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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, Maki-Thompson terms have logarithmic divergence $\lnε$ in the limit $ε=(T- T_{\mathrm{c}})/T_{\mathrm{c}} \rightarrow +0$ in two-dimensional systems for both extrinsic and intrinsic spin Hall effects except the Maki-Thompson terms in extrinsic effect, which are proportional to $(ε-γ_φ)^{-1}\lnε$ with a cutoff $γ_φ$ 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, while those on the intrinsic spin Hall effect have the same sign.