The Quantum Hall Effect

Michael Richardson

In 1985, Klaus von Klitzing was awarded the Nobel Prize for his discovery of the quantized Hall effect. The

quantum Hall effect has provided an amazingly accurate method for calibrating resistance. Some of the

successful explanations of the effect are summarized in the following.

I. INTRODUCTION

The classical Hall effect was discovered in 1879 by

Edwin Hall. He correctly speculated that the charge

carriers in a wire would be deflected to one side of the

wire if a magnetic field were set up perpendicular to

the flow of the current. In other words, the magnetic

field led to a transverse voltage in the wire. This

transverse voltage could be measured experimentally

and was seen to increase as the magnetic field

increased.

It is possible to demonstrate the same effect in a

two dimensional electron gas. The electrons are

trapped at the interface between a metal and a

semiconductor. Then a current is established in one

direction and a magnetic field is applied perpendicular

to the electron gas.

In 1980, von Klitzing did experiments to test the

Hall effect on a two dimensional electron gas at very

low temperature and high magnetic field. He found

that the Hall conductivity (current divided by the

transverse voltage) had plateaus of constant

conductivity as the magnetic field was varied. More

astonishingly, the values of the conductivity at these

steps were integer multiples of e2/h that were measured

to an accuracy of one part in 10 million.

Ever since, theorists have attempted to explain the

phenomenon. A fairly simple explanation involving

Landau levels can be made for an ideal system. Yet

more complicated explanations are needed to explain

real systems.

II. A SIMPLE APPROACH

If the electron gas is subjected to a strong magnetic

field, then the electron orbitals split into Landau

levels. The number of orbitals in each level is

𝐷 = (2𝜋𝑒𝐵/ℏ𝑐)(𝐿/2𝜋)2 = 𝑒𝐵𝐿2/ℎ𝑐.

If the voltage is adjusted so that the Fermi level of the

electron gas falls right between two Landau levels,

then the total number of electrons in the electron gas is

given by

𝑁 = 𝑠𝐷

One can rearrange these equations to determine that

𝜎 = 𝑠𝑒2/ℎ.

No elastic collisions are possible from one state to

another state in the same Landau level because all the

states in the level are filled. Because the magnetic

field is strong and the temperature is low, ℏ𝜔𝑐 ≫ 𝑘𝐵𝑇.

Therefore, the electrons are unlikely to absorb the

necessary energy to move to the next Landau level

from a phonon. As a result, the conductivity remains

quantized while the Fermi energy is between the

Landau levels.

III. LAUGHLIN’S EXPLANATION

According to the above explanation, the occurrence

of the quantum Hall effect would not be expected for

systems with partially filled Landau levels. Yet it does

occur in such systems.

Robert Laughlin used the principle of gauge

invariance to explain the quantization of the

conductance. He imagined a 2D electron system bent

into a cylinder so that the current moves in a loop.

The surface of the cylinder experiences a strong

magnetic field that is everywhere normal to the

cylinder. Now consider a magnetic flux through the

cylinder. Any change in this flux will create an

electromotive force that will cause the charges to

move around the loop and, due to the magnetic field,

deflect to one side of the cylinder. If the change in the

magnetic flux is equal to an integral multiple of the

elementary quantum of magnetic flux ℎ𝑐/𝑒, then the

Hamiltonian describing the system will remain

unchanged. This is because, according to the

Aharonov-Bohm principle, the system is gauge

invariant under flux changes of integral multiples of

ℎ𝑐/𝑒. To calculate the Hall conductance, one divides the

transverse current by the Hall (transverse) voltage.

𝜎 = 𝐼/𝑉𝐻

The current is found by using the electromagnetic

relationships

𝜕𝑈

𝜕𝑡= −𝑉𝐻𝐼 = 𝐼𝑐

𝜕𝜑

𝜕𝑡, 𝐼 = 𝑐

𝛿𝑈

𝛿𝜑

The energy 𝛿𝑈 is equal to 𝑁𝑒𝑉𝐻, where N is the

number of electrons that are moved to the other side of

the cylinder due to the emf from the change in flux.

