Imaging magnetoelectric subbands in ballistic constrictions
A. A. Kozikov, D. Weinmann, C. Roessler, T. Ihn, K. Ensslin, C. Reichl, W. Wegscheider
IImaging magnetoelectric subbands in ballisticconstrictions
A. A. Kozikov , D. Weinmann , C. R¨ossler , T. Ihn ,K. Ensslin , C. Reichl and W. Wegscheider Solid State Physics Laboratory, ETH Z¨urich, CH-8093 Z¨urich, Switzerland Institut de Physique et Chimie des Mat´eriaux de Strasbourg, Universit´e deStrasbourg, CNRS UMR 7504, 23 rue du Loess, F-67034 Strasbourg, FranceE-mail: [email protected]
Abstract.
We perform scanning gate experiments on ballistic constrictions in the presence ofsmall perpendicular magnetic fields. The constrictions form the entrance and exit of acircular gate-defined ballistic stadium. Close to constrictions we observe sets of regularfringes creating a checker board pattern. Inside the stadium conductance fluctuationsgoverned by chaotic dynamics of electrons are visible. The checker board patternallows us to determine the number of transmitted modes in the constrictions formingbetween the tip-induced potential and gate-defined geometry. Spatial investigationof the fringe pattern in a perpendicular magnetic field shows a transition fromelectrostatic to magnetic depopulation of magnetoelectric subbands. Classical andquantum simulations agree well with different aspects of our observations. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y
1. Introduction
Quantum point contacts are the basic building blocks of many mesoscopic circuits.Quantized conductance requires ballistic transport with an elastic mean free pathexceeding the dimensions of the quantum point contact. In a series arrangement oftwo quantum point contacts [1] the total conductance depends on how electron wavescouple into and out of the region between the quantum point contacts. If this regionis a cavity, then interference of electron waves can also play an important role. Inmesoscopic devices the gate arrangement and therefore the dominant features of thepotential landscape are usually fixed. A moving gate, as it is provided by the metallictip of an atomic force microscope (AFM), allows us to vary the potential landscape inreal space and enables the investigation of ballistic and coherent transport properties ofnetworks of quantum point contacts (QPC). Not only the tip-sample bias voltage, butalso the tip-surface distance and the in-plane tip position can be changed in order tocontrol the strength and gradient of the tip-induced potential and thereby influence theelectron wave pattern. A local study of ballistic and coherent transport properties ofnetworks of QPCs is the focus of this paper.Using the metallic tip of an AFM and measuring the conductance change hasalready proven to be a powerful tool to locally investigate physical transport phenomena.It was successfully used to image and study the branching behaviour of electrons injectedfrom a quantum point contact [2, 3, 4], fractional edge states [5] and tunneling betweenedge channels [6] in the integer quantum Hall regime, localized states in graphenequantum dots [7] and nanoribbons [8, 9], carbon nanotubes [10], InAs nanowires [11],single electron quantum dots [12] and interference in quantum rings [13].In this work we study the conductance through a stadium formed by two ballisticconstrictions and perturbed by an AFM tip, at small perpendicular magnetic fields.We observe fringe patterns close to the constrictions when imaging conductance versustip position. We interpret these fringes as resulting from bottlenecks in the potentialbetween the tip and the gates leading to quantized conductance. The fringes form achecker board pattern, which allows us to exactly determine the number of transmittedmodes in the channels between the tip and the constriction edges. When the tip isscanned inside the stadium, large fluctuations of the conductance as a function of thetip position occur. Moreover, at certain tip positions the conductance is increased withrespect to the conductance of the unperturbed structure. Those features cannot beinterpreted in terms of the local current flow in the structure, and we have performedclassical and quantum model calculations in order to compare and understand theexperimental findings.While the fluctuations inside the stadium appear already in classical transmissioncalculations based on electron trajectories, pointing at an origin in the chaotic dynamicsof the tip-perturbed stadium, the checker board patterns result from conductancequantization. They can be qualitatively understood in terms of a network of quantizedresistors and quantitatively reproduced within fully coherent model calculations usingthe recursive Green function method. The appearance of regions with a tip-inducedincrease of the conductance above the unperturbed value is consistent with results ofthe coherent calculation as well, and is interpreted as a signature of the coherent natureof the transport processes in our experiment.When applying a perpendicular magnetic field we observe a transition from purelyelectrostatic subbands to Landau quantization while scanning across the constrictionsof the stadium and thereby changing the width of the channel. We propose a modelfor the resulting constriction potential that is created by the combined action of the tipand the fixed gates. We fit our experimental data with this model, which is based onRef. [14] and find for a large number of occupied subbands a good agreement betweenthe model and the experiment.
