  In this Letter 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
$\sigma_{xy}^{SJ}/\sigma_{xy}^{SS} \sim (\hbar/\tau)/\epsilon_F$, with $\tau$
being the transport relaxation time. Motivated by recent experimental work we
apply our theory to n- and p-doped 3D and 2D GaAs structures, obtaining
$\sigma_s/\sigma_c \sim 10^{-3}-10^{-4}$ where $\sigma_{s(c)}$ is the spin Hall
(charge) conductivity, which is in reasonable agreement with the recent
experimental results of Kato \textit{et al}. [Science \textbf{306}, 1910
(2004)] in n-doped 3D GaAs system.