The change in flux is the elementary quantum of

magnetic flux. So

𝐼 = 𝑐𝑁𝑒𝑉𝐻ℎ𝑐/𝑒

Given that the Hall conductance is 𝜎 = 𝐼/𝑉𝐻, the

above equation yields

𝜎 = 𝑁𝑒2

ℎ

Because the system returns to its original state after the

flux change, one assumes that if the system were to

undergo the same flux change again, then the same

number of electrons would be transferred. Yet

quantum mechanically, two systems in the same state

do not have to produce exactly the same results under

the same measurement. Therefore, something more

must ensure that the average number of electrons

transferred is also an integer.

IV. TOPOLOGICAL QUANTUM NUMBERS

After the system undergoes the cycle of changing

its flux by ℎ𝑐/𝑒, it returns to the same state. Its wave

function is the same as before, except that the phase of

the wave function may be different. This accumulated

phase is known as the Berry phase. When it is

compared to the accumulation of phase by vectors

undergoing parallel transport on a curved surface, the

Hall conductance can be compared to the curvature of

the surface. In other words, just as the curvature of a

surface is linked to the accumulation of phase in

parallel transport, so is the Hall conductance linked to

the Berry phase that arises under the adiabatic cycle of

the wave function.

Doing so allows a formula from topology to be

used in order to explain the integral quantum Hall

effect. The formula, by Gauss and Charles Bonnet, is

1

2𝜋 𝐾𝑑𝐴 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

𝑆

where the integral is over a surface with no boundaries

and K is the local curvature of the surface.

Shiing-shen Chern generalized this formula to the

geometry of eigenstates that can be parameterized by

variables over a toroidal surface. The “adiabatic

curvature” is found by taking the limit of the Berry

phase mismatch divided by the loop as the loop size

goes to zero.

This formula shows that as long as the adiabatic

curvature is the same for each cycle of increasing

magnetic flux, the same integer will be produced every

time. This information, combined with Laughlin’s

explanation demonstrates why the Hall conductance is

quantized in integer multiples of 𝑒2/ℎ.

V. CONCLUSION

The incredible precision with which one may

measure the value of 𝑒2/ℎ has allowed for a new

standard for resistance (ℎ/𝑒2). The quantum Hall

effect also allows a precise calculation of the fine

structure constant. The theory behind the effect is still

being researched by many scientists.

References

C. Kittel, Introduction to Solid State Physics. 498-503

(2005)

J. Avron, D. Osadchy, R. Seiler, Physics Today. 56, 8

(2003)

http://en.wikipedia.org/wiki/Quantum_Hall_effect

http://nobelprize.org/nobel_prizes/physics/

laureates/1985/

Left-handed Materials

Josh Holt

March 5, 2008

1 Theoretical Introduction

In the 1960’s, Victor Veselago investigated the optical repercussions of a hypothetical mediumwith simultaneously negative and real electric permittivity ε and magnetic permeability µ.11

At a flat interface of his conjectured material with “normal” material, the direction of wavepropagation is opposite to the Poynting vector of energy propagation; thus electromagneticwaves obey the left-hand, instead of the right-hand, rule. Veselago coined such media ex-hibiting this effect left-handed materials (LHMs). One must then choose the negative root

of the index of refraction given by n = ±√

εµ/ε0µ0 (where the naught subscript indicate

respective free-space quantities). Although the ensuing consequences of a negative refractiveindex, including negative refraction, reverse Doppler shift and reversal of Cherenkov radia-tion, are peculiar, they do not violate any fundamental physical laws. For example, negativerefraction allows creation of a so-called a “Veselago lens,” a flat, parallel slab of LHM insidea right-handed material (RHM) with the condition that both media have the same isotropicrefractive index and the same impedance. Rays from a point source impinging on a Veselagolens would be refocused to a point on the opposite side of the material.

