Proximity-induced topological phases in bilayer graphene

We study the band structure of phases induced by depositing bilayer graphene on a transition-metal dichalcogenide monolayer. Tight-binding and low-energy effective Hamiltonian calculations show that it is possible to induce topologically nontrivial phases that should exhibit the quantum spin Hall effect in these systems. We classify bulk insulating phases through calculation of the $Z_2$ invariant, which unequivocally identifies the topology of the structure. The study of these and similar hybrid systems under an applied gate voltage opens the possibility for tunable topological structures in real experimental systems.

Figure 1

(a) $\mathrm{\Gamma }-K-M-\mathrm{\Gamma }$ band structure of a coupled bilayer graphene-TMD from tight-binding calculations where the TMD is ${\mathrm{WS}}_2$. (b) Side view of atomic arrangement for the BLG-TMD system. (c) Bands at the $K$ valley near the Fermi level ($E_F=0$). Red (blue) lines indicate $s_z$ spin up (down) projection of each band.

Figure 2

Low-energy band structure of a BLG-TMD multilayer system under a gate voltage. Red (blue) lines describe spin down (up) states obtained from tight-binding calculations; circles show the results of an effective Hamiltonian fit. (a) The neutrality point of BLG is shifted close to the valence bands of the TMD by the gate, resulting in large spin splitting in the valence band of the structure. (b) When the neutrality point is brought close to the conduction band of the TMD, the overall band gap is similar ($\simeq 8$ meV here) but spin splitting is larger in the conduction band of the structure. Details of the parameters for both panels are given in Ref. [30].

Figure 3

Map of the bulk spectral gap in BLG-TMD heterostructure as a function of $R_1$ and $R_2$ for $V=0$. Notice that only two regions $R_1{R}_2<0$ are topologically nontrivial phases, characterized by index $Z_2=-1$ (blue labels). All other insulating regions are topologically trivial, $Z_2=+1$. Inset: Corresponding map of spectral gap for $V=10$ meV. The application of an opposite voltage drives the entire region into different trivial insulating phases, all with $Z_2=+1$. The dispersion relation of points A, B, and D, are shown explicitly in Fig. 4.

Figure 4

Band structure near the $K$ point of different phases in BLG-TMD. (a) shows dispersion relations for trivial insulating phase labeled for $R_1,R_2$ values indicated by point A in Fig. 3. (b) Band structure for the topologically nontrivial insulating phase for point B in Fig. 3. (c) Semimetallic phase separating the nontrivial phase in (b) from the trivial phase (d). This gapless phase evolves from point B under the application of an opposite voltage $V=3.3$ meV. The inset in (c) shows how the gap at B has nonmonotonic dependence with opposite voltage, and yet trivial phases for $V>3.3$ meV. (d) Trivial insulating phase for $R_1,R_2$ given in (b) but gate voltage $V=10$ meV (point D in the inset of Fig. 3). The colors of the dispersion curves indicate $s_z$ projection for each state as per the legend in (a).