Area dependence of interlayer tunneling in strongly correlated bilayer 2D electron systems at ν T =1
A.D.K. Finck, A.R. Champagne, J.P. Eisenstein, L.N. Pfeiffer, K.W. West
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Area dependence of interlayer tunneling in strongly correlatedbilayer 2D electron systems at ν T = 1 A. D. K. Finck , A. R. Champagne , J. P. Eisenstein , L. N. Pfeiffer , and K. W. West Condensed Matter Physics, California Institute of Technology, Pasadena CA 91125 Bell Laboratories, Alcatel-Lucent, Murray Hill, NJ 07974 (Dated: October 30, 2018)The area and perimeter dependence of the Josephson-like interlayer tunneling signature of thecoherent ν T = 1 quantum Hall phase in bilayer two-dimensional electron systems is examined.Electrostatic top gates of various sizes and shapes are used to locally define distinct ν T = 1 regionsin the same sample. Near the phase boundary with the incoherent ν T = 1 state at large layerseparation, our results demonstrate that the tunneling conductance in the coherent phase is closelyproportional to the total area of the tunneling region. This implies that tunneling at ν T = 1 is abulk phenomenon in this regime. PACS numbers: 73.43.Jn, 71.10.Pm, 71.35.Lk
Bilayer two-dimensional electron systems (2DESs) ina large perpendicular magnetic field B support an un-usual collective phase when the separation between thelayers is sufficiently small, the temperature is sufficientlylow, and the total density n T = n + n of electronsin the system equals the degeneracy eB/h of a sin-gle spin-resolved Landau level created by the magneticfield . Interlayer and intralayer correlations are of com-parable importance in this Landau level filling factor ν T = ν + ν = n T / ( eB/h ) = 1 phase. The sys-tem exhibits a number of striking physical properties,including Josephson-like interlayer tunneling and van-ishing Hall and longitudinal resistances when equalelectrical currents are driven in opposition (counterflow)in the two layers. The electronic correlations responsi-ble for these properties may be understood in a numberof equivalent languages, including that of itinerant fer-romagnetism and that of excitonic Bose condensation .Indeed, the remarkable transport properties observed incounterflow are very suggestive of excitonic superfluidity,although experiments have so far always detected smallamounts of residual dissipation.Interlayer tunneling provides particularly dramatic ev-idence that bilayer 2DESs at ν T = 1 are quite differentat large and small interlayer separations. When the ef-fective layer separation (defined as the ratio d/ℓ of thecenter-to-center quantum well spacing d to the magneticlength ℓ = (¯ h/eB ) / ) is large, the interlayer tunnelingconductance dI/dV is strongly suppressed around zerointerlayer voltage . This suppression reflects the stronglycorrelated nature of single layer ν < N -particle system and thus creates a highly excited N +1-particle state which can only slowly relax to equilib-rium. Similar considerations attend the rapid extractionof an electron from a 2DES at high magnetic field. Sinceinterlayer correlations are negligible at large d/ℓ , the nettunneling conductance spectrum of the bilayer is a simpleconvolution of the suppression effects in the individuallayers. In contrast, at sufficiently small d/ℓ the inter- layer tunneling conductance dI/dV at ν T = 1 displays astrong and extremely sharp peak centered at V = 0. Thesharp peak in dI/dV reflects a near-discontinuity in thetunneling I − V curves at V = 0. This is reminiscent ofthe dc Josephson effect, but it remains unclear how closethe analogy is.The tunneling features observed at ν T = 1 and small d/ℓ are widely believed to reflect the development ofspontaneous interlayer quantum phase coherence amongthe electrons in the bilayer . Electronsare just as strongly correlated with their neighbors intheir own layer as they are with electrons in the oppo-site layer. Within the exciton condensate language, everyelectron in the bilayer is bound to a correlation hole inthe opposite layer and there is no energy cost associatedwith transferring an electron from one layer to the other.Crudely speaking, the Coulombic energy penalties asso-ciated with injection and extraction of electrons in thetunneling process are cancelled by the excitonic attrac-tion of the electron to the hole in the final state.The purpose of the present paper is to report on aninvestigation of how interlayer tunneling in the coherent ν T = 1 phase is distributed across the sample area. Viaelectrostatic gating we are able to study regions of differ-ent size and shape in the same sample. For effective layerseparations d/ℓ not too much smaller than the criticalvalue, approximately ( d/ℓ ) c ≈ .