Interest in LHMs irrupted after the work by Pendry which argued that the Veselagolens is a “perfect lens” in the sense that it gives a perfect image of the point source.5 Hisstatement is based upon the observation that the evanescent waves of a form exp(ikyy−κx)that usually decay in the near-field region are amplified by the LHM. Pendry claimed that theamplified evanescent waves restore a perfect image in both the near-field and far-field regions,but he did not present a solution in coordinate space. That the cherished diffraction limitcould be violated did not set well with many physicists. A real-space solution was offered byZiolkowski and Heyman,12 revealing that Pendry’s solution diverges exponentially at eachpoint of a 3D domain near the focus, just where the fields of the evenescent waves increase dueto amplification. Simultaneous attacks against so-called “superlensing” followed from manygroups.2,8,3 Eventually the debate settled until it was generally agreed that superlensing canbe found in the near-field regime while resolution in the far-field remains on the order ofwavelength.6 Even so, implications of negative refraction and the possibility of beating thediffraction limit could not be ignored. The theoretical controversy amplified the need toreveal new experimental evidence of negative refraction.

1

Left-handed Materials Solid State II

2 Experimental Evidence

Since large magnetic response, in general, and a negative permeability at optical frequen-cies, in particular, do not occur in natural materials, metamaterials have been contrived tofacilitate ε < 0 and µ < 0 simultaneously.

The earliest probe for negative refraction used a two-dimensional array of split-ring res-onators (which couple to and alter the local magnetic field) to produce negative µ over aparticular frequency region and wire elements to produce negative ε, showing microwavetransmission at a positive from normal angle using a teflon prism and transmission at a neg-ative angle using a manufactured LHM metamaterial prism.10 An immediate debate ques-tioning the veracity of the results arose1 but, ultimately, the effects of negative refractionwere repeatably verified.

Permeability, permittivity, and refractive index are bulk, effective medium properties.Although metamaterial consist of discrete scattering elements, it may be approximated as aneffective medium for wavelengths that are larger than the unit cell size. This approximationis analogous to the effect of the periodic Bloch potential on band electrons in condensedmatter. The advantage in these metamaterials, of course, is that they can be scaled to anyparticular wavelength of light, whereas the dispersion relation of any solid-state electronicdevice is effectively pinned to a fixed lattice spacing. Photonic crystals and structures havenaturally followed to implement negative refraction within optical frequencies. It was shownthat a dielectric photonic crystal made of non-magnetic materials can behave as a LHM withnegative ε and µ if it has a negative group velocity in the vicinity of the Γ-point of the secondBrillouin zone both theoretically7 and experimentally.4,9 However, the exsistance of surfaceplasmons in photonic crystals make interpretation of experiments and simulations difficult.

3 Conclusions

Physical understanding since Veselago’s initial theory has bloomed into applications proba-bly beyond what he originally imagined (see, for example, Ref. 12). Applications founded onnegative refraction include beam steerers, modulators, band-pass filters, and lenses permit-ting subwavelength point source focusing. The primary players which have stakes in LHMtechnology include the telecom industry (which relies on the ultrafast switching of the optical1.5 µm wavelength for high-throughput communication), data storage and imaging technol-ogy (who are continually trying to work around the diffraction limit). The non-intuitiveeffects of LHMs have driven research in metamaterial and led to their much improved un-derstanding.

References

[1] N. Garcia and M. Nieto-Vesperinas. Is there an experimental verif ication of a negativeindex of refraction yet? Opt. Lett., 27:885–887, 2002.

Left-handed Materials Solid State II

[2] N. Garcia and M. Nieto-Vesperinas. Left-handed materials do not make a perfect lens.Phys. Rev. Lett., 88:207403, 2002.

[3] F. D. M. Haldane. Electromagnetic surface modes at interfaces with negative refractiveindex make a ”not-quite-perfect” lens. cond-mat/0206420, 2002.

[4] P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar. Negativerefraction and left-handed electromagnetism in microwave photonic crystals. Phys. Rev.Lett., 92:127401, 2004.

[5] J. B. Pendry. Negative refraction make a perfect lens. Phys. Rev. Lett., 85:3966–3969,2000.

[6] V. A. Podolskiv and E. E. Narimanov. Near-sighted superlens. Opt. Lett., 30:75–77,2005.

[7] A. L. Pokrovsky and A. L. Efros. Sign of refractive index and group velocity in left-handed media. Solid State Comm., 124:283–287, 2002.

