Spin accumulation in the extrinsic spin Hall effect

The drift-diffusion formalism for spin-polarized carrier transport in semiconductors is generalized to include spin-orbit coupling. The theory is applied to treat the extrinsic spin Hall effect using realistic boundary conditions. It is shown that carrier and spin-diffusion lengths are modified by the presence of spin-orbit coupling and that spin accumulation due to the extrinsic spin Hall effect is strongly and qualitatively influenced by boundary conditions. Analytical formulas for the spin-dependent carrier recombination rates and inhomogeneous spin densities and currents are presented.

Figure 1

Calculated spin density $s$, electron density $n$, and polarization $\alpha$ for BC II. We have used $\zeta ={10}^{-4}$ and applied electric field strengths 0.125, 0.250, and $0.500\mathrm{kV}∕\mathrm{cm}$. While a signature of the extrinsic SHE is the opposite spin accumulation at the edges of the sample, spin accumulation is large also inside.

Figure 2

Calculated spin current $J_s$ in $x$ (top) and $y$ (bottom) directions for BC II and parameters as in Fig. 1.

Figure 3

Calculated spin polarization $\alpha$ at $x=0$ as a function of electric field $E$ for BC II and $\zeta ={10}^{-4}$, ${10}^{-2}$, and 1. Only in the last (unphysical) case $\alpha$ starts to saturate for large $E$, as the dependence of $L_1$ and $L_2$ on $E$ sets in.