Current-induced spin-wave excitation in Pt/YIG bilayer

We develop a self-consistent theory for current-induced spin-wave excitations in normal metal/magnetic insulator bilayer structures. We compute the spin-wave dispersion and dissipation, including dipolar and exchange interactions in the magnet, the spin diffusion in the normal metal, as well as the surface anisotropy, spin-transfer torque, and spin pumping at the interface. We find that (1) the spin-transfer torque and spin pumping affect the surface modes more than the bulk modes; (2) spin pumping inhibits high-frequency spin-wave modes, thereby redshifting the excitation spectrum; (3) easy-axis surface anisotropy induces a new type of surface spin wave, which reduces the excitation threshold current and greatly enhances the excitation power. We propose that the magnetic insulator surface can be engineered to create spin-wave circuits utilizing surface spin waves as information carriers.

Figure 1

An electrically insulating magnetic film of thickness $d$ with magnetization $\mathbf{m}(\mathbf{r},t)$ ($\parallel \widehat{\mathbf{z}}$ at equilibrium) in contact with a normal metal of thickness $d_N$, with translational symmetry in the y-z plane. A spin current ${\mathbf{J}}_{\mathrm{s}}$ polarized along $\widehat{\mathbf{z}}$ is generated in the normal metal due to the spin Hall effect from the applied charge current $J_{\mathrm{c}}$ and absorbed by the ferromagnet. ${\mathbf{J}}_{\mathrm{sp}}$ is the SP current due to the magnetization dynamics at the interface.

Figure 2

The contour plot of ${\tau }_{\mathrm{stt}}$ (at $J_c={10}^{11}$ A/m${}^2$, left) and ${\tau }_{\mathrm{sp}}$ (right) in Eq. (8) vs film thickness $d_N$ and spin diffusion length $\lambda$ for parameters given in Table and $g_r={10}^{18}$/m${}^2$. The dashed curve on the left panel shows the maximum of ${\tau }_{\mathrm{stt}}$ for fixed film thickness $d_N$.

Figure 3

Left: $J_{\mathrm{th}}$ in Eq. (24) vs the mixing conductance $g_r$ (log-log scale) for $k_s=25/\mu$m with $d_N=10$ nm and $\lambda =2$ nm. The dots are the threshold current obtained from numerical calculations below for $k_s=25/\mu$m and $k_s=0$. Right: The magnetization profiles for the EASA surface wave for various $k_s$ values. The solid curves are plotted using Eq. (26) for a semi-infinite film. The dots are obtained by numerical calculations for $d=0.61\mu$m.

Figure 4

Spin-wave dispersion (left) and profiles (right) in the absence of surface anisotropy ($k_s=0$) at $\theta =\pi /2$ (or $\mathbf{q}\perp \mathbf{m}$). Left: Spin-wave dispersion (a); the solid lines (different colors denote different bands) are calculated from the numerical solution of Eq. (16), and the dashed lines are plotted using the analytical expressions given by the real parts of Eqs. (18) and (19). Right: Spin-wave profiles ($m_x$ component) at $qd=0.09$ (b) and $qd=3.78$ (c). The colors in (b) and (c) match that in (a). The thick purple/green mode in (b)/(c) is for the point enclosed with circle in (a) on the purple/green band.

Figure 5

Same as Fig. 4 but with easy-axis surface anisotropy ($k_s=25/\mu$m). Left: Spin-wave dispersion (a); the solid lines are calculated from the numerical solution of Eq. (16), while the dashed lines and the $▸$ symbol are plotted using the analytical expressions given by the real parts of Eqs. (18) and (23). A new (black) band appears due to the easy-axis surface anisotropy. Right: Spin-wave profiles ($m_x$ component) at $qd=0.09$ (b) and $qd=3.78$ (c). The colors in (b) and (c) match that in (a). The thick black/red mode in (b)/(c) is for the points enclosed by a circle in (a) on the black/red band.

Figure 6

Spin-wave dissipation at $\theta =\pi /2$ (or $\mathbf{q}\perp \mathbf{m}$) with $g_r=5.8\times {10}^{17}$/m${}^2$ ($k_p=0.01/\mu$m). Top row: No surface anisotropy ($k_s=0$), bottom row: with easy-axis surface anisotropy ($k_s=25/\mu$m). The first column is the total dissipation with current injection of $J_c=2.3\times {10}^{11}$ A/m${}^2$ ($k_j=0.35k_c$). The second column to fourth column are the contributions from the Gilbert damping, STT, and SP, respectively. For all panels, the solid lines (different colors denote different bands) are calculated from the numerical solution of Eq. (16), and the dashed lines (and the $▸$) are plotted using the analytical expressions given by the imaginary parts of Eqs. (18, 19, 20, 21, 22, 23).

Figure 7

Power spectrum (resolution $\delta \omega /{\omega }_M=0.01$) for different combinations of surface anisotropy and mixing conductance at ten current levels (increasing by $\delta k_j=0.01k_c$) above threshold current.