2. Experimental methods
Local studies of the conductance through a ballistic stadium are carried out usingscanning gate microscopy (SGM). The nanostructure is fabricated on a high-mobility(800 m /Vs) GaAs/AlGaAs heterostructure. Electrons in the 2DEG located 120 nmbelow the surface of GaAs have an elastic mean free path of 49 µ m and a Fermi energy, E F , of 4.4 meV. This corresponds to an electron density of 1 . × cm − and a Fermiwavelength λ F = 72 nm. The stadium with a lithographic diameter of 3 µ m is shown infigure 1(a) (bright yellow gates, SG1 and SG2). The entrance and exit of the stadiumare constrictions, each W = 1 µ m wide, which corresponds to N ≈
27 transmittedmodes. Two narrower quantum point contacts (QPC) 300 nm wide each (shown bydark yellow gates, qpcGL and qpcGR) are fabricated in order to be able to reduce thenumber of transmitted modes in the stadium constrictions from 8 to 0. However, theseQPC gates are not used in this work and therefore they are grounded for all the datashown in this paper.The conductance measurements are performed in a two-terminal configuration byapplying a bias of 100 µ V between the source and drain contacts at 300 mK. We apply abias of -8 V between the tip and the electron gas, which creates a local depletion regionwith a diameter of about 1 µ m [4] at a tip-surface separation of 70 nm. By scanningthe tip at a constant height and simultaneously recording the conductance across thesample, 2D conductance maps, G ( x, y ) are obtained.Figure 1(b) shows G ( x, y ) through the stadium when -1 V is applied to the stadiumgates (the stadium is formed at a voltage smaller than -0.4 V applied to the gates).This voltage reduces the electronic width of the constrictions from 1 to 0.7 µ m (seeSupplementary Material) and therefore the number of transmitted modes to N = 19.The stadium gates, SG1 and SG2, are outlined by black solid lines. One can see twodistinct and symmetric regions labeled I of suppressed conductance close to the entranceand exit of the stadium. In their centers lens shaped regions are seen, in which theconductance is suppressed to zero. Here, the tip-induced potential closes the respectiveconstriction thereby blocking electron flow through the structure. When the tip is in thecenter of the stadium, region II, the conductance is about 13 × / h, i.e. smaller thanin the absence of the tip, G ≈ × / h (e.g. in the corners of the image). In regionIII G increases to more than 16 × / h, a value that is larger than the unperturbedconductance G . The decrease of the conductance in regions I and II can be explainedclassically. Electrons entering the stadium from either electron reservoir are scatteredby the tip back into the respective reservoir. This backscattering of electrons will reducethe conductance. The increase of G in region III can not be explained in the same way.The enhanced conductance exists in this region even at low tip-sample bias voltageswhen the tip does not induce a depletion area. As the tip voltage is made more negativeregion III decreases in size and moves in the direction away from the center of thestadium.We have performed numerical calculations of the non-interacting zero-temperatureSGM-response within a fully coherent two-dimensional tight-binding model [20] usingthe recursive Green function algorithm [18, 19]. With the aim to describe a situationthat is as close as possible to the experiment, we apply the method to an open ballisticcircular cavity with a diameter of 2.5 µ m. The cavity is connected to wide quasi-1D leadsby openings that are similar to the quantum point contacts treated in Ref. [20]. Thehard-wall potentials defining the QPC-like constrictions have the shape of two fingerswith circular ends of diameter 0.15 µ m, facing each other with a minimal distance of0.8 µ m. The tip induced potential is represented by a hard-wall depletion disk with adiameter of 0.8 µ m. The Fermi energy is chosen such that the Fermi wavelength λ F = 73nm corresponds to the experimental value. In order to keep lattice effects negligible, wehave used a 2D square lattice with a lattice constant of 5 nm, much smaller than λ F and the length scales of the cavity.The quantum conductance through the model cavity computed within this approachas a function of tip position exhibits many of the experimentally observed features.Those include the lens-shaped suppression of the conductance close to the constrictions,the large conductance fluctuations as a function of the tip position when the tip isscanned inside the cavity, and tip positions where the conductance is increased aboveits unperturbed value G .Calculations of the transmission based on ballistic electron trajectories injectedin the stadium under different angles in the presence of the tip show that at any tipposition the conductance across the sample