Edge-induced spin polarization in two-dimensional electron gas

We characterize the role of the spin-orbit coupling between electrons and the confining potential of the edge in nonequilibrium two-dimensional homogeneous electronic gas. We derive a simple analytical result for the magnitude of the current-induced spin polarization at the edge and prove that it is independent of the details of the confinement edge potential and the electronic density within realistic values of the parameters of the considered models. While the amplitude of the spin accumulation is comparable to the experimental values of extrinsic spin-Hall effect in similar samples, the spatial extent of edge-induced effect is restricted to the distances on the order of Fermi wavelength $\left(\sim 10\text{nm}\right)$.

Figure 1

(Color online) The edge’s confinement potential of depth $\Delta V$ leads via the SO coupling [Eq. (3)] to attractive potential ($V^{\text{SO},\downarrow }$, in green) for electrons with spin down in the vicinity of the edge. The initially compensated spins of electrons approaching the edge becomes locally uncompensated due to the difference in the scattering shift $l^a$.

Figure 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, exhibits Friedel-type oscillations. Its first dominant local minimum (at $x\approx 0.2$) 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 $\sim A\text{sin}\left(2k_{F}l+\varphi \right)/l$.

Figure 3

(Color online) The dependence of the self-consistent value of the potential dip $V_0$ on the electronic density shows monotonic behavior. While the value of the dip is comparable to the Fermi energy $\left(E_F=3.01\right)$, the considered values do not lead to appearance of the bound state at the edge.

Figure 4

(Color online) The difference in the spin densities for noninteracting (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 differences due to the self consistency, 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 counteracting spin-orbit coupling induced by the self-consistency correction in the region $x∊\left(0,{\lambda }_F/2\right)$.

Figure 5

(Color online) 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 $k$.