Physics Today Half‐Filled Landau Level Yields Intriguing Data and TheoryBertram Schwarzschild Citation: Physics Today 46(7), 17 (1993); doi: 10.1063/1.2808957 View online: http://dx.doi.org/10.1063/1.2808957 View Table of Contents:http://scitation.aip.org/content/aip/magazine/physicstoday/46/7?ver=pdfcov Published by the AIP Publishing

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SEARCH & DISCOVERY

HALF-FILLED LANDAU LEVEL YIELDSINTRIGUING DATA AND THEORY

Surprises abound when strong mag-netic fields are imposed on two-di-mensional electron gases at low tem-perature. When the electron densityapproaches an integral multiple v ofthe density of magnetic flux quantathreading the plane, the resistivity ofthe electron system begins to vanishand its Hall resistance takes on thequantized value hive1 to a spectacularlevel of precision. That's the integralquantum Hall effect, for whose seren-dipitous discovery at Grenoble in 1980Klaus von Klitzing won the 1985Nobel Prize in Physics.

The integral quantum Hall effect isnow well understood in terms of thefilling of successive Landau levels ofthe quantized energies of the elec-trons circling in tight little cyclotronorbits. The first Landau level, forexample, is full when there's preciselyone electron for every flux quantum.If, at that point, one wants to add orscatter an electron, one must promoteit to the next cyclotron energy level orelse add more magnetic flux.

That was only the beginning. Inthe succeeding years the availabilityof ever more perfect semiconductorinterfaces has unearthed a panoply offurther surprises that continue tochallenge the theorists. Looking tosee what happens in very strongmagnetic fields where the flux quantaoutnumber the electrons, Daniel Tsui,Horst Stormer and Arthur Gossard atBell Labs in 1982 found states ofquantized Hall resistance when the"filling factor" v had the fractionalvalues V3 and %. The theory of theintegral quantum Hall effect hadgiven no inkling of such fractionallyfilled states. Many more fractionalquantum Hall states have been foundsince then, almost all of them withodd-denominator filling factors. Ateach of these fractional filling factors,it seems, the two-dimensional elec-tron gas forms an incompressiblequantum liquid with an energy gap inthe density of states. But the wave-function proposed by Robert Laughlin

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6 8 10APPLIED MAGNETIC FIELD B (tesla)

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Acoustic wavespropagating in a two-dimensional electrongas exhibit variablevelocity (top curve)and attenuation(bottom curve) as afunction of themagnetic fieldstrength B. Arrowsindicatecorrespondingfractional and integralfilling factors v. Bothcurves are close towhat's predictedfrom the resistivitymeasured in dctransportexperiments, exceptat v = '/,,'/4 and %.At those filling factorsthe velocity andtransmission dipunexpectedly.Dotted curvesindicate behaviorpredicted from dcresistivity. (Adaptedfrom ref. 1.)

in 1983 to explain the fractionalquantum Hall effect puts only theoriginal V3 and % states on a reallyfirm theoretical foundation. (SeePHYSICS TODAY, July 1983, page 19.)Though interactions between elec-trons play almost no role in theintegral quantum Hall effect, theyare thought to be crucial to theexistence of the quantum Hall statesat fractional filling factors.

In recent months the spotlight hasbeen on the V2 state, even though atrue quantum Hall state is neitherseen nor expected at that fillingfactor. The elementary flux quantum

embodies a magnetic flux of hie. Atv = V2 the first Landau level is halffull; there are two flux quanta per-pendicular to the plane for everyelectron in it. The Fermi-statisticsrequirement that Laughlin's multi-particle wavefunction be antisymme-tric under the exchange of a pair ofelectrons restricts the incompressiblequantum liquids it describes to odd-denominator filling factors. Nonethe-less, the V, state turns out to be full ofenlightening surprises. Three recentexperiments1'2'3 have pointed outreally striking behavior of two-dimen-sional electron systems at v = V2, and

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a bold new theory of the half-filledLandau level is putting the entirefractional quantum Hall regime intoa new perspective.4

Very clean interfacesThe fractional quantum Hall effectcan be observed when a very cleaninterface in a high-mobility semicon-ductor heterostructure at tempera-tures below 1K is subjected to aperpendicular magnetic field on theorder of 10 tesla. A concentration ofconduction electrons is confined nearthe interface by a potential wellcreated by modulation doping and thediscontinuity between the conductionbands of the two different semicon-ducting materials—typically GaAsand AlGaAs. The electrons are freeto move in the plane, but at thesetemperatures they are confined to theground state with respect to motionperpendicular to the interface. Be-cause the two-dimensional electrondensity n is usually fixed in theseheterostructures, one varies the fill-ing factor v = (nlB)x(hle) by varyingthe magnetic field strength B.

