Spin-orbit coupling in transition metal dichalcogenide heterobilayer flat bands

The valence flat bands in transition metal dichalcogenide (TMD) heterobilayers are shown to exhibit strong intralayer spin-orbit coupling. I show that symmetry constrains the spin-dependent complex phase of hopping terms in an effective tight-binding model of the valence flat bands. A perpendicular electric field causes interlayer hybridization, such that the effective model becomes equivalent to the Kane-Mele model of topological insulators. The proposed model can be used as a starting point to understand interactions and the experimentally observed topological transitions.

Figure 1

(a) When two layers with inequivalent lattice constants are combined, in general the $\mathbf{K}$ points of the monolayers map onto the $\mathit{\kappa }$ points of the mini-Brillouin zone. This is specifically the case for aligned ${\mathrm{WS}}_2$/${\mathrm{WSe}}_2$ and ${\mathrm{MoTe}}_2$/${\mathrm{WSe}}_2$. (b) In the case of an incommensurate moiré pattern, for example due to a twist angle $\theta$, the natural mapping is similar. (c) Only when the moiré length satisfies $a_M=p{a}_1$ with $pmod3=0$, do the $\mathbf{K}$ points of layer 1 map onto the center of the mini-BZ $\mathit{\gamma }$, as was done in Ref. [4].

Figure 2

(a) Flat valence band structure of aligned AA-stacked ${\mathrm{WS}}_2$/${\mathrm{WSe}}_2$. The colors red/blue indicate the spin $\downarrow$/$\uparrow$ of the bands, showing significant spin-orbit coupling. The thin lines indicate the monolayer valence band states in the absence of the moiré potential. Upon Wannierization, the nearest-neighbor hopping equals $|t_1|=1.8$ meV. (b) Flat valence band structure of aligned AB-stacked ${\mathrm{MoTe}}_2$/${\mathrm{WSe}}_2$, with a similar spin-orbit splitting. The nearest-neighbor hopping equals $|t_1|=4.0$ meV.

Figure 3

(a) The effective model of a plain heterobilayer contains orbitals at triangular lattice sites with nearest-neighbor hopping $t_1=\left|t_1\right|e^{i\varphi }$ where the phase is spin dependent, $\varphi =\frac{2\pi }{3}{\sigma }^z$. The states live completely within one monolayer. In the case of ${\mathrm{MoTe}}_2$/${\mathrm{WSe}}_2$, the states are centered at the $M{M}^{\prime }$ sites in the ${\mathrm{MoTe}}_2$ layer. (b) By tuning the vertical displacement field the second layer becomes relevant. The second layer, in the case of AB-stacked ${\mathrm{MoTe}}_2$/${\mathrm{WSe}}_2$, is described by the same triangular lattice model with opposite spin-orbit coupling phases and a different moiré unit cell center $X{X}^{\prime }$. As a result, the combination of in-plane spin-orbit coupling Eq. (5) and interlayer hopping Eq. (6) leads to an ideal realization of the Kane-Mele model.

Figure 4

(a) The flat valence bands in the ${\mathrm{MoTe}}_2$/${\mathrm{WSe}}_2$ heterobilayer for the $\mathbf{K}$ valley (the opposite valley is shown in thin lines). The top flat band contains exclusively states in the ${\mathrm{MoTe}}_2$ layer, whereas the bottom band contains exclusively states in the ${\mathrm{WSe}}_2$ layer. (b) By tuning the perpendicular electric field $V$ through a critical value, a band inversion occurs around the ${\mathit{\kappa }}^{\prime }$ point. The color scale indicates whether states live in the ${\mathrm{MoTe}}_2$ layer (red) or ${\mathrm{WSe}}_2$ layer (blue). The resulting bands carry a nonzero Chern number with opposite Chern numbers for opposite spin. This transition to a topological insulator band structure is described by the Kane-Mele model of Eqs. (5, 6, 7).