Transport and tunneling within a compressible electron ...

PHYSICAL REVIEW B 15 DECEMBER 1998-IIVOLUME 58, NUMBER 24

Transport and tunneling within a compressible electron liquid in wires and ringsof GaAs/AlxGa12xAs heterostructures

J. Wrobel and T. DietlInstitute of Physics, Polish Academy of Sciences, aleja Lotniko´w 32/46, PL-02 668 Warszawa, Poland

K. Reginski and M. BugajskiInstitute of Electron Technology, aleja Lotniko´w 32/46, PL-02 668 Warszawa, Poland

~Received 25 February 1998!

A detailed study of low-temperature magnetoconductance in betweenn52 andn53 quantized Hall pla-teaus is presented. The data are obtained for disordered two-terminal submicron wires and rings defined inGaAs/AlxGa12xAs heterostructures modulation doped by Si. In the region between the plateaus, electrondensity of compressible liquid in the central current strip increases gradually when lowering the magnetic field.At low densities of the electrons in the strip, a sequence of peaks is detected in the dependence of conductanceon the magnetic field. This unexpected effect is interpreted in terms of the Coulomb blockade associated withthe charging of the current strip. In the same range of the magnetic fields, a temperature dependence ofconductance, together with the absence of the Aharonov-Bohm oscillations for transport along the central stripin the ring, are taken as the evidence for an enhancement of the electron-phonon scattering rate at the perco-lation threshold. The frequency doubling of the Aharonov-Bohm oscillations in a ring with an additionalpotential barrier is discussed in terms of the phase concept. Slow time evolution of conductance is observed forhigher concentrations of the electrons in the current strip. This surprising noise is attributed to glassy dynamicsof localized electrons in the wire center, and to the corresponding time dependence of the impurity-assistedtunneling probability between the current carrying regions.@S0163-1829~98!03548-6#

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I. INTRODUCTION

Soon after the discoveries of the integer quantum Heffect in a two-dimensional~2D! electron gas and the conductance quantization of a one-dimensional~1D! quantumpoint contact, it was realized that these remarkable effeare essentially a single-electron phenomena. In particutransport properties of 2D and 1D systems in the high mnetic fields were successfully described in terms ofskipping-orbit/edge current concept.1,2 The idea of ancurrent-carrying edge state with almost zero width provedbe very useful, and made it possible to describe a vamount of complex data obtained for mesoscopic devicethe so-called adiabatic transport regime.3

In the last decade, however, the role of many-electeffects has became more and more appreciated. Particuthe total electrostatic energy and nonlinear screening hturned out to play an important role in the carrier redistribtion in the vicinity of edges and impurity potentials.4 Theconcept of the edge currents with nonzero width was initiaapplied to the fractional quantum Hall effect,5 and then ex-tended to the integer case.6 Self-consistent calculations of thmany-electron energy were presented in a large numbepapers7–13 and, gradually, their results have began to be eployed for the interpretation of experimental data. Mostthe experimental studies have been devoted to edge traneither in large 2D systems14–17 or in quantum point contactand zero-dimensional~0D! quantum dots.18–20Less attentionhas been given to the quantum wires, for which the theoical models should apply most directly.

The total electrostatic energy and electron distributionan infinite quantum wire was calculated by Chklovsk

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Matveev, and Shklovskii~CMS! ~Ref. 9! adopting theThomas-Fermi approximation. By minimizing the total eergy of the wire those authors showed that edge cur‘‘strips,’’ containing the compressible electron liquid, aformed in the high magnetic fields. Such strips are charterized by the position dependent filling factorn, and behavelike a metal — they can effectively screen external potetials. The strips are separated by insulatorlike incompressliquid regions. More recently, this model was extendedtaking into account Hartree21,22 and Hartree-Fockinteractions.23 Generally, more rigorous methods give resuwhich are comparable with the CMS model, especially inlimit of the high magnetic fields and low temperatures.

In addition, CMS argued that the two-terminal condutance of an ideal wire is given by

G5~e2/h!nB~0!,

wherenB(0) is the electron filling factor in the center of thwire. Hence, if the middle strip in the channel isincompress-ible, the conductanceG becomes quantized.

In our previous study24,25 we examined the magneticfield–induced conductance quantization in a two-termiballistic wire of lengthL51.6 mm, fabricated from a highmobility (;106 cm2/Vs! modulation-doped GaAsAl xGa12xAs heterostructure. Rather narrow quantized pteaux were observed at 1.5 K, and their positions and widwere in good quantitative agreement with the CMS theoThe width of the plateaus was small because the curstrips in the center of the channel arecompressiblein a widerange of the magnetic fields. In such a case,nB(0) is nonin-teger, andG is not quantized.

