Dynamical spin Hall conductivity in a disordered two-dimensional system calculated in a self-consistent Born approximation

The dynamical spin Hall conductivity is calculated in a two-dimensional system with spin-orbit interaction within a self-consistent Born approximation. The wave-vector dependence of the spin splitting plays an important role in the high-frequency region corresponding to inter-spin-band transitions while being unimportant in the low-frequency region. The relaxation time characterizing the low-frequency behavior of the spin Hall conductivity is twice as large as the transport relaxation time and is independent of the electron concentration in the weak-scattering limit. The cancellation of positive interband and negative intraband contributions in the imaginary part leads to vanishing static conductivity.