Ferromagnetic Resonance in Thin Films. III. Theory of Mode Intensities

A theory of surface-spin pinning and its effects on the ferromagnetic-resonance mode intensities is presented. The pinning by a surface inhomogeneity (e.g., a demagnetization field from surface imperfections or an inhomogeneous saturation magnetization) of thickness $\epsilon$ is considered. Roughly speaking, the modes are nearly unpinned for a thin-surface inhomogeneity (${\epsilon }^2\ll \frac{\Lambda }{\pi }$, where $\Lambda$ is the exchange constant in the exchange field $\Lambda {\nabla }^2\mathrm{M}$), while the low-order modes are pinned by a thick-surface inhomogeneity (${\epsilon }^2\ll \frac{\Lambda }{\pi }$ not satisfied). The theory indicates that the low-order modes should be pinned unless great care is exercied in the film preparation. In 80% Ni-20% Fe permalloy, ${\left(\frac{\Lambda }{\pi }\right)}^{\frac{1}{2}}\cong 90$ Å; thus, the surface region would have to be only a few lattice constants thick in order for there to be no pinning. These results are obtained by considering the equation of motion of the magnetization in the surface region as well as the bulk region. The intensities and frequencies of magnetostatic modes (negligible exchange energy) are relatively independent of surface-spin pinning, in contrast to the result for exchange modes (negligible microwave demagnetization energy) that pinning the surface spins gives rise to large intensities of even modes.