Spin Hall Conductivity in the impure two-dimensional Rashba s-wave superconductor

Based on the Kubo formula approach, the spin Hall conductivity (SHC) of a two-dimensional (2D) Rashba s-wave superconductor in the presence of nonmagnetic impurities is calculated. We will show that by increasing the superconducting gap, the SHC decreases monotonically to zero, while by decreasing the concentration of impurities at zero gap, the SHC closes to the clean limit universal value . As a function of the impurity relaxation rate at and (is the spin-orbit coupling in unit of), we will show that in the dirty limit () the SHC vanishes, and by increasing the relaxation time () the SHC depends on the value of superconducting gap (), is changed from zero for full gap to in zero gap. At low temperatures, the SHC goes to zero exponentially and near the critical temperature depending on the concentration of the scattering centers, the SHC will tend to the value of normal state. We will also show that the SHC is independent of spin-orbit coupling () in the clean limit.