Spin-Hall conductivity due to Rashba spin-orbit interaction in disordered systems

We consider the spin-Hall current in a disordered two-dimenisonal electron gas in the presence of Rashba spin-orbit interaction. We derive a generalized Kubo-Greenwood formula for the spin-Hall conductivity and evaluate it in an systematic way in lowest order (Boltzmann limit) using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime the spin-Hall conductivity is universal and given by $\sigma_{yx}^z=|e|/4\pi$. Due to vertex corrections this value is twice as large as the one found for ballistic systems.