Damping of linear spin-wave modes in magnetic nanostructures: Local, nonlocal, and coordinate-dependent damping

Abstract

A general perturbation theory for the description of weak damping of linear spin-wave modes in magnetic nanostructures is developed. This perturbative approach allows one to account for the usual uniform Gilbert damping, as well as for the spatially nonuniform (coordinate-dependent) and nonlocal (magnetization-texture-dependent) Gilbert-like dissipation mechanisms. Using the derived general expression, it is possible to calculate the damping rate of a particular spin-wave mode if the frequency and the spatial profile of this mode, along with the relevant parameters of a magnetic material, are known. The examples demonstrating the applications of the developed general formalism include (i) generalization of the damping rate of a spin-wave mode propagating in a magnetic sample for the case of a nonuniform static magnetization or/and bias magnetic field, (ii) calculation of a damping rate of a gyrotropic mode in a vortex-state magnetic nanodot, (iii) evaluation of the spin diffusion influence on the damping rate of spin-wave modes in a conducting ferromagnet, and (iv) calculation of damping rates of spin-wave modes in a ferromagnetic film in the presence of a spin pumping into an adjacent nonmagnetic metal layer. The developed formalism is especially useful in micromagnetic simulations, as it allows one to calculate damping rates of spin-wave modes based on the numerical solution of a conservative eigenmode problem.