[8] A. L. Pokrovsky and A. L. Efros. Diffraction theory and focusing of light by a slab ofleft-handed material. Physica B, 338:333–337, 2003.

[9] Vladimir M. Shalaev, Wenshan Cai, Uday K. Chettiar, Hsiao-Kuan Yuan, Andrey K.Sarychev, Vladimir P . Drachev, , and Alexander V. Kildishev. Negative index ofrefraction in optical metamaterials. Opt. Lett., 30:3356–3358, 2005.

[10] R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a negative indexof refraction. Science, 292:77–79, 2001.

[11] V. G. Veselago. Properties of materials having simultaneously negative values of thedielectric(ε) and magnetic (µ) susceptibilities. Sov. Phys.-Solid State, 85:3966–3969,1967.

[12] R. W. Ziolkowski and E. Heyman. Wave propagation in media having negative permit-tivity and permeability. Phys. Rev. E, 64:056625, 2001.

1

1

Graphene: Carbon, Now in Stunning 2D

Jon Paul Johnson, University of Utah, Physics 5520

If thin is in, graphene has a bright future as the thinnest stable material known. If you were to remove a single atomic layer from graphite, one of the more traditionally-built forms of carbon, you would have graphene—a sheet of carbon atoms in a honeycomb array with the thickness of a single carbon atom. The discovery of stable graphene sheets in 2004 came as a surprise1, since it was assumed that it was energetically unfavorable for carbon to be in this configuration and not rolled up in a carbon nanotube, a fullerene, or some other more stable three-dimensional folded shape. Now this intriguing material with its predicted bizarre electrical properties is accessible, which has attracted the attention of electrical engineers looking for a replacement for silicon transistor channels as the limits of that material are expected to be reached within a few decades. Graphene also provides an unlikely sandbox to study quantum effects because its electronic structure yields charge carriers that are massless fermions. Graphene was first discovered by a group at the University of Manchester which, in an effort to study thin sheets of graphite, presumably got more than they bargained for when they discovered the presence of individual flakes of graphene in their samples. Sample preparation was (and is) rather more hunter-gatherer than agricultural in nature: “micromechanical cleavage” of a piece of graphite involves either peeling layers off of it with adhesive tape or drawing with it on a solid SiO2 surface. Then a search is performed with an optical microscope to find the few thin graphene flakes scattered

among the forest of thicker graphite flakes. The thin flakes have a slightly different appearance from the substrate, and the contrast is maximized for certain SiO2 thicknesses. Much of the interest in graphene comes from the electrical dispersion relation of electron waves in a honeycomb lattice, which at the Fermi level (also known as the neutrality point in graphene) is gapless and linear. See Figure 1. This had been predicted long before stable graphene was discovered2,3. Sp2-hybridized bonds connect the carbon atoms in the lattice, with the last electron per carbon almost completely delocalized. Since effective mass depends on the curvature of the dispersion relation, its linearity at the Fermi level implies charge carriers with zero effective mass, and these carriers are quasiparticles that behave as relativistic particles with a speed that is lower than the normal speed of light by about a factor of 300. This enhances some quantum electrodynamic effects, which depend on the

Figure 1. The dispersion relation for graphene.

2

2

inverse of the speed of the particles involved, to the point that they are measurable even at room temperature. Specifically, the quantum Hall effect has been observed in graphene at room temperature. The electron transport properties graphene make it a candidate as a conductive channel in electronic devices. Graphene is conductive without doping, and its carriers exhibit very high mobility and even ballistic transport on small enough scales at room temperature. The carrier velocity (~106 m/s) and the fact that an applied electric field in either direction creates charge carriers could make graphene useful for ultra fast-switching field effect transistors. One drawback for the use of graphene in a more traditional transistor design is actually the lack of a gap at the neutrality point, which creates charge carriers for conduction regardless of whether or not an electric field is applied to the sheet. One way round this is to artificially create a semiconducting gap by introducing quantum confinement effects on a sheet of graphene. Limiting one of the two remaining spatial dimensions by creating ribbons of graphene4 could open a gap that would make graphene suitable as an alternative to silicon as the channel material in field effect transistors. Another way a semiconducting gap could be manufactured is by using the lattice mismatch between graphene and the crystal used as a seed to grow graphene epitaxially (for example, 5). The discovery of graphene nicely rounds out the collection of available forms of atomically thin carbon, taking its place with 0D fullerenes and 1D nanotubes. Graphene has the potential to make waves in electronic device design by swapping massless wavelike quasiparticles for the massive

