Effect of a Surface on Magnon-Magnon Interactions in Ferromagnets

Abstract

In recent years there has been considerable interest in the surface properties of Heisenberg ferromagnets, and it has been shown that in addition to the bulk spin wave excitations at T ≪ T_c there may be surface spin waves (for reviews see [1–3]). These latter excitations are localised near the surface, having amplitudes which decay (usually exponentially) with distance from the surface. At non-zero temperatures interactions between the spin waves will occur, leading to an energy renormalisation and damping of the modes. In infinite Heisenberg ferromagnets the interactions between bulk spin waves have been treated rigorously by DYSON [4]. However, the presence of a surface is expected to give rise to richer and more complicated schemes of spin wave interactions, since scattering processes may occur involving surface spin waves as well as bulk spin waves. In this paper we use a Green’s function method to study spin wave interactions in semi-infinite Heisenberg ferromagnets at low temperatures T ≪ T_c. Specifically we deduce expressions for the energy renormalisation and damping of surface spin waves due to their interaction with either bulk spin waves or surface spin waves. The results are illustrated by numerical examples, and comparisons are made with the renormalisation for bulk spin waves.