Edge transport in InAs and InAs/GaSb quantum wells
Susanne Mueller, Christopher Mittag, Thomas Tschirky, Christophe Charpentier, Werner Wegscheider, Klaus Ensslin, Thomas Ihn
EEdge transport in InAs and InAs/GaSb quantum wells
Susanne Mueller, Christopher Mittag, Thomas Tschirky, ChristopheCharpentier, Werner Wegscheider, Klaus Ensslin, Thomas Ihn
Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland (Dated: June 2, 2017)We investigate low-temperature transport through single InAs quantum wells and broken-gapInAs/GaSb double quantum wells. Non-local measurements in the regime beyond bulk pinch-offconfirm the presence of edge conduction in InAs quantum wells. The edge resistivity of 1-2 kΩ /µ mis of the same order of magnitude as edge resistivities measured in the InAs/GaSb double quantumwell system. Measurements in tilted magnetic field suggests an anisotropy of the conducting regionsat the edges with a larger extent in the plane of the sample than normal to it. Finger gate sampleson both material systems shine light on the length dependence of the edge resistance with the intentto unravel the nature of edge conduction in InAs/GaSb coupled quantum wells. I. INTRODUCTION
Topological insulators have been predicted [1, 2] toshow the quantum spin Hall effect based on dissipation-less transport in edge states separated by an insulatingbulk. Experimentally such a situation was first realized inan inverted HgTe/(Hg,Cd)Te quantum well by the properchoice of quantum well thickness [3]. While a non-localmeasurement proved the existence of edge transport [4],the confirmation of spin-polarized transport demandedmore complex transport experiments [5]. For the cou-pled quantum well system InAs/GaSb, double gating waspredicted to tune density and band alignment indepen-dently [6, 7] resulting in a tunable two-dimensional topo-logical insulator. According to theory, conductance inhelical edge states is switched on or off when crossing theboundary between the topological and trivial insulatorby a proper change of front- and back-gate voltage.In a series of pioneering experiments the group ofR. R. Du reported evidence for edge modes in invertedInAs/GaSb quantum wells [8, 9]. In addition, quantizedconductance close to charge neutrality was reported [10].Subsequently, edge conduction was confirmed by a num-ber of groups in the regime of inverted band alignment bynon-local transport measurements [9, 11, 12], by scanningSQUID [13] and via the detection of edge-mode supercon-ductivity [14]. So far there is no experimental report inthe literature directly demonstrating the helical natureof these edge states. Several publications [8, 11, 12] re-ported on the relevance of bulk conduction and limitedgate tunability which prohibit to study edge conductionin the full phase diagram and hamper, for example, a de-tailed survey of the length dependence of the edge con-ductance.Recent findings of edge conduction in the non-invertedregime [15, 16] raised additional questions. The physicalorigin of these edge states is under debate. In addition,such trivial edge states may possibly co-exist with helicaledges in the inverted regime and it is unclear how suchedge states of different origin could be distinguished bytransport experiments.Here we report edge transport experiments on InAs quantum wells, having in mind that this material is aconstituent of InAs/GaSb double quantum wells. Weconsistently find edge conduction with a resistivity of1.3-2.5 kΩ/ µ m with different techniques. Tilted magneticfield measurements suggest an anisotropy of the conduct-ing edge channels with a larger extent in the plane of thesample. Analogous transport experiments in InAs/GaSbdevices allow us to study the dependence of conductanceon edge length in the inverted regime.This paper is structured as follows: After an introduc-tion to wafer material, sample fabrication and measure-ment setup in section II, we present the results on InAsdevices of different geometries in section III. In section IVwe then compare to measurements on InAs/GaSb dou-ble quantum wells. Measurements in tilted magnetc fieldsare jointly presented for both material systems in sectionV, before we finally attempt to find a consistent inter-pretation of all the presented data and critically discussthe conclusions that can be drawn in section VI. II. WAFER MATERIAL, SAMPLEFABRICATION AND MEASUREMENT SETUP
