Realistic modelling of quantum point contacts subject to high magnetic fields and with current bias at out of linear response regime
S. Arslan, E. Cicek, D. Eksi, S. Aktas, A. Weichselbaum, A. Siddiki
RRealistic modelling of quantum point contacts subject to high magnetic fields andwith current bias at out of linear response regime
S. Arslan , E. Cicek , D. Eksi , S. Aktas , A. Weichselbaum , and A. Siddiki Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience,Ludwig-Maximilans-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany and Trakya University, Faculty of Arts and Sciences, Department of Physics, 22030 Edirne, Turkey
The electron and current density distributions in the close proximity of quantum point contacts(QPCs) are investigated. A three dimensional Poisson equation is solved self-consistently to obtainthe electron density and potential profile in the absence of an external magnetic field for gate andetching defined devices. We observe the surface charges and their apparent effect on the confinementpotential, when considering the (deeply) etched QPCs. In the presence of an external magneticfield, we investigate the formation of the incompressible strips and their influence on the currentdistribution both in the linear response and out of linear response regime. A spatial asymmetry ofthe current carrying incompressible strips, induced by the large source drain voltages, is reportedfor such devices in the non-linear regime.
PACS numbers: 73.20.-r, 73.50.Jt, 71.70.Di
I. INTRODUCTION
The new era of quantum information processing at-tracts an increasing interest to investigate the intrinsicproperties of small-scale electronic devices. One of themost interesting of such devices is the so called quan-tum point contacts (QPCs), where a quantized currentis transmitted through it under certain conditions .They are constructed on two dimensional electron sys-tems (2DES) either by inducing electrostatic potentialon the plane of 2DES and/or by chemically etching thestructure. The essential physics is that, the small sizeof the constraint creates quantized energy levels in onedimension (perpendicular to the current direction) there-fore, transport takes place depending on whether the en-ergy of the electron coincides with this quantized energyor not. In the ideal case at low bias voltages, if the energyof the electron is smaller then the lowest eigen-energy ofthe constraint, no current can pass through the QPC.Otherwise, only a certain integer number of levels (chan-nels) are involved, therefore conductance is quantized .Beyond being a useful play ground for the basic quantummechanical applications many other interesting featuresare reported in the literature such as the 0.7 conductanceanomaly , which became a paradigm since then. An-other adjustable parameter which induces quantizationon the 2DES is the magnetic field B applied perpen-dicularly to the system. Such an external field changesnot only the density of states (DOS) profile of the 2DESbut also the screening properties of the system drasti-cally. The interesting physics dictated by this quanti-zation is observed as the quantum Hall effects . Re-cent theoretical investigations point out the importanceof the electron-electron interactions in explaining the in-teger quantum Hall effect , believed to be irrelevantin the early days of this field , which we discuss brieflyin this work. The basic idea behind the inclusion of theinteraction is as follows: due to the perpendicular mag- netic field the energy spectrum is discrete, known as theLandau levels (LLs), and is given by E n = (cid:126) ω c ( n + 1 / n is a positive integer and cyclotron energy is de-fined as (cid:126) ω c = (cid:126) eB/m ∗ , with effective electron mass m ∗ = 0 . ∗ m e . Taking into account the finite sizeof the sample, i.e. the confinement potential, and themutual interaction (Hartree) potential, one obtains thetotal potential. In the next step for a fixed average elec-tron density one calculates the resulting electron densitydistribution and from this distribution re-calculates thepotential distribution iteratively. This self-consistent cal-culation ends in formation of the compressible and incom-pressible regions. In a situation where Fermi energy E F is pinned to one of the LLs, then the system is compress-ible. Otherwise, if E F falls in between two consecutiveLLs, the system is known to be incompressible and, sincethere are no available states at the E F for electrons to beredistributed, screening is poor. However, within theseincompressible regions the resistivity vanishes, hence allthe applied current is confined to these regions. We willbe discussing the details of this model in Sec.IV.Most recently, the experiments performed at the 2DESincluding the QPCs, in the presence of an external mag-netic field, manifested peculiar results . In thefirst set of experiments electron interference, such asMach-Zehnder (MZ) and Aharonov-Bohm(AB) , was investigated. The MZ interference experi-ments exhibit a novel and yet unexplained behavior, re-garding the interference contrast (visibility) at the inter-ferometer in the nonlinear transport regime (finite trans-port voltage). As a function of voltage, the visibilitydisplayed oscillations whose period was found to be in-dependent of the path lengths of the interferometer, instriking contrast to any straightforward theoretical model(e.g. using Landauer-B¨uttiker edge states (LBES) ). Inparticular, a new energy scale, of order of µ eV, emerged,determining the periodicity of the pattern. This unex-pected behavior of interfering electrons is believed to berelated to electron-electron interactions . The most sat- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r isfying model existing in the literature is proposed by I.P. Levkivskyi and E. V. Sukhorukov . However, otherschemes also including interactions are present andmodels, which consider a non-Gaussian noise as a sourceof the visibility oscillations , without interaction. Thenovel magnetic focusing experiments concerning QPCshas revealed the scattering processes taking place nearthese devices . It was explicitly shown that, the exper-imental realization of the sample and the device itselfstrongly effects the transport properties. It is reportedthat, the potential profile generated by the donors (im-purities) and the gates deviates strongly from the ideal”point” contact. Even in each cool down process, sincethe impurity distribution changes, the quantum interfer-ence fringes differ considerably. Hence a realistic mod-elling of a QPC is desirable, which we partially attack inthis work. Another interesting set of experiments withinthe integer quantum Hall regime is conducted by S. Rod-daro et.al , where the transmission is investigated as afunction of the gate bias. The findings show that currenttransmission strongly deviates from the expected chiralLuttinger liquid behavior, since the transmission is ei-ther enhanced or suppressed by changing the gate bias.This effect was attributed to the particle hole symme-try of the Luttinger liquid and is discussed in detail inRef.[26]. However, the explicit treatment of the QPCwas left unresolved. Since the essential physics can bestill governed by considering a finite size QPC opening,therefore assuming formation of a (integer) filling regionsufficient.The theoretical investigation of QPCs covers a wide va-riety of approaches, which can be grouped into two: i) themodels that include electron-electron interactions and ii)the models that do not. At the very simple model in de-scribing the QPCs, one considers a potential barrier per-pendicular to the current direction quantizing the energylevels. Therefore the electrons are considered to be planewaves before they reach to the QPC and transmissionand reflection coefficients are calculated from this poten-tial profile. A better (2D) approximation is to model theQPCs with well defined functions , such as ellipseswhich lead to analytic solutions for the energy eigenfunc-tions and energies. About a decade ago J. Davies andhis co-workers developed the ”frozen charge” model to calculate the potential profile induced by the gatesdefining the QPC. This approach is still one of the mostused technique to obtain the potential profile, however,it is not self-consistent and completely ignores the donorsand surface charges. There exists many theories whichtakes the potential profile from the frozen charge modelas an initial condition, and provides explicit calculationschemes to obtain charge, current and potential distri-butions . One of the most complete scheme,even in the presence