Spin Hall Edge Spin Polarization in a Ballistic 2D Electron System

Universal properties of the spin Hall effect in ballistic 2D electron systems are addressed. The net spin polarization across the edge of the conductor is second order, $\sim {\lambda }^2$, in spin-orbit coupling constant independent of the form of the boundary potential, with the contributions of normal and evanescent modes each being $\sim \sqrt{\lambda }$ but of opposite signs. This general result is confirmed by the analytical solution for a hard-wall boundary, which also yields the detailed distribution of the local spin polarization. The latter shows fast (Friedel) oscillations with the spin-orbit coupling entering via the period of slow beatings only. Long-wavelength contributions of evanescent and normal modes exactly cancel each other in the spectral distribution of the local spin density.

Figure 1

Geometry of the system. Ideal leads filled by equilibrium electrons up to the chemical potentials shifted by the applied bias. The edge is formed by a confining potential $U(x)$ vanishing for $x\to \infty$.

Figure 2

Dependence of the local spin polarization (16), in units of $eVm/8{\pi }^2$, on the distance to the boundary for different values of spin-orbit coupling constant. Solid (red) line: $\lambda /v_F=0.1$, dotted (blue) line: $\lambda /v_F=0.2$, solid (black) line utilizes the approximate formula (2) for $\lambda /v_F=0.2$. The plot of Eq. (17) is indistinguishable from the exact Eq. (16) on this scale.

Figure 3

Spectral distribution (17) of spin density in units of $eV/4\pi v_F$ for different values of spin-orbit coupling constant. Solid (red) line: $\lambda /v_F=0.1$, Solid (black) line: $\lambda /v_F=0.2$. Dotted (blue) line shows Fourier transform of the exact Eq. (16) for $\lambda /v_F=0.2$.