16 252 ©1998 The American Physical Society

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PRB 58 16 253TRANSPORT AND TUNNELING WITHIN A . . .

Another manifestation of the many-body interactionsthe position dependent spin splitting at the edges of theelectron gas.8,26 The magnitude of spin splitting in the casof quantum wires has been considered theoretically,27–29andalso experimentally in our earlier work.30 In particular, it hasbeen predicted27 that the electrong factor is not enhancedover its one-electron value~in contrast to the 2D case!, pro-vided that the kinetic energy of the edge-skipping electris large compared to the exchange energy. Recently, aconsistent treatment of exchange and correlations in quanwires has been presented.31 The results concerning the lacof the enhancement of the spin splitting were found31 to be invery good agreement with our experimental findings.30

In the present paper we examine the two-terminal condtanceG betweenquantized plateaux for moredisorderedde-vices. When G is not quantized the middle strip oflow-mobility channel is a mixture of incompressible~‘‘insu-lator’’ ! and compressible~‘‘metal’’ ! components. Furthermore, the compressible liquid in the channel center shobreak up into localized and delocalized regions, which foa percolation network.32 Our purpose is to determine thproperties of such an intermediate state by studying theneling and phase coherence of electrons below and abovquantization threshold.

Samples, in the form of wires and rings were fabricaby means of electron-beam lithography of a modulatiodoped GaAs/AlxGa12xAs:Si heterostructure. Additionallyone wire and one ring contained a transverse potentialrier, also patterned lithographically. The measurements wcarried out down to 30 mK in the magnetic fieldsB up to 90kG.

Obviously, in order to study transport properties of a dordered compressible liquid, the device lengthL has to begreater than the mean free pathl . At the same time, how-ever,L should not be too large since in the long wires tconductance is strongly affected by backscatteringacrossthesample between the two current strips. Localized states onating from the upper Landau level serve as intermedstates for such a process. As a result, the conductance olow-field side of the plateaus is reduced below the corsponding quantized value. Such a conductance minim~antiresonance! was theoretically predicted11,33–35 and ob-served experimentally30,36in two-terminal wires. In the backscattering process, the impurity assisted acoustic-phoscattering is presumably involved.37 The efficiency of suchan electron-phonon interaction in quantum wires is intereing of its own. In particular, it has been suggested thatcorresponding scattering rate drops to zero when the elecdrift velocity within the 1D subband becomes smaller ththe sound velocity.38 A detailed temperature study of thconductance antiresonances observed for a 20-mm quantumwire in the quantized magnetic field has been reported inprevious work,39 and shown to corroborate the above theretical prediction.

The organization of the present paper is following. In SII, experimental details and sample characterization arescribed. Experimental data concerning electron transwithin the compressible electron liquid for the filling facto2<n<3 in quasiballistic wiresL*l are discussed in SecIII. The results suggest that Coulomb blockade effectsimportant for charge transport through a just-formed per

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lation path — a sequence of conductance peaks, correspoing to a single-electron addition spectrum is observedon thequantized plateaus. A temperature dependence of contance indicates, however, that inelastic processes affectcharge transport near the percolation threshold. This is sported by experimental results for the ring samples presein Sec. IV. Electron transmission along the central currstrip is by no means coherent since the Aharonov-Bo~AB! oscillations with the fundamental frequency, clearvisible nearB50, arenot observed for the ring-shaped devices in the percolation regime. We interpret these findinaccording to a recent theoretical model,40 which suggeststhat in contrast to the vanishingly small electron-phonscattering rate near the bottom of 1D magnetoelectric sbands in ideal wires,38 the electrons at the percolation thresold are strongly coupled to the acoustic phonons.40 At thesame time, the ring containing the barrier shows frequedoubling of the AB oscillations in the plateau region,n52.These results appear to demonstrate the validity of the phrigidity concept41,42 in disordered two-terminal nanostructures under quantized magnetic fields. Section V contaresults obtained for the high-magnetic-field onset ofn53plateau~i.e., for high-density compressible liquid, far abovthe percolation threshold!, which reveals that, surprisinglythe conductance is time dependent. It is suggested thatnoise is brought about by glassy dynamics of localized etrons in the wire center, and by the corresponding timependence of the impurity-assisted tunneling probabilitytween the current carrying regions.

In summary, the low-temperature properties of a disdered compressible liquid can be outlined as follows~seealso Sec. VI!: ~i! the magnetic field induced changes of ttotal number of percolating electrons lead to the Couloblockade peaks, observed on the low field side of the cductance plateaus. The period of the single-electron addispectra is in agreement with the calculations based onCMS model;~ii ! transport within a percolating path is nocoherent, presumably because the central current stripwhole interacts with the acoustic phonons, as predictedRef. 40; ~iii ! on the high-field side of the conductance plteauG is time dependent for the disordered samples. Snoise, observed for the first time, is dominated by the prence of Lorentz-type fluctuators with only few characterisfrequencies.