of an external B field, is the localspin density approximation (LSDA) within the densityfunctional theory (DFT) . The LSDA+DFT ap-proaches are powerful to describe the essential physics ofdensity distribution and even 0.7 anomaly phenomeno- logically, however, the description of the current dis-tribution is still under debate. The scattering prob-lem through the QPCs is usually handled by the ”wavepacket” formalism and is very successful in explainingthe magnetic focusing experiments. However, the poten-tial profile is not calculated self-consistently and there-fore, the effect of the incompressible strips resulting fromelectron-electron interaction is not taken in to account.Back to early days of the theories that account for elec-tron interactions, i.e. Chklovskii, Shklovskii and Glaz-man (CSG) and Chklovskii, Matveev and Shklovskii (CMS) models, the influence of the formation of the in-compressible strips has been highlighted. In the CMSpaper, it was even conjectured that, ’the ballistic con-ductance of the QPC in strong magnetic field is given bythe filling factor at the saddle point of the electron den-sity distribution multiplied by e / π (cid:126) ’, which is quantizedonly if an incompressible strip (region) resides at the sad-dle point. In one of the recent approaches, in the pres-ence of a strong B field, the electron-electron interactionis treated explicitly within the Thomas-Fermi approx-imation (TFA) self-consistently, meanwhile the currentdistribution is left unresolved . In this model, similarto other approaches, the bare confinement potential isobtained from the ”frozen charge” approximation, whichin turn lead to discrepancies due to its non-self-consistentapproach. Here, we improve on this previous work in twomain aspects: i) the electrostatic potential is obtainedself-consistently in 3D, which allows us to treat also theetched structures ii) the current distribution is calculatedexplicitly using the local version of the Ohm’s law, alsoin the out of linear response regime. We organize ourwork as follows: In Sec.II, we briefly describe the numer-ical scheme to calculate the potential profile at B = 0following Ref.[39], which is based on iterative solution ofthe Poisson equation in 3D. In particular, we study theeffect of different gate geometries and focus on the com-parison of the potential profiles of gate and etch definedQPCs. The numerical scheme to calculate potential anddensity profiles at finite temperature and magnetic fieldis introduced in Sec.III. Here, we review the essential in-gredients of the TFA and discuss the limitations of ourapproach. Sec.IV is dedicated in investigating the currentdistribution within the local Ohm’s law , where weconsider both the linear response (LR) and out of linearresponse (OLR) regimes. In the OLR, we show that thelarge current bias induces an asymmetric distribution ofthe incompressible strips, due to the tilting of the Lan-dau levels resulting from the position dependent chemicalpotential. We conclude our work by a summary. II. ELECTROSTATICS IN 3D
The realistic modelling of 2DES relies on solving the3D Poisson equation for given boundary conditions, setby the hetero-structure (GaAs/AlGaAs in our calcula-tions) and the gate pattern, which describes the chargeand potential distribution. The hetero-structure, shownin Fig.1a, consists of (metallic) surface gates (dark semi-elliptic regions on surface), a thin donor layer (denotedby light thin layer and δ Silicon doping) which provideselectrons to the 2DES and the 2DES itself indicated byminus signs confined to a thin area. The 2DES is formedat the interface of the hetero-structure. The average elec-tron density n el (and its spatial distribution n el ( x, y )) isdictated by the donor density n and the metallic gates.Once the number of donors and the gate voltage V G arefixed, the potential and charge distribution of the sys-tem can be obtained by solving the Poisson equation,self-consistently.For typical nanoscale devices with many (or at leasta few) electrons in each of the electrostatically-definedregions, the charge distribution and the major energyscales are described to a good approximation by classicalelectrostatics. Due to the strong electric fields generatedby segregating charge in a 2DES, the Coulomb energyis the dominant energy scale. In this sense, it is desir-able to have a self-consistent electrostatic description ofthe system if one expects a good quantitative descriptionthereof.For solving the electrostatics of the system in three di-mensions we used a code developed and successfully ap-plied in previous studies . It is based on a 4 th orderalgorithm operating on a square grid. The code allowsflexible implementation of many boundary conditions rel-evant for nanoscale electrostatics: standard boundariessuch as conducting regions at constant voltage (poten-tial gates), of constant charge (large quantum dots) orcharge density (doping), but also boundaries such as adepletable 2DEG, dielectric boundaries and surfaces ofsemiconductors with the Fermi energy pinned due to sur-face charges. Since the calculation is constrained to afinite volume of space including the surface of the sam-ple, the outer boundary is considered open and is alsoobtained self-consistently along with the rest of the cal-culation.Overall the code provides a reliable description of thepotential landscape and thus the electric field as well asthe charge distribution for the sample under considera-tion.As an illustrating example in Fig.1 we show the hetero-structure under investigation together with the chargedistributions at different layers. Area of the unit cell is1 . × . µ m , whereas the hight is chosen to be 156nm. The donor and the electron layer lies 43 nm and 56nm below the surface, respectively. The metallic gatesare deposited on the surface of the structure and arebiased with − . FIG. 1: (Color online) (a) The Silicon doped hetero-structure, the 2DES is formed at the interface of theGaAs/AlGaAs (denoted by minus signs) and the metallicgates are deposited on the surface. At zero gate bias, the elec-tron density is determined by the number of donors, which wechose to be 4 × cm − . Charge distribution at differentlayers, at the gates (b), the dopant layer (c) and the 2DES (d).It is clearly seen that not all the excess electrons are capturedby the 2DES, rather a significant amount is accumulated onthe surface. The electrostatic quantities are normalized withthe relevant scales, i.e. charge (density) is normalized withthe average electron density (e.g. Q ( x, y ) = n el ( x, y ) / ¯ n el )and electrostatic potential (energy) with the potential energyof a single electron. potential profile is assumed. The influence of these in-duced charges become more important when consideringan external B field. Since, the steepness of the externalpotential profile determines the effective widths of thecurrent carrying incompressible strips. We should notethat, our self-consistent model enables us also to handlethe (side) surface charges which becomes important whenconsidering chemical etching. In the following part weinvestigate systematically, the effects of the gate voltageand the device geometry on the electrostatic quantities. FIG. 2: (Color online) Spatial distribution of the electronsas a function of the gate voltage. At zero bias (a) more elec-trons are populated under the gate which changes till deple-tion starts (b-d), where the gate voltage is set to be (b) -0.3V(c) -0.7V (d) -1.0 V. Almost no electrons are left beyond -1.5Vapplied to the gates, (e) -1.5V (f) -2.0 V.
A. Gate defined QPCs
In this subsection we compare the electron density pro-files calculated for different QPC geometries applyingvarious bias voltages, which exhibits strong non-linearbehavior in contrast to many models used in the litera-ture. We start our discussion with a rather smooth con-figuration, where the distance between the gates ( W ) ischosen to be 200 nm (see Fig.1a). In Fig. 2 we showthe cross section of the electron density profile for dif-ferent gate voltages. Interestingly, at V G = 0 we seethat more electrons are residing beneath the gates. Thiseffect is due to inhomogeneous (induced) charge distri-bution at the metallic gates similar to the distributionshown in Fig.1d. The induced charges are mostly ac-cumulated near the gate boundaries, whereas the centerof the gates has almost a constant charge profile. In-creasing V G to -0.3 Volts, already starts to depopulateelectrons under the gates and the depopulation rate re- mains linear to the applied gate potential until the 2DESbecomes depleted. In the [ − . , − .
