Angular dependence of ferromagnetic resonance in exchange-coupled Co/Ru/Co trilayer structures.

Abstract

The angular dependence of the dispersion relation is calculated for a system consisting of two ferromagnetic layers exchange coupled through a nonmagnetic spacer layer. Special attention has been focused on the resonance behavior in the unsaturated state of an antiparallel coupled system. The variation of both the acoustic mode and the optic mode with the external-field orientation is significantly modified from that in a noncoupled system and can be used to accurately evaluate the interlayer exchange-coupling strength A(${\mathrm{\ensuremath{\theta}}}_{12}$) as a function of the angle between the magnetization vectors in the two magnetic layers. Based on the exchange-coupled resonance theory, the angular dependence of ferromagnetic resonance (FMR) measurements has been performed on several series of symmetrical and asymmetrical Co/Ru/Co structures at X-band and K-band frequencies with the temperature ranging from 10 to 300 K. Only the bilinear exchange-coupling coefficient ${\mathit{A}}_{12}$ was observed in these systems. The biquadratic contribution is more than two orders of magnitude smaller than ${\mathit{A}}_{12}$. For the symmetrical Co(32 \AA{})/Ru(${\mathit{t}}_{\mathrm{Ru}}$)/Co(32 \AA{}) series, oscillatory interlayer exchange coupling was observed as a function of the Ru thickness ${\mathit{t}}_{\mathrm{Ru}}$. The oscillation period (\ensuremath{\sim}12 \AA{}) and phase do not vary with temperature. However, the oscillation amplitude is significantly enhanced at low temperatures, following roughly the relationship ${\mathit{A}}_{12}$\ensuremath{\propto}(T/${\mathit{T}}_{0}$)/sinh(T/${\mathit{T}}_{0}$) predicted by the theoretical models. For the asymmetrical Co(32 \AA{})/Ru(${\mathit{t}}_{\mathrm{Ru}}$)/Co(${\mathit{t}}_{2}$) structures, variation of the exchange coupling strength as a function of ${\mathit{t}}_{2}$ has also been observed for several series within which ${\mathit{t}}_{\mathrm{Ru}}$ is constant. The variation length \ensuremath{\Delta}${\mathit{t}}_{2}$ between maximum and minimum coupling strength is rather large (about 10 \AA{}) and consistent from series to series.