II. EXPERIMENTAL DETAILS AND SAMPLECHARACTERIZATION

The present study has been carried out on two-termwires and rings fabricated by means of electron beam lithraphy and wet etching from a modulation-dopGaAs/AlxGa12x:Si heterostructure. The Al contentx50.3and the spacer width of 15 nm resulted in an electron ccentration and mobility of 431011 cm22 and 2.53105

cm2/Vs, respectively. The epilayers were grown by moleclar beam epitaxy and no special procedure for obtaininghigh electron mobility was applied. The deep-level transispectroscopy studies, carried out on layers grown at the sconditions, revealed the presence of residual impurities inconducting channel, most probably related to Te, Se, or43

The wires had lengthL52.1 mm ~comparable to the mea

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16 254 PRB 58WROBEL, DIETL, REGINSKI, AND BUGAJSKI

free path in 2D! and lithographic widthWlith.0.7 m @seeFig. 1~a!#. The ring-shaped structures had a total length omm, and consisted of a central ring with an average diamof 1 mm with two ‘‘leads’’ of 1 mm @see Fig. 1~b!#. The twodevices~one wire and one ring! had an additional transverspotential barrier of lithographic size 50350 nm, located in-ward at one of the mesa edges, as shown in Fig. 1~c!. Stan-dard AuGeNi Ohmic contacts were formed by annealquite close to the devices, typically 8mm apart.

Conductance measurements were conducted in an Hcryostat and He-3/He-4 dilution refrigerator by employingstandard low-frequency lock-in technique with current extation of 1 nA. Special care was taken to avoid heating ofelectron gas by external noise — the cryostat was carefgrounded and the samples were connected through a sdiscrete and distributed high-frequency rejecting filters.

In submicron structures the true~physical! width of elec-tron distribution is considerably smaller thanWlith . In theCMS model it is assumed that the distribution of electrdensity across the wire, in a zero magnetic field, is given

n~x!5n0A~b22x2!/~d22x2!.

FIG. 1. Scanning electron microscope images of wet etctwo-terminal quantum devices:~a! wire ~b! ring, and~c! asymmetricring with an artificial defect — a narrowing located in the uppesemicircular arm~pointed out with an arrow!. All pictures weretaken with the same magnification — the scale is indicated inlower left corner.

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The density reaches (b/d)n0 at the wire center and vanishefor x56b. This formula was derived44 for a split-gate de-vice with a gate distance of 2d. Parameterd determines theshape of the electron distribution. Ford→b, n(x)[n0

whereas ford@b the carrier distribution forms a semicirclelike profile. There are two arguments that this model canlikewise applied to the etched devices, where the electdensity is also depleted near the mesa walls and is the larat the wire center. First, ford@b and x→b the above for-mula agrees well with the density profile calculated by Gfand and Halperin26 for a mesa-etched sample. Second,width of the mesa edge depletion studied experimentally45 isin a reasonable agreement with calculations based on a sgate model.7 For our structures, the depletion region is almuch wider than the true physical width of the carrier distbution ~see below!. In this limit, the shape of the electrodensity profile across the device is less sensitive to the evalue ofd. Therefore we assumedd5Wlith /2.

The physical width of the electron distribution,W52b,has to be independently determined. For this purpose welow-field magnetotransport data. In the case of wires, asistance maximum is observed when the cyclotron radiul cis approximately two times greater than the wire widthW.This maximum is attributed to a nonspecular componenboundary scattering. Following a previous study,46 we adoptthe formula W50.553 l c , and obtain for our wiresW50.28 mm. In the case of rings we determineW52b50.17 mm from the period of the Aharonov-Bohm oscillations in the magnetic-field range21 kG to 11 kG, see Fig.3. The presence of the oscillations demonstrate directlythe phase coherence length is longer than 3mm nearB50.

The last unknown quantityn0 is treated as a fitting parameter to the CMS model~see next section!. By substituting itsvalue to the formula describingn(x) and averaging over thewidth, a 2D electron concentrationn2D can be calculatedThe resulting value is in a good agreement with that demined from the period of the Shubnikov–de Haas oscitions in the magnetic field rangel c!W. KnowingW andn2Das well as assuming the parabolic confinement we areposition to estimate 1D electron densitynL , the number ofthe occupied 1D levelsN, and the interlevel energy distanc\v0 . The results are summarized in Table I. Note thatboth types of the devices we obtain almost the same widtthe depletion region ('0.2 mm!, which is of a typical mag-nitude for the wet etched GaAs/AlxGa12xAs devices. Notealso that sinceW is significantly smaller thanWlith , the es-timations based on the parabolic confinement model sholead to reasonable results. Moreover, a slowly varying pa

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TABLE I. Samples parameters inferred from the low-magnetic field magnetotransport:W, physical width
of the device,n2D , 2D electron concentration;EF , Fermi energy,nL , 1D electron density; parameterestimated from the parabolic confinement model:N, number of occupied 1D subbands~at B50), \v0 , 1Denergy level separation.Wlith is the lithographical width of the device,n0 is the CMS model parameter~seetext!.

