Terahertz Dynamics of a Superlattice in Crossed Electric ... Thesis_q… · Quantum Dot”, SPIE...

UNIVERSITY of CALIFORNIASanta Barbara

Terahertz Dynamics of a Superlattice in

Crossed Electric and Magnetic Fields

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Physics

by

Naser Qureshi

Committee in charge:

Professor S. James Allen, ChairProfessor John RuhlProfessor Andreas Ludwig.

March 2002

The dissertation of Naser Qureshi is approved:

Chair

March 2002

ii

Terahertz Dynamics of a Superlattice in

Crossed Electric and Magnetic Fields

Copyright 2002

by

Naser Qureshi

iii

To my father, too late.

iv

Acknowledgements

I am very grateful to Jim Allen for patiently supporting me and for sharing

in the explorations that led to this work.

I have benefitted greatly from Mark Sherwin’s advice over the years, and

the cleanroom part of this work would not have been possible without Andrew

Cleland’s help.

To David Enyart I owe a chunk of my remaining sanity, because the FEL

user lab would have been unusable without his good humor and dedication. I

would like to thank Mike Wrocklage and Andy Weinberg for their help in the

machine shop, Jack Whaley for his help in the cleanroom and Dean White for

helping me out with countless lab details. I am indebted to Kevin Maranowski

and Prof. Art Gossard for growing the superlattice material used in this work.

Jeff Scott taught me how to do experiments, and did the groundwork for

most of the FEL techniques used here. Dan Schmidt’s ideas on mesa fabrication

were the origin of this project and his comments have kept me honest along

the way.

I am infinitely grateful to my classmates Xomalin Peralta, Toby Eckhause,

Susan Najita and Adriano Batista for their ideas friendship and companion-

ship; to Carey Cates for helping me with a Mac when it really mattered; to

Smitha Vishveshwara, Davide Castelvecchi and Ignacio Wilson-Rae for their

good energies and to my brother Andy Lau for his unrelenting music.

Vast amounts of support and advice came from my sisters Mahmooda and

Sajda, and from my brothers Gert-Jan and David. My parents Shireen and

Wahab showed me the academic path and supported me until the end.

Liliana shared in every detail of this work, in more ways than I can list.

v

Curriculum Vitae

Naser Qureshi

Personal

Birthdate January 26, 1971

Kampala, Uganda

Education

March 2002 Ph.D. in Physics University of California, Santa Barbara.

January 1998 M.A. in Physics University of California, Santa Barbara.

June 1994 A.B. in Physics, Magna Cum Laude Princeton University.

Academic Employment

1994-2002 Graduate Student Researcher, Physics Department, Quan-

tum Institute and Center for Terahertz Science and Tech-

nology, University of California, Santa Barbara.

Graduate Researcher, Japan Science and Technology Cor-

poration, Quantum Transition Project (1995-1999).

Advisor: S. James Allen, Jr.

1994-1995 Teaching Assistant, Physics Department, University of

California, Santa Barbara.

Supervisors: Jean Carlson, David Cannel.

1993-1994 Research Assistant, Physics Department, Princeton Uni-

versity. Advisor: N. P. Ong.

vi

1992-1993 Research Assistant, Gravity Group, Physics Department,

Princeton University. Supervisors: David T. Wilkinson

and Lyman Page.

Awards and Honors

June 1994 Shenstone Award for an undergraduate thesis in Physics,

Princeton University.

Publications

1. “Imaging and probing electronic properties of self-assembled InAs quan-

tum dots by atomic force microscopy with conductive tip”, Applied Physics

Letters, 8 Feb. 1999, vol.74, (no.6):844-6.

2. “Terahertz Excitation, Transport and Spectroscopy of an AFM-Defined

Quantum Dot”, SPIE Vol. 3617, San Jose, CA, January 1999.

3. “Terahertz Excitations in Electrostatically defined Quantum Dots”, Phys-

ica E Vol.2, p.p. 701 - 703, Elsevier, Amsterdam.

vii

Abstract

Terahertz Dynamics of a Superlattice in Crossed Electric and

Magnetic Fields

by

Naser Qureshi

This dissertation presents an experimental investigation of photon assisted

transport in a GaAs/AlGaAs superlattice in the presence of an in-plane mag-

netic field. In striking contrast to numerous works published over the last two

decades, we observe the d.c. saturation current in a superlattice to be very

strongly dependent on the strength of an in-plane magnetic field. Further-

more, when the structure is excited at terahertz frequencies, the broadening of

the I-V induced by the magnetic field appears to quench the photon assisted

transport features.

An intuitive model describing the effects of the magnetic field is developed

that accounts for some of the observations semi-quantitatively. It fails to model

the large current increases we observe with magnetic field and suggests there

is more physics involved.

viii

Contents

1 Introduction 41.1 Past Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background: Superlattice Transport 92.1 Miniband Structure . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Current-Voltage Characteristic . . . . . . . . . . . . . . . . . . . 11

2.2.1 Miniband Transport . . . . . . . . . . . . . . . . . . . . 112.2.2 Sequential Tunneling . . . . . . . . . . . . . . . . . . . . 14

2.3 a.c. Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Classical Rectification . . . . . . . . . . . . . . . . . . . 142.3.2 Photon Assisted Transport . . . . . . . . . . . . . . . . . 15

2.4 In-plane Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 19

3 Experimental Approach 233.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Photoresponse Measurements . . . . . . . . . . . . . . . . . . . 25

4 Results and Analysis 344.1 D.c. Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Temperature Dependence . . . . . . . . . . . . . . . . . 344.1.2 Size Dependence . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Photoresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.1 Frequency Dependence . . . . . . . . . . . . . . . . . . . 434.2.2 Power Dependence . . . . . . . . . . . . . . . . . . . . . 45

4.3 Effect of an In-plane Magnetic Field . . . . . . . . . . . . . . . . 50

ix

4.3.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . 504.3.2 Comparison with Theory . . . . . . . . . . . . . . . . . . 534.3.3 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . 544.3.4 Zero Magnetic Field . . . . . . . . . . . . . . . . . . . . 574.3.5 In-Plane Magnetic Field . . . . . . . . . . . . . . . . . . 58

4.4 Photoresponse in a Magnetic Field . . . . . . . . . . . . . . . . 63

5 Conclusion 66

A Numerical Estimates 68A.1 Some Superlattice Properties . . . . . . . . . . . . . . . . . . . . 68

A.1.1 Level Broadening . . . . . . . . . . . . . . . . . . . . . . 68A.1.2 Diamagnetic Shifts . . . . . . . . . . . . . . . . . . . . . 68

A.2 I-V Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B Cleanroom Process Details 74B.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.2 Sample Clean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.3 Top Ohmic Contact . . . . . . . . . . . . . . . . . . . . . . . . . 76B.4 Mesa Etch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77B.5 Bottom Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.6 Bottom Contact Isolation . . . . . . . . . . . . . . . . . . . . . 79B.7 Wire Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C Experimental Details 82C.1 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C.1.1 Storage Dewar Setup . . . . . . . . . . . . . . . . . . . . 82C.1.2 Magnetic Cryostat . . . . . . . . . . . . . . . . . . . . . 83

C.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 85

x

CONTENTS 1

.

Prologue:

To Probe a Nanostructure

This work originates from a search for a practical way to excite a wide

variety of nanostructures with terahertz radiation and study their terahertz

transport properties. It began with a simple idea: given a large number of

randomly scattered, self-assembled InAs dots grown on a GaAs substrate, why

not use a conducting atomic force microscope tip to image, locate, contact,

couple in radiation from free-space, and measure currents in one single dot of

our choice? Such an approach would allow us to study not only self-assembled

dots, but could be adapted to study any nanostructure on a substrate, regard-

less of how it was fabricated, and provide some remedy for the present dearth of

existing experimental data on terahertz dynamics in mesoscopic and nanoscle

devices.

The idea produced results involving d.c. transport [30] from which it be-

came clear that a mechanical contact on the nanometer scale between a metal

and semiconductor does not necessarily give rise to a good electrical contact.

Even under exquisitely controlled atmospheres, the properties of the tip-dot

contact were usually not ohmic or Schottky-like, were not reproducible and ap-

peared to be dominated by surface conditions. Different choices of metal and

CONTENTS 2

semiconductor produced vastly different results. In short, the idea may sound

simple, but it is not realistic. The nanometer scale tip-sample contact often

dominated the transport and was diffcult to control.

We did, however, find that a cobalt-coated tip made a good Schottky contact

with a clean GaAs surface. This led to an experiment [31] where an undoped

quantum well structure underlying a GaAs surface was contacted with a tip

in such a way that the tip’s own electric field provided the lateral confinement

needed to create a quantum dot. The tip was then used as an antenna to couple

free-space terahertz radiation into the dot (i.e. provide a terahertz a.c. field)

and at the same time serve as a probe of the photoresponse. It was thus possible

to perform a terahertz photoresponse experiment on a 150nm resonant tunnel-

ing diode produced in situ. The problem arose that the resonant tunneling

features in the I-V’s were very broad and the physics in the subsequent results

was limited to classical rectification of the terahertz field. It became necessary

to perform the experiment at cryogenic temperatures, and we developed a 4K

scanning AFM probe for the purpose.

At low temperatures, it became apparent that even after creating quantum

dots less than 100nm in lateral dimension and relatively sharp resonant tun-

neling features in the I-V (<5meV), the photoresponse results amounted to a

quenching of resonant tunneling and did not shed much light on the dynamics

within the dot. In fact there were indications that the observations were, once

again, dominated by the contact and not the quantum dot itself.

A remedy was to lithographically define the dots with gold top contacts and

bring the tip into contact with this gold pad, rather than the semiconductor

itself. This gave us the means to reliably produce large arrays of resonant tun-

neling nanostructures and systematically probe their electrical characteristics.

Not only did this give us some lithographic control over the lateral size of these

CONTENTS 3

devices, but also allowed us to choose one, among all the random variations

inherent in any nanoscale fabrication, that fit our specifications. An attempt to

exploit this to produce a terahertz quantum single electron turnstile using an

appropriately designed multiple quantum well structure indicated transport of

approximately 1 electron per cycle. This was, however, only a partially success-

ful demonstration because of the practical impossibility of creating a dot with

mechanical means and leaving it unchanged for the many hours or days nec-

essary to perform a frequency-swept terahertz spectroscopy experiment with a

tunable free electron laser (FEL).

The search for a clean and simple approach led us back to a standard

lithography-based technique. Using electron beam lithography rather than an

AFM tip to produce and contact a device, we lose the privilege of making vast

numbers of devices and choosing one that happens to fit the exacting specifica-

tions needed to isolate a piece of interesting physics. In return, the devices are

more reliably produced, albeit larger, and robust enough to undergo the treat-

ment of an FEL experiment, the changes in conditions such as temperature

and magnetic field that yield important information and the tight practical

constraints of the FEL user facility. The upshot is an investigation of terahertz

dynamics in a relatively large structure, where electrons are quantum mechan-

ically confined in only one dimension and perturbed by a magnetic field, the

subject of this thesis.

Chapter 1

Introduction

1.1 Past Work

Electron transport in semiconductor superlattices has generated a contin-

uous stream of interest in the physics community for three decades. Shortly

after the advent of precise semiconductor growth by molecular beam epitaxy it

was demonstrated by Esaki and Chang in the 1970’s that the conduction band

of a layered semiconductor material can be engineered at will to mimic the

periodic potential that binds electrons in a natural crystal structure [2]. The

resultant one dimensional ’artificial crystal’, composed of a series of quantum

wells separated by finite potential barriers, is known as a superlattice.

