The effect of disorder within the interaction theory of integer quantized Hall effect
S. E. Gulebaglan, G. Oylumluoglu, U. Erkarslan, A. Siddiki, I. Sokmen
TThe effect of disorder within the interaction theoryof integer quantized Hall effect
S. E. Gulebaglan , G. Oylumluoglu , U. Erkarslan ,A. Siddiki , and I. Sokmen Dokuz Eyl¨ul University, Physics Department, Tınaztepe Campus, 35100 ˙Izmir,Turkey Mu˜gla University, Physics Department, Faculty of Arts and Sciences, 48170-K¨otekli,Mu˜gla, Turkey Istanbul University, Faculty of Sciences, Physics Department, Vezneciler-Istanbul34134, TurkeyE-mail: [email protected]
Abstract.
We study effects of disorder on the integer quantized Hall effect withinthe screening theory, systematically. The disorder potential is analyzed consideringthe range of the potential fluctuations. Short range part of the single impuritypotential is used to define the conductivity tensor elements within the self-consistentBorn approximation, whereas the long range part is treated self-consistently at theHartree level. Using the simple, however, fundamental Thomas-Fermi screening, wefind that the long range disorder potential is well screened. While, the short rangepart is approximately unaffected by screening and is suitable to define the mobility atvanishing magnetic fields. In light of these range dependencies we discuss the extendof the quantized Hall plateaus considering the “mobility” of the wafer and the width ofthe sample, by re-formulating the Ohm’s law at low temperatures and high magneticfields. We find that, the plateau widths mainly depend on the long range fluctuations ofthe disorder, whereas the importance of density of states broadening is less pronouncedand even is predominantly suppressed. These results are in strong contrast with theconventional single particle pictures. We show that the widths of the quantized Hallplateaus increase with increasing disorder, whereas the level broadening is negligible.
This work focuses on the disorder effects on the integer quantized Hall effect within thescreening theory. Since the early days of QHE, disorder played a very important role,however, interactions were completely neglected. Here we present our results which alsoincludes interactions in a self-consistent manner and show that even without localizationone can obtain the quantized Hall plateaus. We investigated different aspects of theimpurity potential and suggested a criterion on mobility at high magnetic fields. Wethink that our work will shed light on the understanding of the QHE and is interest tocondensed matter community.
Keywords : Article preparation, IOP journals
Submittedto:
J. Phys. C: Solid State Phys. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b isorder in screening theory
1. Introduction
The integer quantized Hall effect (IQHE), observed at two dimensional charge systems(2DCS) subject to strong perpendicular magnetic fields B , is usually discussed within thesingle particle picture, which relies on the fact that the system is highly disordered [1, 2].These quantized (spinnless) single particle energy levels are called the Landau levels(LLs) and the discrete energy values are given by E N = (cid:126) ω c ( n + 1 / n isthe Landau index and ω c = eB/m ∗ c is the cyclotron frequency of an electron with aneffective mass m ∗ ( ≈ . m e , m e being the bare electron mass at rest) and c is thespeed of light in vacuum. In single particle models the disorder plays several roles,such as Landau level broadening [3], leading to a finite longitudinal conductivity [4, 5],spatial localization [6] etc . Disorder can be created by inhomogeneous distribution ofdopant ions which essentially generates the confinement potential [7] for the electrons.In the absence of disorder, the density of states are Dirac delta-functions D ( E ) = πl (cid:80) ∞ N =0 δ ( E − E N ), where l = (cid:112) (cid:126) /eB is the magnetic length, and the longitudinalconductivity ( σ l ) vanishes. For a homogeneous two dimensional electron system (2DES),by the inclusion of disorder and due to collisions, LLs become broadened. Therefore thelongitudinal conductance becomes non-zero in a finite energy (in fact magnetic field)interval. Long range potential fluctuations generated by the disorder result in the socalled classical localization [8], i.e. the guiding center of the cyclotron orbit moves alongclosed equi–potentials [9]. In contrast to the above mentioned bulk theories, the edgetheories usually disregard the effect of disorder to explain the (quantized) Hall resistance R H and accompanying (zero) longitudinal resistance R L . However, the non-interactingedge theories still require disorder to provide a reasonable description of the transitionbetween the plateaus. The Landauer-B¨uttiker approach (known as the edge channelpicture) [10] and its direct Coulomb interaction generalized version, i.e. the non-self-consistent Chklovskii picture [11], also needs localization assumptions in order to obtainquantized Hall (QH) plateaus of finite width (see for a review e.g. Datta’s book [12]and Ref. [9] for the estimates of plateau widths at the high disorder limit).In contrast to above discussions very recent experimental [13, 14, 15, 16] andtheoretical [17, 18] results point the incomplete treatment of the disorder potentialand scattering mechanisms. Fairly recent theoretical approaches [19], the QH plateausare obtained by the inclusion of direct Coulomb interaction self-consistently [20] and theeffect of the disorder was handled via conductivity tensor elements [21], however, thesource of the disorder and its properties was left unresolved [22]. Whereas, the influenceof potential fluctuations on the QH plateaus were discussed briefly [23, 24].This work provides a systematic investigation of the disorder potential and itsinfluence on the quantized Hall effect including direct Coulomb interaction. Theinvestigation is extended to realistic experimental conditions in determining the widthsof the quantized Hall plateaus. We, essentially study the effect of disorder in two distinctregimes, namely the short range and the long range. The short range part is includedto the density of states (DOS), thereby influences the widths of the current carrying isorder in screening theory screened disorderpotential within a pure electrostatic approach, by considering an homogeneous twodimensional electron system (2DES) without an external magnetic field and show thatthe long range part is well screened, whereas the short range part is almost unaffected.Section 2.2 is devoted to investigate impurities numerically, where we solve the Poissonequation self-consistently in three dimensions. The numerical and analytical calculationsare compared, considering the estimations of the disorder potential range and itsvariation amplitude. We finalize our discussion with Sec. 3, where we calculate theplateau widths under experimental conditions for different sample widths and mobilities.
