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 $ln\epsilon$ in the limit $\epsilon =(T-T_{\mathrm{c}})/T_{\mathrm{c}}\to +0$ in two-dimensional systems for both extrinsic and intrinsic spin Hall effects except the Maki-Thompson terms in extrinsic effect, which are proportional to ${(\epsilon -{\gamma }_{\phi })}^{-1}ln\epsilon$ with a cutoff ${\gamma }_{\phi }$ 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.

Figure 1

The diagram for the AL term in the presence of side jump.

Figure 2

The diagram for the AL term in the presence of skew scattering.

Figure 3

Diagrams for DOS term in the presence of side jump process.

Figure 4

Diagrams of DOS terms in the presence of skew scattering.

Figure 5

Diagrams of MT terms in the presence of side jump.

Figure 6

Diagrams of MT terms in the presence of skew scattering.

Figure 7

Feynman diagram for fluctuation propagator.

Figure 8

Feynman diagram for the AL term for intrinsic spin Hall effect.

Figure 9

Feynman diagram of DOS term in intrinsic spin Hall conductivity.

Figure 10

Feynman diagram of MT term in intrinsic Hall conductivity.

Figure 11

Estimate of fluctuation effect based on the parameters for Nb.

Figure 12

Estimate of fluctuation effect based on the parameters for clean Al, where skew scattering is considered to be dominant.