# Spin-wave resonance in gradient ferromagnets with concave and convex variations of magnetic parameters

The theory of spin-wave resonance in gradient ferromagnetic films with magnetic parameters varying in space described by both concave and convex quadratic functions is developed. Gradient structures such as a potential well, a potential barrier, and a monotonic change in potential between the film surfaces for both quadratic functions are considered. The waveforms of oscillations mn(z), the laws of the dependence of discrete frequencies ωn, and relative susceptibilities χn=χ0 1 of spin-wave resonances on the resonance number n are studied. It is shown that the law ωn / n for n , nc, where nc is the resonance level near the upper edge of the gradient inhomogeneity, which is well known for a parabolic potential well, is also valid for the potential barrier and for the monotonic change in potential, if these structures are formed by a concave quadratic function. It is shown that the law ωn / (n 1=2)1=2 , which we numerically derived and approximated by the analytical formula, is valid for all three structures formed by a convex quadratic function. It is shown that the magnetic susceptibility χn of spin-wave resonances for n , nc is much greater than the susceptibility of resonances in a uniform film. An experimental study of both laws ωn(n) and χn(n) would allow one to determine the type of quadratic function that formed the gradient structure and the form of this structure. The possibility of creating gradient films with different laws ωn(n) and the high magnitude of the high-frequency magnetic susceptibility χn(n) at n , nc make these metamaterials promising for practical applications.