Wlith W n2D EF \v0 nL N n0

mm mm 1011 cm22 meV meV 106 cm21 1011 cm22

Wire 0.7 0.28 2.84 10.2 0.7 8.0 14 7.75Ring 0.6 0.17 2.59 9.3 1.2 4.4 8 11.46

bolic potential favors the formation of wider edge currentstrips than those expected in the case of the rectangular po-tential well.9

In Figs. 2 and 3 the two-terminal conductanceG versusthe magnetic fieldB is presented for the wire and ring, re-spectively. Always a small (;20 V) resistance was sub-tracted from the measured values, since the same pair ofcontacts was used as a current and voltage terminals. The

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used for bonding the current and voltage leads to the samemetal pad of two-terminal wire. Note that for both devicesthe magnitude of the conductance inB50 is about one halfof the corresponding Landauer-Bu¨ttiker conductance(2e2/h)N, whereN is the number of 1D subbands, displayedin Table I. In the range of low magnetic fields, weak local-ization minima followed by universal conductance fluctua-tions ~UCF’s! are visible. These findings point to the pres-ence of relatively large disorder, partly originating from theS impurities and partly from the mesa wall roughness, asdiscussed above. A magnetic field suppresses backscatteringand thus reduces the role of imperfections—in the high-fieldrange, well developed and exactly quantized conductanceplateaus are seen. Butin betweenthe plateaus, a rich fluc-tuation pattern is preserved. Such fluctuations are not ob-served in the ballistic devices,47 and are again connectedwith disorder.

Nevertheless, results presented in Fig. 2 differ in one im-portant respect from typical data for the long and disorderedwires. Namely, the conductance ‘‘antiresonances’’ are notobserved. As shown in Fig. 2, there is no evidence for thepresence of the antiresonances in the vicinity of eitherG51 (n52) or G52 (n54) plateaus. We conclude that ourstructures are in the quasiballistic regime,l &L.

However, near those plateaus that correspond to the spinsplitting, weak antiresonances are seen. In addition, the pla-teaus with odd-filling factors are not as exactly quantized asthe ones corresponding to the even values ofn. The latter isparticularly evident for the ring structure. These observationsindicate that backscattering across the sample, which in-volves localized states of the higher spin subband is rathereffective. This points, in turn, to the existence of spin non-conserving processes as well as to a small magnitude of the

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FIG. 2. The wire conductance as a function of the magnetic fieldat two temperatures, 0.04 K and 4.2 K. The dashed box in the mainfigure circumscribes the data region betweenn53 andn52 pla-teaus, which is shown in more detail in the inset.

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FIG. 3. The conductance for the symmetric ring measured0.04 K as a function of the magnetic field. The inset showsoscillating part of the magnetoconductance, recorded aroundmagnetic field~the corresponding Fourier spectrum is shown in F7!.

eryspin splitting. The latter is corroborated by a quick disapearance of the plateaus corresponding to the oddn valueswith the increasing temperature. Indeed, Eqs.~12! and ~13!from Ref. 27 predict a small magnitude of the spin splittiup to 180 kG and 150 kG for our wire and ring, respectiveTo emphasize a minor importance of the spin splitting instudied structures we present the conductance in the 2e2/hunits.

III. TRANSPORT WITHIN A COMPRESSIBLE STRIP

Figure 4 shows an expanded view ofG51 (n52) pla-teau, as measured for the wire-shaped structure at low-high-temperature limits. The values of (e2/h)n(0) versusmagnetic field, calculated according to CMS model,9 are alsoshown. In the upper part of Fig. 4, the calculated positionsthe current strips are depicted.

CMS theory predicts a very narrow plateau~of the widthof ;5 kG! sinceG is quantized only in the case when eletron liquid in the center of the wire is incompressible. Thoccurs in a rather limited range of the magnetic fields. Fthe device under consideration, when the magnetic fieldlowered, the compressible metal-like strip starts to devein the middle of the wire already inB'61 kG according toCMS theory. In practice, however,G remains exactly quantized in the lower fields because the compressible chanhas initially a very low density so that the electrons arecalized by disorder. With a further decrease of the magn

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field, the density of the compressible liquid increases aeventually, the central strip starts to conduct. AtT54.2 K,this happens inB'57 kG, when the electron strip haswidth of 0.085mm and the density of the compressible liuid reaches 3.031010 cm22 in the wire center.

