Spin waves in ferromagnetic insulators coupled via a normal metal

Herein, we study 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.

Figure 1

(a) A cross section of the $\mathrm{FI}1|\mathrm{N}|\mathrm{FI}2$ heterostructure. The ferromagnetic insulators FI1 and FI2 are in contact via the normal metal N. The transverse coordinate $\xi$ is indicated along with the thicknesses $L_1,d_{\mathrm{N}}$, and $L_2$ of FI1, N, and FI2, respectively. (b) The coordinate system of the internal field (blue) with respect to the coordinate system of the $\mathrm{FI}1|$ $\mathrm{N}|\mathrm{FI}2$ structure (red). $\theta$ denotes the angle between the film normal and the internal field, and $\varphi$ is the angle between the in-plane component of the magnetic field and the in-plane wave vector.

Figure 2

Two coupled spin waves with amplitude ${\mathbf{m}}_{1Q}$ in ferromagnet FI1 and amplitude ${\mathbf{m}}_{2Q}$ in ferromagnet FI2. The spin waves inject a spin current into the normal metal (NM) via spin pumping. In the NM, the spins diffuse and partially relax, inducing a spin accumulation therein. In turn, the spin accumulation causes spin-transfer torques to arise on FI1 and FI2. The combined effect of spin transfer and spin pumping leads to a dynamic exchange coupling that, together with the dynamic demagnetization field, couples the spin waves in the two FIs.

Figure 3

FI(100 nm)$|\mathrm{N}$(50 nm)$|\mathrm{FI}$(101 nm): (a) Spin-pumping-enhanced Gilbert damping $\Delta \alpha$ as a function of $Q{L}_1$ of the uniform modes and the $n=1$ modes; inset shows the corresponding dispersion relation; (b) relative phase; and (c) amplitude between the out-of-plane magnetizations along $x$ at the edges of $\mathrm{FI}1|\mathrm{N}$ and $\text{FI}2|\mathrm{N}$; the apparent discontinuity in the green line in (c) results from defining the phase over the interval from $-\pi$ to $\pi$.

Figure 4

FI(1000 nm)$|\mathrm{N}$(200 nm)$|\mathrm{FI}$(1000 nm), $l_{\mathrm{sf}}=350\mathrm{nm},K_S=0.05\mathrm{erg}/{\mathrm{cm}}^2$: (a) The dispersion relation as a function of $Q{L}_1$ for the six lowest positive-real-part modes. (b) The renormalization of the damping attributable to spin pumping for the four lowest modes with frequencies with positive real parts as a function of $Q{L}_1$. At large $Q{L}_1$, the computation becomes increasingly demanding, and the point density of the plot becomes sparse. We have therefore individually marked the plotted points in this region.

Figure 5

FI(1000 nm)$|\mathrm{N}$(200 nm)$|\mathrm{FI}$(1000 nm), $l_{\mathrm{sf}}=350\mathrm{nm},K_S=0.05\mathrm{erg}/{\mathrm{cm}}^2$: (a) and (b) present the real parts of the $x$ components of the out-of-equilibrium magnetization vectors for the acoustic and optical surface modes, respectively, for several values of $Q{L}_1$. For values of $Q{L}_1\gtrsim 1$, the modes decouple and become localized in one of the two layers. For large values of $Q{L}_1\sim 100$, the two modes are strongly localized at one of the two $\mathrm{FI}|\mathrm{N}$ interfaces, which correspond to the peaks in the damping that are apparent in Fig. 4.

Figure 6

FI(100 nm)$|\mathrm{N}$(50 nm)$|\mathrm{FI}$(101 nm), BVMSW without spin pumping, showing relative amplitude of the out-of-plane magnetizations along $x$ at the edges of $\mathrm{FI}1|\mathrm{N}$ and $\mathrm{FI}2|\mathrm{N}$ as functions of $Q{L}_1$; blue and yellow (red and green) lines show acoustical (optical) modes.