2. Impurity potential
The disorder potential experienced by the 2DES, resulting from the impurities has quitecomplicated range dependencies. Since, the potential generated by an impurity is (i) damped by the dielectric material in between the impurity and the plane where the2DES resides (ii) is screened by the homogeneous 2DES depending on the density ofstates, which changes drastically with and without magnetic field. It is common totheoreticians to calculate the conductivities from single impurity potentials, such asLorentzian [19], Gaussian [25] or any other analytical functions [26, 27]. However, thelandscape of potential fluctuations is also important to define the actual mobility of thesample at hand, in particular in the presence of an external magnetic field.
We first discuss the different range dependencies of the Coulomb and Gaussian donors,assuming open boundary conditions. Next, the effect of the spacer thickness on thedisorder potential is discussed, namely the damping of the external (Coulomb) potential,and is compared with the Thomas-Fermi screening. The different damping/screeningdependencies of the resulting potentials are discussed in terms of range.The Coulomb potential presents long range part, which leads to long rangefluctuations due to overlapping if several donors are considered. Whereas, the Gaussianpotential decays exponentially on the length scale comparable with the separationthickness. Since the Gaussian potential is relatively short ranged, no overlapping ofthe single donor potentials occur. Hence, the external potential experienced by theelectrons can be approximated to a homogeneous potential fairly good. Thus one canconclude that approximating the total disorder potential by Gaussians is not sufficient torecover the long range part. Similar arguments are also found in the literature [6, 9, 24].In order to overcome the difference observed at the long range potential fluctuations isorder in screening theory q components in each direction,hence only the long range part of the potential is left [24]. Then we add the long rangepart of the Coulomb potential to the potential created by donors, i.e. to the confinementpotential. We take this as a motivation to simulate the short range part of the impuritypotential by Gaussian impurities, and calculate the Landau level broadening and theconductivities, described within the self-consistent Born approximation (SCBA) [25].Here we point to the effect of the spacer thickness on the impurity potentialexperienced in the plane of 2DES. It is well known from experimental and theoreticalinvestigations that, if the distance between the electrons and donors is large, the mobilityis relatively high and it is usually related with suppression of the short range fluctuationsof the disorder potential. These results agree with the experimental observations ofhigh mobility samples and are easy to understand from the z dependence of the Fourierexpansion of the Coulomb potential, V (cid:126)q ( z ) = (cid:90) d(cid:126)re − i(cid:126)q · (cid:126)r N (cid:88) j e / ¯ κ (cid:112) ( (cid:126)r − (cid:126)r j ) + z = 2 πe ¯ κq e −| qz | N S ( (cid:126)q ) , (1)where S ( (cid:126)q ) contains all the information about the in-plane donor distribution and N is the total number of the ionized donors. We observe that if the spacer thickness isincreased, the amplitude of the potential decreases rapidly. We also see that the shortrange potential fluctuations, which correspond to higher order Fourier components, aresuppressed more efficiently.Next, we discuss electronic screening of the external potential created by the donorsdiscussed above. For a dielectric material the relation between the external and thescreened potentials are given by, V q scr = V q ext /(cid:15) ( q ) , (2)where (cid:15) ( q ) is the dielectric function and is given by (cid:15) ( q ) = 1 + πe D ¯ κ | q | , with the constant2D density of states D = m/ ( π (cid:126) ) in the absence of an external B field, and is knownas the Thomas-Fermi (TF) function. The simple linear relation above, together withthe TF dielectric function essentially describes the electronic screening of the Coulombpotential given in Eq. 1, if there are sufficient number of electrons [9] ( n el > . · m − ). Consider a case where the q component approaches to zero, then the external(damped) potential is well screened, hence the long range part of the disorder potential.Whereas, the short range part remain unaffected, i.e. high q Fourier components. Nowwe turn our attention to the second type of impurities considered, the Gaussian ones.As well known, the Fourier transform of a Gaussian is also of the form of a Gaussian,therefore, similar arguments also hold for this kind of impurity.We should emphasize once more the clear distinction between the effect of thespacer on the external potential and the screening by the 2DES, i.e. via (cid:15) ( q ). The isorder in screening theory Figure 1.
