Measuring the fractional charge and its evolution

M. HeiblumBraun Center for Submicron Research, Department of Condensed Matter Physics,Weizmann Institute of Science, Rehovot 76100, Israel

S u m m a r y : The charge of the quasiparticles was measured as the~; tun-neled through a narrow constriction embedded in a fractional quantumHall (FQH) liquid. The determination of the charge was made via detailedmeasurements of the shot noise generated by the partially reflecting con-striction. In the v = 1/3 FQH state, a quasiparticle charge of e* = e/3was deduced when the reflection at the constriction was weak. Similarly,when only the second edge channel in the v = 2/5 FQH state was weaklyreflected, allowing the ~, = 1/3 channel to pass uninerrupted, a quasipar-ticle charge of e" -- e/5 was deduced. These results agree with Laughlin'spredictions. For a progressively stronger reflection of the 1/3 state at theconstriction, the charge of the quasiparticle, deduced from the measurednoise, evolved toward e near pinch off. However, when the constriction waspinched off to reflect strongly only the higher laying channel in the 2/5state, the charge of the quasiparticle remained small, nearly e/5, and lessthan e/3. This counter-intuitive result presently lacks theoretical under-standing.

1 I n t r o d u c t i o n

Ever since Milliken's famous experiment it is well known that the free electri-cal charge is quantized in units of the electronic charge - e. For this reason,Laughlins theoretical prediction of the existence of fractionally charged quasi-particles, put forward in order t o explain the Fractional Quantum Hall (FQH)effect, is very counter intuitive. The FQH effect is a phenomenon that occurs ina Two Dimensional Electron Gas (2DEG) subjected t o a strong perpendicularmagnetic field. This effect results from the strong interaction a m o n g the elec-trons and consequently the current can be described as a result of the motion ofthe above mentioned quasiparticles. We measured this elusive fractional chargeby utilizing measurements of shot noise introduced artificially into the systemvia a tunnelling barrier. Shot noise results from the discreteness of the currentcarrying charges and is thus proportional t o their charge, e*, and t o the averagetransmitted current I. The spectral density in the classical limit is S = 2 e ' I ,

22 M. Heiblum

derived by Schottky in 1918. Our shot noise measurements show unambiguouslythat current in a 2DEG in the FQH regime, at fractional filling factors v = 1/3and 2/5, is carried by fractional charge portions e / 3 and e/5, respectively - inagreement with Langhlin's prediction. We also measured the evolution of thequasiparticle charge across the full r ange o f tunnelling transparency.

2 T h e Q u a n t u m H a l l e f f e c t

The energy spectrum of a 2DEG subjected to a strong perpendicular magneticfield, B, consists o f highly degenerate Landau levels with a degeneracy per unitarea d = B/@o, with ~o = h / e the flux q u a n t u m (h being the Plank constant).Whenever the magnetic field is such that an integer number v (the filling factor)of Landau levels are occupied, that is ~ = n s / d equals an integer (ns being the2DEG area l density), the longitudinal conductivity of the 2DEG vanishes whilethe Hall conductivity equals v e 2 / h with very high accuracy. This phenomenon isknown as the Integer Quan tum Hall (IQH) effect [1, 2]. A similar phenomenon,the FQH effect, occurs a t fractional filling, namely, when the filling fac to r equalsa rational fraction, with an odd denominator 2p + 1 [2, 3]. In contrast to theIQH effect, which is well understood in terms o f non-interacting electrons, theFQH effect can not be explained in such terms and is believed to result frominteractions among the electrons, brought about by the strong magnetic field.

Laughlin [4] had argued t h a t the FQH effect could be explained in terms o fquasiparticles, each having a fractional charge q = e / ( 2 p + 1). Although his the-ory is consistent with most of the experimental d a t a , more definite experimentswere needed to substantiate the existence of fractional charges. Fo r example,early Aharonov-Bohm type measurements [4] were proven to be inadequate, inprinciple, to reveal the fractional charge [5]. A more recent experiment by Gold-man and Su [6], based on resonant tunneling of quasiparticles into a n isolatedisland, measured the fractional charge. It was, however, interpreted differentlyby Franklin et al. [7]. Moreover, the inherent difficulty in these experiments isthe determination o f the isolated islands capacitance - a crucial parameter insuch experiments. Shot noise measurements, on the other hand, p robe the tem-pora l behavior of the current and, thus, offer a ra the r direct way o f measuringthe charge. Indeed, as early as in 1987, Tsui [8] suggested t h a t the quasiparticlecharge could, in principle, be determined by measuring the induced shot noise inthe FQH regime. However, no justifying theory was available unti l Wen [9] rec-ognized that transport in the FQH regime could be t rea ted within a frameworkof One Dimensional ( ld) interacting electrons, propagating along the edge o f thetwo dimensional plane, making use of the so called Luttinger liquid model. Basedon this model subsequent theoretical works [10] predicted that shot noise in theFQH regime should mimic the noise of partitioned non-interacting particles that

