Effective temperature and Gilbert damping of a current-driven localized spin

TL;DR

This paper derives a microscopic model for a localized spin driven by current, revealing how bias voltage influences damping and fluctuations, and introduces an effective temperature concept for non-equilibrium conditions.

Contribution

It provides a microscopic derivation of the Langevin and Landau-Lifschitz-Gilbert equations for current-driven spins, including bias-dependent damping and fluctuation characteristics.

Findings

Bias voltage affects Gilbert damping and fluctuation strength.

An effective temperature describes spin fluctuations at low frequencies.

Fluctuations and damping are not related by fluctuation-dissipation theorem at nonzero bias.

Abstract

Starting from a model that consists of a semiclassical spin coupled to two leads we present a microscopic derivation of the Langevin equation for the direction of the spin. For slowly-changing direction it takes on the form of the stochastic Landau-Lifschitz-Gilbert equation. We give expressions for the Gilbert damping parameter and the strength of the fluctuations, including their bias-voltage dependence. At nonzero bias-voltage the fluctuations and damping are not related by the fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible to introduce a voltage-dependent effective temperature that characterizes the fluctuations in the direction of the spin, and its transport-steady-state probability distribution function.