As expected, the width of the plateaus increases withlowering of temperature. This is mainly due to the fact thlocalization is more effective at low temperatures, at whthe inelastic scattering length becomes longer than the loization radius. At the same time, our data are in good qutative agreement with recent self-consistent numerical calations of the magnetoconductance of short quantum w(L50.5,W50.33 mm! with an additional centraconstriction.48 The plateau widths calculated in that paper alarger than those expected from the CMS model. Moreothe widening increases with decreasing temperature, anoccurs on the low-field side of the plateaus, as indeedserved experimentally in our structures.

Nevertheless, even at the lowest temperatures (T50.04K! the conductance somewhat increases with decreasinBbut in a nonmonotonic way. A sequence of sharp condtance peaks is observed on the low-field side of the plateThis feature is entirely determined by transport within tcompressible liquid as the antiresonances are absent aG>1.

It is possible that the observed peaks result from sequtial tunneling along the wire between closed current loopnear the percolation threshold. The most efficient are pcesses involving resonant tunnelingvia an intermediate state

FIG. 4. The conductance in the plateau region (G;1,n52) forthe wire, presented already in Fig. 2. The width of compresselectron liquid as a function of the magnetic field, calculated witthe Chklovskii, Matveev, and Shklovskii~CMS! model ~Ref. 9! isshown in the upper part of the figure. The conductance of an idwire, G5(e2/h)nB(0), in shown by the thick gray line. The electron distribution across the wire~for B;57 kG! is depicted sche-matically.

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having character of a dot with the edge current circulataround.3,49 If the magnetic field is changed, the zerdimensional states of such a dot cross sequentially the Fenergy any time the magnetic flux through its diameterchanged by one flux quantum.50 The resulting single-electronresonant tunneling produces conductance peaks, indeedserved experimentally in quantum wires in high magnefields.51,52 In our sample this corresponds to the field ranbetween 42 kG and 45 kG in Fig. 2.

However, resonant tunneling alone cannot account folarge number of the conductance peaks superimposedon then52 plateau, forB.45 kG. The peaks, shown in more dtails in Fig. 5, are quasiperiodic inB with an average periodequal approximately to 110 Gs. This value would correspoto an average ‘‘dot’’ diameter of 0.7mm, which appears tobe quite unrealistic. We attribute the conductance maxshown in Fig. 5 to the field-induced change in the numberthe ‘‘compressible’’ electrons one by one. In other words,assume that a charging energy related to the single electransfer trough the wire modulates the conductance ofjust-formed percolation path — similarly to the case of tCoulomb blockade for tunneling through a quantum do53

Such an effect is absent in both ballisticL!l and longL@l wires since in the former case a large transmissionefficient T51 reduces the probability of a simultaneopresence of several electrons in the wire while in the loconductors the charging energy is small.

Since its discovery by Scott-Thomas,54 the phenomenonof the Coulomb blockade in semiconductor nanostructuhas attracted a considerable attention~for recent reviewsee55!. In the case of disordered wires in magnetic field, tcharging effects were studied by Staringet al.56 for devicesdefined by a split gate on a GaAs/AlxGa12xAs heterostruc-ture. Those authors reported on the observation of perio

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FIG. 5. Coulomb blockade peaks observed on the low-field aof G51,n52 plateau for the wire, measured at 0.4, 0.2, and 0.0~top to bottom!. Inset shows the estimated number of spin polarizelectrons in the central compressible strip. We assumedNcomp

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'(1/2)Ncomp. The temperature dependence of conductance inmagnetic fields labeledA, B, andC is shown in the next figure.

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conductance oscillations as a function of the gate voltaobserved close to the pinch-off, and attributed them to tneling modified by the effect of the Coulomb blockade.

In the self-consistent picture, the peaks observed bymay correspond to the population of the central striptreated as a whole—by a single electron. In the CMS mothe number of ‘‘compressible’’ electrons ‘‘floating’’ on toof the incompressible liquid~with n52, which acts as a sorof a reservoir! is given by

Dncomp5nL„nB~0!22…

in the center of the wire, wherenL5eB/h. Since the widthof the central current strip is known~see Fig. 4!, the totalnumber of spin-polarized compressible electrons canevaluated as a function ofB. The results are shown in thinset to Fig. 5. According to such a calculation, the numof the compressible electrons changes by twelve per kGtween 47 and 50 kG, which is in remarkable agreement wthe experimental period of 110 Gs.