electrons and holes used to carry charge in today’s devices, but first there must be advances made in epitaxial growth of this intriguing material6. Without a way to create graphene layers compatible with the parallel processing techniques used by the semiconductor industry, graphene is likely to remain a condensed matter research tool—though a very interesting and, as of now, unique one. 1. A. K. Geim & K. S. Novoselov, The rise of

graphene. Nature Materials 6, 183-191 (2007).

2. P. R. Wallace, The band theory of

graphite, Phys. Rev. 71, 622-634 (1947). 3. J. C. Slonczewski & P. R. Weiss, Band

structure of graphite. Phys. Rev. 109, 272-279 (1958).

4. L. Brey & H. A. Fertig, Electronic states of

graphene nanoribbons. Phys. Rev. B 73, 235411 (2006).

5. S.Y. Zhou et al., Substrate-induced band

gap opening in epitaxial graphene. Nature Materials 6, 770-775 (2007).

6. A. K. Geim & A. H. MacDonald,

Graphene: Exploring carbon flatland. Physics Today 60, 35-41 (2007).

March 5, 2008

Living Green With Thermoelectricity

Ben MangumDepartment of Physics, University of Utah

Thermoelectric devices capable of converting heat directly into electricity and thermoelectriccoolers, which do not contain any moving parts or ozone depleting gases may become a major playerin reducing carbon footprints. While thermoelectric devices have traditionally been plagued by lowefficiencies (ZT < 1), recent technological advances allowed for the fabrication of new nanostructuredmaterials with much higher figures of merit (ZT ∼ 3). Such advances are demonstrated in materialsthat maintain an high electric conductivity and low thermal conductivity by minimizing phononiccontributions. By introducing inclusions at various length scales, phonon contributions can beminimized. The development of high-efficiency thermoelectric devices will play an increasinglylarger role in society as environmental costs of current technologies are considered more carefully.

I. INTRODUCTION

For almost two centuries now, scientists have possessedboth the understanding and the ability to make gen-erators capable of converting heat directly to electric-ity and refrigerators with no moving parts to wear out.Thomas Seebeck discovered in 1823 that two dissimi-lar metals joined together with a temperature gradientacross the junction was capable of deflecting a compassneedle. With later clarification from Ørsted, this discov-ery meant that such a simple device was capable of gen-erating ‘thermoelectricity’ [1]. Working independentlyin 1834 Jean Peltier discovered that applying a voltageacross such a junction was capable of establishing a tem-perature gradient across the junction [4]. Lord Kelvin(William Thompson) described the heating or cooling ofa current-carrying conductor with a temperature gradi-ent in 1851, showing that as some metals exhibit cool-ing while others produce heat as current flows from highto low potential with an applied temperature gradient.These reversible processes (the Seebeck effect, the Peltiereffect, and the Thompson effect) are collectively knownas the thermoelectric effect [1].

By 1838 Heinrich Lenz had demonstrated the use of athermoelectric (TE) device; by placing a drop of waterbetween a bismuth wire and an antimony wire and ap-plying a current, the water would freeze and then meltby reversing the direction of the current [3]. In the 1950’sa renewed interest in the thermoelectric effect arose as aconsequence of the discovery that doped semiconductorswere much exhibited a large thermoelectric effect, thusmuch of the early semiconductor research was in pursuitof thermoelectric refrigerators [3]. Despite the fact thatthese thermoelectric technologies are now centuries old,there has been very little, or at least very slow, progressin putting TE devices to practical use. Currently, dueto low efficiencies TE devices are used for niche mar-kets such as laser diode cooling, and power generation forspace bound devices like the Voyager deep space probe[2]. However, with a new arsenal of tools and techniques,coupled with the green awakening sweeping the globe,thermoelectric technologies are ripe for more investiga-tion and implementation into everyday living.

