Spin Hall effect and spin swapping in diffusive superconductors

We consider the spin-orbit-induced spin Hall effect and spin swapping in diffusive superconductors. By employing the nonequilibrium Keldysh Green's function technique in the quasiclassical approximation, we derive coupled transport equations for the spectral spin and particle distributions and for the energy density in the elastic scattering regime. We compute four contributions to the spin Hall conductivity, namely, skew scattering, side jump, anomalous velocity, and the Yafet contribution. The reduced density of states in the superconductor causes a renormalization of the spin Hall angle. We demonstrate that all four of these contributions to the spin Hall conductivity are renormalized in the same way in the superconducting state. In its simplest manifestation, spin swapping transforms a primary spin current into a secondary spin current with swapped current and polarization directions. We find that the spin-swapping coefficient is not explicitly but only implicitly affected by the superconducting gap through the renormalized diffusion coefficients. We discuss experimental consequences for measurements of the (inverse) spin Hall effect and spin swapping in four-terminal geometries. In our geometry, below the superconducting transition temperature, the spin-swapping signal is increased an order of magnitude while changes in the (inverse) spin Hall signal are moderate.

Figure 1

Nonlocal geometries for measuring the spin Hall effect (a) and the spin swapping (b). In both cases, a particle current flowing in the left normal metal generates via the spin Hall effect a spin current that flows into the superconductor. Inside the superconductor, the inverse spin Hall effect converts the spin current into a particle current, and spin swapping swaps the spin current polarization and flow directions. In (a), the electrochemical potentials in the normal metals measure the inverse spin Hall effect. In (b), the spin-polarized contacts can be switched between a parallel and antiparallel configuration to detect the spin-swapping effect.

Figure 2

The temperature dependence of the spin-swapping signal.

Figure 3

Self-energy diagrams. (a) Self-consistent Born approximation. (b) Third-order contribution that determines skew scattering.