Localised modes of elementary excitations in the system of two ferromagnetic layers separated by a nonmagnetic spacer

The elementary excitations (spin waves) in the system of two ferromagnetic films of different thicknesses exchange coupled through a nonmagnetic metallic spacer are investigated. The Hamiltonian of the system consists of the exchange, Zeeman and both the surface and interface anisotropy terms. The oscillatory exchange coupling between two ferromagnetic layers resulting from the interaction via itinerant electrons is employed to investigate spin wave profiles. In presented contribution the discussion of the influence of the anisotropies at the external surfaces and at the interface is extended to the case of ferromagnetic layers of different thicknesses and interaction parameters. The solutions of a set of difference equations for the amplitudes and wave vector k lead to real (bulk modes) and imaginary (localised modes) solutions for k. The imaginary solutions represents modes localised at surface or (and) interface. The conditions necessary for appearance of both types of modes are discussed. It is shown that that the coexistence of localisation at the surface and interface depends on the ratio of thicknesses of constituent layers. The results obtained as a function of the spacer thickness are illustrated by the ferromagnetic resonance (FMR) spectra. The examples of the systems in which one type of surface anisotropy decides on the character of resonance picture are presented. Analysis of the spectra allows one to estimate the experimental error connected with the thicknesses of magnetic and nonmagnetic layers in FMR experiments.