within the meso- and nano-scopic quantum transport. It gives identical result for the first order expansion in terms of $_{E}$; higher order terms in E differ by a numerical prefactor but are negligibly small for any realistic situations. Using the expression for the asymmetric spatial shift, Eq. 12, we find for the induced spin polarization linear in the drift momentum (and hence the current density) where k$_{F}$ $_{=}$√$_{2}$$_{E}$$_{F}$ is the Fermi momentum, n$_{2}$$_{D}$ = E$_{F}$ / is the density of the 2DEG. (See Fig. 5 for the explanation of the integration domain for k , q .) On the other hand, the current density is j = q$_{d}$n$_{2}$$_{D}$ so that we find the final result In a typical situation, all except the first term of this expan - sion are negligibly small so that the resulting spin polarization is independent of the height of the confining potential V and, according to the numerical results presented in the following section, practically independent of other gentle modificat ions of the model so that the first term should be useful as a general estimate of the magnitude of the current induced spin polariza- tion in heterostructures based on 2DEG. Expressing the first order result in the S.I. units we obtain the central result of our work, stated in the introduction in Eq. 1. The aim of this section is to explore the rigidity of the result in Eq. 1 with respect to changes in the model of the confining potential at the edge. We will first address the dependence of our result on finite slope of the edge potential on the distanc e d using the potential energy Qualitatively we can expect this dependence to be similarly weak as the dependence on V obtained in the previous sec- tion (Eq. 14). The reason behind this is a mutual cancellation of two effects: (1) increasing V increases the strength of the SO coupling in the Hamiltonian and hence the spin polariza- tion, (2) increasing V decreases the amplitude of the wave- functions in the region of the nonzero gradient of the confini ng edge potential and hence its sensitivity to the SO coupling. To characterize this dependence quantitatively we have evaluated the expression Eq. 7 numerically, where the spin- dependent contribution to the density, n k , q ( x ) , was obtained from the eigenfunctions of the effective 1D Hamiltonian con- taining the potential energy Eq. 15 and, according to Eq. 3, Integrating the difference in the spin densities along the whole x axis we obtain the spin-polarization per unit length of the sample. The numerical result of this integration naturally depends on the upper limit of the integration; a reliable result, is easily obtained extrapolating the upper limit to infinity(in- set of Fig. 2). Using this methodology we find the numerical For d = 0 we confirm our analytical results for abrupt step potential, Eq. 14. The difference between the spin-up and spin-down densities, shown in Fig. 2, exhibits Friedel-like oscillations with the first period having a pronounced nega- tive amplitude giving the major contribution to the non-zero spin polarization. The proportion of this amplitude to the electron is ∼ 10 − 4 for the current density j = 0 . 1. Since the current density in the experimental situation is typically j = 0 . 1 − 0 . 01 (the amplitude scales linearly with the current density for these values of j ), this value is comparable to the amplitude of the spin polarization in the extrinsic spin Hall effect$^{4}$. Unfortunately, in contract to the extrinsic spin Hall ef- fect, the nonzero spin-density is located only within few Fermi wavelengths away from the edge. from it derived spin-orbit potential energy. The eigenfunc- tions were obtained by matching the plane-waves in regions x < 0 and x > d with the Airy functions in the region of the potential energy with linear slope, 0 < x < d . The integration over k in Eq. 7 could be done analytically, whereas the inte- gral over q was done numerically on the regular mesh with N$_{k}$ points. We used model parameters that are typical for 2DEG created within GaAs quantum wells$^{2}$, electronic den- sity n = 0 . 985 corresponding to the Fermi energy E$_{F}$ = 3 . 01, the current density j = 0 . 1 and the value of the spin-orbit pa- rameter E = 5 . 53 × 10 − $^{4}$. FIG. 2: (Color online) The difference in spin densities (red-full) and the density (green-dashed) of electrons close to the edge of the sample (located at x = 0). The former, similarly to the density, ex- hibits Friedel-like oscillations. Its first dominant localminimum (at x ≈ 0 . 2) dominates the contribution to the nonzero spin-polarization obtained by integrating the difference in spin densities over whole x axis. The inset demonstrates the extrapolation of this integration with respect to its upper limit, l , using a fit to a functional form ∼ A sin ( 2 k$_{F}$l + ) / l . 4 fit numerical l [a ∗ $_{B}$] m z × 10 5 9 8 7 6 5 -5.48 -5.52 -5.56 -5.6 n$_{↑}$ ( x ) + n$_{↓}$ ( x ) n$_{↑}$ ( x ) − n$_{↓}$ ( x ) x [a ∗ $_{B}$] ( n ↑ + n ↓ ) [a.u. ∗ ] ( n ↑ − n ↓ ) × 10 4 [a.u. ∗ ] 1.2 1 0.8 0.6 0.4 0.2 0 10 8 6 4 2 0 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 m$_{z}$ = ∫ k$_{F}$ − k$_{F}$ dq 2 ∫ √ 2 E$_{F}$ − q 2 0 dk [ l q + q$_{d}$ a ] = ($_{−}$ E − 3 3 E VE$_{F}$ + 2 3$^{α}$$^{3}$ $_{E}$E 2 F ) q$_{d}$n$_{2}$$_{D}$ (13) m$_{z}$ = − E j − 3 3 E VE$_{F}$ j + 2 3$^{α}$$^{3}$ $_{E}$E 2 F j + O ( 5 $_{E}$, j $^{3}$) (14) V$_{d}$ ( x ) =    V x < 0 V − ( V / d ) x 0 < x < d 0 x > d (15) IV. EFFECTS OF THE FINITE SLOPE OF THE EDGE AND THE PARTIAL SELF-CONSISTENCY