Three different wafers were grown by molecular beamepitaxy. Wafers A and B host a two-dimensional electrongas in an InAs quantum well. Wafer C contains a two-dimensional electron and hole gas in a InAs/GaSb doublequantum well. Wafer A is grown on a GaAs substrate andcontains a 15 nm InAs quantum well confined by AlSbbarriers. The layer sequence of wafer A was also usedfor wafer C, differing only by the additional 8 nm GaSbquantum well on top of the InAs quantum well and bythe use of a Ga-source with reduced purity as describedin Ref. 17. Wafer B is grown on a GaSb substrate andhas a layer sequence of Al x Ga − x Sb/InAs/AlSb with a24 nm InAs quantum well showing an improved mobilityas reported in Ref. 18. The information on the epitaxialgrowth of wafers A, B and C are summarized in Table I.Hall bar structures were patterned by optical lithog-raphy combined with wet chemical etching deep into thelower barrier material as described in Ref. 19. Ti/Aupads separated from the wafer surface by a 200-nm-thick a r X i v : . [ c ond - m a t . m e s - h a ll ] J un Wafer name Quantum well material(s) Well thicknesses Barrier materials Substrate Ga-source purity
A InAs 15 nm AlSb and AlSb GaAs highB InAs 24 nm Al x Ga − x Sb and AlSb GaSb highC InAs/GaSb 15 nm/8 nm AlSb and AlSb GaAs lowTABLE I. Wafer details including information on layer sequence, substrate wafer and Ga-source purity used during the epitaxialgrowth.FIG. 1. Transport results on InAs two-dimensional electrongases, Wafer A: The longitudinal resistivity ρ xx as a functionof top gate voltage V tg of a Corbino device (a) and of an asym-metric Hall bar (b). Arrows indicate the gate-voltage sweep-direction. The optical microscope pictures together with themeasurement configuration are shown as insets. (c) Finitenon-local resistances R measured in the configuration indi-cated in the inset. (d) Measurement results from (c) normal-ized by the respective edge segment length L i [dotted linesin the lower inset of (d)]. The resistor network model in theupper inset is explained in the text. Si N dielectric are used as gates. Ohmic contacts weremade by a Au/Ge/Ni eutectic.If not mentioned otherwise, the measurements wereconducted at 1.5 K. Four-terminal resistance measure-ments on wafer A and B were performed by applying anac current of 10 nA with a frequency of 31 Hz. On devicesfrom wafer C, four-terminal DC measurements were con-ducted because of high contact resistances (of the orderof 10 kΩ) in these devices. The Corbino device was mea-sured by applying an alternating voltage and measuringthe AC current with an IV-converter. III. TRANSPORT IN InAs DEVICES OFDIFFERENT GEOMETRIES
Figure 1 shows the top-gate ( V tg ) dependent resistanceat zero magnetic field of different device geometries fab-ricated on wafer A. The Corbino device [Fig. 1(a), inset]allows us to deduce the bulk resistivity from the measuredconductance, in the absence of any edges connecting thetwo ohmic contacts. For positive gate voltages V tg the longitudinal resistivity is ρ xx ≈
30 Ω indicating that theFermi energy is deep in the conduction band. Around V tg ≈ − . ρ xx increases rapidly signal-ing the depletion of the two-dimensional electron gas. Itreaches approximately 10 MΩ for V tg < − . ρ xx ≈ V tg < − V tg > − . V tg < − . V tg is indicated with black arrows. The hysteresis, i.e. thedifference in resistivities between the two sweep direc-tions, depends strongly on the gate voltage range. Mea-surements above V tg = − . V tg < − . V tg > V tg < − . R = V /I shown in Fig. 1(c) (red and blues traces) by the respec-tive gated edge lengths L i between the measurement con-tacts [see inset of Fig. 1(d)] to demonstrate the linearlength dependence. This scaling suggests that the sam-ple can be modeled with a resistor network as shown inthe upper inset of Fig. 1(d). All possible four-terminalmeasurement configurations on this device are consis-tent when scaled with respective edge segment lengths L i leading to the edge resistivity ρ edge = R i /L i = V /I i L i =2 . ± . /µ m (current flow along the edge I i calculatedwith the model). This consistency confirms in retrospectthe resistor network model used for the analysis.Similar measurements and analysis were conducted onsix Hall bar devices fabricated by wet etching [19] on thetwo different wafers A and B. All measurements agreewith the above findings and result in edge resistivities inthe range of ρ edge = 1 . − . /µ m. The wafers usedhave different quantum well thicknesses, are confined bydifferent lower barriers and were grown on different sub-strates, each requiring a different growth procedure