Electronic localization in twisted bilayer ${\mathrm{MoS}}_2$ with small rotation angle

Moiré patterns are known to confine electronic states in transition metal dichalcogenide bilayers, thus generalizing the notion of magic angles discovered in twisted bilayer graphene to semiconductors. Here, we present a revised Slater-Koster tight-binding model that facilitates reliable and systematic studies of such states in twisted bilayer ${\mathrm{MoS}}_2$ for the whole range of rotation angles $\theta$. We show that isolated bands appear at low energy for $\theta \lesssim 5^{\circ }–6^{\circ }$. Moreover, these bands become “flatbands,” characterized by a vanishing average velocity, for the smallest angles $\theta \lesssim 2^{\circ }$.

Figure 1

Atomic structure of bilayer ${\mathrm{MoS}}_2$ at a twist angle $\theta =13.{17}^{\circ }$. (a) Side view. (b) Top view.

Figure 2

DFT conduction and valence bands in tb-${\mathrm{MoS}}_2$: $(1,2)\theta =21.{79}^{\circ }, (2,3)\theta =13.{17}^{\circ }, (3,4)\theta =9.{43}^{\circ }$, and $(4,5)\theta =7.{34}^{\circ }$. For every rotation angle, the origin of energy is fixed at the energy of the state ${\mathrm{S}}_0$.

Figure 3

TB and DFT bands around the gap in tb-${\mathrm{MoS}}_2$: (left) $(1,2)\theta =21.{79}^{\circ }$ and (right) $(3,4)\theta =9.{43}^{\circ }$.

Figure 4

Dependence of valence bands on rotation angle $\theta$: (a) Valence band dispersion of $(4,5)$ tb-${\mathrm{MoS}}_2, \theta =7.{34}^{\circ }$. (b) Energy $E\left({\mathrm{S}}_2\right)$ of the state ${\mathrm{S}}_2$ [see panel (a)] versus ${\theta }^2$. (c) Energy difference between the states ${\mathrm{S}}_4$ and ${\mathrm{S}}_2, \mathrm{\Delta }{E}_{24}=E\left({\mathrm{S}}_4\right)-E\left({\mathrm{S}}_2\right)$, versus $\theta$. A negative value of $\mathrm{\Delta }{E}_{24}$ means that a gap $|\mathrm{\Delta }{E}_{24}|$ exists between the band below the gap and the other valence bands. (d) Average slope of $E\left(\vec{k}\right)$ of the band between states ${\mathrm{S}}_2$ and ${\mathrm{S}}_3$.

Figure 5

TB band dispersion and local density of states (LDOS) of $d_0=d_{z^2}$ Mo atoms at the center of the AA stacking region and the center of the AB region: (a) for $(10,11)$ tb-${\mathrm{MoS}}_2\theta =3.{15}^{\circ }$, and (b) for $(20,21)$ tb-${\mathrm{MoS}}_2\theta =1.{61}^{\circ }$. In a moiré cell, two symmetrically equivalent AB stacking regions are located at $1/3$ and $2/3$ of the longest diagonal of the cell (see Sec. I A of the Supplemental Material [52]). Each AB stacking region contains two types of Mo atoms: (AB-A) Mo atom of a layer lying above an S atom of the other layer; (AB-B) Mo atom of a layer not lying above an atom of the other layer.