Abstract
Band engineering in twisted bilayers of the five generic two-dimensional Bravais networks is demonstrated. We first derive symmetry-based constraints on the interlayer coupling, which helps us to predict and understand the shape of the potential barrier for the electrons under the influence of the moir\'{e} structure without reference to microscopic details. It is also pointed out that the generic constraints becomes best relevant when the typical length scale of the microscopic interlayer coupling is moderate. The concepts are numerically demonstrated in simple tight-binding models to show the band flattening due to the confinement into the potential profile fixed by the generic constraints. On the basis of the generic theory, we propose the possibility of anisotropic band flattening, in which quasi one-dimensional band dispersion is generated from relatively isotropic original band dispersion. In the strongly correlated regime, anisotropic band flattening leads to a spin-orbital model where intertwined magnetic and orbital ordering can give rise to rich physics.
Highlights
Having control over electron propagation is a key to derive new functionalities of materials
When we focus on the states from the M point in the original Brillouin zone, the nearest-neighbor pairs are occupied by orbitals with different C4 rotation eigenvalues, which we refer to as an s-type orbital and a dxy-type orbital. [See Fig. 8(c) in Appendix C for the schematic picture.] The different symmetry of the orbitals suppresses the nearestneighbor hopping in moiré cells, and the leading term will be the nearest-neighbor hopping between the same species of the orbitals, which results in a nested network of the two decoupled square lattices, whose lattice constant is 2 of the one of the moiré pattern
We derive generic constraints on the interlayer coupling that works as an effective potential for electrons in moiré patterns for bilayers of the all five Bravais networks
Summary
Having control over electron propagation is a key to derive new functionalities of materials. A moiré pattern induces spatially varying interlayer coupling/interaction, which affects the electron propagation and enables the band engineering [5,6,7,8,9]. Behaviors in twisted bilayer graphene are associated with effective mass reduction, or band flattening, caused by the moiré pattern [4,7,8,10,11]. Knowing how interlayer coupling is modulated under a moiré pattern [12,13] is essential to predict electronic structure in the stacked systems. The derived constraints help us to understand and predict how electrons are trapped in the effective potential given by the moiré pattern.
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