Valley-dependent spin-orbit torques in two-dimensional hexagonal crystals

We study spin-orbit torques in two-dimensional hexagonal crystals such as graphene, silicene, germanene, and stanene. The torque possesses two components, a fieldlike term due to inverse spin galvanic effect and an antidamping torque originating from Berry curvature in mixed spin-$k$ space. In the presence of staggered potential and exchange field, the valley degeneracy can be lifted and we obtain a valley-dependent Berry curvature, leading to a tunable antidamping torque by controlling the valley degree of freedom. The valley imbalance can be as high as 100% by tuning the bias voltage or magnetization angle. These findings open new venues for the development of current-driven spin-orbit torques by structural design.

Figure 1

Schematics of the device based on graphenelike materials with fieldlike and antidamping-like SOT. (b)–(d) Energy dispersion of graphenelike materials with (b) $U=0.03\text{eV}$ and $J_{\mathrm{ex}}=0$, (c) $U=0$ and $J_{\mathrm{ex}}=0.03\text{eV}$, and (d) $U=0.03\text{eV}$ and $J_{\mathrm{ex}}=0.03\text{eV}$. The pink solid lines and the blue dashed lines denote the spin-up and spin-down subbands, respectively. The current flows from left to right. Magnetization is assumed to be directed along the $z$ axis.

Figure 2

(a) Intraband and (b) interband contributions to spin density as a function of Fermi energy $E_f$ for various Rashba spin-orbit coupling in the absence of intrinsic spin-orbit coupling ${\lambda }_{\mathrm{so}}$. (c) Intraband and (d) interband spin density as a function of Fermi energy $E_f$ for various intrinsic spin-orbit coupling at ${\lambda }_R=0.03$ eV. Inset (b) Band structure of graphenelike materials with ${\lambda }_R=0.03$ eV and ${\lambda }_{\mathrm{so}}=0$ eV. Inset (c) Band structure with ${\lambda }_R=0$ eV and ${\lambda }_{\mathrm{so}}=0.03$ eV. Inset (d) Same as inset (b) but with ${\lambda }_{\mathrm{so}}=0.03$ eV. The current is injected along the $x$ axis.

Figure 3

(a) Intraband and (b)–(c) interband spin density as a function of Rashba spin-orbit coupling for different intrinsic spin-orbit coupling with $J_{\mathrm{ex}}=0.01\text{eV}$ and $E_f=0.1\text{eV}$. The magnetization is directed along the $z$ axis.

Figure 4

(a) Intraband and (b)–(c) interband spin density of two valleys as a function of Fermi energy for different intrinsic spin-orbit coupling with $U=0.01\text{eV}$ and $J_{\mathrm{ex}}=0.01\text{eV}$. Valley polarization for intraband (d) and interband (e)–(f) components for different intrinsic spin-orbit coupling.

Figure 5

Intraband and interband spin density as a function of the magnetization direction with (solid lines) and without (dashed lines) staggered potential for the different valleys when $U=0.03$ eV. Inset (b) Valley polarization of interband spin density for $x$ component.

Figure 6

Intraband and interband spin density as a function of intrinsic spin-orbit coupling for a different staggered potential for the $K$ valley (a)–(c) and $K^{\prime }$ valley (d)–(f). The parameters are $E_f=-0.16\text{eV}, J_{\mathrm{ex}}=0.01\text{eV}$, and ${\lambda }_R=0.03\text{eV}$.

Figure 7

Contour of valley-polarized Berry curvature distribution for different intrinsic spin-orbit coupling in $k_x-k_y$ plane with $U=0.03$ eV. [(a) and (d)] ${\lambda }_{\mathrm{so}}$ = 0.005 eV. [(b) and (e)] ${\lambda }_{\mathrm{so}}=0.01$ eV. [(c) and (f)] ${\lambda }_{\mathrm{so}}=0.015$ eV. Others parameters are the same as in Fig. 6.

Figure 8

Schematics of the realization of valley-dependent antidampinglike SOT: (a) top view and (b) side view. The current is injected into the vertical arm. The presence of both magnetization and staggered potential results in a nonequivalent spin density for the valleys. This leads to a different valleys on the horizontal sidearms.