Theory of spin Hall magnetoresistance

We present a theory of the spin Hall magnetoresistance (SMR) in multilayers made from an insulating ferromagnet F, such as yttrium iron garnet (YIG), and a normal metal N with spin-orbit interactions, such as platinum (Pt). The SMR is induced by the simultaneous action of spin Hall and inverse spin Hall effects and therefore a nonequilibrium proximity phenomenon. We compute the SMR in F$|$N and F$|$N$|$F layered systems, treating N by spin-diffusion theory with quantum mechanical boundary conditions at the interfaces in terms of the spin-mixing conductance. Our results explain the experimentally observed spin Hall magnetoresistance in N$|$F bilayers. For F$|$N$|$F spin valves we predict an enhanced SMR amplitude when magnetizations are collinear. The SMR and the spin-transfer torques in these trilayers can be controlled by the magnetic configuration.

Figure 1

(a) N$|$F bilayer and (b) F$|$N$|$F trilayer systems considered here, where F is a ferromagnetic insulator and N a normal metal.

Figure 2

Normalized ${\mu }_{sx}$, ${\mu }_{sy}$, $j_{sx}$, and $j_{sy}$ as functions of $z$ for magnetizations (a) $\widehat{m}=\widehat{y}$, (b) $\widehat{m}=\left(\widehat{x}+\widehat{y}\right)/\sqrt{2}$, and (c) $\widehat{m}=\widehat{x}$ for a sample with $d_N=12$ nm. We adopt the transport parameters $\rho =8.6\times {10}^{-7}\mathrm{error}m$, $\lambda =1.5$ nm, and $G_r=5\times {10}^{14}{\mathrm{error}}^{-1}{m}^{-2}$. For magnetizations $\widehat{m}=\widehat{y}$ and $\widehat{m}=\widehat{x}$, both ${\mu }_{sx}$ and $j_{sx}$ vanish.

Figure 3

Calculated $\Delta {\rho }_1/\rho$ as a function of $\lambda$ for different spin Hall angles ${\theta }_{\mathrm{SH}}$ with (a) $G_r=1\times {10}^{14}{\mathrm{error}}^{-1}{m}^{-2}$, (b) $G_r=5\times {10}^{14}{\mathrm{error}}^{-1}{m}^{-2}$, (c) $G_r=10\times {10}^{14}{\mathrm{error}}^{-1}{m}^{-2}$, and (d) the ideal limit $G_r\gg \sigma /\left(2\lambda \right)$. The Pt layers are 12 nm thick with resistivity $8.6\times {10}^{-7}\mathrm{error}m$ (sample 1, solid curve) and 7 nm thick with resistivity $4.1\times {10}^{-7}\mathrm{error}m$ (sample 2, dashed curve). Experimental results are shown as horizontal lines for comparison (Ref. 25).

Figure 4

Calculated $\Delta {\rho }_1/\left(\rho {\theta }_{\mathrm{SH}}^2\right)$ in an F$|$N$|$F spin valve as a function of spin-diffusion length $\lambda$ with $d_N=12$ nm, $G_r=5\times {10}^{14}{\mathrm{error}}^{-1}{m}^{-2}$, and $\rho =8.6\times {10}^{-7}\mathrm{error}m$ chosen from sample 1 in Ref. 25. $\Delta {\rho }_1/\left(\rho {\theta }_{\mathrm{SH}}^2\right)$ in an N$|$F bilayer is plotted as a dotted line for comparison.

Figure 5

The ratio $\beta \left(\alpha \right)$ which characterize how ${\vec{\tau }}_{\mathrm{STT}}^{\left(\mathrm{B}\right)}$ changes with respect to the relative orientation between $\widehat{m}$ and ${\widehat{m}}^{\prime }$. We adopt the transport parameters $d_N=12$ nm, $\rho =8.6\times {10}^{-7}\Omega m$, and $G_r=5\times {10}^{14}{\Omega }^{-1}{m}^{-2}$.