Compressibility stripes for mesoscopic quantum Hall samples

We numerically investigate the interplay of disorder and electron–electron interactions in the integer quantum Hall effect. In particular, we focus on the behaviour of the electronic compressibility as a function of magnetic field and electron density. We find manifestations of nonlinear screening and charging effects around integer filling factors, consistent with recent imaging experiments. Our calculations exhibit g-factor enhancement as well as strong overscreening in the centre of the Landau bands. Even though the critical behaviour appears mostly unaffected by interactions, important implications for the phase diagram arise. Our results are in very good agreement with the experimental findings and strongly support the relevance of electron–electron interactions for understanding integer quantum Hall physics.


Introduction
The integer quantum Hall effect (IQHE)-observed in two-dimensional electron systems (2DES) subject to a strong perpendicular magnetic field [1]-has been well explained based on singleparticle arguments [2]- [11]. Crossing the centre of each disorder-broadened Landau level, the Hall conductivity σ xy jumps by e 2 / h and the longitudinal conductivity σ xx is finite. The accompanying localization-delocalization transition is governed by a critical exponentν = 2.34 ± 0.04 [12,13], a universal quantity independent of microscopic details of electron motion, the disorder realization, and thus the chosen material. In fact, it is precisely this astonishing resilience of the quantum Hall (QH) effect that makes it an ideal metrological standard [14].
However, recent experiments on mesoscopic MOSFET devices have questioned the validity of such a simple single-particle picture. Measurements of the Hall conductance as a function of magnetic field B and gate voltage [15] exhibited regular patterns along integer filling factors. It was argued that these patterns should be attributed to Coulomb blockade effects. Similar patterns have been found recently also in measurements of the electronic compressibility κ as a function of B and electron density n e in the IQHE [16] as well as the fractional QH effect (FQHE) [17]. From these measurements it turns out that deep in the localized regime between two Landau levels, stripes of constant width with particularly small κ can be identified. These stripes consist of a collection of small-κ lines, identifiable with localized states, and their number is independent of B. This is inconsistent with a single-particle picture where one expects a fan-diagram of lines emanating from (0, 0) in the (B, n e )-plane. The authors argue that their results may be explained qualitatively by nonlinear screening of the impurity charge density at the Landau level band edges. Clearly, such screening effects-explained within a Thomas-Fermi approach [16]-are beyond the scope of a non-interacting theory. This immediately raises a question on the status of the aforementioned universality of the QH transition [15,18,19], which was obtained largely within a single-particle approach [9,12], [20]- [25]. Indeed a recently developed theory of the IQHE for short-ranged disorder and Coulomb interactions shows that universality is retained due to F invariance [26]- [28].
Further qualitative support for the screening arguments where recently given in [29,30], where a Hartree-Fock (HF) approach lead to a non-fan structure for κ in the (B, n e ) diagram; a further Thomas-Fermi-type investigation showed that charging lines with the desired slopes can indeed be found within a simple model for the impurity potentials [31].
In the present paper, we quantitatively investigate the effects of Coulomb interactions on the compressibility in the (B, n e )-plane within a HF approach. HF accounts for Thomas-Fermi screening effects while at the same time leading to a critical exponent ν whose value is consistent with the results of the non-interacting approaches [32,33]. We find that the observed charging lines in the compressibility can be well reproduced and that the width of each group of lines is well estimated by a force balance argument. Thus we can quantitatively explain the lines and stripes in the compressibility as a function of (B, n e ). We note that these results fully support the qualitative picture proposed in [16], [29]- [31].

