Spin-Wave Resonance in Films Magnetized Parallel to the Surface

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Abstract
A theoretical analysis is presented for spin‐wave resonance in thin ferromagnetic films magnetized in the plane of the film by a dc field. It is shown that, at very high frequencies, a spin‐wave resonance spectrum occurs if a suitable surface anisotropy is present. The analysis is based on the Landau—Lifshitz equation and Maxwell's equations. Both surfaces of the film are assumed to possess uniaxial surface anisotropies of equal magnitude with the axes normal to the plane of the film. Magnetization precession is excited by a weak rf field applied in the plane of the film but perpendicular to the dc field and equal at both surfaces of the film. The boundary conditions for the magnetization1 are different for the oscillating component Mn normal to the film and the component Mp in the plane of the film. The component Mn is influenced by the surface anisotropy and, therefore, more or less pinned at the surfaces. However, the surface anisotropy has no influence on the component Mp, and accordingly, the derivative of Mp with respect to the normal direction vanishes at the surfaces. To fulfil these and the electromagnetic boundary conditions, three linearly independent wave solutions of the Landau—Lifshitz equation and Maxwell's equations are required. The amplitude of the spin‐wave modes and their spacing depends not only on the surface anisotropy constant Ks but also on the frequency ω of the rf field. At frequencies ω, much smaller than 4πMγ the precessional cone is very flat with Mn<<Mp. In this case the influence of the surface anisotropy is weakened and the rf field can strongly excite only one spin‐wave mode, the slightly disturbed uniform precession. The other modes are weak. At frequencies equal to or higher than 4πMγ, the cone of precession becomes nearly circular with Mn ≈ Mp, and the pinning action of the surface anisotropy increases correspondingly. At these higher frequencies, two modes are excited strongly. Theoretical absorption curves for Permalloy films with 4πMγ ≈ 2π·30 Gc are calculated for two frequencies and a range of thicknesses. For the surface anisotropy constant a value of 1 erg/cm2 is chosen. At 10 Gc only one strongly excited mode is predicted. At 30 Gc two modes have large amplitudes. The amplitude of one mode increases with increasing film thickness, whereas the amplitude of the other mode remains approximately constant. This theoretical result is similar to the experimental findings of Chen and Morrish2 at 24 Gc which, however, the authors explained in terms of stratified layers within the film.

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