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

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.