Fluctuation contribution to spin Hall effect in superconductors

We theoretically study the contribution of superconducting fluctuation to extrinsic spin Hall effects in two- and three-dimensional electron gas and intrinsic spin Hall effects in two-dimensional electron gas with Rashba-type spin-orbit interaction. The Aslamazov-Larkin, density of states, and Maki-Thompson terms have logarithmic divergence lnϵ in the limit ϵ=(T-Tc)/Tc→+0 in two-dimensional systems for both extrinsic and intrinsic spin Hall effects except the Maki-Thompson terms in extrinsic effect, which are proportional to (ϵ-γφ)-1lnϵ with a cutoff γφ in two-dimensional systems. We found that the fluctuation effects on the extrinsic spin Hall effect have an opposite sign to that in the normal state and thus suppress the spin Hall conductivity. In contrast, those on the intrinsic spin Hall effect have the same sign and enhance it.