When a current I flows in theinterface plane in the presence of amagnetic field normal to the plane,the Lorentz force generates a Hall-effect voltage VH across the plane inthe direction IxB. The Hall resis-tance RH is defined as VH/I. Thegeneral trend is for RH to increaselinearly with increasing B (whichmeans decreasing v). But in thevicinity of integral and certain fa-vored fractional values of v, thislinear climb levels off to form "quan-tum Hall plateaus" on which RHtemporarily remains constant ath/ve2. These plateaus, each accompa-nied by a deep dip in the ordinarylongitudinal resistivity of the system,are the characteristic signatures ofthe quantum Hall states.

No such plateau is seen at v = V2.In 1989 a Bell Labs-Princeton-MITcollaboration did find a dip in longitu-dinal resistivity at v = \ > but itwasn't really deep enough for a quan-tum Hall state, and its weak tempera-ture dependence was more suggestiveof the behavior of an ordinary metalin the absence of a magnetic field.Having seen quantum Hall states atso many rational filling factors, in-cluding the even-denominator %state (PHYSICS TODAY, January 1988,page 17), one might well have expect-ed the resistivity at v = V2 to exhibitan exponential temperature depend-ence indicative of an energy gap.There was at the time no theory of thehalf-filled Landau level.

The apparent absence of an energygap at v = V2 despite the strong mag-

netic field led Yale theorist NicholasRead to point out that the system wasbehaving as if two flux quanta weresomehow fastened onto each electron.At v = V2 that would just preciselyuse up all the magnetic flux.

Composite fermionsThe magnetic field of a flux quantumis a delta function in the plane, andone assumes that an electron wouldnot feel the field of a flux lineattached to itself. Therefore theseputative composite particles wouldmove as if they were in a field-freeenvironment. Furthermore, an elec-tron dressed with an even number offlux quanta remains a fermion. Soone would have, in effect, a gas offermions wandering around a field-free plane, with a Fermi surface andno energy gap. That would explainthe metallic temperature dependenceof the resistivity at filling factor V2.

Jainendra Jain at Stony Brook hadalready applied just such compositefermions, each one sporting two fluxquanta, to a broad problem of thefractional quantum Hall regime:Even though the Laughlin wavefunc-tion yields fractional quantum Hallstates only at filling factors 1/p and1 — 1/p, where p is any odd integer,Jain pointed out that the most promi-nent plateaus are seen at fillingfactors of p/(2p ± 1).

The best the theorists had been ablecome up with to generate fractionalstates other than the Laughlin stateswas the rather cumbersome "hierar-chy" scheme. It offered no readyexplanation for the special promi-nence of the p/{2p ± 1) "principalsequence" states. The hierarchy the-ory regards all the fractional statesfor which there are no Laughlinwavefunctions as daughters, grand-daughters and more distant descen-dants of the Laughlin states.

In the Laughlin theory, the incom-pressibility of the quantum liquid atv = 1/p is a manifestation of the gapenergy required to create "quasiparti-cle" vortex excitations carrying frac-tional electric charge + e/p when onetries to add or subtract electrons.More and more of these quasiparticlesare generated as one pushes the fillingfactor further and further away from1/p. Eventually, according to thehierarchy scenario, the quasiparticlesthemselves form an incompressiblequantum liquid. If, for example, onestarts out at v = V3, one arrives at aLaughlin-analog state of e/3-chargedvortices when v reaches %. Playingthe same game again, one generatesexcitations of the % daughter stateuntil there are enough of them tocreate a granddaughter quantum Hall

state at v = %. Eventually one cangenerate all the fractional quantumHall states by this scheme.

Seeking an alternative to the hier-archy theory in 1989, Jain pointed outthat the principal-sequence fractionalstates could be thought of as integralquantum Hall states of the compositefermions. The 2/5 state, for example,with 5 flux quanta for every 2 elec-trons, could be viewed as two filledLandau levels of composite fermions.Jain argued that the analytic form ofthe Laughlin multi-electron wave-function was very suggestive of theidea that every electron in a Laughlinstate comes with two flux quantaattached.