2] Volt interval, thedensity distribution changes relatively smooth, since theelectrons can still screen the external potential quite well.It is important to recall that, in the absence of an ex-ternal B field, the DOS of a 2DES is just a constant D (= m ∗ /π (cid:126) = 2 . × meV − cm − for GaAs),which is set by the sample properties, therefore screen-ing is nearly perfect. Whereas this changes considerablywhen the electrons are depleted under the gate. This isobserved by the strong drop of the potential when thedepletion bias is reached around V G = − . y = 350 nm) wherethe V G becomes larger than -1.5 V. Therefore, the simplepicture describing the QPCs as a smooth function of theapplied gate voltage fails. In that picture it is assumedthat the Fermi energy of the system remains constant andthe potential profile, given by a well defined function, ofthe constriction is simply shifted by the amount of po-tential applied to the gates. Such a model is reasonablein the regime where the gate voltage is small enough thatno electrons are essentially depleted. However, as men-tioned above, when the barrier hight is larger than theFermi energy there exists no electrons to screen the ex-ternal potential and the potential distribution must becalculated self-consistently.Another adjustable parameter which can be accessedexperimentally is the geometry of the structure. Ofcourse, in the simplistic models describing QPCs thisdoes not play an important role, since the constriction isassumed to be isotropic in the current direction, in con-trast to the experimental findings. It is well known that,the shape of the QPCs, as well as the cooling and biassingprocedure, is important when measuring interference ormagnetic focusing. In Fig.3, we compare two differentgate patterns considering typical gate separations W fora fixed gate voltage, V G = − . W is changed from 200 nm (black dashed line) to 300 nm(red dashed line). This relies on the fact that, at the firstorder approximation, the screening is better when moreelectrons are accumulated at the opening of the QPC.However, since the screened potential V scr ( x, y ) can beobtained from the external potential V ext ( x, y ) via theThomas-Fermi screening, V q scr = V q ext /(cid:15) ( q ) , (1)where (cid:15) ( q ) = 1 + 2 / ( a ∗ B | q | ), is the Thomas-Fermi dielec-tric function with q being the momentum and a ∗ B (= 9 . q ), compared to a ∗ B , are less screenedwhereas the short range fluctuations are predominantlyscreened. Considering the sharp configuration (C2) thisobservation becomes more evident, since the potentialprofile across the QPC varies smoother than of the con-figuration (C1), when varying W . From an experimental-ists point of view, therefore, drawing shaper QPC struc-
200 400 600 800 1000 12000.600.620.640.660.680.70 C2 V D ES ( x = n m , y ) Y (nm) E F C1, W=200 C1, W=300 C2, W=200 C2, W=300 C2, W=400 C2, W=500C1 V (x,y)
X ( m) Y ( m ) Y ( m ) X ( m) -0.1 0.06 0.2 0.4 0.5 0.7
FIG. 3: (Color online) Potential profile across the QPCs fortwo different geometrical patterns. Insets depict the smooth(C1) and the sharp edged (C2) patterns. Dark (blue) regionsare electron depleted, i.e. the local potential is larger than the E F , denoted by the solid thick horizontal line. White contoursdenote the depletion boundary, where the width of the QPCis set to be 200 nm and the curvature is changed. The dashedlines stand for the first configuration and two different W values (= 200 nm (black) and 300 nm (red)), whereas for thesharper configuration four W values are selected. tures by electron beam lithography may lead to a better(linear) control of the potential profile which is closer tothe ideal potential profile. This is certainly in contrastto what we would expect from an non-interacting model,however, it is known by the experimentalists that defin-ing the QPCs with sharper edges increase the quality ofthe visibility signal . For the C2 configuration we alsoobserve that, the potential profile becomes almost insen-sitive to the width W above 300 nm, which coincides withour previous finding of better screening of the long rangefluctuations. It is worth to note that, our calculationscheme is beyond the simple Thomas-Fermi screeningscheme in obtaining the bare confinement potential . Itfully takes into account the interaction effects, however,does not include any quantum mechanical effects. In abetter approximation, of course, one should also solve theSchr¨odinger equation self-consistently in 3D. This proce-dure is known to be costly in terms of computationalcost even only if the 2DES is treated quantum mechani-cally. Since we are interested in either zero or very strongmagnetic fields, representing electron as a point charge isstill a reasonable and valid approximation. We will dis-cuss the justification of this assumption in the presenceof an external magnetic field in more detail in Sec. III,where we also discuss the limitations of our model.The different configurations of the gate patterns arealso important when investigating the scattering pro-cesses by magnetic focusing experiments . It is apparentthat the scattering patterns of the electron waves will notonly depend on the impurity distribution but also on the -0.10.00.1
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400 600 800 1000 V D ES ( x = n m , y ) V g =-0.5 V V g =-0.6 V V g =-0.7 V V g =-0.8 V V g =-0.9 V V g =-1.5 V V g =-2.0 V V g =-3.0 V Y (nm) b) V D ES ( x = n m , y ) Y (nm) D e = 38.2 nm D e = 47.7 nm D e = 57.3 nm D e = 66.9 nm D e = 76.4 nm a) Y (nm) D e = 4.7 nm D e = 9.4 nm D e =19.1 nm FIG. 4: Potential cross-section of (a) gated and respectively(b) etched QPC’s at selected gate biases and etching depthsfor C1 at a fixed W = 150 nm. Insets focus on the high biasor shallow etching profiles structure of the QPC’s. We expect that for the sharperdefined QPC (C2) the scattering should depend weaklyon the QPC opening, since the width does not changealong the constraint. Whereas, for C1 configuration smallchanges at W should affect the scattered waves drasti-cally. Another comment on the experimental setups is tothe Roddaro experiments, since the formation of (sta-ble) integer filling region is important to explain thefindings, we believe that C2 type configurations wouldbe leading to a better resolution of the transmission am-plitudes. The choice of the structure and the width ap-parently depends on the experimental interest, which isbelieved to be irrelevant when modelling QPCs as idealpoint contacts, and the QPC’s are not only defined bygates but alternatively also by chemical etching. Thegated structures are of course more controllable, how-ever, at high gate voltages required for depletion, electri-cal sparks can occur, therefore the structure can even bedestroyed. In such situations etching defined QPCs arepreferred, although without further gates one loses thefull control of the potential profile. In the next section,we will compare the potential profiles of etched and gatedefined QPC’s, to show that in some cases etch definedQPC’s may be more useful to obtain steeper potentialprofiles. B. Etching versus gating
For simple calculation purposes, QPCs are modelledeither as a finite potential well or with a parabolic con-finement potential, perpendicular to the current direc-tion. Starting from the early experiments , usually theconductance is measured as a function of the applied gatevoltage which presents clear quantized values. This quan-tization can be well explained by the Landauer formula G = e π (cid:126) N c (cid:88) n,m =1 | t n,m | , (2)where ballistic transport is assumed to take place, i.e.the transmission is given by | t n,m | = δ n,m , and no chan-nel mixing is allowed. The (integer) number of chan-nels N c is defined by the Fermi energy and the widthof the constriction, in general. The gate defined QPCs,at a first order approximation, can be represented byparabolic or finite well potential profiles. However, it isknown that the chemical etching process creates (side)surface charges, which in turn generates a steeper poten-tial profile at the edges of the sample. In this situationit is apparent that, the confinement potential can notbe assumed to be parabolic, rather a steeper potentialshould be considered. In this section we compare thesetwo different constriction profiles, namely the gated andthe etched ones.The self-consistent potential across the QPC at thecenter is plotted versus the lateral coordinate in Fig. 4.The 2DES under the gates is depleted at the gate volt-ages larger than − . . D e > . . It is known that, such an improvementwill also cover some of the quantum mechanical aspects(such as the wave functions), which brings extra oscilla-tions to the potential profile . However, for our presentinterest we neglect this correction knowing that the self-consistency of the calculation scheme already takes intoaccount the occupation and the 1D electron density atthe QPC satisfy the validity condition n el a ∗ B >> .We summarize our findings in Fig. 5, where we showthe electron density (left) and potential profiles (right)for typical gate biases V g and etching depths D e versusthe spatial coordinate. We choose a representative cross- section of the obtained profiles along the current direc-tion x , where the y coordinate is fixed at 450 nm. Figure5a depicts the density profile for selected depths of etch-ing varying from shallow ( D e = 4 . − . D e = 38 . − . D e ’s, the2DES is not depleted beneath the pattern and the densityprofile is rather smooth. Depletion is observed when thedepth is larger than 19 . ∼
