Beenakker, C.W.J.

Blom, F.A.P.

Fontein, P.F.

Giling, L.J.

Hendriks, P.

Wolter, J.H.

NARCIS 

Spatial potential distribution in GaAs/AlGaAs heterostructures under quantum Hall conditions studied with the linear electro-optic effect

Leiden University Scholary Publications

Surface Science 263 (1992) 91-96North-Holland surface scienceSpatial potential distribution in GaAs/AlxGa.l_xAs heterostructuresunder quantum Hall conditions studied with the linearelectro-optic effectP.F. Fontein a, P. Hendriks a, F.A.F. Blom a, J.H. Weiter a, L.J. Giling band C.W.J. Beenakker c" Eindhoven Unwersity of Technology, 5600 MB Eindhoven, Netherlandsb Unwersity of Nymegen, 6525 ED Nijmegen, Netherlands'Philips Research Laboratones, 5600 JA Eindhoven, NetherlandsReceived 28 May 1991; accepted for publication 26 August 1991We apply the linear electro-optic effect (Pockels effect) to mvestigate the spatial potential distribution in GaAs/AlA rGa1_A.Asheterostructures under quantum Hall conditions. With this method, which avoids electrical contacts and thus does not disturb thepotential distribution, we probe the electrostatic potential of the two-dimensional electron gas locally. Scanning across the width ofthe sample inside a quantized Hall plateau we observe a steep change of the Hall potential at the edges of the two-dimensionalelectron gas. This steep change occurs over a distance of about 70 μ,ιη, which is the lateral resolution of the expenmental set-up.More than 80% of the total Hall voltage is concentrated near the edges. The remainder of the Hall potential is distnbuted in theinferior of the sample and vanes linearly with the position. The results are interpreted in terms of unscreened Charge at the edgesIf the plateau region is left or if the quantized Hall conditions are violated by increasing the temperature or current level the Hallpotential becomes a linear function of position.1. IntroductionMeasurements of an electrostatic potential dif-ference are usually carried out by attaching elec-trical contacts or potential probes to the Systemunder study. It is generally accepted, though, thatthe presence of these electrical contacts disturbsthe potential distribution. This certainly holds formeasurements of the potential distribution intwo-dimensional electron gases under quantizedHall conditions.The contact introduces an equipotential, whichgives rise to the so-called Corbino effect, it ther-malizes the electron distribution and finally, byattaching an electrical contact, the electrochemi-cal potential rather than the electrostatic poten-tial is measured. With these problems in mind itis not clear whether the effects of current bunch-ing reported in refs. [1,2] are due to the presenceof the electrical contacts or due to an intrinsiceffect in the two-dimensional electron gas.Fortunately the properties of the GaAs/AI.,.Gaj_^As heterostructure allow for an opticaltechnique to determine the spatial potential dis-tribution under quantized Hall conditions. Thistechnique makes use of the effect that GaAsbecomes birefringent when an electric field isapplied, the linear electro-optic effect or Pockelseffect. The application of the Pockels effect is notuncommon in the field of testing of GaAs chips[3], but has until recently never been appliedunder quantized Hall conditions. Since it is atechnique which does not involve electrical con-tacts we avoid the problems mentioned above.2. Details of the experimental setupThe beam of a 1.3 μ m, l mW semiconductorsolid-state laser is focused, with a focal diameterof 70 μπι, on a GaAs/Al.cGa1_.cAs heterostruc-ture with a two-dimensional electron gas in the0039-6028/92/$05.00 © 1992 - Eisevier Science Pubhshers B.V. and Yamada Science Foundation. All nghts reserved92 PF Fontein et al / Spatialpotentiell distnbution m GaAs /AlxGaj_ xAs heterostiucturesFig l Expenmental set-up, the electncal Circuit is indicatedschematically(001) plane, see fig. 1. The light is polarized alongthe (100) axis and travels in the (001 > direction.Since the GaAs is transparent to the wavelengthof 1.3 μ m, the light exits on the back of theSubstrate, on which we evaporated a thin (8 nm)semi-transparent Au-layer acting äs an equipo-tential plane. When a potential difference V ispresent between the two-dimensional electron gasand the Au-layer, the components of the lightpolarized along the fast and slow axes obtain aphase difference. It was shown [3] that this phasedifference Δ Γ is equal toΔΓ= i Ez(x, y, z) dzJ30r4lV(X,y), (1)where n0 and r41 are the refractive index and thecomponent of the electro-optic tensor of