91 as T → G (0) ≡ dI/dV at V = 0, is essentially proportional tothe sample area. This suggests that tunneling at ν T = 1is a bulk phenomenon in our samples, at least close to thecritical layer separation. In spite of the intuitive nature ofthis result, it is not obvious a priori . Indeed, the strongsimilarity between bilayer 2DESs at ν T = 1 and Joseph-son junctions suggests that tunneling currents may berestricted to micron-scale regions close to the source anddrain contacts and not exist in the bulk of the sample .We discuss this and other theories of interlayer tunnel-ing following the description of our experiment and itsresults.The bilayer 2DES samples used in this experiment areGaAs/AlGaAs double quantum wells grown by molecularbeam epitaxy (MBE). Two 18 nm GaAs quantum wellsare separated by a 10 nm Al . Ga . As barrier. Modula-tion doping with Si populates the ground subband of eachwell with a 2DES of nominal density n , ≈ . × cm − and low temperature mobility µ ≈ cm − /V s.The tunnel splitting ∆ SAS between the lowest symmetricand antisymmetric states in the double well potential isestimated to be less than 100 µ K; this is roughly sixorders of magnitude smaller than the mean Coulomb en-ergy e /ǫℓ at ν T = 1. Independent electrical contact tothe individual layers is achieved via a selective depletionmethod , thus allowing the measurement of interlayertunneling via standard lock-in techniques.In order to minimize systematic errors in the tunnel-ing conductances due to sample-to-sample variations, weemploy a technique which allows multiple, independentlycontrollable tunneling regions to be established on a sin-gle sample. Two such samples, A and B, (both takenfrom the same parent MBE wafer) have been studied.In both cases a large electrostatic gate is deposited onthe back side of the sample. This gate allows for globalcontrol of the electron density in the lower 2D electrongas in the active region of the sample. Smaller gates, ofvarying sizes, are deposited on the sample’s top surface.These gates allow for local control of the top 2DES den-sity. By adjusting the gates appropriately it is possibleto establish a density balanced ν T = 1 / / ν T = 1 configuration.Since tunneling is strongly enhanced at ν T = 1 for d/ℓ < ( d/ℓ ) c , but strongly suppressed where ν T = 1, it is notdifficult to separate out the conductance of the ν T = 1region. In fact, as we show below, this separation canalso be done at zero magnetic field. In sample A, theactive region of the device is a 200 µ m wide bar, fullyunderlaid by the large back gate. This bar is crossed byfour rectangular top gates, with lengths 100, 50, 20, and10 µ m. The top gates thus define four different regionswith four different areas that are proportional to the topgate length. In sample B the mesa bar has an inner sec-tion of width 100 µ m and two outer sections of width 200 µ m. A 100 µ m top gate crosses the inner section whiletwo 50 µ m top gates cross the outer sections. This sam-ple allows the comparison of regions with the same areabut different mesa widths and gate lengths. Schematicdiagrams of samples A and B are shown in Figs. 1a and3a, respectively.At zero magnetic field momentum and energy con-serving tunneling between parallel 2DESs can only oc-cur when the subband energy levels in the two quantumwells line up . If the two 2DESs have equal electrondensities this occurs at zero interlayer voltage, V = 0.The width of the tunnel resonance is set by the lifetimeof the electrons in the 2DESs; at low temperatures thisis dominated by impurity scattering . When the bilayersystem is density imbalanced the tunnel resonance movesto a finite interlayer voltage proportional to the density G ( ) ( m S ) -0.2 -0.1 0Top Gate Bias (V) (b) P ea k H e i gh t ( m S ) (c) (a) FIG. 1: (color online) (a) Schematic diagram of sample A.White rectangle represents 200 µ m-wide mesa containing thebilayer 2DES; ohmic contacts (not shown) to the individual2DES layers are attached near both ends of the mesa. Greyrectangle under the mesa is the global back gate; hatchedrectangles crossing above the mesa are the various top gatesunder which tunneling is established. (b) Zero voltage tun-neling conductance G (0) versus top gate bias taken at B = 0and with an applied back gate bias of − .
