Effect of biquadratic coupling and in-plane anisotropy on the resonance modes of a trilayer system

Abstract

Ferromagnetic resonance modes are investigated for a trilayer system consisting of two ferromagnetic films interacting through a nonmagnetic interlayer. Included in the model are the bilinear ${J}_{1}$ and biquadratic ${J}_{2}$ couplings and in-plane uniaxial magnetocrystalline anisotropies with anisotropy axis directions in the two layers making a \ensuremath{\delta} angle. An analytical expression for the mode intensity is derived. The saturation ${(H}_{\mathrm{sat}})$ and critical ${(H}_{\mathrm{crit}})$ fields, the resonant frequency, and the mode intensity are discussed as functions of ${J}_{1},$ ${J}_{2},$ and \ensuremath{\delta}. For a given positive ${J}_{1},$ an additional ${J}_{2}{(J}_{2}<0)$ will lead to an increase (a decrease) of the optical (acoustic) mode intensity. For fixed ${J}_{1}<0,$ and if the magnetizations are parallel $(H>{H}_{\mathrm{sat}})$ only the acoustic mode will appear with constant mode position and intensity for all ${J}_{2}$ values, making it difficult to detect any additional biquadratic coupling. On the other hand, and for the same parameters, if the magnetizations are antiparallel $(H<{H}_{\mathrm{crit}}),$ then two modes are predicted; as $|{J}_{2}|$ increases the intensity of the acoustic (optical) mode will increase (decrease), while the resonant frequency of both modes decrease.