When the temperature is raised, the width and heighthe conductance peaks increase monotonically but theirsitions remain unchanged, as shown in Fig. 5. A similarhavior was reported56 for the gate-induced Coulombblockade oscillations. Furthermore, when the sampleheated, more peaks of the electron addition spectrum apon the high-field side of the plateau. In the spirit of the CMmodel, this can be explained as an increase in the numbdelocalized electrons with temperature. It is also possthat the percolation path is more strongly coupled toleads whenT is raised.57 The self-capacitanceC and charg-ing energye2/2C of the central compressible strip can bestimated from its size. On the low-field side of the platee2/2C'0.2 meV. This value is consistent with the disappeance of the peaks atT*2 K.

Figure 6 shows the conductance of compressible liquDG5G21, as a function of temperature for three charactistic values of the magnetic field~denotedA, B, C in the

FIG. 6. The amplitude of the three selected conductamaxima~see previous figure! as a function of temperature.

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previous figure!. DG is seen to increase monotonically witT. Recently, it was shown that the percolation path can gbally emit or absorb acoustic phonons in strong magnfields, with the rate proportional toT, if only the drift veloc-ity is very small, i.e., just on the percolation threshold.40 Webelieve that this assumption corresponds to the experimeconditions under consideration. It appears, therefore, thatelectron-phonon interaction is the dominant inelastic procleading to apparent delocalization of compressible liquand thus to the increase with temperature of its conductivAs discussed in the next section, this interaction constitualso an efficient phase breaking mechanism.

IV. AHARONOV-BOHM EFFECTIN QUANTIZED MAGNETIC FIELDS

It is natural to ask whether it would be possible to obsethe Aharonov-Bohm~AB! oscillations in a ring-shaped structure in the quantizing magnetic fields. When the curreflows through a percolation path in the channel centercould expect the AB oscillations to have the same frequef 0 as those nearB50 presented in Fig. 3. Figure 7 shows thring conductance on the low-B side of the plateau that corresponds toG51 (n52). The oscillations, with the amplitude greater than that assigned to the Coulomb blockadevisible. In the inset to Fig. 7 the Fourier spectrum is dpicted; it was computed after filtering out slowly varyinbackground conductance. The oscillations are periodictheir frequencyf B is approximately equal to (1/2)f 0 .

An unexpected doubling of the AB period was recenreported by Bird and co-workers58,59 for a quantum dotcoupled to large 2D regions. In our case, however, the pedoubling appears to be purely accidental. As suggestedTimp et al.60 and sketched in Fig. 8a, the observed AB ocillations are very likely associated with resonant tunnel

e FIG. 7. The low-magnetic field side of theG51,n52 plateauobtained for the ring at 0.03 K. The inset shows the Fourier sptrum of the oscillating part of the conductance. For comparison,Fourier amplitude of weak-field (uBu,1 kG! Aharonov-Bohm os-cillations is also shown~dashed line!.

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between the central strips in the two current leadsvia thoseedge states that correspond ton52 and circulate around thinner hole. The transition probability through such a dotlistate is known49,3,50to undergo the AB oscillations. The spatial position of these circulating states evaluated withinCMS model agrees very well with that deduced from tmeasured AB frequencyf B .

From the data presented here we conclude also that etron transport through the central strip isnot coherent as theAB effect of the frequencyf 0 is not observed. This meanthat the inelastic mean free path is particularly short for tstrip as the coherence is maintained for electron transthrough the inner shell as well as nearB50. Following the-oretical considerations40 we note that particularly efficienelectron-phonon scattering is expected at the percolathreshold, where a complex spatial form of the carrier wafunctions allows for the coupling to phonons from a wirange of the wave vectors. The latter shows whyinelastic-scattering length is the shortest for the central crent strip, and explains a decrease of the AB frequency wthe magnetic field, an effect already reported by Timpet al.60

for multiterminal devices.A further insight into tunneling processes can be gain

from data collected for a similar ring but having an artificnarrowing in one of the semicircular arms. Figure 9 shothe low-temperatureG(B) data obtained for such a structurAs expected, the conductance atB50 is smaller as com-pared to the ‘‘ideal’’ ring, and the exactly quantizedG51plateau is observed in weaker fields. On the low-field armthe plateau~from 30 to 35 kG!, strong conductance oscillations with the frequencyf B are observed, as before.

FIG. 8. A schematic representation of two possible tunnellprocesses, responsible for the high magnetic field Aharonov-Booscillations with the frequencyf B and 2f B . Light gray areas represent low-density compressible liquid, darker lines correspond tohigh-density edge currents. The arrows indicate the directioncurrent flow.~a! Tunnelling at the entrance and at the exit to tring. ~b! Additional tunnelling that occurs at the vicinity of an artficial narrowing.

