Spin-wave nonreciprocity on magnetization-graded ferromagnetic films

A theoretical approach has been developed to study the spin-wave dynamics of magnetizationgraded ferromagnetic films, where the magnetic properties change along the film thickness. The theory is based in a multilayer approach, where the influence of both long-range dipolar interactions and interlayer exchange coupling between sublayers is analytically derived, allowing for instance to describe films with continuous variation of the magnetization saturation. A systematic study is carried out in order to analyze different profiles of the magnetic properties along the thickness. It is found that the spin-wave dispersion is significantly modified when the magnetic properties change on the film's bulk, in such a way that a notable frequency nonreciprocity of two counterpropagating spin waves is predicted. Interestingly, the frequency difference exhibits a nonmonothonic behavior that can be positive or negative depending on the wave vector. This is accompanied with heterosymmetric mode profiles and a modification of the conventional quantization condition associated to the 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 channelize and control the spin waves, promoting different kinds of applications for magnon-based devices.