PHYSICAL REVIEW 8 VOLUME 44, NUMBER 23 15 DECEMBER 1991-I

Resistance fluctuations in the integral- and fractional-quantum-Hall-effect regimes

J. A. Simmons, * S. W. Hwang, D. C. Tsui, H. P. Wei, L. W. Engel, and M. ShayeganDepartment of Electrical Engineering, Princeton University, Princeton, ¹wJersey 08544

(Received 22 April 1991)

We report on our measurements of resistance fluctuations as a function of magnetic Geld 8 in anA1„Ga& As/GaAs heterostructure of etched width w=2. 5 pm in the integral- and fractional-quantum-Hall-effect regimes. High-frequency fluctuations are observed near the longitudinal resistance(R„„)minima for v=1, 2, 3, 4, and 3. The quasiperiods 68(v = integer) of the fluctuations for integer v

are all -0.016 T, while for v= 3, the quasiperiod 58(v =—,'

) is -0.05 T, or a factor of 3 larger. Thefluctuations at integer v are consistent with inter-edge-state tunneling via magnetically bound statesencircling a potential hill of a diameter roughly equal to the conducting width of the channel. A similarmodel, with the difference that the tunneling is by quasiparticles of fractional charge 8* = e/q, predicts ascaling of the quasiperiod as 68(v=1/q) =q b8(v = integer). Interpreted in terms of this model, thedata provide direct evidence of the existence of quasiparticles of charge e*=e/3 in the v=

3fractional

quantum Hall effect. For both v=3

and v= integer, the individual fluctuation patterns for different

pairs of voltage probes are strongly correlated only if the pairs share a length of the channel, indicatingthat the source of the fluctuations is local, as predicted by the model. A Coulomb blockade as the originof the fluctuations is ruled out by the fact that for v= 1 and 2 the fluctuation amplitudes saturate at tem-peratures T,(v=1)=—66 mK and T,(v=2)=—121 mK, and also saturate at currents I,(v=1)=—0.5 nAand I,(v=2)-=1.7—3.0 nA. These results indicate that for integer v, the bound-state-energy spacingb, c.(v) scales as v or 8 ', inconsistent with a Coulomb blockade.

I. INTRODUCTION

The integral quantum Hall effect' (IQHE) is under-stood in a single-particle picture in terms of the peculiarnature of two-dimensional (2D) electronic states in strongmagnetic fields (B) in the presence of disorder. Namely,extended states exist near the center of the. Landau levels,and localized states exist between Landau levels. Whenthe Landau-level 611ing factor v has integer values, theFermi energy lies in the region of localized states. If thetemperature kz T &(fi~„ the Landau-level energy separa-tion, the current-carrying extended states cannot scatterinto one another and longitudinal transport occurswithout dissipation, giving a longitudinal resistanceR„„=O. Gauge-invariance arguments then give thequantization of the Hall resistance to R„=(1/v)h /e .

The fractional quantum Hall effect (FQHE) is themanifestation of a series of many-body ground states of2D electrons at special fractional filling factors v=p/q,where p is any integer and q is an odd integer. Theground state is an incompressible Quid which can flowwith no dissipation. Experimentally, the phenomena aresimilar to those for the IQHE, except that the Hall resis-tance plateau for the state at v=p/q is quantized toR,s=(q/p)h/e . This fractional quantization is under-stood to be a consequence of the existence of quasiparti-cles of fractional charge e *=e /q, as predicted byLaughlin's theory. Some evidence for the existence offractionally charged quasiparticles, independent of theFQHE itself, has been obtained. Clark et al. have ob-served that, at 611ing factors v=p/q, the temperaturedependence of the longitudinal conductance o. „ in an

Arrhenius plot extrapolates to (1/q )e /h as T~ao.Chang and Cunningham have observed the quantizationof longitudinal resistance 8 „ for transmission of elec-trons from regions of filling factor v=1 across barriers atfilling factor v= —', , and from regions at v= —', across bar-riers at v= —,', and suggest an interpretation of their re-sults in terms of e*=e/q. However, evidence aftordedby the more direct technique of interference of the frac-tionally charged carriers has remained unobserved untilnow.

Conductance fluctuations due to electron interferencein samples of sizes on the order of the phase coherencelength I+ have received much attention in recent years.The fluctuations arise from interference between all pathsthe electron can take in traversing the sample, and theiramplitude and quasiperiod (in 8 field or Fermi energy)give a good measure of I+. If the electron's availablepaths are restricted to a ring-shaped region, as in theAharanov-Bohm eA'ect, the fluctuations become period-ic. ' If lz, is known independently, the amplitude andperiod of the fluctuations trivially give a measure of thecharge of the carriers.

Transport in the strong-8-field limit of the IQHE andFQHE, however, it less well understood. In small sam-ples experimental evidence"' is increasingly supportingthe notion of edge states' ' —wave functions followingequipotential contours along the edges of the sample-and their importance in transport. While large resistancefluctuations in mesostructures have been observed in theIQHE regime, ' no clear relation between the fiuctua-tions and v was observed. Jain and Kivelson' andButtiker' have proposed that inter-edge-state tunneling

12 933 1991 The American Physical Society

12 934 SIMMONS, HWANG, TSUI, WEI, ENGEL, AND SHAYEGAN

via magnetically bound states —or closed equipotentialcontours —can give rise to sharp resistance fluctuationswhose pattern is characteristic of the sample's particulardisorder potential. Chang, Owusu-Sekyere, and Changhave observed nearly periodic resistance fluctuations nearthe v=2 longitudinal resistance minimum due to suchscattering through bound states. Kivelson and Pokrov-sky ' have suggested that a comparison of the fluctuationquasiperiods near integer and fractional v resistanceminima be used to measure quasiparticle charge in theFQHE. Understood with this model, our data provideevidence that the charge of quasiparticles of the v=