Formulation of the QH model in Landau basis
In order to model a high-mobility heterostructure in the QH regime, we consider a 2DES in the (x, y)-plane subject to a perpendicular magnetic field B = B e z . The system can be described by a Hamiltonian of the form where σ = ±1 is a spin degree of freedom, V I is a smooth random potential modelling the effect of the electron-impurity interaction, V C represents the electron-electron interaction term and m * , g * , and µ B are the effective electron mass, g-factor, and Bohr magneton, respectively. In order to avoid edge effects we impose a torus geometry of size L × L onto the system [34]. The electronimpurity interaction is modelled by an electrostatic potential due to a remote impurity density separated from the plane of the 2DES by a spacer-layer of thickness d, as found for instance in modulation-doped GaAs-GaAlAs heterojunctions. Within the plane of the 2DES, this creates a random, spatially correlated potential with a typical length scale d. We use N I Gaussian-type 'impurities', randomly distributed at r s , with random strengths w s ∈ [ − W, W ], and a fixed width where q x,y = 2πj/L and j = −N φ , −N φ − 1, . . . , N φ . The areal density of impurities therefore is given by n I = N I /L 2 . The limit d → 0 yields a potential of δ-type that would be more adequate for modelling low-mobility structures. We will highlight a few differences between the two cases in the following sections. The electron-electron interaction potential has the form The parameter γ will allow us to continually adjust the interaction strength; γ = 1 corresponds to the bare Coulomb interaction. Choosing the vector potential in Landau gauge, A = Bx e y , the kinetic part of the Hamiltonian is diagonal in the Landau functions [35] ϕ n,k ( r) = 1 where n labels the Landau level index, k = 2πj/L with j = 0, . . . , N φ − 1 labels the momentum, H n (x) is the nth Hermite polynomial, and l c = √h /eB the magnetic length. These functions are extended and L-periodic in the y-direction and localized in the x-direction. For the system's many-body state, | , we use the usual ansatz [36,37] of an anti-symmetrized product of single particle wavefunctions ψ σ α ( r ) (Slater determinant), which we choose as a linear combination of Landau states with N LL being the number of Landau levels and the periodic Landau functions in order to meet the boundary conditions. The number of flux quanta piercing the 2DES of size L × L is given by N φ = L 2 /2πl 2 c , yielding a total number of M = N LL N φ states per spin direction. The filling of the system is characterized by the filling factor ν = N e /N φ , with N e being the number of electrons in the system and areal density n e = N e /L 2 . The total Landau level density is given by n 0 = eB/ h.

HF equation in the Landau basis and its numerical solution
We are left with finding the correct expansion coefficients C α,σ n,k [36]- [39]. The Hamiltonian is represented in matrix form using the periodic Landau states |nk and we have H σ n,k;n ,k = nk|H σ 2DES |n k = n + 1 2 + σg * 4 m * m e hω c δ n,n δ k,k + V n,k;n ,k + F σ n,k;n ,k , with the cyclotron energyhω c =heB/m * . The disorder matrix elements are given by V n,k;n ,k = q V I ( q )S n,k;n ,k ( q ), where mixing of Landau levels is included. The explicit form of the plane wave matrix elements S n,k;n ,k ( q ) = nk| exp(i q · r)|n k is computed in the appendix.
The elements of the Fock matrix F are F σ m,l;m ,l = σ k,n,k ,n G m,l;m ,l n,k;n ,k − δ σ,σ G m,l;n ,k n,k;m ,l D σ n ,k ;n,k , where the first term is the Hartree and the second the Fock contribution. The bielectronic integrals G m,l;m ,l n,k;n ,k = q =0 V C ( q )S n,k;n ,k ( q )S m,l;m ,l (− q ) can be further simplified as given in the appendix. A homogeneous, positive background is assumed that neutralizes the charge of the electrons and thereby prevents the Coulomb term from diverging as | q| → 0. In fact, this interaction with the background can be shown to cancel with the term | q| = 0 in F up to a contribution of the order of L −1 due to the finite system size [40]. The density matrix is given by with Tr(D) = N e and D σ D σ = D σ . Here f( ) denotes the Fermi function. The total energy E tot in terms of the above matrices is given as 2h σ n,k;n ,k + F σ n,k;n ,k D σ n ,k ;n,k .
