Ferromagnetic resonance of an heterogeneous multilayer system with interlayer exchange coupling: an accessible model

We present a general model for the coupled magnetic resonances of an exchange interacting multilayer system, which can be implemented without complex analytical calculations or numerical simulations. The model allows one to study the spin wave modes of a multilayer structure with any number of layers, accounting for individual uniaxial and cubic anisotropies, and (static and dynamic) demagnetizing and external fields as well, assuming that only the interlayer exchange coupling mechanism is relevant between such magnetic layers. This scheme is applied to recent measurements of a NiFe/CoFe bilayer, and to studying the influence of the strength of ferromagnetic and antiferromagnetic exchange interactions and the applied field orientation on the spin wave modes and intensities of the ferromagnetic resonance response. We find that the acoustic oscillation mode tends to stabilize in frequency if the magnetizations of the layers are parallel to each other, while the optical mode stabilizes when the magnetizations are antiparallel. Furthermore, we find that each oscillation mode is governed by either the NiFe or the CoFe. The modes swap the governing layer as the perpendicular field increases, inducing a gap between their frequencies, which appears to be proportional to the exchange coupling. Finally, we find that the field linewidth of the bilayer due to Gilbert damping has a dependence on the frequency very similar to the linear dependence of the linewidth in single layers. The theoretical scheme presented here can be further used to explore magnetization dynamics in different multilayer architectures-such as exchange springs, structures with perpendicular magnetic anisotropy, and complex compositions of layer stacks-and can be useful as a basis to study multilayers with chiral and dipolar interactions.