decreases below G . Such a behavior wouldalso be expected in the case of incoherent diffusive transport. Interestingly, the largeconductance fluctuations in region II also appear in the classical transmission, pointingat their possible relation with underlying chaotic dynamics.The conductance increase observed in region III is reproduced only within thecoherent calculations, but it does not appear in classical transport. It can thus beviewed as a signature of coherent transport through the stadium structure.In a quantum transport picture, the behaviour of the conductance in region III canbe explained as being due to a tip-induced transition from Ohmic to adiabatic transport[1]. In the so-called Ohmic regime the wide cavity leads to a mixing of the modes andacts like an effective reservoir such that the total resistance is close to the series resistanceof two classical resistors representing the QPCs. With the tip in regions III the widthof the cavity (and thus the channel mixing) is reduced such that the conductance ofthe two QPCs in series approaches that of the narrowest QPC in the adiabatic regime,becoming larger than the G observed when the tip is outside the stadium in figure 1(b).A significant tip-induced increase of the conductance does not occur in SGMexperiments on a single QPC on a conductance plateau as the ones of Refs. [2, 3]where only negative conductance changes are reported. However, regions of positive andnegative conductance change have also been observed in SGM experiments on small rings[13] where the unperturbed conductance is not quantized. Though the experiments usetip voltages that represent a quite strong perturbation of the system, those observationsare consistent with a recent prediction [20] based on perturbation theory in the tippotential. The lowest order conductance change can be positive or negative, butits prefactor vanishes when the unperturbed structure is on a quantized conductanceplateau. The second order conductance change that dominates on such a plateau isalways negative. Consistently, the unperturbed conductance of our experiment is notexactly a multiple of 2 e /h , and regions with tip-increased conductance occur. (a)(c) I II IIII
S D
SG1 (b)
III
SG2 qpcGL qpcGR ac current 1 (cid:109) mTip-depletedregion
Figure 1. (a) A room temperature AFM image of the sample. The stadium gates,SG1 and SG2, are biased, which is indicated by bright yellow metal. At the entranceand exit of the stadium there are two narrow QPCs (dark yellow metal) formed by thegates on the left, qpcGL, and on the right, qpcGR. The QPC top gates are groundedin this experiment. A double arrow in the center of the stadium corresponds to the accurrent flowing between source (S) and drain (D). The tip-depleted disc is schematicallyrepresented by a filled yellow circle and placed as an example in the region (III) ofenhanced conductance. (b) Conductance, G , through the stadium in units of 2e /h asa function of tip position, ( x, y ), at zero magnetic field. The stadium is outlined byblack solid lines. White and orange contour lines divide the area inside the stadiuminto three regions of interest, I, II and III, and correspond to the constant conductanceof 12 and 15 × e /h (without the tip), respectively. (c) Numerical derivative of theconductance in (a), d G ( x, y ) / d x , as a function of tip position. The voltage applied tothe stadium gates is -1 V. Insert: The structure of the left-hand side fringe patternwhen -2 V are applied to the stadium gates. The color scale is the same as in the mainplot. The black arrow in the insert and in (b) indicates the lens shaped region of zeroconductance. In order to emphasize the fine structure on top of these coarse conductance changes,we plot in figure 1(c) the conductance numerically differentiated with respect to the x -direction, d G ( x, y ) / d x . One can now clearly see sets of regular fringes near theconstrictions of the stadium (region I) and conductance fluctuations everywhere elseinside it (regions II and III). In this paper we focus on the investigation of the fringepattern in region I. These fringes are local and roughly periodic in space: they appearwhen the tip is close to the constrictions formed by the top gates.The size of the lens shaped zero conductance region depends on the electronic widthof the constrictions given that the size of the tip-depleted region is fixed. The smallerthe width the larger is the area of zero conductance. In the insert of figure 1(c) we showhow the left-hand side fringe pattern and the lens shaped region change when -2 V areapplied to the stadium gates (the fringe pattern close to the other constriction of thestadium changes in the same way). When the gate voltage is decreased from -1 to -2 V,the width of the depleted region around the gates increases, which reduces