Magnetoexchange Branches and Spin-Wave Resonance in Conducting and Insulating Films: Perpendicular Resonance

Abstract

Spin-wave modes in thin ferromagnetic metallic plates or films are characterized by a wave vector $k_{\parallel }$ parallel to the sample surface in addition to the standing spin-wave mode number $n$ which designates the number of half-wavelengths across the sample. The dependence of the spin-wave frequency on $k_{\parallel }$ is principally due to dipolar effects similar to those encountered in magnetostatic theory. Analytic expressions for the frequencies ${\Omega }_n\left(k_{\parallel }\right)$ of the spin-wave modes including the effects of exchange, dipolar interactions, eddy currents, and phenomenological relaxation are derived for the case in which the applied magnetic field is perpendicular to the film surface and the spins are pinned. The dispersion branches ${\Omega }_n\left(k_{\parallel }\right)$ (for fixed $n$) are referred to as magnetoexchange branches since they are both magnetostatic and exchangelike in nature. The magnetoexchange dispersion for an insulating sample is also obtained and shown to be quite different from that of the metallic case. A general theory for the surface impedance $Z(\omega ,k_{\parallel })$ is developed which reduces to the usual surface impedance when $k_{\parallel }\to 0$. Simple analytical expressions for the spin-wave resonance power absorption peaks of thin metallic films are obtained. The intensity $I_n$ of the spin-wave peak absorption is shown to be independent of $n$ in the absence of phenomenological relaxation. For nonvanishing relaxation, $I_n\propto \frac{1}{n^2}$. The "peak-to-valley" difference of the derivative $\frac{\partial I_n}{\partial H_{\mathrm{app}}}$ behaves in the same way. Small variations in film thickness lead to the result that $\frac{\partial I_n}{\partial H_{\mathrm{app}}}\propto \frac{1}{n^4}$ instead of $\frac{1}{n^2}$. Simple analytical expressions are also obtained for the surface impedance of thick films. The ferromagnetic resonance (FMR) peak is shifted upward from the magnetostatic result $\Omega ={\Omega }_H$. Expressions are given for the dipolar dispersion of the FMR peak as a function of $k_{\parallel }$. The curve is linear for small $k_{\parallel }$ with a slope much smaller than that of an equivalent magnetostatic branch in an insulator. The power absorption associated with the right-hand circularly polarized $h_{\mathrm{rf}}$ component has a deep minimum at $\omega =\gamma H_{\mathrm{app}}$ which is associated with the phenomena of transmission resonance in metallic films.