Spin generation by strong inhomogeneous electric fields

Motivated by recent experiments [1], we propose a model with extrinsic spin-orbit interaction, where an inhomogeneous electric field ${\bf E}$ in the x-y plane can give rise, through nonlinear effects, to a spin polarization with non-zero $s_z$, away from the sample boundaries. The field ${\bf E}$ induces a spin current ${\bf j}_s^z= \hat{z} \times(\alpha{\bf j}_c+\beta{\bf E})$, where ${\bf j}_c=\sigma {\bf E}$ is the charge current, and the two terms represent,respectively, the skew scattering and side-jump contributions. [2]. The coefficients $\alpha$ and $\beta$ are assumed to be $E$- independent, but conductivity $\sigma$ is field dependent. We find the spin density $s_z$ by solving the equation for spin diffusion and relaxation with a source term $\nabla \cdot {\bf j}_s^z$. For sufficiently low fields, $j_c$ is linear in $E$, and the source term vanishes, implying that $s_z=0$ away from the edges. However, for large fields, $\sigma$ varies with $E$. Solving the diffusion equation in a T-shaped geometry, where the electric current propagates along the main channel, we find spin accumulation near the entrance of the side channel, similar to experimental findings [1]. Also, we present a toy model where spin accumulation away from the boundary results from a nonlinear and anisotropic conductivity.\\{ }[1] V. Sih, et al, Phys.\ Rev.\ Lett. {\bf 97}, 096605 (2006).\\{ } [2] H.-A. Engel, B.I. Halperin, E.I.Rashba, Phys.\ Rev.\ Lett. {\bf 95}, 166605 (2005).