amp s^\

"Py*?

As mentioned above, the spin resistance of the Py particle is 0.08 Q. When we use the parameters determined in previous experiments [7], we obtain the critical injected spin current as 158 |oA for 1^ = 14 mA.

The value can be compared with the estimated spin current for conventional nano-pillar structures consisting of Py nano-elliptical particles configured in CPP structures with the dimensions similar to the present Py nano-particle [13, 14]. The estimated spin current contribution for switching the free Py nano-particle from the AP to P states in the nano-pillars is about 200 DA, comparable to our present experiment.

As mentioned above, the obtained spin signal is however smaller than that in the NLSV experiment with the same injector-detector distance. We believe that this is caused by the lower quality of the Py particle than the Py element in our previous lateral spin-valve experiments. In the Py particle, the effect of the surface oxidation is more pronounced than the conventional devices because of the small sample dimensions. Such effects reduce the spin polarization and the spin diffusion length in the Py particle. This causes the reduction of the spin resistance and thus lowers the estimation of the spin splitting voltage at the interface. The real injected spin

175

Fig. 3 Schematic illustrations of (a) the induced chemical potential with the negative current injection in the anti-parallel configuration and (b) that with the positive current injection in the parallel configuration, (c) Spin-dependent chemical potential inside the Py particle in parallel (anti-parallel) to the Py pad in the negative current. The direction of the injected spin current depends on the polarity of the current and does not depend on the magnetization configuration between the Py pad and the particle.

current may thus be larger than the above calculation.

Similar analysis for the positive current reveals that the spin current with the same magnitude and the opposite polarity is injected into the Py particle as in Fig. 3(b). The spin current induced by non-local spin injection with the positive current should lead the Py particle magnetization into the AP state. This transition however is not observed in the present experiment.

Although the concrete reason has not been clarified yet, conceivable explanations for this discrepancy may be as follows. One is a bias dependent spin polarization of the Py pad. In general, the spin polarization should be independent of the current passing through the interface in the ohmic junction. However, it is not true once very thin oxide layer is formed at the interface between the Py pad and Cu cross during fabrication process. Note that our device is exposed to the air during the process. Such an oxidized layer may provide an asymmetric barrier especially at high current density. Therefore, we have to take into

account an asymmetric spin injection into or out of the Py pad.

When the electron is injected from the Cu wire into the Py pad, the spin polarization drastically reduces compared to the zero bias value with increasing the applied bias voltage [15], thus diminishing the injected spin current into the Py particle. On the contrary, the spin polarization exhibits only small reduction when the electron is injected from the Py pad into the Cu wire. In this way, our asymmetric reversal of the Py nano-particle can be explained.

Another possibility is tiny magnetic impurity in the Cu wire near the interface. The spin diffusion length of the Cu wire with the magnetic impurity is known to depend on the angle between the direction of the impurity magnetic moment and that of the injected spin [16]. When the direction of the injected spin is anti-parallel to the moment of the magnetic impurity, the spin diffusion length is shorter than that at the parallel alignment because of the reorientation of the magnetic moment of the conduction electron spin to the direction of that of the magnetic impurity. The magnetic impurity in the Cu wire may be parallel to the magnetization of the Py pad because of the exchange interaction. In this case, when the electron is injected from the Cu wire to the Py pad (corresponding to the positive current), the spin diffusion length is shorter than that for the negative current. This asymmetric transport also explains our experimental results.

4. Conclusions

In conclusion, we have fabricated the multi-terminal lateral spin injection devices configured for the non-local spin-injection-induced magnetization switching of the Py particle. We succeeded in switching the magnetization of the Py particle from the AP to the P states by non-local spin injection although we could not realize the P to AP

176

switching. The conceivable reasons for this asymmetry are discussed. The value of the switching spin-current obtained from the experiment was reasonable compared with the values estimated from the conventional pillar devices. In order to flip the magnetization of the Py particle in both directions, further optimization of the device structure is required.

References

1. J. C. Slonczewski, J. Magn. Magn. Mater. 159 (1996) LI

2. J. Grollier, V. Cros, A. Hamzic, J. M. George, H. Jaffr'es, A. Fert, G Faini, J. B. Youssef and H. Legall: Appl. Phys. Lett. 78 (2001) 3663.

3. F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman Phys. Rev. Lett. 89 (2002)226802

4. F. J. Albert, J. A. Katine, R. A. Buhrman and D. C. Ralph: Appl. Phys. Lett. 77 (2000) 3809.

5. T. Kimura, J. Hamrle, and Y. Otani, Phys. Rev. B 72 (2005) 014461

6. T. Kimura, J. Hamrle, Y Otani, et ai, Appl. Phys. Lett. 85 (2004) 3795

7. T. Kimura, J. Hamrle, Y Otani, in preparation

8. M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55 (1985) 1790

9. F.J. Jedema, A.T. Filip, and B.J. van Wees, Nature (London) 410 (2001) 345; F. J. Jedema, M. S. Nijboer, A. T. Filip and B. J. van Wees, Phys. Rev. B 67 (2003) 085319

10. S. Takahashi and S. Maekawa, Phys. Rev. B 67 (2003) 052409

11. B. Ozyilmaz, A. D. Kent, D. Monsma, J. Z. Sun, M. J. Rooks, and R. H. Koch Phys. Rev. Lett. 91 (2003) 067203

12. P. C. van Son, H. van Kempen, and P. Wyder, Phys. Rev. Lett. 58 (1987) 2271

13. I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, R. A. Buhrman, Science 307 (2005) 228

14. S. Urazhdin, Norman O. Birge, W P. Pratt, Jr., and J. Bass Phys. Rev. Lett. 91 (2003) 146803

15. S. O. Valenzuela, D. J. Monsma, C. M.

Marcus, V. Narayanamurti, and M. Tinkham Phys. Rev. Lett. 94 (2005) 196601

16. C. Heide, Phys. Rev. B 65 (2002) 054401

177

THEORY OF CURRENT-DRIVEN DOMAIN WALL D Y N A M I C S

G. TATARA*

PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan and

Graduate School of Science, Tokyo Metropolitan University 1-1 Minamiosawa, Hachioji, Tokyo 192-0397, Japan

* E-mail: tatara@phys. metro-u. ac.jp

H. KOHNO

Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

J. SHIBATA

Frontier Research System (FRS), The Institute of Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

E. SAITOH

Department of Physics, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Motion of a planar domain wall in a nanoscale ferromagnetic wire under electric current is studied based on microscopic description. In the adiabatic case, domain wall is driven by spin torque (spin transfer), and there is an intrinsic pinning arising from hard-axis anisotropy energy. Extrinsic pinning does not affect the threshold current in this case. A resonating oscillation of domain wall occurs under AC current, and this was recently used for a spectroscopy of a single "domain wall particle". Nucleation of domain walls by spin torque is discussed. Fundamental mechanism of spin transfer effect is understood in terms of spin Josephson effect.

Keywords: Spin torque; Spintronics; Domain wall.

1. Introduction

The integration of magnetic and electric properties is highly important in the present technology context, since most of the mag­netic devices are nowadays based on the mag-netoresistive effect arising from modification of the electron transport by the magnetic configuration. In particular, the manipula­tion of magnetization by use of an electric current1'2 is promising for the next genera­tion magnetic memories, where the informa­tion is written by an electric current. Switch­ing by use of domain wall motion has been carried out intensively recently3'4'5'6'7. The­oretically, current-driven motion of a domain wall was studied in a series of pioneering works by Berger 8>9'10. Here we reformulate

the problem from microscopic point of view, and explore some new features such as intrin­sic pinning11 '12 '13 and resonance of domain wall7.

2. Effective Hamiltonian under current

We start from a microscopic Hamilto­nian with an exchange interaction be­tween conduction electrons (denoted by two-component operator c = (c+,c-), ± being the spin index) and local spins, S,

# e x = - A f dxS • ( c W ) x , (1)

A being a coupling constant. We consider a case where local spin texture is slowly vary­ing compared with kp1, the inverse of Fermi

178

wavelength, sometimes called the adiabatic limit. In this case, a gauge transform in the spin space is useful. The transformation is defined by c(x,t) = U(x,t)a(x,t), where a is the new electron operator and U — m • <r, m = (sin | cos 4>, sin | sin </>, cos | ) , where (6, <f>) are polar coordinates of local spin di­rection. Because of this transformation, a gauge field, A^ = —iU~1dflU, appears. For a slowly varying texture we consider, A^ can be treated perturbatively. The lowest contri­bution of the current is represented by the following effective Hamiltonian,

# ( a d ) = j< j s - V 0 ( l - c o s 0 ) . (2)

Here j s = j + —jB_ is the spin current density defined as the difference of the current car­ried by electron with spin + and —, and thus in the adiabatic limit, current affects the lo­cal spin only if the current is spin-polarized. The total Lagrangian of the local spin under current is written as

j . -v L(ad) =I**™[(*-3±P)+Yi ^xhs

-Hs

COS0)

(3)

where the first term is a spin Berry phase, which gorvens the spin dynamics, and Hs is a spin Hamiltonian we specify below. It is seen that spin current favors spin texture flow­ing along its direction. This effect is inter­preted as exchange of spin angular momen­tum between the conduction electron and lo­cal spin and so is called spin-transfer effect. The equation of motion derived from this Lagrangian is a modified Landau-Lifshitz-Gilbert equation,

damping term, assuming damping is domi­nated by non-electron contribution.)

The role of spin transfer effect, Eq. (2), can be understood in terms of the effective field it creates. In fact, the field is obtained as B

( a d ) = ^ ^ - hsm6{eejs • V<f> -e<f,js • W ) , indicating that field perpendicu­lar to the spatial variation of local spin arises, thereby translating the spin texture.

3. Equation of motion of domain wall

So far we considered general spin texture which is adiabatic. We go on now with a par­ticular case of domain wall. The spin Hamil­tonian we consider is a general one with an easy axis and a hard axis (chosen as z and y axis, respectively), the energy denoted by —K and K±, respectively. We consider a wire with width less than domain wall thick­ness, A = ^/J/K, J being the exchange cou­pling between local spins. Static domain wall is a solution with cos 6 = tanh ^ ^ ^ and 4> = 0, where X is a constant. We as­sume K± <C K, in which case the low en­ergy dynamics of domain wall is well desribed by its position X(t) and average polariza­tion <j>o(t) (defined as <f>o = j Jdxsin2 &</>), both being now dynamical variable (collec­tive variable)15 '16 '11. From Eq. (4), in the absence of magnetic field and pinning, the equation of motion of adiabatic domain wall is derived in terms of X and <f>o as1 1 , 1 2 '1 3

U>Q + aj^\ = 0 and

h(dt + 2eS

(4) where Bes = and we added a damping term (Gilbert damping) by use of phenomenolog-ical parameter a. (It was shown recently14

that the electron contribution to damping is not represented under spin current by the last term of Eq. (4). Here we use standard

X — a\(po = vc sin 20o + ve, (5)

where ve = ^ %- j s is a drift velocity of elec­tron spin and where vc = K±XS/(2H).

4. Correction from non-adiabaticity

Due to finite wall thickness, non-adiabatic correction such as reflection of the electron exists in reality. This effect (called momen-

179

turn transfer effect) was shown11 to be repre­sented by a force Fe = eni?wJ.A,where i?w is the resistance due to a wall, J is the current, A is the crosssectional area of wire and n is the density of electron. This force was shown to affect the first equation of motion to be

X

^ + aj) = Ws (Fe + Fext), (6)

where F e x t is external force due to pinning and magnetic field. Non adiabaticity affects also the spin transfer (the last term of Eq. (5))11. In most cases, adiabatic approxima­tion applies quite well as long as the current is DC or long pulse. The exception is the dynamics under AC current discussed below.

5. Examples of wall dynamics

Here we show two typical examples of wall dynamics.

5.1. Intrinsic pinning under DC spin current

In the case of steady spin current, there exists a threshold value for wall motion11,

,cr(l) = ^1K. X Js a3h -1 (7)

Below this threshold, j s < js , the spin torque from spin current can be absorbed by the transverse magnetization, <fi, and so the wall does not move. Domain wall starts to move when cj> > w/2 (i.e., j s > js '). The time-averaged velocity of wall as function of spin current is plotted in Fig. 1). Very im­portant feature of spin torque-induced mo­tion is that the dynamics is quite insensi­tive to extrinsic pinning such as impurities as long as the impurities are spinless11. This has been observed in experiments and would be a great advantage in device application.

Fig. 1. Time-averaged wall velocity as a function of spin current, j s .

5.2. Resonant motion of "domain wall particle"

AC-current can drive domain walls quite ef­fectively if the frequency is tuned to be close to the resonance with the pinning frequency. This resonance was realized in recent exper­iment by Saitoh et af. They applied a small AC current in a wire with a domain wall in a weak pinning potential. The amplitude is well below the threshold, but the wall can shift slightly (for about a distance of /xm). Equation of motion (5) indicates that <j> re­mains small under small current, and as is easily seen, the wall then satisfies a simple equation of motion of a "particle";

mX + —X + mfl2X = f(t), (8) T

where m = h2N/(K±X2) is the wall mass, r ~ H/(aKj_) is a damping time, Ct is the pinning frequency, and f(t) ~ enAIRw — ih2uIs/(eKj_X) (Is = jsA) is a generalized force due to current 7 '17. By measuring the energy dissipation (complex resistance), a resonance peak was observed when the fre­quency of the current, w, is tuned to be Q. From the resonance spectra, mass, friction constant, and resistivity of a single domain wall were identified to be m = 6.6 x 10~23kg, T = 1.4 x 10 -8sec (corresponding to a ~ 0.01), Rw = 3 x 10_4fi. What is more, driv­ing mechanism of domain wall turned out to be the momentum transfer force, which is in contrast to the motion under DC current.

180

This is quite surprisng considering a thick­ness of wall of ~ lOOnm, but is due to strong enhancement of momentum transfer force by resonance (spin torque is, in contrast, sup­pressed in MHz range)7.

Domain wall, which is a soliton in fer-romagnet, has been theoretically known to behave like a particle. This experiment is the first one to measure physical properties of this soliton "particle".

5.3. Themally assisted motion

At finite temperature, threshold current dis­cussed above disappears due to thermally as­sisted motion. We here show that a charac­teristic universal behavior is predicted in this regime. We consider a domain wall at posi­tion X(t) in the presence of a pinning poten­tial of harmonic type

Vpin(X) = ±Mwn2(X2-\2)6(\-\X\), (9)

where Mw = £ ^ is the wall mass, A the wall thickness, fi the pinning frequency (which serves as an attempt frequency), and 0(x) the step function. In the case of adia-batic wall and small current, the wall motion is described in terms of <j>o moving in a tilted washboard potential (Fig. 2)18. Estimating

where non-linear terms in Is are neglected and C{T) = - ^ f ^ + In i s indepen­dent of 7S. What is notable here is an univer­sal dependence on Is (i.e., it does not involve material parameters) At low enough temper­ature, e%

IsT » 1, the expression reduces to a

standard thermal activation type,

lnv~lJIl-+c'(T), 2 e * f l r ' — ( 1 1 )

where C'(T) = C(T) - In 2. This universal behavior in the thermally activated regime would be used to identify the driving mech­anism (i.e., spin torque or momentum trans­fer).

6. Domain nucleation by spin torque

The role of spin torque represented by Eq. (2)19 '20 '21 indicates that spin current favors spatial variation of magnetization. Thus the energy of domain wall can be lowered under spin current. In fact, the enegy of domain wall is estimated to be21

2"

Edw ~ NKS2 1 I. ha" WS*]

\yKK± (12)

Fig. 2. Potential for <f>0 under spin current.

Hence, for j s > j sc r ( 2 ) = 2 e S 2 ^ P ? l A , domain

wall has a lower energy than uniformly fer­romagnetic state, resulting in domain forma­tion from single domain state. The result is more rigorously confirmed by calculating spin wave gap21. This domain formation by current would be useful in device applica­tions.

the potential height for <f>0, the velocity of (f>o induced by thermal fluctuation is ob­tained, which is related to wall velocity by (f>o — - a j , after depinning. From these con­siderations, the Logarithm of wall velocity is given by18

Ms C(T), (10) l n t ; • : In sinh — 2ekBT

7. More on spin torque

7.1. Spin torque and Berry phase

Let us here look more into the mechanism of spin torque from the fundamental viewpoint. The role of spin torque was represented by the Hamiltonian, (2). It is seen that this term is a spatial version of "the spin Berry phase", and thus the role of spin current is

181

to induce spatial spin Berry phase along the current. The spin torque has a particularly interesting consequence when the sample ge­ometry is topologically non-trivial as in a ring. In fact, if we apply a spin current in a ferromagnetic ring, an umbrella-like struc­ture of magnetization is favored rather than uniformly ferromagnetized state. The angle of tilt and resultant Berry phase is deter­mined by the spin current.

7.2. Spin Josephson effect

Here we can pose an opposite question. Does a spontaneous current arise if we prepare a ring with umbrella structure? The answer is yes, as shown in the adiabatic limit by Loss et al22. Physics here would be more eas­ily grasped if we think perturbatively23,24,25. When a conduction electron in a conduc­tor is scattered by local spin S, the elec­tron wave function is multiplied by an am­plitude A(S) = AS • a, which is generally spin-dependent and is represented by Pauli matrices. (S's are treated as classical con­stant vectors.)

A(S2)\ ss/ \ /

A(S,)

Fig. 3. A closed path contributing to the amplitude of the electron propagation from x to x. At X{, the electron experiences a scattering represented by an SU(2) amplitude, A(Si). The contribution from one path (left) and the reversed one (right) are different in general due to the non-commutativity of A(Si)'s.

Let us consider two successive scatter­ing events represented by A(S\) and A(S2) (Fig.3). Due to the non-commutativity of ai, the amplitude depends on the or­der of the scattering event; A(Si)A(S2) ^ A(S2)A(Si) in general. Various features in spin transport, which is under inten­

sive pursuit recently, arise from this non-commutativity. It, however, does not af­fect the charge transport, since the charge is given as a sum of the two spin compo­nents (denoted by tr), and tv[A(S\)A{S2)} -tr[A(S2)A(Si)] = 0. Anomaly in the charge transport arises at the third order. We have

tt[A(S1)A(S2)A(S3) - A(S3)A(S2)A(S1)}

= 4 A 3 S 1 - ( S 2 x S 3 ) . (13)

This equation indicates that in the pres­ence of fixed Si's with finite spin chirality, Si • (S2 x S3), the contribution from one path, x—>X\—>X2—>Xa—>x (Fig.3, left), and its (time-) reversed one, x—>X$—*X2—*X\—>x (Fig.3, right), are not equal. This difference results in a spontaneous electron motion in a direction specified by the sign of spin chi­rality, namely, a permanent current if the electron is coherent. We thus see that the essence of spin torque represented by Eq. (2) is the non-commutativity of the SU(2) alge­bra, purely quantum mechanical. The cur­rent induced by this spin non-commutativity can be said as a Josephson effect of spin. The application to quantum logic gate was dis­cussed in Ref. 24.

8. Anomalous Hall effect

The phenomenon predicted above will be present in metallic frustrated spin systems such as pyrochlore ferromagnets and spin glasses, where finite spin chirality is often realized. The spin chirality was recently pointed out 26 , in the adiabatic limit, to be the origin of the peculiar anomalous Hall ef­fect observed in experiments 27. The spin Josephson effect affords an intuitive interpre­tation to it. The circulating current starts to drift when the electric field is applied, in the direction perpendicular to the electric field (Fig. 4), just as in the normal Hall effect. The frequency of the circulating motion is

182

estimated as

n=^(£)V(s a xs s ) , d4) we may estimate the Hall conductivity by

crxy = (JQSIT. Here <TO is the classical (Boltz-

mann) conductivity, r is the elastic lifetime,

and the dirty case QT -C 1 is assumed. If

the spin chirality is located uniformly on ev­

ery triangle of size of inter-atomic distance

(i.e., S i • (S2 x 5 3 ) = xo and L ~ l/kF), we

have crXy/cr0 ~ XOJ3TI€F- This result agrees

with the result based on the linear response

theory 2 8 . The Hall conductivity was also

shown to be wri t ten by a winding number in

real space2 9 .

Fig. 4. Schematic picture of the chirality-induced Hall effect. Circulating permanent currents drift un­der applied electric field. The crosses denote impu­rity scattering.

A c k n o w l e d g m e n t s

We thank A. Yamaguchi, M. Yamanouchi,

T. Ono, H. Ohno, Y. Otani , H. Miyajima, H.

Fukuyama, S. Maekawa, J. Ferre for valuable

discussion.

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183

MAGNETIC IMPURITY STATES A N D FERROMAGNETIC INTERACTION IN DILUTED MAGNETIC SEMICONDUCTORS

M. ICHIMURA*-1'2, K. TANIKAWA1, S. TAKAHASHI1, G. BASKARAN1-3, and S. MAEKAWA1'4

Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan * E-mail: ichimura@imr. tohoku. ac.jp

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama, 350-0395, Japan

Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India

CREST, Japan Science and Technology Agency, Kawaguchi, Saitama, 332-0012, Japan

The electronic structure of diluted magnetic semiconductors is studied, especially focusing on the hole character. The Haldane-Anderson model is extend to a magnetic impurity, and is analyzed in the Hartree-Fock approximation. Due to the strong hybridization of an impurity d-states with host p-states, it is shown that holes are created in the valence band, at the same time the localized magnetic moments are formed. These holes are weakly bound around the impurity site with the characteristic length of several lattice constants. For the case of the two impurities, it is found that when the separation of two impurities is of the order of the length of holes, the overlap of these holes favors the ferromagnetic interaction. The spatially extended holes play an essential role for the ferromagnetic exchange interaction.

Keywords: Diluted Magnetic Semiconductors; Haldane-Anderson Model; Hartree-Fock Approxi­mation.

1. Introduction

The spin dependent transport has attracted great interest from the technological point of view as well as the physical point of view x. For the requirement to control and manipu­late the spin transport by external perturba­tions, including applied electric and/or mag­netic fields and optical excitation, the diluted magnetic semiconductors (DMS) are consid­ered to be the most important candidates for such promising properties. However, we have not been able to identify the DMS with Curie temperature (Tc) above room temperature, and there exists great discrepancy among the experiments for mechanism of ferromagnetic ground state in DMS.

In various types of DMS, (III,Mn)V fam­ily, (Ga,Mn)As 2 has been extensively stud­ied both from the experimental and theo­retical sides. Matsukura et al. have shown

that the carrier in (Ga,Mn)As is p-type holes by Hall effect 3. These holes are considered to play an important role for the ferromag-netism in (Ga,Mn)As. Ohno et al. have re­alized to control the ferromagnetic proper­ties such as Tc

4 and coercive field (Hc) 5

by the control of hole density in (Ga,Mn)As with the field-effect transistor (FET) de­vice. These results have made it clear that (Ga,Mn)As is hole-mediated ferromagnet.

There are many experimental studies for hole states in (Ga,Mn)As. The Mn 2p core level photoemission for (Ga,Mn) with the configuration interaction analysis has sug­gested that a Mn ion is the neutral (Mn)+ 3

with d5 + lhole configuration and that holes are localized at the Mn site 6. On the other hand, the valence band photoemission exper­iment has shown that holes are occupied in the host valence band (VB) just below the

184

Fermi energy (Ep) 7. The experiment of an­gle resolved photoemission spectra (ARPES) have shown the existence of impurity band below Ep without dispersion in the direc­tion along k = (001) 8 . The infra-red (IR) absorption experiments have shown the hop­ping conduction due to the localized holes, and spectral transfer from interband to in-traband 9. This is interpreted that the local­ized holes do not provide the RKKY inter­action between magnetic moments, and that the spectral transfer is caused by the dou­ble exchange (DE) interaction. The X-ray absorption spectral (XAS) experiment has shown that the holes are mainly in As Ap or­bital 10. There exists disagreement over the interpretation of the hole character, localized or delocalized.

2. Model description

We start with the Haldane-Anderson model, which has treated the charge state of the im­purity d-state in semiconductor hosts, and has succeeded to describe the multiple charge state n . Here, we extend the Haldane-Anderson model to a magnetic impurity, and simplify the model with no orbital degen­eracy. The degenerate-orbital case is dis­cussed in the following section. A standard Hartree-Fock (HF) approximation applied to the Haldane-Andersonas leads to

n*F = E Edifn° + E ekciacka k<?

+ y E( c U + fc-t)> a) ka

where

K" =Ed + U(n-„). (2)

Here na = d\d„, and c'k(7 and d^ are the cre­ation operators for the p-state of host semi­conductor energy band and for d-state at im­purity site, respectively. We assume that in the host band structure, the band widths of the VB and conduction band (CB) is semi-infinite and the density of states (DOS) of

both bands are constant (po), since the in­terested energy region is near the CB bottom (ec) and VB top (ev). U and V represent the onsite Coulomb energy and the hybridization between d- and p-states, respectively. It is convenient to define A = 7r|V|2/»o- E^J is the effective energy level of d-state. In this work, we restrict ourselves to the symmetric case, Ed — —{U — A G ) / 2 . A G is the gap in the host energy band.

The HF single particle spectrum is ob­tained as shown schematically in Fig. 1. It is shown analytically that the Friedel sum rule is satisfied between virtual bound states (VBS) and p-hole, and between d- and p-components in the split-off state. The num­ber of states in VBS is equal to the one in p-hole state. The sum of the d- and p-components in the split-off state is always 1. This means that the split-off state carries the unit charge e and spin 1/2. These two sum rules are convenient to consider the spatial distribution of p-hole spin.

In the situation of Fig. 1, we assume the deep d-level in VB, i.e. negatively large Ed- Under this assumption, when the split-off state is empty, the electron configuration is dl + lhole. When we set the split-off state to be singly occupied, the d1 electron config­uration is realized.

By using the above relation, we can es-

r-' •• L_—u_'-_i

%

%''

©N \

_. .<™>

&,,v^ »'.f •J

K :. ii)

Fig. 1. The schematic picture of the single parti­cle spectrum for up-spin. (i) DOS of VBS due to p-d hybridization, (ii) DOS of host p-band. The DOS de­viates from rectangular shape, corresponding to the formation of VBS. This deviation is regarded as the p-hole with down-spin, (iii) Near the VB top in the gap, the split-off state appears at u>p, which mainly consists of the host p-state.

185

timate the spatial distributions of the charge and spin for the cases of d1 + lhole and d1. For d1 + lhole, the charge and spin dis­tributions are determined by p-hole, since the d-state is well localized like 6-function. We obtained that the distributions of p-holes spread over several lattice constants, then the induced magnetic moment in host is coupled antiferromagnetically with the im­purity spin. For d1, the distribution of p-hole is cancelled by the occupation of the split-off state. The p-component density of the split-off state is calculated analytically as ~ poAexp(-2r/gp)/(r/£p), where fp = 1/\]2m*wv and m* is the effective mass of host. From this result, for d1 + lhole, it is understood that the p-hole is weakly bound around the impurity spin with the spreading of several lattice constants.

3. Result and Discussion

Here, we extend the above result to the im­purity treatment in (Ga,Mn)As. For this purpose, the three degenerate d-orbitals are considered as impurity states since the tetra-hedral crystal field splits the d-orbitals into eg and t^g orbitals and the lower energy eg orbitals with two-hold degeneracy form non-bonding states. We assume these im­purity rf-levels deep enough in the host VB band. Then three p-holes and three split-off states are formed. The substitution of Ga 3 +

ion by Mn3 + can be considered to occupy two electrons in t2g orbitals, and this corre­sponds to the doubly occupied split-off states in our model. We can interpret this state as d5 + lhole configuration.

The single impurity case with three de­generate orbitals is analyzed in HF approx­imation with the Coulomb and exchange J terms on an equal footing. The spin density of the p-holes as a function of distance from the impurity site is plotted in the inset of Fig. 2. The spin density distribution in neg­ative sign in Fig. 2 represents the antiferro-

magnetic couling with impurity spin. Here, £G = l/y/2m*AG which is about 0.6nm in GaAs. Figure 2 also indicates the spin den­sity of p-hole for various occupation number of the split-off states (np). The spin den­sity of p-hole for np = 0,6 is about three times of that for np = 2,4. This is because the split-off states for various np are nearly degenerate for large A due to the multiple charge state by the Haldane-Anderson sce­nario, then these split-off states give the al­most same spreading.

Based on the spin density distribution of p-holes around the impurity, the exchange in­teraction mediated by the holes is evaluated using the Caroli's formula 12, which was orig­inally used to analyze the exchange interac­tion of two impurities in metallic host. We extend this formula to semiconductor host. The exchange energy (Eex) defined as the energy difference between ferromagnetic and antiferromagnetic configurations of two im­purity spins are shown in Fig. 2 as a function of distance between two impurities. Since the spreading of ferromagnetic coupling over sev­eral lattice constants shows well agreement with that of p-holes, these holes plays an es­sential role for the ferromagnetic exchange interaction.

Figure 2 also shows the dependence of the exchange energy on np. Since np can

Fig. 2. The exchange energy as a function of dis­tance between two impurities for various occupation number of the split-off states (np). Inset: The spin density distribution of p-hole as a function of the dis­tance from the impurity site.

186

be interpreted as the tig occupation as dis­

cussed above, the material dependence of the

exchange energy is examined for various £2g

occupation. Under the strong p-d hybridiza­

tion and deep impurity states in the host VB

band, the states with occupations np = 1,2,

and 3 can be considered as hosts Ge, GaAs,

and ZnSe, respectively. The comparison be­

tween these Mn doped semiconductors are

shown in Fig. 3 as a function of impurity

concentration. Since Tc is proportional to

the absolute value of exchange energy within

H F approximation, (Ge,Mn) is expected to

have a higher Tc due to a large amplitude of

p-hole spin density around the impurity site.

On the other hand, (Zn,Mn)Se cannot be ex­

pected higher Tc due to no induced p-hole

only by intrinsic doping. This material de­

pendence of the exchange energy is roughly

similar to the results for the specific doping

rate by first-principles electronic structure

calculation done for the periodic supercells 1 3

and nano-crystals 1 4 . However, these first-

principles results 1 3 '1 4 have shown the anti-

ferromagnetic ground s ta te for (Zn,Mn)Se.

4 . S u m m a r y

To summarize this work, the magnetic im­

puri ty states and magnetic interaction be­

tween d-impurities are studied using ex­

tended Haldane-Anderson model in DMS.

Due to the strong hybridization between d-

states and p-states in the host, p-holes cou-

;> 1)

0.04

0.03

0.02

0.01

0

Ed=-7.5eV U=3.0eV J= 2.3eV A=2.1eV

— / • • » - - — . — , -

(GeJVlnX-----

(Ga,Mn)As .

-(Zn,Mn)Se

I . I .

4 6 x[%]

10

Fig. 3. The exchange energy as a function of impu­rity concentration for various host materials.

pie antiferromagnetically to impurity spins

and the split-off s tates in the gap are formed.

When the occupation of the split-off states is

different from the number of p-holes, the spin

density of p-holes spreads over several lattice

constants around the impurity site. When

two impurities are so close to overlap their

spin density, ferromagnetic interaction is fa­

vorable. The weakly bound holes around the

impurity site plays an essential role for the

ferromagnetic exchange interaction.

A c k n o w l e d g m e n t s

The par t of this work is supported by

CREST, IFCAM, and MEXT. One of the

authors (M.I) thanks the NAREGI project

of M E X T Japan.

R e f e r e n c e s

1. Spin Dependent Transport in Magnetic Nanostructures , edited by S. Maekawa and T. Shinjo (Taylor and Francis, London and New York, 2002).

2. H. Ohno, et al., Appl. Phys. Lett. 69, 363 (1996).

3. F. Matsukura, Y. Ohno and H. Ohno, Phys. Rev. B57, R2037 (1998).

4. H. Ohno, et al., Nature 408, 944 (2000). 5. D. Chiba, M. Yamanouchi, F. Matsukura,

and H. Ohno, Science 301, 943 (2003). 6. J. Okabayashi, et al., Phys. Rev. B58, R4211

(1998). 7. J. Okabayashi, et al., Phys. Rev. B59, R2486

(1999). 8. J. Okabayashi, et al., Phys. Rev. B64, 125304

(2001). 9. K. Hirakawa, et al., Phys. Rev. B65, 193312

(2002). 10. Y. Ishikawa, et al., Phys. Rev. B65, 233201

(2002). 11. F. D. M. Haldane and P. W. Anderson,

Phys. Rev. B13, 2553 (1976). 12. B. Caroli, J. Phys. Chem. Solids 28, 1427

(1967). 13. T. C. Schulthess and W. H. Butler, J. Appl.

Phys. 89, 7021 (2001). 14. X. Huang, et al., Phys. Rev. Lett 94, 236801

(2005).

187

GEOMETRICAL EFFECT ON SPIN C U R R E N T IN MAGNETIC NANO-STRUCTURES

M. ICHIMURA*'1'2 S. TAKAHASHI1 , and S. MAEKAWA1

Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan * E-mail: ichimura@imr. tohoku. ac.jp

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama, 350-0395, Japan

The spatial distribution of spin accumulation and spin current in the non-local geometry is studied taking into account the finite size effect of the contact areas by applying a finite element method in two-dimension. It is shown that , when the contacts are tunneling-like, the sign and the value of spin accumulation signal are consistent with the experimental results, and the one-dimensional analysis is appropriate to analyze the experimental results.

Keywords: Spin Accumulation; Non-local Geometry; Finite Element Method.

1. Introduction

Much attention has been focused on spin-dependent transport in magnetic nano-structures 1. In order to manipulate the spin degree of freedom of an electron, it is necessary to inject and accumulate the spin efficiently. The central issues of recent in­terest are to create nonequilibrium spin cur­rent and spin accumulation using the con­tact between ferromagnetic and nonmagnetic metals. In recent experiments in the geome­try of non-local measurement, spin accumu­lation has been observed at room tempera­ture 2. Subsequently, the spin accumulation is efficiently improved with using tunneling contacts 3 . The spin accumulation signal in the non-local geometry has been theoret­ically evaluated based on a one-dimensional model 4, where the large spin accumulation signals with highly resistive contacts are ob­tained and a strong enhancement of spin ac­cumulation signal is suggested using a super­conductor.

2. Model

We consider a spin injection and detection device as shown in Fig. 1. This device con­sists of a nonmagnetic metal (N) connected

to two ferromagnetic metals (Fl and F2), where F l is a spin injector and F2 is a de­tector. In this work, we study the spatial distribution of spin current taking account of the finite size of the contact area by apply­ing a finite element method (FEM) in two-dimension.

For this purpose it is necessary to charac­terize the contact regions, in addition to each metallic region. The spin-dependent current in the metallic regions has been formulated using the spin-dependent electrochemical po­tential (ECP) Us (s =T, J. ) 4 as follows,

V2(crfMT + c r ^ ) = 0 , (1)

V 2 ( / / T - m) = A~2(/iT - / i A ) , (2)

with the spin dependent conductivity ers, and the spin-diffusion length A = ^/TSJD, where

Fig. 1. Schematic view of a spin injection and de­tection device. The charge current I flows from one edge of F l to the left edge of N.

188

rsf and D denote the spin relaxation time and the effective diffusion constant, respec­tively 4. The spin-dependent current is de­rived as j s = — (as/e)Vps. From this, the charge and spin currents are defined as j j + Ji and j j —Ji, respectively. Here, we do not dis­tinguish the normal from ferromagnetic met­als, since these equations are satisfied solely in each metal. By introducing the contact re­sistivity pCon in units of fim2 5, we can define the length scale characterizing the current distribution on a contact in a ferromagnetic side i2

sF = d(-F">cr(f)pcs°

n, where pfn is the spin-dependent contact resistivity expressed as pc°n = 2pcon/(l + sPj), SF"> the thick­ness of a ferromagnetic metal, Pj the con­tact current spin polarization (SP) and set to be 0.4 6. Then, we obtain the following equations for ECP at the contact in a ferro­magnetic side;

-af^(^-^)=0, (3)

V2(MP-MP)

— l^F I ^ | — p ^ I

-^( /4F ) - /T) - A - ( ^ - ^ ) = 0 . (4)

We note that the second and third terms in Eqs.(3) and (4) are the contribution from the current across the contact, and represent the coupling of the spin and charge currents in the contact region. The equations for ECP at the contact region of the non-magnetic side are obtained by exchanging F and N in Eqs.(3) and (4).

For applying FEM, we set the width of each metal as follows; wp\ = 400nm, Wf2 = 200nm, WN = 250nm as described

in Ref. 3. The spin-diffusion lengths are 50nm and 650nm in Co and Al, respec­tively 3. For establishing a sufficient decay of the spin current at the edge of each lead, we set the length of ferromagnetic and non­magnetic leads to be 400nm and 2400nm, re­spectively. The length of the normal metal between Fl and F2 is assumed to be the same as the spin diffusion length of Al. In this work, we take the following parameter val­ues, pcon = 0.66Clpm2, £-\N = 440nm, ^JV = 714nm, t^F = 372nm, and tiF = 254nm. We note that the values of ISF and (,SN are comparable to the spin-diffusion length of Al, and that the value of pcon is hundred times smaller than the experimental value 3. The boundary conditions are set as follows; the charge-current density is 5mA//xm2 and —8mA//im2 at the edge of F l lead and the left edge of N lead, respectively. These val­ues correspond to 100mA in the experimen­tal condition 3. At the other edges of the nonmagnetic and ferromagnetic metals, the charge-current density is set to be zero. The mesh size is lOnm. When the magnetizations in F l and F2 are parallel (P), the equations for ECP are represented in a symmetric ma­trix. Then, we can apply a standard conju­gate gradient (CG) method. With the anti-parallel (AP) case, the matrix is not symmet­ric, and the Bi-CG method is applied.

3. Resul ts

Figure 2 shows the spin and charge currents in F l , where the contact region is indicated by the dotted line, when the magnetization of F l and F2 are P. The charge current in­jected at the edge of F l lead flows across the contact to the N lead. The corresponding spin current whose bulk SP is taken to be p =0.7 7 8 flows to the N metal. By com­paring the spatial distribution of spin and charge currents, the effect of spin-diffusion is clearly observed. The spin current flowing across the contact changes gradually relative

189

to the charge current. Figure 3 shows the spin and charge cur­

rents in N, where the two contact regions are indicated by dotted lines, when the magne­tization of F l and F2 are P. The charge cur­rent injected from the contact with F l flows to the left edge of N lead, and no charge cur­rent flows in the other part because of the charge conservation. The spin current in­jected from the contact with F l flows in N to both sides of the contact, and decays with the spin-diffusion length outside the contact. The spin current flowing to the right side is modulated by the contact with F2 since F2 plays a role of sink and some of the spin cur­rent flows from N into F2. The rest of the spin current passes over the contact to the right edge of N.

Figure 4 shows the spin and charge cur­rents in F2, when the magnetization of F l and F2 are P. The spin current injected from N into F2 via the contact induces the split­ting of spin-dependent ECP in the contact re­gion of F2, and this splitting becomes smaller with the spin-diffusion length of F2 away from the contact region. The corresponding spin current quickly decays outside the con­tact.

Since the spin dependence of ECP no longer exists at the edges, we define the spin accumulation signal as ^SF2^ — JJS^ in the

geometry in Fig. 1, where / / F 2 ' is the value of ECP at the lower edge of F2 and n{N) is the right edge of N. The value of spin accu­mulation signal in P is 7.3/xeV. The sign of the spin accumulation signal agrees with the experiment, and the order of magnitude is consistent with the experimental value 3.

4. Discussion

As mentioned above, the characteristic length of the current distribution on the con­tact is comparable to the dimensions of the contact area. Then we obtain slightly two-dimensional inhomogeneous distribution of the spin and charge currents in the contact

-2

•A

-6

-8

-

charge -spin

i

-24 -16 -8 0 8 16 24 32 x(*100nm)

Fig. 3. The spin and charge currents in N as a func­tion of x coordinate. The vertical dotted lines indi­cate the two contact regions.

_ 5

I' •=- 3

so ™

S 2

o 1

Jo

•

•

,

Y\

charge

•

0 2 y(*100nm)

10

1 - -10

•

\ charge spin

. . i . . . i

-4 -2 0 2 y(*100nm)

Fig. 2. The spin and charge currents in F l as a function of y coordinate. The vertical dotted lines indicate the contact region.

Fig. 4. The spin and charge currents in F2 as a function of y coordinate. The vertical dotted lines indicate the contact region.

190

region. As the contact resistivity becomes smaller, the characteristic length of contact reduces smaller than the contact dimension, and the spatial dependence should become relevant.

We have seen in Fig. 2 that the spin cur­rent flowing across the contact changes grad­ually relative to the charge current. This gradual behavior is also seen in the charge curent in the spin accumulation region. The charge and spin currents in the spin accu­mulation region shown in Fig. 2 are enlarged in Fig. 5. We can see the finite charge cur­rent around x =4 (400nm) at the end of the spin accumulation region closed to the junc­tion between F l and N. This is caused by the slow decaying of spin current with AJV-

5. Summary

To summarize this work, we study the spa­tial distribution of the spin current in the non-local geometry taking into account the finite size effect of the contact area by apply­ing finite element method in two-dimension. It is shown that the sign and the order of magnitude of the spin accumulation signal are consistent with the experiment 3. The global results show that, when the contacts are tunneling-like, the two-dimensional in-homogeneous current distribution is slightly

x(*100nm)

Fig. 5. The enlarged charge and spin currents in the spin accumulation region shown in Fig. 2. The up-and down-componets of spin current are also shown.

observed in contact area, and the one-dimensional analysis 4 is applicable to evalu­ate the experimental results.

Acknowledgments

This work was supported by NAREGI Nanoscience Project, Ministry of Education, Culture, Sports, Science and Technology, Japan.

References

1. Spin Dependent Transport in Magnetic Nanostructures , edited by S. Maekawa and T. Shinjo (Taylor and Francis, London and New York, 2002).

2. F. J. Jedema, A. T. Filip, and B. J. Van Wees, Nature 410, 345 (2001).

3. F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Baselmans, and B. J. Van Wees, Nature 416, 713 (2002).

4. S. Takahashi and S. Maekawa, Phys. Rev. B67, 052409 (2003).

5. R. J. M. van de Veerdonk, J. Nowak, R. Meservey, J. S. Moodera, and W. J. M. de Jonge, Appl. Phys. Lett. 71, 2839 (1997).

6. R. J. Soulen Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny, T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J. Nowak, J. S. Moodera, A. Barry, and J. M. D. Coey, Science 282, 85 (1998).

7. S. D. Steenwyk, S. Y. Hsu, R. Loloee, J. Bass, and W. P. Pratt, J. Magn. Magn. Mater. 170, LI (1997).

8. S. Dubois, L. Piraux, J. M. George, K. Ounadjela, J. L. Duvail, and A. Fert, Phys. Rev. B60, 477 (1999).

191

FERROMAGNETISM IN ANATASE TiOz CODOPED WITH Co AND Nb

T. fflTOSUGI*, T. SHMADA and T. HASEGAWA

Department of Chemistry, University of Tokyo, Tokyo 113-0033, Japan Kanagawa Academy of Science and Technology (KAST), Kawasaki 213-0012, Japan

E-mail: hitosugi@chem.s. u-tokyo.ac.jp

G. KINODA, K. INABA, Y. YAMAMOTO, Y. FURUBAYASHI and Y. FflROSE

Kanagawa Academy of Science and Technology (KAST), Kawasaki 213-0012, Japan

Magneto-optical Faraday rotation angle of ferromagnetic anatase Co-doped Ti02 films is enhanced and can be controlled by Nb doping. A carrier density of films can be varied by Nb doping in this room-temperature ferromagnetic degenerate semiconductor. A Ti0 spNbo CMCCD 05O2 film showed a Faraday rotation angle of 1.5^ 104

deg/cm at 400 ran. An anomalous Hall eflfect, showing that the charge carriers are spin polarized, is observed in this film.

Keywords: diluted magnetic semiconductor, oxides, ferromagnetism, Ti02

1. Introduction

Discovery of room temperature ferromagnetic anatase Co-doped Ti02

(Co:Ti02) [1] has opened up a new research field of diluted magnetic oxides, and attracted considerable attention because of possible applications to spintronics [2] and optoelectronics. Recent studies of Co:Ti02 has revealed strong coupling interaction between charge carriers and spins. Toyosaki et al. [3,4] reported on the anomalous Hall effect of rutile Co:Ti02 with varying carrier density controlled by oxygen vacancies, and revealed the interplay between charge and spin degrees of freedom. Very recently, Griffin et al. [5,6] reported that ferromagnetism in Co:Ti02 film is not carrier mediated, but coexists with a dielectric state. As can be understood from the above results, mechanism of magnetic ordering in Co:Ti02 is still controversial [7].

In this paper, the carrier density is controlled by Nb content, and relationship between the magnetic properties and carrier density is discussed. We report on magneto-optic Faraday rotation angle (8F) and transport properties of Nb-doped anatase Co:Ti02 by varying Nb content. Furubayashi et al. [8, 9] revealed that doping Nb into anatase

efficiently introduce charge carriers. In anatase Ti02, Nb ions exist as Nb5+, and release conduction electrons with an efficiency of 90%.The X-ray diffraction (XRD) results indicate that the films consist of single-phase (OOl)-oriented anatase Ti02. Superconducting quantum interference device (SQUID) measurements show the films are ferromagnetic at room temperature. In Nb doped Co:Ti02, anomalous Hall effect is observed in magnetotransport measurements, showing that the charge carriers are spin polarized.

2. Experiment

Epitaxial thin films of anatase Tio.95-xNbxCo00sO2 (x = 0, 0.03, 0.06) on LSAT ((LaA103)03(Sr2AlTa06)o7) (100) substrate were grown by using the pulsed laser deposition (PLD) method. An excimer laser (wavelength 248 nm) with a laser fluence of 1-2 J/cm2 at a target was used for ablation. Substrate temperature and oxygen partial pressure rate were kept at 500°C and lxlO"5

Torr, respectively. Most results reported here are for 50 - 100 nm-thick films. Standard six-terminal geometry (Hall bar) was used to measure transport properties (dc resistivity,

192

carrier density, and Hall mobility). The magnetic field was applied along the out-of-plane direction at temperatures from 4K to 300K. Magnetic properties were measured using a Faraday rotation spectrometer, a SQUID magnetometer, and a scanning SQUID magnetometer (SSM).

3. Results and Discussion

Figure 1 shows the XRD pattern of Tio.89Nbo.06Coo.05O; epitaxial thin film. The peak can be indexed with (004) plane of anatase phase. No impurity peak was observed up to x = 0.2. The c-axis lattice parameter increased on doping, almost linearly, with x up to 0.06, following Vegard's law (not shown in figure). Further doping resulted in deviation of c-axis from Vegard's law, and thus, we focus on Nb doping content of x = 0, 0.03, and 0.06. The above results indicate that Nb ions are soluble into Ti02 at least up to x = 0.06. Further, cross-sectional transmission electron microscopy showed no Co metal particle formation in the x = 0.06 sample.

Doping Nb atoms into anatase Co:Ti02

resulted in dramatic resistivity (p) reduction (Fig. 2), associated with semiconductor to metal transition. The undoped anatase Ti02 is characterized by semiconductive temperature dependence, typically having 5.3 Qcm at room temperature, while x = 0.03 film show metallic behavior, that is, dp/dT> 0 down to 30 K. The x = 0.06 film exhibited resistivity of 2.7* 10"4

§ 1000 0 0 " 100

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0 r 0

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Tio.s9Co0.osNb00602

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x = 0.06

x = 0.03 x = 0 ^ ^

l

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0 100 200 300 Temperature [K]

0 20 40 60 80 100 2<9[degree]

Figure 1. XRD pattern of Ti0 89Nb0 06Co0 05O2, showing single phase anatase phase without any impurity peak.

10"

0 100 200 300 Temperature [K]

Fig. 2 Resistivity vs. temperature curves. Inset shows carrier density n, vs. temperature curves for various Nb concentrations x.

Qcm at 300 K, and reduced to 6.4x105 Qcm at 6K, less than one-fourths of the resistivity observed at 300 K.

Concentration of n-type carrier in Nb doped films does not depend on temperature (Fig. 2 inset). The carrier density showed 4.9xl017 cm"3 (x = 0), 6.5xl020 cm"3 (x = 0.03), and 1.4xl021 cm"3 (x = 0.06) at room temperature, showing monotonical increase on Nb doping. Resistivity and carrier density of x = 0 below 200 K could not be measured because of the sample's high resistance. The carrier densities of Ti0 gs Nb^O; are comparable to those of Tii_xNbxCo005O2 which showed 8.0xl020 cm"3 and 1.6xl021 cm"3 for x = 0.03 and x = 0.06, respectively. Incorporation of Co resulted in carrier density decrease of about 10-20 %, possibly due to the compensation of carrier by Co2+ dopant.

Magneto-optic Faraday rotation angle vs. magnetic field (9p-H) profiles at 300 K are shown in Fig. 3. The diamagnetic background originating from a substrate is subtracted. A well-defined hysteresis loop is observed in all samples. The value for saturated 6F has increased nearly three times on doping Nb from x = 0.03 to x = 0.06, indicating the effect of Nb doping, that is, increasing carrier density [10].