Schematic representation of the crystal, which we investigate numerically.The crystal is grown on a thick GaAs substrate, where the 2DES is formed at theinterface of the AlGaAs/GaAs hetero-junction. The top AlGaAs layer is dopedwith Silicon 30 nm above the interface. The crystal is spanned by a 3D matrix(128 × × former depends on the Fourier transform of the Coulomb potential and the importanteffect is the different decays of the different Fourier components (see Eq. 1), so that theshort range part of the disorder potential is well dampened, whereas the latter dependson the relevant DOS of the 2DES and the screening is more effective for the long rangepart.We continue our investigation by solving the 3D Poisson equation iteratively forrandomly distributed single impurities, where three descriptive parameters ( i.e. thenumber of impurities, the amplitude of the impurity potential and the separationthickness) are analyzed separately. Next, we discuss the long range parts of the potentialfluctuations investigating the Coulomb interaction of the 2DES, numerically. The rangeis estimated from these investigations by performing Fourier analysis and is related tothe samples used in experiments [15, 16] (Sec. 4). In the previous section we took a rather simple way to study the effect of interactionsby assuming an homogeneous 2DES and screening is handled by the TF dielectricfunction. Here, we present our results obtained from a rather complicated numericalmethod. We solve the Poisson equation in 3D starting from the material propertiesof the wafer at hand, the typical material we consider is sketched in Fig. 1. Namely,using the growth parameters, we construct a 3D lattice where the potential and thecharge distributions are obtained iteratively assuming open boundary conditions, i.e. V ( x → ±∞ , y → ±∞ , z → ±∞ ) = 0. For such boundary conditions, we chose a latticesize which is considerably larger than the region that we are interested in. We preservethe above conditions within a good numerical accuracy (absolute error of 10 − ). A forthorder grid approach [28] is used to reduce the computational time, which is successfullyused to describe similar structures [29].Figure 1 presents the schematic drawing of the hetero-structure which we areinterested in. The donor layer is δ − doped by a density of 3 . × m − (ionized) Silicon isorder in screening theory Figure 2. (a) Electron density fluctuation considering 3300 impurities 30 nm abovethe electron gas. (b) The long-range part, arrows are to guide the distance betweentwo maxima. The calculation is repeated for 50 random distributions, which lead to asimilar range. atoms, ∼
30 nm above the 2DES, which provide electrons both for the potential well atthe interface and the surface. It is worthwhile to note that most of the electrons ( ∼ %90)escape to the surface to pin the Fermi energy to the mid-gap of the GaAs. In any case,for such wafer parameters there are sufficient number of electrons ( n el (cid:38) . × m − )at the quantum well to form a 2DES. To investigate the effect of impurities we placepositively charged ions at the layer where donors reside. From Eq. 1 we estimate theamplitude of the potential of a single impurity to be e κ V imp z D = 0 .
033 eV and assume that some percent of the ionized donors are generating the disorder potential, that defines thelong range fluctuations. In our simulations we perform calculations for a unit cell withareal size of 1 . µ m × . µ m which contains 3 . × donors per square meters, thus with10 percent disorder we should have N I ∼ √ N I . Such a statisticalinvestigation, sufficiently ensembles the system to provide a reasonable estimation ofthe long range fluctuations. We also tested for larger number of random distributions,however, the estimation deviated less than tens of nanometers. We show our main resultof this section in Fig. 3, where we plot the estimated long range part of the disorderpotential considering various number of impurities N I and impurity potential amplitude V imp . Our first observation is that the long range part of the total potential becomesless when N I becomes large, not surprisingly. However, the range increases nonlinearlywhile decreasing N I , obeying almost an inverse square law and tend to saturate at highlydisordered system. When fixing the distributions and N I , and changing the amplitudeof the impurity potential we observe that for large amplitudes the range can differ as isorder in screening theory Figure 3.