60 nm) we do notsee the surface charges (the spike like point, indicated bythe arrow) at the level of the 2DES. The inset of Fig. 5bshows the electron density distribution in a color codedcontour plot together with the corresponding potentialprofile across the white (dashed) line. The thin (green)lines contouring the depleted (red) region indicates thespatial distribution of the surface charges. The potentialis steeper compared to that of the gated one (Fig. 5d)and the profile does not show any considerable variationonce the etching depth reaches the plane of the 2DES.This behavior clearly exhibits the uncontrollability of theetched samples, since the corresponding potential profileobtained for the gated samples vary slowly on the lengthscales of the Fermi wavelength, even if the 2DES is com-pletely depleted beneath the gates. Moreover, the am-plitude of the potential strongly depends on the appliedgate voltage. The slow variation of the potential is notthe case for the etched sample, for example consider thecase when D e is changed from 57.3 to 78.4 nm, and com-pare it with that of the gated sample when voltage ischanged from − . − . V g . For the etched samples, poten-tial profile becomes very steep when the etching depthexceeds the depth of the 2DES, since the (side) surfacecharges pin the Fermi level at the mesa surface to themid gap of GaAs forming a Schottky like barrier . Weshould also note that, at zero bias, more electrons arepopulated under the gates, which is not the case for theetched samples. As a rule of a thumb, when a steeperpotential profile is required one should consider chemi-cal etching where the etching depth is deeper than theelectron layer and one should keep in mind that biasinggates with high potential does not necessarily imply thatthe electron density profile is also changed considerably.The outcome of the self-consistent solution of the 3DPoisson equation considering QPCs at zero magnetic fieldis two fold: i) the geometrical properties (i.e. consider-ing C1 or C2 type patterns) strongly change the potential
200 400 600 8000.00.20.40.60.8200 400 600 8000.00.20.40.6 200 400 600 8000.00.30.6200 400 600 8000.00.30.6 D e =38.2 nm D e =57.3 nm D e =76.4 nm d)c) Q D ES ( x , y = n m ) X (nm) D e = 4.7 nm D e = 9.4 nm D e =19.1 nm Q D ES ( x , y = n m ) X (nm) V g =-0.5 V V g =-0.6 V V g =-0.8 V V g =-1.5 V V g =-2.0 V V g =-3.0 V a) b) V D ES ( x , y = n m ) X (nm) V D ES ( x , y = n m ) X (nm) Y ( m ) -0.70 -0.35 0 Q (x,y) Y ( m ) FIG. 5: Spatial distribution of the electron density (a,c) andscreened potential (b,d) for etched (upper panel) and gatedsamples (lower panel), y = 450 nm. landscape in the close vicinity of the QPCs. We foundthat the smoother constrictions with a larger width W can be modelled better with the ideal point contacts, i.e.parabolic confinement. The sharper constrictions canbe considered as finite well profiles, up to a first orderapproximation and potential profile remains unchangedwhen considering W >
300 nm. ii) Due to surface pin-ning the etched samples, generate steeper potential pro-files and the density profile remains unaffected once etch-ing is deeper than the depth of the 2DES. These numer-ical results, show a strong deviation from the the widelyused ”ideal point contact” and ”frozen charge” models,proposing that depending on the experimental interest itis important to reconsider the geometrical (C1, C2) andstructural (gated/etched) factors defining the QPC un-der investigation. As final remark, the artifacts, such aslocal minima and maxima at the potential and densityprofiles, resulting from the previous non-self consistentThomas-Fermi and ”frozen charge” models areresolved by considering the 3D calculation scheme. III. FINITE B FIELD
The aim of this and the next section is to calculate,and compare, the density and subsequently the currentdistributions, in the close vicinity of the QPCs, withininteracting and non-interacting models in the presenceof an external perpendicular magnetic field. Here wetake into account electron-electron interaction withinthe Thomas-Fermi theory of screening also consideringstrong magnetic fields using the potential profile calcu-lated in the previous section, as an initial configurationof the landscape. We compare the spatial distributionof the Landauer-B¨uttiker edge states (LB-ES) with thedistribution of the incompressible edge states, where the applied external current is confined . Next wediscuss the limitations of the TFA, and suggest improve-ments on the calculation scheme based on i) quantummechanical considerations, such as the finite extensionof the wave functions, and ii) replacement of the globaldensity of states with the local one.
A. Thomas-Fermi-Approximation (TFA)
The enormous variety of the theories describing thedensity and current distributions at the quantum Hallsystems already show the challenge in giv-ing a proper prescription to these quantities. These the-ories can be grouped into two: the current is carried ei-ther (i) by the compressible regions or (ii) by theincompressible regions . Moreover (and confus-ingly) these regions can reside at the bulk or at theedge of the sample , depending on the modelconsidered and the magnetic field strength . For the sakeof completeness, we start with a generic Hamiltonian de-scribing an electron subject to high magnetic fields. H σ = H + V σ int + V ext + V σ Z , (3)where σ (= ± /
2) is the spin degree of freedom, H thekinetic part, V ext and V int the external and the inter-action potentials, respectively, and V σ Z Zeeman term .Our first assumption is to neglect the spin degener-acy, knowing that the effective band g − factor for theGaAs/AlGaAs hetero-structures is a factor of four lessthan the one of a free electron gas, and therefore Zeemansplitting is much smaller than the Landau splitting ( (cid:126) ω c ,with ω c = eB/m ∗ , i.e. | g ∗ µ B B | / (cid:126) ω c ≈ . µ B is the Bohr magneton. However, Zeeman splitting can beas large as the Landau splitting if exchange and correla-tion effects are taken into account at significantly highmagnetic fields, hence filling factor ν (= E F / (cid:126) ω c ) oneplateau can be observed experimentally. On the otherhand, for higher filling factors ν > H eff = H + V ext ( x, y ) + V int ( x, y ) . (4)The kinetic part, H , can be solved analytically usingthe Landau gauge which yield plane wave solutions inone direction ( y ) and Landau wavefunctions in the otherdirection ( x ). Here, implicitly, a long Hall bar (ideally in-finite) is assumed, which is justified while the Fermi wavelength is ( ∼ −
40 nm) much smaller than the samplelength under consideration ( L y ∼ H can be expressed asΦ n,k y ( x, y ) = 1 (cid:112) n n ! √ πlL y exp ( ik y .y ) × exp [ − ( x − Xl ) / × H n ( x − Xl ) , (5)where k y is the quasi-continuous momentum in y direc-tion, n the Landau index, X = − l k y a center coordi-nate, and H n ( ξ ) the n th order Hermite polynomial withthe argument ξ , whereas the eigen energies are E n = (cid:126) ω c ( n + 1 / . (6)The essence of the TFA relies on the fact that the poten-tial profile varies smoothly on the quantum mechanicallength scales. Through out this work we will only con-sider the 6 − x direction (of the groundstate, i.e. ν = 2), or the magnetic length l = (cid:112) (cid:126) /mω c ,will be always similar or less than 10 nm, therefore inalmost all cases neglecting the finite extend of the wave-functions is still reasonable. However, we have alreadyseen that for the etched samples the potential is quitesteep and the results obtained from the TFA may bedoubted, which we will address in the next section.At the moment let us consider a case that the con-dition of TFA holds, i.e. the total potential V tot ( x, y ) = V ext ( x, y )+ V int ( x, y ) varies smoothly on the quantum me-chanical length scales and the sample is long enough (i.e. k F L y (cid:29) X , and at y and the center coordinate dependent eigenenergy E n ( X )can be approximated to E n + V tot ( X, y ). It follows thatthe spatial distribution of the electron density within theTFA is given by the expression , n el ( x, y ) = (cid:90) D ( E, ( x, y )) f ( E + V tot ( x, y ) − µ ∗ ) dE (7)with