Figure 4 shows a clear signature of spin-

193

Magnetic field [kOe]

Fig. 3. Magneto-optical Faraday rotation angle vs. magnetic field curves (0r - H) at 300 K.

polarized carriers. The Hall resistance vs. magnetic field at 200 K for x = 0.03 shows clear hysteresis witch is attributed to anomalous Hall effect (AHE). Linear background (ordinary Hall resistance) is subtracted from the Hall resistance data. The shape of the hysteresis curve is similar to that of M-H loop. However, relationship between M-H loop and the AHE is under investigation.

0.0021 . 1 . 1

2. 0.001 - ({

1 °- // = -0.001 • ^ (0

x -0.0021 . 1 •

-10 0 10 Magnetic field [kOe]

Fig. 4. Anomalous Hall resistance as a function of the magnetic field for x = 0.03 at 200 K. Linear background (ordinary Hall resistance) is subtracted from the Hall resistance data.

4. Conclusions

In conclusion, we have investigated carrier dependent magnetic property of Nb doped anatase Co:Ti02 epitaxial films grown by using the pulsed laser deposition method. The carrier density of Tio.95-xNbxCoo.05O2 (x = 0 -

0.06) thin films can be controlled from 4.9xl017 cm"3 to 1.4xl021 cm"3. Temperature dependence of the resistivity shows metallic behavior which implies a semiconductor to metal transition induced by doping Nb. The n-type carrier density does not depend on temperature, which implies that the films are degenerated semiconductors. These films show ferromagnetism at room temperature, and magnetization increased with increasing doped Nb. The anomalous Hall effect is observed in magnetotransport measurements, showing that the charge carriers are spin polarized. These results indicate the carrier induced ferromagnetism in Ti02 based diluted magnetic oxides.

Reference 1. Y. Matsumoto, M. Murakami, T. Shono, T.

Hasegawa, T. Fukumura, M. Kawasaki, P. Ahmet, T. Chikyow, S. Koshihara, and H. Koinuma, Science 291, 854 (2001).

2. H. Ohno, Science 281, 951 (1998). 3. H. Toyosaki T. Fukumura, Y. Yamada, K.

Nakajima, T. Chikyow, T. Hasegawa, H. Koinuma, and M. Kawasaki, Nature Mater. 3, 221 (2004).

4. H. Toyosaki, T. Fukumura, Y Yamada, and M. Kawasaki, Appl. Phys. Lett. 86, 182503 (2005).

5. K. A. Griffin, A. B. Parkhomov. C. M. Wang, S. M. Heald, and K. M. Krishnan, Phys. Rev. Lett. 94, 157204 (2005).

6. K. A. Griffin, A. B. Pakhomov, C. M. Wang, S. M. Heald, K. M. Krishnanan, J. Appl. Phys. 97, 10D320 (2005).

7. T. Dietl, Nature Mater. 2, 646 (2003). 8. Y. Furubayashi, T Hitosugi, Y Yamamoto,

K. Inaba, G Kinoda, Y Hirose, T. Shimada, and T. Hasegawa, Appl. Phys. Lett. 86, 252101 (2005).

9. T. Hitosugi, Y Furubayashi, A. Ueda, K. Itabashi, K. Inaba, Y Hirose, G Kinoda, Y Yamamoto, T. Shimada, T. Hasegawa, Jpn. J. Appl. Phys. 44, L1063 (2005).

10. Kinoda et al. in preparation.

194

NONLINEAR Q U A N T U M EFFECTS IN N A N O S U P E R C O N D U C T O R S

CARLOS CARBALLEIRA, GERD TENIERS and VICTOR V. MOSHCHALKOV*

INPAC-Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism Group, K. U. Leuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium

* E-mail: victor. moshchalkov@fys.kuleuven. be

ARNOUT CEULEMANS

INPAC-Institute for Nanoscale Physics and Chemistry, Quantum Chemistry Group, K. U. Leuven, Celestijnenlaan 200 F, 3001 Leuven, Belgium

We have investigated superconductivity in thin mesoscopic regular polygons in a perpendicular magnetic field. For that, we have used an analytical gauge transformation for the vector po­tential which takes into account the superconductor/vacuum boundary condition. As reported previously, this has opened the way to investigate the nucleation of superconductivity in nanosu-perconductors at the normal-superconducting phase boundary in the framework of the linearized Ginzburg-Landau (GL) equations. We report now on the extension of this work to the super­conducting phase beyond the phase boundary, by using the full non-linear GL equations. The stability in the superconducting phase of the symmetry induced vortex-antivortex molecules has been examined in detail for squared and equilateral triangle geometries.

Keywords: Superconductivity; mesoscopic structures; vortices.

1. Introduction

The confinement of a superconducting con­densate in a nanostructured superconductor results in a variety of interesting phenomena. For instance, the superconducting critical pa­rameters can be modified by using nanostruc-turing to optimize the condensate and flux confinement1. The vortex matter in these systems undergoes some peculiar changes. For example, in a mesoscopic disk the tri­angular vortex lattice is replaced by a sin­gle giant vortex in the center near the phase boundary 2 '3.

Recently, the nucleation of supercon­ductivity in symmetrical polygons has been studied extensively4-8. We know now that their critical temperatures (Tc) show a cusp­like behavior superimposed onto a linear background as a function of magnetic field. Every cusp on this TC(H) phase bound­ary corresponds to a change in vorticity by one. Additionally, we determined that it is possible that the discrete symmetry of

the triangle6 and square5 induces a vortex-antivortex patterns built up from an antivor-tex in the center surrounded by three or four vortices, respectively, giving a total fluxoid of two and three flux quanta ($o)- These vortex patterns are in contrast to the often expected giant 2$o- or 3<&o-vortices.

However, what is the temperature-region of stability for these vortex-antivortex mole­cules? To this end, we have studied possible symmetry-breaking and symmetry-switching transitions that may affect the stability of the vortex patterns4 '9. Here we further investi­gate these transitions by extending our calcu­lations to temperatures well below the phase boundary. In particular, we will discuss in detail our results for a microsquare4'10 with side length a and preliminary calculations for a mesoscopic triangle.

2. Method of solution

To investigate superconductivity in a meso­scopic structure it is necessary to minimize

195

the Ginzburg-Landau (GL) free energy func­tional (relative to the normal state),

/ **L* + /3 |* | 4 +a|* | dr , (1)

where L = 1-ihV - ^A\ /2m*, a and /? are the GL parameters, and ^ is the order parameter. This minimization is performed while taking into account the superconduc­tor/vacuum boundary condition,

0, (2)

where A is the vector potential and n indi­cates the normal to the boundary line. Note that in Eq.(l) it has been already considered the particular case of a thin superconductor [i.e., with thickness d much smaller than the penetration depth A(T)], so we can safely as­sume that the field in the superconductor co­incides with the applied field. In addition, if the coherence length £(T) is considerably larger than the thickness, we can altogether neglect the z-dependence of the solution per­pendicular to the square cross section.

Nevertheless, a major problem remains due to the presence of the vector potential in the boundary condition (2). However, by making sure that the vector potential A has a zero normal component through an analyt­ical gauge transformation4'11, the supercon­ductor/vacuum boundary condition reduces to the Neumann boundary condition,Vtp\n = 0, and Eq.(l) becomes easier to minimize. Using this gauge, it is also convenient to expand \I> into the eigenfunctions (j>i of the linearized Ginzburg-Landau (LGL) equation, Lcj)i = Ci<f>i. This leads to * = ^ C t ^ i , with Ci the complex expansion coefficients.

We have already solved the LGL equa­tion when investigating the nucleation of su­perconductivity in symmetrical mesoscopic structures such as the rectangle7, triangle6

and square5 at the TC(H) phase boundary. We have ordered the eigenfunctions accord­ing to the irreducible representations (irreps)

corresponding to the appropriate C2, C3 or Ci symmetry. Using these previously found solutions of the LGL equation as a basis set includes from the start the superconduc­tor/vacuum boundary condition in our cal­culations. Moreover, the free energy can be rewritten into a more manageable form 12

G/3 = J > + ejcfcl + \Y1 4 H * < C W " i ijkl

(3) in which A^j is defined as A^\ — / <j>*<t)*j<j)k<t>idr and c\ = vT^i- For thin struc­tures, this last normalization of the coef­ficients c^ removes the K-dependence from equation (3), which will be then valid for any kind of superconductor. The problem is now reduced to finding the c^-values that minimize the expression (3) and, for that, we have used a standard Monte Carlo method.

3. The vortex patterns at low fields

In figure 1 the TC(H) phase diagram for the square is shown for a broad range of tem­peratures and magnetic fields.4'10 First of all we notice that the number of flux quanta in the square is reduced with decreasing tem­perature (decreasing coherence length £(T)) as expected. However, not only the fluxoid quantum number (L) changes. The vortex patterns also change with decreasing tem­perature, even when L does not necessarily change. To illustrate this, the vortex pat­terns are shown schematically in figure 1.

From previous work4'5 we already know that every cusp in the TC{H) phase bound­ary corresponds to an increase in L by one. So from the second cusp on, two flux quanta penetrate the square. This happens in the form of a giant vortex with vorticity two. When lowering the temperature just after the second cusp the giant 2$o-vortex changes to a $o-vortex. However, when we decrease the temperature at higher magnetic fields below the third cusp, the giant 2$o-vortex splits

196

into two <3>o-vortices positioned on one of the diagonals (fig. 1). The vortices place them­selves on a diagonal because this allows them to sit as far from each other as possible. Mel'nikov et al. reported that it is possible for the two vortices to align themselves along one of the axes, a vortex pattern which we, however, did not find in our calculations.

* / * o

Fig. 1. Schematic TC(H) plot of the different vortex patterns that can be observed in a microsquare.4 ,10

•i>o vortices, giant 2 $ 0 vortices, and antivortices are represented respectively by black dots, large blue dots and red circles. The letters show the irreps needed to construct the found symmetry. The black lines separate regions with different vortex patterns in the square. Lines with number one divide two pure symmetrical states with different L-values and cor­respond to phase transitions of the first order. Lines with number two separate regions with pure and bro­ken symmetries but equal L. The lines with number three divide regions between pure and broken sym­metry and with a different L. Lines with number two and three correspond to phase transitions of the second order. The background colors indicate the vorticity of the vortex in the center, which can be 0 (yellow), 1 (green), 2 (blue), -1 (red).

4. The vortex-antivortex patterns

Between the third and the fourth cusp at the TC(H) phase boundary the vortex pat­tern consists of an antivortex in the center and four vortices on the diagonals leading to L — 3 (see fig. 2). Lowering the tem­perature close to the third cusp leads to a decrease in L to two, which is carried by a

• A #

(a) £ = 0.162a (b) £ = 0.139a (c) £ = 0.116a

Fig. 2. The pictures show the order parameter | * | , with the darkest colors corresponding to the lowest |\P|. When we reduce the temperature (and £,(T)) at a magnetic flux of $ = 6.25$o> we see that the antivortex in the center annihilates with one of the vortices and three singly quantized vortices appear.

giant 2$0-vortex in the center. In turn, this giant vortex splits into two <3>o-vortices when lowering the temperature even further. How­ever, away from the third cusp quite a differ­ent evolution can be observed, since lower­ing the temperature leads to the annihilation of the antivortex and the conservation of L as shown in figure 2. Moreover, in the first stage near cusp three, L = 3 can only be attained by a vortex-antivortex pattern. So there exists a range of magnetic fields where the vortex-antivortex pattern is always pre­ferred over three singly quantized vortices or a giant vortex with vorticity three. We can conclude then that there is a considerable re­gion of stability for the vortex-antivortex pat­tern. This result is being supported by our recent calculations for a mesoscopic triangle with surface S. In this case, we have found that at L = 2 a vortex-antivortex pattern formed by one antivortex in the center sur­rounded by three vortices in the bisectors is also stable in a wide region of the phase di­agram. The evolution with temperature of this pattern when L is conserved is shown in fig. 3). A more detailed analysis of the sta­bility of these vortex-antivortex patterns can be done by using our two-state model4,9.

Looking at the region between the fourth and the fifth cusp at the TC(H) phase bound­ary (fig. 1), we observe four vortices on the diagonals. As previously, when lower­ing the temperature shortly after the fourth

197

/ 4 \ /

/

A . \ • \ . \

(a) £ = 0.182>/S (b) £ = O.lSSv'S (c) £ = 0 .129^5

Fig. 3. Evolution with decreasing the temperature of the vortex-antivortex pattern calculated for a mesoscopic triangle at L = 2. The applied magnetic flux (<I> = 5.54>o) has been chosen to preserve the L-conservation and the center of the sample has been zoomed eight times. Similarly to the square, the an-tivortex annihilates with one of the vortices and two singly quantized vortices remain.

cusp L is reduced. Remarkably, the transi­tion from L = 4 to L = 3 three takes place by introducing an antivortex in the center. So it is possible to create vortex-antivortex patterns away from the TC(H) phase bound­ary even when this pattern is not present at the TC{H) boundary for that field. This is in direct contradiction with the idea that vortex-antivortex patterns can only be sta­ble extremely close to the phase boundary. Results reported by Misko et al. 14 show that for thicker samples, where demagne­tization effects can not be neglected, the vortex-antivortex pattern can be more sta­ble in type-I superconductors. Moreover, the vortex-antivortex pattern can be resolved better in this situation with experimental vi­sualization techniques.

5. T h e vor tex pa t t e rn s a t higher fields

Between the fifth and the sixth cusp we found again four vortices in the diagonals, but with an extra vortex in the center which even­tually disappears by reducing the temper­ature. The next vortex pattern along the phase boundary is build up from four vortices on the diagonal and a giant vortex with vor-ticity two in the center (figs. 1 and 4). De­creasing the temperature just after the sixth cusp leads to the transformation of the giant vortex into a singly quantized vortex. How-

(a) £ = 0.125a (b) £ = 0.0890a (c) £ = 0.0712a

Fig. 4. When lowering the temperature just above the seventh cusp at the TC(H) phase boundary for $ = 10.25$o. the vortex-antivortex pattern is trans­formed into a vortex pattern with a giant vortex (2$o) in the center and four vortices on the diagonals (a). For even lower temperatures the giant vortex splits into two vortices and the resulting six vortices arrange themselves on two parallel lines (b) which evolve into a triangular-like pattern (c).

#

1

! • •

i

•

* •

II

il ' I I

(a) £ = 0.118a (b) f = 0.102a (c) £ = 0.085a

Fig. 5. When lowering the temperature at a mag­netic flux * = 11.25*o, the vortex-antivortex pat­tern (panel a) is first destroyed and a II-pattern of seven vortices is created. Eventually for even lower temperatures there are only six *o-vortices present which are positioned in two parallel lines.

ever at higher fields the giant vortex splits into two vortices and the six resulting vor­tices rearrange themselves in two lines paral­lel to one of the axes (fig. 4). Whereafter, L is reduced to five and the vortex pattern is build up from four vortices on the diagonals and one in the center.

Since the vorticity at the center of the sample is repeated every four cusps (four ir-reps) at the TC(H) phase boundary, increas­ing the field further on we return again to a vortex-antivortex configuration, now build up from eight vortices on the axes and an antivortex in the center (fig. 1). The vortex-antivortex pattern exists for a TC(H) re­gion of about the same size as the vortex-antivortex pattern at lower fields.

Once the antivortex has disappeared with reducing temperature, a competition

198

1 ~rxTiii-ft * (a) £ = 0.133a

* # # :

* • # *

( b ) * =

#

•

•

= 0.116a

»

»

•

(c) £ = 0.100a (d) £ = 0.083a

Fig. 6. At * = 11.75*o the vortex pattern at the TC(H) phase boundary is constructed out of four vor­tices on the diagonals and four vortices on the axes (a). When lowering the temperature L is reduced by one through the generation of an antivortex in the center(b). This antivortex annihilates with one vor­tex when lowering the temperature further on. The remaining seven vortices form a Il-like pattern (c) and at lower temperatures a H-like pattern quite similar to a conventional triangular lattice (d).

* ^ # # # #

# * * * • #

(a) £ = 0.130a (b) £ = 0.097a (c) £ = 0.081a

Fig. 7. For magnetic flux $ = 12.25*o we find at the TC(H) phase boundary four vortices on the di­agonals and four vortices on the axes. Lowering the temperature leads to an outward motion of the vor­tices and eventually for the lowest temperatures a deformed triangular lattice is formed.

arises between the symmetry of the bound­aries and the natural triangular ordering of the vortices, resulting in some peculiar vor­tex patterns as shown in figures 4-6. Again we have to distinguish between lower and higher fields. For instance just after the sev­enth cusp in the TC(H) curve (fig. 4) the antivortex quickly disappears with reducing temperature and six vortices remain on two parallel lines and for even lower temperatures

the vortices are inclining towards a triangu­lar configuration. For higher fields it is pos­sible for the vortices to arrange themselves into a II-pattern (fig. 5), once the antivortex has left the sample. With reducing tempera­ture a flux quantum is further removed from the sample, leaving again two parallel lines of vortices. However at the highest fields just after the eighth cusp (shown in fig. 6), we observe a change of the ordering of the vor­tex pattern from a II-pattern to a if-pattern. At this magnetic field $ = 11.75$0 the vor­tex pattern at the TC(H) phase boundary is a pattern with L = 8. With decreasing tem­perature L is lowered to seven by the intro­duction of an antivortex. We also notice in fig. 6 that in contrast with the situation at lower fields, where close to the TC(H) phase boundary all the vortices sit in the center or in the diagonals, now it is possible for the in­ner four vortices to place themselves on the axes. This is caused by the presence of the outer four vortices which resist the addition of extra vortices on the diagonals.

(a) £ = 0.182V/S (b) £ = 0.U9VS (c) £ = 0.1%/S

Fig. 8. Evolution with temperature of the vortex pattern at the TC(H) phase boundary for L = 6 in a mesoscopic triangle. The symmetry of the sam­ple and of the Abrikosov lattice coincide, and by de­creasing the temperature the three inner vortices first rotate and then migrate along the bisectors until a perfect triangular lattice is formed.

As a final point, let us consider the sit­uation where L — 8 (fig. 7). When decreas­ing the temperature we observe the migra­tion of the four central vortices to the sides and eventually a distorted triangular lattice is formed. This tendency towards a triangu­lar lattice, already observed in figs. 4 and 6 is enhanced for higher fields and lower temper-

199

atures, since the sample will start to behave like a large thin bulk sample under these cir­cumstances. In addition, the symmetry of the sample may also favor this tendency to recover the Abrikosov lattice. As shown in fig. 8, in a mesoscopic triangle the vortex pat­tern at the TC(H) phase boundary for L = 6 evolves with decreasing the temperature by first rotating and then migrating along the bisectors the three inner vortices, until a per­fect triangular lattice is formed.

6. Conclusions

We have presented an extension of our method to describe superconductivity at the Tc{H) phase boundary in mesoscopic reg­ular polygons to temperatures well inside the superconducting state. We have found that a variety of novel vortex-antivortex pat­terns in a square and triangle is stable for a broad region of temperatures and magnetic fields. To observe these vortex molecules close to the phase boundary, local vortex-imaging techniques sensitive to the \^\2 den­sity are needed.

Furthermore, we were able to investigate the evolution of the vortex patterns as a func­tion of temperature and magnetic field for square and triangle. We noticed that there is a tendency of the system to recover even­tually the triangular lattice with increasing magnetic field and decreasing temperature, since the system will resemble a thin bulk sample under these conditions.

Finally, we can conclude that supercon­ducting nanostructures should be treated as genuine examples of confinement problems dominated by the properties and the symme­try of the boundary conditions. This symme­try can however be broken by the non-linear term in the GL equation. As a consequence the properties of the superconducting state in these structures can be quite different from the bulk case.

Acknowledgements

This work has been supported by the Re­search Fund K.U. Leuven GOA/2004/02, the Belgian Interuniversity Attraction Poles, and the Fund for Scientific Research Flanders (FWO). C.C. acknowledges financial sup­port from Fudacion Ramon Areces (Spain) through Post-Doctoral Studentship Grants and from the Ministerio de Education y Ciencia (Spain) through MAT2004-04364. We would like to also thank Liviu F. Chibo­taru for fruitful and stimulating discussions.

References

1. V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck and Y. Bruynseraede, Nature 373, 319 (1995).

2. D. Saint-James, Phys. Lett. 15, 13 (1965). 3. H.J. Fink and A.G. Presson, Phys. Rev. 151,

219 (1966). 4. L.F. Chibotaru, A. Ceulemans, M. Morelle,

G. Teniers, C. Carballeira and V.V. Moshchal­kov, J. Math. Phys. (in press).

5. L.F. Chibotaru, A. Ceulemans, V. Bruyn-doncx and V.V. Moshchalkov, Nature 408, 833 (2000).

6. L.F. Chibotaru, A. Ceulemans, V. Bruyn-doncx and V.V. Moshchalkov, Phys. Rev. Lett. 86, 1323 (2001).

7. G. Teniers, L.F. Chibotaru, V.V. Moshchal­kov and A. Ceulemans, Europhys. Lett. 63, 296 (2003).

8. B.J. Baelus, S.V. Yampolskii and F.M. Peeters, Phys. Rev. B66, 833 (2002).

9. L.F. Chibotaru, G. Teniers, A. Ceulemans and V.V. Moshchalkov, Phys. Rev. B70, 094505 (2004).

10. G. Teniers, V.V. Moshchalkov, L.F. Chibo­taru and A. Ceulemans, to be published.

11. L. F. Chibotaru, A. Ceulemans, G. Teniers, V. Bruyndoncx and V. V. Moshchalkov, Eur. Phys. J. B27, 341 (2002).

12. V.A.Schweigert and F.M. Peeters, Phys. Rev. Lett. 83, 2409 (1999).

13. A.S. Mel'nikov et al., Phys. Rev. B65, 140503 (2002).

14. V.R. Misko, V.M. Fomin, J.T. Devreese and V. V. Moshchalkov, Phys. Rev. Lett. 90, 147003 (2003).

200

COALESCENCE A N D R E A R R A N G E M E N T OF VORTICES IN MESOSCOPIC SUPERCONDUCTORS

AKINOBU KANDA*, NATSUMI SHIMIZU, KUMIKO TADANO and YOUITI OOTUKA

Institute of Physics and Tsukuba Research Center for Interdisciplinary Materials Science (TIMS), University of Tsukuba, Tsukuba 305-8571, Japan

* E-mail: kanda@lt.px. tsukuba. ac.jp

BEN J. BAELUS** and FRANQOIS M. PEETERS

Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium **E-mail: ben.baelus@ua.ac.be

KAZUO KADOWAKI

Institute of Materials Science, University of Tsukuba, Tsukuba 305-8573, Japan

We report an experimental observation of coalescence and rearrangement of vortices in mesoscopic superconductors by using the multiple-small-tunnel-junction method. The former corresponds to a second-order multivortex state (MVS) - giant vortex state (GVS) transition and the latter to a first-order MVS-MVS transition with a fixed vorticity. The experimental results are in good agreement with the theoretical simulations based on the nonlinear Ginzburg-Landau theory.

Keywords: Mesoscopic superconductor, giant vortex states, multivortex states, Ginzburg-Landau theory.

1. Introduction

Mesoscopic superconductors have attracted much attention because of the feasibility of the vortex-state manipulation.1 They have sizes comparable to the superconducting co­herence length and/or the magnetic penetra­tion depth. The vortex configuration in them is strongly affected not only by the vortex-vortex interaction but also by the boundary, since the shielding supercurrent along the boundary repels vortices. The competition between these interactions causes corruption of the Abrikosov triangular lattice, which is the lowest energy configuration in bulk type-II superconductors. Theoretical studies have revealed that for mesoscopic disks the novel vortex configurations are divided into two categories; (i) multivortex states (MVSs) with a spatial arrangement of singly quan­tized vortices, and (ii) giant vortex states (GVSs) with a single multiply quantized vor­

tex in the center.2'3'4'5 Note that because of the disk geometry, the spatial distribution of the Cooper-pair density, supercurrent and magnetic field for a GVS is axially symmetric with respect to the disk center. Interestingly, theories have predicted that vortex states can drastically change with a fixed vorticity L as a function of applied magnetic fields3; vor­tices merge into one (a MVS-GVS transition) or arrange differently (a MVS-MVS transi­tion) as the magnetic field changes.

Experimentally, one measured the resistivity6 and the magnetization7 of the different vortex states. Since these quanti­ties do not provide any information on the vortex configuration, there was no proof for the existence of the different types of vortex states in mesoscopic superconductors. Re-

a27rL is the change in the phase of the superconduct­ing order parameter along a closed path around the superconductor.

201

B C>

Fig. 1. Scanning electron micrograph of a disk sam­ple. The superconducting coherence length £ is esti­mated to be 0.15 to 0.19 fjtm from the residual resis­tance of the Al films prepared in the same way. The superconducting transition temperature was 1.3-1.4 K.

cently, we have developed the multiple-small-tunnel-junction (MSTJ) method, in which transport properties of several small tunnel junctions attached to a mesoscopic supercon­ductor are measured simultaneously in or­der to detect a change in the supercurrent distribution.8 By taking into account the ax­ial symmetry of the disk, we are able to dis­tinguish directly MVSs and GVSs in a thin mesoscopic superconducting disk.

In the present paper, we report an ob­servation of the vortex coalescence and rear­rangement in a mesoscopic superconducting disk.

2. Exper imenta l

The MSTJ method is based on a clas­sical technique for estimating the super­conducting density of states (DOS) from the current-voltage (I~V) characteristics of a superconductor-insulator-normal metal (SIN) tunnel junction. Since the supercon­ducting DOS depends on the magnetic field and the supercurrent density, spatial distri­bution of the DOS provides information on the vortex configuration.

Instead of measuring the DOS every-

m 2§

Fig. 2. Variation of voltages for junction pairs at symmetrical positions A and D in a decreasing mag­netic field. The current through each junction is 100 pA. The temperature is 0.03 K. (b) Differential volt­age dV/dB for junctions A and D.

where over the sample surface, we just mea­sure the DOS at the symmetrical positions in the sample, e.g., at the periphery for a disk sample. When the DOS at the symmetrical positions takes the same value, the state is likely to be a GVS, and otherwise, it is a MVS.

Figure 1 shows a scanning electron mi­crograph of a disk sample for the MSTJ-measurement. Four normal-metal (Cu) leads are connected to the periphery of a supercon­ducting Al disk (indicated by the dashed cir­cle) through highly resistive SIN tunnel junc­tions A, B, C and D. Note that the sample structure (including the junction positions) is symmetric with respect to the central axis, so that the junctions A and D (B and C) are at symmetrical positions. The radius of the disk is 0.75 fjm and the disk thickness is 33 nm. The disk is directly connected to an Al drain lead. This structure was fabricated us­ing e-beam lithography followed by double-angle evaporation of Al and Cu. Details of the process are described elsewhere.9

202

> E

0.04

0.03

0.01

0.00

-0.01

: ifll

• i i . i

• * dV /dB - • a «

ijft&^gk ' dVVdB

a • n D

1 i I . 1

^ ^ a

• ^ ^

• i

tjx * 1 , 1 .

(a)

f , p 3

(b)

L •

r9^

I CD

> •a

12 13 14 15 16 17 18

B(mT)

Fig. 3. (a) Vaxiation of voltages for junction pairs at symmetrical positions A and D for the L = 8 state. The arrows indicate the direction of the magnetic-field sweep, (b) Differential voltages dVa/dB and dVd/dB in decreasing magnetic field.

11 12 13 14 15 16 17 18 B(mT)

(b) iiw (c) mm*, (d)

Fig. 4. (a) Calculated free energy of the (1,7) state (solid curve), the (8) state (dotted curve), and the L = 8 GVS (dashed curve) as a function of the ap­plied magnetic field. A circular hole (defect) with radius 0.1£ is inserted at a distance 0.2£ from the center. Insets show the transition between the (1,7) state and the (8) state in more detail, (b)-(d) shows the Cooper-pair density of the (1,7) state, the (8) state, and the GVS state, respectively. Dark (bright) region correspond to high (low) Cooper-pair density.

In the measurement, we fixed the cur­rent flowing through each junction to a small value, typically 100 pA, and measured si­multaneously the voltages between each of the four Cu leads and the drain lead, while sweeping the perpendicular magnetic field at a typical rate of 20 mT/min.

3. Resu l ts

Figure 2 shows the change of the voltages at / = 100 pA in decreasing magnetic fields. Va and Vd denote the voltages in symmetric junctions A and D, respectively. To elimi­nate the influence of small resistance differ­ence between these junctions, dV/dB is also shown in Fig. 2 (b). The voltage variation as a function of magnetic field results from two origins: (i) smearing of the energy gap due to pair-breaking by the magnetic field, and (ii) a decrease of the energy gap because of the supercurrent. The former leads to a

moderate monotonic decrease in voltage as the strength of the magnetic field increases, so the rapid change in voltage comes from the latter. In particular, each voltage jump corresponds to a transition between differ­ent vortex states with a vorticity change of _ 2 3,4,5 This a n o w s u s to identify the vor­ticity L as shown in the figure. In Figs. 2 (a) and (b), the difference in V and dV/dB is remarkable for L = 2, 4 - 11, indicating that the state is a MVS for these vortici-ties. For increasing magnetic fields, the dif­ference in V and dV/dB is relatively large between L = 4 and 6 (not shown), which is also due to the MVS formation. These exper­imental results are in good agreement with a numerical simulation based on the nonlinear Ginzburg-Landau theory. Detailed analysis is described in Ref. 9.

We applied this technique to the en­tire L = 8 state, obtained by changing the

203

sweep direction of magnetic field. The re­sults for the entire L = 8 state are shown in Fig. 3.b The difference between dVa/dB and dVd/dB is remarkable below 15.8 mT, indicating that the state is an MVS, while at larger fields dVa/dB and dVd/dB coin­cide, indicating a GVS. Note that the ob­served MVS-GVS transition is a continuous one and is not accompanied by hysteresis, corresponding to the theoretical prediction of the second-order transition.2,5 Moreover, small voltage jumps with hysteresis observed around 13.7 mT indicate a first-order transi­tion between two MVSs with the same L. For comparison, we show the free energy of the vortex states with L = 8 [Fig. 4 (a)], as cal­culated within the framework of the nonlin­ear Ginzburg-Landau theory.3 Here, we as­sumed the existence of a defect near the disk center, which was indicated by the experi­mental result.9 At low fields, we find two stable MVSs with L = 8; one is the (1,7)-state, in which 1 vortex exits near the center pinned by the defect and 7 vortices are lo­cated on a shell [Fig. 4 (b)], and the other is the (8)-state, in which all 8 vortices are arranged on a shell [Fig. 4 (c)]. At higher fields the GVS with L = 8 [Fig. 4 (d)] is found to be most stable. With decreasing field the GVS transits into the (8)-state at B = 14.1 mT and then into the (l,7)-state at B = 11.7 mT. With increasing field the (1,7)-state transits into the (8)-state at B = 12.9 mT, and then into the giant vortex state at B = 14.1 mT. Note that the MVS-GVS tran­sition at B = 14.1 mT is a continuous one (second order transition), in agreement with the experimental observation, while the tran­sition between the two MVSs is discontinu­ous and hysteretic (first order transition) [see insets of Fig. 4 (a)], as was also the case in

Note that the data for the highest and lowest mag­netic fields are the same as the L = 8 da ta obtained in monotonically increasing and decreasing magnetic fields, respectively.

the experiment.

4. Conclusion

By using the MSTJ method, we experimen­tally verified the MVS-GVS and MVS-MVS transitions. The former is of second order and the letter of first order. The results are well reproduced in numerical results based on the nonlinear Ginzburg-Landau theory.

Acknowledgments

This work was supported by the Univer­sity of Tsukuba Nanoscience Special Project, Grant-in-Aid for Scientific Research (B) (17340101) and CTC-NES Program of JSPS, the Flemish Science Foundation (FWO-V1) and the Belgian Science policy. B. J. Baelus acknowledges support from FWO-V1.

References

1. See, for instance, Physica C404 (2004) (Proceedings of the Third European Confer­ence on Vortex Matter in Superconductors at Extreme Conditions).

2. V. A. Schweigert and F. M. Peeters, Phys. Rev. B57, 13817 (1998).

3. V. A. Schweigert, F. M. Peeters, and P. S. Deo, Phys. Rev. Lett. 81, 2783 (1998).

4. J. J. Palacios, Phys. Rev. B58, R5948 (1998).

5. J. J. Palacios, Phys. Rev. Lett. 84, 1796 (2000).

6. V. V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck and Y. Bruynseraede, Nature (London) 373, 319 (1995).

7. A. K. Geim, S. V. Dubonos, I. V. Grig-orieva, K. S. Novoselov, F. M. Peeters, and V. A. Schweigert, Nature (London) 407, 55 (2000).

8. A. Kanda and Y. Ootuka, Microelectron. Eng. 63, 313 (2002); A. Kanda and Y. Ootuka, Physica C404, 205 (2004).

9. A. Kanda, B. J. Baelus, F. M. Peeters, K. Kadowaki and Y. Ootuka, Phys. Rev. Lett. 93, 257002 (2004).

204

SUPERCONDUCTIVITY IN TOPOLOGICALLY NONTRIVIAL SPACES

M. HAYASHI, T. SUZUKI, H. EBISAWA

Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki, Aoba-ku, Sendai 980-8579, Japan

and JST-CREST, 4-1-8, Honcho, Kawaguchi, Saitama 332-0012, Japan

M. KATO

Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Sakai, Osaka 599-8531, Japan

and JST-CREST, J^-l-8, Honcho, Kawaguchi, Saitama 332-0012, Japan

K. KUBOKI

Department of Physics, Kobe University, Kobe 657-8501, Japan

Superconducting states in topologically nontrivial spaces, such as Mobius strip and networks, are discussed. Attention has been paid especially to the effects of finite system size or boundaries on the superconducting phase transition under a magnetic field. We show that topology can affect the superconducting transition temperature and the vortex structure strongly and various types of novel vortex states can be generated, which may be called "topologically (or geometrically) induced vortices".

Keywords: superconductivity; vortex; network; phase transition.

1. Introduction

Recently, various types of superconducting (SC) systems with mesoscopic sizes has been produced due to the developments in fab­rication technology. Most of the develop­ments are classified into two types; the self-organized method and the micro-fabrication. The former utilizes the natural property of matter at low temperatures, which tends to form various meso-scale structures, other than the ordinary crystal lattices. On the other hands, the latter is based on artificial control of matter, using fine processing tech­niques, such as lithography. The representa­tive example of the self-organized fabrication is, probably, recent developments in carbon materials, such as FuUerenes and nanotubes 1,2. Tanda and coworkers3 have succeeded in creating ring-shaped single crystals of tran­sition metal calcogenides (NbSe3, TaS3 etc.), which are the materials well-known to show superconductivity and charge-density-wave

order. Especially, they also succeeded in the synthesis of Mobius strip, that may open new possibility to examine the physical properties of superconductivity or charge-density-wave in topologically nontrivial spaces.

As for superconductivity on a Mobius strip, Hayashi, Kuboki and Ebisawa 4 '5 stud­ied s-wave superconductivity on Mobius strip based on the Ginzburg-Landau (GL) theory and found that the Little-Parks oscillation, which is characteristic to the ring-shaped su­perconductor, is modified in Mobius strip and a new state, which does not appear for ordinary ring, appears when the number of the magnetic flux quanta threading the ring is close to a half odd integer. A related physics in eight-figure strip is analyzed in Ref. 6 .

Fine processing technologies have also applied to the investigation of SC properties in mesoscopic scales. Various types of net­work systems are studied both theoretically

205

and experimentally7'8'9'10'11'12'13. Another important development in this field may be the discovery of new types of vortex states in mesoscopic systems, such as disk or square shaped superconductors 14-15>16. So-called giant vortex state has been found for disk and square of superconductors under a mag­netic field.

In this paper, we give discussion on the properties of SC transition in topologically nontrivial spaces, (1) SC Mobius strip, and (2) SC network system, and investigate for a new types of vortex state. Here we especially pay attention to the vortices, whose vortic-ity is not parallel to external magnetic field. They may be called "topologically (or geo­metrically) induced" vortices.

Fig. 1. Structure of the Mobius strip. The bold arrow shows the direction of the magnetic flux. The setting of x- and y-axis is also indicated.

2. Superconducting Mobius strip: node and embedded vortex line

We consider the strip as shown in Fig. 1. The width, circumference and inter-chain spac­ing are denoted by W, L and a, respectively. The thickness is assumed to be zero here and the strip is considered to be two-dimensional (2D). Then we solved the Gizburg-Landau equation on the Mobius strip. From an ap­proximate calculation 4 '5, one can obtain the phase diagram of the SC Mobius strip as shown in Fig. 2.

Here we find two important parameters which determine the SC behaviors of the

Mobius strip. They are defined by r± = t±(0)/W and ry = £||(0)/L where £||(0) is the coherence length parallel to the chains and £j_(0) is that perpendicular to the chains. When 2 ^ r | | < r-Li t n e magnetic phase dia­gram of the system is essentially the same as that for an ordinary SC ring. In this case, the critical temperature shows well-known Little-Parks oscillation. However, when r± < ^sr\\, a series of new states ap­pears when the flux is close to a half-odd-integer times the flux quantum, as shown in Fig. 2 (b) by hatched regions. The minimum temperature of the stability regions of these

states is given by h = 1 - (gj£M - The

energy gap in these states has a real-space line-node at the center the strip along the circumference. Thus we call these states the nodal states. Order parameter configuration in the nodal state (</> ~ 0n/2) is given in Fig. 3.

The meaning of the "node" in 2D Mobius strip is better understood by generalizing the Mobius geometry to 3D system. Such

<t>l<t>0 (a)

(b)

Fig. 2. The phase diagram of a SC Mobius strip based on GL free energy, (a) for the case of ^ f rll <

rx and (b) r± < ^sf\\ (see text for details).

206

(a) (b)

(c)

Fig. 3. Order parameter configuration in the nodal state at </> ~ 0o/2. (a) real part, (b) imaginary part and (c) amplitude are shown.

a generalization is previously suggested by Volovik17. Here we consider 7r-M6bius geom­etry, which is actually the Mobius ring with a non-zero thickness. This is obtained twisting the bar in to a ring, before closing it into ring-shape. Now we put a singly quantized vortex line at the center of the bar as shown in Fig. 4. The phase shift around the vortex line in­duces an additional phase shift along the bar when we twist it. In case of 7r-M6bius, we obtain ir additional phase shift between the points, A and A', in Fig. 4 (b). Then, in order to make the phase at A and A', equal (mod 27r), we have to add additional phase change "along" the bar by —n This means that fractional winding number of the phase is generated along the ring. Then we obtain stable states near <fr ~ (f>o/2 as well as <fr ~ 0 or <fr ~ <fro- Therefore the state with an em­bedded vortex line corresponds to the nodal state obtained for 2D Mobius strip.

It is interesting that the embedded vor­tex line is not parallel to the external field but perpendicular to it. Therefore, we con­sider it is generated "topologically".

3. Superconducting networks: vortex stripes

SC networks are also interesting objects in which topology, geometry and ordering in-

Vortex line / \ (a)

phase change *^~— along the bar

phase shift due to vortex

Fig. 4. (a) Untwisted and (b) 7r-twisted bar with a square cross section. (Cross sections are shown by squares.) After closing the both ends so that A and A', (B and B') are overlapped, we obtain generalized Mobius geometries: (a) an ordinary ring and (b) n-Mobius geometry. A singly quantized vortex line is embedded in the bar (bold arrow).

terplay between each other. This property enables several intriguing vortex states in the system. We especially pay attention to the "vortex stripe" structure which appears near the transition temperature along the straight boundary. This phenomenon is first pointed out by Sato and Kato based on de Gennes-Alexander equation18, which is valid only at transition temperature 13. We extend this result by studying Ginzburg-Landau theory to lower temperatures.

To put an emphasis on the effects of the boundaries, we have considered two differ­ent boundary conditions (BC's). (a) isolated square lattice with open boundaries and (b) half-periodic BC (periodic in x-direction but open in y-direction). In Fig. 5 we have de­picted the vortex pattern formed at the tran­sition temperature, which is obtained based on de Gennes-Alexander equation18. Appar­ently, the vortex stripe structure is dominant in BC (b) as compared to BC (a). This means that the existence of long straight boundary enhances the formation of vortex stripes. This result is in agreement with those obtained by Sato and Kato13.

We further studied lower temperatures

207

by numerically solving Ginzburg-Landau

equation in case of BC (b). The results are

shown in Fig.6. It can be seen tha t the vortex

stripes are stable not only near the transit ion

tempera ture bu t also in a finite tempera ture

range below.

We conclude t h a t the vortex stripe struc­

ture is characteristic t o a finite size SC net­

works with straight boundaries. Since vor­

tices usually repel each other, this structure

is strongly geometrically driven. In such

sense, the vortex stripes are compared with

the giant vortex s ta te which is characteristic

to small continuous superconductors.

T/Tc = 0.918

I I I I I I I I I I I _•_•_•___•_ I : I : I : I : : I :

T/Tc = 0.885

l" l l" l"l"l" - 1 . . 1 . 1 . 1 . 1 I : I : : I : I : I :

:nn: i : i : i 1 1 .1 .1 .

T/Tc = 0.776

Fig. 6. Vortex patterns of half-periodic network at lower temperatures. (0 = O.250o per cell.) Even at T/Tc = 0.885, vortex stripes are stable.

4 . S u m m a r y

We have studied several finite size supercon­

ducting systems and clarified several topolog-

ically or geometrically induced vortex states.

(a) ) 0/0„=O.l

1 - 1

M m " i z z z z i z z

0/0o=O.3

iv 3 1 If • ™

# • •

•••• fl TJ • c • • #

(b) 0/0=0.274 '„=0.3

periodic periodic

Fig. 5. Vortex patterns at transition temperature in (a) isolated and (b) half-periodic network. The cell in black contains a vortex.

R e f e r e n c e s

1. H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley, Nature 318, 162 (1985).

2. S. Iijima, Nature 354, 56 (1991). 3. S. Tanda, T. Tsuneta, Y. Okajima, K. Ina-

gaki, K. Yamaya and N. Hatakenaka, Nature 417, 387 (2002).

4. M. Hayashi and H. Ebisawa, J. Phys. Soc. Jpn. 70, 3495 (2002).

5. M. Hayashi, K. Kuboki and H. Ebisawa, Phys. Rev. B72, 024505 (2005).

6. G. R. Berdiyorov, B. J. Baelus, M. V. Mi­losevic, F. M. Peeters, Phys. Rev. B68, 174521 (2003).

7. B. Pannetier, J. Chaussy, R. Rammal, J.C. Villegier, Phys. Rev. Lett. 53, 1845 (1985).

8. S. Teitel, C. Jayaprakash, Phys. Rev. Lett. 51, 1999 (1983).

9. Y. Xiao, D. A. Huse, P. M. Chaikin, Mark J. Higgins, S. Bhattacharya, D. Spencer, Phys. Rev. B65, 214503 (2002).

10. R. Rammal, T. C. Lubensky, G. Touless, Phys. Rev. B27, 2820 (1983).

11. Y. lye, E. Kuramochi, M. Hara, A. Endo, S. Katsumoto, Phys. Rev. B70, 144524 (2004).

12. T. Ishida, M. Yoshida, S. Okayasu, K. Hojou, Supercond. Sci. Technol. 14, 1166 (2001).

13. O. Sato, M. Kato, Phys. Rev. B68, 094509 (2003).

14. V.V. Moshchalkov et al., Phys. Rev. B55, 11793 (1997).

15. B. J. Baelus et al., Phys. Rev. B63, 144517 (2001).

16. L. F. Chibotaru, A. Ceulemans, V. Bruyn-doncx, V. V. Moshchalkov, Nature 408, 833 (2000).

17. G. E. Volovik, The Universe in a Helium Droplet (CLAREDON PRESS, OXFORD 2003).

18. S. Alexander, Phys. Rev. B27, 1541 (1983).

208

DC-SQUID RATCHET USING ATOMIC POINT CONTACT

Y. OOTUKA*, H. MIYAZAKI and A. KANDA Institute of Physics and Tsukuba Research Center for Interdisciplinary Material Science (TIMS),

University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan * E-mail: ootuka@lt.px. tsukuba. ac.jp

We argue that an asymmetric dc SQUID composed of a tunnel junction and an atomic point contact when it is threaded by a magnetic flux can produce a ratchet potential for the superconducting phase. Preliminary experiment shows the rocking ratchet effect, i.e., generation of dc voltage by ac current biasing.

Keywords: Ratchet effect; SQUID; Mechanical break junction

1. Introduction

The nonequilibrium dynamics of a particle in an asymmetric periodic potential is attract­ing intense attentions. One of the charac­teristic effects of such a system is the rock­ing ratchet, that is, unidirectional motion induced by a time periodic uniform exter­nal perturbation.1 This kind of motion is thought to be the essential mechanism of the molecular motors in living cells, and a large amount of work has been done on the the­ory of ratchets. On the experimental side, artificial ratchet systems have been realized by, for example, electrons in a semiconduc­tor heterostructure2, superconducting flux quanta in a thin superconducting film with ferromagnetic trap array3, but the demon­strations of the effect are still insufficient.

Recently, ratchet systems using Joseph-son junctions were proposed as a promising system for the experimental investigation of the effect under various conditions. Zapta et al.A argued a SQUID ratchet with three junc­tions threaded by an external flux. The key element that gives asymmetric potential is two junctions connected in series. If they are identical, they behave like a single junction, with the only difference that the current-phase relation is sin(</?/2) instead of sirup. Analyzing the dynamics of the supercon­ducting phase across the SQUID numerically,

they showed that the rocking ratchet mecha­nism operates under an external ac current. However, to our knowledge, the experimental confirmation of this type of ratchet has not been reported yet. Sterck et al.5 proposed another type of SQUID ratchet, where asym­metry exists in the critical current and/or in the inductance of the two arms forming the SQUID loop. They showed the rocking ratchet effect by a numerical simulation and demonstrated it experimentally.

In this paper, we propose a novel ratchet structure, that is, a dc-SQUID composed of a tunnel junction and a superconducting atomic point contact (Fig. 1) . The Joseph-

¥>t X^P

Fig. 1 composed of a contact

Schematic picture of asymmetric dc-SQUID tunnel junction and a atomic point

209

son current through the latter junction is car­ried via Andreev bound states formed in the contact region, and the current-phase rela­tion deviates significantly from simp when the transmission coefficient approaches unity. Thus such a SQUID has asymmetric poten­tial when we apply an appropriate external flux to the SQUID loop. Preliminary exper­iment has indicated that we could make the asymmetric potential and the rocking ratchet effect, i.e., the voltage rectification in the presence of ac current, has actually been ob­served.

Fig. 2. Calculated current-phase relation Ii(if,r) via single mode with transmission probability r = 1 (solid), 0.99 (dotted), 0.90 (dashed) and 0.70 (dash-dotted) at T = 0 K.

2. Asymmetric dc-SQUID

According to the RCSJ model for a Joseph-son junction, dynamics of the junction is de­scribed by a second-order differential equa­tion for the superconducting phase <p. As the equation has the same form as the New­ton equation, we can regard the Josephson system as a fictious particle moving in one-dimensional space. Here, the capacitance C, the shunting ohmic conductance 1/R and the Josephson coupling energy U(tp) of the junc­tion correspond to the mass, friction and the potential for the particle, respectively. Also, the current I bias biasing the junction gives the additional external force to the parti­cle, and the voltage means the velocity of the particle. We note that U(<p) can be de­rived from the current-phase relation I(ip) as /(¥>) = (2e/h)(dU/d<p).

A superconducting atomic point contact is a junction where two superconductors are connected via one or a few atoms. According to theories6, electron transport through such a contact is described in terms of the An­dreev bound states that are localized around the contact. Energy of the bound state de­pends on the phase difference <p and the transmission probability r of the contact. In­terestingly, the states generally carry electric current and the net current in thermal equi­

librium, i.e., the superconducting current is6

eAr sin <p H<fi)

2h ^ l - T s i n 2 ( ^ / 2 )

x tanh rsin2(y>/2)

In Fig. 2,1(<p)for several values of r at T = 0 is plotted. The dependence on <p is generally non-sinusoidal. It is the most evident when r is unity at T = 0 K, where I(ip) ~ sin(<^/2) for — n < ip < TT. We should note that it is resemble to what Zapata et al. expected to realize using two tunnel junctions in series.4

A real contact has multiple conduction chan­nels and the supercurrent is the sum of the contributions from all modes.

Now we will consider a SQUID shown in

Fig. 3. Potential U(x) experienced by the phase x of the SQUID with r = 1, Ej = A / 2 and / = 1/4.