Statistically estimated range of the density fluctuations as a function ofnumber of impurities, considering various impurity strengths (a) and spacer thicknesses(b). The calculations are done at zero temperature considering Coulomb impurities.The long range potential fluctuations become larger than the size of the unit cell if oneconsiders less than %5 disorder. large as 200 nm at all impurity densities. We found that for impurity concentrationless than %3, the range of the potential is larger than the unit cell we consider, i.e
R > . µ m. In contrast to the long range part, the short range part is almost unaffectedby the impurity concentration, however, is affected by the amplitude. Therefore, whiledefining the conductivities we will focus our investigation on V imp . Another importantresult is that the estimates of long range fluctuations does not depend strongly onthe spacer thickness, if one keeps the amplitude of single impurity potential amplitudefixed, Fig. 3b. All of the above numerical observations coincide fairly good with ouranalytical investigations in the previous section. However, the range dependency onthe impurity concentration cannot be estimated with the analytical formulas given. Weshould also note that, similar or even complicated numerical calculations are present inthe literature [6, 7]. A indirect measure of the screening effects on the potential can alsobe inferred by capacitance measurements, supported by the above calculation schemein the presence of external field [14].Next section is devoted to investigate the widths of the quantized Hall plateausutilizing our findings. We consider mainly two “mobility” regimes, where the long rangefluctuations is at the order of microns (high mobility) and is at the order of few hundrednanometers, low mobility. However, the amplitude of the total potential fluctuationswill be estimated not only depending on the number of impurities but also dependingon the spacer thickness, range and amplitude of single impurity potential.
3. Quantized Hall plateaus
The main aim of this section is to provide a systematic investigation of the quantizedHall plateau (QHP) widths within the screening theory of the IQHE [20], thereforehere we summarize the essential findings of the mentioned theory. In calculating the isorder in screening theory σ l ( x, y ) and thetransverse σ H ( x, y ). To determine these quantities it is required to relate the electrondensity distribution n el ( x, y ) to the local conductivities explicitly. Here we utilizethe SCBA [25]. However, the calculation of the electron density and the potentialdistribution including direct Coulomb interaction is not straightforward, one has tosolve the Schr¨odinger and the Poisson equations simultaneously. This is done withinthe Thomas-Fermi approximation which provides the following prescription to calculatethe electron density n el ( x, y ) = (cid:90) dED ( E ) 1 e ( E F − V ( x,y )) /k B T + 1 , (3)where D ( E ) is the appropriate density of states calculated within the SCBA, where k B is the Boltzmann constant and T temperature. The total potential is obtained from V ( x, y ) = 2 e ¯ κ (cid:90) dxdyK ( x, y, x (cid:48) , y (cid:48) ) n el ( x, y ) , (4)and the Kernel K ( x, y, x (cid:48) , y (cid:48) ) is the solution of the Poisson equation satisfying theboundary conditions to be discussed next.In the following we assume a translation in variance in y -direction and implementthe boundary conditions V ( − d ) = V ( d ) = 0 (2 d being the sample width), proposedby Chklovskii et.al. [11], such a geometry allows us to calculate the Kernel in a closedform. Hence, Eqs. (3) and (4) forms the self-consistency. For a given initial potentialdistribution, the electron concentration can be calculated at finite temperature andmagnetic field, where the density of states D ( E ) contains the information about thequantizing magnetic field and the effect of short range impurities. Here we implicitlyassume that the electrons reside in the interval − b < x < b (where, d l = | d − b | /d is calledthe depletion length), and is fixed by the Fermi energy, i.e. the number of electrons,hence donors. As a direct consequence of Landau quantization and the locally varyingelectrostatic potential, the electronic system is separated into two distinct regions, whensolving the above self-consistent equations iteratively: i) The Fermi energy equals to(spin degenerate) Landau energy and due to DOS the system illustrates a metallicbehavior, the compressible region, ii) The insulator like incompressible region, where E F falls in between two consequent eigen-energies and no states are available [11, 30].It is usual to define the filling factor ν , to express the electron density in terms ofthe applied B field as, ν = 2 πl n el . Since all the states below the Fermi energy areoccupied the filling factor of the incompressible regions correspond to integer values( e.g. ν = 2 , , ... ), whereas the compressible regions have non-integer values, dueto partially occupied higher most Landau level. The spatial distribution and widthsof these regions are determined by the confinement potential [11], magnetic field [31],temperature [32] and level broadening [19, 20]. For the purpose of the present work wefix the confinement potential profile by confining ourselves to the Chklovskii geometryand keeping the donor concentration (and distribution) constant. Moreover we performour calculations at a default temperature given by k B T /E F = 0 .
02, where E F is the isorder in screening theory Figure 4.