Acoustic wave anomaliesFor a time Jain's approach met withconsiderable resistance among thetheorists. But then experimentersunearthed a striking anomaly atv = V2 that cried out for explanation.In 1990 a Bell Labs group led byRobert Willett set out to investigatethe fractional quantum Hall regimeby means of surface acoustic waves atgigahertz frequencies. Most of theprevious information had come fromstatic measurements of dc transportparameters. But in fact much can belearned about the two-dimensionalelectron system by examining thefrequency and wavelength depend-ence of its resistivity in a time-varying field. In Willett's experi-ments the acoustic waves were gener-ated at the surface of a GaAs/AlGaAsheterostrucure only about 1000 Aabove the interface. The strong piezo-electric character of the materialstranslates an acoustic surface waveinto an electromagnetic wave at theinterface below.

In two years of experiments, Willettand Mikko Paalanen measured theattenuation and velocity of the acous-tic waves as a function of fillingfactor, wavelength and temperature.1

At all of the odd-denominator fillingfactors at which one sees prominentquantum Hall plateaus, Willett's ex-periments show sharp decreases inthe attenuation of the acoustic waveaccompanied by sharp increases insound velocity. That's equivalent tosaying that the high-frequency resis-tivity in the fractional quantum Hallstates is very close to the dc valuemeasured in ordinary transport ex-periments.

That's also what one sees in theintegral Hall states, but not at v = V2.The modest drop in the dc resistivityat V2 suggests that the acoustic at-tenuation should also drop, and thesound velocity increase. But Willettand Paalanen found just the opposite:

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The attenuation increases rathersharply at V2, and the sound velocityshows a distinct dip. That is to say,the ac resistivity at short wavelengthsshows a marked enhancement atv = V2, with similar enhancements atother even-denominator filling fac-tors such as V4 and %. (See the figureon page 17.) They found that thisanomalous enhancement increaseslike the reciprocal of the wavelengthand that it persists to surprisinglyhigh temperature. The robustness ofthese even-denominator peaks sug-gested a series of states complemen-tary to the fractional quantum Hallstates. But these new states, unlikethe real quantum Hall states, wouldhave no energy gap between theground state and its excitations.

A gouge transformationHarvard theorist Bertrand Halperinwas on sabbatical leave at Bell Labs inthe spring of 1992 when the surface-acoustic-wave experiment was yield-ing up some of its best surprises atv = V2. Stimulated by Jain's elucida-tion of the principal-sequence odd-denominator states, Halperin, Readand MIT theorist Patrick Lee hadundertaken to develop a detailed the-ory of the half-filled Landau level.After all, V2 is the limit of theprincipal sequences p/(2p + 1), fromabove and below. Why then is thereno energy gap at v = V2 ?

Halperin and his collaboratorstreated Jain's composite fermion as amathematical artifact generated by agauge transformation. The physicalpredictions of a gauge-invariant fieldtheory remain unaltered when anarbitrary gauge transformation of the

particle wavefunction is accompaniedby a compensatory transformation ofthe field. That's essentially what oneis doing when one takes flux linesfrom the external field and attachesthem to the electrons.

In their paper published back-to-back with Willett's experimental pa-per, Halperin, Lee and Read exam-ined in detail the consequences ofperforming such a gauge transforma-tion (known as a Chern-Simons trans-formation) on a half-filled Landaulevel.4 At v = V2, the transformedelectrons, now sporting two flux quan-ta apiece, see a transformed magneticfield that is, on average, zero. Al-ready at that "mean field" level onehas an explanation of why the V2 statelooks so much like a field-free Fermisea topped by a Fermi surface ratherthan an energy gap. But Halperinand his collaborators went on tohigher-order, time-dependent ap-proximations that take into accountfluctuations of the gauge field aboutits mean value. One reason for thisexercise was to assure themselvesthat the mean-field solution doesn'tblow up in the face of small perturba-tions. The time-dependent treatmentwas also essential for explaining theanomalous response of the V2 systemto high-frequency acoustic waves.

