Dynamical spin Hall conductivity in a magnetic disordered system

We investigate the intrinsic spin Hall effect in a quantum well semiconductor doped with magnetic impurities, as a means to manipulate the carriers' spin. Using a simple Hamiltonian with Rashba spin-orbit coupling and exchange interactions, we analytically compute the spin Hall conductivity. It is demonstrated that using the appropriate order of limits, one recovers the intrinsic universal value. Numerical computations on a tight-binding model, in the weak disorder regime, confirm that the spin Hall effect is preserved in the presence of magnetic impurities. The optical spin conductivity shows large sample to sample fluctuations in the low-frequency region. As a consequence, for weak disorder, the static spin conductivity is found to follow a wide Gaussian distribution with its mean value near the intrinsic clean value.