the width ofthe constrictions of the stadium from 700 to 600 nm. The fringes move together with thedepleted region and the size of the region where the conductance is blocked increases.Figure 2(a) shows a zoom of the fringe pattern at the right constriction of thestadium. The fringes appear as resulting from crossing linear structures that follow thetwo edges of the gate fingers forming the constriction. The superposition of the twosets of lines leads to the checker board patterns to the left and to the right of the lensshaped region. Figure 2(b) shows the left checker board pattern in detail. The triangleclose to the right lower corner of the image corresponds to the zero conductance region(see black arrow; part of the lens shaped region).The absolute value of the conductance, which corresponds to the differentialconductance in figure 2(b), is shown in figure 3(a). In these two 2D plots one cannotice that when the tip moves outside the checker board pattern, from the lens shapedregion perpendicular to the fringes in the upper and lower part of the plots, G changesapproximately in steps of 2e /h. This change seems to occur at the dark blue stripes(fringes) in figure 2(b) where d G/ d x is higher than that between the stripes whered G/ d x is close to zero. This behaviour is reminiscent of how the QPC conductancedepends on its width: outside the checker board pattern a QPC is formed between thetip and the upper or the lower top gate of the stadium producing a set of parallel fringesbelow or above the lens shaped region, respectively. When the tip is inside the checkerboard pattern, two QPCs in parallel are formed between the tip and the top gates ofthe stadium (for a schematic see figure 3(b)).That is why in order to explain the origin of the fringes, we consider a simple model(figure 3(b)), in which we describe constrictions formed at the entrance and exit of thestadium as well as those formed between the tip and the boundaries of the stadiumas four incoherently coupled ballistic resistors labeled a-d. The potential of the twoconstrictions a and b is created by the top gate-induced potential on one side and bythe tip-induced potential on the other. Electron waves bouncing off the walls of thispotential self-interfere, which results in a standing wave pattern – a wave function for (a) (b) (cid:109) m) x ( (cid:109) m) y ( (cid:109) m ) Figure 2. (a) A zoom-in of a fringe pattern at the right-hand side constriction of thestadium. (b) A separate SGM measurement of the area shown in (a) by a rectanglecarried out with a different AFM tip. The voltage applied to the stadium gate is -0.8V. The arrow indicates the lens shaped region of zero conductance. each mode. The number of transmitted modes in the i -th constriction is estimated tobe N i = int[2 W i /λ F ], where W i is the width of constriction i and i = a , b , c , d. Theconductance of each of the constrictions is given by the relation G i = 2 e /h × N i . QPCsa and b are connected in parallel, but in series with c and d giving the total conductance G Total = (cid:34)(cid:18) e h N a + 2 e h N b (cid:19) − + (cid:18) e h N c (cid:19) − + (cid:18) e h N d (cid:19) − (cid:35) − . (1)When the tip moves, only the number of modes N a and N b changes in our model.Beyond that the model assumes that the temperature is zero and that the transmissionfor each mode is a step function of energy, i.e. N i is an integer. The dependence of G Total on tip position resulting from the model is plotted in figure 3(c). The value ofthe Fermi wavelength was determined from classical Hall effect measurements, i.e. fromthe 2DEG density, and the widths of the constrictions W c and W d are both taken to beequal to 0.8 µ m (for the stadium gate voltage of -0.8 V the size of the constriction isabout 0.8 µ m).The model describes the experimental results quite well on a qualitative level,which is seen by comparing the calculated checker board pattern in figure 3(c) withthe measured one in figure 3(a). To check the quantitative agreement between the twoimages, we plot in figure 3(d) the measured (blue) and calculated (red) conductancealong the red line shown in (a) and (c). In addition to these two curves, a finitetemperature of 300 mK (black curve) and a smooth energy-dependent transmission (a) (b)(c) (d) W d W a W c W b TipStadium (0,0)(1,1)(2,2)(3,3) (1,0)(2,0)(3,0)(4,0) (0,1)(0,2)(2,1)(3,1)(3,2)(2,3) (1,2)(1,3) G (2e /h) G (2e /h) r ( (cid:109) m) 0.30.15 Figure 3. (a) G in units of 2e /h plotted as a function of tip position close to the lensshaped region (see figure 2(b) for its numerical derivative). Dashed lines are guidesto the eye. Numbers in brackets correspond to the number of transmitted modes inthe constrictions formed between the tip and the stadium walls. The voltage appliedto the stadium gates is -0.8 V. (b) A model used to explain the results shown in (a).