A variational minimization of |H 2DES | with respect to the coefficients yields the Hartree-Fock-Roothaan equation [41], a self-consistent eigenvalue problem which in compact form can be written as with C σ = ( C σ 1 , . . . , C σ M ) being the matrix of eigenvectors and E σ = diag( σ 1 , . . . , σ M ) the diagonal matrix of the eigenvalues σ Following the aufbau principle [42], the density matrix is constructed starting from the energetically lowest lying state up to the Fermi level F . In our calculations, we keep N e fixed and compute F as the energy of the highest occupied state afterwards. Since the Fock matrix depends on the density matrix, which in turn depends on the full solution of problem, (11) has to be calculated self-consistently which is numerically quite challenging. In the first step we use the solution of the non-interacting Hamiltonian h σ = nk|h σ |n k as a starting guess for the coefficients C σ . From this solution, C (0) , we construct the density and Fock matrices and finally the full Hamiltonian. Diagonalization yields an improved solution, C (1) . The process continues until convergence of the density matrix has been achieved. In practice, we compute the norm of the difference between successive density matrices ||D (n+1) − D (n) || < ε. Here || · || denotes the Hilbert-Schmidt norm defined as ||A|| = Tr(AA * ) 1/2 .
However, convergence of this Roothaan algorithm [41] is not always assured. Especially for fillings close to zero or an integer value it may run into an oscillating limit cycle. Subtracting a small multiple of the density matrix, bD σ , from the Hamiltonian H σ = h σ + F σ favours already occupied states, thereby suppressing such oscillations. The eigenvalues shift F , but eigenstates are not affected. This algorithm is known as the level shifting algorithm [43] and b is the level-shift parameter. We choose b ≈ 1.76e 2 /L which is exactly the contribution of the neglected V C ( q = 0) term that does not cancel with the background charge due to the finite system size. This prevents the occupied and unoccupied states from mixing and stabilizes the convergence of the algorithm, although at the cost of slower convergence. A further improvement of convergence is achieved by constructing the new density matrix D (n+1) as a mixture of the updated and the previous one, i.e.
We adjust λ in each step automatically such that the fastest global convergence is guaranteed [42].
In each HF step, assembling the dense Fock matrix F σ scales as O(N 4 LL N 3 φ ) and is clearly very time-consuming. An improved scheme, even though generally possible, is of little advantage since the diagonalization is of similar complexity. For the calculation of a particular disorder configuration and magnetic field, a self-consistent run has to be made for each of the M possible filling factors. Hence, the complexity of a complete HF calculation is of the order O(2KN 5 LL N 4 φ ) with K the number of iterations until convergence. The dependence on the system size is therefore O(L 8 ). For system sizes of L ∼ 300 nm, we find K ∼ 100-1000. In all results present here, convergence of the HF scheme is assumed for ε 10 −6 .

Chemical potential and electronic compressibility
The electronic compressibility κ = (∂n e /∂µ)/n 2 e reflects the ability of the 2DES to absorb electrons when changing the chemical potential.
Hence, for finite sample calculations, we can obtain κ from E tot (N e ) using Alternatively, at T = 0, we can compute the change in the chemical potential for N e electrons by noting that the Fermi energy F (N e ) = µ(N e ). Thus we immediately have This turns out to be numerically more stable than (10) and shall be used in the following. We would like to remark that for obvious reasons the measurement of the electronic compressibility is experimentally realized by changing the chemical potential instead of changing the electron number density. In our calculation, changing the number density is easier to control and therefore we make use of the inverse relation between the two functions. When the Fermi energy lies in a region of highly localized states, it takes a considerable energy to accommodate another electron, and thus the compressibility is low. On the other hand, in a region of delocalized states a newly added electron is much more easily absorbed and κ is high. For a non-interacting system, κ is proportional to the tunneling density of states (DOS) [44] and expected to exhibit a fan-like structure in the (B, n e ) diagram. In particular, the resonances in κ need not align with slopes equal to integer filling factors [45]. In the interacting case, κ is proportional to the thermodynamic DOS (TDOS) and the inverse screening length. We note that experimentally the change of the chemical potential is detected when changing the back gate voltage and hence the electron density at constant B [16,17].

