A model of damping due to spin–lattice interaction

Abstract

The dynamic behavior of a spin magnetic moment is often described in terms of the Landau–Lifschitz–Gilbert (LLG) equation. This contains two terms, the first describing the precession of the spin, and the second providing a damping of the precessional motion. Whereas the precession part is well understood at a fundamental level, only an intuitive knowledge of the damping term exists. The damping term represents the interaction between the spin system and the heat bath, and is included in a phenomenological way in the LLG equation. In order to understand the latter mechanism at a more basic level, we have developed a model in which the heat bath variables are introduced explicitly, with the damping introduced by a term which couples the spins to the heat bath. Specifically, we solve two sets of coupled dynamical equations, one representing the spin dynamics, and the second the underlying mechanical oscillations of the lattice. The former consists of the precessional term only, while for the latter we use a Brownian Dynamic approach, which excites the phonon modes of the lattice. The equations are solved numerically to give the time evolution of the magnetization. It is found that, although in principle the coupling mechanism does not affect the motion of the FMR ( k=0) mode, damping of this mode does appear due to non-linearities which scatter energy into magnon modes with non-zero k. We demonstrate that the model gives results similar in form to the LLG equation, i.e. a damped precessional motion of the magnetization into the local field direction. We also present the results of a study of finite size effects, which shows that the effective damping constant is dependent on the system size because of changes in the phonon spectrum.