Theory of electric reactance emerging from spin Hall effect

Abstract

The spin Hall effect in a heavy metal intercorrelates an AC electric current to the magnetization dynamics in an adjacent ferromagnet, which manifests as an electric reactance in the system's current-voltage response. We present a comprehensive theoretical analysis for this emergent reactance contribution in the frequency regime relevant to transport measurements up to a few GHz. Our analysis reveals that the reactance becomes inductor-like at low frequency below the ferromagnetic resonance. Crucially, we find that the sign of the reactance is directly governed by the spin transfer mechanism at the interface, which depends on the competition between its damping-like and field-like components parametrized by the spin mixing conductance. This characteristic behavior in the reactance offers a powerful transport observable in distinguishing the interfacial spin transfer processes in spintronic materials.

Figures (6)

• Figure 1: Schematic picture of the hypothetical setup. A heterostructure of the films of heavy metal (HM) and insulator of ferromagnet (FM) is taken. We consider a linear response between an alternating electric field $\boldsymbol{E}(t)$ and current $\boldsymbol{j}_c(t)$ involving the magnetization dynamics $\boldsymbol{m}(t)$.
• Figure 2: Frequency dependences of the (a) Real and (b) Imaginary parts of the longitudinal $(\Delta \rho_\omega^{xx})$ and transverse $(\Delta\rho_\omega^{yx})$ impedances, with out-of plane $\boldsymbol{m}_0 = \hat{\boldsymbol{z}}$. Inset of (a) shows schematic of the system, and inset of (b) shows ${\rm Im}[\Delta\rho_\omega^{xx,yx}/\rho_0]$ at low frequency. Parameters are taken as $\rho_0 = 1\;{\rm k\Omega} \; {\rm nm}$, $\theta_{\rm SH} = 0.01$, $\lambda_s = 1 \; {\rm nm}$, $\mu_0 M_s = 0.1 \; {\rm T}$, $\Omega_0 = 1 \; {\rm GHz}$, $\alpha = 0.01$, $d_H = d_F = 10 \; {\rm nm}$, and $(g_{\rm r}, g_{\rm i}) = (2,0)\times 10^{14} e^{-1} {\rm \Omega^{-1} m^{-2}}$.
• Figure 3: Dependences on the HM film thickness $d_H$ for the (a) real and (b) imaginary parts of the longitudinal $(\Delta \rho_\omega^{xx})$ and transverse $(\Delta\rho_\omega^{yx})$ impedances. All the parameters except for $d_H$ are taken same as those employed in Fig. \ref{['fig:fig2']}, with the frequency fixed at $\omega = 0.7 \Omega_0$.
• Figure 4: Dependences on the in-plane magnetic field direction $\boldsymbol{m}_0 = (\cos\phi, \sin\phi, 0)$ for the (a) real and (b) imaginary parts of the longitudinal $(\Delta \rho^{xx}_\omega)$ and transverse $(\Delta\rho^{yx}_\omega)$ impedances. All the parameters except for $\boldsymbol{m}_0$ are taken same as those employed in Fig. \ref{['fig:fig2']}, with the frequency fixed at $\omega = 0.7 \Omega_0$. Inset of (a) shows schematic of the system.
• Figure 5: (a) Frequency dependence of the longitudinal reactance ${\rm Im}[\Delta\rho_\omega^{xx}]$ for three different ratios of the spin mixing conductance $(g_{\rm r}, g_{\rm i})$, with $|g_{\uparrow\downarrow}| = 2\times 10^{14} e^{-1} {\rm \Omega^{-1} m^{-2}}$ fixed. Inset shows their behaviors at low frequency. (b) Color map of ${\rm Im}[\Delta\rho_\omega^{xx}]$ evaluated at $\omega = 0.7 \Omega_0$, with $(g_{\rm r}, g_{\rm i})$ varied. All the parameters except for $g_{\uparrow\downarrow} = g_{\rm r} + i g_{\rm i}$ are taken same as those employed in Fig. \ref{['fig:fig2']}.