Measuring the fractional charge and its evolution 23

propagate in a ld system, in the absence of an applied magnetic field.

3 S h o t n o i s e a n d e d g e c h a n n e l s

At zero temperature (T = 0), the contribution to the shot noise of the pthpropagating channel, in a few-channel ballistic conductor, is [11]

ST=O = 2 e ' Y g Q p ( 1 - tp), (3.1)

where S is the low frequency ( f << e V / h ) spectral density o f current fluctuations( S A f = (i2)), V the applied source-drain voltage, gp the conductance o f the fullytransmitted pth channel in the QPC, and tp is the transmission coefficient o f thepth channel. This reduces to the well known classical Poissonian expression forshot noise, when tp << 1 (the 'Schottky equation'), ST=o = 2 e ' I , with I = V g p t pthe DC current carried by the pth channel.

The recent theoretical studies of shot noise in the FQH regime, based on thechira l Luttinger liquid model, are applicable only to Laughlin fractional states,e.g., v = 1/3, 1/5, etc. [10], namely, when the edge is composed o f one channelonly. They predict spectral density of noise like that of Eq. 3.1 a t its limits

ST=o = 2 e ' V g p ( 1 - tp) = 2 e * I r ( tp ~ 1)

ST=0 = 2 e V g p t p = 2eIt ( tp ~ 0), (3.2)

where Ir and It are the reflected and transmitted DC currents, respectively.The most important outcome of Eq. 3.2 is the prediction that the tunneling o fquasiparticles, with charge e/3, e/5, etc. at weak reflection (tp ~, 1), contributeto the shot noise. No easily accessible formulation for the transition region exists.

One can gain insight into the characteristics of the expected shot noise inthe FQH regime by considering the Composite Fermion (CF) model [12]. Inits simplest form the CF model identifies the fractionally filled f i r s t electronLandau level, v = p / ( 2 p + 1), as integer p filled Landau levels of CFs (VCF = p).Each CF consists of an electron and two magnetic flux quanta (~0 = h / e ) ,opposing the original magnetic field, are attached to it. Hence, the n e t magneticfield sensed by the CFs is B - 2 n s h / e (equals to zero a t v = 1/2). U n d e rthis weaker effective magnetic field the CFs can be approximated by weaklyinteracting quasiparticles, flowing in separate and non-interacting edge channels- hence, justifying the application o f the above mentioned formulae for the noise.When the quasi ld constriction is being pinched off and the conductance is ina transition between two different FQH plateaus of the series p / ( 2 p + 1), onlythe highest edge channel is being partitioned. The other, lower lying channels,can be approximated as being fully transmitted. Consequently, in Eqs. 3.! and3.2, p designates the CF edge channel that is being partitioned. An example is

24 M. Heiblum

the transition between v = 1/3 and the insulator, namely, p = 1; gl = go~3 andtl = 3g/go. Similarly, in the transition between u = 2/5 and u = 1/3 namely,p = 2; g2 = ( 2 / 5 - 1/3)go and t2 = (g/go - 1/3)/(2/5 - 1/3). Here g is the t o t a lconductance and go = e2/h is the q u a n t u m conductance. Using a simplifiedmodel o f CFs de Picciotto predicted [13] that the quasiparticle charge, deducedfrom shot noise measurements, would vary from e* = e/2p + 1 a t tp ~ 1 toe* = e / 2 p - 1 a t tp ~ 0, depending linearly o n t p .