The Hall resistances versus magnetic field, calculated at default temperatureand considering a 10 µ m sample for different ranges of the single impurity potential.Inset depicts a larger B field interval, where ν = 4 plateau can also be observed. Fermi energy calculated for the electron concentration at the center of the sample andis typically similar to 10 meV.The next step is to calculate the global resistances, i.e. the longitudinal R L and Hall R H resistances, starting from the local conductivity tensor elements. Such a calculationis done within a relaxed local model that relates the current densities j ( x, y ) to theelectric fields E ( x, y ), namely the local Ohm’s law: j ( x, y ) = ˆ σ ( x, y ) E ( x, y ) . (5)The strict locality of the conductivity model is lifted by an spatial averaging process [20]over the quantum mechanical length scales and an averaged conductivity tensor ˆ σ ( x, y )is used to obtain the global resistances. It should be emphasized that, such an averagingprocess also simulates the quantum mechanical effects on the electrostatic quantities. Tobe explicit: if the widths of the current carrying incompressible strips become narrowerthan the extend of the wave functions, these strips become “leaky” which can notdecouple the two sides of the Hall bar and back-scattering takes place. Therefore, tosimulate the “leakiness” of the incompressible strips we perform coarse-graining overquantum mechanical length scales.Now let us relate the local conductivities with the local filling factors. Since thecompressible regions behave like a metal within these regions there is finite scattering isorder in screening theory d l = 70 nm R g = 10 nm 20 nm 40 nm 80 nm2d= 2 µ m 0.120 0.120 0.100 0.0503 µ m 0.135 0.125 0.090 0.0355 µ m 0.140 0.115 0.070 0.0208 µ m 0.135 0.095 0.050 0.01010 µ m 0.130 0.085 0.040 0.010leading to finite conductivity. In contrast, within the incompressible regions theback-scattering is absent, hence, the longitudinal conductivity (and simultaneouslyresistivity) vanishes. Therefore, all the imposed current is confined to these regions.The Hall conductivity, meanwhile is just proportional to the local electron density.The explicit forms of the conductivity tensor elements are presented elsewhere [20].Having the electron density and local magneto-transport coefficients at hand, we performcalculations to obtain the widths of the quantized Hall plateaus utilizing the abovedescribed, microscopic model assisted by the local Ohm’s law at a fixed external current I . Further details of the calculation scheme is reviewed in Ref. [23]. Since the very early days of the charge transport theory, collisions played an importantrole. Such a scattering based definition of conduction also applies for the system athand, i.e. a two-dimensional electron gas subject to perpendicular magnetic field.Among many other approaches [33, 19, 27] the SCBA emerged as a reasonable model todescribe the DOS assuming Gaussian impurities, considering short range scattering. Asingle impurity has two distinct parameters that represents the properties of the resultingpotential, the range R g (at the order of separation thickness) and the amplitude of thepotential (in relevant units), (cid:101) V imp . However, these two parameters are not enough todefine the widths of the Landau levels (Γ), another important parameter is the numberof the impurities, N I . In the previous section we have already investigated these threeparameters in scope of potential landscape, now we utilize our findings to define thelevel widths and the conductivities. It is more convenient to write the single impuritypotential of the form, V g ( r ) = (cid:101) V imp πR exp ( − r R ) . (6)Together with the impurity concentration, the relaxation time is defined as τ = (cid:126) N I (cid:101) V m ∗ and in the limit of delta impurities ( i.e. R g →
0) the Landau level width Γ takes theform Γ = (cid:113) N I (cid:101) V πl . It is useful to define the impurity strength parameter to investigatethe effect of disorder by γ I = (Γ / (cid:126) ω c ) = 2 N I (cid:101) V mπ (cid:126) ω c , (7) isorder in screening theory d l = 150 nm R g = 10 nm 20 nm 40 nm 80 nm2d= 2 µ m 0.140 0.140 0.125 0.0753 µ m 0.160 0.150 0.120 0.0555 µ m 0.180 0.150 0.095 0.0358 µ m 0.180 0.130 0.070 0.02010 µ m 0.175 0.120 0.060 0.015 Table 1.
The ν = 2 plateau widths obtained at default temperature for two depletionlengths d l (left 75 nm, right 150 nm), while γ I = 0 .
05 is fixed (defined in Eq. 7 andthe related text below). The widths are given in units of (cid:126) ω c /E F = Ω c /E F . given in units of magnetic energy (cid:126) ω c = (cid:126) eBm = Ω c and as a normalization parameter wefix the magnetic energy at 10 T.At this point we would like to make a remark on the concepts short/long rangeimpurities and short/long range potential fluctuations, which is commonly mixed. Byshort range impurity potential we mean that R g (cid:46) l , however, by short range potentialfluctuation a length scale of the order of 200 −
300 nm is meant. The long rangeimpurity potential corresponds to R g > l and long range potential fluctuation is ofthe order of micrometers. Thus, when considering short range impurities the potentialfluctuations may be long range, if N I is not large ( < %5 of the donor concentration).We have also observed that, the long-range potential fluctuations are more efficientlyscreened by the 2DES and their range can be at the order of 500 nm at most, whenassuming large impurity concentration, i.e. N I > %10. In light of the above findingsand formulation we now investigate the widths of the quantized Hall plateaus. Figure 4presents the calculated Hall resistances at a fixed temperature for typical single impurityranges. We observe that, when increasing R g the plateau widths remain approximatelythe same, with a small variation, which is in contrast to the experimental findings, i.e. if the system is low mobility (small R g ⇒ highly broadened DOS) the plateauare larger. In fact changing R g from 10 nm to 20 nm should increase the zero B field mobility almost an order of magnitude, when fixing the other parameters (see e.g table I of Ref. [20]). The contradicting behavior is due to the fact that the levelsbecome broader when increasing the single impurity range, therefore the incompressiblestrips become narrower, which results in a narrower plateau. However, the long rangepotential fluctuations are completely neglected, therefore the effect(s) of disorder onthe quantized Hall plateaus cannot be described in a complete manner. To investigatethe effect of the single impurity range we systematically calculated the plateau widths;table 1 depicts the calculated widths of the Hall plateaus considering different samplewidths, depletion lengths, filling factors and R g . One sees that the plateau widths areaffected by the increase of impurity range, however, in a completely wrong direction, i.e. plateaus become narrower when decreasing the mobility. As we show in the next section,it is not sufficient to describe mobility only considering the range of a single impurity.Moreover, we also show that the other two parameters defining B = 0 mobility are either isorder in screening theory Figure 5.