3

state is e*=e/3.Portions of this work have appeared previously. We

summarize our results as follows. In a Hall bar of etchedwidth 2.5 pm, (1) resistance Auctuations are observed onthe high-B shoulder of the R minima for filling factorsv= 1, 2, 3, 4, and —,'. (2) For integer v, the quasiperiod ofthe fluctuations is roughly the same and corresponds tomagnetically bound states of a diameter roughly equal tothe conducting width of the channel, while (3) for v= —,

'

the quasiperiod of the fluctuations is about a factor of 3larger. (4) For v=2 the fluctuations nearly disappear attemperatures of a few hundred mK, consistent with themodel of Jain and Kivelson. ' (5) Upon cycling of tem-perature T to 300 K and back to 25 mK, and upon per-mutation of current and voltage probes, the individualfluctuation patterns change, but the quasiperiods for eachrespective v remain the same to within 30%%uo. (6) Near theresistance minima, fluctuation patterns obtained fromdifferent pairs of voltage probes show strong correlationsif the two pairs measure lengths of channel which over-lap, but not otherwise. No correlations are observed be-tween any two pairs for fluctuations far from the Rminima. After another cycling of T to 300 K, more ex-tensive measurements of the T and current (I) depen-dence of the fluctuations for v = 1 and 2 show that (7) thefluctuation amplitudes clearly saturate at low T, indica-tive of tunneling. (8) The saturation temperaturesT, (v=2)-=121 mK and T, (v= 1)—=66 mK scale as v or8 '. (9) Saturation also occurs at low currentsI,(v=2) —= 1.7 —3.0 nA and I,(v= 1)—=0. 5 nA, which isconsistent with I, (v) scaling as v or 8 . Thus (10) thesaturation Hall voltage VH, (v) =(h /ve )l, (v) is con-sistent with a scaling as B ', as expected from the model.

II. EXPERIMENT

The sample was a GaAs/Al Ga, As heterostructurewhich after illumination with a red light-emitting diode(LED) at 4.2 K had a density nzD = 1.2 X 10" cm and amobility p= 1.4X 10 cm /V s. Two Hall bars connectedin series were defined using standard photolithographyand wet-etching techniques. One Hall bar is of etchedwidth 2.5 pm (shown in Fig. 1) and is the device of exper-imental interest. (We expect that the conducting widthis substantially less, as sidewall depletion lengths of up to0.8 pm were previously observed. ) Various combina-tions of probes were used to supply current and measurevoltage. The center-to-center spacing of probe pairs 6,7

20 p, m

FIG. 1. Sample geometry and probe assignment. The widthof the central narrow channel is 2.5 pm.

and 7,8 is 6.5 pm; for pair 2,3 the spacing is 13 pm; andfor pair 3,4 the spacing is 65 pm. Because the voltageprobes are much wider than the channel —the narrowestprobe is 5 pm wide —the geometry is not rectilinear butrather more similar to a constriction. In series with thisnarrow Hall bar and a large distance away from it is a300-pm-wide Hall bar, which serves to monitor thesample s 2D density, mobility, and homogeneity duringthe cooldown and LED illumination process, and as acheck that the phenomena we observed in the narrowHall bar were due to its small size. For each cooldown,the sample is placed in a top-loading He- He dilution re-frigerator and cooled to 25 mK by lowering directly intothe mixing chamber, and then briefly illuminated by theLED. The process is repeated until the 2D electron gas isin a condition of good homogeneity, as determined by thequality of the IQHE and FQHE. Measurements wereperformed using an ac resistance bridge lock-in techniqueat 17 Hz over the temperature range 25 —450 mK and Bfield range 0—15.5 T. Excitation voltages across the sam-ple were typically in the range 0.01—1 pV.

In Fig. 2(a) we show R and R,~ over the full experi-mentally available field range 0—15.5 T for the 300-pm-wide channel, indicating the state of the 2D electron gas.Fractional states

3 3 5 3and —', are all well developed,

with quantized plateaus in R, and R„„minima close tozero. Fractional states

5 7 7 5and —', are also

present, through their respective Hall plateaus are lesswell developed and their R minima do not reach zero.

On the initial cooldown and measurement the generalcharacteristics of the fluctuations in the narrow channelfor different v were explored using only one probeconfiguration, and a qualitative investigation of theirtemperature dependence was performed. The longitudi-nal resistance R„„was measured using probe pair 1,5 tosupply current and pair 7,8 to measure voltage (using thenotation Ri 5.7.s). For all measurements, R was mea-sured as R, 5. 3 7. In Fig. 2(b) we show R =R, , 7 s andR =R i 5 3 7 at 25 mK for the 2.5-pm-wide channel overthe full 15.5-T field range. A number of differences be-tween the narrow and wide channel data are immediatelyapparent. (1) The general robustness of the fractional(and large integer) states is significantly reduced. The —', ,

53 and —', minima no longer approach zeroclosely, andthe width and sharpness of the plateaus in R are alsogreatly reduced. Further, the

7 7 5and —', states are no

NS IN THE INTEGRALRESISTANCE FLUCTUATION 12 935

20

15-(

x 1Q-CL

(a) wide channe

4/3

2

II

I I

2/5

80

-60

- 4Q

-20

0.8

0.6

0.2Ll)

CC

8(b) narrow c

080 0-

2.0I

2.2 2A

OQ

4LA

Ct4/3

1/3 - 6033

CJi

GD

-40

-20

00 2 4 6

I I I I Q8 10 12 14 16

B (T)

l and Hall resistances at 25 m, pmK lottedge a) for the 300-pm-wi e aHall bar, measured as ] 5 ~ 7 g

t the same field scaf r the 2.5-pm-wide Ha

hannel the robustness o t ef he fractional andFor the narrow channe ed and resistance fluctua-s is reatly reduce, anhigher integer states is g

tions are present.

f r different temperatures.FIG. 3. near v=2 for our i

0.8i(a) v=2

0.4

havin almost disappeare at T =200o rwh'hth h' h-f

widt oresistance peaks deve op, gr

ear B=2.3 and 2.5 T.The general be aehavior describe a ove

'

R minima for otherfor v=2 is also observed near t e „m'how R near theFi s. 4(a) and 4(b we s owintegral v. In igs.