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However, a different behavior is detected whenG valuesare very close to 1. Figure 10 shows the conductance~for35,B,42 kG! and the Fourier spectrum of its oscillatincomponent. In contrast to the data for the ideal ring, weoscillations of a nonmonotonic amplitude and the frequenf B are observedon the quantized plateau~see also Fig. 11!.The ‘‘on-plateau’’ oscillations are seen to lower an averamagnitude ofG below its quantized value,G51. This im-plies that the effect is produced by backscattering. A straigforward interpretation of this observation is that the constrtion provides an additional channel for tunneling betweencompressible liquids, which is efficient even in the fie

gm

ef

FIG. 9. The conductance for asymmetric ring~with an artificialnarrowing!, at 0.05 K.

FIG. 10. The low-field side of conductance plateauG51,n52measured for the asymmetric ring~with an artificial narrowing! at0.03 K. Fourier spectrum of the oscillating part of the conductais shown in the inset.

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range, for which no tunneling occurs in the ideal ring. Nosince the electrons in one of the involved strips circularound the ring, the tunneling probability is modulatedthe corresponding Aharonov-Bohm frequency.

As already mentioned, the amplitude of the oscillatiocorresponding to on-plateau backscattering is highly nmonotonic, and is also found to decrease with temperatStrong additional conductance minima, marked with a ssymbol in Fig. 11, are attributed to single-electron chargof the inner edge state. The observed Aharonov-Bohm efis, therefore, superimposed on the Coulomb-blockade pebut, in contrast to the situation discussed in Sec. III,electrons are removed or added to the central circulastrip, not to the just-formed percolation path.

Moreover, again in contrast to the symmetric deviweak oscillations with double frequency;2 f B are found insomewhat lower fields, just at the onset of conductaquantization. Frequency doubling has been noted in sevstudies of the AB effect in dots and antidots.61–64It has beeninterpreted as originating from spin-resolved oscillatiolocked in the anti-phase. It is not clear, however, howsuperposition of two independent periodic signals could lto the frequency doubling. More probably, in two-terminAharonov-Bohm devices, the frequency doubling may ocas a result of a constrain imposed by the Onsager-Bu¨ttikersymmetry relations,65 G(B)5G(2B). Experimentally, suchan effect was demonstrated by inserting a potential barfor the electrons in a two-terminal AB ring.41,42 In particular,Yacoby, Schuster, and Heiblum41 examined the evolution olow-field AB interference as a function of the barrier heigcontrolled by a gate voltage. They found that indeedphase of theh/e oscillations assumes only the two valuthat are allowed by symmetry: 0 andp. At the same time,for the range of the gate voltage corresponding to the tration region between the above two cases, only a weakh/2e

FIG. 11. A highly enlarged view of conductance plateG51,n52, obtained for asymmetric ring~with an artificial narrow-ing!. Data for T50.2 K are shifted upward for clarity. Note thpresence of narrow peaks~marked with stars!, which we attribute tothe Coulomb blockade of charging of the inner current strip.

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harmonic of the oscillations was detectable, i.e., the fquency doubling was observed. Of course, the effect ofphase rigidity should be preserved in the quantized magnfields. In the case of our sample, the magnetic field—controlling the population of the strips—might play a rolethe gate voltage. In more microscopic terms, the presenc2 f B harmonic in the oscillation spectrum means that eitprocesses involving double encircling of the ring remain cherent or that a sequential tunneling, described by a prodof two transition probabilities, is operating.

V. THE TIME DEPENDENT CONDUCTANCE

So far we have considered the conductance in the hfield range of the transition region betweenG51.5 andG51 plateaus, where low-density compressible liquid, locain the center of the wire, was responsible for the transpproperties. In this section, we look at the low-field portionthe intermediate region, where strips of compressible liqreside on the two sides of the wire center. Consequentldeviations from the quantized value ofG51.5 are caused byelectron transitions between the strips. The sound velocitgreater than that of the electrons in the strips, so that phonassisted tunnelling is not efficient. If, however, the stribecome sufficiently close, elastic scattering betweenstrips via states localized in the wire center may operaSuch a resonant tunneling is supposed to be much moreficient than either direct or two-step incoherent tunnellibetween the current carrying strips. Since the relevant inmediate states have to lie within the rangekBT around theFermi level, their number increases, and thusG decreases,with the temperature. Such a temperature dependence isposite to that observed in the higher fields, at which bthermal spread of electron energies and inelastic scatteenhancescharge transport, as discussed in the previous Stion.

Surprisingly, in the resonant tunneling region,G changesas a function of time. The normalized noise magnitude r(DG) is rather large, reaching 0.033(2e2/h) at the lowesttemperatures and for the time constant of the lock-in amfier set at 0.1 s. The range of the filling factors, wheretime dependence occurs, corresponds to 3&nB(0)&2.4~compare again data presented in Fig. 2!. A similar magni-tude of noise was observed for all wire samples in the abrange ofn values. The appearance of noise in the optiproperties of 2D electrons was reported by Kukushet al.,66 however, for much smaller filling factors (n,0.28). Therefore, the observed fluctuations of the lumincence signal were attributed to the depinning of the Wigsolid.