A superlattice can be designed with periodicity an order of magnitude

greater than the atomic scale and gives rise to its own bands, known as mini-

bands, of allowed electron (or hole) states within the material’s natural con-

duction (or valence) band. This miniband is analogous to the conduction band

found in a natural semiconductor, except that it is typically two orders of mag-

nitude narrower in energy. Control over the width of the barriers and wells in

4

CHAPTER 1. INTRODUCTION 5

a superlattice makes it possible to vary some of the defining properties of this

artificial crystal, such as the wavefunction overlap between adjacent wells and

widths in energy of the minibands. The degree to which electrons are localized

to the individual wells can have a significant effect on the transport mechanism,

which is why superlattices have been a favorite system among those interested

in various aspects of electron transport [43, 44, 45, 46].

Investigations of electron transport in semiconductor superlattices since the

1970’s have led to a detailed understanding of the essential physics that controls

current in these devices. In superlattices with strongly coupled wells, transport

in small electric fields is essentially ohmic and known as miniband transport.

Electrons responsible for the current are spatially delocalized throughout the

structure but energetically confined to the miniband. At large electric fields,

the miniband picture evolves into a Wannier-Stark ladder. One observes with

increasing electric field either a drop or a saturation in current as well as in-

stabilities in the current arising from a negative differential conductance. An

uneven accumulation of charge in different wells and the consequent forma-

tion of domains greatly complicate the physical picture [61]. As a result, the

high-field behavior of superlattices is generally not as well understood as the

low-field behavior.

In weakly coupled superlattices, the miniband is very narrow in energy and

transport at low electric fields can be understood in terms of sequential resonant

tunneling between distinct states localized in adjacent quantum wells [57, 58,

59]. This case lends itself nicely to the study of resonant tunneling mechanisms

and has been a fruitful alternative to resonant tunneling diodes. Transport here

is controlled primarily by tunneling between identical wells and not between a

well and its external contacts as would be the case in a single quantum well

structure [9].


The question of a.c. or high frequency behavior has historically received

much less attention than d.c. transport. At low frequencies the behavior is

what one would expect from any device driven with an a.c. bias: classical rec-

tification, which is not at all interesting in an ohmic device. It is only when the

superlattice is driven at high enough frequency and amplitude, when ω > 1/τ

where τ is the inelastic scattering time, that we begin to see quantization in

time, or the emergence of Floquet States [21, 26]. This occurs at terahertz

frequencies for a typical GaAs/AlGaAs material system used to construct a

superlattice; such frequencies and amplitudes have become accessible to semi-

conductor science only in the last decade or so with the advent of free electron

lasers [27].

In weakly coupled superlattices, tunneling through Floquet states, better

known as photon assisted transport, was observed six years ago by Keay et al.

and Zeuner et al. [22, 23]. A relatively simple theoretical model developed by

Tien and Gordon [20, 26] in the 1960’s for tunnel junctions was used by Keay et

al. to provide a qualitative understanding of the essential physics of sequential

photon assisted tunneling between adjacent wells in a superlattice, but there

is as yet no close quantitative agreement between theory and experiment [29].

Although there is theoretical work in the literature that attempts to refine the

basic model [24] and in some cases capture new physics [26], experiments are

still limited to a handful.

With a magnetic field parallel to the electric field, transport experiments

have shown interesting effects resulting from increased localization of electrons

[9, 49]. This is generally a well understood area and will not be addressed

further here.

When a magnetic field is applied perpendicular to the electric field, how-

ever, the story becomes much more interesting. From a semiclassical point of


view, electron motion is directly modified by the Lorentz force perpendicular

to the motion and the standard one-dimensional picture of superlattice trans-

port falls apart. From a more correct quantum mechanical standpoint [33],

miniband energies are shifted, the superlattice crystal momentum k ceases to

be a good quantum number and transport properties of the system changes in

non-trivial ways. A large body of literature, both experimental and theoretical,

exists for strongly coupled superlattices in this configuration (chapter 2 or Ref.

[43]). As one would expect, the magnetic field can be exploited to understand

or characterize a superlattice: it has been shown, for example, that magne-

toresistance measurements can be used to determine the electron mobility in a

wide miniband superlattice [53]. The case of a weakly coupled superlattice in a

magnetic field perpendicular to the electric field, on the other hand, remains

a neglected one. Both theoretically and experimentally, there is as yet in the

literature no detailed understanding of the transport process for this situation.

1.2 Objective

The goal of this thesis is to address two questions that arise naturally in

the context described above, but remain as yet unanswered in the literature.

Firstly: how is d.c. transport in a weakly coupled superlattice affected by a

strong magnetic field perpendicular to the electric field, and how can we under-

stand this effect? Secondly: what can we learn about photon assisted transport

in such a system with and without the magnetic field? Part of the motivation

behind this work is to add to the sparce volume of data currently available

on photon assited transport in superlattices and contribute to a quantitative

comparison with existing theories.

We present and analyze new transport measurements that show surprisingly


dramatic features in a narrow miniband superlattice I-V with the application of

a magnetic field, a behavior very different from what has been observed many

times over the years in wider miniband transport experiments. We also present

a study of the effect of the magnetic field on transport in the presence of a

strong terahertz a.c. electric field. The transport consists essentially of a d.c.

part and a photon assisted part and we find, somewhat surprisingly, that the

a.c. part falls in line with existing theory more naturally than does the d.c.

part.

This thesis is organized as follows. An introduction to existing works and

the essential background physics is covered in Chapter 2. Chapter 3 gives a

summary of the experimental approach used in our investigations and the main

presentation of new results and interpretations is given in Chapter 4.

Chapter 2

Background: Superlattice

Transport

2.1 Miniband Structure

To begin with, consider a semiconductor heterostructure consisting of a

number of alternating layers of two different materials. The resultant conduc-

tion band profile perpendicular to the layers then becomes a periodic structure

containing a series of wells in the narrower bandgap material, and barriers in

the wider bandgap material. In a common material system, with GaAs wells

and Al.3Ga.7As barriers, the band offset gives rise to 275meV barriers (fig. 2.1).

This thesis will focus entirely on electron transport (ignoring hole transport in

the valence band) in this simple realization of a superlattice. The only addi-

tional structures are ohmic contacts on either end of the superlattice that serve

as a source and drain of electrons.

The most common way to calculate the allowed states of a superlattice is

to use the Kronig-Penny model for a periodic potential of infinite extent. A

9

CHAPTER 2. BACKGROUND: SUPERLATTICE TRANSPORT 10

z

E

V

Lb Lw

Figure 2.1: An idealized superlattice.

simple treatment found in textbooks [11] makes use of the Bloch theorem to

obtain an implicit equation for the energy E and reciprocal wavevector K

cos(KD) = cos(αLw) cosh(βLb) +1

2(β/α− α/β) sin(αLw) sinh(βLb) (2.1)

where

α2 =2mE

h2 (2.2)

and

β2 =2m(V − E)

h2 . (2.3)

V is the band offset, D the superlattice period, m the effective mass (one can,

if needed, account for differing masses in the wells and barriers[12]), Lb the

barrier width and Lw the well width.

The result is that the right hand side of Eq. 2.1 is forced to lie between -1

and 1 which implicitly limits the allowed values of energy. The energy E falls

into discrete bands, known as minibands, whose widths can be tuned between

a millivolt or so to about 100meV for common superlattice dimensions. The


lower limit represents the broadening due to well-width fluctuations in a real

sample and the upper limit is set by the height of the barriers and the miniband

energy. Experimental measurements of the superlattice structure have included

X-ray measurements and scanning electron microscopies [12], and the existence

of actual minibands in agreement with eq. 2.1 has been shown unequivocally

by photoluminescence spectroscopy (ref. [13] is only one example).

2.2 Current-Voltage Characteristic

2.2.1 Miniband Transport

The first treatment of miniband transport dates back to Esaki and Tsu

[1] who in 1970 proposed creating semiconductor superlattice in order to ex-

plore Bloch oscillations. The so-called Esaki-Tsu model assumes a semiclassical

equation of motion

eF = hdk

dt, k = k(0) +

eF t

h, (2.4)

with F the electric field and a group velocity vg = h−1∂E/∂k. Scattering is

introduced phenomenologically: the probability that an electron’s motion is

free of collisions decays exponentially with time, P ∝ e−t/τ , with a collision

time τ . The drift velocity v for the electron gas with a distribution of group

velocities is then

v =∫ ∞

0exp(−t/τ)dvg =

eF

h2

∫ ∞

0

∂2E

∂k2e−t/τdt. (2.5)

The dispersion relation

E(k) =∆

2(1− cos(kd)), (2.6)

where ∆ is the miniband width and d is the spatial periodicity, can be sub-

stituted above and the integral can be performed using (2.4) to obtain an


expression for the drift velocity:

v =µF

1 + (F/Fc)2. (2.7)

Here Fc = h/(eτd) is a critical field, µ = (e∆τd2)/(2h2) is the mobility.

At low fields, we have v = µF , just like a bulk ohmic material, and is

a feature of all superlattices. But the current peaks as a function of voltage

and above some field Fc the drift velocity begins to fall as the electric field

is increased. This is known as negative differential velocity (NDV) and the

prediction led to much interest in realizing a superlattice [2]. Actual observa-

tions of this phenomenon in convincingly miniband transport in a superlattice

came many years later in the 1980’s from Sibille[15], Palmier [4] and others.

The main obstacle was the fact that domain formation (i.e. discrete inho-

mogeneous charge distributions within the structure) can greatly complicate

the picture[19]. Although earlier works [17, 2, 6] did show NDV, controversies

loomed.

After very high quality superlattice materials were produced in the late

1980’s and 1990’s, a large number of experimental papers emerged [14]. The-

oretical interpretations in these works have typically relied on more realistic

treatments of the scattering and the distribution of electrons. In one example

[18], an analytic treatment using the classical Boltzman equation in the relax-

ation time approximation provides a formula for the drift velocity v, and hence

the I-V, as a function of bias electric field F

v =I1(∆/(2kBT ))

I0(∆/(2kBT ))

µF

1 + (F/Fc)2(2.8)

where Io and I1 denote Bessel functions of the second kind and ∆ is the mini-

band width. The scatterting time τ comes into this through the mobility

µ = eτ/m and the critical field Fc = h/(eτd). The temperature dependence


-0.4 -0.2 0 0.2 0.4

Cur

rent

Electric field F

Fc

0

Figure 2.2: A predicted superlattice current-bias characteristic.

encompassed by the Bessel function term simply scales the current: at low tem-

peratures compared to the miniband width, the drift velocity is the same as

in (2.7) and then falls with increasing temperature. This comes from the fact

that at high enough temperatures the entire miniband has an equal probability

of being populated and an applied electric field cannot have any effect. Figure

2.4 shows the shape of this predicted superlattice I-V. Some experiments have

shown good agreement with this [43] although there are invariably a number

of discontinuities in the NDV region indicative of domain formation. Others

have revealed a far more subtle picture in very similar systems [18] where the

miniband picture may actually break down at low temperatures and give way

to a ’hopping’ behavior that has nothing to do with miniband transport.


2.2.2 Sequential Tunneling

When coupling between adjacent wells in a superlattice is weak, and the

width of the miniband becomes smaller then the widths of the individual states

that constitute it, there is no miniband. Instead, current is a result of sequential

resonant tunneling between adjacent wells. The early observations of Esaki

and Chang [2] in 1974, where current oscillations were observed as a function

of voltage in a superlattice, were in fact an observation of resonant tunneling

(combined with charge inhomogeneities, or domains) rather than miniband

transport.

Ten years later, measurements by Furuta et al. [57] and Tarucha et al. [58]

directly addressed the question of sequential resonant tunneling in GaAs/AlGaAs

superlattices. In order to avoid charge buildup and the subsequent formation

of charge inhomogeneities (domains), undoped superlattices were used and car-

riers were excited optically. Photocurrent-voltage characteristics (rather than

I-V’s) showed peaks where confined states in individual wells lined up. This

established the ’sequential tunneling’ mode of transport.

In the I-V characteristic of a doped superlattice, it is not always obvious

whether transport is due to a true miniband or to sequential tunneling.