0.2

er cent resistanceat all. (2) Large (a few perce

ma. A close examinination of the gure revtails, ust on the shoul-are also Auctua

'ations on the R tai s, us

R background is sma 1a. Because the acders of the minim .fl ctuations relative toin these regions, ethe size of these uc ua

er than for far betweenthe R„backgrout remains relativelythe maxima.

ein of much smaller relativesmooth, any fluctuations being of muc ssize.

0.8—

0.6-

0 4-

~ 0.2-

0-5.1

I

5.2

0I

2.3 2.4 2.5

5.3

2.6

I

5A

I

2.7

I

5.5

2.8

I

5.6

III. RESISTANCE FLUCTUATIONSNEAR R„„MINIMA

R near t e v=h =2 minimum for four1:(1)h h-ures. We note severa poind e ent te peratures. e

shoulders of, t e minimhf d fm the fluctuations s ifrom the minimum,drop to below the0.05 —0.1 T; and (3) the fluctuations ro

f r a broad field range inR o o o A T

r measurement for a roathe center of the rn'minimum, where g

' t nce peaks remainlow-frequency resistancei h' h-f n aks dt-relatively unchanged, w iwhile the ig - r

1.2-1.0-0.8-0.6-0.4-0.2.p =--

0.2+ ==== - ~

14.4 14.5 'f4.68

14.7 14.8 14.9

e hi h-B sides of R „miniminima for (a)g-v=2 at 25 mK, (b) v=

~ Insets: Fourier powerK, all plotte w'd ith the same field scale. Insets:spectra o t e uf h fluctuation regions for eac v.

12 936 SIMMONS, BHANG, TSUI, WEI, ENGEL, AND SHAYEGAN

high field side of v=2 and 1 at 25 mK. The Fourierpower spectrum of the Auctuation region for v= 1 gives adominant frequency of -70 T ', corresponding to aperiod of -0.014 T, while for v=2 the Fourier spectrumgives a dominant frequency of -60 T ', correspondingto a period of -0.016 T. Due to the limited number ofAuctuations and rapidly changing background, this deter-mination has an uncertainty of +25%. Resistance Auc-tuations are also observed in the v= 3 and 4 minima, butso few Auctuations are present that their Fourier spectrado not show distinct peaks. Nevertheless, their quasi-period is the same as for v= 1 and 2 to within -30%.

In Fig. 4(c) we show R near the high-8 side of v= —,'

at 25 and 100 mK. Again, high-frequency Auctuationsare present on the shoulder of the minimum, giving wayto lower-frequency Auctuations further away from theminimum, and no Auctuations are present for a broad 8range in the center, where R approaches zero. TheAuctuations closest to the minimum become larger atT=100 mK, growing out of the background of the Rminimum. The overall T behavior of the Auctuations isqualitatively similar to that observed for v=2. Here,however, the period is 0.05 T (+25%).

0.07

0.06—

0.05—

E oo4-CQ

0.03—

0.02—

0.0 I—I tt I

1/3

FIG. 5. Data on quasiperiods of resistance Auctuations at 25mK on the high-8 sides of R„minima for v=1, 2, 3, 4, and 3.from first T cycle, R& 578 (C) ); and from the second T cycle,Ri 5 j 8 ( )f R2, 5;j,8 ( W))& Ri 56 j (X)y Rp 567 {+)y R],5;3,4(& ), R25.34 ( &), and R&,.23 ( «). Uncertainties are all-25%. Points for v= 3, 1, and 2 represent Fourier power spec-tra; v=3 and 4 are estimates by eye.

IV. FLUCTUATIONS FROM DIFFERENT PROBES

After cycling the sample to 300 K and back to 25 mKwe measured R for several different current and voltageprobe combinations. The probes were always configuredso that the R voltage drop was measured along a lengthof channel of 2.5 pm width. Current was drawn throughprobe pairs 1,5 and 2,5 and voltage measured across pair7,8; pair 2,3; pair 6,7; and pair 3,4. (See Fig. 1.) ForR, 5.7 8 the particular pattern of Auctuations was differentfrom that observed on the first cooldown, but the periodof the fluctuations for each respective filling factorremained the same, 0.016 T (+30%) for integer v, and0.05 T (+30%) for v= —,'.

For other combinations of probes, the Auctuation be-havior is also similar to that observed in R

~ 5 7 8 on thefirst cooldown. If Auctuations of measurable amplitudeare present on the high-B sides of the v=1, 2, 3, and 4minima, as is usually the case, their quasiperiods are0.016 T (+30%). For v= —,', on the other hand, the quasi-periods are 0.05 T (+30%). Thus for four different pairsof voltage probes, irrespective of the choice of currentleads, and irrespective of T cycling for the one pair mea-sured over two T cycles, the periods of high frequency R,„-Jluctuations near the integral minima do not depend on 8or v, while for the v= —,

' minimum we find an approximatetripling of the period Asummary of.the quasiperiods b,Bversus v for the different probe combinations is given inFig. 5.

The typical amplitudes of the high-frequency Auctua-tions are -25 Q near v=3, 4; —100—200 Q near v=2;—50—100 Q near v=1; and -200—400 0 near v= —,'.(There are exceptions to this trend, however; for example,R 2 5 ~ 3 4 exhibited Auctuations of —200 0 at v = 1, butonly —50 0 at v =2). Only occasionally do high-frequency resistance Auctuations appear on the low-8

v = 1/3

0-0-

12.0

0.3 rAI I

12.5 13.0 13.5

PIG 6 R i 5- j 8 near v =3 at 25 mK for three different

currents. Below 0.3 nA the fluctuations do not changesignificantly.

sides of the minima. Notable is the fact that Auctuationsnear the minima in the long section of the channel,R

$ 5 3 4 have amplitudes of the same order as those of theother pairs, even though the voltage probe spacing is afactor of 10 greater than that of the shorter channelsR, 5.7 8 and R, 5.6 7. In contrast, the resistances far fromthe minima do scale roughly as the distance between volt-age probes.

