Ferromagnetic resonance in a system composed of a ferromagnetic substrate and an exchange-coupled thin ferromagnetic overlayer

The theory of Rado and Ament and the general exchange boundary conditions of Rado and Weertman have been used to discuss the magnetic field dependence of the absorption of microwave radiation by a composite system consisting of a thin ferromagnetic overlayer exchange coupled to a thick ferromagnetic substrate. The overlayer and substrate are assumed to interact through a surface exchange interaction of the form $E_{\mathrm{ex}=\mathrm{-}\mathrm{J}{\mathrm{M}}_A\mathrm{\cdot }{\mathrm{M}}_B}$. In the limit of weak to moderate coupling strengths, the overlayer ferromagnetic resonance (FMR) is shifted by an effective field of the form ${\mathrm{JM}}_B$/d, where $M_B$ is the substrate equilibrium magnetization and d is the overlayer thickness. Pronounced effects on the strength and position in magnetic field of the substrate absorption occur when the coupling parameter J is such that the overlayer and substrate FMR’s occur at nearly the same value of applied magnetic field: The microwave absorption exhibits two peaks having comparable strengths. In the limit of very strong coupling (J∼${10}^{\mathrm{-}4}$ cm) the magnetizations in the overlayer and in the substrate precess together to yield one absorption peak at a field value which is shifted from that corresponding to the isolated substrate FMR for an unpinned surface. The shift in peak position, as well as changes in the linewidth, is caused by an effective surface pinning due to the presence of the overlayer. This pinning can be described by an effective surface energy which contains contributions from the surface pinning energy at the free surface plus contributions which are proportional to the overlayer thickness and which depend on the difference in magnetization and on the difference in volume magnetocrystalline anisotropy fields between the substrate and overlayer materials.