Spin-wave renormalization in exchange- and dipolar-coupled ferromagnets: Bulk spin waves and the Damon-Eshbach surface spin wave

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Abstract
We have calculated the temperature variation of the bulk and the surface spin-wave frequencies in exchange- and dipolar-coupled ferromagnets. We emphasize the long-wavelength limit where dipolar coupling between the spins strongly influences the excitation energies. We find that leading terms in the finite temperature dispersion relation give results equivalent to the zero-temperature relation, with the saturation magnetization and the exchange stiffness constant replaced by the appropriate temperature-dependent forms. There is, in addition, a contribution to the spin-wave frequency that varies linearly with temperature. Although, this term is smaller in magnitude than the previous one, under conditions prevailing in conventional resonance experiments, it should be of measureable magnitude. For the Damon-Eshbach surface spin waves, we find that the renormalization is affected by the spatially nonuniform magnetization profile near the surface. Consequently, the leading temperature corrections to the frequency of this mode depends on both the magnitude and the direction of propagation of the wave vector. We calculate this quantitatively, in the limit that exchange contributions to the frequency of the Damon-Eshbach wave are small. The results are compared with the data reported by Grunberg and Metawe for the light scattering off EuO surface.