Skew-scattering-induced giant antidamping spin-orbit torques: Collinear and out-of-plane Edelstein effects at two-dimensional material/ferromagnet interfaces

Heavy metal/ferromagnet interfaces feature emergent spin-orbit effects absent in the bulk materials. Because of their inherent strong coupling between spin, charge, and orbital degrees of freedom, such systems provide a platform for technologically sought-after spin-orbit torques (SOTs). However, the microscopic origin of purely interfacial antidamping SOT, especially in the ultimate atomically thin limit, has proven elusive. Here, using two-dimensional (2D) van der Waals materials as a test bed for interfacial phenomena, we address this problem by means of a microscopic framework accounting for band structure effects and impurity scattering on equal footing and nonperturbatively. A number of unconventional and measurable effects are predicted, the most remarkable of which is a giant enhancement of antidamping SOT in the dilute disorder limit induced by a robust skew scattering mechanism, which is operative in realistic interfaces and does not require magnetic impurities. The newly unveiled skew scattering mechanism activates rich semiclassical spin-charge conversion effects that have gone unnoticed in the literature, including a collinear Edelstein effect with nonequilibrium spin polarization aligned with the direction of the applied current.

Figure 1

(a) Schematic of a TMD/thin-film-FM (top) and TMD/graphene/2D-FM bilayer (bottom). (b) Electronic structure around $K$ points in reference heterostructure (plotted along a path with $k_x=0$). (c) Geometry of 2D material SOT-operated device. Direction of SOTs is indicated for current applied along positive $\widehat{x}$ axis assuming $\epsilon >0$. (d) Current-induced distortion of the Fermi surface in the relaxation-time approximation (gray) and full distortion accounting for skew scattering events, $\delta f_{\mathbf{k}}\propto {\tau }_{\parallel }\widehat{\mathbf{k}}·\mathbf{E}+{\tau }_{\perp }(\widehat{\mathbf{k}}\times \mathbf{E})·\widehat{z}$ (purple). 3D arrows depict the net nonequilibrium spin polarization density. (e) Fermi energy dependence of SOT efficiencies for reference $C_{6v}$-invariant monolayer system. Parameters: $\lambda =20$ meV, ${\mathrm{\Delta }}_{\text{xc}}=15$ meV, $\varphi =\pi /11$, $n={10}^{11}{\mathrm{cm}}^{-2}$, and $u_0=1.2\mathrm{eV}{\mathrm{nm}}^2$.

Figure 2

Diagrammatic expansion of the response function. (a) Bethe-Salpeter equation for the charge current vertex in the $R\text{−}A$ sector. (b) Skeleton expansion of the $T$-matrix ladder. Full (open) square denotes a $T$ ($T^{†}$) matrix insertion, while circles represent electron-impurity interaction vertices. The red $\times$ stands for impurity density insertion ($n$).

Figure 3

(a) Giant enhancement of dampinglike SOT due to skew scattering for proximitized graphene. Dashed (solid) lines are calculated in the weak (unitary) scattering regime with $u_0=0.1\mathrm{eV}{\mathrm{nm}}^2$ ($u_0\to \infty$). Other parameters as in Fig. 1. (b) Highly anisotropic SOT generated by TMDs in the unitary limit. Orbital-gap dependence of $\mathcal{T}$-odd SOTs is evaluated at fixed carrier density $n_e\simeq 4.7\times {10}^{13}{\mathrm{cm}}^{-2}$ for ${\varphi }_{\mathbf{m}}=\pi /10$ (squares) and ${\varphi }_{\mathbf{m}}=\pi /18$ (circles). Inset: Angle dependence of ${\tau }_{\text{e}1\left(2\right)}$ for $\mathrm{\Delta }=0.75$ eV. Shaded area indicates nonperturbative region where high-order harmonics in $t_{ei}\left(\varphi \right)$ become prominent. Other parameters: ${\mathrm{\Delta }}_{\text{xc}}=0.1$ eV, $\lambda =60$ meV, ${\lambda }_{\text{sv}}=$ 3 meV, and impurity density $n={10}^{11}{\mathrm{cm}}^{-2}$.

Figure 4

Electronic structure of a TMD|graphene|FM (a) and TMD|FM (b) heterostructure plotted along $\mathbf{p}={(0,p_y)}^{\text{t}}$ (full lines) and $\mathbf{p}={(p_x,0)}^{\text{t}}$ (dashed lines) directions. Shaded areas highlight spectral region II. Parameters: for (a) and (b) $\lambda =40$ meV, ${\mathfrak{m}}_x=20$ meV, ${\mathfrak{m}}_z=30$ meV, and for (b) $\mathrm{\Delta }=0.4$ eV, ${\lambda }_{\text{sv}}(\epsilon <0)=150$ meV, and ${\lambda }_{\text{sv}}(\epsilon >0)=5$ meV.

Figure 5

Emergent pseudospin-spin texture of spin-majority band for a 2D material|FM interface in the presence (thick lines) and absence (thin lines) of orbital gap. The pseudospin-spin textures are plotted along the path $k=(k_x,0)$ (top panels) and $k=(0,k_y)$ (bottom panels). Band structure parameters: $\lambda =20$ meV, $\mathrm{\Delta }=400$ meV, ${\mathrm{\Delta }}_{\text{xc}}=15\mathrm{meV}$ (with $\varphi =\pi /8$), and ${\lambda }_{\text{sv}}=0$.

Figure 6

Angular dependence of m-odd response functions (top) and associated SOT efficiencies (bottom). System parameters: ${\mathrm{\Delta }}_{\text{xc}}=0.1$ eV, $\lambda =60$ meV, ${\lambda }_{\text{sv}}=\mathrm{\Delta }=0$, $n={10}^{11}{\text{cm}}^{-2}$, and $\epsilon =0.4$ eV.