Moiré induced topology and flat bands in twisted bilayer ${\mathrm{WSe}}_2$: A first-principles study

We study the influence of strong spin-orbit interaction on the formation of flat bands in relaxed twisted bilayer $\mathrm{W}{\mathrm{Se}}_2$. Flat bands, well separated in energy, emerge at the band edges for twist angles $\theta$ near $0^{\circ }$ and ${60}^{\circ }$. For $\theta$ near $0^{\circ }$, the interlayer hybridization together with a moiré potential determines the electronic structure. The bands near the valence band edge have nontrivial topology, with Chern numbers equal to $+1$ or $-1$. We propose that the nontrivial topology of the first band can be probed experimentally for twist angles less than a critical angle of $3.5{}^{\circ }$. For $\theta$ near ${60}^{\circ }$, the flattening of the bands arising from the $K$ point of the unit cell Brillouin zone is a result of atomic rearrangements in the individual layers. Our findings on the flat bands and the localization of their wave functions for both ranges of $\theta$ match well with recent experimental observations.

Figure 1

(a) Band structure of $2.28{}^{\circ } \mathrm{t}\mathrm{W}{\mathrm{Se}}_2$ along the path ${\mathrm{\Gamma }}_{\text{M}}-M_{\text{M}}-K_{\text{M}}-{\mathrm{\Gamma }}_{\text{M}}$ with the VB maximum set to zero. (b) Bands near the VB edge along the path $K_{\text{M}}-M_{\text{M}}-K_{\text{M}}^{\prime }$. The spin orientations of the first two bands, each doubly degenerate, are indicated by red and blue arrows. The pink dashed lines represent the band structure of the relaxed-MSL monolayer. For ease of viewing, the VB maximum of the monolayer has been set to $-4$ meV. (c) Variation of ${\mathrm{E}}_W$ (brown line) and ${\mathrm{E}}_{12}$ (purple line) with $\theta$. (d) Variation of ${\mathrm{S}}_{ij}$ with energy. The red and blue lines represent spin-up and spin-down states, respectively.

Figure 2

(a) Schematic of the folding relation of the UBZ to the MBZ for $\mathrm{t}\mathrm{W}{\mathrm{Se}}_2$ with $\theta$ near $0^{\circ }$. The small black hexagon represents the MBZ, and the red and blue hexagons represent the UBZ for the two layers, respectively. The $K$ and $K^{\prime }$ points in all the Brillouin zones are shown. The subscripts 1 and 2 stand for the two layers. (b) Same as above for $\mathrm{t}\mathrm{W}{\mathrm{Se}}_2$ with $\theta$ near ${60}^{\circ }$.

Figure 3

(a), (b), and (c) [(d), (e), and (f)] Distribution of $|{\psi }_{{\mathrm{\Gamma }}_{\text{M}}}{\left(\mathbf{r}\right)|}^2 \left[\right|{\psi }_{K_{\text{M}}}\left(\mathbf{r}\right){|}^2]$ averaged along the out-of-plane ($z$) direction for the V1, V2, and C1 bands for $\theta =2.{28}^{\circ } \mathrm{t}\mathrm{W}{\mathrm{Se}}_2$. (g) Moiré potential (in eV) of the relaxed-MSL monolayer. (h) LDOS (normalized) of the first flat band arising from the $\mathrm{\Gamma }$ point of the UBZ. (i) LDOS of the V1 states arising from the $K$ point of the UBZ. The high-symmetry stacking regions are as marked in (a).

Figure 4

(a) Band structure of $57.72{}^{\circ } \mathrm{t}\mathrm{W}{\mathrm{Se}}_2$. The VB maximum is set to zero. Inset: Bands near the VB edge along the $K_{\text{M}}-M_{\text{M}}-K_{\text{M}}^{\prime }$ path. The spin characteristic of the first two bands is shown in red and blue. (b) and (c) Bands near the VB and CB edges in (a) for the relaxed-MSL bilayer (black lines) and the relaxed-MSL monolayer (pink dashed lines). The VB maxima and CB minima are set to zero as appropriate. (d) ${\mathrm{E}}_W$ and ${\mathrm{E}}_{12}$ vs the twist angle $\theta$. (e) ${\mathrm{S}}_{ij}$ vs energy for spin-up (red) and spin-down (blue) states.

Figure 5

(a), (b), and (c) [(d), (e), and (f)] Depiction of $|{\psi }_{{\mathrm{\Gamma }}_{\text{M}}}{\left(\mathbf{r}\right)|}^2$ averaged along the out-of-plane ($z$) direction for V1, V2, and V3 (see Fig. 4) bands from the relaxed-MSL bilayer (relaxed-MSL monolayer) for $\theta =57.{72}^{\circ } \mathrm{t}\mathrm{W}{\mathrm{Se}}_2$. (g) Depiction of $|{\psi }_{{\mathrm{\Gamma }}_{\text{M}}}{\left(\mathbf{r}\right)|}^2$ averaged along the $z$ direction for the C1 band of the relaxed-MSL bilayer for $\theta =57.{72}^{\circ } \mathrm{t}\mathrm{W}{\mathrm{Se}}_2$. (h) Moiré potential (in eV) for the relaxed-MSL monolayer. (i) LDOS for bands arising from the $\mathrm{\Gamma }$ point of the UBZ.