18 V. Peaks corre-spond, in order of decreasing height, to the 100, 50, 20, and 10 µ m top gates. (c) Background-subtracted peak heights from(b) versus top gate length. The line fitted to this data has an x -intercept of 3 ± µ m. difference. We exploit this fact to allow for clean sepa-ration of the tunneling conductance under the top gateson our samples from the “background” tunneling comingfrom other portions of the device. Figure 1b illustratesthis with data from sample A. For the data shown, theback gate bias has been set to -15.18 V; this lowers thedensity in the bottom 2DES and thereby imbalances thebilayer. With the top gates set to zero bias a measure-ment of the net tunneling conductance at zero interlayervoltage, i.e. G (0), yields a relatively small backgroundvalue; the main resonance having been displaced to a fi-nite interlayer voltage by the imbalance. However, if thebias voltage on one of the top gates is swept, a resonanceappears when the density of the top 2DES under thatgate matches the density of the lower 2DES. This is whatFig. 1b displays, there being one curve for each of thefour top gates. Each gate produces a clear resonance in G (0) when biased to about -0.17 V . These resonancessit atop a background conductance of about 50 nS whicharises from off-resonant tunneling from those regions ofthe device not under the swept top gate.Figure 1c shows the background-subtracted peak tun-neling conductance for each of the four top gate tracesshown in Fig. 1b. The background subtraction is nec-essarily slightly different for each top gate. The tunnel-ing conductance observed with all top gates groundedgives the amount of tunneling that comes from the en-tire tunneling region when imbalanced. When a bias isapplied to one of the top gates to create a balanced re-gion underneath, the net area of the device which re-mains imbalanced is reduced by the area of the biasedtop gate. We use this fact to calculate the correct back-ground for each top gate. The corrected peak heights vs. top gate length are plotted in Fig. 1c. The data arewell-fitted by a straight line with a small x -intercept ofapproximately 3 ± µ m. For the data shown in Fig. 1, n = n = 3 . × cm − at the resonance; but thissame linear dependence on gate length is observed at allmatched densities ranging from 2.9 to 3 . × cm − .We therefore conclude that tunneling is proportional toarea at B = 0, as expected. The positive x -interceptin the peak height versus top gate length relation mayindicate that boundary effects are reducing the effectivetunneling area. For example, fringe fields along the edgeof the gated regions could produce a density gradientover length scales comparable to the distance betweenthe upper 2DES and the top gate (0.5 µ m), thus reduc-ing the area in the upper 2DES that has electron densitymatching that of the lower layer.At ν T = 1 the situation is both simpler and morecomplex than at B = 0. The strong suppression oftunneling in the unbalanced regions of the sample where ν T = 1 and the strong enhancement at small d/ℓ of thetunneling in the balanced ν T = 1 regions under the topgates are both helpful in keeping the background tun-neling conductance (typically ∼ ν T = 1 (typically ∼
10 - 1500nS). In contrast, it is much more difficult to ensure thatthe bilayer is density balanced at ν T = 1 than it is atzero magnetic field. At ν T = 1 the tunnel resonance re-mains centered at zero interlayer voltage, regardless ofthe individual layer densities, provided ν + ν = ν T = 1.Furthermore, the robustness of the ν T = 1 state againstsmall antisymmetric shifts in the individual filling factors( ν → ν + δν , ν → ν − δν ) complicates the determi-nation that a balanced ν T = 1 system is present. Inorder to establish the gate voltages required for balanced ν T = 1 state underneath any given top gate, we first usean empirical calibration of the gates to choose approx-imate top and back gate voltages ( V top and V back ) whichwill create ν ≈ ν ≈ / V = 0, thus proving that the system is at least closeto the desired balanced ν T = 1 configuration. Next,the zero bias tunneling conductance G (0) is recorded asboth the V top and V back are scanned across a window involtage space. Examination of the resulting data clearlyreveals contours of constant total filling factor. Smallrefinements of the magnetic field and the gate voltagesare then made to first maximize G (0) vs. B and then P ea k H e i gh t ( m S ) Gate Length (mm) (b)
60 mK 85 mK125 mK d I/ d V ( m S ) -50 0 50 Interlayer Voltage (mV) (a)
FIG. 2: (color online) Tunneling vs. gate length at ν T = 1 insample A at d/ℓ = 1 .