Led by the proposal of Kane and Fishe r and la te r others [10], previous studiesconcentrated on the range of the weak back-scattering limit specified in Eq. 3.2[14, 15]. But since the transmission coefficient in the experiments ranged between0.6 and 1 t h e approximation tp ~ I was not valid. A more general expression forshot noise due to non-interacting particles [16], applicable a t finite temperatures,was utilized

[ ( 2kB ]ST = 2e*Vgttp(1 - tp) coth \ 2kBT ] e*V J + 4gksT. (3.3)

This equation leads t o a finite noise a t zero applied voltage, ST = 4 k s T g - theJohnson-Nyquist expression. When V > VT ~ 2 k s T / e * the noise approachesthe l inear behavior predicted by Eqs. 3.1 and 3.2. This expression served as thebasis for charge determination.

4 S h o t n o i s e e x p e r i m e n t s

In o rde r to realize such noise measurements we utilized a q u a n t u m poin t contact(QPC) - a constriction in the plane o f the 2DEG - that partially reflects thecurrent. The 2DEGs, embedded in a GaAs-A1GaAs heterostructure, some 100nm beneath the surface, had carr ier densities around ns = 1 • 1011 cm-2 anda mobility of about It = 4- 106 cm2/Vs a t 1.5 K. The QPC was formed bytwo metallic gates evaporated on the surface of the structure, separated by anopening of some 300 nm, which is a few Fermi wavelengths wide (see inset inFig. 1). By applying negative voltage to the gates with respect to the 2DEG,thus, imposing a loca l repulsive potential in the plane o f the 2DEG, one maycontrollably reflect the incoming current. The sample was inserted into a dilutionrefrigerator with a base temperature of a b o u t 50 mK. Noise measurements wheredone by employing an extremely low noise, home made, preamplifier, placed ina 4.2 K reservoir. The preamplifier is manufactured from GaAs-A1GaAs t ran -sistors, grown in our Molecular Beam Epitaxy system. The preamplifier has avoltage noise as low as 2.5 • 10-19 V 2 / H z and a current noise of some 3 • 10 -29(or 1.1- 10 -2s) Amp2/Hz a t 1.7 (or a t 4) MHz.

Measuring the fractional charge and its evolution 25

F i g u r e 1 The total current noise inferred to the input of the preamplifier asa function of the input conductance at equilibrium (full circles). The measurednoise is a sum of t h e r m a l noise, 4 kBTg (leading to a straight line) and theconstant noise of the amplifier. This measurement allows the determination ofthe temperature of the 2DEG. Insert: The quantum point contact in the twodimensional electron gas is shown to be connected to a LCR-circuit at the inputof a cryogenic preamplifier.

Current fluctuations, generated in the QPC, were fed into a JLCR resonantcircuit, with most of the capacitance contributed by the coaxial cable, whichconnects the sample at 50 mK t o the preamplifier at 4.2 K. Outside the cryostatthe amplified signal was fed into an additional amplifier and from there t o aspectrum analyzer whi~:h measured the current fluctuations within a band of30 - 100 kHz a b o u t a central frequency chosen in the range of 1.6 - 4 MHz. Sincethe absolute magnitude of the noise signal is of outmost importance, a carefulcalibration of the total gain, from the QPC t o the spectrum analyzer, was doneby utilizing a caiibra~ed current noise source. This allowed the translation ofthe spectrum analyzer output t o the spectral density of shot noise in the QPC.Although the cooled pre-amplifier had excellent characteristics it still introducedsubstantial current fluctuations into the circuit. This noise and the rest of theunwanted noise of the whole amplification chain must be subtracted from thetotal measured noise. By measuring the total noise of the unbiased QPC whilevarying the QPC conductance g, we deduced both the electron temperature,T = (~ST/~g)/4kB, and the contribution of our amplifier t o the total noise,extracted from the extrapolated total noise to zero conductance, as shown in

26 M. Heiblum

Figure 2 Shot Noise as a function of DC current, I, through the QPC with-out an applied magnetic field (full circles). The solid line is Eq. 3.3 with thetemperature deduced from Fig. 1.

Fig. 1. This procedure was always followed a t the beginning o f each experiment.Shot noise measurements, as a function o f the DC current through t h e partially

pinched off QPC (partly transmitted 1s t channel), were performed f i r s t in theabsence o f a magnetic field. The results, a f t e r calibration and subtraction of theamplifier's noise, are shown in Fig. 2. The transmission of the lowest laying quasild channel was simply deduced from the measured conductance normalized by2e2/h (the factor 2 accounts for spin degeneracy). Our data fits a lmos t perfectlythe expected noise predicted by Eq. 3.2 using the measured electron temperatureand the electronic charge without any fitting parameters.

