Ferromagnetic Resonance in Thin Films. I. Theory of Normal-Mode Frequencies

TLDR: In this article, a theory for the frequencies of the general ferromagnetic normal modes of a sample of arbitrary shape and size with both exchange and demagnetization energies included.

Abstract: A theory is developed for the frequencies of the general ferromagnetic normal modes of a sample of arbitrary shape and size with both exchange and demagnetization energies included. The frequencies of the modes of rectangular and circular films are calculated by casting the linearized equation of motion of the magnetization into the form of an eigenvalue equation, which is solved by a variational method. The results explain the experiments of Dillon, Besser, Sparks et al., Freedman and Brundle, and Voltmer in detail qualitatively and typically to within \ensuremath{\sim}5-10% quantitatively for the mode spacings, with possible exceptions for the first few low-order modes in some samples, for which several contributions to the line spacings are difficult to estimate accurately. Pinning the surface spins has little effect on the frequencies and intensities of magnetostatic modes (with negligible exchange energy). The theory has implications concerning the main-resonance position in finite films, and together with experiments, further verifies Portis's modespacing theory. A simple physical explanation of the results is given, and the relation of the results for for finite films to those for infinite films is given.