2.3 a.c. Behavior

2.3.1 Classical Rectification

If, in addition to the d.c. bias V , we apply a small a.c. excitation A sin ωt

from a radiation source and the device responds adiabatically, then the resul-

tant current is easy to calculate. An intuitive way to look at this is to expand


the current

I(V + A sin ωt) = I(V ) +dI(V )

dVA sin ωt +

1

2!

d2I(V )

dV 2(A sin ωt)2 + . . . (2.9)

for all t. What we observe in a typical photoresponse experiment is a d.c

current, or the actual current averaged over a period much longer than 1/ω.

When we take the time average of the actual current

〈I(V + A sin ωt)〉 = I(V ) +1

2!

d2I(V )

dV 2

A2

2+ . . . (2.10)

the terms with odd powers of A sin ωt average to zero. The change in observed

d.c. current is therefore proportional to the second derivative of the d.c. I-V

for small excitations, and the fourth and higher derivatives become important

for larger excitations. Such an experiment does not give us any information

beyond the d.c. I-V.

This is only true if the excitation δV = A sin ωt is slow enough that the

instantaneous I(V ) is unchanged by the a.c. part. When the excitation fre-

quency is increased, we eventually come to a point where the system can no

longer be treated as a quantum mechanically time-independent problem and we

need to solve a more general problem. It was shown by an explicit experiment

about seven years ago that this transition occurs in the terahertz region for a

GaAs/AlGaAs superlattice, when ω ≈ 1/τ , where τ is the scattering time in

the material [28]. Theoretically, this fact was treated by Tucker [21].

2.3.2 Photon Assisted Transport

The quantum description of rectification in the presence of an a.c. drive

was treated in a simple yet powerful manner by Tien and Gordon [20] in 1963.

Suppose we have a confined system such as a quantum well with stationary


Hamiltonian H0 and a wave function

ψ(z, t) = f(z)e−iEt/h. (2.11)

We apply with spatial uniformity a time varying potential, so that

H = H0 + eVac cos ωt. (2.12)

The interaction part of the Hamiltonian eV cos ωt does not change the spatial

part of the wave function; only the time dependence changes:

ψ(z, t) = f(z)e−iEt/h(∞∑

n=−∞Bne

−inωt) (2.13)

and substituting this into Schrodinger’s equation tells us that

Bn = Jn(eVac/hω) (2.14)

with Jn the nth Bessel function of the first kind. The wave function takes the

form

ψ(x, t) = f(z)e−iEt/h[∞∑

n=−∞Jn(α)e−inωt] (2.15)

where

α = eVac/hω. (2.16)

This is nothing more than a sum of weighted replicas of the original wave

function. The total energy therefore contains components which have energies

E, E± hω, E± 2hω, etc., which means we can think of this as the appearance

of an infinite number of new confined states in the system, all spaced out in

multiples of the photon energy. These new energies are referred to as photon

sidebands, photon replicas, and sometimes Floquet states.

Another interpretation[21] of this wave function is to think of Jn(eVac/hω)

as the probability amplitude for the stationary state at E to be displaced


in energy by hω. In a real device consisting of a one-dimensional confined

state surrounded by contacts, these displacements in energy are equivalent to

d.c. voltages of nhω applied across the confining potential with probability

amplitude Jn(eVac/hω). The current is then just a sum of d.c. I-V’s, displaced

in voltage and weighted by this probability amplitude squared. This is an

intuitive way to arrive at the general expression for the current at d.c. bias Vdc

through the device[20, 21]

I(Vdc, Vac) =∞∑

n=−∞J2

n(eVac/hω)Idc(Vdc + nhω/e). (2.17)

The process whereby the a.c. part of the Hamiltonian opens up new conduction

channels in a tunneling device is usually referred to as photon assisted transport

(PAT). When an electron absorbs photons in the resonant tunneling process,

it is called photon assisted tunneling; when the electron emits photons, the

process is known as stimulated emission.

Although photon assisted tunneling was first observed in superconduct-

ing tunnel junctions in the early 1960’s (ref. [20] and references therein),

it was later observed in resonant tunneling diodes[25], quantum dots[32] and

superlattices[22] in the 1990’s. Qualitatively, the observation amounts to the

appearance of replicas of any sharp feature in the d.c. I-V at photon energy

intervals, and the oscillation of these replicas as a function of Vac (see Fig.2.3).

In practice, resonant tunneling features in the d.c. I-V cannot be sharper than

approximately h/τ , where τ is the scattering time associated with the trans-

port in question. So photon replicas spaced by energy hω cannot possibly be

resolved unless ω À 1/τ . The quantum mechanical description (2.17) goes to

the classical description (2.10) when ω ¿ 1/τ .

In semiconductor superlattices, the first observations of PAT came in 1995

from Guimaraes et al. [27], Keay et al.[22] and then Zeuner[23] which took


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5α

0

12

3

4

Jn

2(α)

Figure 2.3: The relative strengths of photon replicas: 0hω, 1hω, 2hω, 3hω, 4hωas a function of α = eVac/hω.

advantage of a free electron laser capable of reaching the terahertz frequencies

required to surmount the picosecond scattering times in GaAs-based systems.

Many interesting but unpublished efforts to identify PAT in resonant tunneling

diodes made of various III-V material systems failed before and after these two

works, even with a free electron laser providing the necessary frequency and

power (see Prologue for examples). Unrelated effects such as electron heating

or changes in the external contacts usually turn out to be more important than

the time-coherent behavior of the device. For this reason, the idea of putting

many resonant tunneling diodes in series to form a superlattice has proven to

be a very fruitful one. This way, transport is controlled by a series of identical

and sequential resonant tunneling processes between quantum wells, and not

by tunneling between contacts and the well. Indeed, PAT was observed in

superlattices before it was found in resonant tunneling diodes [27, 22].


The approximately 100µm wavelengths used in the early superlattice PAT

experiments were much larger than the actual device dimensions and one would

at first glance expect the Tien-Gordon (TG) assumption of a spatially uniform

a.c. field to be reasonable. Later theoretical work has provided diffrerent for-

mulations of PAT [24, 26] but sufficiently quantitative experiments are lacking,

and a satisfactory connection with these works is missing. The oscillatory

Bessel function behavior was qualitatively identified [22], but there never was

enough experimental information to quantify the agreement with TG behavior.

2.4 In-plane Magnetic Field

The late 1980’s saw the emergence of a great deal of interest in the effect

of a magnetic field parallel to the superlattice layers on transport through the

layers (refs. [43, 44, 45, 46, 56, 50] are good examples). Most of the works

published in that period address the same phenomenlology: the magnetic field

causes a shift in the voltage at which the NDV region begins, and this shift is

proportional to the square of the magnetic field. The current itself typically

changes by a modest amount depending on the particulars of the structure over

a field range of eight to ten Tesla. In some cases the current rises with magnetic

field and then falls again at the highest fields. The maximum current typically

rises as the temperature is lowered down to about 77K.

Some authors have used a semiclassical approach (refs. [9, 47] and sec. 3 of

ref [33]) to understand the energy shifts in transport experiments as wells as

in photoluminescence experiments that show similar shifts [39, 40]. The idea

is that, given a periodic superlattice potential in the z direction under a small

voltage bias, and a magnetic field in the x direction, there is a Lorentz force in

the y direction F = evzBx on every electron, where vz is the velocity in the z


direction. The in-plane momentum gained is

∆py = h∆kx = eB∆z (2.18)

where ∆z is the distance traveled by the electon through the superlattice. In

the case of an electron transported from one well to the next by resonant

tunneling [9], there is a gain in energy

∆E =B2(∆z)2e2

2m. (2.19)

This implies that the electric bias needed to maintain the resonant tunneling

condition must be shifted by ∆V = ∆E/e. Such arguments, heuristic though

they may be, have in some cases been used to explain real transport data [9, 45]

and photoluminescence results [33] using (2.19). This approach is misleading,

though, because it assumes that the electron always starts off with zero in-plane

momentum in the first well, which is not true. Quantum mechanically, vz and

kz are not very meaningful and the approach is not satisfactory.

A large number of theoretical works have treated the problem in a quantum

mechanical framework. Some works from the 1980’s [33, 7, 55] made use of a

piece of physics described around 1970 by Stern [34], Bienvogl [37] and Tsui

[35] in relation to accumulation layers in silicon that predates superlattices.

Treating the magnetic field as a perturbation (see section A.1.2), it is found

that the energy of a particle in a well is also modified by a rigid diamagnetic

shift that depends on the spread of the wave function [33]. This provides a

measure of the energy shift of the miniband in a magnetic field, but turns out

not to be useful in calculating I-V’s in superlattices.

More detailed papers have calculated I-V’s in crossed electric and magnetic

fields [51, 52]. Shchamkhalova and Suris [36], for example, have shown that

the I-V should shift with increasing magnetic field and the peak current should


x

hω

z

∆E

F

yk∆

∆z

B=Bx

(a) (b)

Figure 2.4: A heuristic picture of the effect of the Lorentz force on resonanttunneling seen (a) spatially and (b) as a conduction band diagram.

uniformly decrease with B. Miller and Laikhtman [53] have shown, in agree-

ment with some experiments [41, 44], that in the case of miniband transport

and semiclassical behavior within the miniband, there is not only a peak in the

I-V curve but also in the I-B curve. They critical field is, as before,

Fth = h/edτ (2.20)

where d is the superlattice period and τ the scattering time. At high temper-

atures (EF ¿ kBT ), the critical magnetic field is

Bth =√

2hm

epkBT dτ(2.21)

where pT =√

(2mkBT ) is the width of the electron distribution. The shift in

critical electrical field (i.e field at which current saturates) is

∆Fth ≈ eB2d

2m

kBT

h/τ, (2.22)


with the familiar B2 dependence. This has provided a useful way to deter-

mine the scattering time and therefore mobility of a superlattice that exhibits

miniband transport.

All of the magnetotransport works mentioned above have concentrated on

d.c. behavior. Most of the experiments and all of the detailed quantum mechan-

ically correct theories have concentrated on understanding miniband transport

in crossed electric and magnetic fields. The question of transport by sequential

resonant tunneling in crossed electric and magnetic fields lacks any detailed ex-

periments and theory. High frequency transport is even less understood. There

is as yet no experimental work in the published literature on a.c. transport, or

PAT, in the presence of an in-plane magnetic field.

Chapter 3

Experimental Approach

The experiments described in the next two chapters follow an incremental

approach to understanding photon assisted transport (PAT) in resonant tun-

neling structures. We take precisely the same superlattice material in which

Zeuner et al. [23] observed up to seven photon sidebands at 1.5THz and pat-

tern it into devices of smaller lateral dimension. Using essentially the same Free

Electron Laser photoresponse technique used in many recent terahertz trans-

port experiments [25, 23, 22], we recover the known photon assisted transport

behavior and measure in greater detail its dependence on parameters such as

temperature, excitation frequency and size, and then compare to theory. The

new step we take is to add an in-plane magnetic field to the problem, measure

its effect on the d.c and a.c. transport and seek to understand this effect in

detail.

In order to observe photon assisted tunneling, it is necessary to have res-

onant tunneling transport in the device, an a.c. drive frequency larger than

1/τ , and a large enough drive amplitude (see Section 2.3.2). Terahertz pho-

toresponse experiments are seldom ”clean” enough to show just one transport

23

CHAPTER 3. EXPERIMENTAL APPROACH 24

process, and PAT is often missing. The narrow miniband GaAs/AlGaAs super-

lattices studied in refs. [22, 23] are in fact striking in that their photoresponse

shows a close qualitative resemblance to the relatively simple theory of Tien and

Gordon [20]. It thus makes sense to use such a superlattice as a starting point

for an investigation of deviations from known PAT behavior that may either

be present naturally in a sample or induced by the application of a magnetic

field.

The purpose of this chapter is to give a concise summary of the experimental

methods used in this work. For more specific experimental details, the reader

is referred to Appendicies B and C.

