Boltzmann Equations for Spin and Charge Relaxations in Superconductors

TL;DR

This paper develops a theoretical framework using Boltzmann equations to analyze spin and charge imbalances in superconductors, revealing how these imbalances are characterized and relaxed via spin-orbit scattering.

Contribution

It extends the quasiclassical Green's function method to derive linearized Boltzmann equations for nonequilibrium quasiparticle distributions in superconductors with spin and charge imbalances.

Findings

Four distribution functions are needed to describe nonequilibrium states.

Spin imbalance causes a chemical potential shift for quasiparticles.

Spin-orbit scattering relaxes the spin imbalance.

Abstract

In a superconductor coupled with a ferromagnetic metal, spin and charge imbalances can be induced by injecting spin-polarized electron current from the ferromagnetic metal. We theoretically study a nonequilibrium distribution of quasiparticles in the presence of spin and charge imbalances. We show that four distribution functions are needed to characterize such a nonequilibrium situation, and derive a set of linearized Boltzmann equations for them by extending the argument by Schmid and Sch\"{o}n based on the quasiclassical Green's function method. Using the Boltzmann equations, we analyze the spin imbalance in a thin superconducting wire weakly coupled with a ferromagnetic electrode. The spin imbalance induces a shift $\delta\mu$ ($- \delta \mu$) of the chemical potential for up-spin (down-spin) quasiparticles. We discuss how $\delta \mu$ is relaxed by spin-orbit impurity scattering.