210

<

1UU

50

0

-50

i nn

/=0.15 • i • i . • i

\ : - \ / A. / \ • V : >• / \ / \

- 1 . 0 -0.5 0.0 /

(a)

0.5 1.0 100 200 /AC (nA)

(b)

300

Fig. 4. (a) Switching current Jgw as a function of magnetic flux / . (b) Generated dc Voltage Vbc as a function of amplitude of bias ac current / A C -

Fig. 1, which has a tunnel junction in one of the arms and a superconducting point con­tact in the other arm. We define the phase difference across the tunnel junctions and the point contact as <pt and <pp, respectively. When the SQUID loop is threaded by the external magnetic flux $, the fluxoid quanti­zation yields

ft ~ <PP + 2 7 r / = 2 7 r n i

where / = <&/<&o ($0 = h/2e being the flux quantum). Here, we have assumed that the self inductance L of the loop is so small that the self-induced magnetic flux LI can be ne­glected. Letting x = (<pt + ipp)/2, we ob­tain the resultant potential experienced by the phase x as

U(x) = Up(x + TT/) + Ut(x - irf).

Thus, by tuning the external flux we can se­lect shape of the potential experienced by the phase, and by virtue of the non-sinusoidal I-<p relation in point contact it can be a ratchet potential. An example is depicted in Fig. 3. Here, we have taken Up((p) =

- A y i ^ r s i n 2 ( ^ / 2 ) , Ut(<p) = -EjCos<p,

T = 1, E3 = A/2 and / = 1/4. Evidently it lacks reflection symmetry.

In conventional ratchet systems, a parti­cle moves under the influence of thermal fluc­tuation. Or rather, the influence is the main subject of the ratchet problem. According to the theories,6 the characteristic scale of the potential should be ~ (RQ/RT)A, where .RT a n ( l A are the junction resistance in the normal state and the superconducting energy gap. So, the thermal fluctuation is impor­tant when the junction resistance is compa­rable to or larger than the quantum resis­tance RQ, and the temperature is not far from the transition temperature Tc- In ad­dition to the thermal fluctuation, quantum fluctuation has to be taken into account if mass of the particle is very light and the po­tential is weak. Such effect has been ana­lyzed by Reinmann et al.7 As for Josephson junction, lots of experiments have been done so far on the quantum fluctuation of phase that causes the superconductor-to-insulator transition.8 Therefore, it should be feasible to make a quantum ratchet in this kind of structures.

211

3 . E x p e r i m e n t

In order to investigate the current-phase re­

lation in superconducting atomic point con­

tacts , we have fabricated a dc-SQUID com­

posed of a tunnel junction and a point

contact . 9 The mechanical break junction

method allowed us to realize the atomic point

contact. As the tuning of the contact was

very precise, the situation considered in the

above section could also be realized. Figure

4(a) shows the magnetic flux dependence of

the switching current 7sw- Asymmetry in

i s w means the maximum potential force act­

ing upon the phase in one direction is not the

same as tha t in the other direction. Namely,

the potential in this case lacks the reflection

symmetry. Then we tuned the magnetic flux

at / = 0.15, where the positive switching

current was 45 nA and the negative one was

67 nA, and measured dc voltage across the

SQUID by applying 1 kHz ac current. A re­

sult is depicted in Fig. 4(b), where the dc

voltage VDC is plotted vs. the ampli tude of

ac current / A C • We observed generation of dc

voltage. The dc voltage VQC appears above

58 nA and reaches a peak at 80 nA. These re­

sults are semi-quantitatively consistent with

the static characteristics of the SQUID (Fig.

4(a)) . Detailed discussions will be described

elsewhere.10

In conclusion, we have proposed a novel

SQUID ratchet system composed of a super­

conducting atomic point contact and a tun­

nel junction threaded by a magnetic flux.

The non-sinusoidal current-phase relation of

the former junction is the most essential fea­

ture to build the ratchet potential tha t lacks

the reflection symmetry. Preliminary experi­

ment has actually shown the rocking ratchet

phenomena, tha t is, voltage rectification un­

der an oscillating current bias. The system

could be a model system to investigate effects

of quantum fluctuation in ratchet problem.

A c k n o w l e d g m e n t s

This work was supported by a Grant-in-

Aid for Scientific Research from the Japan

Ministry of Education, Culture, Sports, Sci­

ence and Technology, and by the Univer­

sity of Tsukuba Nanoscience Special Project.

One of the authors (H.M.) acknowledges the

support of the research fellowship from the

Japan Society for the Promotion of Science

for Young Scientists.

R e f e r e n c e s

1. For review, see P. Reimann, Phys. Rep 361, 57 (2002).

2. H.Linke, et al, Science 286, 2314 (1999). 3. J.E. Villegas, et al, Science 302, 1188

(2003). 4. I. Zapata, et al, Phys. Rev. Lett. 77, 2292

(1996). 5. A. Sterck, et al, Appl. Phys. A75, 253

(2002). 6. C.W.J. Beenakker, Phys. Rev. Lett. 67, 3836

(1991)., ibid. 68, 1442 (1992), A. Furusaki and M. Tsukada, Phys. Rev. B43, 10164 (1991), P.F. Bagwell, Phys. Rev. B46, 12573 (1992).

7. P. Reinmann et al, Phys. Rev. Lett. 79, 10 (1997)

8. For example, T. Yamaguchi et al, Phys. Rev. Lett. 85, 1974 (2000)

9. H. Miyazaki, A. Kanda, Y. Ootuka and T. Yamaguchi, in this procedings.

10. H. Miyazaki, T. Yamaguchi, A. Kanda, and Y. Ootuka, in preparation.

212

SUPERCONDUCTING WIRE NETWORK UNDER SPATIALLY MODULATED MAGNETIC FIELD

HIROTAKA SANO, AKIRA ENDO, SHINGO KATSUMOTO AND YASUHIRO IYE

Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha Kashiwa, Chiba 277-8581 Japan

A two-dimensional (2D) superconducting square network under spatially modulated magnetic field is studied. The super/normal phase boundaries were measured with field modulation varied. The dependence on the strength of field modulation exhibited the behavior reproducing the calculation we had done before. In addition, /- V characteristics measurements were also conducted.

Keywords: superconducting network; modulated magnetic field; Little-Parks oscillation; Hofstadter butterfly; vortex glass transition

1. Introduction

The super/normal phase boundary of a two-dimensional (2D) superconducting wire network exhibits characteristic Little-Parks oscillation1'2 as a function of the magnetic frustration a (perpendicular magnetic flux piercing the unit cell). The fine structure of the TC(H) curve reflects the edge of the so-called Hofstadter butterfly spectrum3

which represents the Bloch band of tight binding electrons on a corresponding 2D lattice under a perpendicular magnetic field. Thus the TC(H) curve has local maxima at simple fractional values of a, which correspond to stable vortex configurations.4 In this work, we address the case in which superconducting network is subjected to both a spatially modulated magnetic field and a uniform magnetic field. We measured the dependence of the TC{H) curve on the field modulation amplitude /? (expressed in terms of flux per plaquerte) and compared the results with the corresponding Hofstadter butterfly spectra calculated by the matrix diagonalization method.5

The nature of super/normal transition in 2D superconducting wire networks and Josephson junction arrays has been a subject of extensive studies. The transition at zero magnetic field is generally understood in terms of

Kosterlitz-Thouless (KT) transition6, the hallmark of which is a "universal jump" at T=TKT of the exponent in the power law current-voltage (I-V) characteristics. The super/normal transition in magnetic fields is governed by subtle competition between the vortex-vortex interactions and the interaction of a vortex with the pinning potentials both periodic and random. In particular, the overall interaction of the vortex system with the periodic potential should be sensitive to the value of a. Various models of phase transitions including, melting and floating of a vortex solid, vortex glass (VG) phase transition, and formation commensurate domains have been theoretically proposed7

and experimentally pursued6'8. However, there still remain ambiguities in the comparison of the experimental data and theoretical models, so that the nature of superconducting transition in networks has so far been elusive. In order to shed light on the issue, we conducted I-V characteristics measurement in the present system with spatially varying magnetic field which is different from the usually studied case of uniform applied field.

2. Experiment

Figure 1 shows the scanning electron microscope (SEM) image of the sample used in this study. The sample is a square network

213

of Al wire decorated with an array of ferromagnetic Co dots. The Al wire network consists of 120x120 unit cells with lattice period 500nm. The Al wires are 70 run wide and 35 nm thick. The Co dots are oval shape with lateral size 130nm><200nm and thickness 80nm. The Al wires and Co dots are separated by an intervening layer of 35nm thick Ge. The role of this Ge layer is to prevent oxidation of Al network and to keep it from direct contact with Co dots. Thus the effect of ferromagnetic Co dots in the present system is solely through the local magnetic field generated by them (no pair-breaking effect).

creates a checkerboard-patterned magnetic field whose strength /? can be controlled by changing the direction of magnetization by changing the azimuthal angle cp of the parallel field. In addition to the checkerboard field /?, a uniform perpendicular magnetic field a is applied, creating the flux pattern shown in Fig.2(b).

(a)

^A^HlA (b).

%a~p a+pSa-p

<*+m«-p

Ua~P

<**m <x+Ma~p

a+ma-p

a+pma-p

a+P\

<*+P^~P

Figure 2. (a) Schematic picture of sample from side view, (b) Arrangement of Co dots and checkerboard patterned field modulation.

Figure 1. SEM image of the sample, (a) The whole network consisting of 120 120 squares, (b) Enlarged image showing arrangement of Co dots.

Figure 2(a) shows the schematic side view of the sample structure. When a sufficiently strong magnetic field is applied parallel to the sample, the Co dots are fully magnetized and the stray field from the Co dot array creates a spatially alternating magnetic field. The arrangement of the Co dots shown in Fig. 2(b)

Measurements were conducted with a cross coil superconducting magnet system which enabled us independent control of the parallel and perpendicular field components. The experiment consisted of two steps. The first step was the determination of the super/normal phase boundary Tc(a) for different values of /?. The magnetoresistance was measured by a standard ac method for different settings of the azimuthal angle <p, and the data was converted to TC{H). The second step was the measurement of the I-V characteristics with a programmable dc current source at different temperatures in the vicinity of the transition point. Temperature stability throughout the experiment was better than 1 mK.

214

3.1.

Result

Critical temperature

Figure 3(a) shows the Little-Parks oscillation of Tc as a function of the uniform perpendicular field for different values of /?. Each trace is vertically offset by 0.05K for clarity, and the azimuthal angle <p was rotated by 5 for each trace. Horizontal shift of the traces is a spurious effect due to a small angular misalignment of the sample relative to the horizontal plane. Figure 3(b) is the comparison between the experiment (solid curves) and the calculation (dotted curves) over a single period, which corresponds to the region between the dotted lines in Fig. 3(a). The traces cover the range from /?=0 (bottom) to =1/2 (top).

(a) 2 - 1 .

Figure 3. (a) TC(H) of the Al wire network as a function of the perpendicular magnetic field for different settings of <p which controls the amplitude of the checkerboard field. Each trace is vertically offset by 0.05K. (b) Comparison between the experiment (solid curves) and the calculation (dotted curves). The value of the parameter /? is changed from 0 (bottom) to 1/2 (top).

It is seen that as /? is changed from 0 to 1/2, the peak at a=l/2 become pronounced relative to those at integer a5. They crossover at /?=l/4. These features are in common between both results. But the fine structures seen in the calculated curves disappeared in the experimental results. This discrepancy is presumably attributable to lithographical irregularity, in particular imperfect registration between the Co dot array and the Al network.

3.2. /- V characteristics

Figure 4 shows the I-V curves at different temperatures for a=0 and (a) /H) and (b) /?=0.5. We conducted measurements for a=0, 0.618, 0.5. But, contrary to our expectation, the I-V characteristics for these different values of a have turned out to be virtually indistinguishable.

(a)

>

10

10"'

2 4 6 8 2 10°

6(A)

(b) i o 5

> 5"

10 1

10

/;=o.5 1.42 K< 7~<1 5K

2 4 6

10" m

10°

Figure 4. I-V curves for a=0 and (a) fi=0 and (b) =0.5.

The overall behavior of the /- V characteristics, i.e. the transition from upward-curvature to downward one with decreasing temperature, is reminiscent of the VG transition9. Figure 5

215

shows the results of scaling analysis for ct=0 and (a) P=Q and (b) /?=0.5. It is seen that in both cases, reasonably good scaling plot is obtained. Thus, the superconducting transition at these values of parameters a and /? is consistent with the VG transition, although it does not constitute an exclusive proof. The most puzzling feature of the present result is that the I-V characteristics for different values of a do not differ very much and it appears to fit the VG scaling even in the et=/?=0 case where the KT transition is expected. The overwhelming VG-like behavior suggests importance of disorder. It is conceivable that the requirement for the lithographical perfection is more stringent in the study of dynamics than that of the phase boundary, because ever increasing length scale is involved as the transition is approached. It is conceivable that imperfect registration between the Al network and the Co dot array over long range distorts the I-V characteristics and masks the features of the underlying transition.

4. Conclusion

(a) 1031 :

1028 :

1025d — 22

"E10

1 1 0 " ^ 1016

10"

10'°

107

/

fif(A°T ,m)

(b) 1 0 "

1015

- 1 0 1 3

E 11

N^. 109

107

105

103

10" I f (Am1!0 10'

In conclusion, the superconducting wire network under a spatially modulated magnetic field exhibits behavior reflecting the corresponding Hofstadter spectra. The observed I-V curves are consistent with the vortex glass scaling over the whole parameter range investigated, leading us to suspect overwhelming influence of disorder.

Acknowledgements

The work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sport, Science and Technology (MEXT), Japan.

References

1. W.A.Little and R.D.Parks, Phys. Rev. Lett. 9, 9 (1962).

2. S.Alexander, Phys. Rev. B 27, 1541 (1983).

3. D.R.Hofstadter, Phys. Rev. B 14, 2239 (1976).

4. B.Pannetier, J.Chaussy, R.Rammal and J.C.Villagier, Phys. Rev. Lett. 53, 1845 (1985).

5. Y.Iye, E.Kuramochi, M.Hara, A.Endo and S.Katsumoto, Phys. Rev. B 70, 144524 (2004).

6. H.S.J, van der Zant, M.N.Webster, J.E. Romijn, and J.E.Mooij, Phys. Rev. B 50, 340(1994).

7. S.Teitel and C.Jayaprakash, Phys. Rev. Lett. 51,1999 (1983): Q.Niu and F.Nori, Phys. Rev. B 39, 2134 (1989): T.C.Halsey, Phys. Rev. Lett. 55, 1018 (1985); M.Franz and S.Teitel, Phys. Rev. Lett. 73, 480 (1994).

8. X.S.Ling et al., Phys. Rev. Lett. 76, 2989 (1996).

9. D.S.Fisher, M.P.A.Fisher and D.Huse, Phys. Rev. B 43, 130(1991).

Figure 5. VG scaling plot of the I-V curves in Fig.4. (a) ^=0and(b)y?=0.5.

216

SIMPLE A N D STABLE CONTROL OF MECHANICAL BREAK JUNCTION FOR THE STUDY OF SUPERCONDUCTING ATOMIC

POINT CONTACT

H. MIYAZAKI*, A. KANDA and Y. OOTUKA

Institute of Physics and TIMS, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan * E-mail: hmiya@lt.px. tsukuba. ac.jp

T. YAMAGUCHI

National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan

Aiming at determining the current-phase relation of a superconducting atomic point contact (SAPC), we made an interference experiment using a dc-SQUID consisting of an SAPC and a tunnel junction. In this paper, the experimental method, especially the control of SAPC by the mechanical break junction method is described in detail.

Keywords: Mechanical break junction; Atomic point contact

1. Introduction

Scanning probe microscopes such as STM and AFM are very powerful tools to in­vestigate the electron transport in nano-structures. However, the installation of STM or AFM in a cryostat is rather difficult be­cause of the complexity of the system, and poor stability of its contact to the nano-object. Mechanical break junction (MBJ) is an alternative method.1 In this method, a thin metal wire is fixed onto an insulating elastic substrate, and is broken by bending the substrate in vacuum. By relaxing the bend, the fracture surface can be brought back into contact again, and thus it pro­vides a controllable nano-electrode pair. Be­cause of its simple structure and a small ra­tio of gap- to bending-displacements, MBJ is strong against external vibrations and a drift. Especially, if one uses a microfabri-cated bridge, the displacement ratio is as small as 10~4, so that the drift of the elec­trode distance can be less than 1 pm per hour.

A number of experiments on atomic point contact (APC) have been done. Elec­tron transport in such point contacts is well

described by a quasi one-dimensional pic­ture. In this picture, transport proper­ties are determined by a set of transmission probabilities of conduction channels, {T^} = {ji, 72, • • • 7"JV}- When the material is super­conducting, i.e., Josephson junction, one can determine each r from the non-linear I-V characteristics, which was demonstrated by Scheer et al.2

The current-phase relation /(</?) is an­other fundamental property of a Josephson junction. Koops et al.3 made an experiment to determine the I((p) by measuring the self-induced flux of a superconducting loop closed by an superconducting APC as a function of external magnetic field, and demonstrated that the /(</?) is distinctly different from the sinusoidal relation. The measured amplitude of I(<f) was larger than the theoretical value for a single-channel APC. Thus, the existence of multiple channels is necessary for the ex­planation of the experimental results. How­ever, the independent determination of {r^} was impossible for their sample, and the ex­periment was unsatisfactory in a quantitative sense.

In this study, in order to investigate the

217

relation between I(<p) and {TJ}, we have per­formed an experiment, in which both quan­tities can be determined independently. For this sake, we have made a MBJ working in a dilution refrigerator, which we will report in detail in this paper. The results of the experiment will be given elsewhere.4

2. Experimental Setup

To make an MBJ experiment in a dilu­tion refrigerator, we need a motion mech­anism working at low temperature and in vacuum. This mechanism must have fol­lowing features: (1) Influx and generation of heat should be low. (2) For the first break of the MBJ, the deformation range of substrate must be larger than about 0.5 mm. (3) For control of the contact with subatomic-scale accuracy, the resolution of bending-displacement must be finer than 1 /Ltm. (4) To make precise measurement, we need to keep the contact constant for a long period. Scheer et al.2 made an experiment of atomic-size aluminum contacts using MBJ at dilution refrigerator temperature. They used a cryogenic gear transmission mecha­nism driven by an electric motor at the am­bient temperature, and reported that the frictional heating is generated with rapid motion.5

To improve this drawback, we tried a hydrostatic system using helium pressure as shown in Fig. 1. A bending tip is fixed to an end of a small phosphor-bronze bellows 8 mm in outer diameter and 19 mm in length (Bel­lows Kuze Co.), which is anchored to a mix­ing chamber of a dilution refrigerator. The bellows is connected to a He gas handling sys­tem via a thin cupronickel tube. The tube is long enough to prevent heat inflow via super-fluid He, and is anchored at several points in the cryostat.8 The He pressure is mon­itored by a semiconductor pressure gauge (PT-7011A2O, Ohkura Electric). For coarse control of the pressure, He gas is added from

bellows (anchored to the mixing' chamber)

substrate He cylinder

Fig. 1. Hydrostatic system of He for MBJ installed in a dilution refrigerator.

a He gas cylinder, or removed to a rotary pump via needle valves. On the other hand, fine control is realized by varying the inner volume of the gas line. For this sake, a cylin­der (Slim Cylinder DA40x50-A, Koganei6) is added in the line and is driven by a servo lin­ear actuator (Mechatronics Cylinder SCN6-050-150-B, Dyadic Systems7). A PC-based feedback loop allows us to control the pres­sure finely.

A schematic draw of the MBJ stage is shown in Fig. 2. A phosphor bronze sub­strate upon which the junction is fabricated is set on a pair of counter supports 20 mm distant from each other, and pushed by the bending tip. Elimination of high-frequency external noise is of essential importance in

anchored to the mixing chamber

T T

Cu supporting block —

bellows-(phosphor bronze)

driving tip 1 (brass)

counter support-(stainless)

substrate (phosphor bronze)

cupronickel tube (O.3id70.5od)

resistive coaxial cables

Fig. 2. Schematic draw of a mechanism of MBJ in a dilution refrigerator.

218

the experiment of small junctions; the sam­ple is placed in a /i-metal case which shields not only electromagnetic wave but also the static magnetic field. The residual magnetic field is about 0.6 fxT. Electrical leads are fil­tered by a two-stage low-pass filter consisting of an RC filter and a resistive coaxial cable. The RC filter is made of a 1 kfi metal-film chip resistor and a 1 nF metalized polypheny-lene sulphide chip capacitor, while the resis­tive coaxial cable (150 O, 370 pF) is made by threading a manganin wire 0.05 mm in di­ameter into a CuNi capillary tube 0.1 mm in inside diameter and 70 cm in length. To re­alize stronger attenuation, the space between the wire and the tube is filled with colloidal carbon. In order to apply magnetic fields, a small superconducting coil is installed in the /i-metal case.

3. Performance of M B J

In order to evaluate the performance of the MBJ mechanism, we made experiments us­ing a gold MBJ (Fig. 3). We used a phos­phor bronze substrate 0.47 mm thick. At first, we spin-coated polyimide resin (PI2610, HD Microsystems) and cured at 350 °C to form a insulating film. Then, by the stan­dard electron-beam lithography and vacuum evaporation, gold film with a narrow neck was defined on the substrate. Finally, the polyimide film was plasma-etched in oxygen gas. As the etching is isotropic, the poly­imide underneath thin patterns is removed to form an air-bridge as shown in Fig. 3.

Figure 4 shows a temporal response of conductance G of the gold MBJ during sev­eral periods of repeated pressure swing. The He pressure Pne was increased and decreased by shuttling the air cylinder. Some plateaus were observed at G ~ Go(= 2e2/h be­ing the quantum conductance), and the last plateau just before the junction broke was nearly equal to the quantum conductance. In a tunnel region (G < Go), conductance

1 |i.m

Fig. 3. Gold MBJ measured in the test run of the MBJ mechanism.

1000

Time (sec)

Fig. 4. Variation of conductance when He pressure is changed repeatedly.

depends exponentially on the pressure, i.e., exp(—Pne/Po) as shown in Fig. 5 where Po was 1.1 kPa. Using the work function for bulk gold, the geometry and the mechani­cal properties of the sample and the bend­ing stage, we calculate P0 as 2.1 kPa within the WKB approximation. Considering the uncertainty in the assumed values, we think the agreement is fairly good. No effect of heating was observed at 100 mK even when we change the pressure at its maximum rate about 102 kPa/sec.

Figure 6 shows a typical trace of (a) Pne, (b) the conductance G and (c) the position of the servo linear actuator. In this run, the target value for Pae was set to a new value at t = 200 sec, and Pne approached it in about 150 sec. The fluctuation amplitude of

219

£3

148 150 152 154 He Pressure (kPa)

156

Fig. 5. Conductance of the gold MBJ in a tunnel region as a function of He pressure.

Pile after the regulation was established was about 20 Pa which corresponded to the noise level of the pressure gauge. In the mean­time, the servo linear actuator moved rather steady to compensate the pressure rise due to the evaporation of liquid helium in the bath (Fig. 6(c)). In Fig. 6(b), we found that the fluctuation of conductance was about 4% during the period of 2000 sec after Pfte be­come stable. It corresponds to gap-distance fluctuation of 4 x 10~12 m. This is compa­rable to the tunnel conductance fluctuation AG/G = exp(APHe/Po - 1) ^ 2% which is caused by Pne fluctuation APue (~ 20 Pa) observed in Fig. 6(a). However, the spectrum of G fluctuation is apparently different from that of Pile, and the major fluctuation seems to be a telegraph-type one.

To summarize, the simple bellows-based bending system for the MBJ experiment has been developed. The bending displacement has a mm-scale dynamic range and a sub-^m resolution. The stability in the gap distance is as good as 4 x 10~12 m for about 2000 sec. Thus, it should be suitable for various ex­periments of nano-conductors at the dilution temperatures.

Acknowledgments

This work was supported in part by a Grant-in-Aid for Scientific Research fromthe Japan

• 165.0

' 164.8

164.6

164.4

' 0.6

0.4

6000

5800

measured ( a ) ' set point

I *°° 5400

0 500 1000 1500 2000 2500 time (sec)

Fig. 6. Time trace of (a) He pressure, (b) conduc­tance and (c) position of the servo linear actuator during a feedback regulation of He pressure.

Ministry of Education, Culture, Sports, Sci­ence and Technology,and by the University of Tsukuba Nanoscience Special Project. One of the authors (H.M.) acknowledges the sup­port of the research fellowship from the Japan Society for the Promotion of Science for Young Scientists.

References

1. For review, see N.Agrait, A.L. Yeati and J.M. van Ruitenbeek, Phys. Rep. 377, 81 (2003).

2. E. Scheer, P. Joyez, D. Esteve, C. Urbina and M. Devoret, Phys. Rev. Lett. 78, 3535 (1997).

3. M. Koops, G. van Duyneveldt and R. de Bruyn Ouboter, Phys. Rev. Lett. 77, 2542 (1996).

4. H. Miyazaki, T. Yamaguchi, A. Kanda, and Y. Ootuka, in preparation.

5. R. Cron, PhD thesis, University of Paris (2001).

6. Koganei Corporation, Tokyo, Japan. (http://ww3.koganei.co.jp/)

7. Dyadic Systems Co., Ltd., Ishikawa, Japan (http://www.dyadic.co.jp/)

8. Y. Ootuka and N. Matsunaga, /. Phys. Soc. Jpn. 59, 1801 (1990).

220

CRITICAL CURRENTS IN QUASIPERIODIC PINNING ARRAYS: ONE-DIMENSIONAL CHAINS AND PENROSE LATTICES

V.R. MISKO * and S. SAVEL'EV

Frontier Research System, Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama, 351-0198, Japan, E-mail: vmisko@riken.jp

Franco NORI

Frontier Research System, Institute of Physical and Chemical Research (RIKEN),

Wako-shi, Saitama, 351-0198, Japan, and Center for Theoretical Physics, CSCS, Department of Physics,

University of Michigan, Ann Arbor, MI 48109-1120, USA, E-mail: fnori@riken.jp

Here we summarize results on our study of the critical depinning current Jc versus the applied magnetic flux O, for quasiperiodic (QP) one-dimensional (ID) chains and 2D arrays of pinning centers placed on the nodes of a five-fold Penrose lattice. In ID QP chains, the peaks in c£(®) are determined by a sequence of harmonics of the long and short segments of the chain. The critical current Jc(<t>) has a remarkable self-similarity. In 2D QP pinning arrays, we predict analytically and numerically the main features of Jc(<t>), and demonstrate that the Penrose lattice of pinning sites provides an enormous enhancement of Jc(Q>), even compared to triangular and random pinning site arrays. This huge increase in JC(<S>) could be useful for applications.

Keywords: Vortex lattice; Critical current; Quasicrystals.

1. Introduction

Recent progress in the fabrication of nanostructures has provided a wide variety of well-controlled vortex-confinement topologies, including different regular pinning arrays [1-5]. A main fundamental question in this field is how to drastically increase vortex pinning, and thus the critical current Jc, using artificially-produced periodic Arrays of Pinning Sites (APS). The increase and, more generally, control of the critical current Jc in superconductors by its patterning (perforation) can be of practical importance for applications in microelectronic devices. A peak in the critical current Jc(<S>), for a given value of the magnetic flux per unit cell, say ®i, can be engineered using a superconducting sample with a periodic APS with a matching field Hi = Oi / A (where A is the area of the pinning cell), corresponding to one trapped vortex per pinning site. However, this peak in Jc(<£>), while useful to obtain, decreases very quickly for fluxes away from <J>i. Thus, the desired peak in Jc(<&) is too narrow and not very robust against changes in <J>. It would be

greatly desirable to have samples with APS with many periods. This multiple-period APS sample would provide either very many peaks or an extremely broad peak in t£(0), as opposed to just one (narrow) main peak (and its harmonics). We achieve this goal (a very broad JC(Q>)) here by studying samples with many built-in periods. Here, we study vortex pinning by ID quasiperiodic (QP) chains and by 2D APS located on the nodes of QP lattices (e.g., a five-fold Penrose lattice) [6]. We show that the use of the 2D QP (Penrose) lattice of pinning sites results in a remarkable enhancement of Jd®), as compared to other APS, including triangular and random APS. In contrast to superconducting networks, for which only the areas of the network plaquettes play a role [7], for vortex pinning by QP pinning arrays, the specific geometry of the elements which form the QP lattice and their arrangement are important, making the problem far more complicated.

2. Simulation

We model a three-dimensional (3D) slab,

221

infinitely long in the z-direction, by a 2D (in the xy-plane) simulation cell with periodic boundary conditions. We perform simulated annealing simulations by numerically integrating the overdamped equations of motion (see, e.g., [8, 9]): rjvi = f; = £ w + i^p + iiT + hd. Here fj is the total force per unit length acting on vortex i, fcw and hvP are the forces due to vortex-vortex and vortex-pin interactions, respectively, iiT is the thermal stochastic force, and fV is the driving force; T] is the viscosity, which is set to unity. The force due to the vortex-vortex interaction is fiw = ZjNr£ Ki{ | n -TJ\ I X) Tjj, where Nv

is the number of vortices, Kx is a modified Bessel function, X is the penetration depth, Tij = (xi - r)/\ n - r, | , and £ = OOVSTTXS.

Here Oo = hc/2e. The pinning force is £IVP = ZJ^P fP-(\ r, - rjW IlrP) ®[(rP - I n - T&> I )/X] Tik(p\ where A^ is the number of pinning sites, fP is the maximum pinning force of each short-range parabolic potential well located at r ^ , rP is the range of the pinning potential, 0 is the Heaviside step function, and Xik(p) = (li - Tk(P>)/\ Ti - T^P> | . All the lengths (fields) are expressed in units of X (Oo IX2). The ground state of a system of moving vortices is obtained by simulating the field-cooled experiments. For deep short-range (8-like) potential wells, the energy required to depin vortices trapped by pinning sites is proportional to the number of pinned vortices, NV

(P\ Therefore, in this approximation, we can define the critical current as follows: jd®) = Jo NV{PK<£>) I Nvi<£>), where Jo is a constant, and study the dimensionless value Jc = jc I jo. We use narrow potential wells as pinning sites, wifhi> = 0 . 0 4 ? U o ( m .

3. Critical current in QP APS

3.1. ID quasicrystal

A ID QP chain [6] can be constructed by iteratively applying the Fibonacci rule (Z, -> LS, S -> L), which generates an infinite sequence of two line segments, long L and short S. For an infinite QP sequence, the ratio of the numbers of long to short elements is the

golden mean x = (1 + V5)/2 [6]. The position of the wth point where a new segment, either L or S, begins is determined (e.g., for as = 1 and aL = T) by: xn = n + [n h] I x, where [x] denotes the integer part of x. To study the critical depinning current Jc in ID QP pinning chains, we place pinning sites to the points where the L or S elements of the QP sequence link to each other. The results of calculating JdN^) for different chains and the same y = 1/x (y = as/ai is the ratio of the length of the short segment, as, to the length of the long segment, aj) show that, for sufficiently long chains, the positions of the main peaks in Jc, to a significant extent, do not depend on the length of the chain. The peaks form a Fibonacci sequence: Nv = 13, 21, 34, 55, 89, 144, and other "harmonics". Being rescaled, normalized by the numbers of pins in each chain, the Jc curves reproduce each other and have many pronounced peaks for golden-mean-related values of O/ Oi (<t>i is the flux corresponding to the first matching field, Hi, when Nv = A ,). The same peaks of c£(0), for different chains, are revealed before and after rescaling because of the self-similarity of JJ&). The self-similarity of Jd<S>) has also been studied in reciprocal ^~space and will be presented elsewhere [10].

3.2. 2D QPAPS: Penrose lattice

Consider now a 2D QP APS, namely, an APS located at the nodes of a five-fold Penrose lattice. This lattice is a 2D QP structure, or quasicrystal, also referred to as Penrose tiling [6]. These structures possess a perfect local rotational (five- or ten-fold) symmetry, but do not have translational long-range order. Being constructed of a series of building blocks of certain simple shapes combined according to specific local rules, these structures can extend to infinity without any defects [6]. The unusual self-similar diffraction pattern of a Penrose lattice exhibits a dense set of "Bragg" peaks because the lattice contains an infinite number of periods in it [6]. It is precisely this unusual property that is responsible for the striking Jc(o)'s obtained here.

222

The structure of a five-fold Penrose lattice is presented in the inset to Fig. 1. As an illustration only, a small five-fold symmetric fragment with 46 points is shown. The elemental building blocks are rhombuses with equal sides, a, and angles which are multiples of 6 = 36°. There are two kinds of rhombuses: (i) those having angles 29 and 36 ("thick"), and (ii) rhombuses with angles 9 and 49 ("thin").

Let us analyze whether any specific matching effects can exist between the Penrose pinning lattice and the interacting vortices, which affect the magnetic-field dependence of the critical depinning current Jc(,$>)-Quasicrystalline patterns are intrinsically incommensurate with the flux lattice for any value of the magnetic field [7], therefore, in contrast to periodic APS, one might a priori assume a lack of sharp peaks in J c(0) for QP APS. However, the existence of many periods in the Penrose lattice can lead to a hierarchy of matching effects for certain values of the applied magnetic field, resulting in strikingly-broad shapes for Jc(0). These could be valuable for applications demanding unusually broad Jc(0) 's.

First, there is a "first matching field" (the corresponding flux is denoted as <J>i) when each pinning site is occupied by a vortex. This matching effect is accompanied by a broad maximum involving three kinds of local "commensurability" effects of the flux lattice: with the rhombus side a; with the short diagonal of a thick rhombus, 1.176a, which is close to a; and with the short diagonal of a thin rhombus, which is all ~ 0.618a.

Another matching is related with the filling of all the pinning sites on the vertices of the thick rhombuses and only three out of four of the pinning sites on the vertices of thin rhombuses. For this value of the flux, matching conditions are fulfilled for two close distances, a (the side of a rhombus) and 1.176a (the short diagonal of a thick rhombus),

but are not fulfilled for the short diagonal, alx, of the thin rhombus. Therefore, this 2D QP feature is related to x, although not in such a direct way as in the case of a ID QP pinning array. This 2D QP matching results in a very wide maximum of the function Jc(0) at <t> = <J>vacancy/thin = Ov/t = 0.7570), (Fig. 1).

For higher vortex densities (<J> = OmterstMal/tiuck = * i / T = 1 . 4 8 2 0 0 » S i n g l e

interstitial vortex is inside each thick rhombus. These interstitial vortices can easily move; thus Jc has no peak at 0,/T. The position of this feature is determined by the number of vortices at G> = Oi, which is JV„(0) = NP, plus the number of thick rhombuses, NIh

hlck = Nlh I x.

Figure 1. The critical current JM>) for a triangular, random and QP (301-sites Penrose-lattice) APS. The Penrose lattice provides a remarkable enhancement of &£(<!>) over a very wide range of values of <J> because it contains many periods in it. Inset: example of a five-fold Penrose lattice, consisting of "thick" and "thin" rhombuses.

In order to better understand the structure of yc(0) for the Penrose pinning lattice, we compare the elastic EA and pinning Epm

energies of the vortex lattice at H\ and at (the lower field) Hv/t, corresponding to the two maxima of Jc. Vortices can be pinned if the gain Epm = Upm/3 npm of the pinning energy is larger than the increase of the elastic energy [11] related to local compressions: Ec\ = Cu

[(aeq - b) I aeq]2. Here, Upm ~fp rp, npm is the density of pinning centers, f3 (H <H\) = H/H\ = 5/(<D0 npm), and /? (H > Hx) = 1 is the

223

fraction of occupied pinning sites (/? = 1 for H = Hu and p = 0.757 for H = # v/t), aeq = [2/31G

P "pm]1/2 is the equilibrium distance between vortices in the triangular lattice, b is the minimum distance between vortices in the distorted pinned vortex lattice (b = a I T for H = Hx and b = a for H = Hv/t), and Cn = B2 I [4:i(l + X2 k2)] is the compressibility modulus for short-range deformations [11] with characteristic spatial scale k « (npm)V2. The dimensionless difference of the pinning and elastic energies is Epm - Ee] = p fm «pin <J>0

2 / (4nk2), where /^ = 4nX2Upm I <D„2 - p x [1 - b (31/2 p npm I if2}2. Near matching fields, Jc

has a peak only if /dlff > 0. Since only two matching fields provide /&& > 0, then our analysis explains the two-peak structure observed in Jc shown in Fig 1. Note that for weaker pinning, the two-peak structure gradually turns into one very broad peak, and eventually zero peaks for weak enough pinning.

For comparison, we show the •/<,(<&) for Penrose-lattice, triangular and random pinning arrays (Fig. 1). The latter is an average over five realizations of disorder. Notice that the QP lattice leads to a very broad and potentially useful enhancement of the critical current Jj(0), even compared to the triangular or random APS. The remarkably broad maximum in J c(0) is due to the fact that the Penrose lattice has many (infinite, in the thermodynamic limit) periodicities built in it [6]. In principle, each one of these periods provides a peak in Jc(<$>). In practice, like in quasicrystalline difraction patterns, only few peaks are strong. This is also consistent with our study.

4. Conclusions

The critical depinning current Jc(<$>) was studied in ID QP chains and in 2D QP arrays (the five-fold Penrose lattice) of pinning sites. A hierarchical and self-similar JA<&) was obtained. We physically analyzed all the main features of Jc(<t>). Our analysis shows that the

QP lattice provides an unusually broad critical current Jc(<t>), that could be useful for practical applications demanding high JcS over a wide range of fields.

Acknowledgments

This work was supported in part by ARDA and NSA under AFOSR contract number F49620-02-1-0334; and by the US NSF grant No. EIA-0130383.

References

1. M. Baert et al., Phys. Rev. Lett. 74, 3269 (1995); V.V. Moshchalkov et al., Phys. Rev. B54, 7385 (1996).

2. J.E. Villegas et al., Science 302, 1188 (2003); Phys. Rev. B68, 224504 (2003); M.I. Montero et al., Europhys. Lett. 63, 118 (2003).

3. A.M. Castellanos et al., Appl. Phys. Lett. 71, 962 (1997); R. Wordenweber et al., Phys. Rev. B69, 184504 (2004).

4. L. Van Look et al., Phys. Rev. B66, 214511 (2002).

5. A.V.Silhanek et al., Phys. Rev. B67, 064502 (2003).

6. Quasicrystals, Ed. J.-B. Suck, M. Schreiber, P. Haussler (Springer, Berlin, 2002).

7. A. Behrooz et al., Phys. Rev. B35, 8396 (1987); F. Nori et al., Phys. Rev. B36, 8338 (1987); F. Nori and Q. Niu, Phys. Rev. B37, R2364 (1988); Q. Niu and F. Nori, Phys. Rev. B39, 2134 (1989).

8. C. Reichhardt, C.J. Olson, and F. Nori, Phys. Rev. B57, 7937 (1998).

9. B.Y. Zhu et al., Phys. Rev. Lett. 92, 180602 (2004); Physica E18, 318, 322 (2003); Phys. Rev. B 68, 014514 (2003); Physica C388-389, 665 (2003); 404, 260 (2004).

10. V.R. Misko, S. Savel'ev, and F. Nori, unpublished; cond-mat/0502480.

11. E.H. Brandt, Rep. Prog. Phys. 58, 1465 (1995).

224

MACROSCOPIC Q U A N T U M TUNNELING IN HIGH-TC

SUPERCONDUCTOR JOSEPHSON JUNCTIONS

SHIRO KAWABATA

Nanotechnology Research Institute and Synthetic Nano-Function Materials Project (SYNAF), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono,

Tsukuba, Ibaraki 305-8568, Japan

International Project Center for Integrated Research on Quantum Information and Life Science

(MILq Project), Hiroshima University, Higashi-Hiroshima, 739-8521, Japan * E-mail: s-kawabata@aist.go.jp

We investigate the macroscopic quantum tunneling (MQT) in d-wave high-Tc superconductor Josephson junctions. Using the path-integral method, we analytically obtain the MQT rate for the c-axis twist Josephson junctions. In the case of the zero twist angle, the system shows the super-Ohmic dissipation due to the presence of the nodal quasiparticle tunneling. Therefore, the MQT rate is largely suppressed compared with the finite twist angle cases. Furthermore, the effect of the zero energy bound states (ZES) on the MQT in the in-plane junctions is theoretically investigated. We obtained the analytical formula of the MQT rate and showed that the presence of the ZES at the insulator/superconductor interface leads to a strong Ohmic quasiparticle dissipation. Therefore, the MQT rate is noticeably inhibited compared with the c-axis junctions in which the ZES are completely absent.

Keywords: Macroscopic Quantum Phenomena; Josephson Effect; Quantum Dissipation.

1. In t roduc t ion

Since macroscopic systems are inherently dis-sipative, there arises a fundamental ques­tion of what is the effect of dissipation on the macroscopic quantum tunneling (MQT). This issue was solved by Caldeira and Leggett by using the path-integral method and they showed that MQT is depressed by dissipation.1 This effect has been verified in experiments on s-wave Josephson junctions shunted by an Ohmic normal resistance.2 As was mentioned by Eckern et. al., the influ­ence of the quasiparticle dissipation is quan­titatively weaker than that of the Ohmic dis­sipation on the shunt resistor.3 This is due to the existence of an energy gap A for the quasiparticle excitation in superconductors. Therefore, in an ideal s-wave Josephson junc­tion without the shunt resistance, the sup­pression of the MQT rate due to the quasi­particle dissipation is very weak at low tem­perature regime.

In this paper, we will consider MQT in

the d-wave high-Tc Josephson junction (see Fig.l). In the d-wave superconductors, the gap vanishes in certain directions,4 hence quasiparticles can be excited even at suffi­ciently low temperature regime. Moreover

(a) f - / > •

c 1

,b

"a

(b) d I d

i n ^ H rn

Fig. 1. (a) Schematic drawing of the c-axis twist Josephson junction. 7 is the twist angle about the c-axis. (b)Schematic drawing of the d-wave Josephson junction along the afr-plane.

225

in the case of d-wave junctions along the afo-plane (Fig. 1(b)), the zero energy bound states (ZES) are formed by the com­bined effect of Andreev reflections and the sign change of the d-wave order parameter symmetry.4 The ZES are bound states for the quasiparticle at the Fermi energy. Therefore, we will investigate the effect of the dissipa­tion due to the nodal-quasiparticles (sec. 3)5

and the ZES (sec. 4)6 on the MQT. Note that the effect of the quasiparticle decoher-ence was recently discussed by Fominov et. al.7 and Amin et. al.& and Kawabata et. al.9

in the context of the d-wave qubit.

2. Effective action

In this section, we will show the calculation of the effective action for the c-axis twist Josephson junction.10,11 In Fig. 1(a), we show schematic of this junction. In this figure, 7 is the twist angle about the c-axis. Such a junction was recently fabri­cated by using the single crystal whisker of E^S^CaC^Og+tf. Takano et.al. mea­sured the twist angle dependence of the c-axis Josephson critical current and showed a clear evidence of the dx2_y? symmetry of the pair potential.10

In the following, we assume that the tun­neling between the two superconductors is described in terms of the coherent tunnel­ing. For simplicity, we also assume that each superconductors consist of single CuC>2 layer, Ai(fc) = A0cos2#, and A2(fc) = Ao cos 2 (6 + 7). Moreover, we consider the low temperature limit (fcsT <C A0). By using the path-integral method, the ground partition function for the system can be given

by

is given by

j DCJ>{T) exp Se«[<t>]

h (i)

where <f> = <f>i — 4>2 is the phase difference across the junction and the effective action

%s{4>\ = / JO

-L dr

M (d(j>[T) dr + U{4>)

h0dTdr'a(r-r')cos^T)-<t>{T'\(2)

In this equation, a ( r ) is the dissipation ker­nel which is given by the product of two Nambu Green's functions, M = C(h/2e)2 is the mass and U((f>) is the tilted washboard potential

V{#) = -Ej (7) cos 0 + <A , (3) Ic{l)

where, Ej = (h/2e) Ic is the Josephson cou­

pling energy, Ic is the Josephson critical cur­

rent, and 7ext is the external current.

3. Effect of nodal-quasiparticles: c-axis junctions

In the following, we will consider the effect of the nodal quasiparticles on MQT. For this purpose, we first calculate the dissipation kernel a{r) for two types of the c-axis junc­tion, i.e., (1) 7 = 0 and (2) 7 / 0 (here we will show the result for 7 = n/8 case only.).

In the case of the c-axis junction with 7 = 0, the nodes of the pair potential in the two superconductors are in the same direc­tion. Therefore, the node-to-node quasipar­ticle tunneling is possible even at very low temperatures. In this case, the asymptotic form of the dissipation kernel at the zero tem­perature is given by

a ( r ) 3fi2JV0

2|*|2 1 16A0 |r | ; (4)

for Ao|r|/fi » 1. This gives the super-Ohmic Dissipation.

On the other hand, in the case of the fi­nite twist angle (7 = 7r/8), the asymptotic behavior of the dissipation kernel is given by an exponential function due to the suppres­sion of the node-to-node quasiparticle tun-

226

neling, i.e.,

a{r) ~ exp 1 A 0 | T |

(5) y/2 K

for A 0 | r | / / i » 1. The MQT rate at the zero temperature

is given by

T = lim — Im In Z « A exp SB_ (6)

where SB = Sefi[4>B] is the bounce exponent, that is the value of the action 5eff evaluated along the bounce trajectory <J>B{T). Using the instanton method, we obtain the analytic expressions for the MQT rate:5

r(o) exp - 5 ( 0 ) - 0.14

tUc(0)

r0(o) lnic(0)f ( fext V

V2e C \ U c ( 0 ) ;

A2

. 5 / 4 -

r(7r/8) r0(7r/8)

exp[-B(7r/8)],

, (7)

(8)

where,

to the node-to-node quasiparticle tunneling is very weak.

4. Effect of ZES: in-plane junctions

In this section we will discuss the influence of the ZES on the MQT in the in-plane junc­tions {e.g., the ramp-edge junctions12 and the grain boundary junctions13). In such junctions, the ZES give a crucial contribu­tion to the Josephson and the quasi-particle current.

In the case of the do/l/do junction, the node-to-node quasi particle tunneling can contribute to the dissipative quasiparticle current. However, the ZES is completely absent4. Therefore the system shows the super-Ohmic dissipation as in the case of the zero-twist angle c-axis junctions or the intrin­sic junctions. On the other hand, in the case of the do/I/dir/4 junction (Fig. 2(b)), we find that the dissipation kernel is given by

3 KRQ 1

flW = I v £' o W C , (f 6M(y) M

x <1

5 /4

Ic{i), (9)

and r 0 (7) is the decay rate without the quasiparticle dissipation. In this equation, 5M is the renormalized mass. As an exam­ple, for A0 = 42.0 meV, Ic(-y = 0) = 1.45 x 10"4 A, C = 10 fF, and I^/Icil) = 0.9, we obtain

T(7) _ J 90 % for 7 = 0 96%

7 :

7 : (10)

r 0 (7) " 196% for 7 = TT/8

As expected, the suppression of the node-to-node quasiparticle tunneling in the case of the 7 = 7r/8 junction gives rise to a subtle re­duction of the MQT rate compared with the 7 = 0 junction cases. Astoundingly, however, we find that even in the case of the 7 = 0 junctions, the reduction of the MQT rate due

a(r) 16v/2 RN

(11)

where RQ = h/Ae2 is the resistance quan­tum and RN is the normal state resistance of the junction. Therefore the system shows the strong Ohmic dissipation due to the ZES.

(a)

(b)

Fig. 2. Schematic drawing of the d-wave Josephson junction along the a6-plane. (a) the do/l/do junction and (b) the do/I/dn/4 junction.

227

The M Q T ra t e of the do/l/d^/i junction is

given by6

r To — « exp

81C(3) RQ

(12)

For a mesoscopic bicrystal YBCO Joseph-

son junct ion 1 4 (A 0 = 17.8 meV, C = 20 x

10~ 1 5 F , RN = 100 n, x = 0.95), the M Q T

rate is est imated as r / r 0 « 25%. There­

fore the M Q T ra te in the do/l/d^/i junction

is strongly suppressed in compared with the

do/I/do and the c-axis junctions.

5. S u m m a r y

To summarize, we have presented the cal­

culation of the M Q T ra te for the d-wave

Josephson junct ion by making use of the

path-integral method and the instanton the­

ory. We find the super-Ohmic dissipation in

the case of the zero-twist angle c-axis junc­

tion and the do/l/do junction. This dissi­

pat ion is caused by the node-to-node quasi-

particle tunneling between the two supercon­

ductors. Therefore, in this case, the suppres­

sion of M Q T is weak. On the other hand, in

the case of do/l/d^/^ junction, the suppres­

sion of the M Q T ra te is very strong in com­

pared with the c-axis twist junctions and the

do/l/do junctions due to appearance of the

Ohmic dissipation.

Finally, we would like to comment about

recent experimental researches. Recently, In-

omata et al. 1 5 and Bauch et al. ie have suc­

ceeded to observe the M Q T in the Bi2212 in­

trinsic junct ion and the Y B C O grain bound­

ary bi-epitaxial Josephson junction, respec­

tively. Inomata et al. have reported tha t

the effect of the nodal-quasiparticle on the

M Q T is negligibly small as shown by our

theory (sec. 3) and the thermal- to-quantum

crossover tempera ture Tco is relatively high

(~ IK) compared with the case of s-wave

junctions.