In order to ensure that there was no current heating ofthe sample, R„„scans were taken at successively lowercurrents (by about a factor of 3) until the measured R „no longer changed. In this way we were able to obtain arough value for the maximum current, I, (v), which wecould use without causing the fluctuations to change, foreach Ualue of v examined In Fig. . 6 we show R

& 5.7 s forv= —,

' taken at several difFerent currents. As the current isincreased, the width of the R minimum decreases, theamplitude of the Auctuations decreases, and new resis-tance peaks grow out of the R =0 background, similar

RESISTANCE FLUCTUATIONS IN THE INTEGRAL- AND. . . 12 937

to the effect of raising temperature. [Below 0.3 nA, thefiuctuations stop changing, so I,(v= —,

') -=0.3 nA for this

cooldown. ] Because the effect of using currents largerthan I, (v) appears qualitatively similar to that of raisingthe temperature, we initially attributed it to resistivecurrent heating. This leaves unexplained, however, theinteresting fact that I, (v) appears to depend not on R „but rather on the Hall resistance R . For the secondcooldown, the approximate (uncertain to within a factorof 3) values of I,(v) are I,(v= —,')—=0.3 nA, I,(v=1)-=0.3nA, I,(v=2) =—1.0 nA, and I,(v=3) =l, (v=4) =3.0 —nA.Thus for integer v, I, (v) appears to be proportional to vto a small power.

V. CORRELATIONS BETWEENFLUCTUATION PATTERNS

One can examine the fiuctuation patterns (magnetofingerprints) from various probe configuration for corre-lation behavior in order to obtain information on the lo-cality of transport. In Fig. 7(a) we show the fluctuationpatterns from the high-B sides of the v=2 minimum forR ] 5 ~ 6 7 and R

& 5 ~ p 3 which share a length of channel.The two patterns exhibit a strong correlation with oneanother, with the peaks and valleys from each patternoccurring at almost identical values of B. Even minor de-tails of the patterns correlate; the large peak at B=—2. 3 Tshows a small splitting for R

& 5.6 7, while the same peakfor R

& 5 ~ 2 3 has a bump on its shoulder corresponding tothe splitting.

1.5' (a

1.0—

0.5—

In Fig. 7(b) we compare the fiuctuation patterns atv=2 for R».67 and R, 5 ~ 78 which are adjacent to oneanother and share probe 7, but do not share a length ofchannel. No correlation is observed here. This behavioris in contrast to that observed at low B by other au-thors, and implies that in the IQHE regime the sourceof the resistance fluctuations near R minima is stronglylocal for each probe configuration. For our geometry,where the channel width is substantially less than thevoltage probe width, the source lies in the region of thechannel between the two voltage probes, but does notpenetrate a significant distance into the probes them-selves.

If we look far from the minima, the correlation behav-ior is different, however. Figure 8 shows R ] 5 6 7 andR

& 5.2 3 in the region between v =2 and 3. These twopairs share a length of channel and are the same twowhich exhibit such a strong correlation in Fig 7(a). Hereit is evident that no such strong correlations exist farfrom the minima. Nor are correlations observed on thelow-B sides of the minima.

The correlation behavior described above for integral vis also observed for v= —,'. In Fig. 9(a) we show R, 5 ~ 6 7

and R& 523 which share a length of channel, on thehigh-8 side of the v= —,

' minimum. A strong correlationbetween the two curves is apparent. In Fig. 9(b) we showR f 5 ~ 6 7 and R

& 5 7 8 which share a voltage probe but donot share a length of channel. No strong correlation isapparent between the two curves.

VI. THEORY: INTER-EDGE-STATESCATTERING AT HIGH B

The motion of 2D electrons in the high-B limitco,~)&1, where co, is the cyclotron frequency and ~ is thescattering time, can be treated semiclassically' ' whenthe length scale of fluctuations in the potential is muchlarger than the magnetic length lz. Under these condi-tions, the guiding centers of the electrons move along

0 =--

(b)1.0— 1.5—

2.0.5—

v=21 0

Q

02.20 2.24 2.28

B(2.32 2.36

FIG. 7. Correlation behavior of fluctuations on the high-Bside of the R„minimum for v=2, at 25 mK. (a) R& 5 ~ 2 3 andR l 5 ~ 6 7 which share a length of channel but do not share anyvoltage probes. The two Auctuation patterns are strongly corre-lated. (b) R$ 5 ~ 67 and Rl 5 ~ 7 8 which do not share a length ofchannel but do share a voltage probe. The two patterns showno correlation. (All traces were taken simultaneously. ) Inset:probe assignment of sample.

01.4

I

1.5I

1.6I

2.0 2.1

FIG. 8. Rl 5 ~ 23 and R& 5.67—the same two probe combina-tions as shown in Fig. 7(a)—here shown between v=2 and 3 at25 mK. No correlation is present far from the high-B sides ofthe R„„minima.

12 938 SIMMONS, HWANG, TSUI, WEI, ENGEL, AND SHAYEGAN

013.1 13.2

I I

13.3 13.4B (T)

I

13.5 13.6

equipotential contours with a drift velocity given by thelocal gradient of the potential divided by B. Current-carrying edge states are thus formed at the confining po-tential gradients existing at the physical edges of the sam-ple. In a suSciently narrow sample transport is dominat-ed by the edge states, and the total current is given by thedifference between the edge currents along the two edgesof the sample.