theGaAs, d is the thickness of the Substrate, Ez thecomponent of the electric field perpendicular tothe two-dimensional electron gas and λ the wave-length. The electric field parallel to the two-di-mensional electron gas does not enter this ex-pression. If we Position a quarter wave plate anda polarizer in front of the detector the transmit-ted light intensity varies almost linearly with theapplied potential difference between the two-di-mensional electron gas and the Au-layer.Since we do not want the incident laser beamto ionize additional donors and thus disturb thepotential distribution, we apply a constant back-ground Illumination which empties all donorstates in the AixGal_xAs. We carefully selecteda GaAs/ AI ̂ Ga^^ As heterostructure to ensurethat even under Illumination there is no parallelconduction in the Al^Gaj^As layer. This is es-sential, because parallel conduction might causea potential drop in the Al^Ga^^As. Since theAl^Gaj^As also shows the Pockels effect, addi-tional unwanted phase shifts in the transmittedlight might then occur. However, äs long äs theAlxGa1_xAs is insulating, the potential drop inthe very thin Al^Ga^^As layer is negligibly small.Our sample consists of a 400 μ m GaAs sub-strate with on one side the 8 nm Au-layer kept atground potential. On the other side a 4 μ m GaAsbuffer layer, a 20 nm Al^Ga^^As spacer layer, a40 nm Al/Ja^As Si-doped (ns, = 2 Χ 1024 nT3)layer (both with χ = 0.3) and a 18 nm GaAs caplayer are grown. The sample has a rectangulargeometry of 5.4 mm length and 2 mm widthwithout side arms. Current contacts (In) werealloyed into the two-dimensional electron gas atboth ends, 5.4 mm apart.To avoid interference effects the sample isslightly tilted from normal incidence ( ~ 7 °). Dueto this tut angle, electric fields parallel to thetwo-dimensional electron gas also enter eq. (1).The impact of the error introduced by this tiltingwill be discussed later on in relation to the pres-ence of fringing fields. As the potential differ-ences to be detected are fairly small we apply analternating current (235 Hz) through the two-di-mensional electron gas and thus modulate thetransmitted light intensity. The detector Output ishence measured with a lock-in technique. Wechecked that the measured Signals had neither anout-of-phase component nor a double frequencycomponent.In order to determine the local potential in thetwo-dimensional electron gas we first perform acalibration measurement. To this end an alternat-ing voltage of 5.6 Vpp is applied between thetwo-dimensional electron gas and the Au-layer,which is at ground potential, and the resultingdetector signal is measured. Next, an alternatingcurrent of known amplitude is sent through thetwo-dimensional electron gas, with one currentcontact and the Au-layer at ground potential, andagain the lock-in signal is measured. Both mea-surements are taken at the same Position of thelaser beam. The ratio of the detected intensitiesin these two measurements yields the unknownPF Fontein et al / Spatial potential distribution m GaAs/AlxGa,_ xAs heterostructures 93Potential at the position of the laser beam for thecase of the alternating current flowing throughthe two-dimensional electron gas. Subsequentlythe laser beam is scanned across the surface ofthe sample step by step. At each spot the calibra-tion procedure is repeated. The results do notdepend on the amplitude of the voltage appliedin the calibration measurement. Further, the useof alternating currents with current reversal inthe sample does not cause any problems, sinceour results are the same if we apply a DC offsetcurrent. With this DC offset current we obtain amodulated current density which is not reversed.Therefore we can rule out that spatial switchingof current paths affects our measurements.3. ResultsThe result of a two-pomt resistance measure-ment äs a function of magnetic field is shown infig. 2. Due to the two-point character of themethod both Hall plateaus and Shubnikov-deHaas oscillations are visible. From fig. 2 an elec-tron concentration of 5.0 Χ 1015 m~2 and a mo-bility of 20 m2/V · s can be derived. In the follow-ing we subsequently present and discuss line scansof the potential made at the magnetic field valuesindicated in fig. 2. Unless indicated otherwise thetemperature at which these scans are made is 1.5K.40B (T)Fig 2 Voltage across the sample versus magnetic field (7 = 5μ, A, T = l 5 K) Arrows indicate magnetic field values atwhich line scans are made2010-05 05 1Ox (mm)Fig 3 Line scans of the potential mside a quantized Hallplateau The solid line result from a model calculationThe first two scans, see fig. 3, are made insidethe plateau with filling factor four. These arescans across the width of the Hall bar in themiddle between the current contacts. The edgesof the Hall bar are at +1 mm. It is obvious fromfig. 