Current-induced spin polarization and spin-orbit torque in graphene

Using the Green-function formalism, we calculate a current-induced spin polarization of weakly magnetized graphene with Rashba spin-orbit interaction. In a general case, all components of the current-induced spin polarization are nonzero, contrary to the nonmagnetic limit, where the only nonvanishing component of spin polarization is that in the graphene plane and normal to the electric field. When the induced spin polarization is exchange coupled to the magnetization, it exerts a spin-orbit torque on the latter. Using the Green-function method, we have derived some analytical formulas for the spin polarization and also determined the corresponding spin-orbit torque components. The analytical results are compared with those obtained numerically. Vertex corrections due to scattering on randomly distributed impurities are also calculated and shown to enhance the spin polarization calculated in the bare bubble approximation.

Figure 1

Schematic of the system under consideration. Graphene is on a substrate which assures a nonzero magnetization and also a spin-orbit interaction of Rashba type. Orientation of the magnetic moment $\mathbf{M}$ is described by the angles $\theta$ and $\xi$. An external electric field is oriented along the axis $y$.

Figure 2

Spin-polarization components induced by a current: (a)–(c) the $x$ component; (d)–(f) the $y$ component; (g)–(i) the $z$ component. (a),(d),(g) The spin-polarization components as a function of chemical potential $\mu$ and relaxation time $\tau$. (b),(e),(h) The spin-polarization components as a function of chemical potential $\mu$ for indicated values of $\tau$. (c),(f),(i) The polarization components as a function of $\tau$ for indicated values of $\mu$. The right parts of (d)–(f) present the corresponding shaded regions in the left parts. The solid and dashed lines in (b) represent the results based on the analytical formulas and numerical integration, respectively. The curves for $\tau \to \infty$ in (e) and (h) correspond to analytical solutions. The other parameters are $\lambda =2$ meV, $E_y=1$ V/cm, $\tilde{M}=0.1$ meV, $\theta =\pi /3$, and $\xi =\pi /2$.

Figure 3

The parameter $\eta$ as a function of (a) the chemical potential and relaxation time and (c) the relaxation time and $n_i{V}^2/{\left(n_i{V}^2\right)}_{\max }$. $\eta$ as a function of (b) chemical potential $\mu$ and (d) $n_i{V}^2/{\left(n_i{V}^2\right)}_{\max }$ for indicated values of the relaxation time. (e)–(h) The same variations as (a)–(d), but for the parameter $\zeta$. The white regions in (a) and (e) are excluded for the assumed value of ${\left(n_i{V}^2\right)}_{\max }=0.4\times {10}^{-2}{\left(\mathrm{eV}\mathrm{nm}\right)}^2$. The other parameters are as in Fig. 2. To find appropriate renormalization coefficients in (a) and (e), one has to draw a vertical line corresponding to a given value of $\mu$, and then find, on the part of this line in the color region, the point which corresponds to a given value of $n_i{V}^2$ [note that the point at the border of the white and color regions corresponds to ${\left(n_i{V}^2\right)}_{\max }]$. Since the range of relaxation time in (a) and (e) is limited, to find the renormalization coefficients for $\mu$ very close to the Dirac point, one needs to extend this range of $\tau$.