Spin-wave non-reciprocity in magnetization-graded ferromagnetic films

A theoretical approach has been developed to study the spin-wave dynamics of magnetization-graded ferromagnetic films, where the magnetic properties change along the film thickness. The theory is based on a multilayer approach, where the influence of both long-range dipolar interactions and interlayer exchange coupling between sublayers is included. This allows for instance to describe films with a continuous variation of the saturation magnetization along the thickness. A systematic study is carried out in order to analyze different profiles of the saturation magnetization, which is checked through a test of convergence. It is found that the spin-wave dispersion is significantly modified when the strength of the magnetization changes in the bulk film, where a notable frequency non-reciprocity of two counter propagating spin waves is predicted. This is associated with heterosymmetric mode profiles and a modification of the conventional quantization condition associated to perpendicular standing spin-wave modes. Micromagnetic simulations have been carried out to validate the model, where a perfect agreement is reached between both methods. These results show that magnetization-graded ferromagnetic films can be used to channel and control spin waves, thus promoting different kinds of functionalities for magnon-based devices.