3 that the Hall potential steeply increases ordecreases at the edges. In the interior a more orless linear dependence on position is observed. Ifthe temperature is raised to 55 K, see fig. 4, theedge effects disappear and a linear dependenceof the Hall potential on position is observed. Thisobservation, in combination with the fact that themeasured potential difference is equal to the Hallvoltage measured electrically on the Hall probes,implies that there are no disturbing fringing fieldsat the edges. Prior to the presentation of furthermeasurements we now first turn to the theoreti-cal Interpretation.100E>D D D-1OOO -5OOFig 4 Line scan at 55 K, the straight line is a least-squares fit94 PF Fontein et al / Spatmlpotentiell distnbution m GaAs / AI xGa j _ x As heterostructuresFar away from the current contacts in a homo-geneous sample also a homogeneous current dis-tribution is expected to occur, äs long äs thediagonal component of the resistivity tensor pxx= 0. This can easily be derived from the Substitu-tion of j = σΕ into div j = 0, with j the currentdensity and σ the conductivity, which leads toaxx(d2V/dxz + d2V/dy2)=0. (2)If we assume the current to flow in the y-direc-tion and if we assume an infinitely long sample,we can show that dV/dy = constant, and henceS2V/dy2 = 0. Thus it follows from eq. (2) that inthe homogeneous case also d2V/dx2 = 0 (if σχκ Φ0) and hence W/'dx = constant, which implies ahomogeneous Hall field and hence a homoge-neous current distribution. This is what we ob-serve at 55 K (fig. 4) where the quantized Halleffect is absent and hence σχχ Φ 0.If σχχ = 0, however, this argument does nothold and the potential distribution has to becalculated by other means. This calculation hasbeen carried out by MacDonald et al. [4] andThouless [5]. They argue that at integer fillingfactor i a possibility exists to accommodate morecharge per unit area in one Landau level. In anelectric field all one electron wavefunctions areshifted in space. If the electric field depends onPosition this shift and hence the electron densitydepends on position too. In this way it is possibleto maintain an integer filling factor throughoutthe sample despite charge redistribution. The po-tential and the excess density are related byCoulombs law. The resulting equation which hasto be solved is [4,5](3)-W/2Xln\x-xwith ξ = ί12/πα*, l the magnetic length, a* theeffective Bohr radius (~ 10 nm in GaAs) and Wthe width of the sample. This can be done numer-ically. For the limit of small ξ Beenakker [6] hasshown that the solution of eq. (3) can be approxi-mated accurately by:X\n\(x-W/2)/(xfor x\ <(W/2) -ξ. (4)In our case ξ is small (ξ = 1.6 X 10~8 m for B = 5T and a relative dielectric constant of 13 forGaAs). The Variation of VH(x) within a distanceξ from the edges can be neglected. Eq. (4) ap-proximates the potential äs a result of line chargewith width ξ at the two edges χ = + W/2 of theHall bar. In fig. 3 we have plotted the potentialdistribution calculated from eq. (4). The agree-ment with the experiment is remarkable in viewof the fact that the theory does not contain anyadjustable parameters.Results of scans outside the plateau region arepresented in figs. 5a and 5b. The almost linearJ>500 100O -500 500 1000χ (/um)Fig. 5 Line scans at a magnetic field of 4.55 T (a) (top) and 4.25 T (b) (bottom), the lines are meant äs a guide to the eyePF Fontein et al / Spatialpotential distnbution m GaAs /AI ^Ga,_ ΛΑ$ heteiostmctures 95-5OOFig 6. Line scans m the centre of a plateau at two currentlevelsincrease of the Hall potential in the interior ofthe Hall bar becomes more pronounced whenone leaves the plateau region. Also, outside theplateau the edge effects, although smaller, re-main present. A similar transition to a linearpotential distribution can be observed inside aplateau region if the current is increased, see fig.6. This can be explained by heating effects whichcause pxx to increase.These heating effects are most likely related tothe strong piling up of electrons at the edges andthe associated high electric fields. From eq. (3)we deduce that at a distance ξ from the edgewith (i = 4, KH = 0.1 V), the Hall electric fieldequals Ex = 3 Χ 105 V/m. This corresponds to apotential drop of 3 mV within a distance of 10nm, which results in a substantial overlap ofwavefunctions of