Spin Hall Effect in Doped Semiconductor Structures

We present a microscopic theory of the extrinsic spin Hall effect based on the diagrammatic perturbation theory. Side-jump (SJ) and skew-scattering (SS) contributions are explicitly taken into account to calculate the spin Hall conductivity, and we show their effects scale as $\sigma_{xy}^{SJ}/\sigma_{xy}^{SS} \sim (\hbar/\tau)/\varepsilon_F$, where $\tau$ being the transport relaxation time. Motivated by recent experimental work we apply our theory to n-doped and p-doped 3D and 2D GaAs structures, obtaining analytical formulas for the SJ and SS contributions. Moreover, the ratio of the spin Hall conductivity to longitudinal conductivity is found as $\sigma_s/\sigma_c \sim 10^{-3}-10^{-4}$, in reasonable agreement with the recent experimental results of Kato \textit{et al}. [Science 306, 1910 (2004)] in n-doped 3D GaAs system.