Heat Source

Cool Side

N P

Cooled Surface

Dissipated Heat

N P

(a) (b)

FIG. 1: Schematic of thermoelectric (TE) devices. Panel (a)shows a TE generator; as heat flows from a heat source to aheat sink, heat carriers move along the temperature gradient.Utilizing both n-doped and p-doped semiconductors for thejunctions the heat carriers are also charge carriers and estab-lish a potential difference across the device capable of drivinga load. This is an example of a Seebeck generator. Panel(b) shows a Peltier cooler, and is essentially the converse ofthe effect just described; an applied voltage will preferentiallymove heat carriers to one side of the device establishing atemperature gradient.

II. THEORY

A schematic of both a TE cooler and a TE genera-tor are shown in Fig. 1. When n-doped and p-dopedsemiconductors are joined to both a heat source and aheat sink by metal contacts, thermal carriers (electronsor holes) move from hot to cold. As seen in Fig. 1(a)this motion of thermal, but also charge, carriers createsa potential difference and thus a current in the materialis created as indicated by the arrows in the figure. Inlike manner a Peltier (TE) cooler is shown in Fig. 1(b).An applied voltage drives the motion of charge carriersas indicated by the arrows, while this net flow of carri-ers from one side of the device to the other establishes athermal gradient.

The efficiency of any device is thermodynamicallylimited to a maximum efficiency (Carnot efficiency) ofTc/(Th−Tc) for refrigerators. Modern compressor based

2

refrigeration systems can reach 30-90% of Carnot effi-ciency, thus if TE devices could reach 30% Carnot ef-ficiency, they could compete with modern compressorbased refrigeration technologies. The efficiency of ther-moelectric devices is often characterized by a dimension-less figure of merit ZT, where Z has units of ◦C−1 and Tis the average temperature. Z is a function of the thermalconductivity (κ), the electrical coductivity (σ), and theSeebeck coefficient (S):

Z =S2σ

κ(1)

The Seebeck coefficient (S) is simply a measure of amaterial’s ability to establish a voltage difference in re-sponse to a given temperature difference: S = ∆V/∆T .An examination of metals and degenerate semiconduc-tors yields [2]:

S =8π2k2

b

3e~2m∗ T

( π

3n

)2/3

(2)

where n is the carrier concentration, and m∗ is the effec-tive mass of the carrier [2]. It must also be borne in mindthat both the electrical conductivity and the thermal con-ductivity are also functions of carrier concentration; σalso being a function of the electric charge (e) and mo-bility (µ), σ = neµ. The thermal conductivity has con-tributions from the motion of electrons and holes, but isalso influenced by phonons in the lattice, thus κ = κe+κl,with κe = neµLT , where L is the Lorentz factor. Notingthat σ and κ are directly proportional to the carrier con-centration n, but that S ∝ n−2/3 the optimum carrierconcentration can then be found. This sensitive balanceof carrier concentration effects, indicate then that met-als, although having high electrical conductivity makepoor thermoelectric materials due to a low Seebeck co-efficient and high thermal conductivity. On the otherhand, insulators, while having large Seebeck coefficientsand small electrical contributions to thermal conductiv-ity, have low electrical conductivities also making thempoor TE materials [3]. The optimum balance betweenSeebeck coefficient and conductivities is typically struckwith highly doped semiconductors: n ∼ 1019cm−3 [2].

III. RECENT ADVANCEMENTS

Optimizing the concentration of charge carriers in amaterial is now really the task at hand for determin-

ing the efficiency of a material for potential use as aTE device, however, to make use of quantum mechan-ical predictions, firstly the electronic band structure ofthe material must be known [4]. The inability to predictcrystal structures is the major bottleneck in predictingwhich new materials might be suitable for TE devices.Recent progress in x-ray diffraction techniques, the abil-ity to carefully grow new materials through the use ofnano-wires and thin film deposition, along with modernadvances in computing technologies all increase the like-lihood of finding new and better TE materials [4].