forthe buffer. The observed edge resistivity was indepen-dent of all of these boundary conditions. Two Hall barswere fabricated with a dry etching technique (also de-scribed in [19]). These devices showed edge resistivitesof the same order of magnitude as well.Edge states, or more generally, conducting surfaces,were already suspected previously to reduce the efficiencyof IR detectors based on InAs/GaSb superlattices [20]and therefore were studied optically. Fermi level pinningin the conduction band of InAs is often mentioned asa possible reason for the enhanced electron density atthe edge, especially because the effect is robust againstchanges of layer sequence and fabrication [21–24]. Oth-ers also add the effect of electric field line concentration,present when gates overlap the sample edges. Edge con-duction may also be a side effect of sample processing,either after long exposure to air [25] or because of con-ducting Sb residues on the surface after etching [26]. Ourexperiments add to these results that edge conductioncan occur in pure InAs quantum well samples and thatit may dominate transport if the bulk is insulating. Itsphysical origin cannot be assessed by the present exper-iments.We continue with an investigation of the length depen-dence of the edge resistance inspired by the experimentsof Nichele et al [15]. The device is shown in the insetof Fig. 2(a). Eight gates, 0.5 to 49 µ m in width, crossa long Hall bar (width W = 4 µ m, length L = 162 µ m).The longitudinal resistance R xx plotted in Fig. 2(a) isthe sum of a gate-voltage independent resistance due to FIG. 2. Finger gate samples (optical microscope picture of adevice in the inset of (a)) to study the longitudinal resistance R xx as a function of all eight gates with gate lengths reaching L tg(i) = 0 . µ m for an InAs two-dimensional electrongas, wafer A (a) and for a coupled quantum well InAs/GaSb,wafer C (c). The longitudinal resistance R xx at V tg(i) = − L tg(i) together with a linearfit (black dashed line) in (b) resp. (d). the ungated sections of the Hall bar and the resistancecaused by the gated section below one of the finger gatesbiased with V ( i )tg (note that the constant resistance alsodepends on the length of the biased gate), i.e., R xx ( V ( i )tg ) = ρ gated ( V ( i )tg ) L ( i )tg + ρ ungated ( L total − L ( i )tg ) . (1)A systematic length dependence is seen in Fig. 2(a),where only V ( i )tg down-sweeps are shown for simplicity.The longitudinal resistance at V ( i )tg = − L ( i )tg in Fig. 2(b). Theblack dashed line is a linear fit to the data according toeq. (1) using ρ ungated and ρ gated as fitting parameters,resulting in an edge resistivity ρ edge = 2 ρ gated ( − ≈ . /µ m (the factor of 2 accounts for the two edges).The fit line also serves as a guide to the eye to demon-strate the obvious proportionality between longitudinalresistance R xx and the gated edge length. IV. COMPARISON TO TRANSPORT INInAs/GaSb
With these insights about InAs in mind we now turn tothe discussion of InAs/GaSb double quantum well struc-tures, which contain a hybridized electron-hole system[27–32]. Despite multiple affirmations of edge conduc-tion in the inverted regime of InAs/GaSb [8–14], a care-ful analysis of the various possible contributions to edgeconduction is missing.Figures 2(c) and (d) show measurements on anInAs/GaSb finger gate sample (wafer C, same sample di-mensions as the InAs sample) analogous to Figs. 2(a) and (a) ρ xx ( k Ω ) B ┴ (T) tg = -5.5V (b) B ┴ (T) ρ xx ( k Ω ) α at V tg = -4V ~~ ~~ μ m μ m μ m FIG. 3. Magnetic field is applied with an angle α to the nor-mal of an InAs two-dimensional electron gas (a) resp. cou-pled InAs/GaSb systems (b) as schematically explained inthe inset of (a). The longitudinal resistivity ρ xx in an edgedominated regime (at a gate voltage of V tg(i) = − . V tg(i) = − B ⊥ . (b). Also in this device the resistance depends linearly ongate segment length, in agreement with Ref. 10. Analo-gous results could be found on a second device with sixgates, 0.5 to 20 µ m in width, crossing a 2 µ m × µ mHall bar (data not shown). For the finger gate sam-ples of both material systems we find an edge resistivity ρ edge = 2 ρ gated ( − . − . /µ m. Note that theslope of the fit (black dashed line) indicated in Fig. 2(b)and (d) is not the edge resistivity ρ edge (see eq. 1). V. TRANSPORT IN TILTED MAGNETICFIELDS