The temporal evolution of conductance is shown in F12. The data were recorded at various temperatures but insame magnetic fieldB544.2 kG. For a comparison, timseries obtained on-plateau, i.e., forB548.4 kG andG>1, ispresented in the upper part of Fig. 12. In the scale of Fig.the latter shows time dependence determined by the insmental noise, as expected for the region, in whichG is quan-tized. This result demonstrates also that the amplitude ofbackground noise is far below the temporal changes founthe fields corresponding to tunneling between the strips.

Conductance noise reveals few characteristic frequenc

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which dominate, during a finite period, the temporal evotion. With the increasing temperature the magnitude of nodecreases and the characteristic frequencies become laThis is illustrated in Fig. 13, where Fourier power spectraconductance noiseS( f ) are presented. As shown, over a limited range of frequencies,P( f )5A(T)/ f g(T), where bothA(T) andg(T) decrease with the temperature. The observalue ofg52.060.2 at the lowest temperature suggests ta sum of high frequency tails of Lorenzian distributionsobserved. This would mean that the noise is driven byensemble of fluctuators with switching times longer th1000 s at 50 mK. When the temperature increases,switching frequencies are expected to increase also, wexplains a gradual transition towards a canonical 1/f noise.At the same time, more and more parallel tunnelling pabecome available for the electrons, which leads to decreof bothG and the amplitude of its temporal variations,A( f ).

A question arises as to what the dominant mechanismaccounts for the observed noise is. First of all, we noteits large magnitude indirectly confirms the importanceresonant tunneling, which should be particularly sensitivethe actual potential landscape and to the occupancy of inmediate states. Secondly, the single-electron shot noise,phenomenon with a flat power spectrum, can be excludFrom a comprehensive study of noise in quantum point ctacts, Liefrink, Dikhuis, and van Houten67 concluded thatGchanges as a result of trapping and detrapping of electronthe vicinity of the constriction. By changing the charge staan electron trap shifts the potential energy of the device,

FIG. 12. Conductance variations,DG5G2^G& as a function oftime, recorded for the wire in the plateau region~uppermost trace!,and in the region corresponding to 2.5,n,3; ^G& is the timeaverage conductance; temperatures and magnetic fields arecated, the latter in parenthesis. Data for 0.04 K~48.4 kG! are shiftedupward and data for 0.3 K~44.2 kG! are shifted downward forclarity. Conductance was measured with a lock-in amplifier ticonstant of 0.1 s.

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thus produces a conductance jump. While such a mechanmay also operate in our case, an effect associated with sglassylike dynamics of the hopping between localized stain the wire center could also be involved. The dynamscreening of a slowly varying potential landscape shouldas well considered, especially at low temperatures.

VI. CONCLUSIONS

Our work has demonstrated that a study of lotemperature two-terminal conductance for disorderedrelatively short (L*l ) wires and rings provides a worthwhile opportunity to probe the dominant mechanisms whunderline electron transport in the regime of low- and higdensity compressible current strips, produced by the quaning magnetic fields. At low densities of the electrons in tstrip, a sequence of peaks has been observed in the dedence of the conductance on the magnetic field. A quantive analysis, particularly evaluation of the distance betwethe peaks, has lead us to conclude that the effect reflectsCoulomb blockade, associated with the charging of the crent strip. In the same range of the magnetic fields, byamining both the temperature dependence of conductaand the period of the Aharonov-Bohm oscillations for tranport along the central strip in the ring, we have put into tevidence the presence of enhancement of the electphonon scattering rate at the percolation threshold. We halso suggested that the frequency doubling of the AharonBohm oscillations in a ring with an additional potential barier provides a support for the concept of the phase rigid

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FIG. 13. The Fourier power spectra of conductance time se(DG data of Fig. 12! obtained over a 4000-s time interval.

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Finally, our findings have revealed the time dependencethe conductance in the regime of the high concentrationthe electrons in the current strip. We have assigned thisprising noise to glassy dynamics of localized electrons inwire center, and to the corresponding time dependence oimpurity-assisted tunneling probability between the currcuring strips.

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ACKNOWLEDGMENTS

The authors thank M. Guziewicz, E. Kamin´ska, and E.Papis for the assistance in making Ohmic contacts. One o~J.W.! acknowledges the kind help of B. Chklovskii in clarfying some aspects of the CMS model. This work was sported by Polish KBN Grants No. PB 2-P03B-6411 and PNo. 28.11/P9.

ag,

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Transport and tunneling within a compressible electron ...

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