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

Abstract

Landau-Lifshitz equations and spin wave damping are derived from first principles by the spin operator diagram technique for the Heisenberg model with magnetic dipole and exchange interactions. It is found that spin excitations, which are determined by poles of effective Green functions, are given by solutions of the linearized pseudodifferential Landau-Lifshitz equations and the equation for the magnetostatic potential. For a normal magnetized ferromagnetic film the spin wave damping has been calculated in the oneloop approximation for a diagram expansion of the Green functions at low temperature. In the framework of the Heisenberg model the magnetic dipole interaction makes a major contribution to the long-wavelength spin wave relaxation in comparison with the exchange interaction. It is found that the damping decreases with increasing film thickness and applied magnetic field and increases directly proportionally to the temperature. For modes of high orders the damping is higher than for the first spin wave mode.

Spin waves in nanosized magnetic films

“…Let us consider the Heisenberg model with the exchange interaction and the MDI on a crystal lattice [17,18]. The exchange interaction is short-ranged and the MDI is long-ranged.…”

“…The coefficients Q µ 1 ,...,µn can be expanded with respect to the interaction J µν ( 1 − 1 ′ ) (2) [17,18,14,15,16,28]. Each term of this expansion is represented by a diagram constructed of propagators, vertices, blocks and interaction lines.…”

“…Coefficients Q µ 1 ,...,µn in the expansion (4) in the frequency representation (6) are the sum of N topologically nontrivial diagrams N Q µ 1 ,...,µn N . The general form of the analytical expression of the diagram in the frequency representation is written as [17,18,14,15,16] Q µ 1 ,...,µn N ( 1, . .…”

“…The first approximation of the diagram expansion (4) is the self-consistent field approximation, in which the effective field acting on spins is derived and the self-consistent field H (c) µ induced by the neighboring spins is taken into account [14,17,18]. This leads to the substitution p 0 → p = βgµ B H (c) z in the propagator D − in relation (7).…”

“…The spin-wave relaxation and the spin-wave dynamics become dependent on the dimensions and shapes of ferromagnetic samples. In order to analyze the Heisenberg model with the MDI and the exchange interaction we use the spin operator diagram technique [14,15,16,17,18]. Advantages of the spin operator diagram technique are: the opportunity to calculate the spin wave damping at high temperatures and more exact relationships describing spin-wave scattering and excitations in comparison with methods based on diagram techniques for creation and annihilation magnon Bose operators [19,20,21,22,23,24,25,26,27].…”