A c k n o w l e d g m e n t s

I would like to thank my co-warkers Satoshi

Kashiwaya, Yasuhiro Asano, and Yukio

Tanaka. This work was part ly supported by

NEDO under the Nanotechnology Program.

R e f e r e n c e s

1. A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981).

2. A. N. Cleland, J. M. Martinis, and J. Clarke, Phys. Rev. B 37, 5950 (1988).

3. U. Eckern, G. Schon, and V. Ambegaokar, Phys. Rev. B 30, 6419 (1984).

4. S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 1641 (2000).

5. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys. Rev. B 70, 132505 (2004).

6. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys. Rev. B 72, 052506 (2005).

7. Y. V. Fominov, A. A. Golubov, and M. Kupriyanov, JETP Lett. 77, 587 (2003).

8. M. H. S. Amin and A. Yu. Smirnov, Phys. Rev. Lett. 92, 017001 (2004).

9. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Physica E 28 (2005), cond-mat/0504274.

10. Y. Takano, T. Hatano, A. Fukuyo, A. Ishii, M. Ohmori, S. Arisawa, K. Togano, and M. Tachiki, Phys. Rev. B 65, 140513 (2002).

11. K. Maki and S. Haas, Phys. Rev. B 67, 020510 (2003).

12. H. Arie, K. Yasuda, H. Kobayashi, I. Iguchi, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 62, 11864 (2000).

13. E. Il'ichev, M. Grajcar, R. Hlubina, R. P. J. IJsselsteijn, H. E. Hoenig, H.-G. Meyer, A. Golubov, M. H. S. Amin, A. M. Zagoskin, A. N. Omelyanchouk, and M. Yu. Kupriyanov, Phys. Rev. Lett. 86, 5369 (2001).

14. A. Y. Tzalenchuk, T. Lindstrom, S. A. Charlebois, E. A. Stepantsov, Z. Ivanov, and A. M. Zagoskin, Phys. Rev. B 68 (2003) 100501.

15. K. Inomata, S. Sato, K. Nakajima, A. Tanaka, Y. Takano, H. B. Wang, M. Nagao, T. Hatano, and S. Kawabata, Phys. Rev. Lett. 95 (2005) 107005.

16. T. Bauch, F. Lombardi, F. Tafuri, A. Barone, G. Rotoli, P. Delsing, and T. Clae-son, Phys. Rev. Lett. 94 (2005) 087003.

228

CARBON N A N O T U B E S A N D UNIQUE TRANSPORT PROPERTES: IMPORTANCE OF SYMMETRY A N D CHANNEL N U M B E R

TSUNEYA A N D O

Department of Physics, Tokyo Institute of Technology,

2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan E-mail: ando@phys.titech.ac.jp

A brief review is given on transport properties of carbon nanotubes mainly from a theoretical point of view. The topics include an effective-mass description of electronic states, the absence of backward scattering except for scatterers with a potential range smaller than the lattice constant, the presence of perfectly conducting channel, and roles of symmetry and channel numbers.

Keywords: graphite; carbon nanotube; impurity scattering; Berry phase.

1. Introduction

Transport properties of carbon nanotubes are extremely interesting. In metallic nano­tubes there is no backscattering even in the presence of scatterers unless their potential range is smaller than the lattice constant of two-dimensional graphite.1 When there are several bands at the Fermi level, interband scattering appears, but a perfectly conduct­ing channel transmitting through the system without being scattered back exists and the conductance is always larger than 2e2/7rh.2

The purpose of this paper is to give a brief review of recent theoretical study on such unique transport properties of carbon nano­tubes.

2. Neutrino on Cylinder Surface

A graphite sheet is a zero-gap semiconduc­tor in the sense that the conduction and valence bands consisting of TT states cross at K and K' points of the Brillouin zone.3

Electronic states near a K point of two-dimensional graphite are described by the k p equation:4'5'6

7(<?-k)F(r)=eF(r) , (1)

where F is a two-component wave function, 7 is the band parameter, k = (kx,ky) = iV is a wave-vector operator, e is the energy,

and ax and ay are the Pauli spin matrices. Equation (1) has the form of Weyl's equation for neutrinos.

The electronic states can be obtained by imposing the periodic boundary condition in the circumference direction \?(r + L) = *(r ) except in extremely thin tubes. The Bloch functions at a K point change their phase by exp(iK-L)=exp(27rw/3), where u = 0 or ± 1 , determined by the chiral vector L specifying the nanotube circumference. Because ^ ( r ) is written as a product of the Bloch func­tion and the envelope function, this phase change should be canceled by that of the envelope functions and the boundary condi­tion for the envelope functions is given by F ( r+L) = F(r) exp(-27rii//3).

The phase appearing in the bound­ary condition corresponds to a fictitious Aharonov-Bohm (AB) flux passing through the cross section. Therefore, an electron in a nanotube is regarded as a neutrino on the cylinder surface with an AB flux, the amount of which is determined by the structure.

Energy levels in nanotubes for the K point are obtained by putting kx = Kv(n) with Kv(n) = (2n/L)[n — (^/3)] and ky = k as ei,±)(n,fc) = ± 7 v / ^ ( n ) 2 + fc2,5 where L = | L | , n is an integer, and the upper (+) and lower (—) signs represent the conduction and va-

229

lence bands, respectively. The Hamiltonian for the K' point is obtained by replacing ky

with — ky and the boundary condition by re­placing v with —v. This shows that the nano-tube becomes metallic for v = 0 and semicon­ducting with gap Eg=4TT'y/3L for i/ = ±l.

An important feature is the presence of a topological singularity at k = 0. A neu­trino has a helicity and its spin is quantized into the direction of its motion. The spin eigen function changes its signature due to Berry's phase under a 2TT rotation. There­fore, the wave function acquires phase — •K when the wave vector k is rotated around the origin along a closed contour.7'8 The sig­nature change occurs only when the contour encircles k = 0 but not when the contour does not contain k = 0. This topological singular­ity at k = 0 is the origin of the absence of backscattering and the presence of a perfect channel in metallic carbon nanotubes.

3. Absence of Backscattering

Most of scatterers including charged centers are expected to be characterized by a poten­tial with range larger than the lattice con­stant. For such scatterers, the same potential appears as the diagonal element of the matrix Hamiltonian and mixing between K and K' points is safely neglected. It has been proved that the Born series for backward scattering vanish identically.1 The absence of backward scattering has been confirmed by numerical calculations in a tight binding model.9

Backscattering corresponds to a rotation of the k direction by ±ir. In the presence of the time reversal symmetry, there exists a time reversal process corresponding to each backscattering process. The time reversal process corresponds to a rotation of the k direction by ±7r in the opposite direction. The scattering amplitudes of these two pro­cesses are same in the absolute value but have an opposite signature because of Berry's phase. As a result, the backscattering ampli­

tude cancels out completely. In semiconduct­ing nanotubes, on the other hand, backscat­tering appears because the symmetry is de­stroyed by a nonzero AB magnetic flux.

An important information has been ob­tained on the mean free path in nanotubes by single-electron tunneling experiments.10'11

The Coulomb oscillation in semiconducting nanotubes is quite irregular and can be ex­plained only if nanotubes are divided into many separate spatial regions in contrast to that in metallic nanotubes.12 This be­havior is consistent with the presence of backscattering leading to a localization of the wave function. In metallic nanotubes, the wave function is extended in the whole nanotube because of the absence of back-scattering. With the use of electrostatic force microscopy the voltage drop in a metallic nanotube has been shown to be negligible.13

Single-wall nanotubes usually exhibit large charging effects due to nonideal con­tacts. Contact problems were investigated theoretically in various models and an ideal contact was suggested to be possible un­der certain conditions.14'15'16'17'18 Recently, almost ideal contacts were realized and a Fabry-Perot type oscillation due to reflection by contacts was claimed to be observed.19

4. Perfect Channel

When the Fermi level moves away from the energy range where only linear bands are present, interband scattering appears be­cause of the presence of several bands at the Fermi level. Let r^a be the reflection coef­ficient from a state with wave vector ka to a state with k^ = —k^ in a two-dimensional graphite sheet. Here, ft stands for the state with wave vector opposite to /?. Apart from a trivial phase factor arising from the choice in the phase of the wave function, the reflec­tion coefficients satisfy the symmetry rela­tion rpa = —r&0.2 This leads to the absence of backward scattering r&a — — r&a = 0 in the

230

single-channel case. Define the reflection matrix r by [r]a@ =

ra /3. Then, we have r = — V, where V is the transpose of r. In general we have det *F = d e t P for any matrix P, where detP is the determinant of P. In metallic nanotubes, the number of traveling modes nc is always given by an odd integer and therefore de t ( - r ) = -det ( r ) , leading to det(r) = 0.

By definition, rg a represents the am­plitude of an out-going mode 0 with wave function V^(r) f° r the reflected wave cor­responding to an in-coming mode a with wave function ^ a ( r ) . Therefore, the van­ishing determinant of r shows the presence of a mode which is transmitted through the system with probability one without being scattered back. In fact, numerical calcula­tions show that the conductance decreases with the length but remains larger than the ideal conductance for a single channel.2

5. Effects of Symmetry and Channel

5.1. Electric Field

The absence of backscattering and the pres­ence of a perfect channel prevail even in the presence of a strong perturbation as long as the time-reversal symmetry is not disturbed. Take, for example, an electric field E perpendicular to the axis. Figure 1 shows an example of the energy bands and density of states in metallic nanotubes for (eEL/2n)(2n~(/L)-1 = 2. They are strongly distorted by the electric field. In an elec­tric field, an electron near a conduction-band bottom tends to be localized in the region of the lowest potential, while an electron in a valence-band bottom in the region of the higher potential, leading to the crossing of the conduction and valence bands.

Figure 2 shows some examples of the conductance as a function of the length. In actual calculations, we consider a system with randomly distributed scatterers with strength ±u and concentration n;. The scat­

tering strength is characterized by the di-mensionless quantity W = riiU2/47ry2. The number of the bands are 3, 1, 3, 3, 5, and 7 for e(27r7/L)-1 = 0.1, 0.5, 0.8, 1.5, 2.5, and 3.5, respectively. The conductance remains always larger than 2e2/irh corresponding to

(eEL/27t)(2;ry/L)-1 = 2.0

Wave Number

Density of States

CM

Energy (units of 2JCY/L)

Fig. 1. The energy bands and the density of states of a metallic tube in a strong electric field.

0.0

W"1 =1000.0

U/27L = 0.01

(eEL/27i)(2iry/L)-1 = 2.0

el_/2jty 0.100 0.500 0.800 1.500 2.500 3.500

50 100

Length (units of LW"

Fig. 2. The conductance as a function of the length in metallic tubes in the presence of a strong electric fields.

231

the presence of a perfect channel (actually two channels exist corresponding to the K and K' points).

5.2. Channel Number

The presence of a perfect channel requires the channel number to be odd as well as the symmetry. In the presence of the AB flux equal to the half of the flux quantum, i.e., (l/2)<t>o with <j)Q = c/i/e, the channel number becomes even without destroying the sym­metry of the Hamiltonian. Figure 3 gives ex­amples of the length dependence of the con­ductance in this case. The conductance de­creases exponentially with the increase of the length and therefore states are strongly local­ized, demonstrating the vital importance of the channel number.

5.3. Inelastic Scattering

Inelastic scattering breaking the phase mem­ory of an electron plays very important role also in destroying the perfect channel. In fact, the Boltzmann equation gives a finite conductivity when several bands coexist at

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Length (units of LW"1)

Fig. 3. The conductance as a function of the length in metallic tubes in the presence of the half of the magnetic flux quantum. (• • •) stands for the sample average.

the Fermi level although the conductivity becomes infinite in the energy range of the metallic linear band.2 This is contrary to the presence of a perfect channel giving the ideal conductance 2e2 /•nh for sufficiently long nanotubes. The difference originates from the absence of phase coherence in the ap­proach based on a transport equation. In fact, in the transport equation scattering from each impurity is treated as a completely independent event after which an electron looses its phase memory. The perfect chan­nel requires the phase coherence throughout the system.

Effects of inelastic scattering can be con­sidered in a model in which the nanotube is separated into segments with length of the order of the phase coherence length L^ and the electron looses the phase information af­ter the transmission through each segment. Figure 4 shows some examples of calculated conductance. As long as the length is smaller than or comparable to L^,, the conductance is close to the ideal value 2e2/irh corresponding to the presence of a perfect channel. When the length becomes much larger than L^,, the conductance decreases in proportion to the inverse of the length and there an approxi­mate conductivity proportional to L$ can be defined.

5.4. Symmetry Breaking Perturbation

The time-reversal symmetry is destroyed by a magnetic field H perpendicular to the axis or a flux <j> due to a field parallel to the axis. The dimensionless quantity character­izing the strength is (L/2nl)2 for the mag­netic field and <j)/<t>o for the flux with I = ^Jch/eH. The symmetry is affected also by scatterers with range smaller than the lattice constant a because the K and K' points can­not be regarded as independent due to mix­ing. Its strength is characterized by 6 which denotes the amount of short-range scatter­ers relative to the total amount of scatterers (0 < 5 < 1). It is destroyed also by the pres-

232

o = (4e2/7tfi) 2U •--. \ X 10"2 I ' i i i i I M 11 \ i \ i i \ i

10' 102 103 104

Length (units of L)

Fig. 4. The conductance in the presence of inelastic scattering as a function of the length for different values of the phase coherence length L^. The number of conducting channels is 3, 5, and 7 from the top to the bottom. W represents effective strength of scatterers and u strength of the potential of each scatterer.

ence of trigonal warping of the bands around the K and K' points.8 The effect of warping can be described by higher order k p terms20

and is characterized by j3a/L where /? is a di-mensionless quantity of the order of unity.

Using numerical results of the geomet­ric average of the conductance, i.e., G oc exp(—2aA), we can determine the inverse lo­calization length a, where A is the length.

This inverse localization length is a good measure describing the magnitude of effec­tive backward scattering.

Resulting Q'S at different values of en­ergy e are shown in Fig. 5. When there are several bands at the energy, the inverse local­ization length exhibits a near-plateau behav­ior above a small critical value of the symme­try breaking parameter. The plateau values are nearly same in all the cases and given ap­proximately by the inverse of the sum of the mean free path of the bands at the energy.2

When the energy is in the range of linear bands, the inverse localization length is much smaller unless the symmetry breaking pa­rameter becomes appreciable. Therefore, the perfect channel can easily be destroyed by a very small perturbation, while the absence of backscattering is very robust in the energy range of linear bands.

It should be noted that various effects can be incorporated as an effective mag­netic flux such as the nonzero curvature present in actual nanotubes21'22 and the lattice distortion.22,23'24 It is highly likely, therefore, that an effective flux of the order of 4>/<f>o ~ 10~2 is always present except in thick nanotubes. It turns out further that ef­fects of the trigonal warping are surprisingly strong among those symmetry breaking ef­fects. Even considering the ambiguity in the exact value of j3, we can expect that its ef­fective strength is well in the near-plateau re­gion for typical tubes. Except in very thick nanotubes, therefore, the perfectly conduct­ing channel is likely to be destroyed almost completely when there are several bands at the energy. The absence of backward scat­tering when the Fermi level lies in the region of metallic linear bands remain quite robust.

Acknowledgments

This work was supported in part by a 21st Century COE Program at Tokyo Tech "Nanometer-Scale Quantum Physics" and by

233

: W1 = 1000.0 (a) : U/2TL = O.OI

10°

.y 10-'

£ 10-2

,e(2iry/L)-'

0.50 1.50 2.50 3.50

(b) (c)

- ^

0.50 1.50 2.50 3.50

3 10 2 10"'

Magnetic Field: (L72itl)2 103 10 2 10"' 10"3 102 10"1

Magnetic Flux (units of <t>o) Short-Range Scatterers: 8 10"4 10'3 10 2 10-1

Trigonal Warping: |3a/L

Fig. 5. The inverse localization length as a function of (a) the magnetic field, (b) the magnetic flux, (c) the amount of short-range scatterers, and (d) the trigonal warping of the band in a logarithmic scale for a zigzag nanotube. The results are independent of the nanotube structure in (a)-(c) and weakly dependent in (d).

Grant-in-Aid for Scientific Research from

Ministry of Education, Culture, Sports, Sci­

ence and Technology, Japan. Numerical cal­

culations were performed in par t using the

facilities of the Supercomputer Center, In­

s t i tute for Solid State Physics, University of

Tokyo.

References

1. T. Ando and T. Nakanishi, J. Phys. Soc. Jpn. 67, 1704 (1998).

2. T. Ando and H. Suzuura, J. Phys. Soc. Jpn. 71 , 2753 (2002).

3. P. R. Wallace, Phys. Rev. 71 , 622 (1947). 4. J. C. Slonczewski and P. R. Weiss, Phys.

Rev. 109, 272 (1958). 5. H. Ajiki and T. Ando, J. Phys. Soc. Jpn.

62, 1255 (1993). 6. T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005). 7. M. V. Berry, Proc. Roy. Soc. London A392,

45 (1984). 8. T. Ando, T. Nakanishi, and R. Saito, J.

Phys. Soc. Jpn. 67, 2857 (1998). 9. T. Nakanishi and T. Ando, J. Phys. Soc.

Jpn. 68, 561 (1999). 10. S. J. Tans, M. H. Devoret, H. Dai, A. Thess,

R. E. Smalley, L. J. Geerligs, and C. Dekker, Nature (London) 386, 474 (1997).

11. A. Bezryadin, A. R. M. Verschueren, S. J. Tans, and C. Dekker, Phys. Rev. Lett. 80,

4036 (1998). 12. P. L. McEuen, M. Bockrath, D. H. Cobden,

Y. -G. Yoon, and S. G. Louie, Phys. Rev. Lett. 83, 5098 (1999).

13. A. Bachtold, M. S. Fuhrer, S. Plyasunov, M. Forero, E. H. Anderson, A. Zettl, and P. L. McEuen, Phys. Rev. Lett. 84, 6082 (2000).

14. J. Tersoff, Appl. Phys. Lett. 74, 2122 (1999). 15. M. P. Anantram, S. Datta, and Y. -Q. Xue,

Phys. Rev. B 61, 14219 (2000). 16. K. -J. Kong, S. -W. Han, and J. -S. Ihm,

Phys. Rev. B 60, 6074 (1999). 17. H. J. Choi, J. Ihm, Y. -G. Yoon, and S. G.

Louie, Phys. Rev. B 60, R14009 (1999). 18. T. Nakanishi and T. Ando, / . Phys. Soc.

Jpn. 69, 2175 (2000). 19. J. Kong, E. Yenilmez, T. W. Tombler, W.

Kim, H. J. Dai, R. B. Laughlin, L. Liu, C. S. Jayanthi, and S. Y. Wu, Phys. Rev. Lett. 87, 106801 (2001).

20. H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 65, 505 (1996).

21. T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000). 22. C. L. Kane and E. J. Mele, Phys. Rev. Lett.

78, 1932 (1997). 23. C. L. Kane, E. J. Mele, R. S. Lee, J. E. Fis­

cher, P. Petit, H. Dai, A. Thess, R. E. Smal­ley, A. R. M. Verschueren, S. J. Tans, and C. Dekker, Europhys. Lett. 41 , 683 (1998).

24. H. Suzuura and T. Ando, Physica E 6, 864 (2000); Mol. Cryst. Liq. Cryst. 340, 731 (2000); Phys. Rev. B 65, 235412 (2002).

234

OPTICAL PROCESSES IN SINGLE-WALLED CARBON N A N O T U B E S THREADED B Y A MAGNETIC FLUX

J. K O N O , 1 ' t S. ZARIC , 1 J. SHAVER, 1 X. W E I , 2 S. A. C R O O K E R , 3

0 . P O R T U G A L L , 4 G. L. J . A. R I K K E N , 4 R. H. H A U G E , 5 and R. E. S M A L L E Y 5

Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, U.S.A. 'E-mail: kono@rice.edu

National High Magnetic Field Laboratory, Florida State University,

Tallahassee, Florida 32310, U.S.A. National High Magnetic Field Laboratory, Los Alamos National Laboratory,

Los Alamos, New Mexico 87545, U.S.A.

Laboratoire National des Champs Magnetiques Pulses, 31432 Toulouse Cedex 04, France Department of Chemistry, Rice University, Houston, Texas 77005, U.S.A.

Single-walled carbon nanotubes threaded by a magnetic flux <p are predicted to posses novel mag­netic and optical properties, critically depending on the value of </>/<po where </>o is the magnetic flux quantum. This is a consequence of the Aharonov-Bohm phase 2n<j>/(j>o influencing the bound­ary conditions on the Bloch wavefunctions. Here we report results of a series of magneto-optical studies of micelle-suspended single-walled carbon nanotubes in aqueous solutions in high magnetic fields. Their exotic magnetic properties manifest themselves in near-infrared magneto-absorption and magneto-photoluminescence spectra, including static and dynamic magnetic linear dichroism, splittings of exciton peaks, and field-induced band gap shrinkage. We show that these observations are quantitatively consistent with existing theories based on the Aharonov-Bohm effect.

Keywords: carbon nanotubes; Aharonov-Bohm effect; magnetic flux quantum.

1. In t roduc t ion

The states of Bloch electrons are affected by a magnetic field in a non-trivial manner when the magnetic length is comparable to, or smaller than, the lattice constant. In such high fields, their energy spectra exhibit in­triguing fine structures as a function of the number of magnetic flux quanta (</>o = e/h) per unit cell.1 In a single-walled carbon nan-otube (SWNT), a similar, but much simpler, phenomenon is expected to occur when a magnetic field is applied parallel to the tube axis.2 Namely, the band gap of a SWNT is predicted to oscillate between zero and a fi­nite value as a function of magnetic flux (j> with period <f>o\ i.e., a SWNT can be either a metal or an insulator, depending on </>/</>o-This exotic behavior is a direct consequence of the Aharonov-Bohm (AB) phase influenc­ing the circumferential boundary condition on the Bloch wavefunction.

Here we present results of our recent magneto-optical experiments on carbon nan­otubes in high DC and pulsed magnetic fields,3'4'5 which provide the first experi­mental evidence for the influence of the Aharonov-Bohm phase on the electronic states of SWNTs. Due to the small diam­eters (~1 nm) and the relatively large room temperature linewidth of SWNTs (~20 meV) used in these experiments, a very high mag­netic field is required to make the predicted phenomena clearly visible.

Through near-infrared absorption and photoluminescence (PL) spectroscopies in DC magnetic fields up to 45 T,3 we ob­served field-induced optical anisotropy (i.e., linear dichroism) as well as peak shifts and splittings. The amounts of shifts and split­tings depend on the value of (f>/<fio a n d are quantitatively consistent with existing theo­ries. In particular, our data clearly demon-

235

strated that the band gap of semiconduct­ing SWNTs with 1 nm diameter monotoni-cally shrinks with magnetic field with a rate ~1 meV / T. In addition, through detailed analysis of the magnetic field-dependence of the shifts and splittings, we determined the magnetic susceptibility anisotropy of semi­conducting nanotubes.4

Furthermore, using pulsed magnetic fields up to 75 T,5 we made a clear obser­vation of absorption peak splittings, which show that the degeneracy between the K and K' points in graphene fc-space can be lifted by the breakdown of the time-reversal sym­metry caused by the application of a mag­netic field. Finally, we present time depen­dent magneto-optical transmission measure­ments in pulsed fields up to 54 T. Polarized optical transmission in the Voigt geometry reveals time dependent optical anisotropy, indicating that the nanotubes can dynam­ically align in response to the millisecond time-scale pulsed fields. Unlike the previous studies with DC fields, however, the trans­mission dynamics show a time lag with re­spect to the field pulse, suggesting that nan-otube alignment/de-alignment is hysteretic. These results imply that while magnetic fields can be used for manipulating nan­otubes, it is necessary to take into account nanotube inertia as well as environmental properties in correctly understanding and predicting the magneto-alignment dynamics.

2. Theory

2.1. The Aharanov-Bohm effect on nanotube band structure

In the presence of a magnetic field B, the generalized periodic boundary condition in the circumferential direction is given by

ip(r + Ch) = ip(r) exp(2ni(f), (1)

where <p = (f>/4>o and Ch = (n, m) is the chiral vector. On the other hand, because of the existence of the periodic potential along the

circumference, the Bloch condition has to be satisfied, i.e.,

Tp(r + Ch) = i>(r)exp(ik-Ch), (2)

where k is the Bloch vector. In order for Eqs. (1) and (2) to be simultaneously sat­isfied, k along the circumference has to be quantized as k • Ch = m + <p, where m is an integer. Thus, allowed k states continuously and periodically change with the Aharonov-Bohm phase 2nip.

Using k p theory, Ajiki and Ando solved an effective-mass equation near the K point to find2: E±,n(k) = ±7 V

/ «(n ,0 ) 2 + k2, where 7 « 0.646 eV-nm, n{n,<f>) = (2TT/L) (n - i//3 - (t>/<t>o), L = \Ch\, n =

0, ± 1 , . . . (subband index), and + (—) cor­responds to the conduction (valence) band. Thus, the band gap is

4> En 27|«(0)| = l p

4>o (3)

At zero field ((j> = 0), Eg is zero for metallic (v = 0) nanotubes while Eg = 4wy/3L for type-I {y = 1) and type-II (u = —1) semi­conducting nanotubes. These simple rules break down at finite magnetic fields (<fi ^ 0), and Eg exhibits oscillations with period <po, i.e., the magnetic flux quantum. Further­more, since a magnetic field breaks the time-reversal symmetry, the degeneracy between the K and K' points is lifted. Specifically, for a type-I semiconducting nanotube, the low­est transition (En) splits into

3</>N

E0

4TT7 / 3 0 \

3L V 4>o) (4)

The magnitude of splitting is given by

A A B = 8TT74>/L<I>Q = 2jLB/<t>0 = vB (5)

Similar splitting is expected for higher sub-band transitions as well.6'7'8'9'10 As a nu­merical example, let us consider a (10,3) tube, for which v = 1, L = 2.936 nm, and Eg(<f> = 0) = 47T7/3L = 0.922 eV. Equa­tion (5) then predicts a splitting of AAB W

41 meV at 45 T.

236

2.2. Magnetic alignment

Carbon nanotubes have anisotropic magnetic properties.11 An interesting consequence of this is magnetic alignment. In fact, it is important to understand the angular distri­bution of nanotubes in the solution at each magnetic field value in order to extract the true AB splitting values from data.

While the splitting rate v [Eq. (5)] is de­fined for a nanotube parallel to the applied field, experimentally we observe an apparent splitting rate of an ensemble of nanotubes with an angular distribution characterized by the probability of finding a nanotube in an angular range relative to the B direction. At 0 T, the nanotubes are randomized in solu­tion. In a magnetic field, the nanotubes align due to their magnetic anisotropy. The angu­lar probability distribution is given (in spher­ical coordinates) by:

v> (a\AO exp(-u2sin20)sin

/ 0 exp(-u2sin 0) sin 9d0

where

IB2N(X\\-X±) kBT

(6)

(7)

where x± and X|| are the diamagnetic sus­ceptibilities per mole of carbon atoms in a nanotube along its two principal axes, and 8 is defined as the angle between the nanotube long axis and the magnetic field.

Using Eq. (6), one can calculate the expectation value of any physical quantity that depends on the angle 6. One use­ful quantity is the so-called nematic order parameter,12 '13 '14 defined as

S = (3 (cos2 6) - l ) /2 . (8)

This parameter is 0 when the nanotubes are completely randomly oriented and 1 when all the nanotubes in the solution are pointing in the magnetic field direction. Figure 1 shows S calculated for DC fields. If a DC 100 T magnet existed, it would be ideal for aligning our small diameter SWNTs.

CO

1.0

0.8

0.6

0.4

0.2

0.0

-l—r-rTTTTn 1—m

i i i 1 1 1 1

10 100 Magnetic Field (T)

1000

Fig. 1. Calculated steady state nematic order pa­rameter, S. The magnetic susceptibility anisotropy4

(1.4 X 10~5 emu/mol), average length (300 nm), di­ameter (1 nm), and temperature (300 K) were used in calculation.

3. Experimental

The samples studied in the present work were aqueous suspensions of individualized HiPco nanotubes produced at Rice. The samples were prepared by high sheer mixing of nan­otubes in a 1 wt. % solution of surfactant (sodium dodecyl sulfate, sodium benzosul-fonic acid, or sodium cholate) in D2O, son-ication, and centrifugation. The resulting supernatant was enriched in individualized SWNTs surrounded by surfactant micelles.15

The nanotubes are thus unbundled and pre­vented from interacting with each other, which leads to chirality-dependent peaks in absorption and PL.15 '16

We used a variety of high-field magnets for performing magneto-optical experiments. For DC field experiments a 10 T supercon­ducting magnet (at Rice), a 33 T water-cooled magnet, and the 45 T hybrid mag­net [both in Tallahassee, Florida] were used. Pulsed field experiments were conducted us­ing capacitor bank driven magnets of 67 T (in Los Alamos, New Mexico), 54 T, and the 75 T ARMS magnet17 (in Toulouse, France).

Samples were mounted on a specially designed stick with integrated optic fibers, lenses, mirrors, and polarizers. The nan-

237

otube solution was contained in a quartz cell with an effective optical length of 0.5 cm for absorption (0.1 cm for PL). Absorption spec­tra were normalized to a 1 wt. % surfactant in D2O solution.

We used a Si charge coupled device (in the visible) and an InGaAs array detector (in the near-infrared). A quartz-tungsten-halogen lamp was used for the absorption measurements, and a Ti:Sapphire laser was used in the PL measurements. Experiments in pulsed fields were recorded with an expo­sure plus readout time of ~1.5 ms, this re­sulted in spectra taken in ~.5 T increments.

We utilized Voigt geometry with two different polarizations (P) for these experi­ments, B || P and B ± P. Due to the sam­ples being in aqueous suspensions, all mea­surements were done at room temperature.

4. Resul t s

Absorption spectra in the near-infrared range are shown in Fig. 2 at various DC fields up to 45 T. The peaks in the specra corre-

1.0 1.1 Energy (eV)

Fig. 2. Magnetic field induced anisotropy. Room temperature absorption of the SWNT sample was measured in Voigt geometry for two polarizations of the probe light: B \\ P (B J. P). Dashed lines repre­sent results of our model based on AB effect and the magnetic alignment. No intentional offset.

0.9 1.0 Energy (eV)

Fig. 3. Absorption spectra at various magnetic fields taken in the Voigt geometry with light polar­ization parallel to the field. The traces are offset for clarity. The inset shows relative change in absorption at 2.3 eV during a 67 T shot.

spond to the lowest-energy transitions (En) in semiconducting nanotubes with different chiralities. As the field increases, the B \\ P (B -L P) absorption increases (decreases) as a result of magnetic alignment; note that the E\\ transitions are allowed only for the light polarized parallel to the tube axis. At 45 T, while the B _L P absorption (sensitive to the nanotubes lying more perpendicular to the B) shows only peak broadening, the B \\ P absorption (sensitive to the nanotubes lying more parallel to B) shows more pronounced spectral changes. This can be successfully explained using our model (see the dotted lines, explained later) that takes into account magnetic alignment and AB splitting. How­ever, clear absorption peak splittings cannot be seen even at 45 T.

238

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Fig. 4. Dynamic alignment. Transmis­sion anisotropy in the second subband during a 54 T shot (a). S, calculated from anisotropy, plotted on a semilog scale shows characteristic relaxation time for nanotubes in solution.

The use of pulsed magnets allowed us to access higher fields. Figure 3 shows B \\ P absorption spectra taken during a 67 T mag­netic pulse. At fields above 54 T, clear ab­sorption peak splittings are observed. In ad­dition, time-dependent polarized transmis­sion allows for the observation of nanotube solution dynamics. Figure 4 shows a de­crease (increase) in transmission for B \\ P (B ± P) as field is applied. As the field in­creases the nanotubes begin to align reach­ing a maximum order parameter of ~0.35 at 54 T. The 45 T DC case reaches a maximum order parameter of ~0.5, indicating that the tubes do not completely align on fractional second time scales. This is due to many fac­tors, including nanotube inertia (dependant on length and diameter) and environment (depending on solution viscosity and prox­imity of other nanotubes).

Figure 5 shows PL spectra at 0 T and 45 T. A 790 nm beam selectively excited four chiralities in the sample. This selec­tivity allows us to see much clearer spectral

0)

0.90 0.95 1.00 1.05 Energy (eV)

1.10 1.15

Fig. 5. PL spectra at 0 T and 45 T showing magnetic-field-induced band-gap shrinkage due to the Aharonov-Bohm effect. The data was taking with 790 nm excitation, and different peaks corre­spond to different chirality nanotubes.

changes in PL data compared to absorption data. Namely, all peaks shift to lower ener­gies as B is increased. The dotted lines are simulations based on our model taking into account the AB effect and magnetic align­ment and explain the observations very well.

5. Discussion

In a magnetic field parallel to a nanotube axis, absorption and PL peaks are predicted to split by an amount proportional to the ap­plied field [see Eq. (5)]. The absorption peaks are predicted to split into two equal height peaks with the separation proportional to B. PL peaks will split with the same splitting rate, but with the relative size of the two peaks determined by the Boltzmann factor.

The 0 T PL spectrum was fit using Lorentzian peaks that correspond to the chi­ralities present in the sample. The 45 T PL spectrum was then simulated by varying two parameters for each Lorentzian, u and v. The first parameter, u, describes the angular dis­tribution of the nanotubes at a given B field [see Eq. (7)]. The second, v, is the rate of peak splitting with the field applied paral­lel to the nanotube [see Eq. (5)]. For any given 6 this rate was multiplied by cose? to

239

account only for the flux component thread­ing the nanotube. The different intensities of the split PL peaks were taken into account through a Boltzmann factor (with tempera­ture T = 300 K). The best 45 T fit is shown as dotted lines in Fig. 5. The average splitting rate obtained in this way is v = 0.9 meV/T.

We use a\f'm\hu,B) and a^'m) {hw, B) to denote the intrinsic absorption spectra (in c m - 1 mole - 1) for (n,m) nanotubes in paral­lel magnetic field B when the light is polar­ized parallel and perpendicular to the long axis of the nanotube, respectively (where hw is the photon energy). Theoretically, a±'m (hw, B) is predicted to be zero for E\\ and 2?22 transitions due to the depolariza­tion effect,18 but we retain this quantity in the following formal derivation for the sake of completeness.

The intrinsic absorption spectra defined above are not directly obtainable from ex­periments due to the angular distribution of nanotubes. For example, the measured B || P absorption for (n, m) tubes is an av­erage over the angular distribution of nan­otubes given by

a\^m)(hw,B) =

(aj" ,m)r2 + <4n'm V + <z2)) • AT(".-) (9)

where ay and af,™ in the brackets are functions of hw and B cos 6, p, q (r) are the directional cosines between the polarization direction and the short (long) principal axes of the nanotube, and N^n'm^ is the number of moles of (n, m) nanotubes. Similarly,

a^'m)(hw,B) =

/ •*• (n,m) • 2 / > , ( i i " > ) / i • 2 / i \ \

( -0 | , sin 0 + a\ (1 — sin 6) )

xN(n,m) ( 1 Q )

We used Eqs. (9) and (10) (with a("'m) = 0) to simulate our absorption data. At 0 T,

Eq. (9) becomes:

a j n , m ) (hw, o r ) = o[,n,TO) (hw, o r ) (cos2 e)

= ^-a(^m)(hw,Vr)-N^m\ (11)

which relates a(^'m)(hw,B cos 9) • JV("-m) at 0 T with the measured 0 T absorption. The 67 T absorption can then be simulated us­ing Eq. (9) by fitting parameters u^n'm^ and v(n,m) a g j n j.jje P L simulations.

When fitting the 0 T absorption data, several closely spaced (n, m) peaks might be fitted as a wider absorption peak. Neverthe­less, this simulation yields data that agrees well with the measured data [see Fig. (3)] and yielded (n, m)-averaged u = 1.9 and v = 0.7 meV/T. The same model was then ap­plied to the 45 T data [see Fig. (2)]. This treatment again yielded an average splitting rate of 0.7 meV/T. Furthermore, keeping the same fitting parameters, the B A. P data was successfully reproduced at 45 T (see Fig. 2).

The inset in Fig. 3, as well as the data in Fig. 4(a), shows that the absorption change does not exactly follow the applied pulse, i.e., there is a lag between the peak field and peak alignment. Thus, while the pulsed field data in Fig. 3 are explained well using the an­gular distribution given by Eq. 6, u is no longer given by Eq. 7, but is now a mere parameter. Finally, in Fig. 4(b), magnetic field and nematic order parameter are plot­ted on a semilog scale. It can be seen that the residual alignment persists even after the field goes to zero. This shows that nanotube inertial effects and solution viscosity must be accounted for when measuring nanotube alignment dynamics.

6. Conclusions and Future work

We have observed the modification of states of Bloch electrons by an Aharonov-Bohm phase in SWNTs by magneto-optical spec­troscopy. We also investigated both DC and dynamic magnetic alignment of SWNT so-

240

lutions. There are still many questions tha t

need to be explored such as solution dynam­

ics, tempera ture , other geometries and po­

larizations, and higher field dependences.

When analyzing dynamic alignment da ta

by plott ing on a semilog scale it becomes

apparent t ha t this is a form of pump-probe

spectroscopy. The short field pulse prepares

the nanotubes in an aligned state (magnetic

pump) , while the transmission probes the re­

laxation dynamics (optical probe). This da ta

can be analyzed to enlighten the inertial and

environmental effects on the thermal motion

of the nanotubes in solution. Analysis of

da ta on samples with different lengths and

bundling s tates is underway.

We are also expanding this work with de-

structuve pulsed magnets up to 150 T in Los

Alamos and up to 300 T at Humboldt Uni­

versity in Berlin. We have already completed

single wavelength solution dynamics work in

Berlin,1 9 and magneto-absorption measure­

ments are currently in progress. Aligned

nanotube gel films, fabricated by several-

groups, are ideal for ultrahigh field and low

tempera ture magneto-optics studies. Film

samples can be aligned, cooled, are solid, and

are very thin. These characteristics bypass

alignment convolution, tempera ture restric­

tions, allow for easy handling and smaller,

simpler sample holders.

A c k n o w l e d g e m e n t s

This work was supported in par t by the

Robert A. Welch Foundation (through Grant

No. C-1509) and the National Science Foun­

dation ( through Grant Nos. DMR-0134058,

DMR-0325474, and INT-0437342). A por­

tion of this work was performed a t the

National High Magnetic Field Laboratory,

which is supported by NSF Cooperative

Agreement No. DMR-0084173 and by the

State of Florida. We thank V. I. Klimov

for the use of his InGaAs detector for mea­

surements in Los Alamos. We also thank

G. N. Ostojic, V. C. Moore, and M. Furis

for technical assistance.

References

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3. S. Zaric, G. N. Ostojic, J. Kono, J. Shaver, V. C. Moore, M. S. Strano, R. H. Hauge, R. E. Smalley, and X. Wei, Science 304, 1129 (2004).

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5. S. Zaric, G. N. Ostojic, J. Shaver, J. Kono, O. Portugall, P. H. Frings, G. L. J. A. Rikken, M. Furis, S. A. Crooker, X. Wei, V. C. Moore, R. H. Hauge, and R. E. Smalley, Phys. Rev. Lett., to appear (cond-mat/0509429).

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haus, and R. Saito, Phys. Rev. B 62, 16092 (2000).

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14. Y. Murakami, E. Einarsson, T. Edamura, and S. Maruyama, Phys. Rev. Lett. 94, 087402 (2005).

15. M. J. O'Connell, S. M. Bachilo, C. B. Huff­man, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weis-man, and R. E. Smalley, Science 297, 593 (2002).

16. S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, and R. B. Weis-man, Science 298, 2361 (2002).

17. H. Jones, P. H. Frings, M. von Ortenberg,

A. Lagutin, L. V. Bockstal, 0 . Portugall, and F. Herlach, Physica B 346-347, 553 (2004).

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19. J. Shaver, J. Kono, S. Hansel, A. Kirste, M. von Ortenberg, C. H. Mielke, O. Por­tugall, R. H. Hauge, and R. E. Smalley, in: Narrow Gap Semiconductors 2005, Pro­ceedings of the Twelfth International Con­ference on Narrow Gap Semiconductors, eds. J. Kono and J. Leotin (Institute of Physics, Bristol, 2005, to be published).

242

NON-EQUILIBRIUM TRANSPORT THROUGH A SINGLE-WALLED C A R B O N N A N O T U B E WITH HIGHLY T R A N S P A R E N T COUPLING TO

RESERVOIRS

P. R E C H E R 1 , 2 ' * , N. Y. K I M 1 A N D Y. Y A M A M O T O 1 ' 3

1 Quantum Entanglement Project, SORST, JST,

E.L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA

Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku,

Tokyo 153-8505, Japan National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,

Tokyo 101-8430, Japan * E-mail: recher@stanford.edu

We consider transport through a single-walled carbon nanotube (CNT) in the ballistic regime where weak backscattering at the two contacts gives rise to Fabry-Perot (FP) oscillations in conductance and shot noise. We include the electron-electron interaction and the finite length L of the CNT within the inhomogeneous Tomonaga-Luttinger liquid (TLL) model appropriate for the CNT and treat the non-equilibrium effects due to an applied bias voltage within the Keldysh approach. At low frequencies, the shot noise is S = 2e\Is\, where e is the elementary charge and Jg is the backscattered current. Interaction effects are apparent via a characteristic power-law and FP-oscillation suppression of Ig at large bias voltage in agreement with our low-frequency shot noise experiments. The extracted TLL parameter g agrees well with theoretical predictions. At frequencies above vp/gL, with vp the Fermi velocity, the theoretical impurity noise as a function of bias voltage and frequency shows clear and distinct signatures of the two velocities of collective modes present in the CNT which distinguish spin from charge degrees of freedom (spin-charge separation).

Keywords: Carbon nanotubes, Tomonaga-Luttinger liquids; shot noise

1. Introduction

In one dimension (ID) electron-electron in­teraction cannot be neglected anymore. Dif­ferent from the higher-dimensional counter­parts described successfully by the Fermi liq­uid theory, these ID systems lack the pic­ture of independent quasiparticles and the low energy excitations are collective charge and spin modes travelling with different ve­locities. The class of systems showing such low-energy behavior is named Tomonaga-Luttinger liquids (TLL)1.

Metallic single-walled carbon nanotubes (CNT) have been predicted to behave as a TLL2 which has been verified in several experiments3. These experiments, however, have been restricted to the tunneling regime where the conductance is much smaller than

the theoretically possible upper limit of 2Go where Go = 2e2/h is the spin-degenerate quantum unit of conductance and 2 comes from subband degeneracy in the CNT. More recently, conductance close to 4e2//i has been measured in CNTs although only at low bias voltage4. In this so called Fabry-Perot (FP) regime the low-energy transport shows in­terference oscillations in conductance due to weak backscattering at the two contacts. As the bias voltage is increased, the TLL effect becomes pronounced and, as a consequence, the conductance oscillations reduce in size. This has been investigated theoretically at zero temperature in the absence of experi­mental results5. In this work we will extend this theory to include a discussion of the shot noise properties. We compare the theoreti­cal predictions of this model with our low-

243

frequency shot noise measurements in CNTs at 4 K. Good agreement between experiment and theory is observed in terms of power-law scaling of shot noise and Fano factor as well as decreased FP-oscillation amplitudes when the bias voltage is increased. The deduced TLL-parameter g agrees well with theoretical predictions. At higher frequency co, the shot noise becomes sensitive to the current frag­mentation at the inhomogeneity of g6. The frequency response of shot noise shows pro­nounced oscillations with frequency vp/Lg = ULI9 which are dominant over the ordinary FP-oscillations exhibited by all four modes5. On the other hand, the bias response of shot noise shows FP-oscillations dominated by the non-interacting modes with frequency w^(due to subband degeneracy and spin). Therefore, the investigation of bias-voltage and frequency dependent shot noise brings out important information about the differ­ent velocities of CNT-collective modes.

2. System and Formalism

We consider a CNT where source and drain contacts are deposited on top defining a cav­ity of length L. Electrons flow from the source contact via the CNT to the drain con­tact due to an applied bias voltage Vas. In addition, a back gate voltage Vg can tune the density of electrons in the CNT. The electron-electron interaction is supposed to be strong in the CNT but weak in the higher dimensional electron reservoirs. We there­fore model the contacts as TLLs with g = 1, whereas g < 1 in the CNT. The Hamiltonian of such a system is5

Ht CNT 1 4 f

a = l

--{dxeaf} 9a

(1)

where 6a and na(x) = —(l/Tr)dx<t>a are conju­gated bosonic variables, i.e. [6a(x),TTt)(x')] = i5abS(x — x'). The flavor index a denotes

total(relative) charge a = 1(3) and tc-tal(relative) spin a = 2(4) of the two bands that cross at the Fermi energy2. The long-ranged Coulomb interaction between elec­trons couples primarily to the total charge, leading to <?i = g ~ 0.2 — 0.3, and ga = 1 for a=2-42. The velocity of the bosonic modes are va = vpj ga. To calculate transport prop­erties through the CNT we use the Keldysh generating functional

a J

<-ijdt>[H+s(t') x exp «oi -iJM'j dx'r){x',t')6\(x',t') (2)

where SQ denotes the Keldysh action of HeNT- The interfaces between the CNT and the two contacts produce the small backscat-tering of electrons described by Hbs(t) =

u^exp{iA^ijs(t)], where m de­notes the two barriers. The phase oper­ator is A^ijs{t) = 6fm - VAst - Vgxm +

Sij)(4>fm + s 4>fm)- The strength of backscat-tering is parametrized by the amplitudes u1^ where i,j label the two bands of the CNT. The bias voltage Vds and gate voltage Vg

(in units of L_1) are incorporated as phase factors in H^s(t)

5 . Finally, s = ± distin­guishes spin-up and spin-down electrons, re­spectively involved in the scattering event (assumed to be spin-independent and spin-conserving). The current operator I(x,t) = —e(2/ir)9i(x,t) couples to the real-valued source field r](x,t) in Eq. (2). It holds that 0l = (0+ +6^)/2 where ± denotes the for­ward and backward time-path of the Keldysh contour, respectively.

3. Current Noise

The current noise is defined as

S(x,w)=2 [ dteiuJt S(x,t), (3)

244

with S(x,t) = (l/2)({SI{x,t),SI(x,0)}) where {...} denotes the anticommutator, Sl(x,t) = I(x,t) - (I) and (...) = Tr^o-with po the initial density matrix before Hba

is switched on. The noise, to leading order in the backscattering amplitudes u^ , can also be written in terms of the Keldysh generat­ing functional denned in Eq. (2) as

c, \ 1 6 2 f . / PZ" |

S{x,U) = ~~e JcL, ^ . ( a .w ) & ? ( a .w / )U (4)

4. Low-Frequency Noise

We evaluate Eq. (4) for low frequencies w and to leading order in the backscattering. At low temperatures or equivalently for high bias voltage we obtain the shot noise

S = 2e\IB\, (5)

which is independent of the measurement point x. IB is the backscattered current

6=1,2

x / d r e C l b ( r ) sin[R16(T)/2] sin(wr).(6)

In Eq. (6) we introduced the retarded functions R W ( T ) = J J J ^ T ) + ^mm'(T)

where I and F refer to interacting and free, respectively. The interacting (free) functions describe the propagation of the total charge (free modes) between the two barriers. They are defined as, e.g. for the charge mode,

R1mm,{r) = -i&{r)({elm{r),9lm>{Q)})-

The corresponding correlation function is < w t o = <{0imM,<W(O)}>/2 and C w ( r ) = Clm,(r) + 3C%m,(T). This states that every backscattering event in­volves the backscattering of four effective modes of the CNT where one is interact­ing and three are non-interacting. These correlation functions have been given in Ref.5 for zero temperature. The general­ization to finite temperatures will be dis­cussed elsewhere7. The terms with m ^ m'

(b = 2 in Eq. (6)) lead to the FP-contribution whereas terms with m = m' (b = 1 in Eq. (6)) contribute to the incoherent part of the backscattered current. The dimension-less effective backscattering amplitudes U\ and U2 depend on the amplitudes u1^. In ad­dition, U2 is modulated by the gate voltage5. We also introduced the ideal current in the absence of backscattering, IQ = (4e2//i)Vds

. and the dimensionless voltage v = eVds /^ i -Due to the non-interacting reservoirs a TLL induced renormalization of the backscattered charge is absent in agreement with recent results on shot noise in TLLs6'8. However, the voltage dependence of shot noise or Fano factor F = S/2eI with I = I0 + IB shows a pronounced TLL effect. We calculate the large voltage behavior IB OC |Vds|

1+a with a = - (1/2)(1 - g)/(l + g) up to small FP-oscillations coming from the C/2-term. F shows a similar behavior with 1 + a —> a.

