Intrinsic Spin Hall Effect in the $s$-Wave Superconducting State: Analysis of the Rashba Model

A general expression for spin Hall conductivity (SHC) in the $s$-wave superconducting state at finite temperatures is derived. Based on the expression, we study SHC in a two-dimensional electron gas model in the presence of Rashba spin-orbit interaction (SOI). 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 noncentrosymmetric superconductors.

Figure 1

An example of terms for ${\sigma }_{\mathrm{SH}}^{\mathrm{I}}$ that is finite under $T_c$. This term is proportional to $\Delta \propto \sqrt{1-T/T_c}$.

Figure 2

(a) $\Delta$-dependence of the SHC; comparison of I, IIa, and IIb terms. (b) $t\equiv T/T_c$ dependence of the total SHC, ${\sigma }_{\mathrm{SH}}$, for $T_c=0.2\sim 0.005$.