Spin-current driven spontaneous coupling of ferromagnets

A theoretical framework is proposed for the spin-current driven synchronized self-oscillations in ferromagnets in the spin Hall geometry. The spin current generated by the spin Hall effect in a bottom nonmagnetic heavy metal excites a self-oscillation of the magnetization in an attached ferromagnet through spin-transfer effect. The spin current simultaneously creates spin accumulation inside the ferromagnet. Therefore, when the top surfaces of two ferromagnets are connected by a nonmagnetic material having a long spin diffusion length, another spin current flows according to the gradient of the spin accumulations between the ferromagnets. This additional spin current excites an additional spin torque leading to a coupled motion of the magnetizations. This coupling mechanism comes purely from spin degree of freedom in the system without using electric and/or magnetic interactions. The additional spin torque acts as a repulsive force between the magnetizations, and prefers an antiphase synchronization between the oscillators. The phase difference in a synchronized state is determined by the competition between this additional spin torque and spin pumping. Eventually, either an in-phase or antiphase synchronization is spontaneously excited in the individual ferromagnets, depending on the current magnitude. These conclusions are obtained by deriving the theoretical formula of the additional spin torque from the diffusive spin transport theory and solving the equation of motion of the magnetizations both numerically and analytically.

Figure 1

(a) Schematic view of a single STO. A ferromagnet F is placed onto a nonmagnet N. The electric current density $J_0$ flowing in the nonmagnet is converted to a pure spin current injected into the ferromagnet by the spin Hall effect. The spin current creates the spin accumulation $\delta {\mathbf{\mu }}_{\mathrm{F}}$ in the ferromagnet. (b) Schematic view of the system having two STOs. Another nonmagnet ${\mathrm{N}}^{\prime }$ is connected to the top surface of the ferromagnets, $\mathrm{F}{}_{\ell }$ ($\ell =1,2$). Spin currents driven by the spin accumulations and spin pumping flow in the ${\mathrm{N}}^{\prime }$ layer.

Figure 2

Time evolutions of $m_{1x}$ (red) and $m_{2x}$ (blue) for (a) $J_0=44$ and (b) 48 $\mathrm{MA}/{\mathrm{cm}}^2$. The intrinsic damping constant is $\alpha =0.005$. Note that the time ranges of (a) and (b) are different.

Figure 3

Time evolutions of $m_{1x}$ (red) and $m_{2x}$ (blue) for (a) $J_0=72$ and (b) 82 $\mathrm{MA}/{\mathrm{cm}}^2$. The intrinsic damping constant is $\alpha =0.010$. Note that the time ranges of (a) and (b) are different.

Figure 4

The synchronized motions of the $x, y$, and $z$ components of ${\mathbf{m}}_1$ (red solid line) and ${\mathbf{m}}_2$ (blue dotted line) at $J_0=28\mathrm{MA}/{\mathrm{cm}}^2$ are shown in (a), (b), and (c), respectively. The coupled and free-running (uncoupled) oscillations of $m_{1x}$ at $J_0=28\mathrm{MA}/{\mathrm{cm}}^2$ are shown in (d) by the red solid and black dashed lines, respectively.

Figure 5

Dependencies of the oscillation frequencies of the free-running (blue circle) and coupled (red square) STOs on the current density $J_0$.

Figure 6

(a) The oscillations of $m_{\ell x}(\ell =1,2)$ and (b) their Fourier transformations, $|m_{\ell x}\left(f\right)|$, in the absence of the coupling. The red solid and blue dotted lines correspond to the $\mathrm{F}{}_1$ and $\mathrm{F}{}_2$ layers, respectively. (c) The oscillations of $m_{\ell x}$ and (d) their Fourier transformations for the coupled STOs.

Figure 7

Dependencies of the oscillation frequencies (red square) and phase difference (blue circle) of the coupled STOs on the current density $J_0$.

Figure 8

(a) Time evolutions of $m_{1x}$ and $m_{2x}$ in the presence of the coupling torque given by Eq. (A2). The current density is $J_0=28\mathrm{MA}/{\mathrm{cm}}^2$. The spin pumping is neglected. (b) The time evolutions of $m_{1x}$ and $m_{2x}$ in the presence of the spin pumping. The current density is $J_0=45\mathrm{MA}/{\mathrm{cm}}^2$. The coupling spin torque given by Eq. (A2) is neglected.