The magnetic field was then increased to the range of 12 - 14 Tesla. The two-terminal conductance exhibited Hall plateaus (not shown), expected in the IQHand in the FQH regimes (with ~ = 2/5, 3/5, 2 /3 and 1/3 clearly visible). Noisewas measured at different filling factors in the bulk and different transmissionsof the QPC. We f i r s t measured the noise in the conductance plateaus. This wasfollowed by noise measurements as the QPC was smoothly pinched-off, reflectinghigher laying channels and revealing lower conductance plateaus. Fig. 3a shows,as an example, the behavior of the conductance as the QPC is be ing pinchedoff for a bulk-filling factor o f /"bulk : 2 / 5 . Fig. 3b shows the measured noise,as a function o f the DC current through the QPC, in the 2/5 p la teau (t2 = 1,

Measuring the fractional charge and its evolution 27

Figure 3 Left: The conductance at a bulk filling factor of 2/5 as a function ofthe QPC gate voltage affecting the constriction width. Two nice plateaus, 2/5and 1/3, are observed. Right: Noise measured at the two plateaus, in points Aand B. The noise is constant and does not depend on the current, accounting forthe thermal and stray noise. No shot noise is observed. The conductance is shownfor reference.

t l = 1) and the 1/3 plateau (t2 = 0, t l = 1). No excess noise above the thermalnoise was found. This observation leads to a few conclusions: The first is t h a tthe sample does not heat up with increasing DC current. The second is t h a tthe impinging current is noiseless. The third is that the two CF channels areindependent. This can be understood by writing down the fluctuations of thetota l , unpartitioned, current supported by the two channels and equating themto zero: {~(il + i2)2) -- (~(il)2) q- {~(i2)2) q-2{~(ii" i2)) -- 0 . Since we also foundthat (~(il)2) = 0, each of the o t h e r two terms must be identically zero, leadingto il and i2 being independent.

The noise was then measured by partially reflecting the current, moving thusdown from the conductance plateau. The noise was drastically suppressed com-pared to the noise measured in the absence of a magnetic field. We star t withmeasurements of noise as a function o f the DC current, for a bulk-filling factorof//bulk ---- 1/3, a t weak reflection. The conductance, and, hence, the transmis-sion, remain fairly constant throughout the range of measurements. Fig. 4 showsnoise results as a function of back-scattered current for two transmission coef-ficients. The predictions of Eq. 3.3, for charge e" = e / 3 and e, are plotted insolid lines. The d a t a , a f t e r calibration a n d subtraction o f the amplifiers' chaincontribution, is in excellent agreement with the prediction of the noise generatedby non-interacting particles with charge e/3. In order to validate these resultsand verify that only the local filling factor within the QPC is important in thedetermination of the charge, different bulk filling factors were chosen for the

28 M . Heiblum

Figure 4 Shot Noise as a function of the back scattered current, I~, in the frac-tional quantum Hall regime at v -- 1/3 for two different transmission coefficientsthrough the quantum point contact (open circles and squares). The solid linescorrespond to Eq. 3.3 with a charge e° -- e/3 and the appropriate t. For compar-ison the expected behavior of the noise for e" --- e and t -- 0.82 is shown by thedotted line.

measurements. In Fig. 5 we give a n example o f bulk filling factor ~'bum = 1/2,with the noise measured when the QPC is fully reflecting all the higher CF chan-nels and only part ly reflecting only the 1st CF channel. Again, the da t a agreesvery well with the predicted shot noise for quasiparticles with charge e" = e/3.

What a b o u t the charge of quasiparticles in higher CF channels? This mea-surement is impor tant , since it proves that the measurement does n o t merelymeasure the conductance (or the filling factor) but the t rue charge of t h e quasi-particles. To do just t h a t , the charge was measured by slightly pinching-off the2nd CF channel, again for two different filling factors in the bulk, allowing the 1stCF channel to be fully transmitted. Fig. 6(a) shows the measured noise and thedifferential conductance as a function o f the DC current, for two values o f thetransmission coefficient o f the QPC, a t ~um ~ 1/2. Using a quasiparticle chargee* = e / 5 and a transmission deduced from the average value of the conductance(which is fairly constant), the measured noise agrees well with the predictionof Eq. 3.2 throughout the whole range of DC transmitted current. Figure 6(b)shows similar data for a bulk filling ~um = 2/5 and t2 = 0.9, but the plotis a function of the reflected DC current. Even though the signal is w@ak and