4.1. Theory vs Experiment

We compare the theoretical predictions with shot-noise experiments on a three-terminal geometry consisting of source, drain and back gate at 4 K. The shot noise mea­surements were performed by placing the CNT device and a weakly coupled light emitting diode (LED) and photodiode (PD) pair, serving as a standard full shot-noise source, in parallel. The input-referred volt­age noise of the circuit was approximately 2.2 nV/x/Hz at 4 K with resonant frequency ~15 MHz. Experimentally, the Fano factor F was obtained by F{I) = SCNT{I)/SPD{I)

for various dc current values J. In Fig. 1 we show a log-log plot of the Fano factor mea­sured for a CNT of length L ~ 360 nm as a function of bias voltage Vds at a particu­lar gate voltage Vg = —7.9 V. The average value of the power exponent in F for this sample over many different gate voltages is ~ —0.34, and the inferred g value is ~ 0.19, which is close to theoretical predictions2. In

Fig. 1. Experimental F (diamonds) is compared with theory using Eq. (6) for g = 0.25 and T = 0 K (dotted line). The full line is a fit to the experiment that follows the power-law F oc V£s with a ~ —0.35. From this we extract a TLL-parameter of g ~ 0.18. The highest voltage is V^s = 40 mV. For comparison we plotted the theoretical curve for g = 1 (triangles) which has a = 0 (full line). We used U\ = 0.14 and XJl = 0.10 for the theoretical curves.

Fig. 1 we also clearly see that FP-oscillation become damped at high bias voltage and that the oscillation frequency is close to the theo­retical curve (dotted line) ~ vp/L with vp = 8 x 105m/s. We also analyzed SPD and SCNT

separately9 and found average power-law be­haviors with a ~ 0 for SPD and a ~ —0.31 for SCNT leading to g ~ 0.25.

5. High-Frequency Noise

In Fig. 2 we present the excess noise Se{V,co) = S(V,u) - 5(0, w). We calculated Eq. (4) for general frequency w, temperature T and position of measurement x7. In addi­tion to the ordinary FP-oscillations induced by the interference of backscattered collec­tive modes at the two barriers we find much stronger oscillations in Se as a function of frequency at high bias voltage and with time period 2g/cji. This is due to momentum-conserving reflections of partial charges due to the inhomogeneity of g when traversing the CNT-contact interfaces10. Comparing FP-oscillations in the bias response of Se (or F) dominated by U>L with the frequency re-

Fig. 2. 3D plot of the excess noise Se in units of fio>x,2Go at T = 4 K and g = 0.23 as a function of dimensionless voltage v and frequency ui (in units of U>L) for a measurement point at one of the two barriers. We used U\ = U2 = 0.12.

sponse of shot noise would therefore allow us to determine g without fitting to a power-law and without referring to any system param­eters. In addition, the clear distinction of the two velocities of collective modes brings out valuable information about spin charge separation.

Acknowledgments

We thank the ARO-MURI grant DAAD19-99-1-0215 for financial support.

References

1. J. Voit, Rep. Prog. Phys. 57, 977 (1994). 2. R. Egger and A. Gogolin, Phys. Rev. Lett. 79,

5082 (1997), C. Kane, L. Balents, and M.P.A. Fisher, Phys. Rev. Lett. 79, 5086 (1997).

3. M. Bockrath et al., Nature 397, 598 (1999), Z. Yao et al, Nature 402, 273 (1999).

4. W. Liang et al., Nature 411, 665 (2001). 5. C.S. Pega, L. Balents and K.J. Wiese, Phys.

Rev. B68, 205423 (2003). 6. B. Trauzettel et al., Phys. Rev. Lett. 92,

226405 (2004), F. Dolcini et al., Phys. Rev. B71, 165309 (2005).

7. P. Recher et al., in preparation. 8. V.V. Ponomarenko and N. Nagaosa, Phys.

Rev. B60, 16865 (1999). 9. N.Y. Kim et al., submitted. 10. I. Sati and H. Schulz Phys. Rev. B52, 17040

(1995).

246

TRANSPORT PROPERTIES IN LOW DIMENSIONAL ARTIFICIAL LATTICE OF GOLD NANO-PARTICLES

S. SAITO* , T . A R M , H. F U K U D A , D. H I S A M O T O , S. K I M U R A , a n d T . ONAI

Central Research Laboratory, Hitachi, Ltd., Kokubunji, Tokyo 185-8601, Japan. *E-mail: ssaito@crl.hitachi.co.jp

Transport properties were examined in monolayer and sub-monolayer films composed of Au nanoparticles covered with organic thiol ligands. In the monolayer film, the observed linear conduc­tance scaled exponentially with the ligand chain length. On the other hand, in the sub-monolayer film, nonlinear current-voltage characteristics were observed as expected for one-dimensional me­andering paths induced by structural defects.

Keywords: nanoparticle; artificial lattice; low dimensionality.

1. Introduction

Strongly correlated electron systems :

are potential candidates for future nano-electronic devices. The basic prerequisite for the application is to control inherent parame­ters, such as, transfer (t) and interaction (U) energies in the Hubbard type model, carrier density (n), and dimensionality (D) 1.

However, these parameters are difficult to control in d-band materials like cuprates or manganites, because a gate control of high n (~ 1015 cm - 2 ) or a high pressure (~ 1 GPa) is required for the Mott transition. By contrast, energy scales and lattice structure may be significantly controlled in an artificial lattice, where lattice sites are quantum dots instead of single atoms or molecules.

Arrays of nanoparticles are considered to belong to artificial strongly correlated sys­tems 2. A breakthrough in chemical synthe­sis 3 enables the mass production of nanopar­ticles, whose size (< 10 nm) is enough small to observe single electron charging at room temperature. However, the basic transport mechanisms of monolayer films are not com­pletely understood.

In this paper, we examined the impacts of low dimensionality and many-body effects on transport properties in an artificial lat­tice of Au nanoparticles. We prepared the

2D monolayer films of Au nanoparticles us­ing the Langmuir-Blodgett method. The lin­ear I-V characteristics were observed for four orders of magnitude, and the sheet conduc­tance was limited by the tunneling between adjacent nanoparticles. By contrast, in a sub-monolayer film, prepared by introducing structural defects, we observed nonlinear I-V characteristics. We discuss the origin of this nonlinearity in the context of a Tomonaga-Luttinger liquid 5 in ID percolation paths.

2. Experiments

A Si substrate with 200-nm-thick Si02 was prepared using thermal oxidation at 1000 °C. We avoided using S13N4 dielectrics, which may well have a quenched charge disorder that induces threshold behaviors in I-V char­acteristics. In fact, according to our esti­mate in a state-of-the-art Si process line, SisN4 film typically contains positive fixed charges with the density of nex > 5 x 1012

cm - 2 , which roughly corresponds to one fixed charge per every Au nanoparticle. In contrast, nfix of Si02 is more than two orders of magnitude less, so the quenched charge density is estimated to be less than 1% of the total nanoparticle density.

Au electrodes were patterned using con­ventional lithography and lift-off. The ac-

247

tive channel length, L, separated by adjacent Au electrodes, was varied from 5 to 100 /tm, and the width, W, of each electrode was 200 //m. After the formation of electrodes, the monolayer film of An nanoparticles (Figs. 1 and 2) was prepared on the substrate using Langmuir-Blodgctt (LB) techniques 2. To examine the impact of structural defects, we also prepared the sub-monolayer LB film by reducing the pressure during the film forma­tion in the trough. In this way, we intention­ally introduced voids in a controlled manner.

depended linearly on L using two-terminal measurements and that the contact resis­tance was small using four-terminal measure­ments. The Ohm's law was also confirmed for all the monolayer films of nanoparticles with other ligands. Then, we obtained the sheet conductance, G, which decreases expo­nentially with I, as shown in Fig. 4. This tendency and the magnitude of G in our 2D films are consistent with previous reports on 3D films, prepared by casting droplets and evaporating solvent.

3. Results and Discussion

3 .1 . Monolayer film

Figure 3 shows the I-V characteristics of the monolayer film of Au nanoparticles with C4 thiols, where the linear I-V character­istics were maintained at least for four or­ders of magnitude at temperatures (T) from 300 K to 50 K. We found that the resistivity

Organic ligand

Fig. 1. Au nanoparticle used as lattice site. Organic chain length, I, determines lattice constant, d ~ 21.

— 10nm

Fig. 2. Transmission electron microscope image of self-organized Au nanoparticles covered with C12 thi­ols.

3.2. Estimation of transfer energy

To understand the dependence of G on I, we analytically calculated the transfer en-

SIO

• I I [ M i l l E - .WOK I "-250K

F-2Q0K IMiK

r- IOOK : - * > K

r ^&

^*^^

I I M i l l

U M U i t n

I I M i l l I I I I J J

^JL^0 •

-T

u

10 II) II)

Vollagc(V)

Fig. 3. Linearity of I-V characteristics for 2D monolayer film of Au nanoparticles with C4 ligands.

0 0.5 1 1.5 2 2.5 Length (nm)

Fig. 4. Observed sheet conductance (G) exponen­tially decreased with the ligand chain length. G is proportional to the calculated transfer energies (t) for both electrons and holes.

248

ergy for electrons (te) and holes (th) using an instanton method. Note that by electron (hole), we mean one extra electron added to (removed from) an Au nanoparticle. Al­though conduction-band electrons with the number comparable to that of Au atoms are present in each nanoparticle, the state with (without) a single extra charge can be clearly distinguished due to single electron charging, as confirmed by voltammetry measurements. We assumed a simple double-well potential, V(x) = m0uj2(x2 - l2)2/(8l2), for the tun­neling along the ligand chain direction, x, where mo is the effective mass and w is the attempt frequency. We used mo = 0.25me

with the bare electron mass, me. We ob­tained w = 2y/2AE/mo/l, where AE is bar­rier height. We estimated AE as AEe = •ELUMO — {EF + Eo) = 4.25 eV for electrons

and as AEh = {EF - E0) - £HOMO = 3.65

eV for holes, where .EHOMO = —9.20 eV (-ELUMO = —0.334 eV) is the energy level of the highest occupied (lowest unoccupied) molecular orbital of C4, Ep = —£AU — —5.07 eV is the Fermi level of Au estimated from the work function of Au (111) -EAU, and

= 0.482 eV (2)

is the single electron charging energy. Then, by using the bounce solution, XB(T) = itanh(wr), along the imaginary time, r , we obtain the bounce action SB = / ^dT{rno±B(T)72 + V(a:B(T))} = 5m0u)l2/6. Finally, the transfer energy is estimated as f = hu)exp(—S&/K) = hujexp(—l/X), where A is the penetration depth : A = m/b/^2m0AE ~ 1 A. The cal­culated dependence of te and th on I agrees reasonably well with the observed G, which means that G is limited by the tunneling be­tween adjacent nanopartides.

The temperature dependence of G ob­tained from Fig. 3 exhibits an activa­tion behavior from 300 K down to 100 K.

The activation energy, Eact = 9.7 meV, is much smaller than EQ. This means that the chemical potential, /i, in the arti­ficial lattice, is moved from EF. In such cases, minority carriers are negligible com­pared to majority ones, because their ratio is estimated as exp(—2|/u — EF\/(kBT)) ~ exp(-2(£b - E&ct)/(kBT)) ~ 10 ' 1 6 < 1 at T = 300 K, where fee is the Boltzmann con­stant.

Now, we consider a Hamiltonian, H, to describe arrays of Au nanoparticles. Dou­ble occupancy is strictly forbidden, because the interaction energy is estimated as U = 4E0 ~ 1.93 eV » te ~ 2.81 meV for the nanoparticles with C4 ligands, so the system is in the strong coupling limit. The exchange interaction energy, J ~ t2/U ~ 4.10 /xeV, is not relevant in the temperature range we considered. Therefore, the system can be de­scribed by a spinless fermion, whose creation (annihilation) operator, c\ (CJ), describes ei­ther an electron or a hole at the i-th lattice site. Assuming the Fermi anticommutation relations, the double occupancy is automati­cally forbidden : (c?)2 = 0. The relevant en­ergy competed with t is the long-range part of the Coulomb repulsion, V(\*i — Xj|), be­tween charges located at x, and Xj. Finally, we obtain as

H=-t^2 (C\CJ + h.c.) - fi^2m (ij) i

where () means the sum is restricted to neigh­boring sites, h.c. stands for the Hermitian conjugate, and m is the number operator. The carrier hoppings would be significantly limited once the structural defects are intro­duced, since they corresponds to the local absence of the lattice sites. Although struc­tural defects due to disorders are not explic­itly written in H, they are implicitly con­sidered by limiting the sum to the available lattice site. We estimated V(|x» — Xj|) from

249

the repulsion energy between charges embed­ded in a dielectric medium of permittivity K as e2/(47r/teo)/|xj — Xj|, where eo is the vacuum dielectric constant. Although H is simple, the properties of the system include highly nontrivial results in low-dimensional systems, as shown below.

3.3. Sub-monolayer film

Next, we examine the sub-monolayer film of Au nanoparticles with C4 ligands, whose nonlinear I-V characteristics are shown in Fig. 5. We attributed this nonlinearity to the lower-dimensional characteristics caused by the formation of ID meandering paths, as schematically shown in Fig. 6. In fact, the atomic force microscope (AFM) image in Fig. 7 shows that the coverage of the film was 69 %, and significant amounts of voids can be observed. These voids limited the conduction paths, and lower dimensional characteristics are expected.

Then, we compared our data with the theoretical predictions by Kane and Fisher 8

Voliage (V)

Fig. 5. Nonlinear I-V characteristics of sub-monolayer films of Obligated Au nanoparticles.

0 0-

. 0

Fig. 6. ID meandering paths induced by structural defects.

100 200 300 400 500

Position (nm)

Fig. 7- Atomic force microscope image of sub-monolayer film.

for ID Tomonaga-Luttinger liquid system connected with Fermi liquid electrodes. In fact, we actually confirmed the power law be­haviors of the differential conductance with the same a = 2.5 for both V and T. We also confirmed the universal scaling 5 behaviors of (dJ/dV)/Ta against eV/kBT.

4. Conclusion

In summary, we observed that the linear con­ductance scales exponentially with the ligand chain length in the monolayer film. In the sub-monolayer film, we observed non-linear current-voltage characteristics, which can be explained by the realization of Tomonaga-Luttinger liquids in a one dimensional me­andering path. However, we cannot rule out alternative scenarios, such as Coulomb block­ade transport, and further works are neces­sary to completely understand the transport mechanism in the nano-particle arrays.

References

1. M. Imada et al., Rev. Mod. Phys. 70, 1039 (1998).

2. C. P. Collier et al., Science 277, 1978 (1997). 3. M. Brust et al., Chem. Commun. , 801

(1994). 4. T. Arai et al., Jpn. J. Appl. Phys. , (to be

published). 5. C. L. Kane and M. P. A. Fisher, Phys. Rev.

Lett. 68, 1220 (1992).

250

FIRST PRINCIPLES STUDY OF DIHYDRIDE-CHAIN STRUCTURES ON H-TERMINATED Si(100) SURFACE

Y U J I SUWA, M. F U J I M O R I , S. H E I K E , Y. T E R A D A a n d T . H A SH IZ U ME

Advanced Res. Lab., Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan

On a H-terminated Si(100)-2xl surface, a dihydride-chain structure parallel to the step edge is often found by Scanning Tunneling Microscopy. The chain is located near an SB step edge, but away from it by more than one Si-dimer's distance. We performed first principles calculations to determine the mechanism of the chain formation. As a result, we found that the rebonded step edge of the clean Si surface turns into a combination of a dihydride chain and a non-rebonded step edge after hydrogen termination. We also found that the chain adjacent to the step edge is energetically less stable than that at one Si-dimer distance away from the step.

Keywords: dihydride, hydrogen termination, rebonded structure, non-rebonded structure, scanning tunneling microscopy, first principles calculation

Self-assemblage/self-organization is a key to realize nanodevices through a bottom-up approach. When the size of the cir­cuit is smaller than the limit of photo­lithography technique, one has to find an al­ternative method to place each part as de­signed. Although the atom manipulation by scanning tunneling microscope (STM)1,2 '3 '4

provides possibilities for nano-device realiza­tion, speed limitation for atom manipulation will be a problem in practical use. For mass production, utilization of self-organization phenomena is necessary.

Fujimori et al.5 showed that there is a di­hydride chain in each terrace on a hydrogen-terminated Si(100)-2xl surface (see Fig. 1). These are on the terraces below the SB step edges and parallel to those edges. These chains are located at a few Si-dimer's dis­tance away from the SB step edges. These self-assembled atomic-scale chains have a possibility of applications in nano-scale de­vices. When the fabrication of a semicon­ductor chain or a metal chain based on this structure becomes possible, one can use it as a wiring of nano-scale circuits and nano­devices. In order to explore such a possibil­ity, we investigate how theses structures are created and what kind of experimental con­

ditions are important in creating these struc­tures.

To obtain these structures, samples were prepared as follows: First, a standard in situ cleaning process was performed in the cham­ber. The cleaning consisted of degassing of the sample at 1070 K for 2 h, followed by flashing the sample several times for 10 s. The temperature was increased up to 1470 K in the final flashing process. Next, hydro­gen termination was performed by supply­ing atomic hydrogen to the clean surface of the sample. During this process (1-10 min), the temperature of the sample was kept con­stant between 550 K and 610 K. Finally, sam­ple heating and atomic hydrogen supply were terminated simultaneously and the surface structure were quenched. ,' When hydrogen termination process was

carried out below 520 K, many dihydride chains were produced to form striped pat­terns with monohydride Si-dimer rows, i.e., the 3x1 phase, was formed. On the other hand, when the temperature was above 610 K during the process, no dihydride chains were observed on the surface.

Several questions arise from these exper­imental results; (1) Why only one dihydride chain exists in each terrace below the SB

Fig. 1. Filled state STM images of dihydride chains on Si(100)-2xl-H surface, obtained at room temper­ature with a sample bias voltage of —2.0V and a con­stant tunneling current of 10 pA. Dihydride chains are (a) marked by arrows and (b) accentuated by colored lines.

step? (2) Why most of the chains are located near the SB step edges but not adjacent to them? (3) Why the temperature of the sam­ple preparation should be between 520 K and 610 K to produce dihydride chain structure? In order to clarify these points, we performed first principles calculations of H-terminated Si(100) surfaces and S B step structures.

Calculations in this paper were based on density functional theory applying gen-

251

Fig. 2, Atomic structures of the S B step, (a) Rebonded structures of the clean surface, (b) H-terminated rebonded structures, (c) H-terminated non-rebonded structures and an adjacent dihydride chain, (d) H-terminated non-rebonded structures and a distant dihydride chain.

eralized gradient approximation (GGA)6

with plane-wave-based ultrasoft pseudo-potentials7,8. The energy cutoff was set at 12.25 Rydberg. The convergence criterion of the geometry optimization was as follows: All forces acting on atoms were less than 1 x 10~3 Hartree/a.u. Slab models of five (partly six) silicon atomic layers were used for the calculation of H-terminated Si(100) surfaces.

Based on our calculation, we explain the formation of the dihydride chain structure: Figures 2(a)-2(d) show atomic structures of the SB step edges during the sample prepara­tion process. First, in the cleaning process of the Si surface, the SB step edge on a Si(100) clean surface takes a "rebonded" structure as

252

shown in Fig. 2(a). Oshiyama9 showed that this structure is more stable than the other, "non-rebonded", structure for a Si(100) sur­face without hydrogen. While the SB step edges form rebonded structures, the terraces below the SB step edges take the width of an even number of Si sites, so that the ter­races are filled with Si-dimer rows parallel to the steps without unstable Si-monomer rows. Re-formation of the width of the terrace is possible because the surface migration of Si atoms is possible at the sample temperature of the cleaning process.

Figure 2(b) shows the H-terminated re-bonded structures of the SB step edges. Here all the dangling bonds of Si atoms in Fig. 2(a) are H-terminated and buckling Si-dimer structures are relaxed. We expect that rebonded SB steps remain until all the dan­gling bonds in terrace are hydrogenated to form this structure because the termination of the dangling bonds is energetically favor­able.

Figure 2(c) shows the H-terminated non-rebonded structures of the SB step edges and adjacent dihydride Si-monomer chain. These structures are formed by acquiring two more hydrogen atoms at each step-edge structure of Fig. 2(b). Total energy for each atomic row perpendicular to the non-rebonded step edge is 0.45 eV lower than that of the re­bonded step edge (Fig. 2(b)) with a hydro­gen molecule. Therefore, at low tempera­tures, the non-rebonded structure is more stable than the rebonded structure for H-terminated surface, in contrast to that for clean surface. The energy gain of the non-rebonded structure comes from the relax­ation of the deformation energy of the re­bonded structure by decoupling Si-Si bond­ing at the edge.

In order to discuss relative stability of structures with different numbers of hydro­gen atoms under the condition of finite tem­perature and atomic hydrogen gas pressure,

one must compare free energies by intro­ducing chemical potential. Higher temper­ature makes the chemical potential lower and stability of the non-rebonded structure weaker. Above a certain critical tempera­ture, the rebonded structure becomes more stable. We do not calculate the critical tem­perature here; however, this explains the experimental results that hydrogen termi­nation process above 610 K produces no dihydride chains (Fig. 2(b)). Jeong and Oshiyama10 discussed relative stability of the rebonded and the non-rebonded structures on H-terminated surfaces by treating the chemical potential of hydrogen atoms as a parameter. Although they have not relaxed the geometry of the terrace below the step, their result is qualitatively consistent with ours.

Figure 2(d) shows the H-terminated non-rebonded structures of the SB step edges and a dihydride chain with a monohydride Si-dimer row in between them. Total energy of this structure is 0.09 eV lower than that of Fig. 2(c) for each non-rebonded structure. This means that the structure in Fig. 2(c) still has deformation energy due to a repul­sion between too close hydrogen atoms at the edge, and it is relaxed when the dihy­dride chain is not adjacent to the step edges. We also calculated the structure with two Si-dimer rows between the step edges and dihy­dride chain, and found that its total energy is the same as that of Fig. 2(d) within the accu­racy of the calculation. We concluded that once a dihydride chain is formed, its origi­nal position adjacent to the step edge is no longer stable, and the dihydride chain takes at least one Si-dimer row distance from the step edge.

Figure 3 shows simulated filled-state STM images of the SB step structures of the H-terminated surface. Figures 3(a)-3(c) cor­respond to the structure shown in Figs. 2(b)-2(d), respectively. By comparing these STM images with experimental images (Fig. 1),

253

(a) rebonded

i

(b) non-rebonded dihydride

(c) non-rebonded Ml]

dihydride

ill

Fig. 8. Simulated filled-state STM images of H-terminated S B steps for (a) rebonded structures, (b) non-rebonded structures and a dihydride chain ad­jacent to it, and (c) non-rebonded structures and a dihydride chain with a Si-dimer row between, (a-c) correspond to Figs. 2(b-d), respectively.

we can determine which structures are ex­perimentally observed. For this purpose, we must look at positions of nionohydride Si-dimer rows adjacent to the SB step. Since those in Fig. 1 are located very close to the step edge, they correspond to the structure in Fig. 2(d) and the image in Fig. 3(c).

In summary, we have clarified the origin of the existence of a dihydride-chain struc­ture parallel to the SB step edge. The re­bonded step structures stable on a clean sur­face are transformed into the combination of the dihydride chain and the non-rebonded

step structures at H-termination. Because the dihydride chain at its original position adjacent to the step edge is not stable ener­getically, it exchanges its position with that of the neighboring Si-dimer row.

Acknowledgments

This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.

References

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2. T. C. Shen, C. Wang, G. C. Abeln, J. R. Tucker, J. W. Lyding, P. Avouris, and R. E. Walkup, Science 268, 1590 (1995).

3. T. Hashizume, S. Heike, M. I. Lutwyche, S. Watanabe, K. Nakajima, T. Nishi, and Y. Wada, Jpn. J. Appl. Phys. 35, L1085 (1996).

4. T. Hitosugi, S. Heike, T. Onogi, T. Hashizume, S. Watanabe, Z.-Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and K. Kitazawa, Phys. Rev. Lett. 82, 4034 (1999).

5. M. Fujimori, S. Heike, Y. Suwa, and T. Hashizume, Jpn. J. Appl. Phys. 42, L1387 (2003).

6. J.P. Perdew, K. Brake and A. Wang, Phys. Rev. B 54, 16533(1996).

7. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).

8. K. Laasonen, A. Pasquarelio, R. Car, C. Lee and D. Vanderbilt, Phys. Rev. B 47, 10142 (1993).

9. A. Oshiyama, Phys. Rev. Lett. 74, 130 (1995).

10. S. Jeong and A. Oshiyama, Phys. Rev. Lett. 81, 5366 (1998).

11. J. E. Northrap, Phys. Rev. B 44, 1419 (1991).

254

ELECTRICAL PROPERTY OF AG NANOWIRES FABRICATED ON HYDROGEN-TERMINATED SI(IOO) SURFACE

M A S A A K I F U J I M O R I 1 ' * , S. H E I K E 1 , A N D T. H A S H I Z U M E 1 ' 2

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan Department of Physics, Tokyo Institute of Technology, Oh-okayama, Tokyo 152-8551, Japan

*E-mail: fuji@rd.hitachi.co.jp

Current-voltage characteristics were measured of atomic-scale Ag wires on the hydrogenated sur­face of Si(100) using four-probe fine electrodes fabricated with scanning-probe nanofabrication technique. Carrier transport was observed even at the initial stage of Ag deposition and conduc­tivity increased as a function of supplied Ag atoms. Ballistic and diffusive transport are discussed as possible mechanisms of carrier transport.

Keywords: scanning probe nanofabrication; nanowire; conductivity measurement.

1. Introduction

An atomic wire can be regarded as an "ul­timate" small wire. Electrical conduction through such a wire is an attractive phe­nomenon, because we can expect manifesta­tion of quantum phenomena such as quan­tized conduction, or even emergence of un­known phenomena.

One of the milestones in a series of ex­periments concerning this issue was made by Takayanagi and his colleagues [1]. They re­ported clear quantum conductance in an iso­lated gold wire between gold electrodes.

In addition to isolated atomic-scale wires, atomic-scale wires fabricated on a sub­strate surface are attractive objects. Because a bulk substrate causes significant changes on electronic states of the atomic-scale wires on the substrate due to coupling of electronics states. Further, we can arrange wire struc­tures. We can control electrical properties of the wire, even constructing artificial struc­tures.

Several types of atomic-scale wires have been fabricated on substrate surfaces. Lyd-ing and colleagues [2] demonstrated atomic-scale dangling-bond wires on a hydrogen-terminated surface of Si(100), followed by Ga wires [3], C6u wires [4], Ag wires [5], and P

wires [6]. Although interconnects for isolable

nanowires can be made by placing the nano-wires on the pre-fabricated fine electrodes, the interconnects for the surface-modified nanowires must be made on the same sur­face of the substrate on which the nanowires are made. In that case, the surface should be maintained atomically clean to manipu­late atoms or molecules. This is usually ac­companied by serious difficulty because clean surface is damaged by conventional lithogra­phy technique. Therefore, only few groups have succeeded in preparing electrodes on the clean surface [7,8].

In order to measure the electrical prop­erties of the surface-modified nanowires, we have fabricated four-probe fine electrodes on a Si(100) surface. In this paper, we report result of current-voltage measurement for an atomic-scale Ag wire fabricated on the hy­drogenated surface.

2. Experiment

We used a commercial Si(100) wafer (P-doped, n-type, resistivity of 0.5 to 1 fi-cm) as the substrate. We applied a conventional photolithography for fabricating large part of electrodes and used a scanning-probe (SP)

255

nanofabrication technique [9] for making fine structure of the electrodes.

Thin films of SiN, W and Ti with thick­nesses of 20, 10 and 5 nm, respectively, were first sputter-deposited on the substrate. By photolithography, patterns for 1mm x lmm contact pads and 20/xm-wide intermediate electrodes were transferred to a resist film. A set of patterns for four-probe fine electrodes were then drawn on the resist film with the SP nanofabrication technique. For process details, see reference [10].

The substrate was placed in a ultra high vacuum (UHV) chamber. Then, high tem­perature heating for surface cleaning and atomic-hydrogen exposure for hydrogen ter­mination of a surface [3] were carried out. We confirmed by scanning tunneling micro­scope (STM) observation that an atomically flat Si(100)-2xl-H surface was obtained. All STM images in this report were taken at room temperature with a sample bias voltage of —2.0 V and a constant tunneling current of 10 pA.

For the four-probe interconnects be­tween the pre-fabricated fine electrodes and atomic-scale Ag wires, we fabricated 10 to 50-nm-wide dangling-bond wires. Se­lective adsorption of Ag atoms, supplied from a tungsten coil-heater surrounded by a thin layer of Ag, was used to metalize the dangling-bond wire [5]. Finally, atomic-scale Ag wires were fabricated in the simi­lar way starting with one to two atomic-size dangling-bond wires connecting through the four-probe interconnects. The amount of the deposited Ag atoms on the atomic-scale Ag wires was controlled through the evaporation time.

When the sample holder was set on the STM stage, electrical contact with the 10mm x 10mm pads on the sample was re­alized with four Ta wires. All the mea­surements were performed by the four-probe method in UHV at room temperature. All data at sampling points were obtained by

averaging 1024 measured values. When a bias voltage was applied to the electrodes, a leak current flowed through paths except the atomic-scale Ag wires. To avoid influence of the leak current, we measured the leak cur­rent before fabricating the atomic-scale Ag wires and subtracted it from the current ob­tained after fabricating the atomic-scale Ag wires.

3. Results and discussion

The fabricated fine electrodes were made of grains (Fig. 1), formed during the high-temperature cleaning process. We observed the metallic feature of the fine electrodes by STM. The center region of the four-probe fine electrodes showed an atomically flat terrace, where an atomic-scale Ag wires were formed.

An example of the four-probe intercon­nects is shown in Fig. 2 (a). We observed that the Ag clusters were densely formed on the interconnects by selective adsorption of Ag atoms on the dangling-bond wire.

The larger clusters of Ag atoms were ob­served at the side of the interconnects. Be­cause the larger clusters were sparsely dis­tributed and spatially separated, they were electrically separated too. The STM data showed that step-bunched surfaces near the fine electrodes were also terminated with hy­drogen atoms. Figure 2(b) shows a formed dangling-bond wire (Fig. 2 (b)).

Fig. 1. STM image of the four-probe fine electrodes. The inset shows a cross-sectional profile along the dotted line.

256

An example of an atomic-scale Ag wire is shown in Fig. 3. We measured electrical conductance of atomic-scale Ag wires as a function of a number of deposited Ag atoms per dangling bond estimated from deposition conditions. No electrical conduction was ob­served for a dangling bond wire before Ag-atom deposition. The conductance was obvi­ously changed after Ag atom deposition, even at the initial stage of the deposition. Thus we concluded that we properly detected electri­cal conduction through the atomic-scale Ag wires.

Conductance in unit of quantum con­ductance, Go = 2e2/h, is summarized in Fig. 4. Quantized conduction is expected for an atomic-scale metallic wire but clear steps

Fig. 2. (a) STM image of four-probe interconnect running from the upper left to the lower right. White protrusions are Ag clusters. Small Ag clusters are densely located on the interconnect and larger clus­ters are sparsely distributed near the interconnect, (b) A dangling-bond wire formed on a step-bunched surface.

Fig. 3. Example of an atomic-scale Ag wire. The Ag wire runs from the upper left to the lower right shown by arrows. White protrusions surrounding the Ag nanowire are isolated Ag-atom clusters.

were not observed. We, however, obtained a weak smeared-out step-like feature. Under the ballistic conduction condition, the step­like feature of quantized steps should appear. Therefore, we conjecture that the broaden­ing is probably due to the influence of tem­perature or the bulk substrate. Since our atomic-scale Ag wires consisted of aggrega­tion of Ag clusters, strong scattering of the carriers is expected at cluster boundaries and rough surfaces of the atomic-scale Ag wires. In this way, ballistic conduction is prevented.

The STM observation showed that the atomic-scale Ag wire developed in three steps as Ag was deposited : (I) nucleation and growth of clusters, (II) growth and coales­cence of the clusters, (III) formation of clus­ter boundaries. Thus the atomic-scale Ag wire consisted of a chain of atomic-scale Ag clusters. At the step (II), the clusters began to touch with the neighboring clusters.

Cluster boundaries were densely formed along the atomic-scale Ag wire. Since spac­ing of neighboring boundaries was less than a typical mean free path of carriers in a metal (~10 nm), the carriers passing through the atomic-scale Ag wire were scattered and flowed diffusively. In addition, the surface of the atomic-scale Ag wire acts as scatter­ing site of the carriers. Therefore, a ballistic conduction in the atomic-scale Ag wire was

5 i — i — i — i — i — i — i — i — i

Fig. 4. Conductance of an atomic-scale Ag wire as a function of the estimated number of deposited Ag atoms per dangling bond. Conductance is normal­ized by unit of 2e2/h.

257

suppressed and the diffusive conduction ap­peared.

We observed the electrical conduction even at the initial stage of the Ag deposi­tion. The Ag clusters on the dangling-bond wire are not in touch with each other at this stage. This reveals that the carriers were supplied from Ag atoms to electronic states of the dangling-bond wire.

According to the result by Henzler and his colleague [11], electronic states of a Ag thin film changes from localized to metallic state at the border of the film thickness being 2 monolayers. We conjecture that the bent in the conductance curve observed in our exper­iments at the number of deposited Ag atom being 2.5 is originated from a metallic transi­tion in the atomic-scale wire. Since the num­ber of deposited Ag atom means a deposited number of Ag atoms per dangling bond, the value of 2.5 means virtually 2 monolayers.

If diffusive metallic conduction assumed, the observed conductance should be smaller than the value obtained by setting typical values of the atomic-scale Ag wire size (width w, height h and length / are 5, 1 and 100 nm, respectively) in the simple estimation, g = awh/l, where g and a are conduc­tance and the conductivity of bulk Ag, re­spectively. This can be explained by the weak localization of carriers due to strong scattering by surfaces and cluster boundaries of the atomic-scale wires and low dimension­ality. Indeed, similar localization was ob­served in a thin Ag film [11]. Further in­vestigation such as temperature dependence measurements will be indispensable to clarify detailed mechanisms of observed conduction behavior.

4. Summary

Electrical conduction of an atomic-scale Ag wire was measured by using the four-probe method. The conductance showed the fol­lowing features: the conduction appears at

the initial stage of Ag deposition, and the bent in the conductance curve. Possible mechanisms of the carrier transport charac­teristics are diffusive conduction with strong scattering at cluster boundaries and surfaces of the atomic-scale Ag wire.

Acknowledgment s

This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.

References

1. H. Ohnishi, Y. Kondo, K. Takayanagi, Na­ture 395, 1998 (780).

2. J. W. Lyding, T. C. Shen, J. S. Hubacek, J. R. Tucker and G. C. Abeln, Appl. Phys. Lett. 64, 2010 (1994).

3. T. Hashizume, S. Heike, M. I. Lutwyche, S. Watanabe, K. Nakajima, T. Nishi and Y. Wada, Jpn. J. Appl. Phys. 35, L1085 (1996).

4. A. W. Dunn, P. H. Beton and P. Moriarty, J. Vac. Sci. Technol. B14, 1596 (1996).

5. M. Sakurai, C. Thirstrup and M. Aono, Phys. Rev. B62, 16167 (2000).

6. S. R. Schofield, N. J. Curson, M. Y. Sim­mons, F. J. Ruess, T. Hallam, L. Oberbeck and R. G. Clark, comd-mat/0307599.

7. O. V. Hul'ko, R. Boukherroub, and G. P. Lopinski, J. Appl. Phys. 90, 1655 (2001).

8. F. J. Ruess, L. Oberbeck, M. Y. Simmons, K. E. J. Goh, A. R. Hamilton, T. Hal­lam, S. R. Schofield, N. J. Curson, and R. G. Clark, Nano Lett. 4, 1969 (2004).

9. M. Ishibashi, S. Heike, H. Kajiyama, Y. Wada and T. Hashizume, Appl. Phys. Lett. 72, 1581 (1998).

10. M. Fujimori, S. Heike, Y. Terada and T. Hashizume, Nanotechnology 15, S333 (2004).

11. R. Schad, S. Heun, T. Heidenblut and M. Henzler, Phys. Rev. B45, 11430 (1992).

258

EFFECT OF ENVIRONMENT ON IONIZATION OF EXCITED ATOMS EMBEDDED IN A SOLID-STATE CAVITY

*M. ANDO, 2C. LEE, and 3S. SAITO

'ARL, 2HM„ and3CRL, Hitachi, Ltd., 'Omika 7-1-1, Hitachi, Ibaraki 319-1292, Japan * E-mail: anmasa@rd. hitachi. co.jp

4Y. A. ONO 4School of Engineering, Univ. Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan

E-mail: yaono@t-adm. t. n-tokyo. ac.jp

By increasing the lifetime of the excited atoms embedded in a solid-state cavity, the field-ionization rate was enhanced approximately three times over that without the cavity effect. The characteristic temperature dependence of the enhanced field-ionization rate suggested the generation process was dissipative quantum tunneling induced by the strong electron-phonon coupling between localized electrons in the excited atoms and the surrounding lattice vibrations of the host in the cavity.

Keywords: Cavity QED; Electroluminescence; Dissipative quantum tunneling.

Introduction

Cavity quantum electrodynamics illuminates the most fundamental aspects of coherence and decoherence in quantum mechanics1, and governs quantum efficiency of optical devices such as light emitting diodes, and solar cells2, where two metallic electrodes sandwiching the active layer act as a cavity. In these solid-state cavities, dynamics of excited atoms is affected by environmental factors such as electric field, temperature, and inevitably electron-phonon coupling with the surrounding atoms as well as the electron-photon coupling, i.e. cavity effect.

This paper describes the effect of environment on the ionization process of a photo-excited atoms embedded in a solid-state cavity. The observation is made possible by inhibiting the visible spontaneous emission of the excited atoms. This phenomena was successfully observed using thin-film electro­luminescent (TFEL) devices with a light-emitting active layer composed of a CaS:Eu2+

thin film3 (Fig.l). The TFEL device was developed for emissive flat-panel displays and CaS:Eu2+ is a red-emitting phosphor material showing a broad-band emission with a peak at

around 660 nm. The emission is due to a partially allowed 4f -5d intra-shell transition of divalent Eu2+ ions embedded in an ionic host crystal CaS. We stress that this system is a new class of materials for exploring dissipative quantum tunneling in solids at low temperature where strong electron-phonon coupling is theoretically predicted to enhance or suppress the tunneling probability4,5.

Electron-phonon coupling (dissipation)

Fig.l. Solid state cavity of CaS:Eu2+ TFEL device.

259

1. Experimental

An 500 nm-thick active layer composed of the rare-earth ion doped ionic crystal CaS:Eu2+ is sandwiched between 500 nm-thick high-dielectric Ta205 insulating layers and these stacks are further sandwiched between a transparent front electrode (200 nm-thick ITO) and a rear electrode composed of 200 nm-thick ITO or Al with reflectivity of 30% and 98%, respectively. The two insulating layers prevent carrier injection from the electrodes to the active layer and high electric field up to 1 MV/cm can be applied to induce the field ionization. The amount of emitted electrons by ionization was measured by photo-capacitance method6. CaS:Eu2+ has a unique energy configuration, where the ground and the excited states of Eu2+ are situated within the bandgap of CaS, and the 5d excited state is located just below the conduction band edge. Gated spectral hole burning due to this specific energy configuration was reported7. Eu2+ ions are excited by a 488 nm continuous wave Ar laser with an output power of 5mW/cm2. The excitation wavelength corresponds to the peak wavelength of the direct excitation spectrum of Eu2+ in CaS. The sample size is 2 mm2 and the photoluminescent (PL) emission with the peak wavelength at 660 nm is guided to a photo-

Wavelength (nm)

Fig.2. Photo-emission, excitation, and ionization spectrum.

v_. . - — ^ ,

— 10- excitatiw! emission.* CM ^-y OLV

I Al . • • * ITO o 0 o O o C P

- o l f l g ^ o ^ c p o ° o 0 r RT T ^ ^ s o T o o Tib

Applied Voltage (V) Fig.3. Field-modulate PL and photoionization.

multiplier (PMT) tube using an optical-flexible fiber bundle of 2 mm in diameter. A red color filter, which only transmits light with wavelengths longer than 600 ran, is inserted between the optical fiber and the PMT in order to detect only the PL emission from CaS:Eu2+. The applied pulsed voltage used for the modulation has 50 ms duration with rise/fall times of 1 ms of alternating polarity at 100 Hz.

2. Results and Discussion

Figure 2 shows the effect of the rear electrode material on the PL emission and excitation spectra, and the photo-ionization spectra; the latter is defined as the number of photo-ionized Eu2+ ions as a function of the excitation wavelength with an applied voltage at 10V. The sample with ITO transparent rear electrode shows a PL emission spectrum almost identical to that for a CaS:Eu2+ thin film deposited on a glass substrate, peaking at 660 nm and full width at half maximum of 60 nm. This indicates there is no noticeable cavity effect taking place in this case. On the other hand,

260

the PL emission intensity of the sample with an Al reflective rear electrode is partially suppressed at the peak at 660 nm, while two peaks, one at shorter, and the other at longer, wavelengths appear. This modulated PL intensity is confirmed to be the cavity effect; the modulation pattern is well reproduced by multiplying the CaS:Eu2+ emission spectrum data with the modulated intensity calculated based on a simple cavity model and normal incidence.

Eu2+ ions are photo-ionized with the excitation wavelength ranging from 400 to 600 nm and the profile of the photo-ionization spectra corresponds to that of the direct excitation spectrum for the PL. The number of the ionized Eu2+ ions of the sample with the cavity effect (an Al rear electrode) is approximately three times larger than that without the cavity effect (an ITO rear electrode). These results strongly suggest that the photoelectrons are emitted from the directly excited Eu2+ ions and the field ionization of the excited Eu2+ ions is enhanced by the cavity effect.

As shown in Fig.3(a), with cavity effect (an Al rear electrode), PL intensity of direct excited Eu2+ was reduced to 97% with applied voltage at 120V (IMV/cm), approximately three times larger than that without cavity effect (an ITO rear electrode). The inhibited spontaneous emission of Eu2+ coincides with the same effect observed in nanospheres8.

Inhibition of the PL emission is larger in the sample with the cavity effect because the field-ionization rate of the excited Eu2+ ions is larger in this sample, as shown in Fig.3(b), which indicates the voltage dependence of the number of field-ionized Eu2+ ions. In both samples, the number of the field-ionized Eu2+

ions increases with applied voltage. However, the number for the sample without the cavity effect is 1/3 of the value for the sample with the cavity effect. The difference in the number of field-ionized Eu2+ ions between the two

samples is quantitatively in good agreement with that obtained through the field-induced inhibition of the PL emission.

100-

g 8 0 -

^ 60-XJ CD

H40-E

I 20-0-

100 -

^ 8 0 -CM u v

=1 LU *S 60-& := 40

. § 2 0

1 T - 1 1

^v • (a) -< ^ t >

a • \ •

- EPtt PeJPq

G J 4

t3 •

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• • . (b) " • •

J •

•

• V=10V

• 1 1 1

15

-ioe

- 5 -D

100 200 300 400 Temperature (K)

500

Fig.4. Temperature effect on PL and photoionization.

PL emission quenching accompanying Eu2+ ionization begins at -200 K with the activation energy of 160 meV (Fig.4(a)). This activation energy of ionization is larger than the LO phonon energy in CaS (42meV) and prevents complete thermal ionization at room temperature (26meV). Field-ionization rate defined by reduction rate of PL intensity with applied field at 0.IMV/cm increases with temperature, reaches the maximum at -300K, and decreases with higher temperature (Fig.4(b)).

In contrast to the usual cavity-atom systems, where atoms are freely placed in the cavity, Eu2+ ions are embedded in a solid-state host composed of ionic-crystal, CaS, in the present system schematically shown in Fig.l. Therefore, a strong electron-phonon coupling exists between the localized electrons in Eu2+

ions and the vibrational mode of the surrounding ions constituting the host lattice.

261

This electron-phonon coupling makes possible energy transfer between the electrons in the localized ions and the surrounding ions during the excitation, the emission, and the ionization processes. These processes accompany the displacement or relaxation of the relevant ions, usually observed in ionic crystals. The strong interaction between the excited electron and the lattice vibration is enhanced with increasing temperature. It should change the field-ionization mechanism and cause the sensitive temperature dependence of the observed field-ionization rate.

* - Q

Fig.5. Configuration coordinate of the field-ionization.

Based on the above argument, these experimental results are explained using one configuration coordinate for the CaS:Eu2+

system under the cavity effect (Fig.5). Curves G and E correspond to the total system energy with a localized electron in either the ground state or the excited state of Eu2+. Curve I corresponds to that with the electron delocalized into the conduction band of CaS by perturbation of electric field or lattice vibration with temperature. It is critical for this diagram that during derealization from E to I the system experience relatively large lattice displacement generally observed in ionic crystal. Ionization of the excited Eu2+ occurs via the thermally assisted quantum tunneling across the potential barrier at the E-I crossing point and it may be theoretically treated as an

inverse process of self-trapping of excitons . Therefore, the ionization is enhanced by an electric field, thermal energy, and the life time of the excited state. Since the spontaneous emission is a bottleneck of for the lifetime, the ionization rate was enhanced by inhibiting spontaneous emission using the cavity effect.

With increasing temperature, the strong interaction between the atom (Eu2+) and the surrounding environment (CaS) should destroy the coherence of the system and may induce the transition of the field-ionization mechanism from quantum to classical10. The field-ionization rate was increased by the thermally assisted quantum tunneling with lower temperature, but was decreased by thermally-suppressed classical diffusion with higher temperature.

References

1. H. Mabuchi et al., Science 298, 1372 (2002). 2. H. Becker et al., Phys. Rev. B56, 1893 (1997). 3. M. Ando et al., Phys. Rev. B58, 13580 (1998). 4. A. J. Leggett, Proc. 4th ISQM(Tokyo, 1992), p.10. 5. K. Fujikawa et al., Phys. Rev. B46, 10295 (1992). 6. M. Segal et al., Phys. Rev. B68, 075211 (2003). 7. S. A. Basun et al., Phys. Rev. B56, 12992 (1997). 8. H. Schniepp et al., Phys. Rev. Lett. 89, 257403 (2002). 9. Y. Toyozawa, Optical Processes in Solids (Cambridge Univ. Press, 2003). 10. W. H. Zurek, Phys. Today 44(10), 36 (1991).

262

DEVELOPMENT OF UNIVERSAL VIRTUAL SPECTRSCOPE FOR OPTO­ELECTRONICS RESEARCH:

FIRST PRINCIPLES SOFTWARE REPLACING DIELECTRIC CONSTANT MEASUREMENTS

T. HAMADA*" , T. YAMAMOTO*, H. MOMIDA", T. UDA4 and T. OHNO"c

N. TAJIMA1, S. HASAKA'', ML INOUE'', and N. KOBAYSAHf

"Institute of Industrial Science, University of Tokyo, 4-6-1,Komaba, Meguro, Tokyo, 153-8605, Japan hAdvancesoft Corporation, 4-6-1, Komaba, Meguro, Tokyo, 153-8601, Japan

'National Institute of Materials Research, 1-2-1, Sengen, Tsukuba, Ibaraki, 305-0047, Japan Taiyo-Nippon Sanso Corporation, 1-3-26, Oyama, Shinagawa, Tokyo, 142-8558, Japan

"Semiconductor Leading Edge Technologies, 16-1, Onogawa, Tsukuba, Ibaraki, 305-8569, Japan E-mail: hamada@fsis. iis. u-tokyo. ac.jp

We have developed a first principles software, universal virtual spectroscope for opto-electronics research (UVSOR), which can calculate dielectric functions of materials at atomistic levels on the basis of the density functional pseudopotential method in the frequency range from the static to ultra-violet region. The UVSOR can calculate separately electronic and lattice components of the dielectric function, by using the random phase approximation and the Berry phase polarization theory, respectively. This makes the UVSOR unique "virtual spectroscope" on computer, covering all frequencies interested in materials science; i. e., radio, Tera-Hz, infrared, visible, and ultra-violet frequencies. Since the UVSOR can quantitatively calculate the dielectric constant of materials, it is quite effective for studying new dielectrics such as high-k and low-k materials discussed in nano-scale semiconductor technologies. The UVSOR is free software and can be downloaded from our web site: http://www.fsis.iis.u-tokyo.ac.jp.

Keywords: First Principles Calculation; Software; Dielectric Function.

