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 m n ( z ), the laws of the dependence of discrete frequencies ω n, and relative susceptibilities χ n / χ 1 0 of spin-wave resonances on the resonance number n are studied. It is shown that the law ω n ∝ n for n < n c, where n c 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 ω n ∝ ( 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 χ n of spin-wave resonances for n < n c is much greater than the susceptibility of resonances in a uniform film. An experimental study of both laws ω n ( n ) and χ n ( 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 ω n ( n ) and the high magnitude of the high-frequency magnetic susceptibility χ n ( n ) at n < n c make these metamaterials promising for practical applications.