Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions

TLDR

A theory is developed to describe the dispersion of spin waves in ferromagnetic films, incorporating both dipole‑dipole and exchange interactions. The authors assume an arbitrary orientation of the internal bias magnetic field, consider mixed exchange boundary conditions (surface spin pinning), and derive simple analytical dispersion equations via classical perturbation theory. The theory shows how surface anisotropy or pinning conditions modify the spin‑wave spectrum.

Abstract

A theory is developed for dispersion characteristics of spin waves in ferromagnetic films taking into account both the dipole-dipole and the exchange interactions. An arbitrary orientation of the internal bias magnetic field is assumed. The general case of mixing exchange boundary conditions (surface spin pinning conditions) is considered. The simple analytical dispersion equations are obtained using the classical perturbation theory. The modification of the spin wave spectrum due to surface anisotropy (or pinning conditions) is discussed.