# Microscopic theory of magnon-drag electron flow in ferromagnetic metals

## TL;DR

This paper provides a microscopic theory explaining how magnons induce electron flow in ferromagnetic metals under a temperature gradient, highlighting the roles of spin transfer, momentum transfer, and spin chemical potential.

## Contribution

It presents a microscopic derivation of magnon-drag electron current using the $s$-$d$ model, connecting it to entropy and damping parameters, and compares it with previous phenomenological results.

## Key findings

- Magnon-drag current proportional to magnon entropy and ($eta - eta$) parameters.
- Almost matches previous phenomenological results, with slight differences in spin-transfer contributions.
- Interprets the effect via nonequilibrium spin chemical potential.

## Abstract

A temperature gradient applied to a ferromagnetic metal induces not only independent flows of electrons and magnons but also drag currents because of their mutual interaction. In this paper, we present a microscopic study of the electron flow induced by the drag due to magnons. The analysis is based on the $s$-$d$ model, which describes conduction electrons and magnons coupled via the $s$-$d$ exchange interaction. Magnetic impurities are introduced in the electron subsystem as a source of spin relaxation. The obtained magnon-drag electron current is proportional to the entropy of magnons and to $\alpha - \beta$ (more precisely, to $1 - \beta/\alpha$), where $\alpha$ is the Gilbert damping constant and $\beta$ is the dissipative spin-transfer torque parameter. This result almost coincides with the previous phenomenological result based on the magnonic spin-motive forces, and consists of spin-transfer and momentum-transfer contributions, but with a slight disagreement in the former. The result is interpreted in terms of the nonequilibrium spin chemical potential generated by nonequilibrium magnons.