Sample mobility and DOS
In order to tune our parameters for the electron-impurity interaction to the experiment, we estimate the zero field mobility, defined as µ 0 = eτ/m * , with τ being the transport scattering time [44,46]. For a short-range δ-impurity potential, τ is identical to the single-particle momentum relaxation time, τ s , which determines the level broadening, ( /2) 2 =hω ch /2πτ s [47]. For longrange potential, however, these two times can be very different [48] and knowledge about the level broadening does not necessarily imply knowledge about the mobility and vice versa. In fact, for a smooth potential with d , which does not depend on B. In that case, we can determine the mobility from the transport cross-section calculated in Born approximation [35,47]. Since the transport scattering time is momentum dependent, we take the low temperature limit, where the relevant scattering time is the one for electrons having Fermi momentum k F = (2πn e ) 1/2 . With a radially symmetric electron-impurity interaction potential (for a single scatterer), u( r ), we obtain for the transport scattering time with the Fourier transform In case of δ-interaction and uniformly distributed strengths, we simply haveũ(k) = W/ √ 3 and the cos(θ) term in (14) vanishes. Without it, (14) becomes the expression for τ s , which proves the equivalence of τ and τ s for short-range potentials. For long-range potentials, however, forward (small θ) scattering receives little weight since it hardly impairs the electron movement and τ/τ s 1. In order to model a situation comparable to the experiments of [16], we use material parameters for GaAs (g * = −0.44 for the effective g-factor, m * = 0.067m e for the effective mass) and impurity parameters of W/nm 2 4 eV with a concentration of n I = 3.2 × 10 11 cm −2 (e.g. N I = 288 for L = 300 nm or N I = 392 for L = 350 nm). Assuming n e ≈ n I , for the δ-potential this yields a mobility of µ 0 10 3 cm 2 Vs −1 , whereas for d = 40 nm (≈ spacer layer thickness) we get µ 0 10 6 cm 2 Vs −1 -a value which is reasonable for a high mobility sample such as a GaAs-GaAlAs heterojunction.
We have calculated the DOS in the lowest Landau level by averaging over at least 1000 samples in the non-interacting and the interacting system. In figure 1 we show results for Gaussian and δ-impurities for three system sizes L = 400, 500 and 600 nm for the non-interacting case as well as the interacting case. Within the accuracy of the calculation, we find the DOS to be independent of the system size, irrespective of interactions, as of course expected for our static disorder model 3 . Furthermore, in the interacting case, we find a strong suppression of the DOS at the Fermi level. The formation of this Coulomb gap and its non-criticality has been studied previously [32,33,40,50]. In case of Gaussian-type impurities we also observe a strong reduction of the band broadening due to screening of the impurity potential, while the bandwidth in systems with δ-type impurities is hardly affected by interactions.  Note the strong Coulomb reduction of the DOS at the critical energy ( α = crit ) in the interacting systems.

Scaling of the participation ratio
The participation ratio P α is defined as the inverse of the variance of the charge density in the state α, Large values of P α correspond to spatially extended states, while low values indicate a confined state [51,52]. This is intuitively understood by the fact that the density of an extended state varies much less over space than a highly localized one. Thus P α is a measure of the degree of localization 4 and may be computed in our model as  It has been shown that unscreened HF-interactions do not alter the critical exponentν while renormalizing the dynamical scaling exponent to z = 1 [32,33,54]. The former has led to the conclusion that the HF theory should be within the same universality class as the non-interacting theory. The underlying reason has been explained [55] with an additional global symmetry named F invariance which is absent in case of short-ranged interactions and non-interacting electrons. As a check to our HF results, we calculate P α of spinless electrons in the lowest Landau level for the same samples as in section 5. The participation ratio is expected to obey the single parameter scaling form [56] with the anomalous diffusion coefficient D(2) ≈ 1.6-related to the multifractal character of the critical states [57]- [59]-and the critical exponentν ≈ 2.3 [12]. Figure 2 shows the scaling function for the non-interacting and an interacting system at filling factor ν = 1/2. The scaling function collapses reasonably well onto a single curve for both, non-interacting and HF-interacting systems. We find D(2) = 1.62 ± 0.10 as typical average over both, HF-and non-interacting systems as shown in figure 3. This demonstrates the irrelevance of interactions and the type of disorder for the multifractal dimension of the critical state, in very good agreement with previous results [30,33,55,59]. We note that this result might be limited to HF interactions since very recent results [60] suggest that multifractality is absent (i.e. D(2) = 2) for spinless electrons with Coulomb interactions. A similar fit in the tails of P α is numerically less  accurate but still yields estimates forν between 2 and 2.4, compatible with the expected value 2.34 ± 0.04 [12,23].

