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

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.