In order to obtain information about the spatial ex-tent of orbital states in the conducting channels alongthe edges we measure the resistance of micron-sized Hallbars in magnetic fields tilted by an angle α with respectto the normal of the sample plane [inset of Fig. 3(a)].Figure 3(a) displays the resistivity of an InAs sample(wafer A, Hall bar width 2 µ m, length 3 µ m) measuredat V tg = − . B ⊥ for various angles α . We find that all curvesscale on top of each other when plotted against B ⊥ . Thissuggests an anisotropy of the conducting regions at theedges with a larger extent in the plane of the samplethan normal to it. Furthermore, the resistance decreaseswith increasing magnetic field. Theories describing sucha trend in narrow channels of two-dimensional systemsare found, for example, in Ref. 33.The measurements shown in Fig. 3(b) were obtainedon InAs/GaSb (wafer C, Hall bar width 2 µ m, length5 µ m and 5 µ m × µ m, respectively), again in the regimedominated by edge conduction at V tg = − B ⊥ , likein InAs. Again, we interpret this finding with a larger extent of the conducting edge-region within the plane ofthe sample than in growth direction.In InAs/GaSb the trend of the edge resistivity withincreasing field depends on the Hall bar size. The edgeresistivity of the larger device is suppressed with field likefor InAs Hall bars. For small devices the trend in mag-netic field is opposite. The reason for such a dependenceon device size remains to be explained.For InAs/GaSb samples in the inverted regime Du et al [10] also find an increasing edge resistance with magneticfield for small four-terminal devices. Nichele et al [34]showed an increase in resistivity with rising field for largedevices, which is not in agreement with the findings here,but due to large bulk conductivity their measurement isnot in an edge dominated regime. The same applies tomagnetic field dependent measurements in Ref. 35. Thedecrease in resistance around charge neutrality was at-tributed to the enhanced anisotropy of the band structurein parallel field. The latter could not be observed for thedisordered material presented here. The magnetic fielddependence on InAs/GaSb samples in the trivial regimeis measured in Ref. 16, but hard to extract from colorplots. It can therefore not be compared with the resultspresented here. VI. CRITICAL DISCUSSION OF EDGECONDUCTION IN InAs/GaSb DOUBLEQUANTUM WELLS.
Based on the new insights obtained from the mea-surements presented in this paper we now try to criti-cally discuss edge conduction and its length-dependencein InAs/GaSb devices. In this endeavor, we take the datapresented in Fig. 2(d) and data from Ref. 12 into account,which were all measured on devices from the same waferC. Figure 4 presents a summary of all the data mea-sured in our lab. The blue and black stars in this figurerepresent the data points from Fig. 2(d). Plotted is theresistance R gated = 2 ρ gated ( − L ( i )tg [c.f. Eq. (1)] ver-sus the gate length L ( i )tg . All the other colored symbolsare non-local resistances R nl from the different devices ofRef. 12 (in Ref. 12 referred as truly non-local resistancesof type 1). In Fig. 3(a) of this reference, only the datapoints of device C were explicitly shown , but all deviceswere analyzed and it was concluded that the edge resis-tance is independent of edge length, in apparent contrastto the data in Fig. 2(d).First we note that the resistance R gated in Fig. 4 is onlya lower bound for the true edge resistance. The measure-ments on the finger gate sample do not distinguish bulk-and edge contributions to the total current. If there wasa bulk current in the InAs/GaSb finger gate devices, then R gated would underestimate the true edge resistance.The resistances for devices A–E from Ref. 12 plottedin Fig. 4 are bare non-local resistances R ( i )nl obtained bydividing particular measured non-local voltages V ( i )nl by FIG. 4. Summary of InAs/GaSb devices from Fig. 2(d) inthis paper, and from Ref. 12 measured in the edge dominatedregime. The black and blue stars represent the calculatededge resistances R gated (derivation explained in text) versusthe respective gate length L gated of the finger-gate sample inFig. 2(d). The other symbols refer to the Hall bar devicespresented in Ref. 12. Plotted is the four terminal non-localresistance R nl against the gated edge between the respectivepair of voltage probes (details in text). I tot , the total current applied. These resistances are plot-ted in Fig. 4 against the lengths L ( i )gated of the gated edge i between the respective pair of voltage probes. Here, thestriking phenomenon is that the bare non-local resistanceis independent of i , and therefore of L ( i )gated . For this rea-son it was concluded in Ref. 12 that the edge transportis ballistic on the investigated length scales. We see inFig. 