(c) Results of the numerical simulations. (d) Conductance plotted as a function ofdistance along the red line in (a) and (c). The blue line is the experimental curve.The red curve corresponds to simulations at zero temperature and a step unit functionof the transmission coefficient. The black and green curves differ from the red oneby taking into account a finite temperature of T = 300 mK and a smooth energy-dependent transmission coefficient, respectively. The number “0” (in red) in (a) and(c) corresponds to “0” in (d). The orange curve corresponds to a numerically computedconductance as a function of tip position using a fully coherent tight-binding model.The modeled curves are slightly shifted to the right for a better comparison with theexperimental curve. coefficient (green curve) are also taken into account in the calculation for comparison.The two additional curves were calculated using a saddle-point model based on theexperimentally determined subband spacings for qpcGR [15, 16, 17] (see figure 1(a)).We then assumed that all constrictions, a-d, behave in the same way in the structureunder investigation. One can see that the absence of clear plateaus in the experimental0
100 mT (0.58 (cid:109) m) 300 mT (0.19 (cid:109) m) 500 mT (0.12 (cid:109) m) (cid:109) m) B ( m T ) (a) (b) (c)(d) Conductance plateaus (cid:109) m) 420-2-4-6r ( (cid:109) m)dG/dr dG/dB (e)
B G
Figure 4. (a)-(c) Evolution of the left-hand side fringe pattern with a perpendicularmagnetic field. The derivative of the conductance, d G ( x, y ) / d x , is plotted as a functionof tip position, ( x, y ), at 100, 300 and 500 mT. The stadium walls are indicated by theblack solid lines. The scale bar corresponds to 500 nm. Numbers in brackets correspondto values of the classical cyclotron radius. Numbers -1, 0 and 1 along the red line in (a)correspond to the distance in microns. They are the same as on the horizontal axis in(d). (d) The conductance along the red line in (a) plotted as d G ( B, r ) / d r as a functionof magnetic field, B , and distance, r . (e) The conductance along the red line in (a)plotted as d G ( B, r ) / d B . The black solid curve shows the conductance, G along a shortblack horizontal axis, (Shubnikov-de Haas) oscillations as a function of magnetic field(black vertical axis) in the range between 230 and 600 mT. The horizontal dashed lineindicates positions of conductance minima in this curve and the filling factors in thebulk. A smooth background was subtracted from the black curve. Red arrows in (d)and (e) indicate transitions between conductance plateaus. Numbers in squares in (d)and (e) correspond to filling factors in the constriction and in the bulk, respectively.In (a)-(e) the voltage applied to the stadium is -1 V. curve is due to the smooth energy-dependent transmission and to the finite temperatureof 300 mK, but the latter is dominated by the first. The slopes of the experimentaland simulated curves are not exactly the same due to differences in the potentials of1the constrictions. In order to eliminate the difference in the plateau conductance, thenumber of transmitted modes in the stadium constrictions, N c and N d , would have tobe taken infinite (see Supplementary Material). It is also important to note that in themodel the lens shaped regions are in the centers of the stadium constrictions. In theexperiment they appear inside the stadium, which is probably due to screening effects.Although the model agrees qualitatively with the experiment, it should be used withcare given the assumptions made. The real potential is not a hard-wall potential aswe assumed, which means that all four resistances change with tip position. The useof only four QPCs is a rough approximation. However, the model does qualitativelyexplain our observations. Thus, the fringes at the stadium constrictions seen in d G/ d x are basically the result of conductance quantization plateaus in G ( x, y ) arising from anetwork of quantum point contacts.The fully coherent calculation using the recursive Green function methodreproduces the checker board-like patterns in figure 2 and 3. In figure 3(d), we show thenumerical results (orange line) for tip positions along a straight line similar to the red linein figure 3(d)) that is parallel to the electron propagation. The qualitative agreementwith the experimental blue curve is striking. However, there is a difference in the valuesof the conductance of the third plateau. The reason can be that the tip being inside ofthe stadium makes the dynamics in the stadium chaotic and thereby affects the totalconductance not only by opening or closing the constriction as it is in the simple modeldescribed previously (figure 3(a)). This makes the system extremely sensitive to detailsof the geometry and other