Charging lines and stripes in the electronic compressibility
Let us first briefly recall the experimental results of [16,17] most relevant to the present investigation. The compressibilities in the (B, n e )-plane (i) exhibit only little variation in regions close to the QH transitions at half-integer filling factors, but (ii) show a strong variation between Landau bands at integer fillings which by virtue of the relation n e = νeB/ h correspond to lines of constant slope. Furthermore, (iii) these regions of strong variation seem to have a width which is B and Landau level index independent and (iv) within these stripes, thin lines of equal slope jeB/ h, j = 0, 1, . . . can be identified. In what follows, we have calculated the electronic compressibility as outlined in section 4 in the lowest two, spin-split Landau levels for a sample of size L = 300 nm at magnetic fields between B = 0.2 T and B = 6 T. and lie almost on top of each other. Hence, we only find a single, very strongly incompressible region between the first and the second orbital Landau level at ν = 2. This broad line is due to the band gap and the highly localized states at the band edges. Other less pronounced lines are visible along different filling factors, seemingly mostly emanating from (0, 0). Some lines even appear to have a varying slope as shown in the inset of figure 4. We interpret these features as the aforementioned fingerprints of scattering resonances in the disorder potential [45] which do not necessarily need to align with constant filling factors. Moreover, we clearly observe an increasing number of those lines with increasing magnetic field. At ν = 0 and 4, the compressibility is again low.

Effects of the HF interaction
We next include interaction with γ = 0.3. This is not yet the full γ = 1 Coulomb term, but the results are numerically more stable while at the same time not being dramatically different from γ = 1. Furthermore, γ < 1 is essentially equivalent to increased disorder with the full Coulomb interaction present. Figures 5-7 show results in the (B, n e )-plane for an interacting system of size L = 300 nm with disorder strengths W/d 2 = 1.25, 2.5, and 3.75 meV, respectively, at fixed impurity range d = 40 nm. We observe that the exchange interaction results in an effective g-factor substantially enhanced from its bare value [61]- [63], leading to a clear separation of the two spin bands. This yields two additional strongly incompressible stripes at ν = 1 and 3, indicated by particularly high κ −1 values. Quite different from the non-interacting case, we find that most of the incompressible lines form groups which align parallel in the (B, n e )-plane along integer filling factors. Above a certain minimal magnetic field, the width of these groups-the number of the lines-is independent of the magnetic field and Landau level, forming incompressible stripes of constant width around integer filling factors. Overall, this behaviour is strikingly similar to the effects observed in the experiments of [16,17]. Outside the stripes, there is hardly any feature in the compressibilities except directly at the QH plateau-to-plateau transitions at half-integer fillings where a small increase in compressibility is discernible. In these areas between incompressible stripes, the inverse compressibility tends to have a very low or even negative value, which relates to a very high or negative TDOS, respectively. This effect has been observed experimentally [64,65] and is a signature of the exchange interaction. From the proportionality between compressibility and the screening length, we can conclude to observe strong overscreening in the areas of negative compressibility. We attribute this to the tendency of the HF-interacting 2DES to form a charge density wave [38,66]. Furthermore, when comparing figures 5, 6, and 7 we find that the width of the incompressible stripes increases with increasing disorder strength W/d 2 .