Landauer-type formulations for transport in the IQHEregime, put forth by several authors, ' ' ' ' give ex-pressions for R„ in terms of the probabilities for elec-trons to scatter between the incoming edge states associ-ated with different contacts to a sample. Because for oursample the width of the voltage probes is substantiallylarger than the 2.5 pm width of the central channel,scattering along paths which cross the voltage probes ismuch less likely than scattering across the central chan-nel, and we use the two terminal formulation of Streda,Kucera, and MacDonald' and of Jain and Kivelson. '

For small currents and zero temperature, when only oneLandau level is occupied, Rxx is given by'

h RXX

FIG. 9. Correlation behavior at 25 mK for the high-8 side ofthe R„minimum for v= 3. (a) R

& 5 2 3 and R J 5 ~ 6 7 which share

a length of channel but do not share any voltage probes, exhibita strong correlation. (b) R

& 5.6 7 and R& 5.7 8, which do not share

a length of channel but do share a voltage probe, exhibit nocorrelation. The traces were taken simultaneously.

path 2path

PotentialEnergy potential

at hill

saddle-pointenergy

0--

potentialfar from hil)

it closely approaches both edges of the channel, produc-ing two saddle-shaped regions. (See Fig. 10.) The edgestates for this geometry will follow the equipotential con-tours defined by the intersection of the first Landau leveland p, and thus the edge state paths will depend on thevalue of p. The possible edge state paths in this potentialare of three types. (1) When p is low, below both saddle-point energies, paths start at one end of the sample and,unable to pass by the potential hill, return to the sameend, but on the opposite side, corresponding to perfectclassical reflection, R =1. (2) When p is higher than thepeak energy of the potential hill, the two edge state pathsremain well separated on opposite sides of the channel,corresponding to perfect classical transmission, R =0. (3)The type of path relevant to the experiment occurs when

p lies between the saddle-point energies and the peak en-ergy of the potential hill. In this case there will be twoedge state paths which follow the two edges of the sam-ple, and in addition a closed magnetically bound statealong an equipotential which encircles the potential hill,and is thus localized. Semiclassically, electrons will beable to resonantly tunnel from one edge state to anothervia the magnetically bound states, in which case thereAection probability will be 0 &R & 1. Because the mag-netically bound states lie along closed equipotential con-tours, however, they are subject to self-interference andwill have their energies quantized according to the Bohr-Sommerfeld condition that an integral number of Auxquanta &b=h/e penetrate their area. Tunneling occurswhen the energy of a bound state coincides with the p ofthe sample edge. Thus, R, and hence R„„,will exhibit

where R is the (reflection) probability that an electronwill be scattered from one edge of the sample to the oppo-site edge.

The existence of potential Auctuations in the samplewill introduce numerous potential hills and valleys, whichwill also be rejected in the Landau levels. Jain andKivelson' have calculated R as a function of the chemi-cal potential p for the case of a single potential hill in anarrow channel. The hill is taken to be large enough that

FIG. 10. Geometry treated by Jain and Kivelson, that of apotential hill in the center of a narrow channel. (a) Perspectiveview of the potential as a function of x and y, showing the threetypes of edge states formed at three different values of p. Path(1) corresponds to perfect reAection, path (2) to perfecttransmission, and path (3) to the intermediate case, with a mag-netically bound state formed around the potential hill. (b) Thepotential as a function of y, both at the potential hill and farfrom the potential hill.

RESISTANCE FLUCTUATIONS IN THE INTEGRAL- AND. . . 12 939

sharp peaks as a function of p.Since tunneling probabilities decrease exponentially

with distance, the height of a given peak in R is propor-tional to exp[(L, L—2)/la], where L, and L2 are thedistances from the corresponding bound state to the twoedge states, and we take 1., I.2. Similarly, the width Iof the peak in R is just equal to the energy width of thebound state. Thus I is given by the leak rate of the elec-tron on the bound state, which also decreases exponen-tially with distance and is proportional toexp( —2L f /l~). Thus the sharpest and largest peaks willbe produced by a potential hill near the center of thechannel; the peak height is reduced and width broadenedexponentially with the distance the bound state is movedoff center. '

For a given bound state of width I, with adjacentbound states differing in energy by Ac. , there are threedifferent temperatures regimes. (1) When ke T ((I, thetemperature is effectively zero. (2) When k~T) I, elec-trons can access the bound states even when p, does notcoincide with the bound-state energy, so the width of thepeak is broadened to approximately kz T. In addition, atmost a fraction —I /kz T of the electrons can access thebound state, so the height of the peak is reduced by a fac-tor of —I /kz T. (3) When kz T ))b, c, , the energy spacingof the bound states, several bound states will be availableto the electrons, and the structure due to individualbound states will become washed out. [Whether regime(2) or (3) is reached first as T is raised depends on the rel-ative sizes of I and b,e.] Finally, because for a given po-tential hill the bound states furthest in energy from thesaddle-point energies will have the smallest I"s, thermalbroadening of their resistance peaks will occur at lowertemperatures.

VII. COMPARISON WITH EXPERIMENT

We expect that there are probably many potential hillsand valleys existing in the 2.5-pm-wide channel. Howev-er, since tunneling through the bound states exponential-ly decreases with the distance to the channel edges, weexpect that when tunneling is present, it will be dominat-ed by a single bound state which closely approaches bothedges. If the bound state is assumed to be roughly circu-lar, its diameter will thus be approximately equal to theconducting width of the channel.

In the experiment, the bound-state energies are variedby sweeping B and peaks are observed in R„. The quasi-period hB of the R„peaks is found by solving

where r is the radius of the (assumed circular) bound stateclosest in energy to p. The first term is the usualAharanov-Bohm term; the second is due to the changingenergy of the Landau level. Assuming that p is constant,(Br/Bv)„=%co, /eE„(p) = AB/m *E„(p,), whe—re A'co, isthe Landau-level energy spacing and E,(p) is the radialelectric field of the potential hill at the bound state. Here

d v/dB is just —(nzDh )/eB . We then have

2 ] —]rh n2DAB= —mr +

e m *eE„(p)(3)

Since the disorder potential is reproduced in each Landaulevel, it is reasonable to assume that E„(p) is the same foreach integer value of v. Hence AB is the same for all in-tegral filling factor R minima, as observed in the data.