"It's quite remakable how much thedc transport properties of the half-filled Landau level look like what wesee when there's no magnetic field,"Halperin told us. "But when you getto the [time dependent] random-phaseapproximation, you find that high-frequency behavior is quite differentfrom the field-free case." That's most-ly due to the surprising slowness of

the relaxation response of the elec-tron gas at v = V2. The random phaseapproximation predicts that theanomalous resistivity of the V2 statein the presence of an accoustic waveincreases linearly with wavenumber(reciprocal wavelength). That's justwhat Willett and Paalanen were find-ing in the laboratory. In fact theoverall agreement between the sur-face-acoustic-wave experiment andthe Halperin-Lee-Read theoryturned out to be surprisingly good.

Weighing composite fermionsThe theory also predicts what hap-pens when the filling factor departsfrom V2. Higher-order perturbationcalculations beyond the random-phase approximation predict that in-teraction between electrons generatesa surprisingly high effective mass forthe composite fermions. This effec-tive mass parametrizes the v depend-ence of the energy gaps of the frac-tional quantum Hall states. "If youstayed at the mean-field approxima-tion," says Halperin, "you'd just usethe ordinary effective mass of conduc-tion electrons in GaAs and you'd endup with gaps that are much too large."

With the advent of the Halperin-Lee-Read theory, it became particu-larly interesting to have good mea-surements of the energy gaps of allthe accessible principal-sequencequantum Hall states. That task wasundertaken last fall by Stormer, Tsuiand Tsui's Princeton colleague RuiDu, using very high-mobility GaAs-AlGaAs heterostructures fabricatedat Bell Labs by Loren Pfeiffer andKenneth West.2 The figure on thispage summarizes their results. Itplots the energy gaps for the promi-nent fractional quantum Hall statesmeasured with one particular hetero-structure against B.

The figure serves to underline astriking general property of the frac-tional quantum Hall states that isembodied in the composite-fermiontheories. If one simply shifts themagnetic field axis of the figure topretend that the field at v = V2 repre-sents B = 0, then the fractional stateslook almost exactly like the integralquantum Hall states around B — 0.Because the A5 between the V2 and V3fractional states is just the B corre-sponding to the integral state at v = 1,the axis shift makes the V3 state looklike the first integal quantum Hallstate. By the same argument the %state looks like the v = 2 state, the %state becomes the v = 3, and so on adinfinitum at the center. The left-hand side of the plot now representsthe integral quantum Hall states withthe magnetic field direction reversed.

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The gap energies of the integralquantum Hall states are just thecyclotron energies. They, of course,scale linearly with magnetic field,and the resulting slope gives theordinary effective mass of electrons inGaAs. As one expects from theHalperin-Lee-Read theory, the fig-ure shows that the gap energies of thefractional states have just the samekind of linear dependence on magnet-ic field around v = V2. "That suggestsa description of the fractional quan-tum Hall effect in terms of the mass ofsome new particle," Stormer argues.From the slope of these straight linesthe Bell Labs-Princeton collaborationgets an effective mass about of 60% ofthe free-electron mass. As the theorypredicts, that's an order of magnitudemore than the usual effective massthat comes from the band structure ofthe semiconductor.

There is, of course, a bit of a fiddlehere. The theory and the experi-ments tell us that the V2 state has noenergy gap. Therefore a straightfor-ward interpretation of these slopes asmeasures of the composite-fermioneffective mass would require all thedata to extrapolate to zero gap atv = V2. But in fact we see that thedata in the figure extrapolate toslightly negative energies at %.Stormer and company, however, re-gard this negative intercept as yetanother analogy with the case of realelectrons: The integral quantum-Hall levels are broadened by thescattering of electrons at impurities—otherwise one wouldn't see plateaus.That broadening reduces the Landau-level energy gaps and thus yields anegative intercept in the integralquantum-Hall case. Therefore,Stormer argues, the negative inter-cept in the plot of energy gaps atfractional filling factors is actually a

measure of the scattering rate of thecomposite fermions. That scatteringrate turns out to be in good agreementwith the Halperin-Lee-Read theory.