Spatial distribution of electronic density and screening
The spatial distribution of the total electronic density is readily calculated in our model. It details the screening mechanism by providing direct insight into the interplay of disorder and interaction. Let us start at the QH transition. Figure 8 depicts the critical charge density at ν = 1/2 for a non-interacting system in units of n 0 . The contour lines show the impurity potential V I ( r ) where the critical energy V I ( r ) = F is highlighted by a thick line. The charge density evidently behaves according to the semiclassical approximation [23] and follows the equipotential lines of V I ( r ). For the interacting case, however, we expect Thomas-Fermi screening theory to apply [49], [67]- [69]. This approximation is appropriate for an impurity potential smooth on the scale of the magnetic length as well as a sufficient separation of the Landau bands, characterized by the conditionhω c / l c > |∇V I ( r )| 2 [70]. The electrostatic potential of the charge density φ( r ) = e 4π 0 d 2 r n( r ) −n | r − r| (20) and the impurity potential V I ( r ) form a screened potential V scr ( r ) = V I ( r ) + eφ( r ). Here,n accounts for the positive background. Since a flat screened potential is energetically most favourable, one expects to find V scr ( r ) = F for the case of perfect screening. However, since fluctuations of the density, δn( r ) = n( r ) −n, are restricted between an empty and a full Landau level, i.e. 0 < δn( r ) < n 0 , the screening is not always perfect but depends on the fluctuations in the impurity potential as well as on the filling factor [49], [67, [68]. The plane can be divided into fully electron or hole depleted, insulating regions-where n( r ) = 0 or n( r ) = n 0 , respectively-and metallic regions-where n( r ) lies in between. Depending on the filling factor, the extent of those regions varies. Close to the band edge, insulating regions dominate. Screening is highly nonlinear and transport virtually impossible. On the other hand, if disorder is weak enough, there exists a finite range of filling factors in the centres of each band where metallic regions cover most of the sample, percolate and render the whole system metallic. The disorder is effectively screened and transport greatly enhanced. In that case, the charge density n scr ( r ) can be obtained by Fourier transforming the screened potential. In 3D, this simply leads to the Laplace equation. For 2D, however, one obtains [71] n scr ( q ) = − 2 0 e 2 | q|V I ( q ) + νn 0 δ q,0 , where the | q| = 0 term is 'perfectly screened' by the positive background and thus does not contribute to screening of the impurity potential. In other words, in our model only the fluctuations δn( r) are essential for screening. Hence, in 2D, a perfectly screening charge density would obey Clearly, the actual charge density is expected to deviate from n scr ( r ) for several reasons. Firstly, the fluctuations of n( r ) are restricted as discussed above. Secondly, (22) is valid for the Hartree case only. Taking the Fock contribution into account will introduce short wavelength fluctuations due to the tendency for crystalization. However, we still expect the charge density to follow (22) in the limit of | q| → 0 5 . Figure 9 shows results for the charge density of interacting electrons at ν = 1/2. Broken lines indicate the regions where n scr ( r ) exceeds the range for δn( r ) either below or above, i.e. areas that cannot be screened at all and thus exhibit insulating behaviour. Otherwise, we find the charge density to follow n scr ( r ) very closely. In this regime, the density is well described by (22) and the screening is very effective. Metallic regions dominate over insulating ones and transport is expected to be good. In figure 10, we depict cross-sectional plots of n( r ) and n scr ( r ) for the sample of figure 9 at x = 100 nm and three different filling factors, demonstrating the discussed effects again very clearly.

