Dynamic Feedback in Ferromagnet–Spin Hall Metal Heterostructures

In ferromagnet–normal-metal heterostructures, spin pumping and spin-transfer torques are two reciprocal processes that occur concomitantly. Their interplay introduces a dynamic feedback effect interconnecting energy dissipation channels of both magnetization and current. By solving the spin diffusion process in the presence of the spin Hall effect in the normal metal, we show that the dynamic feedback gives rise to (i) a nonlinear magnetic damping that is crucial to sustain uniform steady-state oscillations of a spin Hall oscillator at large angles and (ii) a frequency-dependent spin Hall magnetoimpedance that reduces to the spin Hall magnetoresistance in the dc limit.

Figure 1

(a) In a FM-NM bilayer, spin pumping and spin backflow are connected by the SHE and its inverse process. (b),(c) Simulations of a spin Hall nano-oscillator in the presence of the feedback-induced nonlinear damping ${\alpha }_{\mathrm{fb}}$, with $\gamma H=10\mathrm{GHz}$, $d_m=1\mathrm{nm}$, ${\alpha }_0+{\alpha }_{\mathrm{sp}}=0.01$, and other parameters taken from Ref. [33]. The STT strength ${\omega }_s$ is scaled in megahertz.

Figure 2

Frequency dependence of the spin Hall magnetoimpedance scaled by $\mathrm{\Delta }{\rho }_1\equiv {\lim }_{\omega \to 0}\mathrm{Re}[\mathrm{\Delta }{Z}_1]$. $\mathrm{Re}[\mathrm{\Delta }{Z}_2]$ is not shown, since $\mathrm{Re}[\mathrm{\Delta }{Z}_2]=\mathrm{Im}[\mathrm{\Delta }{Z}_1]$. The plot is based on a YIG-Pt structure [33] with ${\alpha }_0=2.3\times {10}^{-4}$, $d_M=1\mathrm{nm}$, and $d_N\gg \lambda$.