D ( E, ( x, y )) (local) density of states, f ( E ) =1 / [exp( E/k b T ) + 1] the Fermi function, µ ∗ the electro-chemical potential, which is a constant in equilibriumstate, k B the Boltzmann constant, and T the tempera-ture. Once the electron density is obtained, the interac-tion potential, i.e. the Hartree potential, can be obtainedfrom V int ( x, y ) = 2 e ¯ κ (cid:90) A K ( x, y, x (cid:48) , y (cid:48) ) n ( x (cid:48) , y (cid:48) ) dx (cid:48) dy (cid:48) . (8)Here, ¯ κ is an average dielectric constant (= 12 . K ( x, y, x (cid:48) , y (cid:48) ) is the solution of the 2D Pois-son equation satisfying the boundary conditions dictatedby the sample. The results reported in this and thefollowing section assume periodic boundary conditions,where a closed form of the kernel K ( x, y, x (cid:48) , y (cid:48) ) can beobtained analytically . Equations (7) and (8) form aself-consistent loop to obtain the potential and the den-sity profiles of a 2DES subject to high perpendicular mag-netic field in the absence of an external current at equilib-rium, which has to be solved iteratively using numericalmethods. The computational effort to calculate electronand potential profiles within the TFA is much less thanthat of the full quantum mechanical calculation proce-dures. The results of both agree quantitatively very wellin certain magnetic field intervals where the widths of the incompressible strips W IS (in which the potential changesstrongly) is larger than l . If W IS (cid:46) l condition is reached,first of all the TFA becomes invalid and the calculationof the electron density should include the finite extend ofthe wavefunctions. The underestimation of the quantummechanical effects lead to existence of artificial incom-pressible strips both in the non self-consistent electro-static approximation (NSCESA) and self-consistentTFA schemes . In fact, as early as the NSCESA, theself-consistent schemes which also took into account finiteextend of the wave functions already pointed out the sup-pression of the incompressible strips in certain magneticfield intervals and also in the recent reports . For asystematic comparison of the calculated widths of the in-compressible strips within the TFA and the full Hartreeapproximations, we suggest the reader to check Ref. [9],where a simpler quasi-Hartree scheme is proposed to re-cover the artifacts arising from TFA. B. Corrections to the TFA
Historically, the first implementation of the TFA, in-cluding electron interactions, to quantum Hall systemsgoes back to the seminal work by Chklovskii et.al .There it was shown that within the electrostatic approx-imation the 2DES is divided into two regions which havecompletely different screening properties. In this model,a translation invariance is assumed in the current ( y − )direction. Due to finite widths of the samples in the x direction the electrostatic potential is bent upwardsat the edges of the sample, hence the Landau levels, E n ( X ) = E n + V ( X ). Inclusion of the Coulomb interac-tion and the pinning of the Fermi energy to the Landaulevels result in two regions (strips): i) The Fermi energyis pinned to one of the highly degenerate Landau levels,then the screening is perfect, effective potential is com-pletely flat (metal like) and electron density varies spa-tially ii) The Fermi energy falls in between two consecu-tive Landau levels, screening is poor, effective (screened)potential varies (the amplitude of the variation is (cid:126) ω c )and electron density is constant over this region. It isapparent that, if the potential varies rapidly on the scaleof l , the TFA fails and the results become unreliable.This condition is realized when considering narrow in-compressible strips having a width smaller than the mag-netic length. Therefore, one should include the effect ofwave functions within these narrow strips. One way is,of course, to do full Hartree calculations. We alreadymentioned the challenges in the computational effort. Asimpler approach is to replace the delta wave functions ofthe TFA with the unperturbed Landau wave functions,i.e. quasi Hartree approximation (QHA). The findingsof the QHA is shown to be more reasonable than of theTFA, which now also includes the finite extend of thewave functions in the close vicinity of the incompressiblestrips. Therefore, as an end result, when the W IS (cid:46) l con-dition is reached the incompressible strip disappear dueto the overlap of the neighboring wave functions. Basedon this fact, in our calculations in the following we willexclude the effects arising from the artifacts of the TFAby considering a spatial averaging of the electron den-sity on the length scales smaller than l , which is knownto be relevant in simulating the quantum mechanical ef-fects .We should also make one more point clear that withthe NSCESA usually a gate defined quantum Hall baris considered. It is more common to define Hall bars bychemical etching and the edge potential profile is muchmore steep compared to gated samples which was shownin the previous section. Therefore, to fit the predictionsof this model concerning the widths and the positions ofthe incompressible strips with the experimental data onehas to assume that i) the 2DES and the gates are onthe same plane and ii) the gate voltage applied shouldbe fixed to the half of the mid gap of the GaAs, i.e. pin-ning of the Fermi energy at the GaAs surface. In fact,after making these two crucial assumptions the experi-mental findings of E. Ahlswede perfectly fits with theNSCESA. However, the widths (and the existence) of theincompressible strips strongly deviate from the predic-tions, since only the innermost incompressible strip canbe observed. We have argued that, the widths of the in-compressible strips strongly depend on the slope of thepotential, i.e. if the external potential is steep the in-compressible strips are narrow. Therefore one can easilyconclude that, since the widths of the outermost incom-pressible strips become smaller than the magnetic length,the outer incompressible strips, i.e. the ones close to theedge, disappear and could not be observed. The overesti-mation of the W IS within the TFA becomes more severewhen an external current is imposed to the system, whichwe will discuss in the next section. Before discussing theresults of the relaxed TFA, we want to touch another lo-cally defined quantity, namely the DOS, and comment onthe implementation of the global DOS to our local TFA.In the absence of impurity scattering the DOS of an in-finite (spin-less) 2DES is given by the bare Landau DOSas D ( E ) = 12 πl (cid:88) n δ ( E − E n ) (9)however, this DOS is broadened by the scattering pro-cesses, which can be described in self-consistent Born ap-proximation accurately for short range impurity po-tentials yielding a semi elliptic broadening. Of course,other impurity models and scattering processes can alsobe considered resulting in Gaussian or Lorentzian broad-ened Landau DOS . In such descriptions of the DOSbroadening an infinite 2DES is assumed and the DOSis calculated for impurity distributions averaged over allpossible configurations. Inserting this (global) DOS inEqn. (7) can be justified again if the TFA condition issatisfied. We have already shown that this condition isviolated when a narrow incompressible strip is formedwhere the external potential is poorly screened. There- fore, the actual distribution of the impurity potential be-comes more effective at these transparent regions. Weshould also note that the effect of screening on the DOSof an infinite system has been investigated in detail inRef. [63] and it has been shown that, since the screen-ing is poor within the incompressible regions, the DOSbecomes much broader than that of the non-interactingcase. Hence, the gap between two consecutive Landaulevels is narrower within the incompressible strips com-pared to compressible strips. Moreover, recently it hasbeen shown that the (local) electric field within the sam-ple also leads to broadening of the (local) DOS . Theidea is basically that one calculates the Greens functionfor the given potential profile, which is a function of theapplied magnetic field and external current, and obtainsthe local DOS from the general expression D ( E, ( x, y )) = (cid:88) n | (cid:101) Φ n,k y ( x, y ) | δ ( E − (cid:101) E n,k y ) , (10)where (cid:101) Φ n,k y ( x, y ) is the n th eigenfunction of the Hamil-tonian given at Eqn. (4) with the eigenvalue (cid:101) E n,k y . Inour above discussion about the formation of the com-pressible/incompressible strips we have mentioned thatthe potential varies locally whenever an incompressiblestrip is formed, where the variation is linear in positionup to a first order approximation. Now let us considera linear potential profile and re-obtain the local DOS(LDOS) following for the k th Landau level, D k ( E ) = 12 k +1 k ! π / l Γ e − E k / Γ [ H k ( E k / Γ)] (11)with the level width parameterΓ = E x l (12)where E x = ∂V ( x, y ) /∂x is the electric field in the x direction and E k = E − Γ / (4 (cid:126) ω c ) + (2 k + 1) (cid:126) ω c . Theimmediate consequence of a strong electric field in the x direction is a broadening of the LDOS, which happensat the incompressible strips. On the other hand, since E x vanishes at the compressible strips, the bare LandauDOS is reconstructed from Eqn. (11) in the Γ → Y ( m) X ( m ) -0.008 0.002 0.01 0.02 V (x,y) Y ( m) X ( m ) d)c) b) V D ES ( x , y = n m ) X ( m) V st LL 2 nd LL 3 rd LL E F a) G ( e / h ) , B (a.u) G FIG. 6: The distribution of the spin degenerate LB-ES at(a) ν ≈ ν ≈