The calculated Hall resistances at default temperature assuming a 5 µ msample considering three characteristic value of broadening parameter. The lowestmobility ( γ I =0.3) shows the narrowest plateau. not important or behaves in the opposite direction when calculating the resistances.Next we investigate the effect of the remaining two parameters, (cid:101) V imp and N I .However, these two parameters both effect the level width simultaneously, thereby thewidths of the incompressible strips. Hence, one cannot to distinguish their influenceon the QHPs separately. Typical Hall resistances are shown in Fig. 5 calculated atdefault temperature considering different impurity parameters. Similar to the rangeparameter, we observe that the plateau widths become narrower when the mobility islow, which also points that our single particle based level broadening calculations arenot in the correct direction. Such a behavior is easy to understand, when we decreasethe mobility either by increasing the impurity concentration or by the amplitude of theimpurity potential, the Landau levels become broader due to collisions. This means that,both the energetic and spatial gap between two consequent levels is reduced, hence theresulting incompressible strips are also narrower and fragile even at low temperatures. Adetailed investigation on the incompressible widths depending on impurity parametersare reported in Ref. [19]. It is known that if there exists an incompressible strip widerthan the Fermi wavelength the system is in the quantized Hall regime [20], therefore, ifthe gap is reduced the incompressible strips are smeared, thus the quantized Hall plateauvanish. As a general remark on the single particle theories, we should note that such a isorder in screening theory i.e. narrowand high mobility samples. Moreover, the universal behavior of the localization lengthdictated by these theories fail [40]. An explicit treatment of the activation energy [41]and critical exponents are left to an other publication. Another important parameter in defining the plateau widths is the depletion length d l . The slope of the confinement potential close to the edges essentially determines thewidths of the incompressible strips [11], which in turn determines the plateau widths.In Fig. 6 we show the ν =2 plateau calculated for two different depletion lengths, wesee that for the larger depletion the plateau is more extended. Since, the larger thedepletion is, the smoother the electron density is. Therefore, resulting incompressiblestrips are wider, hence the plateau. Such an argument will fail if one considers ahighly disordered large sample, which we discuss in Sec. 3.3. Next, we compare theplateau widths of different sample sizes while keeping constant the disorder parametersand depletion length. Figure 6 depicts the sample size dependency of ν = 2 plateauwidth. It is seen that the larger samples present wider plateaus, if the magnetic fieldis normalized with the center Fermi energy, E F . One can understand this by similararguments given above, i.e. if the sample is narrow the variation of the confinementpotential is stronger, therefore the incompressible strips become narrower, hence, theplateaus. The discrepancy between the experimental results and the screening theoryof the IQHE is solved if one considers not only the single impurity potentials but alsothe overall disorder potential landscape generated by the impurities. In the next partof this section, we investigate the effect of the long range potential fluctuations on thequantized Hall plateaus and find that, when the mobility is reduced the plateaus becomewider and stabile, as it is observed in many experiments, (see e.g. Refs. [42, 15, 16]).
So far we have investigated the effect of single impurity potentials on the overallpotential landscape in Sec. 2.2 and on the widths of the plateaus in Sec. 3.1. Wehave seen that, at high impurity concentration the overall potential fluctuates over alength scale of couple of hundred nanometers, whereas for low N I concentration suchlength scale can be as large as micrometers. Now we include the effect of this long rangepotential fluctuations into our screening calculations via modulation potential defined isorder in screening theory Figure 6. a) The calculated Hall resistances at a large B interval at defaulttemperature, setting 2 d = 5 µ m, R g = 20 nm and γ I = 0 .
05, while changing thedepletion length. It is clearly seen that depletion length is much more importantthan the single impurity parameters in determining the plateau widths. (b) Thedirect comparison of the plateau widths considering different sample sizes. Theimpurity parameters and depletion lengths are kept constant. Calculations are doneat k B T /E F = 0 .