3.1 Sample Preparation

The MBE-grown superlattice material used here consists of ten GaAs wells

80A wide and eleven AlGaAs barriers 50A wide all with a uniform n-type

doping of 5x1015cm−3. This superlattice is sandwiched between two 3000A

wide n+ layers doped to 2x1018cm−3. A 1000A GaAs spacer layer n-doped to

5x1015cm−3, separates the superlattice from the n+ layers. A realistic calcula-

tion of the actual band diagram of this structure is given in Chapter 4, Figure

4.2.

The doping structure in this superlattice was designed by Zeuner [23] with

the intent of observing photon assisted transport. The relatively light doping

was chosen with an eye to keeping scattering times as high as possible in order

to keep ωτ low and preserve the coherent behavior at frequencies as low as

possible.

In order to make transport measurements, the material is patterned into

small mesas with ohmic contacts alloyed onto the n+ regions. Figure 3.1 gives a


summary of the cleanroom process flow, and Appendix B provides the practical

details.

In practice, with careful control of the last undercut etch, the physical

diameter of the mesa can be made far smaller than the smallest features written

with electron beam lithography (about 50nm), as is evident in Figure 3.2. The

electrical properties of the device, however, cannot be controlled as well as

the physical dimension. It was found that devices smaller than about 500nm

seemed to be depleted of electrons, and below about 300nm, were essentially

open circuits.

The sample is fabricated with a series of nine devices with top contact

diameters varying from 1µm to 2.6µm (Fig 3.3). Each device has its own top

contact and a shared bottom contact. The nine top contacts connect to 100mm-

long metal leads that fan out to form a rudimentary non-resonant antenna onto

which terahertz radiation can be focused (Fig. 3.4). A gold wire is bonded to

each lead connected to a top ohmic contact, and a wire is soldered onto the

common bottom contact lead using heated indium. The indium spikes through

the TiAu film and forms an ohmic contact to the bottom n+ layer. This way,

each device can be d.c.-biased independently and terahertz radiation can be

focused onto this antenna and coupled into all of them simultaneously.

3.2 Photoresponse Measurements

The sample is mounted onto a standard fiberglass 16-pin chip carrier and

is placed inside a helium-4 cryostat for temperature control between 1.5K and

300K (see Appendix C.1 for details). The cryostat allows a magnetic field to

be swept from zero to 10 Tesla and the orientation is chosen parallel to the

plane of the sample.


(a)

(b)

(c)

(d)

Ni/Au/Ge top cont

superlattice

Ti/Au bottom contact

bottom n+

Figure 3.1: Schematic of the mesa fabrication process. (a) Electron beamlithography and liftoff steps produce a top NiAuGe contact. (b) An isotropicwet etch defines and undercuts the mesa. (c) Another electron beam lithog-raphy and liftoff sequence defines the bottom contact, shadowed by the mesaitself. (d) An isotropic reactive ion etch electrically isolates the top and bottomcontacts, after which a wet etch removes the material under the air-bridge.


Figure 3.2: An SEM micrograph of two typical devices.


Figure 3.3: Sample layout: Nine devices with top contact diameters rangingfrom 1 µm to 2.6 µm have individual top contact leads that fan out. Thephysical diameter of the active region is 0.5 µm less than that of the top contact.The bottom contact is shared.


Figure 3.4: The lead structure close to the mesas serves both for d.c. electricalbias and as a simple antenna. Typical wavelengths are smaller than 100 µm.


Vd.c, applied

RI=Idc+Iac

=VR/R

Eac(THz)

B

n+

substrate

top contactbottom contact

VR

S.L. oscilloscope

multimeter

Figure 3.5: Sample bias and photoresponse measurement scheme. A d.c. volt-age is applied and the d.c. current is measured in the presence of a 5µs pulseof far-infrared radiation and an in-plane magnetic field.

A d.c. bias is applied to each device using a battery connected to a 500Ω

potentiometer driven by a computer-interfaced stepper motor. With adequate

shielding, this simple arrangement turns out to be far less noisy than a standard

programmable voltage source and has a negligible source resistance compared

to the high impedance (roughly 1MΩ) of the superlattice device. D.c. current

through the device is determined by measuring the voltage across a 1KΩ series

resistor using a Keithley 2000 multimeter. This is shown schematically in Fig.

3.5.

The terahertz drive is achieved by focusing radiation from a Free Electron

Laser onto the sample using an off-axis parabolic mirror (Fig. 3.6). The ra-


diation reaching the sample is polarized such that the electric field lines up

with the lead of the bottom contact (this was empirically found to yield good

coupling). The drive intensity is controlled by placing an attenuator in the op-

tical path of the laser. To implement this, we simply rotate the relative angle

between two crossed polarizers (two metal wire grids) with the grid closest to

the sample fixed in order to preserve polarization. The intensity is measured

by splitting off a small fraction of the beam (with an 40µm thick Mylar beam

splitter) and focusing it onto a pyroelectric detector, that provides a signal that

grows linearly with the power (square of the electric field) reaching it. This

gives a relative measure of the square of the a.c. electric field driving the su-

perlattice and is the best we can do in this work. An absolute measurement of

the a.c. electric field would require understanding quantitatively the near-field

coupling between the antenna and the superlattice mesa, clearly a non-trivial

task at far-infrared frequencies.

The Free Electron Laser (FEL) provides 5µs-long pulses of radiation with

a duty cycle of approximately one second. The photoresponse, or d.c. cur-

rent induced by the a.c. drive, appears as a 5µs-long voltage pulse across the

1KΩ series resistor in the circuit in Fig. 3.5. To measure the photocurrent,

we first amplify (using an SRS 560 preamplifier with appropriate band-pass

filtering) and then digitize the voltage change across the series resistor using a

HP54305A oscilloscope. By adding the photoresponse current thus measured

to the d.c. current measured simultaneously using a Keithley 2000 multimeter,

we obtain the total current at a given voltage bias point. The applied d.c bias

is incremented by a few millivolts after every FEL pulse, allowing us to trace

out the I-V of the device in the presence of an effectively continuous terahertz

a.c. drive. This ’irradiated I-V’ is the main product of the experiment.

By far the dominant source of noise in this experiment is RF pickup from


FEL

pyroelectricdetector

cryostat

sample

crossed polarizers

Figure 3.6: Schematic of the optical table. Free-space far-infrared radiationfrom a free electron laser is focused onto a sample inside an optical cryostat.


a nearby radio station transmitting at 1.25MHz, which is unfortunately close

to the bandwidth associated with the FEL pulses. This noise source cannot be

eliminated because it is impossible to implement an effective RF cage around

the experiment since an opening is necessary for the FEL beam. We minimize

its effect by time-averaging the current response signal over an integer number

of cycles of this RF pickup with the hope that the pickup signal over each cycle

will average to zero. In practice, the pickup signal does not have a constant

amplitude over many cycles, so it can never be completely eliminated. With

careful grounding, the noise floor is about 10µV, or 10nA in current at 1MHz.

As will be clear in the results, this is barely adequate to study our superlattice

devices, which carry currents in the range of 1µA.

The only other significant noise limitation is associated with variations in

the power and timing of terahertz radiation pulses from the FEL. These are

beyond our control and manifest themselves, occasionally, as scatter in the

photoresponse data.

In the next chapter we discuss the results of an investigation in which we

measure irradiated IV’s of a series of sub-micron devices at various tempera-

tures, radiation intensities and applied in-plane magnetic fields.

Chapter 4

Results and Analysis

We begin this chapter by considering the temperature and size dependences

of the d.c. I-V, and gain some insight into the actual band structure of the

device and its deviations from the idealized superlattice. The next step is to

examine the terahertz photoresponse and compare it both to previous experi-

mental work and to available theories of photon assisted transport in light of

what we have learned from the temperature dependencies. In the following

section we present the d.c. behavior in crossed electric and magnetic fields and

develop a simple model that attempts to capture the essential physics. The

last result involves the a.c. behavior in the presence of the magnetic field.

4.1 D.c. Characteristics

4.1.1 Temperature Dependence

The I-V at 300K agrees with the form in Sec. 2.2: it is ohmic at low bias

and then saturates at higher biases when the wells go out of resonance (Fig.

4.1). The voltage at which the current ceases to rise is where the wells go out

34

CHAPTER 4. RESULTS AND ANALYSIS 35

of resonance. This voltage is 80mV across the entire superlattice of 10 wells,

or approximately 8mV across each single well (see Section 4.2). There is no

negative differential resistance in the high-field regime above 80mV in this case,

which suggests domain formation. We need not consider this in detail here. The

aim of this analysis is to understand the positions of resonant tunneling regions

of the I-V and the magnitudes of the currents associated with them.

More interesting is the fact that when the temperature is lowered, both

the zero-bias conductivity and the saturation current decrease (Fig. 4.1). This

is opposite to what we would expect from miniband transport in a reasonable

relaxation time approximation (Section 2.2 and refs. [43, 44, 45]). Below 100K,

we see in addition a change in the form of the I-V. The relatively abrupt sat-

uration at 80mV disappears in favor of a more gradual fall in the conductivity

as the bias is increased; the I-V is broadened.

The essential physics behind this discrepancy turns out to be rather simple:

the doping level in our superlattice is below the Mott critical density [23] which

means the donors are not completely ionized. As the temperature is lowered,

fewer donors are ionized and the charge density in the actual superlattice falls.

To quantify this, we begin by numerically computing the band structure

of the superlattice. Figure 4.2 shows the result of a self-consistent numerical

solution to Schrodinger’s and Poisson’s equations [62] for the one-dimensional

band structure in the material’s growth direction. The positions of discrete

confined states are computed and are found to be very close together, about

36meV above the GaAs conduction band. Ten confined states, each 2meV wide

(Section A.1) are packed into a range of 4.0meV. If indeed this does form a

miniband, then we can think of the miniband width as 4meV. When a small

bias is applied to the structure, electrons are injected from an ohmic contact

into the this collection of states giving rise to ohmic behavior.


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

I µA

Voltage (V)

300K

100K77K

5K

Figure 4.1: Temperature dependence of the d.c. I-V. for a 1.5µm device.


Two properties of this superlattice stand out. Firstly, the charge density

in the individual wells is much higher than that in the barriers. Secondly the

width of the miniband is very small, comparable in fact to the width of an

individual state. This suggests that this may not be a miniband, but a series

of individual states in individual wells.

When the temperature is lowered to 100K (Fig. 4.3), not only does the

charge density fall throughout the superlattice, but electrons begin to accumu-

late in the center of the structure giving rise to a significant amount of band

bending at zero bias. Transport is again ohmic at low biases. But when the

bias exceeds the width of the confined states and the high-field sets in, the

ten wells in the superlattice do not all go out of resonance at the same bias.

Instead they go out of resonance a few at a time, which explains qualitatively

the more gradual saturation in the I-V.

At much lower temperatures, the band structure remains essentially that at

100K (Fig. 4.3) according to this Poisson-Boltzman calculation, but the charge

density in the superlattice continues to fall. As can be seen in Fig.4.4, the cal-

culated charge density in a center well falls in step with the current actually

measured in the experiment at 100mV bias, which explains why the current

falls when the temperature is lowered. (Note that the calculated charge density

is a function not only of donor ionization but also band bending and interac-

tion with the contacts, so we should not expect a priori a simple temperature

dependence). In short, the temperature dependence of the I-V is a result of

changes in charge density.

The different form of the I-V at 5K (Fig. 4.1), where a finite bias is required

to turn on the current at low voltages, is not due to a change in the shape of

the superlattice band structure. It is due, rather, to the fact that not enough

electrons in the contacts are thermally excited to the energy of the miniband


-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0

5 1017

1 1018

1.5 1018

2 1018

2000 2500 3000 3500 4000 4500 5000 5500 6000

n (cm-3)

Distance from surface (Å)

(eV)

Figure 4.2: The conduction band profile at 300K and zero external bias (darklines). The ten states that constitute the first miniband are shown in greyand are spread out by 4meV. The Fermi level is shown by the dotted line.The lower curve indicates the charge density in the structure: very high in theohmic contacts, lower in the wells and essentially zero within the barriers.