81. Gate layout as shown in Fig. 1. (a)Tunneling conductance resonances ( dI/dV vs. V ) at T = 85mK. Gate lengths 100, 50, 20, and 10 µ m; tallest to shortest.(b) Peak heights plotted vs. top gate length; open circles 60mK, solid circles 85 mK, squares 125 mK. Straight lines arelinear least squares fits. to perfect the symmetry of the tunnel resonance shape( dI/dV vs. V ).Figure 2a shows typical tunneling conductance reso-nances ( dI/dV vs. V ) at ν T = 1 in sample A. For thesedata the temperature is T = 85 mK and the effectivelayer spacing is d/ℓ = 1 .
81. The four traces (tallest toshortest) correspond to tunneling under the four differ-ent top gates, with lengths 100, 50, 20, and 10 µ m. Asexplained above, there is negligible background tunnel-ing coming from other parts of the sample. Figure 2bshows the height of the tunneling peak at zero interlayervoltage, i.e. G (0), vs. top gate length at three temper-atures, 60, 85, and 125 mK, for this same effective layerspacing. The data reveal a clear linear dependence ongate length. A small x -intercept, 5 ± µ m is obtainedfrom the linear least squares fits. Data very similar tothat shown in Fig. 2 has been obtained at effective layerspacings ranging from d/ℓ = 1 .
70 to 1.88 and at tempera-tures from T = 60 mK to 300 mK. In all cases the heightof the tunneling peak scales linearly with gate length.At smaller d/ℓ deviations from the linear dependenceon gate length begin to appear. However, these devia-tions are most likely artifacts arising from the reducedsheet conductivity of the 2D systems at these lower den-sities and the concomitant enhanced tunneling conduc-tance. When the sheet conductivity of the 2DESs nolonger greatly exceeds the tunneling conductance, volt-age drops develop within the 2D layers. The resonancein dI/dV becomes broadened and suppressed in height.Since this effect is more pronounced in larger tunnelingareas than in smaller ones, the tunneling peak heightbecomes sub-linear in gate length. For this reason werestrict our attention here to effective layer spacings andtemperatures relatively close to the phase boundary d I/ d V ( m S ) -100 0 100 Interlayer Voltage (mV) (c) G ( ) ( n S ) -0.3 -0.2 Top Gate Bias (V) (b) (a)
FIG. 3: (color online) (a) Schematic diagram of the mesa andgates in the active region of sample B. (b) Zero voltage tun-neling conductance G (0) versus top gate bias taken at B = 0and with an applied back gate bias of − .
08 V. (c) Tunnelingconductance resonances ( dI/dV vs. V ) at ν T = 1 with d/ℓ =1.64 and T = 61 mK. In (b) and (c) the solid trace (black)corresponds to the 100 µ m square tunneling region and dot-ted trace (red) to the 200 × µ m rectangular region. Thered trace in (c) has been multiplied by 1.086, as described inthe text. separating the interlayer coherent and incoherent statesat ν T = 1. In this regime we are confident that sheetresistance effects are negligible.The data in Fig. 2 strongly suggest that the interlayertunneling conductance in the coherent ν T = 1 state isproportional to the area of the tunneling region. How-ever, since in sample A it is the length of the gated regionsthat is varied while the width (200 µ m) is constant, it re-mains possible that the tunneling conductance is simplyproportional to gate length, not area. For this reason wenow turn to the results obtained from sample B. This de-vice, illustrated schematically in Fig. 3a, contains threetop-gated regions; two are 200 µ m wide by 50 µ m longrectangles and the third is a 100 × µ m square. Thesethree regions all have the same area but differ by a factorof two in gate length and width. Since the two rectan-gular gates are identical in size and shape, we focus hereon the comparison of the inner square top gate and oneof these outer rectangular gates. Direct comparisons oftunneling in the two rectangles justify this simplification.Figure 3b shows typical tunneling conductance reso-nances in sample B at zero magnetic field. As with sam-ple A, at B = 0 it is most informative to plot G (0), thetunneling conductance at zero interlayer voltage, versustop gate bias at fixed back gate bias, here V back = − . ∼ . Moresignificantly, Fig. 3b shows that the peak tunneling con-ductance of the 100 µ m square region is about 10 per-cent larger, and the resonance width about 10 percentsmaller, than that in the 200 × µ m rectangle. Whilethese differences might be due to differing degrees of den-sity inhomogeneity in the two regions, their exact originis unknown. In the following, when comparing the tun-neling conductances at ν T = 1 in the square and rectan-gular regions of sample B, we correct for this small effectby multiplying the tunneling data from the rectangularregion by the appropriate factor deduced from the zerofield data. Figure 3c illustrates the results of this com-parison of tunneling conductance spectra ( dI/dV vs. V )at ν T = 1 for d/ℓ = 1 .