parameters, in particular in the fully coherent calculationswhere the temperature was taken to be zero. While the observed conductance stepheights are well reproduced by the theory, the tip positions where subbands are openedwhen moving the tip away from the center of the constrictions show small quantitativedeviations due to the hard wall potentials used to model the tip depletion disk andthe gate-defined constrictions. Moreover, the capacitive coupling between the tip andthe top-gates of the sample, not taken into account in the simulation, might have aninfluence in the experiment.The stripes in figure 2(b) or 3(c) and diamonds formed by the crossing stripesshow how many modes are open in the constrictions formed between the tip and thestadium walls. For example, following the red line in figure 3(a) or (c) we start fromthe zero conductance region. Then the tip moves to the first diamond-like area. Thiscorresponds to the situation when the two constrictions ( a and b in figure 3(b)) have oneopen mode each, ( N a , N b ) = (1 , N a (cid:54) = N b . For example, outside the checker board pattern one of the two numbers ofmodes, N a or N b , is equal to zero, whereas the other one increases starting from 0 (thelens shaped region). The rectangular form of these regions in the experiment is dueto slightly different lever arms of characteristic induced potentials of the stadium top2gates.Figure 4(a)-(c) shows the evolution of the fringe patterns in d G/ d x (only one isshown as an example) in a perpendicular magnetic field. As the magnetic field increases,the number of observable conductance plateaus decreases and their width increases. Tosee precisely the effect of the magnetic field on the fringe patterns with higher resolutionin B -field, we plot in figure 4(d) the derivative of the conductance, d G ( B, r ) / d r , as afunction of magnetic field and distance along the red line shown in figure 4(a). At zeromagnetic field a set of fringes, which correspond to the transition between conductanceplateaus, appear close to the constrictions like in figure 1(c). As the magnetic fieldincreases from 0 to 600 mT the plateaus’ width increases. Above 100 mT the numberof observable plateaus decreases. The effect of the magnetic field on the number ofoccupied subbands becomes important when the classical cyclotron orbits do not fitinto the width of the constriction, W i . When the tip is outside the constriction (alongthe red line in figure 4(a)) and the cyclotron radius, r c = (cid:126) k F / ( eB ), is bigger than W c or W d , then the geometric confinement dominates over the magnetic confinement andthe number of conductance plateaus is independent of B . The condition r c = W inour experiment is fulfilled at about 100 mT where r c ≈
600 nm, and W ≈
700 nmwhen -1 V is applied to the stadium gates. When the tip approaches the constriction, W c or W d further decreases becoming equal or even smaller than r c . This means thatbelow 100 mT the number of the plateaus should not change, which is in agreementwith our experiment, figure 4(d). At higher magnetic fields, when r c < W/ r due to themagnetic depopulation of subbands as also seen in the figure. Around ± µ m atlow magnetic fields the fringes disappear completely, because the tip is outside theconstriction. The estimated corresponding distance from the center of the constrictionis | W c /2+ R tip | = 0 .
35 + 0 . . µ m, which is close to 0.7 µ m. The difference mayoriginate from the radius of the tip-depleted region, for which the value of 0.5 µ m is anestimation.Figure 4(e) shows the derivative d G ( B, r ) / d B as a function of distance andmagnetic field along the same red line in (a). In comparison with (d), horizontal linesare now clearly seen. They correspond to regions of the magnetic field between extremain the bulk Shubnikov-de Haas oscillations (an example is illustrated by the black solidline when the tip is outside the constriction). Bulk filling factors (numbers in squares)are assigned to the conductance minima. The conductance along any horizontal cut infigure 4(e) at a constant finite magnetic field exhibits clear plateaus (see figure 4(a)-(c))each of which corresponds to a certain filling factor in the constriction. The boundariesof these plateaus are marked by the red arrows for a particular magnetic field as anexample. For instance, for the uppermost horizontal dashed line, when the tip is outsidethe constriction the filling factor is that of the bulk, i.e. 10. As the tip moves closerto the center of the constriction the edge channels are brought closer to each other andare closed one by one, which is seen as a stepwise reduction of the filling factor from 10to 4 (see figure 4(d)). Stripes between the fringes widening with magnetic field (bent3 N = 1 7N = 5 N = 4N = 3
B (T) r ( m m ) N = 2
Figure 5.