-0.20

-0.10

0.00

0.10

0.20

0.30

0

1 1017

2 1017

3 1017

4 1017

5 1017

2000 2500 3000 3500 4000 4500 5000 5500 6000

n (cm-3)

Distance from surface (Å)

(eV)

Figure 4.3: The conduction band profile at 100K and zero external bias (darklines). The ten states that constitute the first miniband are shown in grey andare spread out by 8meV. The Fermi level is shown by the dotted line and Thelower curve indicates the charge density.


0

5 10-6

1 10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

3.5 10-5

4 10-5

0

1 1015

2 1015

3 1015

4 1015

5 1015

50 100 150 200 250 300 350

Con

duct

ivity

(m

ho) C

harge Density

T (K)

Figure 4.4: The measured conductivity at zero bias compared to the calculatedcharge density.

to enable a current. A bias is required to bring the emitter into resonance with

the miniband in order to turn on the current.

In summary, at higher temperatures we have higher currents and a sharper

step in the I-V. This fact will be useful later on.

4.1.2 Size Dependence

The size dependence of transport is easily understood at 300K. Figure 4.5

shows that saturation current scales closely with area of the device perpendic-

ular to the current. We learn that the current scales as the square of the device

diameter if we subtract 0.5µm from the diameter. We are forced to assume

diameters for the current path that are much smaller than the physical diam-

eters in order to make the fit work with a zero intercept (inset, Fig.4.5). This


means that there is a depletion width of about 0.25µm near the surface of the

mesas.

It is clear that the current density is the same in all devices and we can

conclude that the essential physics that determines the shape of the d.c. I-V

does not change as we go to smaller devices. In other words, d.c. transport is a

one-dimensional process. In the remainder of this chapter, we will concentrate

on the same device 1.5µm in physical diameter with an inferred current path

1.0µm in diameter.

As an experimental note, it was observed that radiation coupled much more

efficiently into smaller devices than large ones. We were able achieve at least

an order of magnitude higher a.c. electric fields by going from a 2.1µm device

to a 1.5µm device. This is because the devices also act as capacitors that short

out the electric field, so smaller devices with smaller capacitances suffer less

from this problem.

4.2 Photoresponse

When radiation at 2.5THz (84 cm−1) is focused onto the sample, there arises

a power-dependent change in the I-V. Figure 4.6 summarizes the response at a

variety of a.c. excitation powers.

Several aspects of this behavior are examined in this section. (1) At low

power (i.e. low a.c. field), the current at all voltages increases in magni-

tude as we increase the excitation power but the overall shape of the I-V does

not change. This is exemplified by the two thin lines in Fig. 4.6(a). (2) At

higher powers, (e.g. the thicker lines in Fig. 4.6(a), and all of (b)), a series of

steps appears in the I-V as in previous works [22], [23]. This is qualitatively

the behavior one would expect from a photon assisted transport process: a


-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

I (µA

)

V(Volts)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 0.5 1 1.5 2 2.5 3(Diameter)2

(µm)2

I µA

Figure 4.5: The d.c. I-V as a function of size. The physical diameters are.7, .9, 1.1, 1.5, 1.7, 1.9 and 2.1µm and the current paths are inferred to be,respectively, .2, .4, .6, 1.0, 1.2, 1.4, 1.6µm. The temperature is 300K


series of replicas of the original I-V appear with the a.c. field (see Sec. 2.3).

Quantitatively, however, how consistent is this with theories of photon assisted

transport? (3) Above a certain power, the steps disappear. Can this be ex-

plained? (4) In all of the above observations, there is a subtle interplay with

frequency and temperature dependent effects.

4.2.1 Frequency Dependence

The positions of the steps in Figure 4.6 are equally spaced in voltage, within

a small experimental scatter. This spacing (i.e. the voltage difference between

consequtive steps) changes linearly with frequency (Fig.4.7) and gives us good

confidence that this is photon assisted transport. The spacing is not exactly

the number of wells multiplied by the photon energy, as we would expect. The

the spacing is, rather, η · 10 · hω where a small lever-arm η = 1.92 accounts for

the fact that there is effectively some distance between the contacts and the

beginning of the superlattice.

The photon assisted transport features emerge gradually as we increase the

frequency to above a terahertz. At low frequencies (0.6THz or 20cm−1), we

see absolutely no sign of equally spaced steps in the irradiated I-V; the voltage

position of the step moves with increasing power (Fig. 4.8 (a)) and can easily

be explained as classical rectification. This is a regime where ωa.c. < 1/τ ; where

τ is the inelastic scattering time. On the other extreme, at 3.4THz (114cm−1),

the steps are very sharply defined, their position is independent of a.c. power

and their amplitudes show the oscillatory behavior predicted by Tien-Gordon

theory. This is where ωa.c. > 1/τ .

These two facts constitute convincing evidence that we are looking at pho-

ton assisted transport behavior, and we have gained a conversion factor η be-


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

I(µA

)

(a)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

I (µA

)

Vd (V)

(b)

01

23

4

Figure 4.6: The I-V of a 1.5µm top contact device cooled to 100K and irradiatedat 84 cm−1 at (a) a series of low powers and (b) high powers. Within each figurethicker lines represent higher powers; the relative magnitudes of a.c. field are,respectively: (a) 0, .05, .16, .20; (b) .22, .33, .44, .48. The arrows show n-photonreplicas.


0

50

100

150

200

0 0.5 1 1.5 2 2.5 3 3.5

Ste

p sp

acin

g (m

V)

Frequency (THz)

Figure 4.7: The zero-photon to one-photon step spacing as a function of fre-quency. The point at zero is only a guide to the eye.

tween the voltage scale in the data and the voltage bias within the superlattice.

4.2.2 Power Dependence

To compare the observed power dependence with Tien-Gordon theory (Sec-

tion 2.3), we begin by plotting the height of each step in the irradiated I-V

versus the a.c. electric field. The step height is a measure of the current due to

each of the n-photon transport processes. Due to experimental limitations, the

absolute a.c. electric field strengths are unknown and we use arbitrary units

for it. The result, Figure 4.9, is similar to theory (Fig. 2.3), but we notice a

few differences.

At low a.c. electric field we see an increase in the zero-photon step height

with a.c. field. This can be understood if we assume that the irradiation serves


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6V (Volts)

I (µA

)

(a)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

V (Volts)

1 0

(b)

Figure 4.8: Emergence of photon assisted transport. (a) I-V at 0.6THz, and(b) at 3.4THz. The arrows point to 0hω and 1hω features. Darker lines denotehigher a.c. power.


-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14 16

Ste

p he

ight

(µA

)

E (Arb. units)

01

2

3

4

Figure 4.9: Photoresponse at 2.5THz. The numbered curves represent stepheights corresponding to 0hω (•), 1hω (...), 2hω (2 ), 3hω (5) and 4hω (-).


to heat the sample locally, increase donor ionization and consequently increase

the charge density in the superlattice. As we saw in Section 4.1.1, an increased

charge density implies a higher current.

Although the zero-photon and the one-photon steps increase at the same

time in our data, the difference between them decreases with a.c. field (between

E = 2.5 and E = 6.5 in Fig. 4.9) after the initial emergence of the one-

photon step. This corresponds to the regime in Tien-Gordon theory where

the one-photon process emerges at the expense of the zero-photon process.

This conclusion remains qualitative since we do not have a convincing way to

calculate or measure the exact relation between the a.c. power and the charge

density in the device (although ref. [29] attempts this).

At high powers, where both the zero- and one- photon steps are clearly

visible (Fig. 4.6(b)), we notice that the steps become much sharper than those

in the d.c. I-V. Following the discussion in Sec. 4.1.1, this lends further cre-

dence to the suggestion that the electron temperature of the device during the

irradiation pulse increases significantly.

The qualitative correspondence between Tien-Gordon theory and Fig. 4.9

continues with the emergence of the two-photon, three-photon and four-photon

features as the one-photon continues to oscillate. If we assume that at these

high irradiation powers (above E = 8 in Fig. 4.9) the donors are fully ion-

ized, rendering the charge density independent of the power, we can begin a

quantitative comparison between theory and experiment.

At E = 10.7, we have the first maximum in the 1-photon current, which

according to Figure 2.3 must correspond to α = eVac/hω = 1.86. This gives

us a relation between our measured E and α, and allows us to predict where

other minima and maxima should be in the data. Figure 4.10 gives an example

of this approach with the two most distinct features in the power dependence.


Experiment Theory1-photon maximum E=10.7, I=1.58 E=10.7, I=1.582-photon maximum E=14.7, I=0.93 E=17.4, I=1.090-photon minumim E=13.1, I=-0.13 E=14.0, I=0

Figure 4.10: A comparison between Tien-Gordon theory and Fig.4.9 where weforce the 1-photon maximum to fit the theory. E deonotes the electric field andI denotes the step height in Fig.4.9. Currents are in µA.

The agreement is clearly not good (taking into account the lever-arm), and the

implication is that Tien-Gordon theory alone cannot account quantitatively for

the data.

When we consider the highest a.c. powers where the three- and four- pho-

ton features in appear, even the qualitative correspondence between data and

theory breaks down. Both these features rise together, with the four-photon

current higher than the three-photon current. This is a more fundamental de-

viation from theory than the ones considered above and has not been reported

in other works.

At just above E = 15, the resonant tunneling features become unmeasur-

able: we observe the onset of huge oscillations in the current between subse-

quent irradiation pulses under identical conditions. In the TG framework, this

regime corresponds to a situation where α ≈ 3.5 (by comparison of Fig.4.10

and Fig.2.3) , or Vac = 36meV; which is well into the high-field region of the

d.c. I-V.

In summary, the low power behavior is thus an interplay between a rising

charge density due to heating, which tends to increase the zero-photon current

and the Tien-Gordon behavior which tends to suppress it. The mid-power

behavior is qualitatively close to theory but quantitatively it deviates. We

conclude that TG theory is not sufficient to describe our superlattices under


terahertz irradiation.

4.3 Effect of an In-plane Magnetic Field

4.3.1 Observations

An in-plane magnetic field has a dramatic effect on the d.c. I-V. At 100K,

Figure 4.11 shows how the conductivity at zero bias is unchanged, but the

saturation current increases by an order of magnitude within five Tesla. In

comparison, previous works [38, 45] have consistently reported a change in

zero bias conductivity and a much smaller increase in current than observed

here. Even in very similarly designed superlattices with narrow minibands,

previous results have shown modest increases in saturation currents and have

been successfully modelled with theories that rely essentially on a relaxation

time approximation [53].

The position of the saturation feature moves with magnetic field. If we

measure this position as the intersection of two lines, one fitted to the low bias

I-V and the other to the saturated I-V, we can see a clear trend, shown in Fig.

4.11, inset. The position is proportional to the square of the magnetic field as

in previous works.

An increase in temperature does not change the observation of a dramatic

increase in current with magnetic field and a shift in the saturation feature. It

does, however, lead to a slight change in zero-bias conductivity with magnetic

field (Fig. 4.12(a)). A decrease in temperature (Fig. 4.12(b)) again shows a

large increase in current with magnetic field, but in this case, carriers are frozen

out and there is no feature in the I-V we can look at to pin-point the onset or

end of the resonant tunneling process.


-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

I (µA

)

V (V)

0T

7T

0.00

0.0500

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0 10 20 30 40 50

Ste

p po

sitio

n (V

)

B2(T2)

Figure 4.11: The d.c. IV. with in-plane magnetic field, 0, 1, 2 ...7T withtemperature at 100K. Inset: magnetic field squared vs. step position.


-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

I (µA

)

V(V)

(a)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

V(V)

(b)

Figure 4.12: The d.c. IV. with in-plane magnetic field, 0, 1, 2 ...7T withtemperature at (a) 200K and (b) 7K.


4.3.2 Comparison with Theory

The most intuitive way to begin is to follow a semi-classical picture (Sec.