8. The potential cross-section at ν = 8plateau (c). Sketch of the conductance (G) and expected co-herence Ξ (d) for C1 considering W = 150 nm with an appliedgate potential V g = 2 . C. Results
The transport through the QPCs in the absence of anexternal magnetic field is well described by the Landauerformalism, summarized in Eqn. (2). The main idea isthat the transport is ballistic and due to the cancella-tion of the velocity and a 1D DOS , the conductance isinteger multiples of the conductance unit, e /h . Theseintegers are just the number of channels N c , i.e. the num-ber of eigen energies below the Fermi energy. A similarpath is followed when considering an external B field,which assumes that the conductance is ballistic within1D channels and neglects the electron interactions. These1D channels are formed whenever the Fermi energy co-incides with a Landau level (Landauer-B¨uttiker picture).Before proceeding with the full self-consistent solutionof the density and current distribution problem, we firstinvestigate the positions of the Landauer-B¨uttiker edgestates. The procedure is simple, to obtain the energydispersion we use the relation E n,k y ( x, y ) = E n + V ( x, y ) (13)and follow the equipotential (energy) lines coincidingwith the Fermi energy. By doing so we would be ableto discuss qualitatively the phase coherence Ξ, taken tobe equal to 1 when there is a full phase coherence and 0when there is none. We neglect all the external sourcesof decoherence and assume that the LB-ES are coherentat the length scales we are interested in, i.e. the wavefunctions of the associated channels do not overlap.In Fig. 6, we show the spatial distribution of the ex-pected positions of the LB-ES at two filling factors. The color scale depicts the self-consistent potential, whereasthe black strips show the LB-ES. The white shaded ar-eas are the electron depleted regions. The ν = 2 and ν = 4 edge states nicely show the expected distributionwhich are spatially ∼
40 nm apart, Fig. 6a. Dependingon the steepness of the potential or the magnetic fieldvalue, however, this distance may become less. For theetched samples (not shown) at the same filling factor thespatial distance between ν = 2 and ν = 4 edge statesbecome almost half the value of the gated samples. For afilling factor ν =8 plateau the outermost (the ones closestto the gates) two edge states are less than 15 nm apartfrom each other, and the wave functions extend over alarger distance. It is apparent that, when the two wavefunctions start to overlap, the coherence Ξ is reduceddrastically. However, for the ideal case (no overlap) Ξshould stay constant for all plateau regions, since by def-inition the edge states can not cross each other. Theconductance quantization is, of course, independent ofthe structure of the ES itself and according to Eqn. (2)one should simply count the number of ES which crossthe constriction. The conductance is shown by the sketchin Fig. 6d, of course the sharp transition between theplateaus is changed when one considers level broadeningor in general scattering. One should note that, althoughthe LBES picture is useful in making qualitative argu-ments, one needs to grasp the actual distribution of theedge states to understand the physics observed at theexperiments .Next we investigate the distribution of the incom-pressible strips calculated self-consistently described byEqns. (7-8). The conductance through the QPC can berewritten G = e h ν center (14)as conjectured in Ref. [27], where ν center is the filling fac-tor at the very center of the QPC. It is apparent that, ifthis value is an integer, i.e. incompressible, the conduc-tance is quantized. Therefore, it is important to studythis condition for a realistic QPC geometry. In the fol-lowing we first calculate the filling factor distribution inthe absence of an external current and then obtain thecurrent distribution in the next section.In order to cure the artifacts arising from TFA i) weconsider a DOS broadened by a Gaussian given by D ( E ) = 12 πl ∞ (cid:88) n =0 exp( − [ E n − E ] / Γ ) √ π Γ imp (15)with the impurity parameter Γ imp , which is chosen largeenough Γ imp / (cid:126) ω c = 0 . and self-consistentbroadening effects and ii) a spatial averaging is car-ried out over the Fermi wavelength ( ∼
30 nm). Fig. 7summarizes our results showing the spatial distributionof the incompressible ES, considering the quantum Hallplateau ν = 2. Pedagogically, starting our investigation1from large magnetic fields is preferable; at large magneticfields (Fig. 7e), the system is mostly compressible (col-ored area) and the two incompressible (white) regions donot merge at the opening of the QPC. Therefore, both theHall resistance and the conductance through the QPC isnot quantized. As soon as one enters to the QH plateaualmost all of the sample becomes incompressible shownin Fig. 7d (in the absence of short range impurities) andboth R H and G becomes quantized. Decreasing the mag-netic field creates two incompressible ES which are spa-tially separated seen in Fig. 7b, however, the quantiza-tion is not affected. At a lower magnetic field value theseIS-ES disappear (Fig. 7a) as an end result of level broad-ening and (simulation) of the finite extend of the wavefunctions, now we are out of the QH plateau and G isno longer quantized. This picture and the LB-ES pictureyield same behavior for the R H and G , however, in thelater one current is carried by the IS-ES, which we willdiscuss in the next section. The qualitative differencebetween the two pictures is the coherence as shown bythe (red) dashed line in Fig. 7c. First Ξ presents min-ima in between two plateau regimes, since the IS die out,second at the higher edge of the QH plateau the systembecomes completely incompressible therefore, it is notpossible to define separate ESs hence coherence is lost(averaged). We believe that, this nonuniform behaviorof the coherence within the QH plateau coincides withthe experimental findings of Roche et.al. , however, weadmit that other explanations are also possible. Anotherinteresting experimental work is carried by the Regens-burg group, where they have investigated the amplitudeof the visibility oscillations as a function of B field .They have reported a maximum visibility at ν ≈ . . However, it is easy to see that theirsample has a homogeneous width all over, which is notthe case for other groups. From self-consistent calcula-tions it known that, if the sample width is narrowerthan 5-6 µ m the center electron density (or filling factor)can differ strongly from that of the average one(s). Anindication of such a case is also shown by numerical sim-ulations . In interconnecting the Hall plateaus and thespatial distribution of the incompressible strips, we haveused the findings of Ref. [9] where the current is shownto be flowing through the incompressible strips. This isin contrast to some of the models in the literaturewhere the opposite is proposed. In the next section, wewill present the general concepts of the local Ohm’s lawand based on the absence of back scattering in the in-compressible strips we will show that the local resistivityvanishes and the external current should be confined tothese regions. FIG. 7: Spatial distribution if the incompressible strips(white areas) for characteristic B values (a) 6.8 T, (b) 7.3 T,(d) 8.3 T (e) 8.8 T calculated at temperatures (cid:126) ω c /k B T (cid:28) W = 150 nm, the gate voltage is chosen such that all theelectrons beneath the gates are depleted. IV. CURRENT DISTRIBUTION WITHINLOCAL OHM’S LAW