adjacent Landau levels. Henceinelastic scattering processes may occur.In between plateaus these high electric fieldsdo not occur. The linear potential distributionwhich should develop if σχχ = constant ¥= 0 is as-sociated by an excess electron concentration of[5]«excess = (2«V¥l/eW)χ/(\W2 -x^/2, (5)with κ the dielectric constant and e the elemen-tary Charge. At a distance ξ from the edge and atKH = 0.1 V, B = 5 T this results in ncxcess = 1.4 X1013 m"2, which seems to be small regarding themagnitude of n = 5.0 Χ 1015 m"2. However, sinceσχχ can depend strongly on n, the conditionσΑ Λ = constant will no longer be fulfilled, even forsuch a small deviation from a homogeneous elec-tron distribution.It is tempting to Interpret the presence of edgeeffects in between the plateaus in terms of theabove mentioned inhomogeneities induced by alarge current. However, there are two major ob-jections to such an Interpretation. First, thereshould be a clear current dependence outside theplateau region äs eq. (5) depends on J/H. This,however, is not what we observe, although thismay be due to our limited ränge of currents used.Second, the potential distribution should beasymmetrical due to an electron excess at oneedge and a shortage at the other edge. This is notthe case in fig. 5.Perhaps the clue to the presence of edge ef-fects in between the plateaus can be found in thecorrespondence between the transition from in-side to outside a plateau and the transition fromlow to high current inside a plateau. Both transi-tions are gradual. This resemblance probably in-dicates that the underlying physics of both transi-tions is similar. If we assume that the sample isinhomogeneous, it is possible that the quantizedHall effect breaks down locally if the current isincreased. In an inhomogeneous sample evenoutside a plateau the quantized Hall conditionsmay still be fulfilled in part of the sample. In thiscase the transition from the Situation in fig. 3 tofig. 4 is no longer abrupt.We now turn to the influence of electricalcontacts. In fig. 7 line scans along the length ofthe sample are presented. These scans are car-ried out at a current of 50 μΑ and B = 5 T. Atthis large current the sample is heated up tosome extent, but the measuring time is consider-ably reduced. The influence of the ends of theHall bar with the current contacts is clearly visi-ble. Fig. 7 shows that the current enters at onecorner of the sample and exits at the oppositecorner, äs expected theoretically.The influence of internal electrical contacts ismuch less clear. In fig. 8 a scan at B = 5 T acrosssuch an internal electrical contact is shown. Wealso show a scan l mm below the internal con-tact. The contact is disconnected. Note, however,96 PF Fontein et al / Spatialpotentiell distnbution m GaAs / AI xGa, _ x As heterostructures> 300J> 2OO1002000 3000 SOOOy (/um)Fig 7 Line scans across the length of the sample, Imes are arather arbitrary fit with polynomials of order four B = 5 T125-10OO -75O -5OO -25O O 25O 5OO 75O 10OOX (AUTOFig 8 Line scan across and below an intenor contact TheImes are a connection of the data points, B = 5 Tthat the result is not changed when we connectthe contact to a lock-m amphfier with an mputimpedance of 100 ΜΩ to ground potential Apartfrom the edge effects at the boundary of thetwo-dimensional electron gas we see a sharpbendmg of the measured potential m the immedi-ate neighbourhood of the contact The Interpre-tation of this effect is yet unclearWe conclude from our expenments that theHall potential distnbution m a plateau region iswell descnbed by the presence of edge Charge Inbetween plateaus and at high current levels theHall potential distnbution becomes a linear func-tion of Position, with a gradual, sometimes m-complete change between both kmds of distnbu-tions This mdicates the coexistence of both re-gions with σχχ = 0 and regions with σχχ Φ 0 underthese circumstancesReferences[1] G Ebert, K von Khtzmg and G Weimann, J Phys C 18(1985) L257[2] H Z Zheng, D C Tsui and A M Chang, Phys Rev B 32(1985) 5506[3] B H Kolner and D M Bloom, IEEE J Quantum Electron 22 (1986) 79[4] AH MacDonald, TM Rice and W F Brmkman, PhysRev B 28 (1983) 3648[5] DJ Thouless.J Phys C 18 (1985) 6211[6] C W J Beenakker and H van Houten, Solid State Physics,Eds H Ehrenreich and D Turnbull (Academic Press,New York, 1991)

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Spatial potential distribution in GaAs/AlGaAs heterostructures under quantum Hall conditions studied with the linear electro-optic effect

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