2.4, ref. [9]) and assume a Lorentz force modifies electron trajectories and

thus the resonant tunneling condition between two adjacent wells. In this case,

we would expect a shift of e2B2∆z2/2m = 5.5meV per well in the saturation

position at 5T. Multiplying by the number of wells we expect a shift of 55meV

compared to a measured value of 175meV (Fig. 4.11). If we include the lever-

arm of η = 1.9 found in the photon assisted transport data, we expect a shift

of 105meV, which is of the right order of magnitude, but different from the

measured value. As far as the current is concerned, this picture assumes any

change in tunneling transmision coefficient is caused by the shift in energy

and consequently a change in effective barrier height [56]. Using the WKB

expression for this coefficient T = exp(−2√

(2mEb)d/h), with Eb the barrier

energy and d the barrier width, a shift of 10meV in the confined state energy

of one well corresponds to a change of only 16% in the tunneling coefficient at

5T. Again, this is in the right ballpark for some previous works, but not our

data.

If we assume for the moment that the theory of Miller and Laikhtman

[53] applies, Eq. 2.22 gives a correction to the bias shift found in the above

semiclassical argument. This correction kBTτ/h = 1.75 implies τ = 0.1ps,

which is clearly not true since we have observed PAT in the device at frequencies

well below the corresponding 1.6 THz. The theory does not apply, and it is

likely that the miniband transport assumption inherent in it is not applicable

here, and that transport is due instead to sequential tunneling.

Another indication of sequential tunneling is the fact that we observe no

change in low-bias conductivity with magnetic field. There is no change in


low-bias current even at 5T, where the cyclotron radius is well below the total

length of the superlattice, which means current is controled by dynamics on a

scale much shorter scale.

4.3.3 A Simple Model

In an attempt to explain our data, let us assume that the essential physics

that controls the current through the superlattice at a small bias is a series of

resonant tunneling process from one well to the next. The current through two

adjacent wells L and R is then proportional to a function that peaks when the

energy of an electron in both wells is the same. We can model this by taking

such a function to be a Gaussian, so that

I ∝ e−(E−E′)2

∆E2 (4.1)

accounts for the energy conservation condition with a spread in energy ∆E

determined by inhomogeneities in the structure (Sec. A.1.1). Each well has

only one discrete energy level in the z direction (the growth direction), E0 and

E ′0 respectively, and each well has the dispersion relation of a free electron in

the x and y directions. We can write out the energies of the two wells to give:

I ∝ e−[E0+h2k2

x2m

+h2k2

y2m

−(E′0+h2k

′2x

2m+

h2k′2y

2m)]2/∆E2

. (4.2)

In the absence of a magnetic field, momentum is conserved in the x and y

directions when tunneling occurs. This can again be accounted for by using

peaked functions that require the momenta to be equal in both wells, within a

spread ∆k determined by the scattering length in the material. Hence

I ∝ e−[kx−k′x]2/∆k2

e−[ky−k′y]2/∆k2

. (4.3)


These two Gaussians representing the two conservation criteria can be multi-

plied together to give us a transmission function for an electron tunneling from

one well to the next. The mass m denotes the effective mass m = 0.067me in

GaAs and the spreads in energy and momentum need to be estimated empiri-

cally.

The forward current from well L to well R is the sum of the transmission

function for every possible value of initial and final momentum, weighted by

the occupancy of the corresponding state in the initial well. This occupancy is

just the Boltzman factor for temperatures of interest to us. Since the k-states

are very closely spaced, we can convert the sum to an integral and use a density

of states which is a constant in k-space. The resonant tunneling behavior is

then captured in one expression:

Iright = ND∫ ∫ ∫ ∫

dkxdkydk′xdk′ye−[E0+

h2k2x

2m+

h2k2y

2m−(E′0+

h2k′2x

2m+

h2k′2y

2m)]2/∆E2

(4.4)

×e−[kx−k′x]2/∆k2


(4.5)

×e−[E0+h2k2

x2m

+h2k2

y2m

]/kBT (4.6)

where N is an appropriate normalization, D is a constant density of states

and the integral is over all values of kx and ky in both wells. Similarly, we can

calculate the reverse current from well R to well L by weighting the transmission

function with a Boltzman factor dependent on the energy in R and integrating

over the possible values of k in both wells:

Ileft = ND∫ ∫ ∫ ∫

dkxdkydk′xdk′ye−[E0+

h2k2x

2m+

h2k2y

2m−(E′0+

h2k′2x

2m+

h2k′2y

2m)]2/∆E2

(4.7)

×e−[kx−k′x]2/∆k2


(4.8)

×e−[E′0+h2k

′2x

2m+

h2k′2y

2m]/kBT (4.9)


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

I (ar

b)

Bias at one well (V)

Figure 4.13: Simulated d.c. I-V with energy width∆E = 2meV, mobilityµ = 0.2× 105cm 2/V-sec.

and the total current is given by

I = Iright − Ileft. (4.10)

The effect of applying an electric bias to the two wells is to change the value

of E0 − E ′0. We can therefore evaluate this rather large integral numerically

for various values of the two energies to yield an I-V curve for the two- well

structure.

An example of such an I-V is shown in Fig. 4.13 and has the essential

features we would expect from the resonant tunneling I-V: ohmic at low bias

when the states in the two wells are aligned, a peak at roughly the width of

the states when they begin to go out of alignment, and a region of negative

differential resistance after that.


4.3.4 Zero Magnetic Field

In the actual superlattice we are trying to model, this resonant tunneling

process happens sequentially, ten times, with E0−E ′0 the same each time. The

only important difference is that at higher biases, in the negative differential

resistance region, this model does not account for domain formation. But the

results of this model are useful inasmuch as they tell us the behavior of the

current at low biases, tell us at what voltage the current should cease to rise

and tell us in relative terms how much the current should rise. In this sense,

the model accounts adequately for the physics we observe in the actual I-V,

Fig. 4.1.

To estimate the spread in energy, we simply look at the position of the

current saturation in the data, Fig. 4.11, divide it by the number of wells and

the lever arm η = 1.92 determined through photon assisted tunneling, and

take this as a measure of when the wells go out of resonance. This is also a

measure of how broad the energy levels in the wells are, since the wells are

exactly at resonance at zero bias. This number turns out to be 2meV, which

is, incidentally close to what we would expect the spread in energy to be if

we assumed a 1-monolayer well-width fluctuation within the sample (see Sec.

A.1.1). It is also consistent with the fact that photon assisted transport appears

above about 1THz, or hω >4mev.

The spread in momentum is a function of the mobility µ of the material.

Following an argument from Choi et al. [41], in the presence of impurity scat-

tering, tunneling is possible within an energy δE ≈ 2h/τ , determined by the

scattering time τ = µm/e which is in turn determined by the mobility. Using

δE = h2∆k2/2m, this allows us to estimate the maximum possible change in

momentum as ∆k = (4m/hτ)1/2 and a scattering length l = 1/∆k. To start


with, we consider a mobility of µ = 0.2 × 105cm2/V − sec corresponding to a

scattering length of 170A. Details of the numerical procedure are given in Sec.

A.2.

These estimates result in a simulated I-V for one well (Fig. 4.13) in which

the ohmic region ends in the right place and the resonant tunneling feature

has the right width to model our data. As expected, increasing the value of

∆E broadens the I-V and increases the maximum current because this means

more states are available to tunnel into. Similarly, increasing the mobility

also sharpens the I-V as one would expect. Increasing the temperature also

increases the current, as it should and we can be confident that the model is

adequate. The normalization N was not included in the calculation because it

does not affect the shape of the I-V we are trying to understand.

4.3.5 In-Plane Magnetic Field

When we introduce a magnetic field in the x direction, we break the mo-

mentum conservation condition in the y direction. Can this fact account for

what we observe in Figure 4.11?

The magnetic field modifies the subband energy in both wells identically, so

E0 −E ′0 is unaffected. It modifies ky by eBz0/h, where z0 is the distance from

one well to the next, so that the momentum conservation condition changes to

I ∝ e−[kx−k′x]2/∆k2

e−[ky+eBz0/h−k′y]/∆k2

. (4.11)

A numerical evaluation of the current then gives us the curves shown in Fig.

4.14. Clearly, the magnetic field increases the maximum current and decreases

the zero bias conductivity, just as previous works have found (compare with

ref.[43]). The relative increase in current, though, is not reproduced.


Exploring the effect of changing the scattering length, we find that there

is no way we can reproduce our data with a physically meaningful scattering

length; Figure 4.15 shows an example with a longer scattering length (and a

scattering time long enough to be consistent with the fact that we observe PAT).

We notice that with higher mobilities, a trend emerges where the maximum

current first rises with magnetic field and then falls. This takes us back to the

behavior seen in previous works [45].

In this model, the magnetic field also shifts the position of the peak, but

again, the predicted shift deviates from what we observe. Although the orders

of magnitude are the same, the theory does not reproduce the values closely

and deviates somewhat from the B2 dependence as shown in Figure 4.16.

The model also fails to capture the observation that low-bias conductivities

are unaffected by the magnetic field. It does, however, agree qualitatively with

what has been observed in other works such as that of Aristone et al.[45] (Fig.

4.17) which have been explained by solving a Boltzman equation.

One must also consider the effect of diamagnetic shifts in the energies E0 and

E ′0 caused by the magnetic field. This is done in Sec. A.1.2 using perturbation

theory and is shown to be small enough to contribute insignificantly to the I-V.

We learn from this section that the dramatic increase in current with mag-

netic field is not simply a result of resonant tunneling in the presence of scat-

tering and inhomogeneities, with the resonant tunneling condition shifted by

the magnetic field as assumed by our model. In our observations there is more

physics involved. We are as yet unable to say what this is, and leave it as an

open question.


0.00

0.500

1.00

1.50

2.00

2.50

3.00

0 0.005 0.01 0.015

Cur

rent

(ar

b.)

Single well bias (V)

0

2

3

4

5

6

7

0.00

2.00

4.00

6.00

8.00

10.0

12.0

14.0

0 5 10 15 20 25 30 35 40

Pea

k bi

as (

mV

)

B2(T2)

Figure 4.14: Simulated d.c. I-V with energy width∆E = 2meV, mobilityµ = 0.2× 105cm2/V-sec. The numbers labels are denote the magnetic field inTesla. Inset: peak position versus magnetic field squared.


0.00

0.500

1.00

1.50

2.00

2.50

3.00

3.50

0 0.005 0.01 0.015

Single well bias (V)

Cur

rent

(ar

b.)

0

2

3

4

5

6

7

Figure 4.15: Simulated d.c. I-V with energy width∆E = 2meV, mobilityµ = 0.5× 105cm2/V-sec. The numbers labels are denote the magnetic field inTesla.


0.00

0.0500

0.100

0.150

0.200

0.250

0.300

0.350

0.400

-10 0 10 20 30 40 50B2(T2)

Ste

p po

sitio

n (V

)

Figure 4.16: Positions of the I-V resonant tunneling feature as a funtion of mag-netic field squared: data (•) and (×) theory: positions in Fig. 4.14 multipliedby the number of wells and lever-arm.


Figure 4.17: Theoretical (left) and experimental (right) I-V’s of a superlatticefrom Aristone et al. [45]. The magnetic fields are 0 (solid line), 2, 4, 6T.

4.4 Photoresponse in a Magnetic Field

The effect of the magnetic field on photon assisted tunneling follows simply

from the preceding observations. The magnetic field broadens the d.c. resonant

tunneling feature, increasing the voltage at which the current saturates. The

photon replicas are consequently broadened and when the width of this feature

comes close to the photon energy, we cannot resolve the d.c. part from the

replica. This phenomenon is vividly illustrated in Figure 4.18. There are very

sharp zero- and one-photon features below 3T, which disappear above 3T. This

magnetic field corresponds to an I-V saturating at about 200meV in Figure 4.11,

which corresponds to an energy of 10meV (taking into account a lever arm of

1.9) at each well; the photon energy of 14meV is comparable.