1. Introduction

1.1. Dielectric function of materials

Accurate measurement of dielectric constant e is a subject of great concern in physics as well as in chemistry because the measurement provides detailed information on the quantum mechanical structure of materials. The measurement in the visible region provides information on the electronic structure of materials, and in the infrared (IR) region provides information on the phononic structure of materials. The measurement is also important in electronics and opto-electronics, because most of electronic and optical devices, such as transistors, lasers, and optical fibers, are made of dielectrics. Recently, nano-scale dielectrics have attracted much attention in electronics, because the recent drastic scale-down of electronics devices reduces the size of the devices to the nanometer (10"9m) scale. In

silicon transistor devices, Si02 has been used as core insulating dielectrics since the born of the devices. However, the scale-down of the devices makes Si02 useless because of the following reason. If Si02 is used for the insulating layers, the layer thickness should be reduced to several nanometers in nano-scale transistors. Then Si02 layers are too thin to be insulating, and the direct tunneling current through the layers becomes substantially large when the transistors are operated. It is currently recognized that the development of new dielectrics replacing Si02 is necessary to overcome this problem, and that quantum mechanical study of dielectrics is effective for this purpose. We have developed first principles software, universal virtual spectroscope for opto-electronics research (UVSOR) [1], enabling to calculate quantum mechanically dielectric functions of dielectrics. In this paper, we describe the theory of

263

UVSOR and its performance, showing e calculation results for new dielectrics for the next generation electronics.

2. Dielectric Phenomena

Dielectric constant e of materials is described by complex numbers and is frequency dependent. Figure 1 shows frequency dispersion of the real part of e. If the local field is neglected, s is given by the sum of the orientation £p", lattice vibration dat, and electronic etk dielectric constants. The £p" is due to the orientation of molecules or atomic groups to an external electric field, and is observed in liquids or polymers having polar molecules or atomic groups at frequencies lower than 1 tera (1012) Hz. The dat is due to the lattice polarizations induced by lattice vibrations and is observed in infrared (IR) active materials at frequencies lower than 10 tera Hz. The dh is due to the electronic structure of materials and is observed in all materials at frequencies lower than 10 peta (1016) Hz. The UVSOR can calculate separately £'* and s!"' of insulators, taking into account frequency dependences, and also can calculate s of non-molecular solids. The calculation is based on the first principles density functional (DFT) pseudopotential (PP) method using the local density approximation (LDA) and generalized gradient approximation (GGA) exchange-correlation functionals. The norm-conserving (NC) as well as ultra-soft

107 10'° 1013 10"

Frequency (Hz)

Fig. 1 Frequency dependence of dielectric constant

(US) PP can be used in the £le and ia

calculations.

3. Theory 3.1 Electronic dielectric constant The d'e is calculated by using the time dependent perturbation theory for the optical property calculation of materials [2]. Imaginary part of tfle, Im[£ffe], of an insulator is calculated from electronic transition moments between its valence and conduction bands at k-points in the Brillouin zone (BZ), using the random phase approximation (Eq. (1)).

Im[£efc] = ^ I J 8 Z | f e | u r | ^ ) | 2

< J ( £ J - ^ - ^ ) d k ( l )

Here, (p'\s the Kohn-Sham wavefunction, k is the wave vector and E is the eigenenergy of cp, c and v are conduction and valence band indices, respectively, and r is the position operator. In addition, u and co are polarization vector and frequency of an external electro­magnetic field, respectively. The BZ means the integration is done in the Brillouin zone. Real part of «"', R e ^ ] , is calculated from \m[i?'e] by using the Kramers-Kronig transformation. The PPs have electron-moment dependent, non-local potentials for electrons and thus influence electronic transition moments, interacting with an external electric field. The UVSOR can calculate ile in the same way as all-electron calculations, compensating the PP effects on electronic transition moments. 3.2 Lattice dielectric constant Lattice dielectric constant d"' (real part) of insulators is given by [3].

E%(a>)--*e yZxa Zxp

Q 2 (2)

O)

Here, a and /? are the Cartesian indices, O is the unit cell volume, and a>x is the frequency of the A-th lattice vibration mode of an insulator. The Zx is the transition moment for the A-th mode given by

1 iA0-Zxa~

'P\rrii ZiftP^is (3)

264

Here, z'and w, are the Born effective charge and mass of the i-th atom, respectively, and £j is the -l-th mode eigenvector. The z* is a second-rank tensor describing the lattice polarization AP, of the insulator induced by the i-th atom infinitesimal displacement Am as

AP/ = ~ ^ Z * A u / , (4)

where AP, is calculated from the Berry phase of valence electrons of the insulator, using so-called the Berry phase polarization theory[4],

Here, u is the periodic part of valence band wavefunction <pv, and superscript (1) and <0)

mean displaced and non-displaced states of the z'-th atom in the insulator, respectively. The a^ and £j are calculated at T-point by solving the eigenvalue problem of the dynamical matrix obtained by numerically differentiating Hellmann-Feynman forces on atoms [3].

4. Dielectric Constant Calculation 4.1. High-k materials High-k materials are a group of insulating materials having a higher e than that of Si02(^=3.9), which has been used for the gate dielectric of complementary metal oxide semiconductor (CMOS) transistors. High-k materials are considered to be suitable for nano-scale CMOSs of the next generation, because they can work as gate dielectrics even in the thicker state and reduce the dielectric tunneling current between the gate electrode and Si substrate of the CMOSs. We used the UVSOR [5] to calculate static £ of typical high-k materials, /. e., crystalline A1203 (hexagonal), Hf02 (monoclinic) and Ce02 and Al203

(amorphous) as a model structure, estimating their £h and iat. The crystal structures of A1203, and Hf02, and Ce02, were energy-optimized by using the first principles DFT PP method. Model structures of amorphous A1203

were generated by using a combination of the

classical molecular dynamics (MD) and the first principles methods: initial guess structures were generated by using the melt and quench MD method and then the generated structure were energy-optimized by using the first principles method [5]. It was characteristic of amorphous A1203 models that they had Al sites with several O-coordination numbers. Table I shows calculated i?le, J"', and s (=/fe+£/a') for these high-k materials, along with experimental values for comparison. The calculated ^'e and fi'"' values for the high-k materials are in good agreement with the experimental £k and d"1

values, respectively. It is remarkable that the UVSOR calculations well reproduces the £le

and dal of amorphous A1203. The results show that UVSOR is effective for theoretical studies of crystalline and amorphous high-k materials, which have a large dal and a moderate ^le.

Table 1. Calculated dielectric constants of high-k materials. Experimental values are shown in parentheses for comparison.

Material ?kc £** g=g""c + e"'

alpha Al203(hexagonal) e=em 2.9(3.1) 6.1(5.8) 9.0(8.9)

A1,0, w

2 3 «zz 2.8(3.0) 8.4(8.1) 11.2(11.1) amorphous ~3 5.9-8.8 8.9-11.8

(2.5-2.8) (5.7-8.2) (8.2-11.0)

Hf02(monoclinic) 4.7(~5) 11.4 15.9(16-25)

C e 0 2 7.5(6) 16.8(17) 24.8(23)

4.2. Low-k materials Low-k materials are a group of insulating materials having an e smaller than 3.9, and are used as the insulator for three-dimensional circuits of semiconductor devices. The materials are required to have a lower e to avoid the signal delay in the circuits and high mechanical strength to tolerate the mechanical stress imposed during the circuit fabrication process. Model structures of a typical amorphous SiOCH low-k material were generated and its IR absorption spectra and static e were calculated by using the UVSOR

265

[6]. The Si-O network topology of generated models was determined by using a network structure modeling technique in classical MD calculations, and then the network structure was energy-optimized by using the first principles GGA DFT PP method. Figure 2 shows the structure of a model most likely to represent the low-k material with the chemical composition of Si10154Co67H2i4. Figure 3 shows calculated IR absorption spectra for the model structure, in comparison with experimental spectra of the material. The theoretical and experimental IR spectra have similar peak positions, and the similarity suggests the validity of the material modeling

Fig.2 Model structure of low-k material SiiOi 54C067H2.M(2X2X2 super-cell structure)

wavenumber fcm ')

Fig.3 (a)Experimental infrared (IR) spectra for a low-k material and (b)theoretical IR spectra for model structure of the low-k material (Fie.2).

and £lat calculation.

Calculated static e for the model structure was

3.12, and was close to an experimental value of

2.9 for the low-k material. The model structure had an t?,e of 2.14 and an ial of 0.98; /. e., the electronic contribution to e was dominant contrary to the high-k material cases. The difference in e component between high- and low-k materials suggests that the material design strategy would differ, though they belong to the same group of materials, dielectric insulators.

5. Conclusions We developed first principle software UVSOR, which can accurately calculate the dielectric function of materials. The UVSOR can work as "virtual spectroscope" on computer, and was shown to be effective for the theoretical study of new dielectrics.

Acknowledgments The UVSOR software was developed in "Frontier Simulation Software for Industrial Science (FSIS)" project supported by IT program of Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

References

1. T. Hamada, T, Yamamoto, H. Momida, T. Uda,

and T. Ohno, UVSOR ver. 1.0, Univ. Tokyo

(2005).

2. G. Harbeke, in Optical Properties of Solids, ed. F.

Abeles (North-Holland, Amsterdam, 1972); p21.

3. X. Zhao and D. Vanderbilt, Phys. Rev. B, 65,

233106(2992).

4. R. Resta, Rev. Mod. Phys. 66, 899 (1994).

5. T. Hamada, H. Momida, T. Yamamoto, T. Uda,

and T. Ohno, 1ECE Technical Report, SDM2005-

58, 1 (2005) (in Japanese).

6. N. Tajima, T. Hamada, T. Ohno, K. Yoneda, N.

Kobayashi, T. Hasaka, N. Inoue, Proceedings of

the IEEE 2005 International Interconnect

Technology Conference, 66 (2005).

266

Q U A N T U M N E R N S T EFFECT

H I R O A K I N A K A M U R A

Theory and Computer Simulation Center, National Institute for Fusion Science, Oroshi-cho, Toki, Gifu 509-5292, Japan

E-mail: nakamura@tcsc.nifs.ac.jp

N A O M I C H I HATANO

Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan E-mail: hatano@iis.u-tokyo.ac.jp

R Y O E N SHIRASAKI

Department of Physics, Yokohama National University,

Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan

E-mail: sirasaki@phys. ynu. ac.jp

We report our recent predictions on the quantum Nernst effect, a novel thermomagnetic effect in the quantum Hall regime. We assume that , when the chemical potential is located between a pair of neighboring Landau levels, edge currents convect around the system. This yields theoretical predictions that the Nernst coefficient is strongly suppressed and the thermal conductance is quantized. The present system is a physical realization of the non-equilibrium steady state.

Keywords: Nernst effect; Nernst coefficient; edge current; quantum Hall effect; thermoelectric power; thermomagnetic effect; non-equilibrium steady state

1. Introduction

The adiabatic Nernst effect arises in a con­ductor bar under a magnetic field B in the z direction and a temperature bias AT in the x direction. The conductor is electrically and thermally insulated on all surfaces, ex­cept that heat baths are attached to the sur­faces facing the +x and — x directions, which produces the temperature bias. A classical-mechanical consideration gives the following: electrons carrying the heat current in the x direction are deflected to the y direction be­cause of the Lorentz force generated by the magnetic field in the z direction, and thereby produce a voltage difference (the Nernst volt­age) VN in the y direction. The Nernst coef­ficient is defined by

~ WBAT' ( ' where W and L are the width and the length of the conductor bar, respectively. We define

the temperature bias such that AT > 0 if the temperature is higher in the heat bath on the —x surface than that on the +x sur­face. We also define the Nernst voltage such that VN > 0 if the voltage is higher on the +y surface than on the —y surface. (We always put B > 0 here and hereafter.) The above classical-mechanical consideration, where no scattering is taken into account, gives a posi­tive Nernst coefficient. In fact, electron scat­tering can make the Nernst coefficient both positive and negative.1

In a recent article,2 we considered the Nernst effect in the quantum Hall regime, that is, the Nernst effect of the two-dimensional electron gas in semiconductor heterojunctions under a strong magnetic field (namely a Hall bar) at low tempera­tures, low enough for the mean free path to be greater than the system size. We theo­retically predicted that, when the chemical

267

potential is located between a pair of neigh­boring Landau levels:

(i) the Nernst coefficient is strongly sup­pressed;

(ii) the thermal conductance in the x direc­tion is quantized.

Hasegawa and Machida are planning an experiment3 in the setup of the present the­ory. In what follows, we review our predic­tions and numerical demonstration.

2. Convection of edge currents

Our argument is based on a simple assump­tion that edge currents4 convect around the system. We here explain our idea (Fig. 1). Since the Hall bar is electrically insulated, edge currents circulate along the edges of the Hall bar when the chemical potential is between a pair of neighboring Landau lev­els. An edge current on the left end of the Hall bar is, while running from the corner C4 to the corner Ci, heated up by the heat bath with a temperature T+ and is equili­brated to the Fermi distribution f(T+,fi+) with the temperature T+ and a chemical po­tential /i+ around the corner Ci. Since the upper edge is electrically and thermally insu­lated, the edge current runs ballistically from the corner Ci to the corner C2, maintaining the Fermi distribution / (T+ , /x+ ) all the way. The edge current, upon arriving at the cor­ner C2, encounters the other heat bath with

T.

Ci

C 4 ^ -

(T+.\U)

(T-,\L-)

->-x

Fig. 1. A schematic view of the convection of an edge current in a Hall bar under the setup for the Nernst effect.

a temperature T_ and is equilibrated to the Fermi distribution / (T_ , /x_), arriving at the corner C3. The edge current along the lower edge runs ballistically from the corner C3 to the corner C4, maintaining the Fermi distri­bution / (T_ ,^_ ) all the way. The Nernst voltage is then given by

VN = -^ Afx _ jj,+ - fi.

(2)

for the temperature bias AT = T+—T- > 0, where e(< 0) is the charge of the electron.

Incidentally, a convecting edge current constitutes the non-equilibrium steady state (NESS), a new concept that attracts much attention in the field of non-equilibrium statistical physics.5'6 The non-equilibrium steady state is almost the first statistical state of a quantum system far from equilib­rium that can be handled analytically. It consists of a pair of independent currents running in different directions with different temperatures and different chemical poten­tials. In most studies, the non-equilibrium steady state has been considered in a one-dimensional non-interacting electron system and hence has been an almost purely math­ematical concept. The pair of the upper and lower edge currents in Fig. 1, however, can be regarded as a non-equilibrium steady state. We consider it valuable to give a physical re­alization to the mathematical-physical con­cept.

3. Nernst coefficient and heat conductance

Let us describe our calculation briefly. We define the electric current and the heat cur­rent in the form

Ie = {ev) and IQ = ((E - /j)v), (3)

where the thermal average is given by

(A) = -Y, AfnM{T{yk), »{yk))dk 7T l-lc n = 0 '

(4)

268

with yk = hk/\e\B. The subscript n stands for the channel of the edge current. The in­tegration limits ±fcm are the maximum and minimum possible momenta. The function /n<fc is the Fermi distribution at the energy of the state of the nth channel with the momen­tum k. The functions T(y) and n{y) denote the temperature and the chemical potential of an edge current running at y. The convec­tion of an edge current shown in Fig. 1 gives (r(3/fc),A*(j/fe)) = (T+,n+) for the upper edge states and (T_, M-) for lower edge states. We can hence express the currents (3) in terms of (T+,n+) and (T_,/x_), and thereby in terms of the first order of the Taylor expansion with respect to AT = T+—T_ and A/i = /JL+ —/X_.

We then put Ie = 0 because the system is electrically insulated. This condition relates Afi to AT, yielding the Nernst coefficient (1) with the Nernst voltage (2), or

1 L AM N: (5)

\e\BWAT' Applying the relation between A/x and AT to the Taylor expansion of the heat current IQ, we obtain the heat conductance

GQ S £ . (6)

For details of the calculation, refer to the original article, Ref. 2.

4. Predictions

We can understand our predictions based on the convection of edge currents. First, the number of the conduction electrons is con­served during the convection. Hence the dif­ference in the chemical potential of the upper edge current and that of the lower edge cur­rent is of a higher order of the difference in the temperature of the upper and lower edge currents; that is, A/x = o(AT). The Nernst coefficient (5) hence vanishes as a linear re­sponse, or N = 0 in the limit AT —> 0.

Second, the heat current IQ in the x di­rection is carried by the ballistic edge cur­rents along the upper and lower edges; the

total heat current is the difference in the heat carried by the upper edge currents and the lower edge currents. The edge currents are quantized as long as the chemical potential remains between a pair of neighboring Lan­dau levels. The heat current hence has quan­tized steps as a function of B. The heat cur­rent is M times a unit current when there are M channels of the edge current. After some algebra, we arrive at the conclusion that the heat conductance GQ is quantized as

2

(7) ^ ~ M ( M = 1,2 ,3 ,

when the chemical potential is located be­tween the Mth and (M + l) th Landau levels.

5. Numerical demonstration

We now present a numerical demonstration of our predictions using a confining potential in the y direction in the form7

V(y) = 0 for \y\ <

^ (|y| - f Y for f < | y | < f . (8)

We used the following parameter values: the effective mass is m = 0.067mo for GaAs, where mo is the bare mass of the electron; the size of the Hall bar is L = 20/zm and W = 20/xm (less than the mean free path at low temperatures8) with w = 16^m; the confining potential is given by V(±W/2) = 5.0eV, the work function of GaAs; the chem­ical potential at equilibrium is n = 15meV, or the carrier density ns = 4.24 x 101 5m - 2 .

We thus evaluated the Nernst coefficient N and the thermal conductance GQ as in Fig. 2. Our predictions N = 0 and Eq. (7) are indeed realized at low temperatures and when the chemical potential is located be­tween a pair of neighboring Landau levels. (The gray curves in Fig. 2 are our new results of the self-consistent Born approximation in the case where we took account of impurity scattering of strength h/r = 1.0 x 10~4eV. See Ref. 9 for details.) We also note that the

269

e 3 .2 "o •H E *§ .a U e *- op

IS (a)

- 7 0

- 6 0

j ^ - 5 0 ^ -40 ^ - 3 0 £ -20

/^ _ /

I / ffli M /2 -- ~——f-

,r=20K / pOK

/ ' ,'5K

/ / '

II

- 1 0

0 1 2 3 4 5 6 7 8 Inverse Dimensionless Magnetic Field

I3 <L> T

I 2

H 1

0.

&' * T=i20K

ilOK :""5K i IK

(b) 0 1 2 3 4

Inverse Dimensionless Magnetic Field

Fig. 2. Scaling plots of (a) N x B and (b) GQ/T X 3H/nk#2, both against mn/h\e\B at T = 1, 5,10 and 20K for IT < B < 20T. The gray curves in each panel indicate results that take account of impurity scattering at T = 1 and 5K; see text.

Nernst coefficient is generally negative in the

present calculation.

6. S u m m a r y

We predicted a novel quan tum effect of the

two-dimensional electron gas, in close anal­

ogy to the quantum Hall effect. When the

chemical potential is between a pair of Lan­

dau levels, the edge currents suppress the

Nernst coefficient and quantize the thermal

conductance. The system is a physical real­

ization of the non-equilibrium steady state .

The precise forms of the peaks and the

steps in Fig. 2 can be different when we

take account of electron scattering. The elec­

tronic states extend over the system when

the chemical potential is close to a Landau

level, namely when m/x/fi,|e|B = n+ \. Then

the heat current is carried mainly by the bulk

states and we have to take account of impuri­

ties as well as electron interactions possibly.9

We comment on other approaches to the

quantum Nernst effect. Kontani derived1 0 '1 1

on the basis of the Fermi liquid theory, gen­

eral expressions of the Nernst coefficient and

the thermal conductivity of strongly corre­

lated electron systems such as high-Tc ma­

terials. Akera and Suzuura1 2 considered

with the use of thermohydrodynamics, the

Ett ingshausen effect, the reciprocal of the

Nernst effect. The quantum behavior pre­

dicted here, however, was not reported in ei­

ther studies.

A c k n o w l e d g m e n t s

The authors express sincere grat i tude to

Dr. Y. Hasegawa and Dr. T. Machida for use­

ful comments on experiments of the Nernst

effect and the quantum Hall effect. This re­

search was partially supported by the Min­

istry of Education, Culture, Sports, Sci­

ence and Technology, Grant-in-Aid for Ex­

ploratory Research, 2005, No. 17654073 as

well as by the Mura ta Science Foundation.

N.H. gratefully acknowledges the financial

support by the Sumitomo Foundation.

R e f e r e n c e s

1. H. Nakamura, K. Ikeda, Y. Ishikawa, A. Suzuki and H. Shirai, Jpn. J. Appl. Phys. 38, 5745 (1999).

2. H. Nakamura, N. Hatano and R. Shirasaki, Solid State Commun. 135, 510 (2005).

3. Y. Hasegawa and T. Machida, private com­munication.

4. B. I. Halperin, Phys. Rev. B 25, 2185 (1982).

5. D. Ruelle, J. Stat. Phys. 98, 57 (2000). 6. Y. Ogata, Phys. Rev. E 66, 016135 (2004). 7. S. Komiyama, H. Hirai, M. Ohsawa, Y. Mat-

suda, S. Sasa, and T. Fujii, Phys. Rev. B 45, 11085 (1992).

8. S. Tarucha, T. Saku, Y. Hirayama, and Y. Horikoshi, Phys. Rev. B 45, 13465 (1992).

9. H. Nakamura, N. Hatano, and R. Shirasaki, in preparation.

10. H. Kontani, Phys. Rev. Lett. 89, 237003 (2002).

11. H. Kontani, Phys. Rev. £ 6 7 , 014408 (2003). 12. H. Akera and H. Suzuura,

cond-mat/0409498.

270

QUANTUM PHENOMENA VISUALIZED USING ELECTRON WAVES

Akira TONOMURA

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395 Japan

Frontier Research System, Riken, Wako, Saitama 350-0198 Japan

Phase information of bright and monochromatic field-emission electron beams, which we had been repeatedly developing for the past 30 years, was used to visualize quantum phenomena, which have recently begun to crop up in many microscopic regions. Recent examples include microscopic quantum phenomena we directly observed in superconductors such as quantized vortices that exhibited strange behaviors peculiar to anisotropic layered high-rc superconductors.

1. Introduction

Quantum mechanics, which was born as a law describing the behavior of electrons inside atoms, now provides the basis for nearly all physical theories. Quantum mechanics has explained many of the microscopic mechanisms not only of natural phenomena but also of electronic devices in fields where practical applications are sought, and has thus helped open up the semiconductor and other industries. Furthermore, recent technological developments since the 1980s have made it possible to experimentally test the fundamentals of quantum mechanics, and all the results obtained up to now have agreed well with quantum mechanics. Quantum phenomena are thus attracting attention from two aspects: as obstacles to further improvements in device performances and as pathfinders in the development of future devices.

We have been continuously developing electron microscopes equipped with field-emission guns that produce ever-more-coherent electron beams with the aim of directly observing such quantum phenomena [1], so that we can precisely measure not only the intensity but also the phase of the transmitted electrons.

In this paper, we describe our direct observation of the behaviors of quantum objects

using the wave nature of electrons after briefly reviewing the development of coherent electron beams.

2. Historical Developments of Coherent Electron Beams

We started our research on electron holography in 1967 as a way of overcoming the saturated state of electron microscopy technology and confirmed after our first year of experiments that optical images could be reconstructed from in-line electron holograms [2]. We intended to overcome the resolution limit, which was restricted by the inevitable aberrations found in axially-symmetric electron lenses, by applying electron holography. However, we clarified from the experiments that bright electron beams, like laser beams, would be needed to practically apply such holographic techniques. We then began developing coherent electron beams that were field-emitted from pointed tips, and we have continued to do so up to now. However, since the electron source is extremely small, typically 50 A in diameter, and it has to be immobile even within a fraction of the source diameter, we had to overcome such technical difficulties to prevent even the slightest mechanical vibration of the tip, the accelerating tube, and the microscope column, or a deflection of the

271

fine beam due to stray ac magnetic fields. Otherwise, the inherent brightness, or luminosity, of the electron beam would deteriorate.

After a decade of work, we developed an 80-kV field-emission electron beam [3], which was two orders of magnitude brighter than the thermal beams then used (see Table 1), and we became the first to observe electron interference patterns directly on the fluorescent screen and to record up to 3000 interference fringes on film. Thanks to this development, information that had previously been inaccessible by conventional electron microscopy could be obtained through using both electron holography and this bright electron beam. In fact, magnetic lines of force inside and outside ferromagnetic samples were directly and quantitatively observed in hie flux units [4] in interference micrographs, which could be obtained in the optical reconstruction stage of off-axis electron holography formed with an electron biprism [5].

In 1978, the existence of the Aharonov-Bohm (AB) effect [6] had been questioned and controversy about it has again

arisen [7]. We attempted to provide definitive experimental evidence, since the AB effect was also the fundamental principle behind our method of observing magnetic lines offeree. At that time, Prof. C. N. Yang of State University of New York kindly visited our laboratory [8] to discuss our proposal to experimentally verify the AB effect, and since then he has continued to show interest in our work on electron interference and to advise us in our experiments. We had to conduct a series of experiments on the AB effect up to 1986 [9-11], since repeated objections arose about our results during the controversy.

In 1992, we observed magnetic vortices in metal superconductors using phase information of electrons transmitted through thin film samples [12], and later vortices in high-Tc superconductors [13-15].

3. Experimental Results

New advanced technologies, such as coherent electron beams, sensitive detectors, and lithography, have made it possible to carry out fundamental experiments in quantum

Year

1968

1978

1982

1989

2000

Electron microscope

100-kVEM (Thermionic

electrons)

80 - kV FEEM

250-kVFEEM

35-kV FEEM

1 - MV FEEM

Brightness

(A/cm2- Ster)

1 x 1 0 6

1 x 1 0 8

4 x 1 0 8

5 x 1 0 9

10

2x10

Applications

Experimental feasibility of electron holography [2]

Direct observation of magnetic lines of force [4]

Conclusive experiments ofAB effect [11]

Dynamic observation of vortices in metal superconductors [12]

Observation of unusual behaviors of vortices in high - Tc superconductors [13-15]

FEEM: field-emission electron microscope

Table 1 History of developments of bright electron beams

272

mechanics that once belonged to the realm of "thought experiments", such as the single-electron build-up of an interference pattern [16], conclusive experiments verifying the existence of the AB effect [9-11], and the observation of static and dynamic behaviors of quantized vortices in superconductors.

Here, we report results that concern the unusual behaviors of vortices peculiar to anisotropic, layered high-rc superconductors observed with our newly developed 1-MV microscope [17], where the beam was four orders of magnitude brighter than that of a conventional 100-kV thermal beam, and the number of biprism interference fringes that could be recorded on film increased from 300 to 11,000 [18].

An incident electron wave is phase-shifted or deflected by the magnetic fields of vortices when passing through them, and a vortex appears as a spot consisting of black-and-white contrast produced by image defocusing (Lorentz microscopy).

Vortices usually form a closely packed triangular lattice. This is the case even for anisotropic high-rc superconductors, as long as the magnetic field is directed along the anisotropy c-axis. When the magnetic field is strongly tilted away from the c-axis, however, Bitter images reveal that the vortices no longer form a triangular lattice. Instead, in YBaCu307 8

(YBCO) [19], they form arrays of linear chains along the direction of the tilting field, and in

, , ; ; • ' . ; . , ; : •H::pj;J:ii:i::::;?;:;:|i::il:::::;:JJii : .* J : '••; ' • " " : -; l^#i:%'Ls»>^^!p[ii:^^i

•r - : • : . '• - . : * JI:lJi*-3$-Sl&|.^;:S-1::*:::«ls: ; . • - \ -,-'••:. \ • &ia$^$^¥&&:$#tfft$i

•*','*•' " ' • • .' S^^as'ss isss iS 5 ^^!^ t , '- - - • ' - • ^«IWMJSS%«*ISS^S>*S * : - • - ; ; , ; : .• • ISs^j^^^^f^psfe^^fel;

' • " : •" • " - • / * ; • ^ ^ ^ ^ ^ K ® |

* (a) " "*" "*"" %)"

Fig.l. Bitter patterns of vortices at tilted magnetic field: (a)YBCO;(b)Bi-2212

Bi-2212 alternating domains of chains and triangular lattices [20,21] (see Fig. 1). While the chain state in YBCO can be explained by the tilting of vortex lines within the framework of anisotropic London theory [22], the chain-lattice state in Bi-2212 has long been an object of discussion. It has been attributed to two sets of vortex lines perpendicular to each other [23], one set forming chains and the other forming triangular lattices.

Koshelev [23] proposed an interesting model for the chain lattice state; Josephson vortices penetrate between the layer planes in Bi-2212 and vortices that perpendicularly intersect the Josephson vortices form chains; the rest of the vortices form triangular lattices.

No direct evidence for such mechanisms, however, was given through experiments because there were few methods for observing the arrangements of vortex lines inside superconductors. In addition, this model was difficult to accept, since no interaction took place between the two perpendicular magnetic fields. However, Koshelev took second-order approximation into consideration and determined there was an energy reduction in this vortex arrangement. As a result of interaction, a vertical vortex line winds slightly in opposite directions, above and below the crossing Josephson vortex, thus forming a stable state, since the circulating supercurrent of the Josephson vortex exerts Lorentz forces onto the vertical vortex line.

Lorentz microscopy with our 1-MV electron microscope has been used to determine whether the vortex lines in the chain states inside high-rc superconductors are tilted or not. In the case of YBCO, we found that vortices tilted together in the direction the applied magnetic field was tilted. This conclusion is evident from the Lorentz micrographs in Fig. 2, in which the vortex images become more elongated and form linear chains together as the tilting angle of the magnetic field increases. In the case of our Bi-2212, Lorentz microscopy observations under various defocusing conditions clarified

273

that neither chain nor lattice vortices tilted, but both stood perpendicular to the layer plane [24]. If vortex lines are strongly tilted at an angle comparable to that of the applied magnetic field, the vortex images in Fig. 3(a) should be elongated.

Our findings that both chain and lattice vortices stand straight perpendicular to the c-plane and do not tilt is clear evidence of the Koshelev mechanism. Although the clearest evidence for this model would of course be the direct observation of Josephson vortices, the magnetic field of a Josephson vortex extends widely between the layers to several tens of urn and, therefore, makes it difficult to detect with our method.

Although we could not observe Josephson vortices directly, we had supporting evidence for their existence. For example, we could observe vertical vortices always beginning to penetrateg into the sample along some straight lines.

7- ' '••".'

(a) (b) (c)

Tilted vortex (d)

Fig.2. Lorentz micrographs of vortices in YBCO film sample at tilted magnetic fields (T= 30 K; B, = 3 G): (a) 6 -75? (b) 0 =82? (c) 0 =83? (d) Schematic of tilted vortex lines. When the tilt angle becomes larger than 75? the vortex images begin to elongate and, at the same time, to form arrays of linear chains. This implies that chain vortices in YBCO are produced by some attractive force between tilted vortex lines.

Josephson vortex

Fig.3. Chain-lattice state of vortices at tilted magnetic field in Bi-2212: (a) Lorentz micrograph (T = 50 K and B = 50 G). (b) Schematic. The chains are indicated by the white arrows in (a). If vortex lines are tilted at an angle comparable to the tilt angle of the magnetic field, i.e., 85°, the vortex images must be elongated and weak in contrast under this defocusing condition. Since the images of chain vortices as well as those of lattices vortices are not elongated but circular, all the vortex lines are not tilted, but perpendicular to the layer plane. If tilted, die vortex line images become elongated as the vortex images indicated by the arrows in the inset in (a)

Fig.4. Series of Lorentz micrographs of vortices in field-cooled Bi-2212 film sample when magnetic field Bp

perpendicular to layer plane begins to be applied and increases at fixed in-plane magnetic field of 50 G at T= 50 K: (a) Bp = 0; (b) B, = 0.2 G; (c) Bp = 1 G; (d) Schematic of vortex lines consisting of vertical and Josephson vortices.

274

These straight lines must be determined by Josephson vortices. There is an example in Fig. 4. When we apply an in-plane magnetic field, no vortex images can be seen in the Lorentz micrograph [see Fig. 4(a)]. However, when a perpendicular magnetic field is applied and slowly increased, images of vertical vortices appear in this field of view.

When a magnetic field is applied parallel to the layer plane as in (a), no vertical vortices are produced. Therefore, no vortex images can be seen. Although there should be Josephson vortices parallel to the layer plane, these cannot be observed by Lorentz microscopy because of the wide distribution of the vortex magnetic field. When the vertical magnetic field increases, vertical vortices begin to appear along straight lines, indicated by the white lines in (b), which are considered to be determined by Josephson vortices. Since vortices are arranged along straight lines even when there are large intervals between vortices, we could find no reason why chain vortices were produced other than assuming that vertical vortices crossing Josephson vortices formed chains as illustrated in Fig. 3 (b). Above 6P

(magnetic field perpendicular to the layer plane) = 1 G, vertical vortices appeared also between chain vortices as shown in (c).

We also found that only images of chain vortices and not of lattice ones in Bi-2212 began to disappear at temperatures much lower than Tc. Figure 5 is a Lorentz micrograph of such an example.

We attribute the disappearance of vortex images to the synchronous oscillation of vortices between "node vortices" A and C along the chain direction, just like a coupled oscillation. This does not mean that the vortices themselves disappear, since vortex images gradually fade away with rising temperatures and also partly do so along the chain. In Fig. 5, vortices A and C in the chain can clearly be seen, but vortices that are far away from these vortices cannot. Vortices A and C are located "stably" in the midst of six surrounding vortices (indicated by the red points in the figure), while vortex B is sandwiched "unstably" between two vortices above and below. This vortex arrangement may be stable at low temperatures, but at high temperatures where vortices vibrate thermally, vortex B begins to oscillate back and forth, like pinballs connected by springs on incommensurate periodic potentials, as in the Frenkel-Kontorova model. We attribute the disappearance of chain vortices to such longitudinal oscillations of vortices along chains.

4. Conclusion

Thanks to recent developments in advanced technologies, such as coherent electron beams, highly sensitive electron detectors, and photolithography, some experiments that were once regarded as "Gedanken" experiments can now be carried out in practice. In addition, the wave nature of electrons can now be utilized to observe microscopic objects that were previously not able to be observed. Examples are the quantitative observations of both the microscopic distribution of magnetic lines of force in hie units and the dynamics of quantized vortices in superconductors. These measurement and observation techniques are expected to play important roles in future research on and developments in nano-science and nano-technology.

• I t I

b nr» 3(im

Fig.5. Lorentz micrograph of disappearing chain vortices inBi-2212(r=57K,Bp = 8G, 6=80° )

275

References

1) A. Tonomura. (1999) Electron Holography (2nd edition, Springer, Heidelberg).

2) A. Tonomura, A. Fukuhara, H. Watanabe, & T. Komoda, Jpn. J. Appl. Phys. 7 (1968) 295.

3) A. Tonomura, T. Matsuda, J. Endo, H. Todokoro & T. Komoda, J. Electron Microsc. 28 (1979) 1-11.

4) A. Tonomura, T. Matsuda, J. Endo, T. Arii & K. Mihama, Phys. Rev. Lett. 44 (1980) 1430-1433.

5) G. MoUenstedt & H. DUker, Z. Physik 145(1956) 377-397.

6) Y. Aharonov & D. Bohm, Phys. Rev. 115 (1959)485-491.

7) P. Bocchieri & A. Loinger, Nuovo Cimento ^(1978)475-482.

8) C. N. Yang, Foreword, Foundations of Quantum Mechanics in the Light of New Technology, eds. S. Nakajima, C. N. Yang, Y. Murayama & A. Tonomura, Advanced Series in Applied Physics Vol. 4 (World Scientific, Singapore, 1996) p. v.

9) A. Tonomura, H. Umezaki, T. Matsuda, N. Osakabe, J. Endo & Y Sugita, Phys. Rev. Lett. 51(1983)331-334.

10) A. Tonomura, T. Matsuda, R. Suzuki, A. Fukuhara, N. Osakabe, H. Umezaki, J. Endo, K. Shinagawa, Y Sugita & H. Fujiwara, Phys. Rev. Lett. 48(1982) 1443-1446.

11) A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, & H. Yamada. Phys. Rev. Lett. 56(1986) 792-795.

12) K. Harada, T. Matsuda, J. Bonevich, M. Igarashi, S. Kondo, G. Pozzi, U. Kawabe & A. Tonomura, Nature 360 (1992) 51-53.

13) A. Tonomura, H. Kasai, O. Kamimura, T. Matsuda, K. Harada, Y. Nakayama, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, M. Sasase & S. Okayasu, Nature 412(2001) 620-622.

14) T. Matsuda, O. Kamimura, H. Kasai, K.

Harada, T. Yoshida, T. Akashi, A. Tonomura, Y. Nakayama, J. Shimoyama, K. Kishio, T. Hanaguri & K. Kitazawa, Science 294(2001) 2136-2138.

15) A. Tonomura, H. Kasai, T. Matsuda, K. Harada, T. Yoshida, T. Akashi, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, T. Masui, S. Tajima, N. Koshizuka, P. L. Gammel, D. Bishop, M. Sasase & S. Okayasu, Phys. Rev. Lett. 88 (2002)237001.

16) A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki & H. Ezawa, Amer. J. Phys. 57(1989)117-120.

17) T. Kawasaki, T. Yoshida, T. Matsuda, N. Osakabe, A. Tonomura, I. Matsui & K. Kitazawa, Appl. Phys. Lett. 76(2000) 1342-1344.

18) T. Akashi, K. Harada, T. Matsuda, H. Kasai, A. Tonomura, T. Furutsu, T. Moriya, T. Yoshida, T. Kawasaki, K. Kitazawa & H. Koinuma, Appl. Phys. Lett. 81(2002) 1922-1924.

19) P. L. Gammel, D. J. Bishop, J. P. Rice & D. M. Ginsberg, Phys. Rev. Lett. 68 (1992) 3343-3346.

20) A. Bolle, P. L. Gammel, D. G. Grier, C. A. Murray, D. J. Bishop, D. B. Mitzi & A. Kapitulnik, Phys. Rev. Lett. 66 (1991) 112-115.

21) I. V. Grigorieva & J. W. Steeds, Phys. Rev. B51 (1995) 3765-3771.

22) A. I. Buzdin & A. Y. Simonov, JETP Lett. 51 (1990) 191-195.

23) A. E. Koshelev, Phys. Rev. Lett. 83 (1999) 187-190.

24) A. Tonomura, H. Kasai, O. Kamimura, T. Matsuda, K. Harada, T. Yoshida, T. Akashi, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, T. Matsui, S. Tajima, N. Koshizuka, P. L. Gammel, D. J. Bishop, M. Sasase & S. Okayasu, Phys. Rev. Lett. 88 (2002)237001.

276

A N OPTICAL LATTICE CLOCK: ULTRASTABLE ATOMIC CLOCK WITH ENGINEERED PERTURBATION

H. KATORI1'2'*, M. TAKAMOTO1, R. HIGASHI1, AND F.-L. HONG3

Department of Applied Physics, Graduate school of Engineering, The University of Tokyo, PRESTO, Japan Science and Technology Agency,

7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, National Institute of Advanced Industrial Science and Technology (AIST),

Tsukuba, Ibaraki 305-8563, Japan *E-mail: katori@amo.tu-tokyo.ac.jp

An optical lattice clock was realized for 8 7Sr atoms and its absolute frequency has been measured. Projected uncertainty in the lattice clock is illustrated for the case of 8 7Sr. We discuss application of the clock scheme to other atom species, in search for better lattice clocks as well as for the variation of fine-structure constant.

Keywords: atomic clock; optical lattice; frequency measurement.

1. Introduction

A singly trapped ion in Paul traps embod­ies an ideal quantum absorber for atomic clocks, which led to the most accurate optical atomic-clocks1 with fractional uncertainty at 10 - 1 5 level2 operated at the quantum projec­tion noise (QPN) limit. One of the impor­tant features of Paul traps that make them attractive for atomic clocks is that they pro­vide smallest possible perturbations for in­terrogated ions1, while they precisely control the ions' motion. Throughout the history of atomic clocks, indeed, such careful removal of electromagnetic perturbations has been con­sidered to be the heart of the atomic clocks.

Recent experiments,2 however, have wit­nessed that conventional approaches, such as Cs fountain clocks, neutral-atom-based opti­cal clocks interrogated in their free fall, and single-ion-based clocks, are confronted with serious limitations in their fractional accu­racy or stability at a level of 10 - 1 6 . In or­der to break through this limitation, we pro­posed an "optical lattice clock"3 that utilizes engineered perturbation to enhance the clock stability without degrading its accuracy. Fig­ure 1 depicts the scheme: Sub-wavelength lo­

calization of a single atom in each optical lat­tice site suppresses the first order Doppler shift4 as well as the collisional frequency shift, which simulates millions of single-ion clocks operated simultaneously. Although this lattice clock interrogates atoms while they are strongly perturbed by an external field, we have shown that this perturbation can be cancelled out to below 10 - 1 7 preci­sion level5 by designing the light shift trap so as to adjust dipole polarizabilities of probed electronic states. In this contribution, we describe our recent experiments and future prospects for the optical lattice clock.

2. Light shift cancellation

The transition frequency v of atoms exposed to a lattice-trap laser with intensity / is given by the sum of the unperturbed transition fre­quency Z/Q and the light shift fac,

"(Az,,e) = VQ + V&

VQ Aa(\L,e)

I + 0(I2) (1) 2eQch

where Aa(XL,e) = ae(XL,e) - ag(Ai,,e) is the difference of ac polarizabilities of the up­per and lower states that depend both on

277

Fig. 1. An optical lattice clock. Atoms localized in a sub-optical wavelength region offer Doppler-free spectrum of the clock transition, and simulate mil­lions of single-ion clocks operated in parallel.

the trap laser wavelength A/, and its polar­ization e. By adjusting both of the polariz-abilities so as to satisfy Aa(Az,,e) = 0, the observed atomic transition frequency v will be equal to v§ independent of the trap laser intensity I,6'7 as long as the higher order cor­rections 0{I2) are negligible. In search for a clock transition in which the Stark shift cancellation condition Aa(Ai,,e) = 0 can be given by a well-definable parameter of XL, the use of J = 0 state that exhibits a scalar light shift is preferable. Specifically, we adopt the 5s2 So -> 5s5p 3Po forbidden transition of 87Sr with nuclear spin of 9/2 as a "clock" transition (Fig. 2), in which hyper-fine mixing of the 3P0 (F = 9/2) state with the 1,3Pi states provides a finite lifetime of rV1 = 150s.

3. Exper iment

3 .1 . The magic wavelength and spectroscopy

The schematics of our experiment8 is shown in Fig. 3. 87Sr atoms were laser-cooled and trapped on the :So — 3Pi transition at a few /xK and loaded into a ID optical lattice that was formed by a standing wave of Ti-Sapphire laser. For atoms trapped in the op­tical lattice, we irradiated a clock laser at Ao = 698 nm, with its wave-vector parallel

35,

Detection yg|

Position / XL

[n[0 0.5 1 1.5 2 ^

sfl ' •' ' ^ III I I I

PI I I I

461 nm

Fig. 2. The ^ o and 3PQ states of Sr are coupled to the upper respective spin states by an off-resonant standing wave light field to produce equal light shifts in the clock transition, where atoms are excited on the pSo) <8> \n) —» |3Po) ® \n) electronic-vibrational transitions.

to the fast axis of the ID lattice potential, for 10 — 40 ms to excite atoms to the 3Po state. The excitation of the clock transition was monitored by the population of atoms in the ground state by driving the xSn — ^ l cyclic transition.

We measured the light shift ^ as a function of the lattice laser intensity of / to determine the differential dipole polariz-ability Aa(Ax,) of Eq. (1), which is given by 6Vac(\L,I)/6I = -Aa(XL)/(2s0ch). Fig­ure 4 plotted the normalized light shift Svac/SI as a function of the lattice laser wavelength Az, to determine the magic wave­length, where the light shift ^ac(Az,,-0 due to the lattice laser is cancelled out, to be 813.420(7) nm.9

The inset of Fig. 5 shows a clock spec­trum (empty circles) measured at the magic wavelength with an interrogation time of 40 ms and a cycle time of 1 s for each point. Nearly Fourier-limited linewidth of 27 Hz was obtained, which demonstrated an order of magnitude reduction of the linewidth mea­sured for neutral-atom-based optical clocks so far.10'12 This measurement inferred the clock stability of <TV(T) ~ 10_14/-\/r with r measured in second. Orders of magnitude improvement of the stability will be expected by the reduction of the clock laser linewidth,

278

in addition to the normalization of the clock excitation rate.12

Cooling & trapping

%£,„ Kiiffl Optical /1L fibre

7H : j ^ ^ g : 4 ^ f ^

Lattice laser (A.m=813.4nm)

Clock laser (^=698 nm)

Synthesizer |

GPS satellite

/

S

International atomic time

(TAI)

/

Cs atomic clock (Agilent 5071A)

JCEO ftep

| Reference cavity |

Nd:YAG/I2

(NMIJ-Y1)

GPS disciplined oscillator

/ 4L &4

Optical frequency

I | comb

f. s

Fig. 3. a. Ultracold Sr atoms were trapped in a one-dimensional optical lattice formed by a standing wave of a Ti-sapphire laser tuned to a magic wave­length. The clock laser was introduced along the fast axis of the lattice potential to guarantee the Lamb-Dicke confinement, b. The frequency of the clock laser was measured by an optical frequency comb ref­erenced to the SI second.

in Table 1. Recently, the optical lattice clock based on 87Sr was also realized by JILA group and its absolute frequency has been reported.13

j i

813.0 813.2 813.4 813.6 813.8

Lattice laser wavelength (nm)

Fig. 4. a. Light shifts on the clock transition were measured as a function of the lattice laser wave­length to determine the magic wavelength to be AL = 813.420(7) nm.

3.2. Absolute frequency measurement

The frequency of the lattice clock was mea­sured with an optical frequency comb ref­erenced to a commercial Cs clock (Agilent 5071A) as depicted in Fig. 3b.9-14 The fre­quency of the Cs clock was monitored by the GPS time during the period of the frequency measurement of over one month, which de­termined the frequency offset between the Cs clock and the GPS time15 to be -1.04(8) x io-13.

Figure 5 summarizes the absolute fre­quency measurements referenced to the Cs clock. Each filled symbol represents a fre­quency measurement of typically 104 s to reduce the fractional instability of the Cs clock down to 6.5 x 10~14 corresponding to an uncertainty of 28 Hz. With total aver­aging time of T « 9.4 x 104 s over 9 days, we determined the clock transition to be 429,228,004,229,952(15) Hz,9 including the systematic correction of —45.7 Hz as given

Clock laser frequency (Hz)

i , * • - * • i f f

1 2 3 4 5 6 7

Data number

Fig. 5. Absolute frequency measurement of the 1So — 3Po transition of 8 7Sr atoms in an optical lat­tice. Filled symbols correspond to a frequency mea­surement over 104 s. The inset shows the typical clock transition resolving a linewidth of 27 Hz.

3.3. Uncertainty budget

Table 1 summarizes the uncertainties for this experiment as well as those expected for fu­ture experiments. The first-order Doppler shift may be in­troduced by the relative motion between the clock laser and the lattice potential. An active phase-noise cancellation by referring

279

Table 1. Systematic corrections and uncertainties for the absolute fre­quency measurement of the 1So—3Po transition o f 8 ' S r atoms in an optical lattice. A typical lattice laser intensity of Io = 10 kW/cm is assumed for attainable uncertainties associated with the light shifts. Frequency shifts are given in the unit of Hz.

Effect

1st order Doppler 2nd order Doppler recoil shift 1st order Zeeman collision shift blackbody shift probe laser light shift Scalar light shift Vector light shift Tensor light shift 4th order light shift Cs clock offset Frequency measurement

Systematic total Total uncertainty 8v

Correction

0 0 0 0

0.6 2.4 0.1

- 3 . 8 0 0 0

- 4 5 0

-45 .7 Hz

Achieved uncertainty

3 X 10~2

2 x IO" 6

0 10 2.4 0.1

0.01 4

IO" 3

10~3

10~3

3 9

15 Hz

Attainable uncertainty

< 10~3

< 2 X 10~6

0 10~3

10~5

3 x 10~3

i o - 3

10~3

i o - 3

10~3

I O - 3

4 x 10~3 Hz

to a reflection from a mirror surface that forms a standing wave for the lattice poten­tial could reduce this Doppler shift to less than 1 mHz.16

The first-order Zeeman shift arises due to the hyperfine mixing in the 3Pn state, which is m x 106 Hz/G for Am = 0 tran­sitions. The reduction or control of the mag­netic field at /zG level would allow 1 mHz uncertainty. The collisional frequency shift may ex­ist in the ID lattice configuration, since tens of unpolarized atoms were typically trapped in a single lattice potential with an atom density of ~ 101 2cm - 3 . This collision shift would ultimately be removed by spin-polarizing 87Sr atoms to activate Fermi sup­pression of collisions or by applying a 3D lat­tice with less than unity occupation. For the latter case, a frequency shift due to the res­onant dipole-dipole interaction at an inter­atomic distance of A/2 « 400 nm is given. The Blackbody shift scales as VB « 3 • 10"10 x (T/K)4 Hz, which would introduce a 3 mHz uncertainty for the temperature imho-mogeneity of AT = 0.1 K at T = 293 K. The vector light shift can be problem­

atic in 3D optical lattices. A phase stable optical lattice, where the 3D laser intersec­tion is formed by a single linearly-polarized standing-wave with folded mirrors,17 could be applied to provide linearly polarized trap­ping light fields. The forth-order light shift in Table 1 is estimated5 without including resonant ef­fects, which require accurate information on the magic wavelength and dipole moments of associated transitions. While the energy dif­ference between the 5s7p1P\ state and the clock upper state 5s5p3P0 (813.359 nm x 2) nearly coincides with the magic wavelength, this transition is only weakly allowed by hyperfine-mixing, hence its contribution to the 4th order shifts should be small. Next closely lying states are the 5s4/3i r

;f and 5s7p3P§ connected by 2-photons with wave­lengths of 818.55 nm and 796.81 nm, respec­tively.

