Spin Hall effect and spin swapping in diffusive superconductors

Physical Review B: Condensed Matter and Materials Physics. 2017. Vol. 95. P. 054509(1)–054509(17).

A.G. Mal'shukov, Espedal C., Lange P., Sadjina S., Brataas A.

We consider the spin-orbit-induced spin Hall effect and spin swapping in diffusive superconductors.
By employing the non-equilibrium 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 superconducting correlations 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.

Keywords: superconductors, spin injection, spin Hall effect