Dipole exchange spin wave modes in multilayered ferromagnetic films*

Dipole exchange spin wave modes are studied theoretically in multilayered ferromagnetic films under different and general configurations using a method that allows to calculate them with ease. The studied geometries correspond to a couple or several ferromagnetic films separated by non magnetic layers, or to periodically repeated configurations. In general there is a magnetic field applied obliquely with respect to the planar directions, boundary conditions of different origin are considered.
We determine the frequency dispersion relations as well as the shape of the modes using an application of the Green-Extinction theorem. This method allows to write eigenvalue integral equations for the modes, written on the edges of the samples. In these multilayered geometries there is translation invariance in the planar directions, meaning that these integral equations are local in wave vector space. Thus, for example, for a couple of films there is an algebraic eigenvalue problem that involves just a 12x12 matrix.
Results for different geometries and boundary conditions are discussed, with emphasis on the effect of coupling between films that may be of dipolar or exchange origin.