Landau–Lifshitz equations and relaxation of spin wave modes in the Heisenberg model withdipole–exchange interaction

Abstract

Landau–Lifshitz equations and spin wave damping are derived from first principles by thespin operator diagram technique for the Heisenberg model with magnetic dipole andexchange interactions. It is found that spin excitations, which are determined by poles ofeffective Green functions, are given by solutions of the linearized pseudodifferentialLandau–Lifshitz equations and the equation for the magnetostatic potential. For a normalmagnetized ferromagnetic film the spin wave damping has been calculated in the one-loopapproximation for a diagram expansion of the Green functions at low temperature. In theframework of the Heisenberg model the magnetic dipole interaction makes a majorcontribution to the long-wavelength spin wave relaxation in comparison with theexchange interaction. It is found that the damping decreases with increasing filmthickness and applied magnetic field and increases directly proportionally to thetemperature. For modes of high orders the damping is higher than for the first spin wavemode.