4. Outlook

4.1. Lattice clock families

The optical lattice clock scheme is applicable to odd isotopes of other divalent atoms18 '19

280

Table 2. The lattice clock scheme is applicable to divalent atoms with nuclear spin / . The ^ o - 3 P 0 clock transition at Xp may be interrogated in optical lattices tuned to the magic wavelength A^,. The sensitivity q on a variation and the fractional frequency variation Su/uo for A a / a = 1 0 - 1 6 are listed.

Species

Nuclear spin (/) Abundance (%) Clock transition, Ap (nm) Magic wavelength, A , (nm) Light shift, vu (kHz/kWcm" 2 ) Atom temperature, Ti (/iK) Catching power, Pi ( k W c m - 2 ) Min. power, Po ( k W c m - 2 ) Photon scattering rate (Hz)

Sensitivity q Sv/vo x l O " 1 6

4 3 Ca

7/2 0.14 660

735.5 11 5.6 106 65.3 0.4

125 0.017

87Sr

9/2 7.0 698

813.42 13 1

16 26.7 0.03

443 0.062

1 7 1 Yb

1/2 14

578 752 20 20

209 16

0.01

2714 0.31

1 9 9 H g

1/2 17

266 358 1.4 28

4480 995

1

15299 0.81

2 1Ne

3/2 0.27 74 613 0.03 100

697 • 103

2.1 • 108

350• 103

that have hyperfine-mixing induced J = 0 —¥ J = 0 transition between long-lived states. The candidate atoms include, Be, Mg, Ca18, Sr, Yb19 '20, Hg21 and rare gases such as Ne, Kr, Xe. Some of them are listed in Ta­ble 2. Furthermore, a multi-photon excita­tion of the clock transition22 may allow using even isotopes of these candidates that exhibit purely scalar nature of J = 0 state. A lat­tice clock with ultimate performance needs to be experimentally explored among these possible candidates because of difficulties in predicting some of uncertainties associated with higher order light field perturbations; such as resonant contribution to the 4th or­der light shifts as mentioned previously and (multi-)photon ionization processes. Once the far-off-resonance condition is satisfied for the magic wavelength, in order to minimize such uncertainties, it is essential to operate the lattice clock with lowest possible lattice laser intensity Po, as long as the tunneling of atoms among lattice potentials does not affect the clock accuracy.21

Table 2 summarizes laser parameters for lattice clocks. Light intensities Pi required to catch laser cooled atoms are given as­suming the lattice depth of Ui = lOksTi with atom temperatures Ti reported so far (Doppler temperature is used for Hg). Once

atoms are trapped, they can be, in prin­ciple, sideband-cooled down to their vibra­tional ground states using narrow "clock" transitions. The lattice potential depth that confines these atoms with the Lamb-Dicke parameter 77 = y/h/(2Mvt)/\p is given by,

(2) •W^Vi. where we assumed the lattice potential pe­riodicity of A L / 2 . ER represents the pho­ton recoil energy associated with the clock transition Ap. M and Vt are the mass of trapped atom and its trapping frequency, re­spectively. Assuming rj = 0.26, the minimum lattice laser intensities Po that provide Uo and associated photon scattering rates are calculated as shown in the Table. These es­timates suggest that Sr and Yb atoms would be favorable candidates for the lattice clock, unless the multi-photon resonant effect is not encountered.

4.2. Search for a variation

The potential high precision and wide appli­cability of the lattice clock scheme to other atom species would allow exploring a possi­ble variation of fine structure constant a.23

So far frequencies of several atom clocks op­erated at 10 - 1 5 uncertainty level were com-

281

pared over a few years to verify the constancy

a t the level of | d / a | = 2.0 • 10" 1 5 /y r - 2 4 As­

suming A a / a = 1 0 - 1 6 , at which level per

year astrophysical determinations gave con­

troversial results,2 3 a fractional change of the

clock frequency25 5V/VQ is listed in Table 2.

Heavier atoms such as Yb and Hg tend to

show higher sensitivity as the relativistic cor­

rections are proportional to a tom number as

Z2. In order to verify Aa/a at 10~ 1 6 , one

may take Sr lattice clock as an anchor to de­

tect the fractional frequency change of Yb or

Hg based lattice clock at 8V/I>Q = 1 0 - 1 6 level.

Such experiment would be feasible in view of

the current laser-stabilization technologies26

and frequency measurement based on the op­

tical frequency comb technique.1 1

A c k n o w l e d g e m e n t

This work received support from the Strate­

gic Information and Communications R&D

Promotion Programme (SCOPE) of the Min­

istry of Internal Affairs and Communications

of Japan .

R e f e r e n c e s

1. H. Dehmelt, IEEE Trans. Instrum. Meas. 31, 83 (1982).

2. See references in P. Gill, Metrologia 42, S125 (2005).

3. H. Katori, Proceedings of the 6th Symposium Frequency Standards and Metrology, edited by P. Gill (World Scientific, Singapore, 2002) p323 .

4. T. Ido and H. Katori, Phys. Rev. Lett. 91 , 053001 (2003).

5. H. Katori, M. Takamoto, V. G. Pal'chikov, and V. D. Ovsiannikov, Phys. Rev. Lett. 9 1 , 173005 (2003).

6. H. Katori, T. Ido, and M. Kuwata-Gonokami, J. Phys. Soc. Jpn., 68, 2479 (1999).

7. H. J. Kimble et al., Laser Spectroscopy, Pro­ceedings of the XIV International Confer­ence, edited by R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler (World Sci­entific, Singapore, 2000) p. 80.

8. M. Takamoto and H. Katori, Phys. Rev. Lett. 91 , 223001 (2003).

9. M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, Nature 435, 321 (2005).

10. F. Ruschewitz et al, Phys. Rev. Lett. 80, 3173 (1998).

11. Th. Udem et al, Phys. Rev. Lett. 86, 4996 (2001).

12. G. Wilpers et al, Phys. Rev. Lett. 89, 230801 (2002).

13. A. D. Ludlow et al, arXive:physics/ 0508041.

14. F.-L. Hong et al, Opt. Express 13, 5253 (2005).

15. The agreement between the GPS time and the TAI was about a few parts in 10 during the period of the frequency measurement.

16. L.-S. Ma, P. Jungner, J. Ye and J. L. Hall, Opt. Lett. 19 1777 (1994).

17. A. Rauschenbeutel, H. Schadwinkel, V. Gomer and D. Meschede, Opt. Commun. 148 45 (1998).

18. C. Degenhardt et al, Phys. Rev. A. 70, 023414 (2004).

19. S. G. Porsev, A. Derevianko, and E. N. Fort-son, Phys. Rev. A 69, 021403 (2004).

20. C. W. Hoyt et al, Phys. Rev. Lett. 95, 083003 (2005).

21. P. Lemonde and P. Wolf, Phys. Rev. A 72, 033409 (2005).

22. T. Hong, C. Cramer, W. Nagourney, and E. N. Fortson, Phys. Rev. Lett. 94, 050801 (2005); R. Santra et al, Phys. Rev. Lett. 94, 173002 (2005).

23. J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003). 24. E. Peik et al, Phys. Rev. Lett. 93 , 170801

(2004). 25. In Table 2, sensitivities q is defined as fol­

lows: the clock transition frequency varies as v = i/o + q(a2/aQ — 1) + 0(a ), where ao is the present-day value of a. See, E. J. Angst-mann, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. A 70, 014102 (2004).

26. M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, Opt. Lett. 30, 1815 (2005).

282

DEVELOPMENT OF MACH-ZEHNDER INTERFEROMETER AND "COHERENT BEAM STEERING" TECHNIQUE FOR COLD NEUTRONS

K. Taketani.*-, H. Funahashi and Y. Seki

Department of Physics, Kyoto University, Kyoto 606-8502, Japan . -E-mail:taketani@nh.scphys.kyoto-u.ac.jp

M. Hino, M. Kitaguchi and R. Maruyama*

Research Reactor Institute, Kyoto University, Kumatori, Osaka 590-0494, Japan

Y. Otake and H. M. Shimizu

RIKEN, Wako, Saitama 351-0198, Japan

We are developing multilayer interferometer of Mach-Zehnder type for cold neutrons. To fulfill severe requirements for aligning the four mirrors, we utilize six solid-etalon-plates and a high precision flat base plate. New devices making use of magnetic birefringence have been developed and demonstrated to be useful for fine adjustment of superposition between the two paths labeled with neutron spin.

Keywords: Neutron Interferometry, Cold Neutron, Mach-Zehnder Interferometer

1. Introduction

One of the most important applications of interferometer is precision measurements of the energy difference. When there is an interaction energy difference AE between two paths of a neutron interferometer, the relative phase is given by the expression

At = 2x^AE (1) h2

where m is the neutron mass, I is the neutron wavelength, and L is the interaction path length. A large interferometer for long wavelength neutrons enables us to measure small interaction energy difference.

The multilayer mirror is one of the most useful devices in cold-neutron optics. A multilayer of two materials with different potentials is understood as a one-dimensional crystal that is suitable for Bragg reflection of long-wavelength neutrons. Since the first successful interferogram of cold neutrons using multilayer mirrors [1], some remarkable experiments [2] have been performed.

The aim of our development is to increase

the spatial beam separation of multilayer interferometer in order to broaden the applicability of neutron interferometry and carry out high precision measurements.

2. Jamin interferometer

The first interferometer using four independent multilayer mirrors was achieved in 2002 [3]. Interference fringes with a contrast of 60% were observed. The interferometer is equivalent to the Jamin type based on a pair of air-spaced etalons. An etalon consists of two parallel planes with high-precision surface finish. The new devices named "beam-splitting etalons" (BSE) in ref[3], which are etalons deposited multilayer mirrors on their parallel planes, splits a neutron beam into two parallel paths spatially. The two waves are superposed each other on the second BSE.

The success of Jamin type has qualified the flatness and roughness of etalon plates. They are good enough for neutron-mirror substrates, which cause no serious distortion of wave front to compose an interferometer.

t Present address JAERI, Tokai, Ibaraki 319-1195, Japan

283

3. Mach-Zehnder interferometer

In order to enlarge an area enclosed by the two paths of interferometer, we are developing Mach-Zehnder interferometer (Fig. 1). To fulfill the severe requirement for aligning the four mirrors, we utilize six solid-etalon-plates. Each of them has a pair of precision parallel planes and an equal thickness to all of the six. We deposit neutron mirrors on four of the six solid etalons, and place the six on a high precision flat base plate using optical contact technique.

To determine the tolerance of etalons and base plate, we have examined the coherence lengths of our neutron beam and also examined moire fringes arising from finite crossing angle of the superposed two waves. The coherence length is here defined as the optical path-difference which gives interferograms with the contrast of e_1 of the maximum.

All our experiments are performed using the cold neutron beam line "MINE2" at the JRR-3M reactor in JAERI. The mean wavelength A is 0.88nm and the bandwidth is 2.7% in FWHM. The longitudinal coherence length is evaluated at 17nm which is consistent with neutron spin-echo spectra [3].

The transverse coherence length lT can be approximately expressed as

I =JL2l±. (2) In #div

where <?div is the maximum angle of beam divergence from the center [4]. The beam collimation consists of the entrance slit of the beam line and the detector slit. The distance between the two slits is 4410mm. When the entrance slit is set to 40mm high and 1.0mm wide, and the detector slit to 14mm high and 0.35mm wide, the vertical and horizontal coherence lengths become 79nm and 3.2 um respectively.

The interval of moire fringes is written as

L e (3)

where 0 is the finite crossing angle of the two beams. The contrast C of interferograms obtained using a neutron counter with the narrow slit of w wide is given by

C = sin nw

L 7CW

L (4)

V moire / /

When w//moire is larger than 0.7, the contrasts are reduced to smaller than e"1. Therefore, the crossing angle of the two beams should be smaller than ±44nrad vertically and +1.8urad horizontally.

Relative inclination or displacement, or both between the two beams arise from dimensional errors of the six etalons and the flat base plate. For example, irregularity in thickness of the six etalons causes both horizontal and longitudinal displacements, and the finite nonparallelism of etalon plates makes the relative inclination. The imperfection of the flat base also causes the relative inclination and displacement. The etalon plates must match in thickness to±0.15um , and must be parallel within 6nm over the clear aperture of 20mm diameter. The surface of base plate must be flat better than 30nm over the clear aperture of 480mm long by 80mm wide. Using etalons and base plate satisfying tolerances on each degree

Analyzer mirror

Phase shifter coil

Top view

* /2 10mm

424mm

Magnetic Non Magnetic' mirror Magnetic m j r r 0 r

mirror

SOO'if i f"--

Fig. 1 Schematic view and photograph of the present neutron multilayer interferometer of Mach-Zehnder type

284

of freedom, we have constructed a pilot unit of Mach-Zehnder interferometer. The beam separation is 10mm and the distance from the splitter mirror to the analyzer mirror is 424mm. We have deposited magnetic mirrors as splitting mirrors and non-magnetic mirrors as reflection mirrors.

We performed a trial run, but the contrast of observed interferograms was not significant enough. Each dimension of the etalons and the base was surely within each individual tolerance in the accuracy of measurements, however, it was impossible to measure exactly the value of actual errors, and more than two simultaneous errors would happen in reality. To adjust the superposition and improve the contrast of interferograms, we have developed "coherent beam steering" technique.

4. "Coherent Beam Steering" technique for cold neutrons

"Coherent beam steering" is a new technique to correct a crossing angle and parallel shift of the two paths using magnetic birefringence. It is possible to label the two paths with spin divided by the magnetic-multilayer splitter of our Mach-Zehnder interferometer.

A quadrupole magnet generates a gradient of magnetic field, which deflects a polarized neutron beam along the direction of gradient. The deflection angle is

A<9 = //-dBr

A T ( 5 )

ax rnv where m , ju, v are mass, magnetic moment and velocity of neutron respectively. BQ is the strength of magnetic field. L is the length of magnetic field along beam.

A triangle solenoid coil bends a polarized neutron beam in the plane perpendicular to the magnetic field just like an optical prism. The bending angle is given by

A mjuBrA a he = 2-^-~—tan —

h2 2

(6)

where BT is magnetic field inside the solenoid.

a is the vertex angle of triangle, h is the

Plank's constant. To avoid the spin depolarization, we

usually apply a weak magnetic field of several Gauss in the vertical direction to the whole of setup. When we operate the steering devices introduced above, we have to arrange their field for the neutron beam not to pass through zero magnetic field regions. For adjustment of the horizontal crossing angle and parallel shift, we use triangle solenoids of vertical axis, and for the vertical steering, quadrupole magnets with narrow collimation slits as shown in Fig.2.

The crossing angle and parallel shift of our pilot unit were estimated from the flatness data of the base and the specifications of etalons. As the error estimation of horizontal parallel shift is less than half of its tolerance, there is no need to adjust it. We have demonstrated the feasibility of adjustment using the present steering devices in the other degrees of freedom as follows.

4.1. Moire"fringes

Optical devices of the MINE2 were arranged in a form of neutron spin-interferometer (NSI) [5]. When the two spin components between the polarizer and analyzer are deflected in the opposite direction slightly even tens of nano radians, an image of moire' fringe appears in the superposition as the evidence of deflection.

We used a quadrupole magnet shown in

Top view of the triangle solenoid ALU.

100mm

Down

50mm Neutron

Side view of i0omm the quadrupole magnet Fig. 2 Schematic view and photograph of the quadrupole magnet and the triangle solenoid

285

Fig.2. The gradient of magnetic field was 3.2G/cm at 2A. The size of magnet along beam was 50 mm. Clear moire fringes were recorded using imaging plate (Fuji film BAS1800) with exposure time of 60min. The interval of them was roughly 10mm. The shrinkage of moire fringes was also observed by increasing the current of the quadrupole magnets. The stronger gradient of magnetic field makes the larger crossing angle between the two spin components. Over the range of about ± 90nrad, we have demonstrated the control of vertical crossing angle.

We prepared a triangle solenoid shown in Fig.2. This triangle solenoid generated 50G inside the triangle when it was applied 4A. The vertex angle was 127deg. Using a scanning detector with a fine slit of 0.25mm wide, we observed clear moire fringes with the interval of approximately 1mm at the current of 4A. This result demonstrates that we can control horizontal crossing angle over the range of about ±lurad.

4.2. Recovery of contrast

We tested the adjustment of vertical parallel shift using a pair of quadrupole magnets. We installed a pair of BSE with the gap thickness of 9.75 um [4] and placed the pair of quadrupole magnets at the downstream of the second BSE. The inter distance of the

200 150

1 15°0°U

° 0 « 200 § 150 0 100 " 50 1 ° 5 200

150 100 50

0

****** 36 ±2%

• V . •T*

0A

a>

**** 5 9 ± 2 ^ + +

V* \/ * vHHI

V ••»•• *•.•**

31 ±2% ••*•.

|i.56nm

QQ

1.75A

3.5A

-M~ 0 0.05 0.1 0.15 0.2 0.25 0.3

Phase shifter coil current [A]

Fig. 3 Demonstration of the recovery of NSI interferograms using a pair of quadrupole magnets

pair of quadrupole magnets was 500mm. The maximum contrast of the NSI interferograms was 60% without tilting the second BSE. When we tilted the second BSE in 0.16deg, the contrast of interferograms decreased from 60% to 36% due to the intentional parallel shift of the two paths caused by twisting the pair of BSEs [4]. Although a quantitative explanation has yet to be produced for the optimum current, we have succeeded in recovering the contrast from 36% to 60% using the pair of quadrupole magnets.

5. Summary

We are developing multilayer interferometer of Mach-Zehnder type to carry out high precision measurements, and have developed "coherent beam steering" technique. The experimental results in this paper show that when we apply the present technique to our Mach-Zehnder interferometer, we can realize the adjustment of superposition accurately enough to achieve clear interferograms.

Acknowledgments

This work was supported by the interuniversity program for common use JAERI and KUR, and financially by Special Coordination Funds for Promoting Science and Technology of the Ministry of Education of Japanese Government. One of the authors (K.T.) was supported by JSPS Research Fellowships for Young Scientists.

References

1. H. Funahashi, et al., Phys. Rev. A54 (1996) 649.

2. Y. Otake, et al., Proc. of ISQM, Tokyo, 98 (1999)323.

3. M. Kitaguchi, et al., Phys. Rev. A67 (2003) 033609.; M. Kitaguchi, PhD thesis, Kyoto University (2004).

4. M. Kitaguchi, et al., JPSJ 72 (2003) 3079. 5. D. Yamazaki, Nucl. Instr. Meth. A488

(2002) 623.

286

SURFACE POTENTIAL MEASUREMENT B Y ATOMIC FORCE MICROSCOPY USING A QUARTZ RESONATOR

SEIJI H E I K E * AND T. H A SH IZ U ME

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan

* E-mail: heike@rd.hitachi.co.jp

Scanning Kelvin probe microscopy is demonstrated by using a noncontact atomic force microscope with a 1 MHz quartz length-extensional resonator as a force sensor. A tungsten probe tip glued onto the end of the quartz rod is electrically connected to the electrode of the resonator. To detect Coulomb force, frequency shift signal is used instead of oscillation amplitude of the resonator. A surface potential on a A u ( l l l ) single crystal with nanoscale carbon dots is measured. The potential difference is observed between the Au surface and the carbon dots.

Keywords: Kelvin probe; atomic force microscopy; surface potential; work function.

1. Introduction

Scanning tunneling microscope (STM) [1] have been expected to be a powerful tool for miniaturization of future electronic devices. Although recent developments have made it possible to manipulate individual atoms and molecules with STM probe tips [2-4], the ac­tual devices should include insulating mate­rials which cannot be observed with an STM. As a scanning probe method with atomic res­olution, noncontact atomic force microscopy (NC-AFM) has been spotlighted in recent years. In particular, the frequency modula­tion (FM) detection technique [5] realized to achieve AFM observation with a true atomic resolution [6-9]. However, in the process of surface modification where electric field is of­ten used, a sudden change in electrostatic force causes bends in cantilever. To avoid this problem, a quartz resonator with a high force constant was used as a resonator for NC-AFM [10-13]. Moreover, a quartz res­onator can sense the deflection electrically by itself.

To investigate electrical characteristics of nanoscale electronic devices, scanning Kelvin probe microscopy (SKPM) based on NC-AFM is a useful method because it can map the surface potential [14, 15]. How­

ever, it is quite difficult to realize Kelvin probe measurement using quartz resonator, because the resonator has a low sensitivity to Coulomb force because of its high force constant. In this paper, we report develop­ment of the SKPM measurement system us­ing a quartz resonator and demonstration of a potential mapping on a Au surface using 1 MHz quartz length-extensional resonator.

2. Exper iments

Figure 1(a) shows a photograph of the res­onator consisting of a rod shaped quartz (2.75 mm x 85 /mi x 85 /*m, force constant k~ 4 x 105 N/m) and two electrodes on both sides of the rod as introduced in [10]. Fig­ure 1(b) shows an electrochemically etched tungsten probe tip (100 to 200 /mi in length) glued onto the end of the rod and electri­cally connected to one of the electrodes by using silver epoxy. Bias voltages are applied between the tip and the surface for SKPM observation.

A schematic diagram of the SKPM de­tection system is shown in Fig. 2. An ac volt­age signal tuned to the resonant frequency /o of the resonator is applied to one of the elec­trodes, and the output current from another electrode is converted to a voltage signal by

287

Fig. 1. (a) 1 MHz quartz resonator, (b) Magnified photograph of end of the resonator. Tungsten tip is glued by silver epoxy to electrically contact with one of the resonator electrodes.

a pre-amplifier. The phase shift data for the AFM feedback are obtained by comparing the phases of the input voltage and the out­put current. The phase shift can easily be converted to a frequency shift A / by using a frequency-phase curve. The oscillation am­plitude AQ is 0.4 to 0.6 nm, estimated from the displacement of the tip height when the amplitude was changed in the STM feedback mode. The bias voltage is applied to the sam­ple and the tunneling current through the sample is detected for STM operation.

For Kelvin probe measurement, an ac voltage fy (~ 100 Hz, 2 V) was added to the bias voltage. The amplitude of the fy com­ponent in the phase shift signal is detected with a lock-in amplifier and used as an elec­trostatic force signal. The surface potential is determined by adjusting the bias voltage so as to make the electrostatic force zero. In potential mapping measurement, the Kelvin probe feedback loop works only when the tip comes to data acquisition points for map­ping. The tip stays at each point for 100 ms until the feedback is completed.

The AFM used in this experiment was based on a commercial Omicron ultra high vacuum (UHV) STM and the whole scanner head was replaced by a homemade one. We used a polished sapphire sample, a Si( l l l ) clean suface sample and a Au( l l l ) single crystal sample. The silicon wafer (n-type,

phase shift

voltage J1

Fig. 2. Schematic diagram of the SKPM detection system. An ac Yoltage /Q (~ 1 mV) is applied to one of the electrodes, and the output current from another electrode is detected by a pre-amplifier. The phase shift data are obtained by comparing the input and output signals. For Kelvin probe measurement, an ac voltage fy is added to the sample bias voltage. The amplitude of the fy component in the phase shift signal is detected with a lock-in amplifier as an electrostatic force signal.

0.005 Ocm) was flushed in UHV to obtain a 7x7 structure. The Au( l l l ) sample was sputtered by Ar to make small hole struc­tures on the surface and was annealed at around 300 °C. The chamber pressure was maintained below IxlO""8 Pa during the SKPM measurements. An electrochemically etched tungsten tip was cut and glued onto the resonator, and then was sputtered by Ar to remove the overlaying oxide layer formed during etching.

3* Resul ts and discussions

To confirm AFM/STM observation, we mea­sured topographic images of polished sap­phire surface and Si( l l l )-7x7 clean surface. Figure 3 (a) shows an AFM image over the 500 nm x 500 nm area taken with Af = —0.5 Hz and AQ = 0.4 nm. Single atomic steps of 0.2 nm high and terraces of 150 nm wide are clearly observed. A magnified image with 100 nm x 100 nm area is shown in Fig. 3 (b). The terraces are not atomi-

288

cally l a t with a roughness of about 0.1 nm. Next, AFM/STM observation was demon­strated on a Si( l l l ) -7x7 surface. Figure 3 (c) shows an AFM image of Si(ll l)-7x 7 sur­face with 8.4 nm x 8.4 nm area obtained with imaging parameters, Af = —0.5 Hz, Vsampie = 0 V and A0 = 0-55 nm. An STM image obtained with a tunneling current of 0.5 nA and Vsampie = 2.0 V is shown in Fig. 3 (d). The scan size is the same as that of Fig. 3 (c), though the imaged region is not the same. Individual Si atoms and corner holes are resolved in both images. However, the resolution of the AFM image is a little bit lower than that of the STM image. These re­sults indicate that sub-nanometer resolution is achieved both on insulating and conduct­ing surface.

Fig. 3. Images of AFM/STM measurements, (a) AFM image of polished sapphire surface observed in 500 nm X 500 nm area. Parameters are Af = —0.5 Hz and AQ = 0.4 nm. (b) Magniied image with 100 nm x 100 nm area, (c) AFM image of S i ( l l l ) -7x7 surface obtained with Af = —0.5 Hz, V3ampie = 0 V and AQ = 0.55 nm. (d) STM image observed using the same tip as (c). Itunnei = 0.5 nA and F 5 0 m p l e = 2.0 V.

The surface potential was measured on the Au surface with 200 nm x 200 nm area. Figure 4 (a) shows the AFM topographic im­

age. The imaging parameters were Af = -0 .2 Hz and Vup = 0 V. Flat terraces with atomic steps are clearly observed. The hole structures, 5 to 20 nm in diameter, may be caused by Ar sputtering. Hillocks with ap­proximately 1 nm in height and 10 to 20 nm in diameter are also observed. Since these structures have conically shaped terraces and seem to pin the steps, they probably consist of hydrocarbon not removed by sputtering. Figure 4 (b) is a simultaneously obtained surface potential image for the same area as (a). The brightness corresponds to the work function. Bright spots indicate approx­imately 100 mV higher work function than that of the dark region. Because the position of these spots agree well with the position of the hillocks in Fig. 4 (a), the spots can be attributed to the hillock structures.

Fig. 4. (a) 200 nm x 200 nm AFM image of A u ( l l l ) single crystal surface after Ar sputtering. Parameters are Af = —0.2 Hz and AQ = 0.6 nm. Atomic steps and hole structures caused by sputter­ing are clearly observed. Hillocks, around 1 nm in height, 20 nm in diameter, are also observed, which may originate from carbon contamination remaining after sputtering, (b) Simultaneously obtained sur­face potential mapping of the same area as (a). The bright spots correspond to the hillock structures in (a), and their work function is 100 mV higher than that of the A u ( l l l ) surface.

Because of the enhanced contrast, the noise signals appear as small dark spots not related to actual surface structures. From the minimum size of the spots corresponding to the hillocks, the lateral resolution of po­tential mapping is estimated to be less than 5 nm. From the potential difference between

289

the Au surface and hillocks, the energy res­olution is estimated to be less than 100 mV. Although surface potential data are averaged in the actual mapping measurement, there still remain noises of around 50 mV, esti­mated from a cross section of Fig. 4 (b). Most presumable origin of the noise is inter­mittent feedback signals for potential mea­surement. At every data acquisition point, the Kelvin probe feedback loop has to ex­perience an unstable transition period where the bias voltage is fluctuated until the feed­back in completed. To remove this problem, a continuous feedback system should be in­troduced.

4. Summary

Surface potential measurement was per­formed by SKPM based on the NC-AFM using a 1 MHz quartz length extension res­onator as a force sensor. A tungsten probe tip glued onto the end of the quartz rod was electrically connected to the electrode of the resonator for the detection of Coulomb force between the tip and the surface. The fre­quency shift signal was used for Coulomb force detection instead of oscillation am­plitude of the resonator used in the con­ventional SKPM. A surface potential was measured on a Au( l l l ) single crystal with nanoscale carbon dots. The potential dif­ference of 100 mV were observed between Au surface and the carbon dots. The lat­eral resolution was less than 5 nm, and the energy resolution was less than 100 mV. It was shown that the resolution of potential measurement was high enough to evaluate nanoscale electronic devices.

Acknowledgments

This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.

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290

BERRY'S PHASES A N D TOPOLOGICAL PROPERTIES IN THE BORN-OPPENHEIMER APPROXIMATION

K A Z U O FUJIKAWA

Institute of Quantum Science, College of Science and Technology

Nihon University, Chiyoda-ku, Tokyo 101-8308, Japan

E-mail: fujikawa@phys.cst.nihon-u.ac.jp

The level crossing problem is neatly formulated by the second quantized formulation, which ex­hibits a hidden local gauge symmetry. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singu­larity) for arbitrarily large but finite time interval T. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. The crucial difference between the Aharonov-Bohm phase and the geometric phase is explained. It is also noted that the gauge symmetries involved in the adaibatic and non-adiabatic geometric phases are quite different.

1. Introduction

The geometric phases revealed the impor­tance of hitherto un-recognized phase factors in the adiabatic approximation1'2'3'4'5'6. It may then be interesting to investigate how those phases behave in the exact formula­tion. We formulate the level crossing prob­lem by using the second quantization tech­nique, which works both in the path integral and operator formulations7'8'9. In this for­mulation, the analysis of geometric phases is reduced to the familiar diagonalization of the Hamiltonian. Also, a hidden local gauge symmetry replaces the notions of parallel transport and holonomy.

When one diagonalizes the Hamiltonian in a very specific limit, one recovers the con­ventional geometric phases defined in the adiabatic approximation. If one diagonalizes the Hamiltonian in the other extreme limit, namely, in the infinitesimal neighborhood of level crossing for any fixed finite time interval T, one can show that the geometric phases become trivial and thus no monopole-like sin­gularity. At the level crossing point, the con­ventional energy eigenvalues become degen­erate but the degeneracy is lifted if one diag­

onalizes the geometric terms. Our analysis shows that the topological interpretation3'1

of geometric phases such as the topological proof of the Longuet-Higgins' phase-change rule4 fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. This analy­sis shows that the topological properties of the geometric phase and the Aharonov-Bohm phase are quite different.

Also, the difference between gauge sym­metries for adiabatic phase and "non-adiabatic phase" by Aharonov-Anandan10

becomes quite clear in this formulation.

2. Second quantized formulation

We start with the generic hermitian Hamil­tonian H = H(p,x,X(t)) for a single par­ticle theory in a slowly varying background variable X(t) = {X1{t),X2{t),...). The path integral for this theory for the time interval 0 < t < T, which is taken to be the period of the slower background system, in the second quantized formulation is given by

fvip*Vipexp{?r J dtd3x[C}}

291

where

8 C = i/>*(t,x)ih—ip(t,x)

-r(t,S)H(^,x,X(t)Mt,x).(l)

We then define a complete set of eigenfunc-tions

H(^ — ,x,X(t))vn(x,X(t))

= £n(X(t))vn(x,X(t)),

Jd3xv*n(x,X(t))vm(x,X(t)) = <5„,m (2)

and expand

iP(t,x) = J2bn{t)vn(x,X(0)). (3) n

The path integral for the time interval 0 < t < T in the second quantized formulation is given by

Z= [l[-Db*n-Dbn J n

xexp{^ f dt[J^b*n(t)ih^-tbn(t) u n

-X>*(*)£«(A-(t))M*)]} (4) n

where the second term in the action stands for the term commonly referred to as Berry's phase1 and its off-diagonal generalization. The second term is defined by

(n\ih—\m) =

h . fl d'xv*n(x,X(t))ih-vm(x,X(t)). (5)

In the operator formulation of the second quantized theory, we thus obtain the effective Hamiltonian

Heff(t) = Y/bt(t)£n(X(t))bn(t)

When we define the Schrodinger picture %eff{t)by replacing all bn(t) ->• bn(0) in Heff(t) we can show7,8

<n(T) |T*exp{-i J H&2,X(t))dt}\n(0))

= (n\T*e: X P { 4 / Heff(t)dt}\n). (7)

-Y,biit)(n\ih^-\m)bm{t).{&)

Both-hand sides of this formula are exact, but the difference is that the geometric terms, both of diagonal and off-diagonal, are explicit in the second quantized formulation on the right-hand side. The state vectors in the second quantization are defined by \n) = 6jj(0)|0), and the state vectors in the first quantized states by (2). If one retains only the diagonal elements in this formula (7), one recovers the conventional adiabatic formula5

e x p { ~ J dt[£n(X(t)) - (n\ih—\n)]}.{8)

The above formula (7) represents the essence of geometric phases: If one performs an exact evaluation one does not obtain a clear physical picture of what is going on. On the other hand, if one makes an adiabatic approximation one obtains a clear universal picture.

The path integral formula (4) is based on the expansion (3), and the starting the­ory depends only on the field variable ip(t, x), not on {bn(t)} and {vn(x,X(t))} separately. This fact shows that our formulation con­tains a hidden local gauge symmetry

v'n(x,X(t))=eia^vn(x,X(t))

b'n(t)=e-^%n(t) (9)

where the gauge parameter an(t) is a gen­eral function of t. By using this gauge free­dom, one can choose the phase convention of the basis set {vn(x,X(t))} at one's will such that the analysis of geometric phases becomes simplest. Prom the view point of hidden local symmetry, the formula (8) is a result of the specific choice of eigenfunctions

292

v„(x,X(0)) — v„(x,X(T)) in the gauge in­variant expression

vn(x;X(0))*vn(x;X(T))x (10)

e x p { - ^ J [Sn(X(t)) - (n\ih-\n)\dt}.

This hidden local symmetry replaces the no­tions of paralell transport and holonomy in the analyses of geometric phases, and it works not only for cyclic but also for non-cyclic evolutions9.

3. Level crossing problem

For a simplified two-level problem, the Hamiltonian is defined by the matrix in the neighborhood of level crossing 7

h(X(t))=^] E

0it^+gal

m(t)(U)

after a suitable re-definition of the parame­ters by taking linear combinations of Xk (t). Here yi (t) stands for the background variable and a1 for the Pauli matrices, and g is a suit­able (positive) coupling constant.

The eigenfunctions in the present case are given by

-<"-(*£?) (12)

by using the polar coordinates, yi = rsin9cos(p, 3/2 = rsmdsiaip, y$ = rcos9. Note that, by using hidden local symmetry, our eigenfunctions are chosen to be periodic under a 2n rotation around 3-axis, which is quite different from a 2TT rotation of a spin-1/2 wave function. If one defines

«m(y)*^«n(l/) = Akmn{y)yk

where m and n run over ± , we have

.k . , . (1 +cos6>) . Ak

++(y)yk = - g *

Ak+_(y)yk =

S^ +l-6 = (Ak_+(y)yky,

AUy)n = ^ ^ V (13)

The effective Hamiltonian (6) is then given by

Heff(t) = (E(t)+gr(t))blb+

+ (E(t) - gr(t))blb-

- hY,blAkmn{y)ykbn. (14)

m,n

with r(t) = y/yl + y\ + y\. The point r(t) = 0 corresponds to the level crossing. In the adiabatic approximation, one neglects the off-diagonal terms in the last geometric terms, which is justified for Tgr(t) ;§> hit, where tm stands for the magnitude of the ge­ometric term times T. The adiabatic formula (8) then gives the familiar result

exp{i7r(l — cos#)}

xexp{-1- [ dt[E(t) - gr(t)]ll5) n JC(Q->T)

for a 2ir rotation in ip with fixed 0, for exam­ple.

To analyze the behavior near the level crossing point, we perform a unitary trans­formation bm = J2n U(0(t))mncn where m,n run over ± with

"<»<«»-(E|"c#)' (16)

which diagonalizes the geometric terms and the above effective Hamiltonian (13) is writ-

Fig. 1. The path 1 gives the conventional geometric phase for a fixed finite T, whereas the path 2 gives a trivial phase for a fixed finite T. Note that both of the paths cover the same solid angle 27r(l — cos8).

293

ten as

Heff(t) a (E(t) + gr cos 6)c\c+ (17)

+ (E(t) — grcos6)c_£- — tupc'+c+

in the infinitesimal neighborhood of the level crossing point, namely, for sufficiently close to the origin of the parameter space (Vi(t),y2(t),y3(t)) but (yi(t),y2{t),y3(t)) ? (0,0,0). To be precise, for any given fixed time interval T, we can choose in the in­finitesimal neighborhood of level crossing Tgr(t) <C Ttkp ~ 2-KH. In this new basis, the geometric phase appears only for the mode c+ which gives rise to a phase factor

exp{i / tpdt} = exp{2j7r} = 1, (18) Jc

and thus no physical effects. In the infinitesi­mal neighborhood of level crossing, the states spanned by (6+,6_) are transformed to a linear combination of the states spanned by (c+,C-), which give no non-trivial geometric phase.

We emphasize that this topological prop­erty is quite different from the familiar Aharonov-Bohm effect 10, which is topolog­ical^ exact for any finite time interval T. Besides, the setting of the Aharonov-Bohm effect differs from the present level crossing problem in the fact that the space is not sim­ply connected in the case of the Aharonov-Bohm effect.

4. Non-adiabatic phase

We comment that the non-adiabatic phase by Aharonov and Anandan 10 is based on the equivalence class (or gauge symmetry) which identifies all the Schrodinger amplitudes of the form

{ e - ^ M } . (19)

This gauge symmetry is quite different from our hidden local symmetry which is related to an arbitrariness of the choice of coordi­nates in the functional space.

5. Discussion

The notion of Berry's phase is known to be useful in various physical contexts11. Our analysis however shows that the topologi­cal interpretation of Berry's phase associ­ated with level crossing generally fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. The notion of "approximate topology" has no rigorous meaning, and it is important to keep this approximate topolog­ical property of geometric phases associated with level crossing in mind when one applies the notion of geometric phases to concrete physical processes.

I thank Prof. N. Nagaosa for stimulating discussions.

References

1. M.V. Berry, Proc. Roy. Soc. A392, 45 (1984).

2. B. Simon, Phys. Rev. Lett. 51, 2167 (1983). 3. A.J. Stone, Proc. Roy. Soc. A351, 141

(1976). 4. H. Longuet-Higgins, Proc. Roy. Soc. A344,

147 (1975). 5. H. Kuratsuji and S. Iida, Prog. Theor. Phys.

74, 439 (1985). 6. M.V. Berry, Proc. Roy. Soc. A414, 31

(1987). 7. K. Fujikawa, Mod. Phys. Lett. A20, 335

(2005). 8. S. Deguchi and K. Fujikawa, Phys. Rev.

A72, 012111 (2005). 9. K. Fujikawa, Phys. Rev. D72, 025009

(2005). 10. Y. Aharonov and J. Anandan, Phys. Rev.

Lett. 58, 1593 (1987). 11. For the review of the subject see, for exam­

ple, A. Shapere and F. Wilczek, ed., Ge­ometric Phases in Physics (World Scien­tific, Singapore, 1989); D. Chruscinski and A. Jamiolkowski, Geometric Phases in Clas­sical and Quantum Mechanics (Birkhauser, Berlin, 2004).

294

SELF-TRAPPING OF BOSE-EINSTEIN CONDENSATES B Y OSCILLATING INTERACTIONS

HIROKI SAITO1 and MASAHITO TJEDA1'2

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan CREST, Japan Science and Technology Corporation (JST), Saitama 332-0012, Japan

We show that the self-trapped condensate can be dynamically stabilized in two and three di­mensional free space by periodic oscillation of the scattering length. The oscillating interaction produces an effective potential that prevents the condensate from collapsing. We take into account the effect of dissipation to simulate realistic situations.

Keywords: Bose-Einstein condensate; Feshbach resonance; soliton

1. Introduction

A raindrop keeps its shape without the help of an external confinement potential because of the interaction between water molecules. The attractive part of the interaction po­tential between molecules prevents a droplet from expanding and the repulsive part pre­vents the collapse.

Is it possible to make a droplet in the gas phase using the ultracold gas? In one di­mension (ID), the answer is yes. Recently, the ENS [1] and Rice [2] groups have suc­ceeded in creating the matter-wave bright solitons using a Bose Einstein condensate (BEC). The soliton in a quasi-lD trap keeps its shape, since the attractive interaction counterbalances the quantum kinetic pres­sure. The effective potential for the char­acteristic size of the condensate R consists of the interaction and kinetic energies, which are proportional to —R~d and R~2 in d di­mensions. Hence the effective potential al­ways has a minimum for d = 1. In 2D and 3D, however, the effective potential for R has no minimum, and the matter-wave droplet is always unstable against collapse or expan­sion, as long as the interaction is constant in time.

We try to stabilize the matter-wave droplet in 2D and 3D by oscillating the in­teraction using the Feshbach resonance. This

idea comes from the "inverted pendulum" in classical mechanics [3,4], in which rapid oscil­lation of the pivot stabilizes the pendulum in an upturned position. Rapid modulation of the motion of the pendulum effectively pro­duces a barrier against falling down. In a similar manner, we will show that rapid os­cillation of the interaction between repulsive and attractive effectively produces a barrier that prevents the condensate from collapsing and stabilizes the "gaseous droplet." [5-7]

2. Gross-Pitaevskii equation with oscillating interactions

We consider the Gross-Pitaevskii (GP) equa­tion given by

at 2m m -H'7/it/>, (1)

where m is the mass of the atom and a(t) is the time-dependent s-wave scattering length. We assume that a(t) oscillates as

a(t) = do + a\ sin fit. (2)

The factor 7 on the left-hand side of Eq. (1) describes the energy dissipation, and the last term on the right-hand side assures the conservation of the norm, where fi is the chemical potential. This type of phenomeno-logical dissipative equation has been used to

295

study the damping of collective modes [8] and vortex nucleation [9]. When 7 = 0, Eq. (1) reduces the usual GP equation in free space.

In order to reach the dynamically stabi­lized state, we start from the noninteracting ground state in a trapping potential. The trapping potential is gradually turned off as

v{t)=(V(0)(l-t/Ttr) t<Ttr K ' \ 0 t > T t r.

v ;

At the same time, the oscillating scattering length is gradually turned on as

a(t){t/T&) t<Ta Or(t)

a(t) t>T^

where a(t) is given by Eq. (2). This gradual change of parameters avoids initial nonadia-batic disturbances and makes the condensate more stable.

3. Two-dimensional case

We first consider a tight pancake-shaped trap, in which the axial frequency co± is much larger than the radial frequency u>±_ and other characteristic frequencies. In this case, the axial dynamics is frozen and the system behaves as 2D with an effective scat­tering length y/mu!z/(2Trh)a.

We numerically solved the GP equation in 2D and obtained the time evolution of the system as shown in Fig. 1 (a).

The initial state is the noninteracting ground state with COZ/LJ± = 50, and the ra­dial trapping potential uj2

L(x2+y2)/2 is grad­ually turned off while the oscillating inter­action is gradually turned on according to Eqs. (3) and (4). We find that even af­ter the radial confinement is completely re­moved, the condensate does not expand, nor does it collapse, and thus the BEC "droplet" is stabilized. The condensate undergoes the rapid oscillation due to the oscillating inter­action and the slow oscillation due to the initial nonadiabaticity. Figure 1 (b) show the density profiles at t = 0 and t = 1 s. The "droplet" state at t = 1 s is almost the

(4) T

Fig. 1. (a) Time evolution of the central den­sity }MT_= 0) |2 and the mean radius (r) = f y/xi + y2\ilj\2dxdy of a 8 5 R b in 2D with 7 = 0.03. The radial frequency of the t rap is u>± = 27r X 10 Hz, and the number of atoms is 103. The scattering length oscillates as — 0.6 + 2.4sin(30ui±t). The ramp functions in Eqs. (3) and (4) with T t r = Ta = 0.3 s are used, (b) The density profiles at t = 0 (dashed line) and ( = 1 s (solid line). The inset shows the semilogarithmic plots, where the dotted line is pro­portional to e~18r.

same as the initial Gaussian profile, but the peak density is higher and the tail is slightly longer. The inset shows the semilogarithmic plots of the density profiles. We find that the tail of the droplet is approximately propor­tional to e _ 1 8 r . It is interesting to note that the Townes soliton also decays exponentially at infinity [10].

We note that the amplitude of the slow oscillation gradually decays, while the ampli­tude is almost constant for 7 = 0 in Ref. 5. The decay of the slow oscillation is therefore due to the energy dissipation. For large t, the slow oscillation decays to vanish, and the

296

2 4 6 8 10

Fig. 2. (a) Time evolution of the central density \tl>(r = 0) |2 and the mean radius (r) = fr\ip\2dr of a 8 5 R b in 3D with 7 = 0.03. The frequency of the isotropic trap is u> — 2n X 10 Hz, and the number of atoms is 5 x 103. The scattering length oscillates as —0.7 + 1.5sin(30a;t). The ramp functions in Eqs. (3) and (4) with Tti-0.3 s and Ta = 0.2 s are used, (b) The density profiles at t = 0 and t = l s .

time evolution becomes "stationary" with the rapid oscillation. This state is the Flo-quet solution of Eq. (1).

4. Three-dimensional case

We numerically solve the 3D GP equation. The initial state is the noninteracting ground state confined in an isotropic trapping po­tential mw2r2 /2 with w = 2n x 10 Hz. The trapping potential is linearly turned off from t = 0 to 0.3 s, while the oscillating scatter­ing length — 0.7+ 1.5sin(30u>£) nm is linearly turned on from t = 0 to 0.3 s. Even after the trapping potential vanishes, the conden­sate neither expands nor collapses, as shown in Fig. 2 (a).

The density profile of the 3D droplet is

isotropic with the radial distribution shown in Fig. 2 (b). We find from Fig. 2 (b) that the density profile of the droplet strongly de­viates from the Gaussian.

Thus, we have demonstrated that the BEC droplet can be stabilized in 3D in the presence of dissipation with 7 = 0.03. How­ever, we have not succeeded in stabilizing it for 7 = 0 yet. Even when we start from the stable state in Fig. 2 and adiabatically decrease 7, unstable oscillations grow and destroy the droplet. The dissipation there­fore suppresses the dynamical instabilities and stabilizes the droplet. We have not un­derstood the origin of the dynamical insta­bilities yet. The stabilization in 3D is much more difficult than in 2D; still, we cannot ex­clude the possibility that there is a parameter regime in which 3D droplets are stabilized.

5. Stabilization mechanism

The physical mechanism of the stabilization of BEC droplets is very similar to that in the inverted pendulum [3,4]. Here we under­stand it by a simple argument. As mentioned in Sec. 1, the interaction energy is propor­tional to —R~d in d dimensions. Hence effec­tive "force" for R is given by -d(R~d)/dR oc R~d~1, which rapidly oscillates. According to the textbook [3], an effective potential pro­duced by rapid oscillation of the force is pro­portional to the square of the amplitude of the oscillating force, and then oc R~2d~2. In 2D, the effective potential produced by the oscillating interaction is oc R~6, which can counterbalance the attractive interaction oc — J R - 2 , and then stabilizes the droplet. In 3D, the effective potential is oc R~8, which can also counterbalance the attractive inter­action oc — R~3. Thus, the oscillating inter­action prevents the condensate from collaps­ing and stabilizes the droplet.

The quantitative analysis of the above argument has been done in Ref. 5, 6 using the variational method with Gaussian trial wave

297

functions. The Gaussian variational method

can predict the parameter dependences of the

size of the droplet and the collective frequen­

cies in 2D [5]; however, in 3D, it fails to pre­

dict even the stability [6]. Reference 11 has

pointed out t ha t the variational and other

approximations used so far have consider­

able deviations from the exact numerical re­

sults. We must therefore find more reliable

approximations to t reat the rapidly oscillat­

ing atomic cloud.

6. C o n c l u s i o n s a n d d i scuss ion

We have shown tha t matter-wave droplets

can be stabilized in 2D [5] and 3D [6] free

space by the oscillating interactions. The

oscillating interaction produces the effective

potential t ha t prevents the droplet from col­

lapsing.

One may think tha t the rapid oscillation

of the interaction induces thermal excitations

and the condensate is destroyed. However,

this is not the case as long as the oscillation

frequency is nonresonant with the excitation

frequencies. In fact, numerically solving the

full 3D G P equation, we find tha t no coherent

excitations above the droplet s tate grow. We

therefore expect tha t incoherent excitations

are also suppressed and the BEC is retained.

