Magnetic resonance in multilayer films

Abstract

The ferromagnetic resonance frequency for films having a nonuniform magnetization is derived. It is shown that for both normal and in-plane polarizations a film in which the exchange coupling extends throughout the whole depth resonates at the same frequency as a uniform film with a magnetization $\frac{〈M^2〉}{〈M〉}$, where $M\left(y\right)\mathrm{}$ is the depth-dependent magnetization and the angular brackets denote averages over depth. Anomalously high apparent magnetizations can result if such a film has regions of reversed magnetization.