Fitting of the experimental data (points) extracted from figure 4(d) to themodel (solid lines) described in the main text. green or dark blue) in figure 4(d) correspond to one particular filling factor (subbandoccupation). The stripes are vertical at low magnetic fields and bent towards larger r at high B -fields (see the model below).Figure 5 shows experimental points for the maxima of the fringe pattern extractedfrom figure 4(d). Each set of points of the same color mark the transitions betweeninteger number of subbands starting from N = 2 (the closest to r = 0 µ m) to N = 17.We propose a model to explain the depopulation of the subbands as a function ofmagnetic field and tip position, which is an extension of the model of Beenakker [14].The constriction is formed by the superposition of the gate-induced and the tip-inducedpotential. The gate-induced potential is fixed in space and that formed by the tip canbe moved. The size of the constriction is changed by moving the tip above the surface.In our model we approximate the potential of the gate and the tip along the narrowestsection of the constriction in confinement direction x by a 1 /x -power law as V ( x ) = (cid:20) x + V ( x − r ) (cid:21) U, (2)where r is the tip position, V ∝ V tip /V g , V tip is the tip bias, V g is the gate voltage, U ∝ V g in units of eVm . By expanding this potential around its minimum, x min , oneobtains in lowest order the resulting parabolic transverse potential V ( x ) = V min + 12 mω ( x − x min ) = Uαx + 12 6 U (1 − α ) x ( x − x min ) , (3)where V min ( r ) is the minimum of V ( x ) at x min , α = x min /r describes a relative motion4of the potential minimum with respect to the tip motion and (cid:126) ω ( r ) is the subbandspacing at B = 0. We can see that this model incorporates the fact that the potentialminimum in the constriction between the tip and the gate is lifted as the tip approachesthe gate. At the same time, the position of the minimum shifts proportionally to thetip position with a lever arm α . Furthermore, the parabolic confinement ω becomessharper as the tip approaches the gate. These three effects are incorporated by the twoparameters α and U .The model takes into account the tip-position-dependent potential minimum inspace, x min , and in energy, V min ( r ). The minima of the magnetoelectric subbands haveenergies E N = ( N + 1 / (cid:126) ω ( r, B ) + V min ( r ), where ω ( r, B ) = ω ( r ) + ω c ( B ) , ω c isthe cyclotron frequency and N > N th magnetoelectric subband is depopulated at tip position r N and magnetic field B N if E N ( r N , B N ) = E F . From this expression the dependence of the magnetic depopulationfield B N on tip position r N is B N = nh e (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − (cid:18) Kr N (cid:19) + − (cid:16) ˜ rr N (cid:17) + ( N +0 . K (˜ r r N ) N + 0 . , (4)where n is the carrier density in the absence of the tip, ˜ r is the tip position at B = 0 atwhich N is an integer for different N , K is a fitting parameter.The results of the fitting procedure using equation (4) with the same value of K for all curves are shown in figure 5 by solid lines. When seven and more subbands areoccupied the agreement between model and experiment is very good. For lower subbandoccupation the modelled curve starts deviating from the experimental points. In ourmodel this deviation becomes stronger as N decreases. A more elaborate model wouldinclude carrier-carrier interactions and electron interference.
3. Conclusion
We have presented experimental results of local studies of the conductance througha ballistic cavity connected to the leads by two constrictions using scanning gatemicroscopy in the presence of small perpendicular magnetic fields. We observed strongtip-position dependent fluctuations of the conductance when the tip was scannedinside the cavity, regions where the conductance was increased above the unperturbedconductance of the structure, and checker board shaped fringe patterns when the tipwas close to one of the constrictions. These patterns allowed to precisely control thenumber of transmitted modes in the channels formed between the tip potential andthe confinement caused by the top gates, which may in the future be used to imageAharonov-Bohm oscillations in the quantum Hall regime. Investigating the fringepatterns in a perpendicular magnetic field we observed a transition from the electrostaticto magnetic depopulation of subbands. This effect was seen in spatially resolved images5where narrow fringes were present at low magnetic fields, which gradually transformedinto wide conductance plateaus at higher B -fields. The plateaus corresponded to integerfilling factors. Classical and quantum simulations described very well most of ourobservations.The observed features in the tip-induced conductance changes and the comparisonwith the model calculations clearly indicate that our device was operated in the coherentquantum transport regime. Moreover, our modelling allowed to understand in detailthe origin of the observed conductance changes. Most of the local features seen inour experimental conductance images cannot directly be interpreted in terms of acorresponding local current flow as it was suggested [3, 2] in the context of single QPCsoperated on a quantized conductance plateau. Though we cannot measure the localcurrent density in the absence of the tip, it seems obvious that the checker boardfringe patterns observed close to the QPCs and the tip-induced enhancement of theconductance above the unperturbed value do not reflect the behaviour of local currentflow in the unperturbed device. Since our data do not show a well-defined conductanceplateau, the absence of an interpretation of the conductance change in terms of localcurrent flow is consistent with a recent theoretical study [21], perturbative in the tippotential, where the conductance change could be unambiguously related to the localcurrent density only in the case of a structure having spatial symmetry and beingoperated on a quantized conductance plateau.Quantitative deviations between the experimental results and model calculationscould be due to interaction effects which were not taken into account in the models.Further experiments and refined modelling are needed to clarify this.
