Spin pumping in a ferromagnet/superconductor/normal-metal trilayer in presence of spin-orbit coupling and Fermi-liquid effects

The injection of spin current into superconducting (S) layer is of primary importance for superconducting spintronics. The precession of a ferromagnet magnetization close to the ferromagnetic resonance (FMR) ^pumps spins from the ferromagnet (FM) layer into adjacent layers. The injected spin current into superconducting (S) layer is expected to decrease below Tc because singlet Cooper pairs do not carry spin and the addition of a metallic layer does not change this result. Here we show that if an FM-S bilayer is capped on the superconducting side by a normal metal with strong spin-orbit coupling (Nso), as for example in a Py/Nb/Pt trilayer, then an additionnal spin current exists in the superconducting state associated with Fermi liquid corrections, and spin-triplet correlations are induced in the entire structure that are long-range within the F layer. We calculate the spin current accross a F/S/Nso structure as a function of temperature. We show that this effect is sensitive to the F/S and S/N proximity effect and depends on the S layer thickness. We discuss the magnitude of the injected spin current as function of the magnitudes of the spin-orbit coupling and of the Fermi-liquid parameters.