Dipole-Exchange Spin Waves in Ferromagnetic Films

The effects of exchange on the magnetostatic surface and bulk spin waves of a ferromagnetic film in the regime where the dipole and exchange fields are both important are determined from the solutions of a sixth‐order differential equation. The eigenstates are admixtures of bulk and surface waves. With an applied magnetic field parallel to the film surface and for small values of the wave vector parallel to the surface, k, the Damon and Eshbach surface state is split into segments which join adjacent bulk branches. No sharp cutoff of the surface branch occurs as the angle between k and the applied field is increased, instead the character of the branch changes continuously from surface‐like to bulk‐like. For values of $\mathrm{|\hspace{0.17em}}\mathrm{k}{\text{\hspace{0.17em}|≳10}}^4$ a number of branches have significant surface character so that no single branch can be identified with the Damon‐Eshbach surface state. When the applied field is perpendicular to the surface a new type of surface spin wave occurs below the bulk manifold and is characterized by a complex wave vector normal to the surface. These surface states exist only when the dipole and exchange energies are comparable. By including microscopic perturbations in the boundary conditions it is shown that these states map smoothly into the microscopic solutions for the Heisenberg model. Detailed calculations of the dispersion curves of the bulk and surface spin‐wave frequencies and eigenvectors are presented for yttrium iron garnet for both the parallel‐ and perpendicular‐field cases.