Proximity Effects in Bilayer Graphene on Monolayer ${\mathrm{WSe}}_2$: Field-Effect Spin Valley Locking, Spin-Orbit Valve, and Spin Transistor

Proximity orbital and spin-orbit effects of bilayer graphene on monolayer ${\mathrm{WSe}}_2$ are investigated from first principles. We find that the built-in electric field induces an orbital band gap of about 10 meV in bilayer graphene. Remarkably, the proximity spin-orbit splitting for holes is 2 orders of magnitude—the spin-orbit splitting of the valence band at $K$ is about 2 meV—more than for electrons. Effectively, holes experience spin valley locking due to the strong proximity of the lower graphene layer to ${\mathrm{WSe}}_2$. However, applying an external transverse electric field of some $1\mathrm{V}/\mathrm{nm}$, countering the built-in field of the heterostructure, completely reverses this effect and allows, instead of holes, electrons to be spin valley locked with 2 meV spin-orbit splitting. Such a behavior constitutes a highly efficient field-effect spin-orbit valve, making bilayer graphene on ${\mathrm{WSe}}_2$ a potential platform for a field-effect spin transistor.

Figure 1

(a) Atomic structure of bilayer graphene on monolayer ${\mathrm{WSe}}_2$ (supercell). (b) Sketch of bilayer graphene with atom labels forming a bare unit cell in Bernal stacking. Orbitals on nondimer atoms $B_1$ and $A_2$ form the low energy valence and conduction bands in the electronic structure of bilayer graphene, with $B_1$ being closer to ${\mathrm{WSe}}_2$.

Figure 2

(a) Calculated electronic band structure of bilayer graphene on monolayer ${\mathrm{WSe}}_2$. Bilayer graphene $\pi$ conjugated states are emphasized by the gray circles. (b) Enlargement to the fine structure of the low energy bands close to the Fermi level. Bands with a positive (negative) $z$ component of the spin are shown in red (blue).

Figure 3

Calculated low energy electronic and spin properties of bilayer graphene on monolayer ${\mathrm{WSe}}_2$. Shown are color map plots centered at the $K$ point representing 0.5% of the first Brillouin zone area for (a) the energy of the bottom conduction band measured from the conduction band edge. The contours correspond to carrier concentrations for 0.05, 0.18, 0.42, and $0.76\times {10}^{12}{\mathrm{cm}}^{-2}$. The inset depicts a cut of the first Brillouin zone, shown by dashed lines, near the $K$ point with directions towards $M$ and $\mathrm{\Gamma }$ points. (b) Color map of the $z$ component spin expectation value with in-plane spin textures shown by the arrows. (c) Energy of the valence band measured from the valence band edge with contours showing a carrier concentration of 0.02, 0.064, 0.22, 0.5, and $0.8\times {10}^{12}{\mathrm{cm}}^{-2}$, and (d) similar as in (b) but for the valence band.

Figure 4

Calculated sublattice resolved band structures around the $K$ valley for a transverse electric field of (a) $-0.5\mathrm{V}/\mathrm{nm}$, (b) $-0.2979\mathrm{V}/\mathrm{nm}$, (c) $-0.25\mathrm{V}/\mathrm{nm}$, (d) a zero field, and (e) $0.25\mathrm{V}/\mathrm{nm}$. The circle radii correspond to the probability of the state being localized on carbon atoms $B_1$ [(red) filled circles] and atoms $A_2$ [(blue) open circles].

Figure 5

Schematics of a spin-field effect transistor of bilayer graphene on a transition metal dichalcogenide with two ferromagnets in antiparallel configuration acting as the injector and detector of spins in the (a) spin-on and (b) spin-off state.