Dynamics of magnetic moments coupled to electrons and lattice oscillations

TL;DR

This paper models the dynamics of magnetic moments in ferromagnets interacting with electrons and phonons, deriving equations of motion and analyzing fluctuation effects across temperature regimes.

Contribution

It provides a comprehensive derivation of spin dynamics equations considering electron and phonon interactions, highlighting the validity of classical equations at high temperatures and the limitations of Markovian approximations.

Findings

Correct high-temperature behavior matches Brown and Bloch equations.

Markovian approximation fails below thermal correlation time.

Fluctuation-dissipation theorem holds across temperatures.

Abstract

Inspired by the models of A. Rebei and G. J. Parker and A. Rebei et. al., we study a physical model which describes the behaviour of magnetic moments in a ferromagnet. The magnetic moments are associated to 3d electrons which interact with conduction band electrons and with phonons. We study each interaction separately and then collect the results assuming that the electron-phonon interaction can be neglected. For the case of the spin-phonon interaction, we study the derivation of the equations of motion for the classical spin vector and find that the correct behaviour, as given by the Brown equation for the spin vector and the Bloch equation, using the results obtained by D. A. Garanin for the average over fluctuations of the spin vector, can be obtained in the high temperature limit. At finite temperatures we show that the Markovian approximation for the fluctuations is not correct for time scales below some thermal correlation time $\tau_{Th}$. For the case of electrons we workout a perturbative expansion of the Feynman-Vernon functional. We find the expression for the random field correlation function. The composite model (as well as the individual models) is shown to satisfy a fluctuation-dissipation theorem for all temperature regimes if the behaviour of the coupling constants of the phonon-spin interaction remains unchanged with the temperature. The equations of motion are derived.