The zero-photon step positions, shown in Figure 4.19 as a function of mag-


-4.00

-2.00

0.00

2.00

4.00

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

I(µA

)

V(Volts)

0

23

45

Figure 4.18: I-V under irradiation at 3.4THz, with magnetic fields 0, 1, 2, 3,4, 5T as numbered in the plot. The temperature is 100K.


-0.350

-0.300

-0.250

-0.200

-0.150

-0.100

-0.0500

0.00

0 2 4 6 8 10

Pea

k po

sitio

n (V

)

B2(T2)

Figure 4.19: Positions of the I-V resonant tunneling feature (•) and the 1-photon replica (×) at 3.4THz.

netic field, behave in the same was as in the d.c. I-V, proportional to the

square of the magnetic field. This lead us to conclude that the same physics

that changes in d.c. I-V when a magnetic field is applied, is at work here. The

magnetic field does not change the nature of the photon assisted transport, it

simply broadens the I-V that is replicated by the PAT process.

Chapter 5

Conclusion

This dissertation has presented an analysis of terahertz transport in a MBE-

grown superlattice. From the d.c. behavior, we have learned that the device

aberrates from the idealized superlattice: there is band bending within the

structure even before the Stark ladder regime, which has its origin in the tem-

perature dependence of the carrier density.

The photoresponse data has not only reproduced the first works on photon

assisted transport, but it has also shown that there is a subtle interplay be-

tween radiation induced ionization of donors (heating?) and photon assisted

transport at low a.c. powers. The result is a quantitative deviation from the

well-understood Tien-Gordon (TG) theory. At high radiation powers we find a

striking qualitative deviation in our data from the power dependence of higher

order photon replicas prescribed by the theory. If this investigation were to

be continued, it would be helpful to increase the doping level to reduce the

interplay with heating and thus test TG theory more directly at all powers. If

the high power qualitative deviations persist, we would have stronger evidence

that a different theoretical understanding is needed.

66

CHAPTER 5. CONCLUSION 67

The transport measurements in crossed electric and magnetic fields have

revealed a new behavior, different from that published in a number of com-

parable works. We find a magnetic field dependence of current that defies

any of the analyses that have been used successfully in the past to explain

superlattice transport in this configuration. We show this quantitatively by

developing a phenomenological model that captures the essential features of

these past analyses, agrees with their predictions, but does not reproduce our

data. The broadening of the I-V in the presence of a magnetic field is, however,

reproduced quantitatively by our model.

Photoresponse data in crossed electric and magnetic fields show that photon

assisted transport persists in low magnetic fields but is quenched at higher

fields. We can understand this in the context of TG theory, from the fact that

the broadening of the d.c. I-V in a magnetic field makes it impossible to resolve

the photon assited transport features.

The main contribution of this work is to present a phenomenon new to the

literature: an in-plane magnetic field causes a large increase in current in a

weakly coupled superlattice, with no change in the low-bias conductivity. To

understand this, a good approach in future works would be to first identify

experimentally the essential physics behind it and later try to model it pre-

cisely if needed. It would be fruitful to investigate the effect of incrementally

increasing the coupling between wells, and to approach the wide miniband limit

in which other works have been done. This would tell us whether or not this

phenomenon is related to the absence of a miniband. Another instructive exer-

cise would be to check if the results are sensitive to the number of wells in the

superlattice. This should yield information on the relative scales of magnetic

and electric confinement at high magnetic fields, and on whether current is

indeed controled by sequential tunneling rather than by miniband transport.

Appendix A

Numerical Estimates

A.1 Some Superlattice Properties

A.1.1 Level Broadening

It is a good assumption that well widths within the superlattice are not

constant: there is typically a 1 monolayer (2.7A) fluctuation due to growth

limitations. Assuming a particle-in-a-box formula for the energy, we can esti-

mate that this translates into an energy width of

∆E =h2

2mπ2(1/L2

1 − 1/L22). (A.1)

Assuming L1 = 80A and L2 = 82.7A, the level width of each well ∆E is 5meV.

A.1.2 Diamagnetic Shifts

Consider the Schrodinger equation for an electron in a magnetic field B in

the x direction and an external potential V (z)

H =1

2m(p− eA)2 + V (z) (A.2)

68

APPENDIX A. NUMERICAL ESTIMATES 69

where the vector potential is chosen to be A = (0,−zB, 0). Using the substi-

tution z = z0 − z′, multiplying out the first term and eliminating terms in z

gives rise to the form

H =p2

x

2m+

p2y

2m+

e2B2

2mz′2 + V (z′ − z0) (A.3)

with z0 = py/eB. For small enough magnetic fields, we can view this Hamil-

tonian as that of a quantum well structure in the z′ direction, free electron in

the x and y directions with a perturbation represented by the third term.

The question is: how large is the effect of this perturbation on the energy

of a confined state E0 in a quantum well? We can estimate this by using the

simplest particle-in-a-box wave functions

ψ0 = N cos(π

Lz) (A.4)

and

ψ1 = N sin(2π

Lz) (A.5)

ranging from z = −L/2 to z = L/2 with N an appropriate normalization

(which turns out to be the same for both functions). The first order correction

to the energy is

∆E1 =∫

dz|ψ0|2 e2B2

2mz′2 = N

e2B2

2m

∫ L/2

−L/2dz cos2(πz/L)(z2 + z2

0). (A.6)

After evaluating the integral, this simplifies to

∆E1 = 2.5× 10−6eV +p2

y

2m(A.7)

at B = 1T, from which it is clear that the correction is very small compared

to the energy associated with motion in the y direction.


The second order correction is

∆E2 =H2

12

E0 − E1

(A.8)

and

H12 = N∫ L/2

−L/2dz cos(πz/L) sin(2πz/L)

e2B2

2m(z2 − 2z0z + z2

0). (A.9)

Noting that only one of the terms in the brackets is non-zero, evaluating the

integral and then noting that z0 = py/eB, we find that

H12 = (−2× 10−9)eBpy

2m. (A.10)

Using Eq.A.8 we find

∆E2 = 2× 10−5 p2y

2m(A.11)

which shows that the diamagnetic correction is tiny compared to the kinetic

energy. Hence we need not consider it in the I-V model.

A.2 I-V Model

The integrals in section 4.3.3 were evaluated by summing the function in

eq. 4.6 over a grid of 1004 elements. This was chosen so that the widths ∆E

and ∆k correspond to at least seven grid points in order to keep the estimate

reasonably accurate. In practice, the range of k values was chosen so that at the

limits, the corresponding value of the Boltzman factor was about 0.001 times

its maximum value. We can thus be confident that the integral is accurate.

The grid size of 1004 elements is clearly far too large to be summed in its

entirely on a common computer. But that is certainly not necessary. Each

integral was performed only in the vicinity of the peak of the function being


summed, which is easily estimated beforehand. This made the computation

time acceptable.

The Mathcad worksheets are shown in Figs. A.1 and A.2. The I-V is

Iright− Ileft and the I-V in a magnetic field is IrB− IlB. r enumerates bias

points. The plot shows the Boltzman distribution, which dictates the range of

kx and ky values used.

APPENDIX A. NUMERICAL ESTIMATES 72M

agnetic fielde

1.610

19.

hb1.05459

1034

.kb

1.3810

23.

bmax

6

me

9.110

31.

.067.

hb2me

hb 2

2m

e.

µ1

10 5.10 4

∆E

.00035e .

b0

bmax

..V

b.275

τµ

mee

.τ

3.81110

12=

Bb

b1

B

1234567

=V

bias.0064

Vsubband

.05

a50

1010

.

∆k

4m

e

hbτ .

.∆

k2.463

10 7=

dzb130

1010

.

l1∆k

l4.059

108

=fk

2ktypical

2m

e.

.01.

e .

hb 2fk

ktypical.

2.64910 8

=∆

k2.463

10 7=

l2l 2

Nenergies

10r

0N

energies..

δkxBb

eB

b.

dzb

hb.

Ntim

es100

n0

Ntim

es..

EoV

subbande .

0 .

Eopr

Eo.01

e .r10

.kxpm

inktypical

fk .

EopL

rEo

.01e .

r10.

kypmin

kxpmin

kxpmax

ktypicalfk .

NN

times

kypmax

kxpmax

i1

N..

dkpkxpm

axkxpm

in

Ndkp

5.29810 6

=dp

ceil2

∆k

.dkpdp

10=

kxpikxpm

ini

dkp.

∆kxp∆

kypdkp 2

dqdp

∆k

dkp4.65

=kypi

kypmin

idkp.

kxp2ikxpi

2dkbb

ceilδkxB

b

dkpE

opL10

e0.01

=kyp2i

kypi2

1

Figure A.1: Numerical evaluation of the I-V: Mathcad definitions.

APPENDIX A. NUMERICAL ESTIMATES 73T

100kbT

kbT .

E0

.002e .

,.1

e ...

kxmin

ktypicalfk

()

.

i1

Ntim

es..

dkkxm

axkxm

in

Ntim

esdk

=∆

kx∆ky

dk 2kym

inkxm

in

kxikxm

ini

dk.

kx2ikxi

2kxm

axktypical

fk(

).

kyikym

ini

dk.

Norm

lπ

lπ

.∆

kxp∆kyp

∆kx∆

ky.

()

.hb 2

2m

e.

kbT.

.kym

axkxm

axky2i

kyi2

IrightrN

orm

1

Ntim

es

p1

Ntim

es

qm

ax1

pdp

()

()

min

Np

dp(

)(

)

im

ax1

qdq

()

()

min

Nq

dq(

)(

)

j

expEo

hb2me

kx2p.

hb2me

ky2q.

Eopr

hb2me

kxp2i.

hb2me

kyp2j.

∆E

2

.

==

==

.

expkxpi

kxp2

l2 .exp

kypjkyq

2l2 .

.exp

hb2me

kx2p.

hb2me

ky2q.

kbT.

IrBb

r,N

orm

1

Ntim

es

p1

Ntim

es

qm

inN

max

1p

dkbbdp

max

min

Np

dkbbdp

1

im

ax1

qdq

()

()

min

Nq

dq(

)(

)

j

expEo

hb2me

kx2p.

hb2me

ky2q.

Eopr

hb2me

kxp2i.

hb2me

kyp2j.

∆E

2

==

==

.

expkxpi

δkxBb

kxp2

l2 .exp

kypjkyq

2l2 .

.exp

hb2me

kx2p.

hb2me

ky2q.

kbT

IleftrN

orm

1

Ntim

es

p1

Ntim

es

qm

ax1

pdp

()

()

min

Np

dp(

)(

)

im

ax1

qdq

()

()

min

Nq

dq(

)(

)

j

expEo

hb2me

kx2p.

hb2me

ky2q.

EopL

rhb2m

ekxp2i.

hb2me

kyp2j.

∆E

2

.

==

==

.

expkxpi

kxp2

l2 .exp

kypjkyq

2l2 .

.exp

hb2me

kx2p.

hb2me

ky2q.

kbT.

IlBb

r,N

orm

1

Ntim

es

p1

Ntim

es

qm

inN

max

1p

dkbbdp

max

min

Np

dkbbdp

1

im

ax1

qdq

()

()

min

Nq

dq(

)(

)

j

expEo

hb2me

kx2p.

hb2me

ky2q.

EopL

rhb2m

ekxp2i.

hb2me

kyp2j.

∆E

2

.

==

==

.

expkxpi

δkxBb

kxp2

l2 .exp

kypjkyq

2l2 .

.exp

hb2me

kx2p.

hb2me

ky2q.

kbT.

2

Figure A.2: Numerical evaluation of the I-V: integrals.

Appendix B

Cleanroom Process Details

In what follows, the sample preparation procedure is given step by step.

Although this was mostly a standard fabrication, the yield was in practice very

small (1 out of 25 working samples). The reason is simply that we are pushing

the limits of the process and equipment to achieve features on the 1000 A

scale; tolerances become tight and there is ample room for random errors by

imperfect people in an imperfect Cleanroom.