Plots of Hall resistivity against Balso exhibit this extraordinary simi-larity between the fractional andintegral quantum Hall states, eventhough the physical bases of the twoeffects look so different at first glance.But it's the second glance, embodiedin the Jain and Halperin-Lee-Readtheories, that clarifies the resem-blance. "It's given us a whole newway of looking at the hierarchy offractional quantum Hall effect," ex-plains Stormer. "The fractionalstates are the integral quantum Hallstates of these composite fermions—the electrons with two flux quantaattached. This new frameworkmakes the whole subject so mucheasier to think about." The paper inwhich the Bell Labs-Princeton colla-boration purports to measure theeffective mass of the composite fer-mions by way of the energy gaps2 isentitled "Experimental Evidence forNew Particles in the Fractional Quan-tum Hall Effect." How real are these"new particles"? "They're as real asCooper pairs," asserts Stormer.

In the midst of this euphoria,Princeton theorist Duncan Haldanesprings to the defense of the muchmaligned hierarchy theory. "Thehierarchy scheme and composite fer-mions are not competing theories," hetold us. "They're just different waysof organizing the same construction.Each has its own particular useful-ness" A recent paper by Jian Yangand Wu-pei Su at the University ofHouston argues that the two theoriesare mathematically equivalent.5

TunnelingPfeiffer, West and their Bell Labs

colleague James Eisenstein have re-cently developed yet another experi-mental means of studying the quan-tum Hall regime. They measure thetunneling current between two het-erostructure interfaces separated by afew hundred angstroms. If the V2state indeed has a Fermi surface onewould expect to see an anomaly in thetunneling current at v = V2 because ofthe absence of an energy gap at thatfilling factor. But Eisenstein andcompany reported last December3

their observation of a broad energygap covering an extensive range offractional filling factors including V2.

In a recent preprint, Halperin andBell Labs theorists Song He andPhilip Platzman invoke the Hal-perin-Lee-Read theory to save thespecial character of the half-filledLandau level.6 They conclude thatthe observed tunneling gap is due tothe energy dependence of the overlapbetween the quantum state of theintruding electron and low-energystates of the half-filled Landau levelinto which it is tunneling. Platzman,He and Halperin calculate that thisoverlap falls off very fast as the low-energy states approach the Fermilevel, thus simulating an energy gap.

—BERTRAM SCHWARZSCHILD

References1. R. Willett, R. Ruel, M. Paalanen, K.

West, L. Pfeiffer, Phys. Rev. B. 47, 7344(1993).

2. R. Du, H. Stormer, D. Tsui, L. Pfeiffer,K. West, Phys. Rev. Lett. 70,2944 (1993).

3. J. Eisenstein, L. Pfeiffer, K. West, Phys.Rev. Lett. 69, 3804 (1992).

4. B. Halperin, P. Lee, N. Read, Phys. Rev.B. 47, 7312 (1993).

5. J. Yang, W. P. Wu, Phys Rev. Lett. 70,1163 (1993).

6. S. He, P. Platzman, B. Halperin, Sub-mitted to Phys. Rev. Lett. (1993).

CRITICAL TEMPERATURE NEARS 135 KIN A MERCURY-BASED SUPERCONDUCTORNo superconductor has broken therecord for the highest critical tem-perature since 1988, when a thallium-bearing compound, exhibiting resist-anceless conduction at temperaturesas high as 125 K, laid claim to thetitle.1 (See PHYSICS TODAY, April,1988, page 21.) But in early May agroup from ETH in Zurich reportedmeasuring a superconducting transi-tion several degrees above 130 K in acompound containing mercury to-gether with barium, calcium, copperand oxygen.2 The researchers werenot able to specify the exact stoichi-

ometry of the new champion, becausetheir sample consisted of severalphases of the compound and theyhave been unable so far to separateout the phase that is responsible forthe high Tc.

High Tc is not the whole story. Theincrement is not that great, especiallygiven that thallium-based materialcan be made, albeit with some diffi-culty, to superconduct at tempera-tures as high as 130 K. But in addi-tion to high Tc, at least one member ofthe mercury-based copper oxide fam-ily appears to have favorable behavior

in magnetic fields. Furthermore thematerial is both simple and novel,offering perhaps some new clues tothe mechanism of superconductivityin the class of copper oxides.

There is a dark side, however.Thallium has frightened away someresearchers because of its extremetoxicity, and mercury is not much ofan improvement. Even if the so-far-unknown phase proves to have favor-able electrical, magnetic and materi-als properties, it will have to be boundinto a stable compound if it is to seewidespread application. Robert Cava

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