64 and T = 60 mK; there is es-sentially no difference in the spectra from these two dif-ferently shaped regions. Very similar results have beenfound at densities corresponding to 1 . ≤ d/ℓ ≤ . T = 60 to 300 mK.Taken together, the data presented in Figs. 2 and 3provide strong evidence that the enhanced tunneling con-ductance characteristic of the coherent ν T = 1 bilayerstate is proportional to the area of the tunneling region,at least over a limited range of d/ℓ and temperature nearthe phase boundary. The possibility that the conduc-tance is instead proportional to the perimeter of the tun-neling region, or simply the length of that region alongthe mesa boundary is not consistent with the presentfindings.The intuitive character of our findings belies the subtlenature of tunneling in the coherent bilayer ν T = 1 state.The long-wavelength effective Hamiltonian for this many-electron state is closely analogous to that of a Josephsonjunction. As pointed out by Stern, et al. this anal-ogy leads to a sine-Gordon equation for the collectivephase variable φ representing the orientation of the pseu-dospin magnetization in the ferromagnetically orderedstate . In an ideal, disorder-free sample this equa-tion leads to the conclusion that interlayer tunnel cur-rents (proportional to sin φ ) are restricted to narrow re-gions near where the current enters and leaves the bilayersystem . The width λ J of these regions is determined bythe pseudospin stiffness and the tunnel splitting ∆ SAS ,and is estimated to be in the few µ m range. In contrast tothe results presented here, this scenario would not resultin a tunneling conductance proportional to area .More realistic models of the distribution of tunneling inthe coherent ν T = 1 state have recently been advanced byFertig and Murthy and by Rossi, et al. . Both modelsattempt to incorporate the crucial effects of disorder onthe tunneling. In Fertig and Murthy’s picture fluctua-tions in the 2DES density lead to a complex network ofchannels and nodes, covering the entire sample, in whichthe coherent ν T = 1 state exists. Tunneling occurs atthe nodes of the network. Since the disorder length scaleis expected to be small ( < µ m, set by the separationbetween the dopant layers and the 2DESs), the numberof nodes, and therefore the net tunneling conductance,is predicted to be proportional to sample area in largedevices, in agreement with our results. In the model ofRossi, et al. disorder along the physical edge of thesample is expected to dominate when the coherent phaseis well-developed (i.e. at small d/ℓ and very low temper-ature). In this regime they predict that the tunnelingconductance will be proportional to the length along thesample edge, not the total area. Although this is in con-flict with our observations, we stress that our results areconfined to relatively large d/ℓ , in fairly close proximity to the phase boundary.In summary, we have found strong evidence that inter-layer tunneling in the coherent bilayer ν T = 1 phase nearthe critical layer separation is proportional to the area ofthe tunneling region, with some evidence for small edgecorrections. We therefore conclude that tunneling in thecoherent ν T = 1 state, much like tunneling at zero mag-netic field, is a bulk phenomenon in our samples.We are pleased to thank Herb Fertig and Allan Mac-Donald for enlightening discussions. This work was sup-ported via NSF Grant No. DMR-0552270 and DOEGrant No. DE-FG03-99ER45766. For an early review, see the chapters by S.M. Girvin andA.H. MacDonald, and by J.P. Eisenstein in
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