02, whereas the donor density is 4 × m − for all sample sizes. isorder in screening theory Figure 7.
Self-consistently obtained Hall resistances for a modulated systemconsidering a sample of 3 µ m. The depletion lengths and other single impurityparameters are kept fixed, whereas the parity of the modulation period is set 5. as V mod ( x ) = V cos ( πxm p d ) where, the modulation period m p , is chosen such that theboundary conditions are preserved. At the moment, we consider two modulation periodsregardless of the sample width and vary the amplitude of the modulation potential. Inthe next section, however, we select these two parameters from our estimations obtainedin Sec. 2 and Sec. 2.2.Figure 7 depicts the self-consistently calculated Hall resistances, consideringdifferent modulation amplitudes V for a fixed sample width (2 d = 3 µ m) and m p = 5.We observe that, the plateaus become wider from the high B field side, when V is increased, i.e mobility is reduced. Such a behavior is now consistent with theexperimental findings. Since the QHPs occur whenever an incompressible strip isformed (somewhere) in the sample and the modulation forces the 2DES to form anincompressible strip at a higher magnetic field, therefore the plateau is also extendedup to higher field compared with the (approximately) non-modulated calculation, V /E F < . ≈ %1the long range part of the potential fluctuation can be approximated to 900 nm.However, note that the amplitude of this fluctuation varies between %5 −
25 of theFermi energy, considering different separation thicknesses, therefore the wafer changesfrom low mobility to intermediate one. Another important parameter is the number ofmodulations within the system: a sample with an extend of 2 µ m and V /E F = 0 . isorder in screening theory m p (10 µ m) m p (2 µ m) V /E F low 19-20 5-6 0.5intermediate 1 9-10 2-3 0.5intermediate 2 19-20 5-6 0.05high 9-10 2-3 0.05 Table 2.
A qualitative comparison of the mobility in the presence of magnetic fieldalso taking into account self-consistent screening. Mobility also depends on the size ofthe sample when screening is also considered. is a high mobility sample with the same m p (only 2 maximum), however, sample witha width of 10 µ m is low mobility (10 maximum). In the next section we study theplateau widths of different mobility samples, while keeping constant the extend and theamplitude of long range potential fluctuations ( i.e. V and m p ) and short range impurityparameters ( (cid:101) V imp , N I and R g ) under experimental conditions.
4. Discussion:Comparison with the experiments
In this final section, we harvest our findings of the previous sections to make quantitativeestimations of the plateau widths, considering narrow gate defined samples. Our aim isto show the qualitative and quantitative differences between “high” and “low” mobilitysamples, by taking into account properties of the single impurity potentials and theresulting disorder potential. The experimental realizations of these samples are reportedin the literature [15, 16]. We estimated in Sec. 2.2 that, the range of the potentialfluctuations is (cid:46)
500 nm for low mobility ( N I > (cid:38) µ m at high mobility.Therefore, the modulation period is chosen such that many oscillations correspond tolow mobility, and few oscillations correspond high mobility. As an specific example letus consider a 10 µ m sample, for the low mobility we choose m p = 19 −
20 and forthe high mobility m p is taken as 9 or 10. The amplitude of the disorder potential isdamped to %50 of the Fermi energy when considering the effect of spacer thickness,however, including screening this amplitude is further reduced to few percents. Inlight of this estimations the low mobility will be presented by a modulation amplitudeof V /E F = 0 .
5, whereas high mobility corresponds to V /E F = 0 .
05. Therefore, wehave 4 different combinations of the disorder potential parameters yielding four differentmobilities considering two sample widths, as tabulated in table 2. The second importantaspect of the disorder is the single impurity parameters, for low mobility set we choose R g = 20 nm and γ I = 0 .
3, whereas for high mobility R g = 10 nm and γ I = 0 .
05 isset. Remember that, the range of the single impurity is much less important than γ I indetermining the plateau width (see sec. 3.1).Figure 8 summarizes our results considering above discussed mobility regimes fortwo different sample widths. In Fig. 8a, we show the calculated Hall resistances for asample of 10 microns with the highest mobility (solid (black) line) and intermediate isorder in screening theory
171 mobility (broken (red) line). The solid line is the highest mobility since the rangeof the fluctuations are at the order of 1 µ m and the amplitude of the modulationpotential is five percent of the Fermi energy. The broken line presents the intermediatemobility considering a modulation amplitude of fifty percent. We observe that thelower mobility wafer presents a larger quantized Hall plateau, which is now in completeagreement with the experimental results. Moreover, our calculation scheme is free oflocalization assumptions in contrast to the known literature and we only considered avery limited level broadening, i.e. γ I = 0 .
