Finite-frequency spin susceptibility and spin pumping in superconductors with spin-orbit relaxation

TL;DR

This paper derives an analytical expression for the finite-frequency spin susceptibility in superconductors with spin-orbit relaxation, advancing understanding of spin dynamics and spin pumping in superconducting spintronics.

Contribution

It generalizes the static spin susceptibility to finite frequencies using the Eilenberger equation, providing a new theoretical framework for superconducting spintronics.

Findings

Derived analytical expression for finite-frequency spin susceptibility.

Analyzed the impact of spin-orbit relaxation on spin response.

Discussed implications for spin pumping and Gilbert damping in superconductors.

Abstract

Static spin susceptibility of superconductors with spin-orbit relaxation has been calculated in the seminal work of A.A. Abrikosov and L.P. Gor'kov [Sov. Phys. JETP, {\bf 15}, 752 (1962)]. Surprisingly the generalization of this result to finite frequencies has not been done despite being quite important for the modern topic of superconducting spintronics. The present paper fills this gap by deriving the analytical expression for spin susceptibility. The time-dependent spin response is shown to be captured by the quasiclassical Eilenberger equation with collision integrals corresponding to the ordinary and spin-orbit scattering. Using the developed formalism we study the linear spin pumping effect between the ferromagnet and the adjacent superconducting film. The consequences for understanding recent experiments demonstrating the modification of Gilbert damping by the superconducting correlations are discussed.