Characterizing Spin-Orbit Torques by Tensorial Spin Hall Magnetoresistance

Abstract

Magnetoresistance (MR) provides a crucial tool for experimentally studying spin torques. While MR is well established in the device geometry of the spin Hall effect (SHE), as exemplified by the magnet/heavy-metal heterostructures, its role and manifestation beyond the SHE paradigm remain elusive. We propose a hitherto unknown form of MR where the underlying charge-to-spin conversion and its inverse process violate the simple geometry of the SHE, calling for tensorial descriptions. This MR can generate a series of unique harmonic responses essential for the experimental characterization of unconventional spin-orbit torques in non-SHE materials. We demonstrate these harmonic signals with semimetal WTe$_2$ in mind but the results are not restricted to specific materials.

Figures (4)

• Figure 1: (a) Illustration of the $t$-SHE and its reciprocal effect. A charge current (density) $\bm{\mathcal{J}}_{c}^{(\rm in)}$ injected along $x$ generates a spin current flowing in $z$, which carries a spin polarization $\bm{s}^{(\rm in)}$ due to the $\textit{t}$-SHE. The reflected spin current carrying $\bm{s}^{(\rm out)}$ converts back into charge currents through the inverse $\textit{t}$-SHE, which produces $\bm{\mathcal{J}}_{c,\parallel}^{(\rm out)}$ and $\bm{\mathcal{J}}_{c,\perp}^{(\rm out)}$, affecting the MR. (b) and (c) plot the longitudinal first-harmonic signal $V_{x}^{1\omega}$ scaled by the non-harmonic background $V_0$ in WTe$_2$-like materials for the $xy$ and $yz$ field scans, respectively, where $\theta_{H}$ and $\phi_{H}$ specify the magnetic field direction while $\theta_{s}$ parametrizes the direction of non-equilibrium $\bm{s}$. The values of the vertical axes are measured in $2t\eta^2\tilde{g}_R$, typically of order $10^{-3}\sim10^{-4}$.
• Figure 2: The coefficients of the symmetric and antisymmetric components of the rectification voltage $V_{\rm rec}/V_0$ with respect to $(H-\omega_0)$ plotted as functions of the angle of magnetic field undergoing the $xy$, $xz$ and $yz$ scans, for different values of $\theta_s$. (a)--(c): $H_{\rm Oe}=0$ while $H_{\rm DL}=H_{\rm FL}$ and the vertical axes scale in $H_{\rm DL}/\Delta$; (d)--(e): $H_{\rm DL}=H_{\rm FL}=0$ while the vertical axes scale in $H_{\rm Oe}/\Delta$.
• Figure 3: The symmetric and antisymmetric components of the rectification voltage with respect to $(H-\omega_0)$ plotted as functions of the angle of the magnetic field undergoing the $xy$ scan, where the red (green) curves correspond to the $t$-SMR (AMR) mechanism. $V_{\rm rec}$ is scaled by its maximum value $V_{\rm max}$, restricting the actual plot range within $\pm1$. The solid red ($\mathcal{S}$) and dashed red ($\mathcal{A}$) curves completely overlap.
• Figure 4: Second harmonic Hall voltage $V_{H}^{2\omega}/V_0$ as functions of the angle of magnetic field undergoing the $xy$, $xz$ and $yz$ scans for different values of $\theta_s$. (a)--(c): $H_{\rm Oe}$ vanishes and the solid (dashed) curves are plotted in the unit of $H_{\rm FL}/H$ ($H_{\rm DL}/H$) with $H_{\rm DL}=0$ ($H_{\rm FL}=0$), highlighting the contrast between the FL and DL torques. (d)--(e): $H_{\rm DL}$ and $H_{\rm FL}$ both vanish and the curves are plotted in the unit of $H_{\rm Oe}/H$.