05. In fact our results also hold for Dirac-deltaLandau levels, however, for the sake of consistency we choose the broadening parametersaccording to the selected disorder parameters. In Fig. 8c, we show two curves for evenlower mobilities, the solid line corresponds to the intermediate 2 case, whereas thebroken line is the lowest mobility considered here. The potential fluctuation range ( i.e. the modulation period) is chosen to present the low mobility wafer. We again see thatfor the lowest mobility the quantized Hall plateau is enlarged considerably from bothedges of the plateau. These results explicitly show that the quantized Hall plateausbecome broader if one strongly modulates the electronic system by long range potentialfluctuations, either by changing the range or the amplitude of the modulation. Similarresults are also obtained for a relatively narrower sample 2 d = 3 µ m, Fig. 8b and 8d,however, we see that decreasing the range of the potential fluctuation is more efficientin enlarging the quantized Hall plateaus when compared to the effect of the amplitudeof the modulation.The last interesting investigation is on the parity of the modulation period, i.e. whether m p is odd or even. Figure 9 presents the different behavior when consideringeven (a) or odd (b) periods. Here, all the disorder parameters are kept fixed, other thanthe parity. We see that for the even parity the plateau is shifted towards the high fieldedge, both for ν = 2 and 4, whereas for the odd parity the plateau is enlarged fromboth sides. This tendency is also observed for the larger sample (not shown here). Weattribute this behavior again to the formation of the incompressible strips, however, thistime only to the one residing at the center of the sample, i.e. the bulk incompressiblestrip. The picture is as follows: If the maxima of the modulation potential is at thecenter of the sample, the incompressible strip is formed at a higher magnetic field value,whereas, the edge incompressible strips become narrower at the lower field side. Hence,due to the larger incompressible strip at the bulk of the sample the plateau is shifted tothe higher field, in contrast, due to the narrower (compared to the unmodulated system)edge strips the plateau is cut off at higher fields. Since, the edge incompressible stripbecomes narrower than the extend of the wave function. For the odd parity, the edgeincompressible strips become wider, therefore, the plateau extends to the lower B fields.The enhancement at the high field edge results from the two maximum in the proximityof the center. For a better visualization of the incompressible strip distribution wesuggest reader to look at Fig.2 of Ref. [24] and Fig.1 of Ref. [43]. Such a shift of thequantized Hall plateaus is also reported in the literature [42] and is attributed to theasymmetrical density of states due to the acceptors in the system [44]. We claim that, isorder in screening theory Figure 8.
Line plots of the Hall resistance as a function of magnetic field consideringtwo sample widths (2 d = 10 µ m left panels, 2 d = 3 µ m right panels) and impurityconcentrations ( ∼ %3 (a) and (b), ∼ %20 (c) and (d)). Here the single impurityparameters are calculated from Eq. 7, otherwise other parameters are the same. the shift due to the modulation parity change observed in our calculations overlap withtheir findings. Note that in our calculations we only consider symmetric DOS, however,replacing a maxima with a minima at the confinement potential corresponds to theacceptor behavior of the dopants. A systematic experimental investigation is suggestedto understand the underlying physical mechanism, where the system is doped with smallnumber of acceptors.
5. Conclusion
In this work we tackled with the long standing and widely discussed question of the effectof disorder on the quantized Hall plateaus. The distinguishing aspect of our approachrelies on the separate treatment of the long and short range of the disorder potential. Weshow that assuming Gaussian impurities is not sufficient to describe long range potentialfluctuations, however, is adequate to give a prescription in defining the density of statesbroadening and conductivities. The discrepancy in handling the long range potentialfluctuations is cured by the inclusion of a modulation potential to the self-consistentcalculations. We estimated the range of these fluctuations from our analytical andnumerical calculations considering the effect of dielectric spacer and the screening of the isorder in screening theory Figure 9.
Even-odd parity dependency of the Hall plateaus at high impurityconcentration. (a) corresponds to a “acceptor” doped wafer, whereas in (b) the ionizedimpurities are positively charged. B = 0 and/or short range impuritydefined mobility is not adequate to describe the actual mobility at high magnetic fields,moreover, one has to include geometrical properties of the sample at hand.A natural persecutor theoretical investigation of the present work should deal withthe activated behavior of the longitudinal resistance within the screening theory. Asit is well known, the properties of the localized states, e.g . the localization length, isusually obtained from the activation experiments [45]. Moreover, spin generalization ofthe screening theory [46] is necessary to describe and investigate odd integer quantizedplateaus also considering level broadening, namely disorder. isorder in screening theory Acknowledgments
One of the authors (A.S.) would like to thank E. Ahlswede, S. C. Lok and J. Weiss for theenlightening discussions on the disorder from “an experimentalist” point of view. TheAuthors acknowledges, the Feza-Grsey Institute for supporting the III. Nano-electronicsymposium, where this work has been conducted partially and would like to acknowledgethe Scientific and Technical Research Council of Turkey (TUBITAK) for supportingunder grant no 109T083.
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