Measuring the fractional charge and its evolution 29

Figure 5 Shot Noise measured as a function of the transmitted current, for abulk filling factor v = 1/2 and transmission t = 0.9 of the 1st CF channel. Theexcellent agreement with a quasiparticle charge of e/3 shows that the bulk fillingfactor is not important.

scattering of the da t a is relatively large, it clearly verifies that the charge of thequasiparticles is e* = e/5. Here again, we see a clear manifestation o f transportin the 2nd CF channel with no observable contribution o f the 1s t CF channel(which is fully transmitted); as one would have expected for non-interacting CFchannels.

In the strong back-scattering limit, though, the quasiparticle charge is ex-pected to be different - as suggested by Eq. 3.2. Measurements in this rangewere lacking thus far due to technical difficulties. As the QPC constriction isbeing pinched-off, in o rde r t o reflect a larger portion of the incident current,the conductance exhibits the familiar impurity resonances as a function o f theconstriction width (see Fig. 7). Moreover, the I-V characteristic becomes highlynonlinear (g and t depend on current), making the analysis difficult. Measuringa large number of samples, and across the full range of the transmission coeffi-cients of the f i r s t two CF channels (u = 1/3 and u = 2/5), we found relatively'resonant-free' samples. Moreover, we extended our analysis o f the noise in o rde rto account for the non-linearity of the I-V characteristic.

Experimental results are presented from four samples: th ree measured in theu = 1/3 FQH s t a t e and two in the u = 2/5 FQH state. The bare Hall samplesof this batch of devices exhibited, as a function of magnetic field, a n accurate

30 M. Heiblum

F i g u r e 6 Left: Sho t Noise measurements as a funct ion of t r a n s m i t t e d c u r r e n tfor bulk f i l l ing f a c t o r v = 1/2 wi th the 2nd C F c h a n n e l b e i n g w e a k l y reflected. Aquasipar t ic le c h a r g e of e/5 i s d e d u c e d . Righ t : S imi lar measurements wi th r e s p e c tto the reflected c u r r e n t and at a bulk f i l l ing factor v = 2 /5 . A g a i n a nice a g r e e m e n twi th a quasipar t ic le c h a r g e of e/5 is f o u n d .

F i g u r e 7 Two-terminal c o n d u c t a n c e as a funct ion of Q P C gate v o l t a g e forsamples # 1 and # 4 . The deviat ions from the quantized values of the c o n d u c t a n c eare due to the longitudinal resistance. So l id poin ts show c o n d u c t a n c ev a l u e s w h e r ec o n d u c t a n c e and no i se measurements were m a d e . R i g h t Insert : C o n d u c t a n c e asa funct ion of a p p l i e d DC c u r r e n t at the poin ts shown. Left Insert : S c h e m a t i c ofs a m p l e and m e a s u r e m e n t sys tem.

Measuring the fractional charge and its evolution 31

v = 1/3 quantization of the resistance but deviated at the v = 2/5 plateau, dueto a finite longitudinal resistance. The measurements in the v = 2/5 channelwere conducted in two different bulk filling factors: /]bulk ~- 2/5 and Vbulk = 1/2(see sample # 1 in Fig. 7), and for the measurements in the v = 1/3 channelthe bulk filling factors were ~ulk = 1/3 and //bulk ~-" 1/2 (see sample # 4 inFig. 7). Typical problems are seen in Fig. 1: sample # 1 shows a single large'resonance' which prohibits further measurement into the 1/3 channel, and theclosure of the 1/3 channel in sample #4, even though much smoother, saturatesa t about O.le2/h. As seen in the inset of Fig. 7 the differential conductance is notconstant and varies with current. While a t point A, where t is relatively large,the conductance is almost constant with changing current (Ag/gl=o = 0.05), a tpo in t C, where t is very small, there is a significant change (Ag/gi=o = 0.3) inthe differential conductance a t large currents.