The local (potential) probe experiments ,brought novel information concerning the Hall potentialdistribution over the sample. The first set of experimentsshow clearly that the potential, therefore the current,distribution is strongly magnetic field dependent. Itwas shown that, out of the QH plateau regime the Hallpotential varies linearly (Type I) all over the sample, asimilar behavior to classical (Drude) result. Whereas, atthe lower edge of the QH plateau the current is confinedto the edges of the sample (Type II), which was shownto be coinciding with the positions of the incompressiblestrips. The most interesting case is observed whenan exact (even) integer filling is approached. In thesemagnetic field values, the potential exhibits a strongnonlinear variation all over the sample, which was at-2tributed to the existence of a large (bulk) incompressibleregion. The explanation of these measurements acquireda local transport theory, where the conductivities andtherefore current distribution can be provided alsotaking into account interactions. In the subsequenttheoretical works the required conditions were sat-isfied and an excellent agreement with the experimentswere obtained . In the second set of experiments a single electron transistor has been placed on top ofthe 2DES and the local transparency, i.e. whetherthe system is compressible or incompressible, and thelocal resistivity have been measured. Comparing thetransparency and the longitudinal resistivity, it has beenconcluded that the resistivity vanishes when the systemis incompressible.Theoretically, if the local electrostatic potential andthe resistivity tensor ˆ ρ ( x, y ) are known the current dis-tribution (cid:126)j ( x, y ) can be obtained from the local versionof the Ohm’s law (cid:126)E ( x, y ) = ˆ ρ ( x, y ) .(cid:126)j ( x, y ) (16)provided that (cid:126)E ( x, y ) = ∇ µ ∗ ( x, y ) /e (17)where the electrochemical potential is position dependentwhen an fixed external current I = (cid:82) A (cid:126)j ( x, y ) dxdy is im-posed. In our calculations we assume a local equilibriumin order to describe the stationary non-equilibrium stategenerated by the imposed current, starting from a ther-mal equilibrium state obtained from the modified TFA.At this point if the local resistivity tensor is known, Eqn.s(16)-(17) should be solved once again iteratively for agiven electron density and potential profile, where theequation of continuity ∇ · j ( r ) = (18)also holds. We assume that the local resistivity is relatedto the local electron density via the conductivity tensor,i.e. ˆ ρ ( x, y ) = ˆ σ − ( n el ( x, y )). For a Gaussian broadenedDOS the longitudinal component of the conductivity ten-sor is obtained from σ l = 2 e h (cid:90) ∞−∞ dE [ − ∂f∂E ] ∞ (cid:88) n =0 ( n + 12 )[ e ( − [ En − E Γimp ] ) ] (19)whereas Hall component is simply σ H = 2 e h ν, (20)where we ignored the self-energy corrections dependingon the type of the impurity scatterers. We should em-phasize that, the above conductivities are used for consis-tency reasons, in principle, any other reasonable impuritymodel like the commonly used SCBA can be consid-ered. Assuming that TFA is valid, we can replace thelocal conductivities with the above defined global ones. FIG. 8: The local current density calculated at different fieldstrengths, same as in Fig.7. The intensity of the current den-sity is chosen such that, the applied current does not effectthe density distribution, i.e. | j ( x, y ) | ∼ . × − A/m.
In the absence of an external current our calculationscheme is as follows, we initialize Eqn. (7) using the to-tal potential obtained from the 3D calculations and ob-tain the electron density at relatively high temperatures( kT / (cid:126) ω c ∼ .
5) and use this density distribution to ob-tain resulting potential from Eqn. (8). Next we keep oniterating until a numerical convergence is reached, wherethe electron density is kept constant. This is followedby the step where the temperature is lowered by a smallamount and iteration process is repeated till the targettemperature is reached. After the thermal equilibrium isobtained, we impose a small amount of external currentand solve Eqns. (16-17) self-consistently. While doingthis second iteration, we fix the constant arising from theintegral equation to a value such that the total numberof electrons is kept constant with and without current.As a numerical remark, if the current loop does not con-verge we increase the temperature by a relevant amountand start the iteration procedure. The whole calculationscheme is composed of three different codes, which arewritten in C++, Fortran and Matlab respectively. In3order to obtain reasonable grid resolution parallel com-putation techniques were used .Since we are interested in current distribution and alsoits effect on the density distribution we find appropriateto present our results in two separate sections i) wherethe applied current is weak enough that the electron andpotential distribution is unaffected, linear response. ii)the applied current is sufficiently large so that the im-posed current induces a considerable change on the po-sition dependent electrochemical potential, out of linearresponse. A. Linear response regime
The crucial part of the local approach is that for a given(large) magnetic field we can calculate the local poten-tials and electron distributions self-consistently. The re-sult of such a calculation is that the 2DES is essentiallyseparated in two regions, i.e. compressible and incom-pressible, therefore for a given (obtained) density we cancalculate the local conductivities via Eqns. (19-20). Letus now discuss the distinguishing conductance propertiesof these two regions starting from a compressible region.At a compressible region the Fermi energy is pinned toone of the Landau levels, screening is nearly perfect, self-consistent potential is flat and filling factor is locally anon-integer. According to Eqn.(20), the Hall conductiv-ity is a non-integer and the longitudinal conductivity isnon-zero meaning finite back scattering. Now the clas-sically defined drift velocity, and also its quantum me-chanical counter part, is proportional to the electric fieldperpendicular to the current direction. We have seen thatat the compressible region the potential perpendicular tothe current direction is flat, therefore the x component ofthe electric field is zero, hence the drift velocity. Mean-while, at an incompressible region the Fermi energy is inbetween two Landau levels, the filling factor is fixed toan integer value and potential presents a variation per-pendicular to the current direction. Due to the Landaugap the longitudinal conductivity vanishes, whereas theHall conductivity assumes its (quantized) integer value.If one calculates the inverse of the conductivity tensor forthe longitudinal component one obtains, ρ l ( x, y ) = σ l ( x, y ) σ l ( x, y ) + σ H ( x, y ) (21)thus the longitudinal resistivity vanishes within the in-compressible region pointing the absence of back scatter-ing. Of course, the simultaneous vanishing of both thelongitudinal resistivity and the conductivity is a resultof applied external (and perpendicular) B field and isobtained only in two dimensions. Moreover, since the x component of the electric field is now non-zero, the driftvelocity is finite and the current is confined to this region.Combining these two one concludes that, if there existsan incompressible region somewhere in the sample all theexternal current is confined to this region otherwise (if there are no incompressible regions and all the systemis compressible) the current is distributed according toDrude formalism, i.e. the current density is proportionalto the electron density.We start our discussion of the current distributionwhen a small current is imposed for which the electro-static and electrochemical potential satisfies the linearresponse relation V (( x, y ); I ) − V (( x, y ); 0) ≈ µ ∗ (( x, y ); I ) − µ ∗ eq . (22)This condition essentially states that, the imposed cur-rent does not modify the electrochemical potential there-fore the electron density remains unchanged. Fig. 8presents the current distribution which is calculated forthe density distribution shown in Fig. 7. The correspon-dence between the positions of the incompressible stripsand the current density is one to one. In the out ofplateau regime the current is essentially distributed allover the sample, where no incompressible regions exist,Fig.8a. As soon as one enters the QHP, i.e. when a large bulk incompressible strip (region) is formed, the essen-tial future of current distribution is not effected strongly,however, in this situation current is flowing in the in-compressible region. Tracking the positions of the ISs inFig. 