Boundary-Value Problem for Magnetoelastic Waves in a Metallic Film

A boundary-value problem using traveling-wave magnetoelastic normal modes is solved for the case of a ferromagnetic metal film attached to a substrate. A specialization is made to certain spin-wave resonance experiments in the vicinity of 57.8 GHz, reported recently by Weber. For the dc magnetic field perpendicular to the plane of the film, the general dispersion relation for coupled electromagnetic, spin-wave, and acoustic modes is described. The effect of damping on the shape of the dispersion curves is discussed. For the choice of conductivity, the eddy-current exchange effects in the magnetoelastic crossover region are shown to be small. An expression for the power absorption by the film as a function of dc magnetic field is calculated assuming that the spins are pinned on the film surfaces. The magnetic field shift of a particular spin-wave resonance peak as a function of frequency through magnetoelastic crossover is computed. Values of the magnetoelastic constant $b$ and the phenomenological phonon relaxation time $\tau$ obtained by Weber are adjusted slightly to take into account the effect of the glass substrate: $b$ is decreased from 6.87×${10}^7$ to 6.66×${10}^7$ erg/${\mathrm{cm}}^3$, and $\tau$ is increased from 0.90×${10}^{-10\mathrm{}}$ to 1.00×${10}^{-10\mathrm{}}$ sec.