6 V. CONCLUSIONS Acknowledgments VII. APPENDIX B VI. APPENDIX A The confining potential close to the edge of a 2D electron gas together with a nonzero current along this edge induces a nonzero spin polarization that is localized within a few Fermi wavelengths from the edge. To characterize this effect quan- titatively we have derived a simple analytical formula for the spin polarization per unit length of the edge. Interestingly, the spin polarization is independent of the height of the confini ng potential as well as the electronic density of the 2DEG. Fur- thermore, using numerical calculations we have also showed that this result is independent on other possible modificati ons of the shape of the confining potential: the spatial extent ofthe confining potential and the partial selfconsistency of the c on- fining potential with respect to charge neutrality of the edg e. Expanding the last result in powers of E to 3 rd oder we obtain the expression Eq. 5. The authors wish to acknowledge fruitful discussions with J. T´obik and L. Kiˇc´ınov´a. P.B. would like to thank Rex God by for many stimulating discussions. This research has been supported by the Slovak grant agency VEGA (project No. 1/0452/09) and the NANOQUANTA EU Network of Excel- lence (NMP4-CT-2004-500198). We have stated that the result for the spin-polarization, Eq. 14, is independent of the considered non-equilibrium oc- cupations of electronic states to the first order in the the sp in- orbit coupling $_{E}$. This is true as long as the current density in the system is the same for all of these considered occupa- tions. Let us consider two simple models of occupations: (1) the “two-Fermi radii model” (2FR), frequently used within t he coherent transport with the occupancies dictated by the left and right macroscopic electrodes with Fermi momenta k R F and k L F respectively 32 attached to the sample at its ends (Fig. 5, right), and (2) the shifted Fermi distribution function (ShF), used within the main text (Fig. 5, left) From this one immedi- ately sees that the contribution to the spin polarization that is linear in E in Eq. 11, and therefore also linear in the momen- tum k$_{y}$ (= q ) , will be independent of the particular form of the occupations as long as the zero-th and the first moments are identical. Apart from this, the 2FR and ShF occupations represent two different situations: the former is suited for very short ballistic systems and it directly facilitates interpretation of the results in terms of the difference in electro-chemical potentials of the electrodes, = ( 1 / 2 ) ( k L $_{F}$) $^{2}$− ( k R $_{F}$) $^{2}$) . The latter is related to the current density inside the sample through The calculation of the spin-dependent phase-shift follows the usual textbook treatment of the wave-function matching method in 1D problems. For the separated x − dependent fac- tor of the eigenstate we need to solve the 1D Schr¨odinger equation where the ‘ + ’ sign is for = ↑ spin and the ’ − ’ sign for = ↓ spin. We are only interested in the solution well below the vacuum, i.e. e < V for which we demand asymptotic forms The reflection amplitude ( r q , ( k ) ) and the coefficient t q , ( ) are then found from the continuity of q , e ( x ) and its derivative at x = 0 with the result The sought phase shift is obtained from the reflection ampli- tude r q , e = | r q , ( k ) | e i q , k , r q , ( k ) = − ± 2 $_{E}$q V + + ik 2 $_{E}$q V + − ik (21) t q , ( ) = − 2 ik ± 2 $_{E}$q V + + ik . (22) q , k = + 2atan k ± 2 $_{E}$q V + . (23) j = ∫ dk$_{x}$dk$_{y}$ 4 2 n ( k$_{x}$ , k$_{y}$ ) k$_{y}$ = 1 3 2 ( k R $_{F}$) $^{3}$− ( k L $_{F}$) $^{3}$) (24) ≈ 1 2 √ 2 E$_{F}$ , (25) q , e ( x ) = t q , ( ) e $^{x}$, x < 0 , = √ 2 ( V − e ) (19) q , e ( x ) = e − $^{ikx}$+ r q , ( k ) e $^{ikx}$, x > 0 , k $_{=}$√$_{2}$$_{e}$ (20) ($_{−}$ 1 2 d 2 dx 2 + V ( − x ) ± $_{E}$q V ( x ) q , e ( x ) = e q , e ( x ) , (18) FIG. 5: The shifted Fermi sphere (left) and the two-Fermi radii (right) occupations in 2DEG. The occupied states are shown as filled areas, have both identical current density and electronic density but differ in higher moments of k . 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Eq. 25 gives the linear ex- pansion in the applied bias . No such simple connection to the applied bias voltage can be made for the ShF occupation. On the other hand, ShF is suitable for longer samples, where electrons’ scattering leads to partial equilibration of the dis- tribution function$^{30}$. This is then characterized with the drift