The process is summarized in Figure 3.1. A NiAuGe top contact is deposited

onto the exposed top n+ layer using electron beam lithography and liftoff, and

then annealed (Fig. 3.1 (a) ). The mesa is then defined by a wet etch, masked

by this top contact (Fig. 3.1(b)). The bottom n+ layer is thus exposed and

most of the area directly underneath a narrow line of metal is etched away

leaving an air-bridge between the large leads and the mesa. Another electron

beam lithography and liftoff step is performed to deposit a Ti/Au film onto

this layer to define a bottom contact (Fig. 3.1(c)). We take advantage of the

mesa undercut in the previous step to self-align the bottom contact in such

a way that it does not touch the active region of the superlattice or the top

74

APPENDIX B. CLEANROOM PROCESS DETAILS 75

contact. A reactive ion etch followed by a wet etch masked by this Ti/Au film

isolates the bottom contact from the top contact (Fig. 3.1(d)). The region

masked by the air-bridge survives this etch, but is removed in a subsequent

step with a brief isotropic wet etch. This etch serves the additional purpose

of undercutting further the mesa to reduce its diameter. This leaves us with a

superlattice device with ohmic contacts on either side.

B.1 Material

The material used is a GaAs/AlxGa1−xAs (x=0.3) superlattice grown on a

2-inch s.i.-GaAs wafer by K. D. Maranowski in 1995 in the UCSB MBE lab.

The layer sequence from the surface is:

[7] GaAs n-doped 2e18 cm−3 3000 A


[5b] (AlGaAs n-doped 8e15 cm−3 50 A

[5a] GaAs n-doped 8e15 cm−3 80 A) x 10

[4] AlGaAs n-doped 8e15 cm−3 50 A



[1] smoothing superlattice; undoped

[0] semi-insulating GaAs substrate.

B.2 Sample Clean

The material used was about five years old. In order to prepare it for

lithography, it was cleaned:

• Hot acetone dip; 10 minutes. Rinse in isopropanol and D.I. water.


• Oxygen plasma clean; 5 minutes at 100mT, 300W.

• Oxide removal in HCl : H2O; 1:10 for 1 minute, blow with dry nitrogen.

Some of the top n+ layer was sacrificed.

Note that an ammonium hydroxide dip, normally used in conjunction with

the acid dip, was not used. In this case, an ammonium hydroxide dip appeared

to reduce the yield of electrically active devices for unknown reasons.

B.3 Top Ohmic Contact

The first step is to fabricate the top metal contact which also serves as a

mask for subsequent etches. A natural choice of lithography is electron beam

lithography since we need to define features smaller than the limits of optical

lithography. The most critical part of this step is to produce an air-bridge less

than 1500Awide. This air-bridge needs to be this narrow because we later rely

on the undercut of wet etches to completely remove material from beneath it.

Since we cannot wet etch more than about 1500A, the air-bridge needs to be

correspondingly narrow.

The resist of choice is a bilayer negative resist with a more sensitive the

lower layer.

• Spin on E-beam resist Layer 1: 495 PMMA A.7 at 4000rpm, 30 seconds.

• Bake at 180 oC for 3 minutes

• Spin on E-beam resist Layer 2: 950 PMMA A.5.5 at 4000rpm, 30 seconds.

• Bake at 180 oC for 3 minutes.

•Write pattern. JEOL620, NPGS pattern generation. Typical dose 100 µC;

Typical current: 60nA.

• Develop in MIBK:ISO; 1:3 for one minute or more. Check development

on a microscope every 30 seconds, repeat as necessary.


• Oxide removal in 1M HCl : H2O; 1:10 for 10 seconds. Blow dry with

nitrogen. This also attacks the PMMA resist, so the dip must be limited to a

few seconds.

• Metallization in E-beam evaporator:

Ni/Ge/Au/Ni/Au : 50/350/800/200/1500 A. This is clearly not the canoni-

cal Nickel-Gold-Germanium recipe, but was found empirically to work on small

scales.

• Liftoff in room temperature acetone, at least 4 hours. Rinse with iso-

propanol, D.I. water.

• Dip in 1M HCl : H2O; 1:1 for a few seconds. This is a cleaning step that

proved necessary, most probably because of the scum deposited from air in the

cleanroom. The next step was done immediately. Features larger than about

1 mum did not need this.

• Anneal at 420 oC for 1 minute, or until a slight color change is observed.

No forming gas (again, an unidentified scum formed when forming gas was

used). Check integrity of smallest features under optical microscope. If broken,

start process again. There is no reliable recipe for this!

• Dip in 1M HCl : H2O; 1:1 until next step. Prevents formation of tiny

deposits on Au surface from cleanroom air and microscope environment.

B.4 Mesa Etch

This step defines and undercuts the mesas and exposes the bottom n+ layer

(Fig. 3.1 (b)). Care must be taken not to etch too deep because the bottom

n+, only 3000Athick, must be preserved as much as possible.


nitrogen.


• Etch: 1M Citric acid: 30 % H2O2 : D.I; 1:10:10. Etch rate: 1000 A/minute.

• Rinse in D.I. water, blow dry gently. The air-bridge is thin enough that

the nitrogen stream can mechanically damage it.

• Check etch depth with Dektak, repeat as necessary.

B.5 Bottom Contact

The bottom contact is just a large metal pad deposited directly onto the

mesas but not onto the air-bridge or leads connecting to the top contact. Be-

cause of the mesa undercut, the metal actually deposits onto the exposed bot-

tom n+ without shorting out the sides of the mesas (Fig. 3.1 (c)).

The metal used for the bottom contact is Ti/Au deposited directly onto

the bottom n+ layer and never annealed. It is important to avoid annealing

at this point to preserve the air-bridges and the shape of both contacts which

are separated by less than 1000A. This Ti/Au will never make ohmic contact

to the bottom n+; it acts purely as an etch mask. It is, however, thin enough

that an indium blob can later ’spike’ through it and make the necessary ohmic

contact.

• Spin on E-beam resist: 495 PMMA A.7 at 4000rpm, 30 seconds. A single

layer resist is sufficient, since this feature is big.

• Bake at 180 oC for 3 minutes.

• Write pattern. JEOL620, NPGS pattern generation. This is a self-aligned

contact shadowed by the existing mesa. Typical dose 100 µC.


nitrogen.

• Metallization in E-beam evaporator: Ti/Au : 60/1000A. Ti deposited

at 0.3A/sec to minimize strain; this is critical, since this layer will eventually


overhang a little beneath the air-bridges, and any curling up of the metal will

short the devices.

• Liftoff: room temperature acetone, at least 10 minutes.

• Rinse in isopropanol, then H2O, then blow dry very gently. It is very easy

to mechanically damage the air-bridges with the nitrogen stream.

• Clean in 1M HCl : H2O; 1:1 for as long as one is waiting for the next step.

Leaving the sample in air at this point results in an exposed surface sufficiently

dirty to make the following reactive ion etch ineffective.

B.6 Bottom Contact Isolation

At this point, the devices are complete, but the only thing preventing them

from being used is the fact that the continuous bottom n+ layer shorts out

the devices. To break it, we etch it away using a reactive ion etch. Since

this etch is essentially vertical, it does not attack the mesas protected by the

top contacts. Unfortunately, some of the n+ beneath the air-bridges is also

protected and need to be subsequently removed with a very brief wet etch.

The wet etching must be minimized because it also undercuts the mesas with

the risk of removing the necessary n+ region immediately surrounding them.

• Oxide removal in 1M HCl : H2O; 1:10 for 10 seconds.

• Reactive Ion Etch: SiCl4 at 10mT, 60W, 3 minutes. Etch rate 1000

A/minute. A very unpredictable etch rate in PlasmaTherm etching system

(RIE 5). Repeat the oxide removal and etch until 3000Ais etched as measured

on Dektak. The laser interference depth monitor was useless in our system.

• Oxide/residue removal in 1M HCl : H2O; 1:10 for 10 seconds. If there

are signs of tougher residues, a dip in more concentrated HCl helps.

• Air-bridge definition etch: 1M Citric acid:30% H2O2 : D.I; 1:10:10, 1


minute. Etch rate: 1000 A/minute.

• Rinse in D.I. water. Look at dummy structures or large contact leads

with Dektak to determine degree of undercut. If undercut more than 1000 A,

then stop, otherwise repeat.

B.7 Wire Bonding

The sample was mounted onto a standard fiberglass DIP chip carrier with

adhesive copper tape and wires were soldered onto the contact pads.

• Top contacts: the standard Au wire bonder proved inadequate for the

50µm top contact bond pads. Instead, a fine blob on indium was soldered onto

each bond pad, and a gold wire from the pin of a chip carrier was smashed into

the indium on each bond pad. This proved to be a very robust contact.

• For the bottom contact, the same process was used. In this case, the

indium was heated a little more in order to spike into the n+ layer.


Figure B.1: Gold wires are pressed into indium blobs to connect the sample tooutside electronics.

Appendix C

Experimental Details

C.1 Cryogenics

C.1.1 Storage Dewar Setup

The d.c. and photoresponse measurements at lower powers were done using

a sample stick inside a stainless steel vacuum can (”dunker”) filled with about

a Torr of helium exchange gas. This can was then inserted inside a helium

storage dewar for 4K operation and a nitrogen dewar for 77K operation. A 1/4

inch internally polished stainless steel light pipe was built in to the can so that

free electron laser (FEL) radiation focused into it was carried down the pipe.

At the end of the pipe, about 4mm from the sample, a parabolic mirror was

used to focus the radiation onto the sample.

The advantage of this setup was that it was extremely stable both thermally

and mechanically, and required relatively little maintenance in time and helium,

so that more effort could be put into babysitting the FEL.

The sample stick was custom made with 1 1/4 inch stainless steel tube so

82

APPENDIX C. EXPERIMENTAL DETAILS 83

that it was rigid enough to hold the alignment with the light pipe. A heater

was placed on it, but was not needed. A Lakeshore carbon-glass thermometer

was used to determine the sample temperature.

C.1.2 Magnetic Cryostat

The higher power photoresponse measurements, and the magnetic filed ex-

periments were done in an Oxford Spectromag cryostat with a variable tem-

perature insert (VTI) for temperatures down to 1.5K. The same sample stick

above was used. This allowed FEL radiation to be focused onto the sample

through windows in the cryostat. We used a custom built aluminum off-axis

parabolic mirror, 4 inch in diameter, to focus the FEL beam.

For temperature control we relied on the internal heater of the VTI, which

proved to be adequate.

C.2 Electronics

All of the electrical wiring in the sample stick was done using coaxial cable

to allow for the megahertz bandwidth needed to do the pulsed measurements.

The only piece of electronics inside the sick itself was a precision 1K metal film

resistor next to the sample across which the voltage was measured to determine

the current.

The voltage was supplied by a battery connected to a precision potentiome-

ter driven by a computer-interfaced stepper motor. The bias voltage and the

d.c. voltage across the 1K resistor were measured with Keithley 2000 multime-

ters set to filter for about 1/2 second. An SRS 560 amplifier was placed on the

cryostat, right next to the sample stick to amplify the measured voltage before

APPENDIX C. EXPERIMENTAL DETAILS 84

it was sent down a 2m coaxial cable.

The a.c. response was measured by first filtering the incoming voltage across

the resistor with a low-pass filter and then amplifying it with another SRS 560

amplifier which fed into an Hewlett Packard oscilloscope synchronized with the

FEL. The scope trace was digitized and the voltages before and during each

FEL pulse were extracted using an appropriately made LabView program. The

greatest difficulty in this step was to eliminate a 1.25MHz pickup omnipresent

in the FEL user lab. The only way was to spend hours optimizing the grounding

configuration (which itself had to be changed from day to day). Although the

FEL was capable of sending out pulses at 2.5Hz, all the measurements done here

were done at 0.75Hz because of the speed limitations of the old HP oscilloscope.

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