In general, we measured two quantities: the differential conductance, g o¢ e't ,and the noise. We used these measurements to extract t and e*. However, theanalysis is complicated due t o the strong dependence of the conductance on thecurrent. To account for this non-linearity, the energy independent Eq. 3.3 wasmodified, by resorting to the integral over energy used to derive Eq. 3.3 (foundin Ref. 14). Still, the range of DC current was restricted to minimize the effecto f the non-linearity. The dependence o f the conductance on the current wasattributed only to t, i.e., the charge e* was assumed not to vary for a fixed gatevoltage of the QPC. We believe that this is a reasonable assumption in the smallrange of current where we measured. Replacing the integration over energy bya sum over discreet points and substituting t in terms of g and e" in Eq. 3.3, weget for v = 1/3

S T ( I ) __ 2e,I N ~-~. (1 gi/go -

i=1 \ 2ksT ]2kuT]e*V J + 4gkBT. (4.4)

Here i runs over the measured points (N) up to current I, and gi is the differen-tial conductance a t each point. The coth term, however, rising from integrationover the Fermi function, was inserted in 'by hand ' . In the v = 2/5 channel wesubstitute only the fraction of the current that flows through that channel (usingthe CF model), I2/5/Ito t = (g/go - 1/3)/(2/5 - 1/3), and for the transmissiont2/5 = (g/go - 1/3)/[(2/5 - 1/3)he*/e]. Indeed, if e* -- e / 5 is substituted in thisexpression, t2/5 is found to be the expected bare transmission of the CF channel.The noise expression now contains a single fitting parameter e*. In o t h e r words,for each width of the QPC constriction we find the best fitting quasiparticlecharge e* and consequently the channel transmission t via g.

32 M . Heiblum

5 Q u a s i p a r t i c l e c h a r g e

The dependence o f the quasiparticle charge on transmission coefficient, for allfour samples, is summarized in Fig. 8. All results collapse into two separatecurves. While in the v = 1/3 case the charge changes smoothly from e / 3 atweak reflection (large t) to a r o u n d e a t strong reflection (at t = 0.1), the chargein the L, = 2/5 case stays nearly e / 5 over a lmos t the full range o f transmission.There is an apparent slight increase of e* at the lower transmission coefficients,which could result from contributions from both channels, due to a possible weakpartitioning also of the 1/3 channel. Although the scattering of the data (dueto the small signal) prevents an accurate determination of the charge, it clearlydoes not experience the steep rise that is observed a t ~ = 1/3. In accordancewith Ref. 13, this difference can be understood by considering how much chargecrosses the constriction when one composite fermion, which is composed of anelectron charge and two flux quanta, traverses it. When the two flux q u a n t a t ra -verse the constriction, they induce a perpendicular component to their motion ofa Chern-Simons electric field, t h a t generates a current between the t ransmi t tedand reflected edges, leading to a net extra charge crossing the constriction. Thecharge of the net tunnelling quasiparticle is the sum of that extra charge and theintrinsic electron charge. In the v = 1/3 case, a strongly pinched-off constrictionis almost an insulator and the extra charge is negligible - leading to a quasiparti-cle charge, approximately e. In contrast, in the t / = 2/5 case only the h ighe redgechannel is strongly pinched-off while the lower laying one is fully t ransmi t ted .Consequently, the constriction is not an insulator and the extra charge t h a t istransferred is finite, leading to a quasiparticle charge different than e. A rigorousanalysis is lacking thus far.

6 C o n c l u s i o n

Shot noise measurements were proven here to be a powerful and useful tool fordeducing the charge o f the quasiparticles in the fractional quantum Hall effectregime. Thus far, they had been useful only when the particles behave indepen-dently. In more general situations, when the independent particle model is notapplicable, the use of such measurements is questionable since it lacks theoreticaljustification. The experimenters are now awaiting new theoretical developmentsin o rde r t o encourage experiments in other highly correlated systems.

Measuring the fractional charge and its evolution 33

Figure 8 Summary of the results of the determined evolution of the chargeof the quasiparticle as a function of transmission, for all four samples, for thev = 1/3 and v = 2/5 channels.

7 Acknowledgments

The work described here was initiated by M. Reznikov, who, together with R.de Picciotto performed the bulk of the experiments a t high QPC transmissions.T. Griffiths and E. Comforti performed the measurements across the full r angeof the transmission. High purity samples were grown by V. Umansky and thesub-micron lithography was done by D. Mahalu. I t h a n k H. Shtrikman for com-ment ing on the manuscript. The work was partly supported by a gran t from theIsraeli Academy o f Science.

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34 M. Heiblum

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