Magnetization damping in noncollinear spin valves with antiferromagnetic interlayer couplings

We study the magnetic damping in the simplest of synthetic antiferromagnets, i.e., antiferromagnetically exchange-coupled spin valves, in the presence of applied magnetic fields that enforce noncolliear magnetic configurations. We formulate the dynamic exchange of spin currents in a noncollinear texture based on the spin-diffusion theory with quantum mechanical boundary conditions at the ferrromagnet/normal-metal interfaces and derive the Landau-Lifshitz-Gilbert equations coupled by the interlayer static and dynamic exchange interactions. We predict noncollinearity-induced additional damping that is modulated by an applied magnetic field. We compare theoretical results with published experiments.

Figure 1

(a) Sketch of the sample with interlayer exchange couplings illustrating the spin pumping and backflow currents. (b) Magnetic resonance in an antiferromagnetically exchange-coupled spin valve with a normal-metal (NM) film sandwiched by two ferromagnets (F1, F2) subject to a microwave magnetic field $\mathbf{h}$. The magnetization vectors $({\mathbf{m}}_1,{\mathbf{m}}_2$) are tilted by an angle $\theta$ in a static in-plane magnetic field $\mathbf{H}$ applied along the $y$ axis. The vectors $\mathbf{m}$ and $\mathbf{n}$ represent the sum and difference of the two layer magnetizations, respectively. (c),(d) Precession-phase relations for the acoustic and optical modes.

Figure 2

(a) ${\alpha }_m$ (dashed line) and ${\alpha }_n$ (solid line) as a function of $\lambda /d_{\mathrm{N}}$ for different values of the dimensionless mixing conductance $g_r$. (b) ${\alpha }_{\mathrm{AC}}$ (dashed line) and ${\alpha }_{\mathrm{OP}}$ (solid line) as a function of tilt angle $\theta$ for $g_r=5$ and different values of $\lambda /d_{\mathrm{N}}$.

Figure 3

(a),(c) ${\alpha }_{\mathrm{A}}\left(\theta \right)/{\alpha }_m$ and (b),(d) ${\alpha }_{\mathrm{O}}\left(\theta \right)/{\alpha }_n$ as a function of $\theta$ and $g_r$ for different values of $\lambda /d_{\mathrm{N}}$. (a),(b) with $\lambda /d_{\mathrm{N}}=1$, (c),(d) with $\lambda /d_{\mathrm{N}}=10$, (e) ${\alpha }_{\mathrm{A}}\left(\theta \right)/{\alpha }_0$ as a function of $\theta$ and $\lambda /d_{\mathrm{N}}$ with $g_r=5$ and ${\alpha }_0=0.02$. (f) ${\alpha }_{\mathrm{A}}\left(\theta \right)/{\alpha }_0$ as a function of $\theta$ and $g_r$ with $\lambda /d_{\mathrm{N}}=1$ and ${\alpha }_0=0.02$.

Figure 4

(a) Resonance frequencies of the A and O modes as a function of magnetic field for $H_s/\left(4\pi M_s\right)=1$. (b),(c) Linewidths of the A (dashed line) and the O (solid line) modes for $H_s/\left(4\pi M_s\right)=1,g_r=5$, and different values of $d_{\mathrm{N}}/\lambda$. (b) ${\alpha }_1/{\alpha }_0=1$ and (c) ${\alpha }_1/{\alpha }_0=10$.

Figure 5

(a) Derivative of the microwave absorption spectrum $dP/dH$ at frequency $\omega /\left(2\pi \right)=9.22\mathrm{GHz}$ for different angles $\phi$ between the microwave field and the external magnetic field for $H_s/\left(4\pi M_s\right)=0.5,\omega /\left(4\pi \gamma M_s\right)=0.35,d_{\mathrm{N}}/\lambda =0.1,d_{\mathrm{F}}/\lambda =0.3,{\alpha }_0={\alpha }_1=0.02$, and $g_r=4$. The experimental data have been adopted from Ref. [31]. (b) Computed linewidths of the A and O modes of a $\mathrm{Co}\left|\mathrm{Cu}\right|\mathrm{Co}$ spin valve (solid line) compared with experiments on a $\mathrm{Co}|\mathrm{Cu}$ multilayer (dashed line) [30].