In order to estimate the bound-state radius r, we needan estimate for E„. Davies and Nixon have calculateddisorder potential due to the random placement of donorimpurities in the dopant layer of an Al Ga, As/CraAsheterojunction. They find typical sizes for potential hillsof a few thousand A across, and typical local electricfields of a few 10 V/m. The sample structure for whichtheir calculation was performed differs from the sampleused in our experiments primarily in that theirAl„Ga& „As spacer layer was only 50 A, while for oursample it is 823 A. Thus we expect the amplitude of thepotential Auctuations in our sample to be a good deallower, and the length scale to be larger. Using an esti-mate of E„(p) = 10 V/m and our measured valuekB =0.016 T in Eq. (3), we obtain r=—0.4 pm, or abound-state diameter of 0.8 pm, in agreement with the es-timated conducting width of the sample of —1 pm.

With our estimate of r, we can also obtain an estimatefor the bound-state energy spacing Ac. The Bohr-Sommerfeld condition gives

eE, (S )hc. = (4)

e 2arB

Thus while AB is independent of B and v, Ac, scales as v.Using r =0.4 pm and E„(p,)=10 V/m, we obtain an en-

ergy spacing of Ac-=7X10 eV for B=2.4 T, the fieldat which the v=2 minimum in R occurs. This corre-sponds to a temperature of -800 mK, of the same orderas the 200 mK at which the v=2 Auctuations decreaseappreciably.

The fact that no Auctuations are present for broadranges of B in the center of the minima, when R„=O, isnot inconsistent with the model. In these regions, peaksin R„may be present at T =0, but their I 's are so smallthat they are destroyed at the smallest experimental tem-peratures. ' As B is increased, p moves to a lower energyrelative to the Landau level, and electrons tunnel throughstates of much larger I, giving rise to peaks on the high-B side of the minimum observable at experimental tem-peratures.

As B is increased yet further, p moves to even lowerrelative energies, and the situation switches from one ofpotential hills or islands in the Fermi sea to one of Fermilakes on insulating land. (See Fig. 11.) This correspondsto the low-B sides of the R „minima. Transport maythen take place by tunneling down the length of the chan-nel from one closed contour to another. No single boundstate will dominate, and their sizes can vary. Thus fluc-tuations on the low-B sides of the minima may not alwaysbe present. And since the tunneling paths may easilypenetrate some distance into the voltage probes, the

12 940 SIMMONS, HWANG, TSUI, WEI, ENGEL, AND SHAYEGAN

AX%&&&&%+ggg&XVj&&

Rxx

i%%%Xm

(a)

Rxx

O a

(b)

Rxx

(c)

FIG. 11. Illustration of edge state configurations as a func-tion of p. The left column of figures shows the relative energiesof p and two Landau levels; the center column the correspond-ing edge state configurations {where shaded regions representenergies above p and dashed lines represent tunneling paths);and the right column the corresponding values of B. In (a) plies well between two Landau levels, corresponding to well-separated edge states and B in an R„„minimum. In (b) p islower in energy, corresponding to islands in the Fermi sea and Bat the high-8 side of an R„„minimum. In (c) p is lower yet inenergy, corresponding to Fermi lakes on insulating land, and Bat the low-B side of an R„minimum. Transport in {c)occursby tunneling along the length of the channel.

strong correlation behavior observed on the high-B sidesof the minima is expected to be absent on the low-B sides,as observed in our data.

VIII. FRACTIONAL CHARGE

The behavior of R fluctuations at the v= —,' minimum

is qualitatively similar in all respects to those appearingat integral v minima, except for the factor of —3difference in their quasiperiod. Because the quasiperiodof the fluctuations is the same for all integral v= 1, 2, 3,and 4, and is consistent with the model of Jain and Kivel-son, one is immediately led to the question of how themodel is modi6ed if the carriers are quasiparticles of frac-tional charge e*=e/3, rather than electrons.

When the 6lling factor v of a bulk 2D electron gas isprecisely —,', all electrons are in the ground state, de-scribed by Laughlin's wave function, which can carrycurrent without any dissipation, giving an R „=0. Whenv deviates slightly from —,', the incompressible groundstate remains rigid and the deviation is accommodated by

the system as quasiparticles of charge e/3, or quasiholesof charge —e/3, depending on the sign of the deviation.In a sample with edges and a disorder potential, the local"filling factor" will change with the potential, and bandsof quasiparticles will form along the sample edges andalong the disorder potential. While this picture of frac-tional edge states arises naturally from the existence of anenergy gap at v= —,', its details remain somewhat contro-versial. For the moment we proceed with the as-sumption that near v= —,

' there exist fractional edge statesalong equipotentials of energy p, analogous to the edgestates in the IQHE, only carrying quasiparticles of chargee/3 rather than electrons.

Under the conditions that (I) the quasiparticle densityis sufficiently dilute that interquasiparticle interactionsmay be neglected (which is increasingly likely to besatisfied as v~ —,

' ), and (2) that the length of the disorderpotential is much larger than the spatial extent of thequasiparticles (roughly lz), the quasiparticles can betreated as independent point particles of charge e*=e /3.The quasiparticles will then be described by an effectiveSchrodinger equation in which p —(e/c) A is replaced byp —(e*/c) A. In this high-8 limit we can also treat thequasiparticles with the same semiclassical methods usedin the IQHE regime. ' If the quasiparticles lie in a po-tential of the type treated by Jain and Kivelson, ' and byKivelson and Pokrovsky, ' the results obtained will beessentially the same as for electrons, except that the Auxquantum @=h/e is replaced by the effective Ilux quan-tum @*=h/e*. The effective Bohr-Sommerfeld relationthen becomes that each bound state enclose a fIux N4,with N an integer, and thus the energy spacing b, s( v= —,')of the bound states becomes three times what it would befor electrons at the same B. Observation of this scalingrelationship then constitutes evidence for fractionalcharge in the FQHE.

