Spin-orbit dependence of anisotropic current-induced spin polarization

Studies of the current-induced spin polarization (CISP) have been recently reinvigorated due to the discoveries of CISP in some burgeoning materials such as oxide interfaces, van der Waals, and topological quantum materials. Here, we investigate the CISP in two-dimensional systems for different types of spin-orbit coupling (SOC) using the Boltzmann transport theory. We find an anisotropic response of CISP to the current direction which strongly depends on the type of SOC. We demonstrate that the CISP is nonlinear with respect to the SOC magnitude, depends on the Fermi energy, and exhibits two different transport regimes for low or high carrier density. Finally, we propose a magnetoresistance device which can exploit the predicted CISP anisotropy.

Figure 1

Schematic Fermi contours ($E_F>0$ case) and spin textures for (a) RSOC, (b) DSOC, (c) WSOC, and (d) PSOC. Arrows indicate the spin directions while the red and blue contour lines represent the $E_{+}$ and $E_{-}$ energy branches, respectively.

Figure 2

Schematic Fermi contours for (a), (b) RSOC, DSOC or WSOC and (c), (d) PSOC. (a), (c) for $E_F<0$, and (b), (d) for $E_F>0$. The gray shaded regions represent the electron occupied states. In (c), ${\varphi }_{1,2}$ (${\varphi }_1+{\varphi }_2=3\pi /2$) denotes the critical angles separating the electron occupied and unoccupied states. Red and blue contour lines represent the $E_{+}$ and $E_{-}$ energy branches, respectively.

Figure 3

$\delta \mathbf{s}$ as a function of the direction of the electric field $\phi$ in polar coordinates for (a) RSOC, (b) DSOC, (c) WSOC and (d) PSOC. Blue lines represent the magnitude while the red arrows indicate the direction of $\delta \mathbf{s}$.

Figure 4

(a) CISP efficiency $\delta s/j_e$ as a function of $E_F$ (in units of $|E_c|$) for RSOC, DSOC, and WSOC (black line) and PSOC when $\phi =0$ (red line), $\phi =\pi /8$ (orange line) and $\phi =3\pi /4$ (blue line). (b) CISP efficiency $\delta s/j_e$ (color) as a function of $E_F$ and the direction of the electric field $\phi$ for PSOC. The SOC parameters are assumed to be $\alpha =\beta =\gamma =2\lambda =1.0\mathrm{eV}\AA$ and electron effective mass is $m=0.5$, under which the band minima $E_c$ are the same for the different types of SOC. The vertical dashed lines indicate the topological transition point of the Fermi contour.

Figure 5

CISP efficiency $\delta s/j_e$ as a function of SOC parameters α and λ for (a) RSOC/DSOC/WSOC type and (b) PSOC type. The other parameters are assumed to be $m=0.5, |E_F|=27.2$ meV and $\phi =3\pi /4$ in (b). The dashed lines denote the critical points, at which $\delta s/j_e$ reaches maximum when $E_F>0$.

Figure 6

Schematic illustration of magnetoresistance device due to the interplay of current-direction (${\mathbf{j}}_e$, white arrow) dependent CISP $\mathit{\delta }\mathbf{s}$ (red arrow) in SOC conductor layer and local magnetization M (yellow arrow) in ferromagnetic (FM) layer.