4. Acknowledgements
We are grateful for fruitful discussions with Markus B¨uttiker, Rodolfo Jalabert, CosimoGorini and Bernd Rosenow. We acknowledge financial support from the Swiss NationalScience Foundation and NCCR “Quantum Science and Technology” and the FrenchNational Research Agency ANR within project ANR-08-BLAN-0030-02.
Appendix
In order to determine the width of the stadium constriction at different top gate voltages,the conductance was measured as a function of voltage applied to the stadium gatesalong the red line shown in the insert to figure A1. The green triangle in the main graphat x ≈ . µ m is the zero conductance region (in real space it is a lens shaped region).Tilted lines on both sides of this region correspond to sets of fringes also observed infigure 1(c). As the gate voltage changes, the fringes move together with the depletedregion around the top gates. They move linearly with distance due to the constant slopeof the lines. Therefore, the slope of the lines allows determining how the depleted regionaround the top gates changes with gate voltage. One can see from the main figure that6the size of the constriction changes with the gate voltage as ∆ r/µ m = 0 . V g / V.The voltage at which the stadium constrictions are formed is -0.4 V. In this casetheir widths are roughly equal to the lithographic width, which is 1 µ m each. In thepresented measurements voltages of -0.8 and -1.0 V were applied to the top gates, whichcorrespond to the size of the constrictions of 1 − · . · (0 . − .
4) = 0 . µ m and1 − · . · (1 . − .
4) = 0 . µ m, respectively.The lever arms of the two top gates are slightly different from each other, seen asdifferent slopes of the lines to the left and to the right of the zero conductance region.For the lower gate ∆ x L /µ m = 0 . V g / V and for the upper gate ∆ x U /µ m = 0 . V g / V.However, the sizes of the stadium constrictions determined with the different and thesame lever arms differ only by 2% for the top gate voltage of -0.8 V and by 4% for-1.0 V. Such errors will not change the results presented in this work. We thereforeconsider the lever arms of the two top gates to be the same in our paper. Changing theorientation of the red line to vertical results in the same lever arms.Figure A2 shows a comparison between the measured conductance trace along aline perpendicular to the fringe pattern and the constriction model similar to the one infigure 3(d). This time only the constrictions formed between the tip and the stadium aretaken into account. The contact resistance subtracted to obtain the experimental curvewas increased by 20% compared to the model discussed in the main text in order toalign the third conductance plateau in the two curves in the figure. This added contactresistance accounts for the small contribution to the total resistance coming from thetwo stadium constrictions. The difference between the curves around the first plateauarises (as mentioned in the main text) from the saddle point potential used in the model.7 r ( (cid:109) m) V ( V ) g dG/dx Figure A1.
Derivative of the conductance with respect to distance as a function ofdistance and top gate voltage. The insert shows the right-hand side fringe pattern.Conductance in the main graph was measured along the red line.
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Conductance plotted as a function of distance along the red line infigure 3(a) and (c). The blue line is the experimental curve. The green curvecorresponds to the model. A smooth energy-dependent transmission coefficient is takeninto account. In the model the resistance of the stadium constrictions R a and R b areneglected.L P 2007 Nano Lett Nano Lett Nature Physics Solid State Physics , , 1[15] Fertig H A and Halperin B I 1987 Phys. Rev. B Phys. Rev. B New J. Phys. Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett.105