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We theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the Gilbert damping in magnetic insulator--normal metal bilayers due to spin pumping as well as the enhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In the long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement for transverse volume modes to that of the macrospin mode is equal to two. With an out-of-plane magnetization, this ratio decreases with both an increasing surface anisotropic energy and mode number. If the surface anisotropy induces a surface state, the enhancement can be an order of magnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation enhancement can be understood by mapping the anisotropy parameter to the out-of-plane case with anisotropy. For shorter wavelengths, we compute the enhancement numerically and find good agreement with the analytical results in the applicable limits. We also compute the induced direct- and alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored in the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin dissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin Hall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is greatest for surface-localized modes.

Herein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator--normal metal--ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and spin transfer lead to collective magnetic excitations in the two thin-film ferromagnets. In addition, the dynamic dipolar field contributes to the interlayer coupling. By solving the Landau-Lifshitz-Gilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as surface spin waves, we compute the effect of the dynamic coupling on the resonance frequencies and linewidths of the various modes. The long-wavelength modes may couple acoustically or optically. In the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damping enhancement of the acoustic mode vanishes, whereas the optical mode acquires a significant Gilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The dynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For intermediate wavelengths, the coupling can be increased by the dipolar field such that the modes in the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The surface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping enhancement. These modes also may acoustically or optically couple, but they are unaffected by thickness asymmetries., Comment: 13 pages, 5 figures

Precessing magnetization in a thin film magnetic insulator pumps spins into adjacent metals; however, this phenomenon is not quantitatively understood. We present a theory for the dependence of spin-pumping on the transverse mode number and in-plane wave vector. For long-wavelength spin waves, the enhanced Gilbert damping for the transverse mode volume waves is twice that of the macrospin mode, and for surface modes, the enhancement can be ten or more times stronger. Spin-pumping is negligible for short-wavelength exchange spin waves. We corroborate our analytical theory with numerical calculations in agreement with recent experimental results.