5 FIG. 3: (Color online) The dependence of the self-consistent value of the potential dip V$_{0}$ on the electronic density shows monotonic be- havior. While the value of the dip is comparable to the Fermi energy ( E$_{F}$ = 3 . 01), the considered values do not lead to appearance of the bound state at the edge. FIG. 4: (Color online) The difference in the spin densities for non- interacting (red-full) and the partially self-consistent (green-dashed) electrons close to the edge of the sample (located at x = 0). The spatially resolved spin density difference exhibits pronounced dif- ferences due to the selfconsistency, the most important effect being the shift of the curve closer to the edge. However, this results only in weak enhancement of the total spin polarization, m$_{z}$ , shown in the inset for an interval of electronic densities, due to counter-acting spin-orbit coupling induced by the self-consistency correction in the region x ∈ ( 0 , $_{F}$/ 2 ) . result for the spin polarization m$_{z}$ = ( 5 . 53 ± 0 . 003 ) × 10 − $^{5}$, which is in perfect agreement with our analytical expression, Eq. 14. Using this extrapolation scheme we can address the changes in the spin-polarization with parameters of the model discussed in the following. First, we confirm the observatio n that the dependence on the magnitude of the confining edge potential comes in the 3rd order of $_{E}$: the value of the slope of the polarization vs. the height of the confinement edge po- tential, V is m$_{z}$ / ( V ) ≈ 1 . 6 × 10 − 10 which is close to the analytical value 1 . 5 × 10 − $^{10}$. The difference comes primarily from the extrapolation of the value of m$_{z}$ with respect to the upper limit of integration in x . Considering nonzero values of the smearing length of the confining potential, d ∈ ( 0 , $_{F}$/ 2 ) , we find that the resulting spin polarization is changing only very little ( m$_{z}$ / d ∼ 3 . 3 × 10 − $^{8}$). The reason for this negligible dependence is, similarly to the dependence on V , the mutual cancellation of the two effects discussed earlier. The second modification to the confining potential that we consider is partial self-consistency of the edge potential that guarantees charge neutrality of the edge of the sample. Pres- ence of the model confining edge (Eq. 15) leads to redistribu- tion of the charge density characterized with excess positive charge (per unit length of the sample) which on a distance $_{F}$/ 2 exhibits small potential dip to attract more electrons. The magnitude of the dip is obtained self- consistently to achieve zero total charge at the edge, i.e. the condition N = 0. While in principle this form of the potential may support new bound states (edge states) for sufficiently large V$_{0}$ , we have checked that for the here, self-consistently found values of V$_{0}$ no such states exist. The dependence of the self-consistent value of V$_{0}$ on the typical values of the density of the 2DEG is shown in Fig. 3. Including the SO interaction, Eq. 3, on top of the potential energy V$_{n}$ ( x ) , we obtain aditional repulsive SO-induced con- tribution for the spin-down electrons due to the linearly rising potential of the dip for 0 < x < $_{F}$/ 2. This is in competition with the character of the edge-confinement potential prefer - ring the spin-down electrons. However, the dip in the potential energy also tends to move the electrons closer to the confinin g edge and, by increasing the density in this region, effectively enhances the effect of the SO coupling there. While the form of the spin-density does change significantly due to all thes e mechanisms (see Fig. 4), the overall spin polarization per unit length remains essentially unaltered over a wide range of elec- tronic densities, shown in the inset of Fig. 4. This result once again confirms the rigidity of the result given by Eq. 1. where n ( x ) is the electronic density and E$_{F}$ / is the density of the positive background charge. The Fermi energy of the electrons guarantees that for large x we have n ( x ) = E$_{F}$ / so that the system is charge neutral in the bulk even through the total charge at the edge is not necessarily zero. The behav- ior of the density at the edge of a 2D sample is in contrast with the typical situation for the edge of 3D metals where the work function and the Fermi energy, mutually comparable in value, lead to leakage of the electronic density into vacuum and thereby to an access of negative charge$^{31}$. However at the edge of a 2DEG, the Fermi energy, typically few meV is much smaller than the work function which is of order of several eV and the situation is reversed. The overall charge of the edge can be neutralized by a sim- ple model potential V$_{n}$ ( x ) =    V x < 0 − V$_{0}$ + ( 2 V$_{0}$ / $_{F}$) x 0 < x < $_{F}$/ 2 0 x > $_{F}$/ 2 , (17) N = ∫ + − dx n ( x ) = ∫ + − dx ( E$_{F}$ / − n ( x )) , (16) n$_{2}$$_{D}$ [a.u. $^{∗}$] V 0 [Ha ∗ ] 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 10 8 6 4 2 0 n$_{2}$$_{D}$ [a.u. $^{∗}$] m z × 10 5 [a.u. ∗ ] 4.5 3.5 2.5 1.5 0.5 -5.54 -5.56 -5.58 partial SCF no SCF x [a ∗ $_{B}$] ( n ↑ − n ↓ ) × 10 4 [a.u. ∗ ] 10 8 6 4 2 0 1 0.5 0 -0.5 -1 -1.5 -2