4 that this holds true for the small devices A–D, butis no longer found for the larger device E. This behaviorof InAs/GaSb is in stark contrast to the non-local resis-tances of the investigated InAs devices [c.f. Fig. 1(c)],where we found scaling with L ( i )gated .However, the R ( i )nl cannot be directly interpreted asedge resistances, because I tot is the sum of two edge cur-rents of possibly unequal magnitude running along thetwo Hall bar edges of unequal lengths between the cur-rent contacts, plus a possible bulk current. Assumingcompletely ballistic edge conduction and zero bulk cur-rent we find that the edge resistance is larger than R ( i )nl by a factor of two. Diffusive edge conduction and also afinite bulk current would raise this factor further. Thiswould move all data points of devices A–E above thelower bound of the edge resistance given by the blackand blue stars of the finger gate sample in Fig. 4. Basedon these considerations, we may state that the edge re-sistances of all the investigated devices are of the sameorder of magnitude and consistent with each other.The question still remains, why the non-local measure-ments on Hall bars give length independent R ( i )nl , whereasthe finger gate sample exhibits a linear length dependencedown to at least 1 µ m. In order to find possible sources of misinterpretations here, we take a critical look at theedge lengths L ( i )gated extracted for the devices in Ref. 12.These lengths were taken from optical microscope imagesassuming that the width of conductive edge regions ismuch smaller than any lithographic width of the samples.However, considering the finding of a finite extent of theedge conducting regions in the plane of the sample (c.f.Fig. 3) , it is conceivable that the conducting regions can-not enter the narrow voltage probes without coupling sostrongly that the gated edge length within these voltageprobes does not contribute to the relevant edge length.This scenario would reduce the spread of the true L ( i )gated in Fig. 4 so strongly that a length-independent resistancecould no longer be deduced from the data with sufficientconfidence.Similarly, one could find reasons why the linear de-pendence of R gated on gate length L ( i )tg arises in spiteof the presence of helical edge modes in the finger-gatesample of this paper. One possible scenario is the pres-ence of bulk or trivial edge conductance shunting the sig-nificantly lower conductance of the helical edge modes,which is expected to be e /h .Summarizing this critical discussion of our own mea-surements, we have to state that, first, the linear lengthdependence in the finger-gate sample only gives a lowerbound for the possible edge resistances which is well be-low the value expected for helical edge states at least upto lengths of 10 µ m. Second, the non-local resistances ofHall-bar devices do not give a robust estimate of edge re-sistances either, because the relevant current along a par-ticular sample edge is not known. Third, the edge-lengthestimates for these samples are based on the assump-tion of narrow edge channels that may well be violated,which would render the length-independent edge resis-tance an illusion. We believe that this discussion bearsimportance also beyond our data for the interpretation ofrelated work on transport in InAs/GaSb double quantumwells by other authors. VII. CONCLUSION
Our experiments show edge conduction in InAs two-dimensional electron gases where no topological effectsare expected. An edge resistivity of ρ edge = 1 . − . /µ m could be confirmed for standard as well asasymmetric Hall bars and finger gate samples. Theseresults have to be compared to investigations in theInAs/GaSb double quantum well system, a topologicalinsulator candidate. The latter also shows a resistancewith linear dependence on edge length of the same orderof magnitude for edge lengths as small as 1 µ m. Addition-ally, both systems show a magnetoresistance in tilted fieldthat is independent of the parallel magnetic field com-ponent with respect to the sample plane. Even thoughstandard InAs/GaSb samples have indications for edgelength independent non-local resistances, an alternative,trivial explanation can not be excluded with the latter re-sults in mind. 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