Surface properties influence on spin wave modes of ferromagnetic films

The frequencies of propagating spin waves in ferromagnetic films may be determined experimentally with some precision, and reflect the surface properties of the films. For example, if the surface properties break symmetries between both surfaces, there will be frequency non reciprocity between spin waves that travel in different directions. Thus, in order to relate films surface properties to the behavior of spin waves, we study theoretically, within the micro magnetic approximation, spin wave modes under obliquely applied magnetic fields to the film, since in this case the surface properties have a more clear influence in the spin wave properties. For this case the equilibrium magnetization is non-uniform, it deflects close to the surfaces due to effective surface boundary conditions that model local surface fields. We use the Extinction-Green theorems to determine the spin wave modes, solving numerically differential equations along the transverse direction for the so called auxiliary functions. We are aided in the obtention and interpretation of these numerical results by use of the WKB approximation and Boundary Layer methods.