Theoretical predictions suggest that the TE efficien-cies can be improved by quantum confinement of theelectron charge carriers. In a quantum confined struc-ture, high confinement and low dimensionality leads tonarrow electronic energy bands. This equates to higheffective masses and thus large Seebeck coefficients [2].Being able to create a ‘phonon glass’ while maintainingan ‘electron crystal’ is seen as another key to creatinghigh ZT materials. The thermal conductivity contribu-tion due to phonons can be reduced by increased phononscattering accomplished by a variety of methods such asscattering phonons within the unit cell by creating pointdefects or alloying, as well as scattering phonons at in-terfaces through the use of multiphase composites withnanometer scale structuring [2]. To date, the most effi-cient TE materials are made from superlattice nanowireswith a ZT of 2.5 - 3 [5].

IV. CONCLUSIONS

Despite relatively low efficiencies of even modern ther-moelectric devices, great improvements have been madein recent years. Several years ago the most efficient de-vices had a figure of merit only as high as ZT ∼ 1,whereas today nanostructured materials are rated as highas ZT ∼ 3. It is predicted that the heretofore elusive bar-rier of ZT > 4 will soon be broken, thus allowing ther-moelectric coolers to compete directly with current com-pressor based refrigeration technologies. The optimiza-tion of carrier concentrations is certainly vital to creatinghigh ZT devices, but the next major advances will likelycome from finding ways to maintain an ’electronic crystal’while simultaneously being a ’phonon glass.’ In an erawhere the environmental cost of any technology or deviceis become ever more scrutinized, thermoelectric powergeneration is also poised to move out of its niche marketas ever more efficient TE materials are made available.

[1] http://en.wikipedia.org/wiki/Thermoelectric effect[2] G.J. Snyder, and E.S. Toberer, “Complex thermoelectric

materials,” Nature Mater. 7, 105–114 (2008).[3] G. Mahan, B. Sales, J. Sharp, “Thermoelectric Materials:

New Approaches to an Old Problem,” Phys. Today, March

1997 pp. 42–47[4] F.J. DiSalvo, “Thermoelectric Cooling and Power Gener-

ation,” Science, 285, 703–706 (1999).[5] K. Walter, https://www.llnl.gov/str/May07/Williamson.html

The Quantum Cascade Laser

February 27, 2008

Nick Borys

Introduction

The quantum cascade laser (QCL) is a novel semiconductor laser device which allows for

the creation of laser sources in a broad wavelength range of approximately 1µm up

through 30µm (near-infrared to terahertz radiation). This wavelength range is

exceptionally important in chemical detection due to the majority of vibrational and

rotational energy levels of molecules within this energy range. Consequently, QCLs have

direct applications in environmental analysis, trace detection, and defense and security

technology. Further, QCLs are the first reliable and low-cost laser source in this

wavelength ranges since material stability and processing complications make the regime

hard to access with conventional semiconductor diode lasers [1].

Although the QCL is based off of semiconductor materials, it is fundamentally different

from conventional diode lasers. A diode laser relies upon the recombination of an

electron in the conduction band and a hole in the valance band within the semiconductor-

based active region. Consequently, the wavelength of a diode laser is defined and limited

by the material dependent bandgap energy. On the other hand, a QCL is based on a

layering of semiconductor materials such that layers of neighboring energy wells and

barriers are created (Figure 1). The physical thickness of the wells and barriers define the

energy levels of the laser. So, rather than relying on the material properties alone, the

wavelength of the QCL can be tuned simply by modifying the thickness of the layers in

the active region consequently opening up the broad wavelength regime of the QCL [1].

Injector Active Region

large DOS

small DOS

n=3

n=2

n=1

distance

energy

n=3

n=2

n=1

g

g

Injector Active Region

large DOS

small DOS

large DOS

small DOS

n=3

n=2

n=1

distance

energy

n=3

n=2

n=1

g

g

Figure 1 Energy diagram of a QCL

Technical Description of the Quantum Cascade Laser The QCL was first demonstrated in 1994 by a group at Bell Labs that consisted of

Federico Capasso, Claire Gmachi, Deborah Sivco, Alfred Cho, Jerome Faist, Carol

Sirtori, and Albert Hutchinson [2]. As mentioned above, it is principally different than a

semiconductor diode laser in that the QCL operates with electrons cascading through a

series of potential wells that make up the conduction band. Since the QCL only involves

conduction electrons, and not valance band holes, it is referred to as a unipolar laser [1].

