Magnon-Phonon Coupling in Metallic Films

Abstract

The method of standing spin-wave resonance, which sensitively detects periodic changes in the transverse magnetization through the film thickness, has been used to study magnetoelastic coupling in several metallic films. The effect was recorded as a function of $\ensuremath{\Delta}{\overline{H}}_{p}$, the deviation of the magnetic field for resonance of the $p\mathrm{th}$ spin-wave mode in the presence of magnetoelastic coupling from the magnetic field for resonance of the same mode in the absence of coupling. The film compositions varied from the 81% Ni-19% Fe nonmagneto-strictive alloy to a 65% Ni-35% Fe alloy. The film thicknesses were of the order of 6000 \AA{}. The frequency range of the experiment was 57 to 62 GHz. A calculation which combined the strain-wave equations with the equations of motion for the transverse magnetization $m$ allowed for the solution of $m$ as a product of a complex susceptibility $\ensuremath{\chi}$ and the magnitude of the cavity-produced rf magnetic field. In the calculation, trigonometric solutions were used for $m$ and for the strain. For the sake of simplicity, the boundary conditions were taken to be $m=0$ (spin pinning) and zero stress at the film surfaces. $\mathrm{Re}\ensuremath{\chi}$ described the dispersion, and was solved for $\ensuremath{\Delta}{\overline{H}}_{p}$. The presence of a phonon relaxation time was crucial in that it kept $\ensuremath{\Delta}{\overline{H}}_{p}$ finite. In fitting the theoretical $\ensuremath{\Delta}{\overline{H}}_{p}$ to the experimental data, values were obtained for the speed of transverse phonons (\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}${10}^{5}$ cm/sec) in the film at the microwave frequency, for the magnitude of the magnetoelastic coupling constant (\ensuremath{\approx}6\ifmmode\times\else\texttimes\fi{}${10}^{7}$ erg/${\mathrm{cm}}^{3}$) at 77\ifmmode^\circ\else\textdegree\fi{}K, and for the phenomenological phonon relaxation time (\ensuremath{\approx}1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}10}$ sec). The comparison of the calculated $\ensuremath{\Delta}{\overline{H}}_{p}$ with the data showed excellent agreement in the region near the crossover point. The agreement was less satisfactory away from the crossover point. Examination of $|\mathrm{Im}\ensuremath{\chi}|$, which is proportional to the power absorbed in the film, explains the qualitative behavior of the resonances in the crossover region, again indicating the important role played by the phonon relaxation time in the explanation of the effect of magnetoelastic coupling in metallic films.