Flat Bands in Twisted Bilayers of Two-Dimensional Polar Materials

The existence of Bloch flat bands provides an facile pathway to realize strongly correlated phenomena in materials. Using density-functional theory and tight-binding approach, we show that the flat bands can form in twisted bilayer of hexagonal boron nitride ($h$BN). However, unlike the twisted graphene bilayer where a magic angle is needed to form the flat band, for the polar $h$BN, the flat bands can appear as long as the twisted angle is less than certain critical values. Our simulations reveal that the valence band maximum (conduction band minimum) states are predominantly resided in the regions of the moir\'{e} supperlattice where the anion N (cation B) atoms in both layers are on top of each other. The preferential localization of these valence and conduction states originate from the chemical potential difference between N and B and is enhanced by the stacking effects of N and B in both layers, respectively, as demonstrated by an analysis of the energy level order of the $h$BN bilayers with different stacking patterns. When these states are spatially localized because regions with a specific stacking pattern are isolated for moir\'{e} supperlattices at sufficient small twist angle, completely flat bands will form. This mechanism is applicable to other twisted bilayers of two-dimensional polar crystals.