Spin-wave oscillations in gradient ferromagnets: Exactly solvable models

Ignatchenko, V. A.; Tsikalov, D. S. Journal Of Magnetism And Magnetic Materials. https://doi.org/10.1016/j.jmmm.2020.166643

The method of searching for the profiles of the gradient dependence of the material parameters of matter on the coordinates that allow the exact solution of wave equations, developed previously for electromagnetic and elastic waves, was generalized to spin waves in gradient ferromagnets. Such profiles were found and exact solutions of the wave equations for a ferromagnet with uniaxial magnetic anisotropy $β (z)$ or exchange $α (z)$ varying in space were obtained. The obtained solutions were used to develop the theory of spin-wave resonance in gradient thin magnetic films. The dependences of the eigenfunctions $m_{n} (z)$ , the frequencies of the discrete spectrum $ω_{n}$ , and the high-frequency susceptibility $χ_{n}$ on the number of spectral levels n were found. The cardinal differences between the spin-wave spectra of films with gradients $β (z)$ and $α (z)$ are shown. The variable anisotropy $β (z)$ changes the shape of the energy potential of the magnetic film and leads to a change in the discrete spectrum for frequencies $ω_{n} (n)$ lower than the frequency of the gradient potential well or potential barrier $ω_{c}$ . The variable exchange $α (z)$ does not change the shape of the energy potential. Spin-wave oscillations occur in a rectangular potential well created by the surfaces of the film, regardless of profile $α (z)$ . The discrete frequency spectrum $ω_{n} (n)$ is quadratic on n, or has negligible deviations from the quadratic, for all n. An analytical expression for the effective exchange parameter is obtained. Exact solutions of the Schrödinger equation with spatially dependent effective mass $m (z)$ were found for the profile of $m (z)$ inverse to the function of $α (z)$ .