The stabilization of the 2D condensate

has also been studied in Ref. 12. Reference 13

has considered the multi-component conden­

sate, and shown tha t the multi-component

BEC droplet can also be stabilized by oscil­

lating interactions, which is called a "vector

soliton." Reference 14 has studied the case of

3D space with I D optical lattice. For strong

lattice potential , the condensate is confined

in each lattice site as 2D confinement, and

a train of droplets is stabilized by oscillating

interactions. The stabilization by oscillating

interactions will be applied to various sys­

tems, and the application o f the matter-wave

droplet will be explored.

A c k n o w l e d g m e n t s

This work was supported by Grant-in-Aids

for Scientific Research (Grant No. 17740263,

No. 17071005 and No. 15340129) and by a

21st Century C O E program at Tokyo Tech

"Nanometer-Scale Quantum Physics," from

the Ministry of Education, Culture, Sports,

Science and Technology of Japan.

R e f e r e n c e s

1. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 (2002).

2. K. E. Strecker, G. B. Partridge, A. G. Tr-uscott, and R. G. Hulet, Nature (London) 417, 150 (2002).

3. L. D. Landau and E. M. Lifshitz, Mechanics, (Pergamon, Oxford, 1960).

4. An illuminating account of this stabilizing mechanism is found in: R. P. Feynman, R. B. Leighton, and M. L. Sands, The Feynman Lectures on Physics, (Addison-Wesley, Read­ing, MA, 1964), Vol. II, Chap. 29.

5. H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403 (2003).

6. H. Saito and M. Ueda, Phys. Rev. A70, 053610 (2004).

7. F. Kh. Abdullaev, J. G. Cupto, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A67, 013605 (2003).

8. S. Choi, S. A. Morgan, and K. Burnett, Phys. Rev. A57, 4057 (1998).

9. M. Tsubota, K. Kasamatsu, and M. Ueda, Phys. Rev. A65, 023603 (2002).

10. C. Sulem and P. L. Sulem, The Nonlinear Schrodinger Equation, (Springer, New York, 1999).

11. A. Itin, S. Watanabe, and T. Morishita, cond-mat/0506472.

12. G. D. Montesinos, V. M. Perez-Garcia, and P. J. Torres, Physica D191, 193 (2004).

13. G. D. Montesinos, V. M. Perez-Garcia, and H. Michinel, Phys. Rev. Lett. 92, 133901 (2004).

14. M. Trippenbach, M. Matuszewski, and B. A. Malomed, Europhys. Lett. 70, 8 (2005).

298

SPINOR SOLITONS IN BOSE-EINSTEIN CONDENSATES — ATOMIC SPIN TRANSPORT

J. IEDA1'2

1 CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

E-mail: ieda@imr.tohoku.ac.jp

Matter-wave solitons in Bose-Einstein condensates of ultracold gaseous atoms with spin degrees of freedom are investigated based on the exact solution method. Combining some theoretical and experimental results, we present the undiscovered spin dependent soliton dynamics such as macroscopic spin precession and spin switching.

Keywords: Bose-Einstein Condensate; Spin Dynamics; Exact Soliton Solution.

1. Introduction

BBC of Atomic Gases. — For a decade, Bose-Einstein Condensation (BEC) of atomic gases has renewed theoretical and experi­mental interests in quantum many body sys­tems at extremely low temperatures1, which stem from two main features: (1) Almost all the parameters of the system, such as the shape, dimensionality, internal states of the condensates and even the strength of the inter-atomic interactions, are control­lable. (2) Due to the diluteness, no? -C 1 (n the particle number density and a the s-wave scattering length), mean-field the­ory explains experiments. In particular, the Gross-Pitaevskii (GP) equation demon­strates its validity as a basic equation for the dynamics of the condensates. The GP equation is a counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear op­tics. Thus, a study based on nonlinear anal­ysis acquires importance.

Matter-Wave Solitons.— Among the nonlinear physics, soliton is a remarkable ob­ject not only for the fact that the exact so­lutions can be obtained but also for the use­fulness as communication tools due to the robustness2. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the for­

mer is attributed to the inter-atomic interac­tions while the latter comes from the kinetic energy. Either dark or bright solitons are al­lowable depending on the positive or negative sign of the inter-atomic coupling constants a, respectively, and indeed have been observed in a quasi-one dimensional (qlD) optically constructed waveguide3. Such matter-wave solitons are expected in atom optics for ap­plications in atom laser, atom interferometry, and coherent atom transport4.

Spinor Condensates.— When one ex­plores those future applications, atomic BEC has another advantage. That is, atoms have spin degrees of freedom liberated under op­tical traps5 '6 '7, giving rise to a multiplicity as signals. Here, "spin" means the hyper-fine spin of atoms; the hyperfine spin / is de­fined by f = s + i, where s and i denote the electronic and nuclear spins. The so-called spinor condensates, being a candidate media for multicomponent solitons, have nontrivial nonlinear terms reflecting the SU{2) sym­metry of the spins. The spin-exchange in­teractions are the sources of the spin-mixing within condensates8 and there is no analogue in conventional nonlinear optics.

We combine the two fascinating proper­ties: matter-wave soliton and spin dynamics of the condensates, considering alkali atoms

299

of / = 1 with i = 3/2, e.g., 7Li, 87Rb, and 23Na, confined in the qlD space by purely optical means. Without external magnetic fields, the substates m/ = 0, ±1 (mj the magnetic quantum number) are degenerate. Based on the multicomponent GP equations, the spin dependent soliton dynamics is ana­lyzed.

2. Integrable Model

The assembly of atoms in the F = 1 state is characterized by a vectorial order param­eter: &(x,t) = [$i(a: , i) ,$oOM),$-i(a; , t)]T

with the components subject to the hyper-fine spin space. The system is quasi-one di­mensional: the trap is elongated in the x direction such that the transverse dynamics is factorized from the longitudinal and stays in the transverse ground state9. The Gross-Pitaevskii energy functional is expressed as6

EGP=Jdx (^dx$*a(x,t)dx$a(x,t)

+ ^n2(x,t) + ^f2(x,t)), (1)

where n(x,t) = &a(x,t)<fra(x,t) the parti­cle density and i(x,t) = <&*a(x,t)fai3$f}(x,t) the spin density (a, (3 = 0, ±1) with / = (fx, P, f z ) T the 3 x 3 spin-1 matrices. In this expression, CQ = (go + 2<?2)/3, c2 = (32—9o)/3, with the effective qlD couplings,9

_ 4ft2 aF 1 9F ma\ (l-CaF/a±y V ;

where ap are the s-wave scattering lengths in the total hyperfine spin F = 0, 2 scatter­ing channels of two atoms, a± is the size of the transverse ground states, m is the atomic mass, and C = —C(l/2) = 1.4603- • •. Note that a resonance occurs in Eq. (2) as oj_ ap­proaches the value of CCLF-

The time evolution of spinor condensate wave function &(x,t) can be derived from the variational principle: ihdt$a(x,t) = 5EGP/6$*a(x,t). Substituting Eq. (1) into

this, we obtain a set of equations:

h2

i/id t$±i = - ^ a * $ ± i + (co " c 2 ) |$ T i | 2 $±i

+ (So + c2)( |$±i |2 + |$o|2)$±i + C2*5=i*o. h2

ihdt$0 = ~^dx®o + c0 |$o|2$o + (co + c2) X ( |$! |2 + | $ _ i | 2 ) $ 0 + 2C2*5*i*_i- (3)

Based on Refs. 10, 11, we go into the case c0 = c2 = — c < 0 which corresponds to the integrable condition of Eqs. (3). Note that c~o < 0 means that bright soliton is stable and dark soliton in the opposite case c"o = c2 = c > 0 was analyzed in Ref. 12. The effective interactions between atoms in the qlD BEC can be tuned with the so-called confinement induced resonance9, which do not affect the rotational symmetry of the spin states, by setting oj_ = 3Caoa2/(2oo + a2) m Eq. (2) when ao > a2 > 0 or a2 > 0 > ao-

By the dimensionless form $ —> (<f>i, \/2</>o> <A-i)T! where time and length are measured in units of ha±/c and Hy/a±/2mc, respectively, Eqs. (3) reduce to a 2 x 2 matrix version of the NLS equation:

idtQ = -d2xQ- 2QQtQ, (4)

« - ( £ £ ) • (5> Since the matrix NLS equation (4) is integrable13, the dynamical problems of this system can be solved exactly. Here we list some conserved quantities of this system: number, NT = J dxn(x,t), n(x,t) = <&1-# = ti{Q^Q}; spin, FT = f dxf(x,t), f(x,t) = $ t f $ = ti{Q^a-Q}, (<r: Pauli matri­ces); momentum, PT = / dxp(x,t), p(x,t) = —ih& • dx& = —ihtv{Q^Qx}; energy, ET = Jdxe(x,t), e(x,t) = (h2/2m)dx& • dx& -c ( n 2 + f 2 ) / 2 = c t r { Q t Q x - Q + Q g t Q } .

3. Spin Dependent Dynamics

Because of the spin conservation, each soli­ton solution can be classified by the spin states10 '11, i.e., the ferromagnetic | F ^ | =

300

IT 0 I

0.1,

0.05,

0,

20

1

V

20

-20

0.1 0.05

0

20

i i • . . % -

•T"

/ - V'

0.1,

0.05,

0,

20

l * ±

0.1,

0.05,

4 20

i Fig. 1. Density plots of |<^o|2 (top) and |<£±i|2 (bot­tom) for a polar-polar collision. Soliton 1 (left mover) carries only 0 component and soliton 2 (right mover) consists of ± 1 components.

Fig. 2. Density plots of \4>o\2 (top) and |<)!>±i|2 (bot­tom) for a polar-ferromagnetic collision. Soliton 1 (left mover) is a polar soliton and soliton 2 (right mover) is a ferromagnetic soliton.

NT, and the polar |FT-| = 0. The ferromag­netic solitons carry the macroscopic spin and the polar solitons do not. According to the classification of the soliton solutions, we an­alyze two-soliton collisions in the following three cases: 1) Polar-polar (PP) solitons col­lision. 2) Polar-ferromagnetic (PF) solitons collision. 3) Ferromagnetic-ferromagnetic (FF) solitons collision.

PP solitons collision.— In Figs. 1, we show a PP solitons collision. These figures correspond to each component of the exact two-soliton solution11 for one collisional run. For simplicity, we choose the parameters to have |(/>i | = |</>-i |. Collisional effects appear only in the position shift and the phase shifts (not shown). In this sense, the polar-polar collision is basically the same as that of the single-component NLS equation.

PF solitons collision. •— In Figs. 2, we have density plots of a PF collision. Again

we set |0i | = |0 - i |. The polar soliton (soli­ton 1) initially prepared in mp — ±1 are switched into a soliton with a large popula­tion in rriF = 0 and the remnant of mp = ±1 after the collision. Through the collision, the ferromagnetic soliton (soliton 2) plays only a switcher, showing no mixing in the internal state of itself outside the collisional region.

FF solitons collision.— In Figs. 3, we give examples of this type of collisions for different soliton velocities: Figs. 3 (a) high speed, Figs. 3 (b) middle speed, Figs. 3 (c) low speed, with the other conditions fixed to illustrate the velocity dependence. The inter­nal shift 0i —+ 0_i, and vice versa, gradually increase by slowing down the velocity of the solitons. We can gain a better understand­ing of the FF solitons collision by recasting it in terms of the spin dynamics. The to­tal spin conservation restricts the motion of the spin of each soliton on a circumference

301

-10 0 10 -10 0 10 -10 0 10 x x x

Fig. 3. Time evolution of |^o|2 (left column), \<pi\2

(middle column), |^<-i|2 (right column) for FF soli-tons collisions with (a) high speed, (b) middle speed, and (c) low speed.

around the total spin axis. It will be inter­

preted as a spin precession around the total

magnetization. For example, the rotation an­

gles for Figs. 3 (a), Figs. 3 (b), and Figs. 3

(c) are calculated as ~ 0.27T, 0.57T and 0.97T,

respectively.

4 . S u m m a r y

We have demonstrated some spin dependent

effects during two-soliton collisions in spinor

Bose-Einstein condensates confined in opti­

cal waveguides. In the three characteristic

collisions, i.e., the PP , P F , and FF-type, the

polar soliton is always "passive" which means

tha t it can not affect the par tner ' s spin struc­

ture while the ferromagnetic soliton rotates

the spin of the other soliton. Thus, in the P F

collision, one can use the polar soliton as a

signal and ferromagnetic soliton as a switch

to realize a coherent matter-wave switching

device. Collision of two ferromagnetic soli-

tons can be interpreted as the spin precession

around the total spin. The rotat ion angle de­

pends on the total spin, ampli tude and veloc­

ity of the solitons. Only varying the veloc­

ity induces drastic change of the population

shifts among the components.

In conclusion, the spin dependent soliton

dynamics predicted in this work should be

observed in experiments and open up variety

of applications in coherent a tom transport .

R e f e r e n c e s

1. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71 , 463 (1999); A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001); R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Rev. Mod. Phys. 77, 187 (2005).

2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981); G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

3. K. E. Strecker et al, Nature 417, 150 (2002); L. Khaykovich et al., Science 296, 1290 (2002); S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999); J. Denschlag et al., Science 287, 97 (2000).

4. P. Meystre, Atom Optics (Springer-Verlag, New York, 2001).

5. J. Stenger et al., Nature 396, 345 (1998); D. M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 (1998); H.-J. Miesner et al., Phys. Rev. Lett. 82, 2228 (1999).

6. T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998); T.-L. Ho, Phys. Rev. Lett. 81 , 742 (1998).

7. H. Pu et al., Phys. Rev. A60, 1463 (1999); C. K. Law, H. Pu, and N.P. Bigelow, Phys. Rev. Lett. 81 , 5257 (1999); M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066 (2000); C. V. Ciobanu, S.-K. Yip, and Tin-Lun Ho, Phys. Rev. A61, 033607 (2000); M. Ueda and M. Koashi, Phys. Rev. A65, 063602 (2002).

8. H. Schmaljohann et al., Phys. Rev. Lett. 92, 040402 (2004); M.-S. Chang et al., Phys. Rev. Lett. 92, 140403 (2004).

9. M. Olshanii, Phys. Rev. Lett. 81 , 938 (1998); T. Bergeman, M. G. Moore, and M. Olshanii, Phys. Rev. Lett. 91 , 163201 (2003).

10. J. leda, T. Miyakawa, and M. Wadati, Phys. Rev. Lett. 93, 194102 (2004).

11. J. leda, T. Miyakawa, and M. Wadati, J. Phys. Soc. Jpn. 73, 2996 (2004).

12. J. leda, Study of Matter-Wave Solitons in Spinor Bose-Einstein Condensates (Ph.D thesis, Department of Physics, University of Tokyo, 2004).

13. T. Tsuchida and M. Wadati, J. Phys. Soc. Jpn. 67, 1175 (1998).

302

SPIN DECOHERENCE IN A GRAVITATIONAL FIELD

HIROAKI T E R A S H I M A * and M A S A H I T O U E D A

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

CREST, Japan Science and Technology Corporation (JST), Saitama 332-0012, Japan

* E-mail: terasima@stat.phys.titech. ac.jp

We discuss a mechanism of spin decoherence in gravitation within the framework of general rela­tivity. The spin state of a particle moving in a gravitational field is shown to decohere due to the curvature of spacetime. As an example, we analyze a particle going around a static spherically-symmetric object.

Keywords: spin entropy; general relativity; quantum information.

1. Introduction

The spin of a particle is an interesting de­gree of freedom in quantum theory. Recently, Peres, Scudo, and Terno1 have shown that in special relativity the spin entropy (i.e., the von Neumann entropy of a spin state) of a particle is not invariant under Lorentz transformations unless the particle is in a momentum eigenstate. Namely, even if the spin state is pure in one frame of reference, it may become mixed in another frame of ref­erence. The origin of this spin decoherence is that the Lorentz transformation entan­gles the spin and momentum via the Wigner rotation2. The entanglement then produces spin entropy by a partial trace over the mo­mentum.

In this paper, we study the spin state of a particle moving in a gravitational field to show its decoherence by the effects of general relativity3. Our result implies that even if the spin state is pure at one space-time point, it may become mixed at an­other spacetime point. This spin decoher­ence is derived from the curvature of space-time caused by the gravitational field. Such a spacetime curvature entails a local descrip­tion of spin by local Lorentz frame due to a breakdown of the global rotational symme­try. The motion of a particle is thus accom­panied by the change of frame, which can in­

crease the spin entropy analogous to the case of special relativity. As an example, we con­sider a particle in a circular orbit around a static spherically-symmetric object using the Schwarzschild spacetime.

2. Formulation

Consider a wave packet of a spin-1/2 parti­cle with mass m in a gravitational field. The gravitational field is described by a curved spacetime with metric in general relativity. Nevertheless, despite the spacetime curva­ture, we can locally describe this wave packet as if it were in a flat spacetime, since a curved spacetime locally looks like flat. More precisely, for any spacetime point we can find a coordinate system in which the met­ric becomes the Minkowski one. The coor­dinate transformation from a general coordi­nate system {x11} to this local Lorentz frame {xa} can be carried out using a vierbein (or a tetrad) ea'

i(a;) and its inverse ea^(x) defined by4

:Vab, ^(x)eb'/(x)gllv(x)

e%(x)eb"(x)=6°b, (1)

where g^ix) is the metric in the general co­ordinate system and rjab = diag(—1,1,1,1) is the Minkowski metric with a, b = 0,1,2,3. The vierbein then transforms a tensor in the general coordinate system into that in the

303

local Lorentz frame, and vice versa. For ex­ample, momentum p^(x) in the general coor­dinate system can be transformed into that in the local Lorentz frame via the relation pa(x) = ea

ll(x)p»(x). Therefore, we describe the wave packet

as in the case of special relativity1 using a local Lorentz frame at the spacetime point X1* where the centroid of the wave packet is located. Since a momentum eigenstate of the particle is labeled by four-momentum pa = ( \ / | P 1 2 +Tn2c2,p) and by the ^-component a (=T> I) of spin5 as \pa,a), the wave packet can be expressed by a linear combination

M = E / d3PN(Pa) C{Pa,<r) | p°, a),

(2) where

*PN{jr) = #p m c 2 2 (3)

is a Lorentz-invariant volume element. From the normalization condition

(p'^a'\pa,a) = J^-)S3(P-p)S^, (4)

the coefficient C(pa,a) must satisfy

£ fd3pN(pa)\C(pa,a)\2 = l. (5)

To obtain the spin state of this wave packet, we take the trace of the density ma­trix p = \i/))(if>\ over the momentum,

pr(a';a) = J d3pN(pa)(pa,a'\p\pa,a).

(6) The spin entropy is then given by the von Neumann entropy of this reduced density matrix:

S=-Tr[Pl(a';a)log2Pl(a';a)}. (7)

Moreover, the spin operator for the wave packet is defined by

^= l E ^ [d3pN(pa)\pa,a)(pa,P\, a,0 J

(8)

with a being the Pauli matrices. Note that this spin is not Dirac spin but Wigner one6. As is well known, Dirac spin, which corre­sponds to the index of 4-component Dirac spinor, is not a conserved quantity in a rela-tivistic regime and thus is not suitable degree of freedom for labelling one-particle states. In contrast, Wigner spin is a conserved quan­tity suitable for labelling one-particle states, because it is defined using the particle's rest frame.

3. Decoherence

Suppose that the centroid of the wave packet is moving with four-velocity u^{x) normal­ized as u' i(x)u / i(x) = —c2; this motion is not necessarily geodesic in the presence of an ex­ternal force. After an infinitesimal proper time dr, the centroid moves to a new point

x>n _ xiJ,+u'i(x)dT and then the wave packet

is described by the local Lorentz frame at the new point. This change in the local Lorentz frame is represented by a Lorentz transfor­mation A.ab(x) = Sa

b + x°b(x)dT, where

Xab(x) = u»(x)[eb»(x)Vlie\(x)]. (9)

In addition to this change, the acceleration by an external force is also interpreted as a Lorentz transformation. Thus, the motion of the wave packet is equivalent to a Lorentz transformation Aa

6(a;) = Sab + Xa

b(x)dT, where7

A°fc(x) = x\(.x)

- ^ [ aa(x) qb(x) - qa(x) ab(x)} (10)

using the momentum and acceleration of the centroid in the local Lorentz frame

qa(x) = e\(x)[mu»(x)}, (11)

aa(x) = e^ix) [uv{x)Vvu»(x) ]. (12)

Note that even if the wave packet moves as straight as possible along a geodesic curve, this Lorentz transformation may be nonzero in general relativity because of the first term.

304

Since spin entropy is not invariant under a Lorentz transformation1, neither is it in­variant during the motion of the wave packet. Note that a Lorentz transformation rotates the spin of a particle through an angle that depends on the particle's momentum; this ro­tation is known as Wigner rotation. The mo­mentum eigenstate |pa,er) thus transforms under the Lorentz transformation (10) as5

U(A(x))\pa,cr)

= J2D„>AW(x)) | A(x)pa,a'), (13) a'

where Daia(W(x)) is the 2x2 unitary matrix that represents a Wigner rotation given by7

Wik(x) = Si

k + Xik(x)dr

+ \i0(x)pk-Xk0(x)p*dT

p° + mc v '

with i,k = 1,2,3. Taking the trace of the density matrix p' = U(A{x))\ip)(ij\U(A(x)y over the momentum, we obtain the spin state p'T((r'; <J) and the spin entropy S' of the wave packet in the local Lorentz frame at the new point ar'M. However, the spin has been en­tangled with the momentum by the Lorentz transformation (10), since the Wigner rota­tion (14) of spin depends on the momentum. Due to this entanglement, the new entropy S" is not, in general, equal to the original one S. This implies that the spin state may de­cohere during the motion of the wave packet by the effects of general relativity.

4. Example

As an example in general relativity, we con­sider the Schwarzschild spacetime, which is the unique static spherically-symmetric solu­tion of Einstein's equation in vacuum. In the spherical coordinate system (t,r,6,<p), the metric is given by

1 giiv{x)dxildxv = -f(r)c2dtz + -dr'

f(r) + r2(d92 + sin2 0 # 2 ) , (15)

Fig. 1. A wave packet (small circle) going around a static spherically-symmetric object (large circle). The 1- and 3-axes of the local Lorentz frame are il­lustrated at the initial and final points.

where f(r) = 1 — (rs/r) with the Schwarz­schild radius rs. In the Schwarzschild space-time, we introduce an observer at each point who is static with respect to the time t using a static local Lorentz frame. The vierbein (1) is then

1 e0 '(x) =

cy/fir)' ei r(x) = Vftr),

e2e(x) = \ ,

r \x) =

1 (16)

rsin0

Suppose that the centroid of the wave packet is moving along a circular trajectory of radius r (> rs) with a constant velocity rd(j>/dt = Vy/JJr) on the equatorial plane 0 = 7r/2 (see Fig. 1). The four-velocity of the centroid is then given by

u*(x) = cosh£ :*M - C S i l l h £ u*(ar) = (17)

where £ is denned by tanh£ = v/c. We as­sume that at the initial point the wave packet has the definite values a = | and p1 = p2 = 0 but is Gaussian in p 3 with width w:

\C(p",a)\2 = ^ 6 ^ ) 8 ^ ) 5 ^

{p3-q3(x))^ y/nw exp

t i r (18)

where q3(x) = mc sinh£ is the momentum of the centroid (11) along the 3-direction.

305

Fig. 2. The spin entropy S at v/c = 0.8, r/rs = 0.9, and w/mc = 0.1 as a function of the proper time T normalized by T3 = mr3/w.

whereas extremely rapid decoherence occurs (r<2 —> 0) near the Schwarzschild radius r —> rs. The spin state does not decohere also at r = 3rs/2, because the first term in Eq. (10) is canceled by the second term.

Of course, this spin decoherence is very slow in the gravitational field of the earth rs ~ 1 cm. For example, when a wave packet is at rest in the International Space Station going around the earth, v ~ 7.7km/s and r ~ 6800 km, the characteristic decoherence time is Td ~ 2.2 x mc/w years.

Fig. 3. The inverse of the characteristic decoherence time TJ , normalized by TJ" , as a function of rs/r at v/c = 0.8.

Clearly, the spin entropy of this wave packet is zero at the initial point, since the spin is separable from the momentum. How­ever, after a proper time r of the parti­cle, spin entropy is generated by the gravity and acceleration, i.e., by the first and sec­ond terms in Eq. (10). Figure 2 shows the generated spin entropy S as a function of the proper time T in the case of v/c = 0.8, r/rs = 0.9, and w/mc = 0.1. The spin state of the wave packet decoheres to a mixed state and becomes maximally mixed (S —> 1) in the limit of T = oo. The characteristic deco­herence time is given by the inverse of

- l _ w(cosh£ — 1)

mr 1 -

2r/(r) vm-- i

(19) Figure 3 shows this value rd

1 as a function of rs/r when v/c = 0.8. No decoherence occurs (jd —> oo) at the spatial infinity r —• oo,

5. Conclusion

We have shown that spin entropy is gener­ated when a particle moves in a gravitational field. The spin state evolves into a mixed state even if the particle moves as straight as possible along a geodesic curve. This deco­herence is due to the spacetime curvature by gravity.

Acknowledgments

This research was supported by a Grant-in-Aid for Scientific Research (Grant No. 15340129) by the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References

1. A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. 88, 230402 (2002).

2. E. P. Wigner, Ann. Math. 40, 149 (1939). 3. H. Terashima and M. Ueda, J. Phys. A:

Math. Gen. 38, 2029 (2005). 4. N. D. Birrell and P. C. W. Davies, Quantum

Fields in Curved Space (Cambridge Univer­sity Press, Cambridge, 1982).

5. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995).

6. D. R. Terno, Phys. Rev. A 67, 014102 (2003).

7. H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004).

306

BERRY'S PHASE OF ATOMS WITH DIFFERENT SIGN OF THE g-FACTOR IN A CONICAL ROTATING MAGNETIC FIELD OBSERVED BY A

TIME-DOMAIN ATOM INTERFEROMETER

Atsuo Morinaga , Hirotaka Narui, Akinori Monma and Takatoshi Aoki

Department of Physics, Faculty of Science & Tech., Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan

E-mail: morinaga@ph.noda.tus.ac.jp

Berry's phase of the atom in the state with a positive or negative g factor for partial cycles of a conical rotating magnetic field was determined using a time-domain atom interferometer. The experimental results show that the solid angle for positive g-factor is $l-cos#) and that for a negative g-factor is ^(l+cos#) addition to the reversing the sign.

Keywords: Berry's phase;Atom interferometry; g-factor.

1. Introduction

In 1984, Berry predicted that a quantal system slowly transported round a circuit C by varying parameters in its Hamiltonian acquires the geometrical phase in addition to the dynamical phase. So far, numerous experiments on Berry's phase have been carried out and it has become well known in many physical systems [1]. As an example, Berry proposed that for a whole turn of the magnetic field round a cone of semiangle 0, the particle with a magnetic quantum number mF will give a phase shift of [2]

y = -2mnF(\-cos0). (1) Recently, the experiment was realized by

us using a time-domain atom interferometer [3]. It was comprised of the two long lived states with different magnetic quantum numbers coupled by two two-photon stimulated Raman pulses with a time interval T. Then the phase of the interference fringes was shifted in proportion to the difference between Berry's phases of the two states. First, we measured the phase shifts between F=l, mp=\ and F=2, nif=2 of the sodium ground hyperfine states under the rotating magnetic field on the equator. The results show that the phase is dependent on the magnetic quantum number multiplied by the rotation angle for partial cycles and that the

sense of the phase shift depends on the direction of rotation of the magnetic field. Finally, the phase was shifted with a slope of 2.9±0.2 as a function of rotating angle <p [4], although the difference between the magnetic quantum numbers AmF is 1. This contradiction will seemed to be solved if we assume that the sense of Berry's phase depends on the sign of the g-factor, because the hyperfine state of F=l has a negative g=-l/2, while the hyperfine state of F=2 has a positive g=l/2. Tycko already demonstrated that it also depends on the sign of the g-factor using a single crystal in a magnetic field [5]. In order to investigate the dependence of Berry's phase on the g-factor in detail, the authors aimed to measure the phase shift under a conical rotating magnetic field.

2. Experimental

The time-domain atom interferometer is composed of two different states of cold atoms whose magnetic quantum numbers are different, and two coherent two-photon stimulated Raman laser pulses whose frequencies are different by a frequency of the transition between two states. The atom is initially in the lower level under the quantization field. After two irradiations of laser pulses which are separated by T in time, the Ramsey resonance occurs in the population of the upper levels.

307

During the pulse separation, if the amplitude of the magnetic field varies, the dynamical phase shift, which is known as the scalar Aharonov-Bohm effect, occurs [6]. While, during the pulse separation, a conical rotating magnetic field with a different frequency v was applied to the atoms and Berry's phase was observed as a function of rotating angle of the field ^ = 2nTlv.

The ground hyperfine states of sodium atom are composed of the F=l state which has a g-factor of-1/2 and F=2 state of+1/2. Their Zeeman splitting is shown in Fig. 1. We could couple only the state of F=l, m^=l and F=2, mp=2 with a circular polarized two-photon Raman transitions.

Br,

it , a

F=2, g=l/2

F=l,g=-I/2

mF=2

mF=l

Fig. 1. Zeeman energy splitting of sodium atom and a two

photon Raman transition.

In order to get interference fringes with a visibility of more than 50 %, we carefully reduced the stray magnetic field to less than 0.2 mG and the magnetic field gradient due to zeeman slower was compensated by applying the anti-Helmholz magnetic field.

The conical rotating magnetic field was produced as a resultant of the rotating magnetic field Brot and the axial magnetic field B ^ i , which is perpendicular to the rotation plane. The rotation of the magnetic field was made by two mutually orthogonal pairs of Helmholtz

*~h$\. B axial Laser pulses

Fig. 2. A conical rotating magnetic field with a semiangle

of 8 from the propagation direction of the laser field..

coils which were driven by alternating currents with a relative phase shift of 90°. The axial magnetic field was applied so that the semiangle of the resultant magnetic field was 60°. The strength of the rotation magnetic field was 183 mG, which corresponds to a frequency shift of 385 kHz. The resultant magnetic field was 212 mG, which corresponds to frequency shifts of 445 kHz. At 3 ms after free expansion of the cold ensemble of atoms, they were initialized perfectly by optical pumping to the F=l state. Then a quantization magnetic field was applied to sodium atoms and two-photon Raman pulses with a pulse width of 20 ms were applied to them with a pulse separation of 180 ms to compose the atom interferometer. During two Raman pulses, the resultant magnetic field was rotated from DC to several kHz, while keeping the amplitude of the magnetic field constant. Then we obtained the Ramsey fringes with a period of 5 kHz and a visibility of 50%.

The experimental results of 0=60° and 120°, where the direction of the axial magnetic field were reversed, shown in Fig.3. The slopes are -2.8±0.2 and 2.7±0.3, respectively. The experimental detail will be written in another paper [7].

3. Discussions

The reverse of the slope between semiangle of #=60° and #=120° will be understood easily as follows. Even if the axial magnetic field is

308

I—r- 1 1 r

[ T -- 0=60° : -( 2.8 + 0.2 X> J I -0=120" :( 2.7 ±0.3)4

_ i 1 1 1 1— -2 -I 0 I 2

Rotation Angle <fr (rad.)

Fig.3 Phase shift between the states of M , niF=l and F=2,

rriF=2 as a (unction of rotation angle <j> for semiangles 0=60

and 0=120.

reversed, the sense of the giromagnetic motion is hold, while the sense of the rotation of the magnetic field is reversed for the direction of the axial magnetic field. Consequently, the reverse of the axial magnetic field corresponds to the reverse of the sense of the magnetic field keeping the sense of the axial magnetic field. Namely, the sense of the rotation is defined according to the direction of the axial magnetic field. This fact is easily ascertained by reversing the sense of the rotation magnetic field.

The fact that the magnitude of the slope is near the 2.5, not but 1.5 will be more important, since the Eq. (1) for #=60° predicts 1.5 only taking into consideration the sign of the g-factor. The fact will be explained by the idea that a solid angle for a positive g-factor is

Q = 0(1-COS 0) , while that for a negative g-factor should be

Q = -0(1 + cos<9), including the reverse of the sign. Then, we could get a slope of 2.5 for #=60°. To confirm this idea, further experiments for a few other semiangle will be necessary.

4. Conclusion

Berry's phase of atoms with magnetic quantum number using time-domain atom interferometer was investigated. The dependency of Berry's phase on the sign of the g-factor was discussed.

Acknowledgments

The authors would like to thank M. Kitano of Kyoto University, F. Riehle of PTB(Germany) and A. Oguchi of Tokyo University of Science for many discussions and comments on Berry's phase.

References

1. Geometric Phases in Physics, ed. A. Shapere and F. Wilczek (World Scientific, Singapole, 1989). L. Maiani, Phys. Lett. B62, 183(1976).

2. M. V. Berry, Proa R. Soc. London, Ser. A 392, 451(1984)., M. Colonna, M. Di Toro and V. Greco, Phys. Rev. Lett. 86, 4492 (2001).

3. M. Yasuhara, T. Aoki, H. Narui and A. Morinaga, IEEE Trans. Instrum. Measur. 54, 864(2005).

4. A. Morinaga, T. Aoki and M. Yasuhara, Phys. Rev. A71, 054101(2005).

5. R. Tycko, Phys. Rev. Lett. 58,2281 (1987). 6. K. Shinohara, T. Aoki and A. Morinaga,

Phys. Rev. A 66, 042106 (2002). 7. H, Narui, T. Aoki, A. Monma and A.

Morinaga, to be published.

LIST OF PARTICIPANTS

Aeppli, Gabriel Department of Physics and Astronomy University College London

Altshuler, Boris Department of Physics Princeton University

Ando, Tsuneya Department of Physics Tokyo Institute of Technology

Ando, Masahiko Advanced Research Laboratory Hitachi, Ltd.

Ashhab, Sahel S. Digital Materials Laboratory Frontier Research System RIKEN

Awschalom, David D. Department of Physics University of California, Santa Barbara

Azuma, Koji Department of Applied Physics The University of Tokyo

Corbett, John V. Division of Information and Communication Sciences Macquarie University

Dietl, Tomasz Institute of Physics Polish Academy of Sciences

Ebisawa, Hiromichi Graduate School of Information Sciences Tohoku University

Edamatsu, Keiichi Research Institute of Electrical Communication Tohoku University

Fujii, Toru Imaging Technology Department Nikon Corporation

Fujikawa, Kazuo Institute of Quantum Science College of Science and Technology Nihon University

Fujimori, Masaaki Advanced Research Laboratory Hitachi, Ltd.

Fujiwara, Akio Department of Mathematics Osaka University

Fukuyama, Hidetoshi Institute for Materials Research Tohoku University

Funahashi, Haruhiko Physics Department Kyoto University

Giddings, Devin Physics & Astronomy The University of Nottingham

Gorman, John Cavendish Laboratory The University of Cambridge

Hansel, Wolfgang Institute for Experimental Physics Innsbruck University

Haffner, Hartmut Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences

Hamada, Tomoyuki FSIS Center of Collaborative Research Institute of Industrial Science The University of Tokyo

Harada, Ken Advanced Research Laboratory Hitachi, Ltd.

Hasegawa, Shuichi Quantum Engineering and Systems Science The University of Tokyo

Hashimoto, Yoshinori Quantum Engineering and Systems Science The University of Tokyo

Hashizume, Tomihiro Advanced Research Laboratory Hitachi, Ltd.

Hatakenaka, Noriyuki Faculty of Integrated Arts and Sciences Hiroshima University

Hatano, Naomichi Institute of Industrial Science The University of Tokyo

Hayakawa, Jun Advanced Research Laboratory Hitachi, Ltd.

Hayashi, Masahiko Graduate School of Information Sciences Tohoku University

Heike, Seiji Advanced Research Laboratory Hitachi, Ltd.

Heydari, Hoshang Institute of Quantum Science Nihon University

Hirai, Yasuharu Advanced Research Laboratory Hitachi, Ltd.

Hirayama, Yoichi Advanced Research Laboratory Hitachi, Ltd.

Hitosugi, Taro Department of Chemistry The University of Tokyo

Hofmann, Holger F. Graduate School of Advanced Sciences of Matter Hiroshima University

Ichimura, Masahiko Advanced Research Laboratory Hitachi, Ltd.

Ieda, Jun 'ichi Institute for Materials Research Tohoku University

Iijima, Sumio Department of Materials Science and Engineering Meijo University

Ishioka, Sachio Advanced Research Laboratory Hitachi, Ltd.

lye, Yasuhiro Institute for Solid State Physics The University of Tokyo

Jain, Jainendra K. Department of Physics The Pennsylvania State University

Johansson, Robert J. Digital Materials Laboratory Frontier Research System WKEN

Kanda, Akinobu Institute of Physics and TIMS University of Tsukuba

Kato, Masanori Institute for Solid State Physics The University of Tokyo

Katori, Hidetoshi Department of Applied Physics The University of Tokyo

Katsumoto, Shingo Institute for Solid State Physics The University of Tokyo

Katsura, Hosho Department of Applied Physics The University of Tokyo

Kawabata, Shiro Nanotechnology Research Institute National Institute of Advanced Industrial Science and Technology

Kimble, H. Jeff Norman Bridge Laboratory of Physics California Institute of Technology

King, Christophers. Physics & Astronomy The University of Nottingham

Kobayashi, Kensuke Institute for Chemical Research Kyoto University

Kobayashi, Shun-ichi Tokyo University of Agriculture and Technology

Kohashi, Teruo Central Research Laboratory Hitachi, Ltd.

312

Koizumi, Hideaki Advanced Research Laboratory Hitachi, Ltd.

Kono, Junichiro Department of Electrical and Computer Engineering Rice University

Liu, Yu-xi Digital Materials Laboratory Frontier Research System RIKEN

MacDonald, Allan H. Department of Physics The University of Texas at Austin

Marcus, Charles M. Department of Physics Harvard University

Maruyama, Eiichi Center for Intellectual Property Strategies RIKEN

Matsuoka, Hideyuki Advanced Research Laboratory Hitachi, Ltd.

Matsuoka, Leo Quantum Engineering and Systems Science The University of Tokyo

Misko, Vyacheslav R. Digital Materials Laboratory Frontier Research System RIKEN

Miyazaki, Hisao Institute of Physics University of Tsukuba

Mooij, J. E. Kavli Institute of Nanoscience Delft University of Technology

Morinaga, Atsuo Department of Physics Tokyo University of Science

Moshchalkov, Victor V. Nanoscale Superconductivity and Magnetism Group Katholieke Universiteit Leuven

Murakami, Shuichi Department of Applied Physics The University of Tokyo

Nagaosa, Naoto Department of Applied Physics The University of Tokyo

Nakajima, Sadao Superconductivity Research Laboratory International Superconductivity Technology Center

Nakamura, Michiharu Hitachi, Ltd.

Nakamura, Yasunobu Fundamental and Environmental Research Laboratories NEC Corporation

Nakano, Hayato Superconducting Quantum Physics Group NTT Basic Research Laboratories

Ogawa, Susumu Advanced Research Laboratory Hitachi, Ltd.

Ohnishi, Takashi Department of Applied Physics The University of Tokyo

Ohno, Hideo Research Institute of Electrical Communication Tohoku University

Ohta, Fumio Systems Development Laboratory Hitachi, Ltd.

Ong, Nai-Phuan Department of Physics Princeton University

Otto, Teruo Institute for Chemical Research Kyoto University

Ono, Yoshimasa A. Center for Innovation in Engineering Education The University of Tokyo

Onoda, Shigeki Department of Applied Physics The University of Tokyo

Onogi, Toshiyuki Advanced Research Laboratory Hitachi, Ltd.

Ootuka, Youiti Institute of Physics University of Tsukuba

Osakabe, Nobuyuki Advanced Research Laboratory Hitachi, Ltd.

Otani, Yoshichika Institute for Solid State Physics The University of Tokyo

Otsuka, Tomohiro Institute for Solid State Physics The University of Tokyo

Ozeki, Akira Science News Department The Asahi Shimbun

Recher, Patrik Edward L. Ginzton Laboratory Stanford University

Saikawa, Kazuhiko

Saito, Hiroki Department of Physics Tokyo Institute of Technology

Saito, Kazuo Advanced Research Laboratory Hitachi, Ltd.

Saito, Shin-ichi Central Research Laboratory Hitachi, Ltd.

Sakurai, Akio Department of Physics Kyoto Sangyo University

Sano, Hirotaka Institute for Solid State Physics The University of Tokyo

Satoh, Tetsuo Tokyo Office Asian Technology Information Program

Savel'ev, Sergey Digital Materials Laboratory Frontier Research System RIKEN

Seki, Yoshichika Graduate School of Science Kyoto University

Seto, Katsunori Central Research Laboratory Hitachi, Ltd.

Shimazu, Yoshihiro Department of Physics Yokohama National University

Shiokawa, Tom Frontier Research System RIKEN

Soderholm, Jonas M.A. Institute of Quantum Science Nihon University

Sugano, Ryoko Advanced Research Laboratory Hitachi, Ltd.

Suwa, Yuji Advanced Research Laboratory Hitachi, Ltd.

Suzuki, Kazuya Institute for Solid State Physics The University of Tokyo

Takagi, Kazumasa Corporate Technology Group Hitachi Europe, Ltd.

Takagi, Shin Fuji Tokoha University

Takayanagi, Hideaki NTT Basic Research Laboratories

Takei, Nobuyuki Department of Applied Physics The University of Tokyo

Shimizu, Fujio The University of Electro-Communications

Taketani, Kaoru Department of Physics Kyoko University

Takeuchi, Masayuki Corporate Communications Division Hitachi, Ltd.

Tanamoto, Tetsufumi Advanced LSI Technology Laboratory Corporate R & D Center Toshiba Corporation

Tarucha, Seigo Department of Applied Physics The University of Tokyo

Tatara, Gen Department of Physics Tokyo Metropolitan University

Terashima, Hiroaki Department of Physics Tokyo Institute of Technology

Teshima, Tatsuya Advanced Research Laboratory Hitachi, Ltd.

Togawa, Yoshihiko Quantum Phenomena Observation Technology Laboratory Frontier Research System RIKEN

Tokura, Yoshinori Department of Applied Physics The University of Tokyo

Tomaru, Tatsuya Advanced Research Laboratory Hitachi, Ltd.

Tonomura, Akira Advanced Research Laboratory Hitachi, Ltd.

Tsai, Jaw-Shen Quantum Information Technology Group NEC Corporation

Tsai, Sheng-Yi Institute of Quantum Science Nihon University

Tsujimura, Tatsuya Kyodo News

Uchida, Fumihiko Central Research Laboratory Hitachi, Ltd.

Uchida, Takahiro Institute for Solid State Physics The University of Tokyo

Uchiyama, Fumiyo Institute of Applied Physics University of Tsukuba

Utsunti, Yasuhiro Condensed Matter Theory Laboratory RIKEN

Williams, David A. Hitachi Cambridge Laboratory Hitachi Europe, Ltd.

Wunderlich, Jorg Hitachi Cambridge Laboratory Hitachi Europe, Ltd.

Xu, Xiulai Hitachi Cambridge Laboratory Hitachi Europe, Ltd.

Yamada, Eizaburo Information Science Meisei University

Yamamoto, Yoshihisa Edward L. Ginzton Laboratory Stanford University

Yamashita, Taro Institute for Materials Research Tohoku University

Yang, Chen N. Tsinghua University

Yasuda, Tomooki Science News Department The Asahi Shimbun

Yoshida, Takaho Advanced Research Laboratory Hitachi, Ltd.

Zeilinger, Anton Institute for Experimental Physics Vienna University

Author Index

Aikawa, H. Ando, M. Ando, T. Aoki, T. Arai, T. Baelus, B.J. Baskaran, G. Becher, C. Benhelm, J. Blatt, R. Carballeira, C. Cexilemans, A. Chek-al-Kar, D. Chwalla, M. Cleaver, J.R.A. Corbett, J. V. Crooker, S.A. Dietl, T. Ebisawa, H. Edamatsu, K. Endo, A. Fazio, R. Fujii, T. Fujikawa, K. Fujimori, M. Fujita, S. Fujiwara, A. Fukuda, H. Fukuyama, H. Funahashi, H. Furubayashi, Y. Furusawa, A. Gorman, J. Grabecki, G. Haffner, H. Hamada, T. Hansel, W. Harrabi, K. Hasaka, S. Hasegawa, T. Hashizume, T.

99 258 228 306 246 200 183 33 33 33

194 194 33 33

105 38

234 159 204

50 109,212

64 88

290 250,254

76 80

246 1

282 191 29 60

159 33

262 33 54

262 191

250,254,286

Hasko, D.G. Hatakenaka, N. Hatano, N. Hatano,T. Hauge, R.H. Hayashi, M. Heike, S. Heydari, H. Higashi, R. Hino, M. Hirano, K. Hirose, Y. Hisamoto, D. Hitosugi, T. Hofmann, H.F. Home, D. Hong, F.-L. Ichimura, M. Ieda, J. Inaba, K. Inoue, M. Inoue, S. Iye,Y. Jungwirth, T. Kadowaki, K. Kanda, A. Kasai, S. Kaestner, B. Kato, Masanori. Kato, M. Katori, H. Katsumoto, S. Kawabata, S. Kim, N. Y. Kimble, H.J. Kimura, S. Kimura, T. Kinoda, G. Kitaguchi, M. Kobayashi, N. Kohno, H.

60 88

266 92

234 204

250,254,286 42

276 282 46

191 246 191 68 38

276 183,187

298 191 262 46

99,109,212 140 200

200,208,216 165 140 109 204 276

99,109,212 224 242

10 246 171 191 282 262 177

Kono, J. Korber, T. Kuboki, K. Kurimura, S. Lee, C. Lee, W.-L. Liu, Y.-X. MacDonald, A.H. Maekawa, S. Mar, J.D. Martinek, J. Maruyama, R. Misko, V. R. Miyazaki, H. Momida, H. Monma, A. Mori, S. Morinaga, A. Moshchalkov, V.V. Murakami, S. Nagaosa, N. Nakamura, H. Nakamura, Y. Nakano, H. Narui, H. Nishida, M. Nomura, K. Nori, F. Nunez, A.S. Ohno, T. Okamoto, R. Onai, T. Ong, N.P. Ono, T. Ono, Y. A. Oohata, G. Ootuka, Y. Osakabe, N. Otake, Y. Otani, Y. Peeters, F.M. Portugall, 0. Rapol, U.D. Recher, P. Riebe, M. Rikken, G.L.J.A.

234 33

204 46

258 121 72

140,150 84,113,183,187

105 113 282 220

208,216 262 306 46

306 194 146 134 266

54 64

306 88

140 72,220

150 262 68

246 121 165 258

50 200,208,216

3 282 171 200 234 33

242 33

234

Roos, C. Saito, H. Saito, Shin-ichi Saito, Shiro Saitoh, E. Sano, H. Sato, M. Savel'ev, S. Schmidt, P.O. Schmidt-Kaler, Schon, G. Seki, Y. Shaver, J. Shibata, J. Shimada, T. Shimizu, H.M. Shimizu, N. Shimizu, R. Shirasaki, R. Sinova, J. Smalley, RE. Sdderholm, J. Stopa,M. Sun, C.P. Suwa, Y. Suzuki, T. Tadano, K. Taguchi, Y. Tajima, N. Takahashi, S. Takamoto, M. Takayanagi, H. Takei, N. Taketani, K. Takeuchi, S. Tanaka, H. Tanamoto, T. Tanigawa, H. Tanikawa, K. Tarucha, S. Tatara, G. Teniers, G. Terada, Y. Terashima, H. Tokura, Y. Tonomura, A.

Tsai, J.S. Uda, T. Ueda, M. Utsumi, Y. Wei, L.F. Wei, X. Williams, D.A. Wrobel, J. Wunderlich, J. Xu,X. Yamaguchi, A. Yamaguchi, T. Yamamoto, T. Yamamoto, Yoshihisa Yamamoto, Y. Yamashita, T. Yang, C.N. Yano, K. Yoshihara, F. You, J.Q. Zaric, S. Zeilinger, A.

54 262

294,302 113,117

72 234

60,105 159 140 105 165 216 262

16,242 191 84 4

165 54 72

234 24

FOUNDATIONS OF OURNTUM MECHANICS •

IN THE LIGHT OF NEW TECHNOLOGY

The goal of the 8th International

Symposium on Foundations of Quantum

Mechanics in the Light of New Technology

was to link recent advances in technology

with fundamental problems and issues in

quantum mechanics with an emphasis on

quantum coherence, decoherence, and

geometrical phase.

The papers collected in this volume cover

a wide range of quantum physics, including

quantum information and entanglement,

quantum computing, quantum-dot systems,

the anomalous Hall effect and the spin-

Hall effect, spin related phenomena,

superconductivity in nano-systems, precise

measurements, and fundamental problems.

The volume serves both as an excellent

reference for experts and a useful

introduction for newcomers to the field of

quantum coherence and decoherence.

3* YEARS OF P U B L I S H I N G

HUH l l lN t l t »»• I I I I

J! f,*c. www wmlcl-.i