Atomic relaxation and flat bands in strain-engineered transition metal dichalcogenide bilayer moiré systems

Abstract

Strain-induced lattice mismatch leads to moiré patterns in homobilayer transition metal dichalcogenides (TMDs). We investigate the structural and electronic properties of such strained moiré patterns in TMD homobilayers. The moiré patterns in strained TMDs consist of several stacking domains which are separated by tensile solitons. Relaxation of these systems distributes the strain unevenly in the moiré superlattice, with the maximum strain energy concentrating at the highest energy stackings. The order parameter distribution shows the formation of aster topological defects at the same sites. In contrast, twisted TMDs host shear solitons at the domain walls, and the order parameter distribution in these systems shows the formation of vortex defects. The strained moiré systems also show the emergence of several well-separated flat bands at both the valence and conduction band edges, and we observe a significant reduction in the band gap. The flat bands in these strained moiré superlattices provide platforms for studying the Hubbard model on a triangular lattice as well as the ionic Hubbard model on a honeycomb lattice. Furthermore, we study the localization of the wave functions corresponding to these flat bands. The wave functions localize at different stackings compared to twisted TMDs, and our results are in excellent agreement with spectroscopic experiments.

Figures (5)

• Figure 1: (a): Top and side views of the relaxed atomic structure of the moiré pattern formed by applying isotropic biaxial strain to the bottom layer of AA stacked bilayer MoS$_{2}$. The Mo atoms of the bottom and top layers are represented by red and blue colors, respectively. We use yellow for the S atoms of both layers. The circles denote the high-symmetry stackings. The arrows in the side view indicates the applied tensile strain to the bottom layer. (b) and (c): Interlayer separation and corrugation distribution of a 2% - strain-induced moiré pattern of MoS$_{2}$. (d) and (e): Distribution of strains in the bottom and top layers of the same moiré pattern.
• Figure 2: (a): Distribution of order parameter in a MSL induced by 2%-strain. (b) and (d): The schematic of the solitons in a strain-induced and twist-induced moiré pattern, respectively. The arrows represent the orientation of order parameters. (c): Variation of domain wall width with applied strain
• Figure 3: (a): Electronic band structure of a 3.3% - biaxially strained bilayer MoS$_2$ with the valence band maximum set to zero. (b)-(e): $|\psi_{\Gamma_{M}}(\textbf{r})|^2$ averaged along the out-of-plane direction for the first four distinct energy levels near the valence band edge. (f)-(g): The same for the first two energy levels at the conduction band edge. (h): The effective planar potential of the relaxed MSL for the same system. The maroon, pink and green dots represent the $A\tilde{A}$, $A\tilde{B}$ and $B\tilde{A}$ high symmetry regions respectively. The colorbar is in meV.
• Figure 4: (a): Top and side view of the relaxed atomic structure of a moiré pattern formed by applying isotropic biaxial strain to the bottom layer of AA$^\prime$ stacked bilayer MoS$_{2}$. The circles denote the high-symmetry stackings. (b) and (c): Interlayer separation and corrugation distribution of a 2% strain-induced moiré pattern of MoS$_{2}$. (d) and (e): Distribution of strains in the bottom and top layers of the same moiré pattern.
• Figure 5: (a): Electronic band structure of 2% biaxially strained anti-parallely stacked moiré pattern of MoS$_{2}$ with the valence band maximum set to zero. (b)-(d) ((e)-(g)): $|\psi_{\Gamma_{M}}(\textbf{r})|^2$ averaged along the out-of-plane direction for the first three distinct flat bands near the valence band edge (conduction band edge) . (h): The effective planar potential of the relaxed moiré pattern for the same system. The colorbar is in meV.