7b, we can readily guess the distribution of the cur-rent density in Fig. 8b. Following our arguments aboutthe smearing out of the narrow incompressible strips, wehave a situation in which, again, the current is spreadover the sample shown in Fig. 8a. Although, the ISvanishes the reminiscence of it still provides a narrowstrip of small longitudinal resistivity and, therefore ahigher amount of current is kept confined to these re-gions. Fig. 8c, presents the corresponding longitudinalresistance, when measured as a function of B togetherwith the conductance across the QPC. The relation be-tween G and R L is interesting, the conductance is quan-tized, as soon as R L vanishes at large fields, however,becomes non-quantized even though the R L = 0 at thelower edge of the zero resistance state. Let us first discussthe B dependence of the R L , it is finite if the system iscompressible and is zero if an incompressible strip perco-lates from one edge of the sample to the other edge in thecurrent direction, i.e. from source to drain. Therefore,existence of an IS percolating is sufficient enough to mea-sure zero longitudinal resistance. However, to have a con-ductance quantization the center of the QPC should beincompressible, which is a stronger restriction . Hence,in the lower edge of the QHP, the IS percolates but thecenter of the QPC is compressible. The implication ofthis fact to the coherence is a bit more complicated, wehave seen that as soon as one enters to the QHP a largebulk IS is formed therefore the phase of the electronsis highly averaged. This implies that the coherence isrelatively less than that of the two well separated ISs.On the other hand, at the lower edge of the conductanceplateau, the ISs become narrower and are less immune todecoherence effects arising from the environment, hence,the coherence is reduced. Our above discussion coincides4with the recent experiments performed in small Mach-Zehnder interference devices (MZI), where the visibilityis measured as a function of the external magnetic field .It is fair to note that, some other mechanisms providing B dependence of decoherence can also account for sucha behavior.In the mentioned MZI experiments and also themeasurements performed at the group of S. Roddaro a finite (and large) source drain voltage V SD is appliedeither to measure the V SD dependency of the visibility orthe transmission. The intensity of the applied current,in these experiments, can not be treated within the lin-ear response regime, where the electrochemical potentialremains constant, i.e. position independent. In the nextsection, we present the current and the density distribu-tion calculated where Eqn. (22) does not hold any more. B. Beyond linear response
In the absence of an external current an equilibriumstate is obtained by solving the Eqns. (7) and (8) self-consistently. Even in the presence of a small current,a Hall potential is generated which, in principle, modi-fies the electrochemical potential, i.e. tilts the Landaulevels. This modification can be compensated by the re-distribution of the electrons, which certainly modifies thetotal electrostatic potential. If the applied current is suf-ficiently small, the modification is negligible, i.e. linearresponse. However, if the current is large, the resultingHall potential is also large and one should re-calculatethe electron density, and therefore the potential distri-bution till a steady state is reached. In this section wepresent the current and density distribution in the pres-ence of a large external current, where a local equilibriumis assumed implicitly. Fig. 9a shows the electron densitydistribution in color scale for B = 7 . B value is chosen such that theHall resistance is quantized, however, the conductanceis not. The general behavior is similar to that of lin-ear response, however, it is clearly seen that the widthsof the ISs are asymmetric with respect to y = 750 nmline, where current is driven in y direction. The asym-metry is induced by the large current, since the electronsare redistributed according to the new (self-consistent)potential distribution. The corresponding current den-sity distribution is plotted in Fig. 9b, once more the oneto one correspondence between the positions of the ISsand the local current maxima is apparent. The conse-quence of the asymmetry and thereby the widening ofthe ISs can be observed in the conductance and the R L ,such that the narrow IS at the upper side is smeared outmuch earlier than the one on the lower side. We shouldnote that, such large currents heat the sample thereforethe local temperature within the ISs is larger comparedto the lattice temperature due to Joule heating . Sucha (local) temperature dependence is treated explicitly by FIG. 9: The local filling factor distribution and the corre-sponding local current density calculated at the default tem-perature. The current intensity is sufficiently high ( | j ( x, y ) | =2 . × − A/m) to induce an asymmetry on the density dis-tribution via position dependent electrochemical potential.
H. Akera and his co-workers and a strong evidence isprovided towards explaining the breakdown of the IQHEwithin this approach. Our present approach lacks such atreatment, therefore the competition between the widen-ing of the ISs and heating due to large currents is thoughtto be more complicated than presented here. As a con-sequence, the discussion of the coherence is far beyondour model, however, we think that the amplitude of thecurrent when measuring visibility ] is assumably still inthe linear response regime.In conclusion, by exploiting the local equilibrium andthe properties of a steady state we have calculated thecurrent distribution near a QPC in the out of linear re-sponse regime. We have shown that a asymmetry is in-duced on the density profile due to the bending of theLandau levels generated by the large Hall potential. Weestimate that, the system can still be considered in thelinear response regime if the 1D current density is smallerthan 0 . × − A/m, which certainly depends on thedetails of the sample geometry.5
V. SUMMARY
In this paper, we provided a self-consistent scheme toobtain the electron density, potential profile and currentdistribution in the close vicinity of a QPC, within theThomas-Fermi approximation. Starting from a litho-graphically defined 3D sample, we calculated the chargedistribution at the surface gates, at the plane of 2DESand for etched samples at the side surfaces. The 3D self-consistent solution of the Poisson equation enabled us topresent the similarities and differences between an etchedand gate defined QPC. We found that, the relatively deepetched samples present a sharp potential profile near theedges of the sample. If the depth of the etching exceedsthe depth of the 2DES, surface charges are calculatedexplicitly. In the presence of a quantizing perpendicularmagnetic field, we have calculated the distribution of theincompressible strips as a function of the field strength.We have argued that, if an incompressible strip becomesnarrower than the magnetic length and/or if the trans-verse electric field is sufficiently large, due to Level broad-ening, the narrow incompressible strip is smeared. In thenext step, the current distribution is obtained both inthe linear response and out of linear response regimesusing a local version of the Ohm’s law assuming a steadystate at local equilibrium. It is shown that the current isconfined to the incompressible strips, due to the absenceof back scattering, otherwise is distributed all over thesample. We have commented on the relation betweenthe existence and percolation properties of the incom-pressible strips and the measured quantities such as thelongitudinal resistance and conductance across the QPC.For the ideal clean sample, i.e. in the absence of longrange fluctuations, it is shown that the QH plateau ex-tends wider than that of the conductance plateau. In theout of linear response regime, a current induced densityasymmetry is presented for the first time in such geome- tries under quantized Hall conditions. The observableeffects of such an asymmetry are not clarified, since it isalso know that large currents increase the temperaturelocally due to Joule heating.A natural extension of the existing model is to includethe spin degree of freedom and thereby exchange and cor-relation effects . A local spin density functional theoryapproach is the much promising one among otherssuch as Monte Carlo and exact diagonalization, fromcomputational and application point of view. In fact,such an approach already exists, however, the current ishandled within the Landauer-B¨uttiker formalism, whichwe think is not reasonable in the presence of large incom-pressible strips. On the other hand, a time dependentspin density functional model would be a good candi-date to describe current for the geometries under inves-tigation. The implementation of the Akera’s theory, i.e.Joule heating, to our model is of course desirable whichis been worked presently.Finally, in order to have a predictive power on the in-terference experiments we would like to utilize the exist-ing coherent transport models in describing the currenttogether with our electrostatic model, which we are notable at the present. Another challenge is to simulatethe real experimental geometries which already includesmore then a single QPC and contacts etc. The numericalroutine we developed is now able to do such large scalecalculations within the linear response regime, however,still lacks describing the exchange and correlation effects. Acknowledgement
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