Topological phases and twisting of graphene on a dichalcogenide monolayer

Depositing monolayer graphene on a transition metal dichalcogenide (TMD) semiconductor substrate has been shown to change the dynamics of the electronic states in graphene, inducing spin-orbit coupling (SOC) and staggered potential effects. Theoretical studies on commensurate supercells have demonstrated the appearance of interesting phases, as different materials and relative gate voltages are applied. Here we address the effects of the real incommensurability between lattices by implementing a continuum model approach that does not require small-period supercells. The approach allows us to study the role of possible relative twists of the layers, and verify that the SOC transfer is robust to twists, in agreement with observations. We characterize the nature of the different phases by studying an effective Hamiltonian that fully describes the graphene-TMD heterostructure. We find the system supports topologically nontrivial phases over a wide range of parameter ranges, which require the dominance of the intrinsic SOC over the staggered and Rashba potentials. This tantalizing result suggests the possible experimental realization of a tunable quantum spin Hall phase under suitable conditions. We estimate that most TMDs used to date likely result in weak intrinsic SOC that would not drive the heterostructure into topologically nontrivial phases. Additional means to induce a larger intrinsic SOC, such as strain fields or heavy metal intercalation, may be required.

Figure 1

(a) Band structure of coupled graphene-TMD continuum model at zero relative rotation angle, shown along the reciprocal space direction indicated in (b). The commensurate continuum model is shown in gray lines; the incommensurate model results are shown in green and orange lines. The small incommensurability in real space makes the TMD $K_{\mathrm{T}}$ point (orange lines) to fold close to the $K_{\mathrm{G}}$ graphene point (green lines). (b) The relative positions of nearby points (green) in comparison to those for the commensurate structure (gray) around the $K_{\mathrm{G}}$ point of interest. (c) Zoom-in into the graphene neutrality points reveals band splitting due to Zeeman-like and Rashba SOC effects, leading here to mass inverted bands. Due to time reversal symmetry, $K_{\mathrm{G}}^{\prime }$ valley shows similar features with spin reversed states (not shown).

Figure 2

Effective parameters for rotation angles $\theta \le 5^{\circ }$ in the continuum model. (a) Phase diagram on the gate voltage and twist angle plane showing the boundary between direct (blue shading) and inverted mass (yellow) band gaps. Right panels show heterostructure effective parameters where the graphene neutrality point is close to (b) the conduction, (c) nearly at the middle, and (d) valence bands. Intrinsic SOC parameter $S$ is nearly zero at all angles (green symbols). The staggered term $\mathrm{\Delta }$ (red squares) is larger (smaller) than the Zeeman-like SOC $\lambda$ (blue circles), in (b) and (c) [(d)] for small twist angles. The system exhibits a direct band gap whenever $\mathrm{\Delta }>\lambda$, and inverted-mass bands for $\mathrm{\Delta }<\lambda$, as indicated by the shading. Notice that for large twists, $\theta >4^{\circ }$, the system in (b) and (c) have switched to an inverted mass regime. In all these panels, $R=0.5$ meV. Structure parameters correspond here to graphene on ${\mathrm{WS}}_2$.

Figure 3

(a) Map of the gap size on the $\lambda$ and $\mathrm{\Delta }$ parameter plane for the effective Hamiltonian ($S=R=0.1$ are kept fixed). Color indicates gap size separating conduction and valence bands in the G-TMD structure. (b) ${\mathbb{Z}}_2$ values for the same parameter regime. Khaki (cyan) shading indicates ${\mathbb{Z}}_2=-1$ (+1) value, which identifies a nontrivial (trivial) phase of the system. Red, blue, and gray curves in both panels represent band degeneracy conditions in Eq. (4). All effective parameters and gap size are in units of the graphene intralayer coupling, $t=2\hslash v_{\mathrm{F}}/a\sqrt{3}=3.03$ eV—see the Supplemental Material [33].

Figure 4

(a) Gap size map for $\lambda$ and $\mathrm{\Delta }$ parameters of the effective Hamiltonian ($S$ and $R$ are fixed as indicated). Color represents gap size separating conduction and valence bands in the G-TMD structure. (b) ${\mathbb{Z}}_2$ values for the same parameter regime. The cyan shading indicates ${\mathbb{Z}}_2=1$, identifying as trivial all the phases over the entire parameter regime, given the small $\left|S\right|$ and relatively large $R$ value, despite the gap closing at $\lambda =-\mathrm{\Delta }$. Red, blue, and gray lines in both panels represent band degeneracy conditions in Eq. (4). All energies in units of the interlayer coupling term, $t=3.03$ eV.