The QCL consists of two fundamental regions (Figure 1): the injector and the active

region. Electrons are injected from the injector ground state (dashed line labeled “g” in

Figure 1) into the n=3 state of the active region. The injected electrons then radiatively

relax into the n=2 state and quickly, further relax into the n=1 state. From the n=1 state,

the electrons can then tunnel into another injector region and leave the active region, thus

completing the fundamental light emission process of the QCL [1, 2].

The ground-state of the injector region combined with the three energy levels in the

active region essentially comprises a four-level laser system. Electrons can efficiently

tunnel through the injector region into the n=3 state of the active layer. The scattering

rate of n=3 state to the n=2 state is quite large (several picoseconds). However, the

scattering rate from the n=2 state to the n=1 state is an order of magnitude smaller (~0.3

ps). This arrangement of scattering rates accomplished by engineering the well and

barrier widths such that there is minimal wavefunction overlap of the n=3 and n=2 states

and maximal overlap between the n=2 and n=1 states. The net consequence of the

scattering rates is that the n=3 state remains populated by electrons from the injector

ground state while the n=2 state is quickly depopulated resulting in population inversion

between the two states – a requirement for laser emission [1, 2].

Additionally, the injector region is engineered so that it promotes population inversion

between the two states in the active region. The wells and barriers of the injector region

are designed such that the ground-state energy is very close to the n=3 energy level of the

active region. However, this equality is only for electrons being injected. Notice, in

Figure 1, the n=3 energy level of the active region falls in an energy range where the

injector has a large density of states. On the right-hand side, however, the n=3 energy

level falls in an energy range of the injector region that has a low density of states. This

engineered configuration helps prohibit electron tunneling into the subsequent injector

region directly from the n=3 state thus making it more likely the injected electrons will

radiatively relax into the n=2 state [1, 2].

As illustrated in Figure 1, a typical QCL consists of several active regions separated by

injectors, referred to as stages. As of 1999, a typical QCL has a 25-75 stages.

Consequently, one single electron in a QCL produces up to N photons for a device that

has N stages as compared to a diode laser where an injected electron will only produce a

single photon (i.e., once an electron combines with a hole in a diode laser, it will not

produce any additional photons). This, combined with the ability to support larger device

currents, leads to QCLs outperforming diode lasers by up to factors of 1000 [1].

Finally, the full QCL device has the numerous semiconductor layers comprising the

stages clad with semiconducting material of a lower refractive index so that the radiation

is guided through the device. In more complicated commercial laser setups this cladding

is also grated such that single mode can be significantly amplified over another [1, 2].

Summary Since its invention 1994, the QCL has quickly found a considerable amount of practical

applications. A primary reason for this is the large wavelength range QCLs can be

engineered for. This wavelength range, combined with the relative ease of production

such that QCLs can be produced in large volumes, opens up laser spectroscopy solutions

for a large amount of industries that cannot otherwise afford expensive laser systems [3].

As an example, QCLs are poised to provide a significantly better solution for the large

gas-sensing market [4].

Furthermore, the QCL is also finding use for theoretical studies. In 2007, a paper was

published in Nature that documented using QCLs to directly probe the phase of laser

emission. This method is claimed to open up the ability to probe the dynamics of the

photons as they traverse the laser cavity and should find possible use in studying laser

dynamics [5]. So in addition to a significant impact in several industries, the QCL has

what appears to be a promising future in scientific fields as well.

References

[1] F Capasso, C Gmachl, D L Sivco, A Y Cho, Physics World, 27-33 (June 1999).

[2] J Faist, F Capasso, D L Sivco, C Sirtori, A L Hutchinson, A Cho, Science 264,

553-556 (1994).

[3] I Howieson, Laser Focus World 41, (2005).

[4] E Normand, Laser Focus World 43, (2007).

[5] D Citrin, Nature 449, 669-670 (2007).