This scaling relationship can manifest itself in a num-ber of different ways, depending on the experimentalconfiguration. Variations of temperature, excitationcurrent, and p through the use of a back gate are all pos-sibilities. ' ' In measurements of R„as a unction ofB, Eq. (3) will be replaced by

2 —1rh n2DAB= m-r +

)jc I*eE„(p)

near fractional minima. Thus our observation of a factorof -3 difference in the quasiperiod of R „ fluctuationsnear the v= —,

' minimum provides evidence for a quasipar-ticle charge of e*=e /3 in the v= —,

' FQHE.Since our initial report, a number of theoretical works

relevant to our experiment have appeared. ' Thou-less; Thouless and Gefen; and Wu, Hatsugai, andKohmoto have studied the role of symmetry breaking inthe gauge invariance of a 2D system of fractionallycharged quasiparticles, and the conditions under whichthe properties of the system exhibit a periodicity of N*.Kivelson has expanded his treatment to include theefFects of the statistical interactions of the quasiparticles.He predicts different periodicities depending on whether

12 941RESISTANCE FLUCTUATIONS IN THE INTEGRAL- AND. . .

1 confi uration holds the number ofthe experimenta con gurnd suggests ain the system constant, an squasiparticles in

hich can measure theodification of our experiment w ic can

tant points wit» regt at eexh the existence of quasiparticles of c arge e

h AB would remain the same for v= —' as orplies that wAs'v= —') is onlyb t that the energy spacing Ac v=

3v= integer, ut a

vation off b, e v= 1). Thus he attributes our observato finite-temperature

her ex eriments, either utilizing a back gater or conducting a more t oroug cto vary p,

f the fluctuations at v= —,the temperatu pure de endences od t '

d t the issue.d to ad u ica esu ests that the observed resistance Auctua-

In the next section we descri e a irDl

'hich more careful in-measurements of ouour samp e, in w ic

nt de endencesns of the temperature and current epen ed2r the v=1 an m'

formed. The results are inconsistent wit eblockade effect as t e south source of the Auctuations in R„.

1.8

1.6—

1.4—

1.2—

1.0—Q

XK

0.4-

0.2-

0 ----—-I

2.0I

2.1l

2.2 2.3

IX. ADDITIONAL T AND I DEPENDENCEMEASUREMENTS

le to 300 K and back to 25 mK,After cycling the samp e osurements were per orme ora third series of measur

=1 and 2 R minima, using R& 5.7 z. xci a i

h hi h-B id of hIn Fig. 12 we show R, 5.7 z on t e ig-m eratures ranging romv= m=2 inimum for several temp

mK. All traces were taken at a cua current or T =33, 37, 62, 88, and 101 mK are all

1 'd tical and have their first peanearly i en ica,taken at tempera--2.02 T. (Other traces not shown, take p

101 mK are also nearly identical. )tur es between 33 and m ave 101 mK, the curves begin to change.

earlier, the Auctuations ur esAs seen ear ier,=2. 13 T, diminish in am-minimum, such as those near

eaks grow out of thede first, and new resistance pea snear 2.00 T at the higher temperatures.

Thus clear saturation behav'oXX

vior is o serve

' hB 'd ofthe= 1 minimum for se

same behavior is ex i ite;T(66 mK are nearly identical, wit e

~ 4. 3 T diminish first with increasing, an1

w' 'Here however, saturation is

e eaks row out o t e~ ~-4.15 T with increasing T. Here, owev

observed below T, (vv= 1 —=66 mK.In Fi . 14, we show traces of R I 5.7 s on t e igv=2 at 25 mK, ta en a iv=, k t different currents. Traces

~ ~ ~I v =2 = 1.7—3.0 nA are virtually Identi-th ff t ofcal. At currents ~ 3.0 nA, t e e ec

X2—

— v=1

3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6

= 1 at several different tempera-FIG. 3. R =RI &78 near v= a s0.5 nA. The behavior is simi-

~ ~ ~ ~tures meaeasured with a current o . nt that the amplitude saturates atlar to that near v=2, except that t e amp

'

lower temperatures, below T (v ——1i=66 mK.

=2 at several different tempera-R 1 5 ~ 7 8 near v= aFIG 12 xx= 15;,

c cle. The current used wass taken on a third temperature cyc e. e ctures, a ention am litude clearly saturates at tempera-p

tures below, v=l T ( =2)—= 121 mK. The onset o e uT.moves to lower 8 with increasing

12 942 SIMMONS, HWANG, TSUI, %'EI, ENG EL AND SHAYEGAN

1.0Q

0.8

0.6

04

0.2

I

2.0I

2.1B(

I

2.2 2.3

0 -----I

3.9 4.0 4.1 4.2 4.3B(

I

45 4.6

FIG. 14. R„~ =1,5;7, 8=R . near v==2 for several different

turates atrrents at T=25 m . eK The fluctuation amplitude satu( = =—. 3.0 A Increasing the currentb o,((v=2)=—1.7— . n

~ 3.0 nA has an effect similar to that o raising, w'

B and their amplitude di-onset of fluctuations moving to lower B anminishlng.

ar v = 1 for several differentFIG. 15. R„„=R15 ~ 7 8 near v=K. The behavior is similar to that nearcurrents, at T=25 m . e ai near

v=2, except t at e uh th 6 ctuations saturate below a ocurrent, I, ( v = 1 }—=0.5 nA.

rature. Innt is similar to that of increasing temperatuh h' h-B side of v=1 at 2SFig. 15 we show, 578R . onte ig-

a' b h vior ism K for different currents. ga', v'ain saturation e av'

nt but this time for currents beloww I (v=1)—=present, u i1 t de with increas-A. The fIuctuations decrease in amp i u

in I, until they are nearly completely gone at I=30 nA.ing, un iIn the Coulom oc ab bl k ge pictUre the allowed energy

ial hill are discretized by the chargingstates on the potentia i areener associate wi ed th the tunneling of single elecgy