Breakdown of linear screening
Thus far we have shown that our results can qualitatively reproduce the structures observed in the (B, n e ) plots of the compressibility. We find stripes of constant width with very similar characteristics as in the experiments. Furthermore, we show that within HF, the impurity potential in the band centre can be quite effectively screened by the charge density. Let us now turn our attention to the stripes. The screening of the impurity potential is nonlinear near the edges of the Landau bands. Most of the sample is thus covered by insulating regions where the Landau band is either completely depleted or filled. Metallic behaviour is confined to small regions around potential extrema, where electron or hole islands are formed. If additional charge is introduced into the system, charging effects will govern the spectrum. These charging effects will manifest themselves in jumps in the compressibility as a function of charge density. The simple dot model of [31] was able to account for this effect and even demonstrated that charging events will take place along lines of integer filling factor, in agreement with our calculations. An estimation for the cross-over from linear to nonlinear screening can be found by very general considerations [72]. An insulating island where n( r ) = n 0 is confined by the force of the impurity potential, ∇V I ( r ), around its edge. Thereby, the Coulomb interactions opposing this force making the edge of the full region metallic. The size of the edges, i.e. the size of the metallic region, is then determined by the Coulomb force n 0 e 2 /2π 0 . Only if the Coulomb force acquires a magnitude comparable to the typical confining potential force, the metallic edges of the full islands will connect and dominate over the insulating regions. The typical force of our impurity potential is given by |∇V I ( r )| 2 = n I w s 2 s /πd 4 . We would like to remark that with N I = 288, the expected standard deviation of w s 2 s is ∼ 2%, which makes the typical force a reliable characteristic of V I for finite sample calculations. From equating the typical force with the Coulomb force we can derive an expression for the minimal required density n 0 which corresponds to a minimum magnetic field B min = n 0 h/e below which linear screening breaks down for any density. Therefore, n 0 determines the width of the charging stripes n and we find Note that n is indeed independent of B and n e as observed in the experiments. In figures 5-7, we have indicated the breakdown of the linear screening regime by dashed white lines. The points at which the lines merge indicate B min . Evidently, (23) nicely estimates the widths of the observed stripes for all three disorder configurations used. Furthermore, we have tested the criterion for breakdown by plotting compressibilities as a function of n e and disorder strength W/d 2 . Figure 12 shows the result for W/d 2 between 1 and 3 meV at B = 3.5 T. The dashed white lines again indicate (23). In order to confirm the dependence of the stripes on the ratio between W and d 2 only, the plot has been divided into two regimes. Between 1 and 2 meV, we kept d = 40 nm as a constant and varied W , and between 2 and 3 meV we kept Wnm −2 = 3.2 meV constant while varying d, accordingly. The results confirm (23). Deviations from the expected behaviour can be Recently, a numerical study on compressible edge channels in split-gate quantum wires was presented [73]. Although edges are absent in our model, extrema in the impurity potential might lead to an edge-like situation in their vicinity. For the quantum wires an increase in the width of the compressible channels upon increasing the magnetic field has been reported. This is in good agreement with our findings for figures 5-7.

Conclusions
We have investigated numerically how electron-electron interactions affect the localization properties of a 2DES under influence of a strong perpendicular magnetic field. We therefore diagonalized the Hamiltonian in a suitable basis and treated interactions as an effective mean field.
Our calculations reveal substantial differences in the electronic compressibility between non-interacting and interacting systems when viewed as a function of magnetic field and carrier density. For interacting systems, we find strongly incompressible stripes of constant width around integer filling factors. We show the dependence of the width of the stripes on the disorder configuration and compute the width based on a force balance argument. These results are in very good agreement with recent imaging experiments. Moreover, we find strong g-factor enhancement as well as negative compressibility in the band centres, also consistent with experiments. We demonstrate that the incompressible patterns can be attributed to nonlinear screening effects in the tails of the Landau bands. For magnetic fields larger than B min , the effects of linear screening-and hence interactions-dominate in the (B, n e )-plane. Thus, our results support the existence of a greater variety of transport regimes due to electron-electron interactions in the IQHE.
Similar compressibility patterns have also been observed around fractional filling factors ν = 1/3, 2/5 and 2/3 [17]. Energy gaps at fractional filling, e.g. ν = p/(2p + 1), with p being an integer, are a consequence of electron correlations which are absent in the HF approximation. However, with the formal analogy [74] between IQHE and FQHE put forward by the composite fermion (CF) model [75,76], let us venture a few statements about compressibility patterns around those fractional fillings. It is argued that the FQHE can be regarded as a manifestation of the IQHE for CFs in an effective magnetic field B * = B ν − B ν=1/2 [77,78]. If we pretend to have obtained results for CFs in an (B * , n CF )-plane, a transformation back to electrons yields an increase in the density of charging lines (per n e ) by a factor of 2p + 1. Indeed, in the above mentioned experiment an increase of 3 has been found for ν = 1/3. Furthermore, such a transformation predicts a dependence of the width of the incompressible stripes on the filling factor as well as a strong increase of B min when fractional filling factors approach ν = 1/2. This remains yet to be explored.