Intrinsic Spin Hall Effect in s-wave Superconducting State: Analysis of Rashba Model

A general expression for the spin Hall conductivity (SHC) in the s-wave superconducting state at finite temperatures is derived. Based on the expression, we study the SHC in a two-dimensional electron gas model in the presence of Rashba spin-orbit interaction (SOI). The SHC is zero in the normal state, whereas it takes a large negative value as soon as the superconductivity occurs, due to the change in the quasiparticle contributions. Since this remarkable behavior is independent of the strength of the SOI, it will be widely observed in thin films of superconductors with surface-induced Rashba SOI, or in various non-centrosymmetric superconductors.