~ ~

onto the potential i . ' hthe addition of a single electron is E, =e, w erethe ca acitance o e pf th otential hill with respect to thep

d tunneling occurs whenever ped e states. As p is varie, u~ ~ ~c ' '

ll ed energy, and oscillations in Rcoincides with an a owe en1 . Finite-temperature effects appear w enwill occur. in'

more than one allowedk T~E, and electrons can access more an

will remain constant wit an v,v. This behavior, however, is nobe independent of v. is, no

la ed b the data. On the ot er an,E (4)' ives an allowed ener-Sommerfeld quantization [ q. , g'

s acin Ac. ~v or, anB ' d thus for integer v the satu-1 s T (v) ~ v, as obserued in theration temperature sca es as, v

data.x lained by a similarH h rrent effects can be exp aine y

hanism. As the current is increased, the Ha gve )I. When the Hall voltageVIt increases as Vtt =(h/ve

eV is reater t an or equth or equal to the allowed energy spac-

nd hi h current effects will occur. If a ou omblockade dominates, then the allowe en g

p d g and hence the Hall voltage VH, atE inde endent o v, anwhich saturation occurs is a so in p

&h) V, in the Coulomb blockade case wewill scale asen ex ect that the saturation current wi s

hand if the discretization of ther is instead determined by the Bohr-Sommer e in-

c' '

n b E . (4) the allowed energyterference condition, then y q.in hc. ~v, and the saturation Hall voltage II, v

= ve jh)V we have in thewill also scale as v. Since I= veBohr-Sommerfeld case that the saturation current sca esas I (v) ~ v . While our values for I, v= 1 ) and I,(v=

a'

than T (v=1) and T, (v=2),have a larger uncertainty ancalin as v than as v, anthey correspond better with a sc g

hence support o r-8 h -Sommerfeld quantization over t eCoulomb blockade.

ue toe that if the high current effect is dueFinally, we note t a ilin throu h more than one energy s a e atunne ing ro

nd eV are indirect measuresthen since both ks T, (v) an e H, are'

thats acin, one also expects t aof the allowed energy spacing,of eV /k~ for, ~k =—T (v). Our measured value of e H, s oeVH, ~ —=, v.

v =2 is —250 —450 K, and our value of e V~,e V /k for=1 is —150 mK. Both of these agree with their respec-

tive values of T, (v) to wet in a amay well be due to the differing electron energy istri u-tions involved.

RESISTANCE FLUCTUATIONS IN THE INTEGRAL- AND. . . 12 943

X. SUMMARY

The measurements discussed in this paper represent asystematic study of the v dependence of resistance Aue-tuations in the quantum-Hall-effect regime. For integerv, the fluctuations are consistent with Jain and Kivelson smodel of inter-edge-state tunneling via disorder-inducedmagnetically bound states, in that (1i the quasiperiod ofthe fluctuations is independent of v and corresponds to abound state of a diameter roughly equal to the conduct-ing width of the channel; (2) the temperatures at whichthe Auctuations begin to disappear are of the same orderas the estimated bound-state energy spacing; and (3) thecorrelation behavior between fIuctuation patterns mea-sured using different pairs of voltage probes clearly showsthat the source of the fluctuations is local. Further, acareful study of the saturation temperatures and satura-tion currents for v=1 and 2 strongly indicates that, forinteger v, the allowed energy spacing scales as v, con-sistent with Bohr-Sommerfeld quantization but not withthe Coulomb blockade picture.

For v= —,', the quasiperiod of the resistance fluctuations

is a factor of 3 larger than for integer v. This is con-sistent with a simple extension of the model to encompassquasiparticles of fractional charge e*=e/3, and there-fore provides evidence for their existence. We note, how-ever, that the data for v= —,

' are not as extensive as for in-teger v, particularly with respect to current and tempera-ture dependence, and so we are unable to critically assessthe suggestions of Lee concerning v= —,'. Further experi-ments in the FQHE regime, particularly those whichmore directly probe the bound-state energy spacing Ac,are needed.

ACKNO&I. EDGMENTS

The authors have benefited from stimulating discus-sions with numerous colleagues, among whom J. K. Jain,S. A. Kivelson, P. A. Lee, and X. G. %en deserve specialmention. This work is supported by ONR through Con-tract No. N00014-88-J-1567, by NSF through Ctyrants No.DMR-8719694 and No. DMR-8921073, and by a grantfrom the NEC Corporation.

Present address: Sandia National Laboratories, Division 1152,Albuquerque, NM 87185.

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23In Fig. 1 of Ref. 22 the sample contacts were mislabeled.Contacts 6 and 8 should have been interchanged. This erra-turn does not in any way afFect the conclusions of that work.K. K. Choi and D. C. Tsui, Appl. Phys. Lett. 50, 110 (1987).

25A contact problem in the wide channel develops for B) -7T, causing a slowly increasing background resistance to beadded to R . Because the —,', —,, and —, minima appear locallyflat, however, we assume that a simple subtraction of thebackground resistance yields the true R, and thus that thecontact problem does not in any way a6'ect the validity of ourconclusions.W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D. Jack-el, and D. M. Tenant, Phys. Rev. Lett. 58, 2347 (1987).

~ J. A. Davies and J. A. Nixon, Phys. Rev. 8 39, 3423 (1989).~8A. H. MacDonald, Phys. Rev. Lett. 64, 220 (1990); A. M.

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12 944 SIMMONS, HWANG, TSUI, WEI, ENGEL, AND SHAYEGAN

P. A. Lee, Phys. Rev. Lett. 65, 2206 (1990).D. J. Thouless, Phys. Rev. B 40, 12 034 (1989).

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