Enumeration of Moiré patterns of a hexagonal twisted bilayer: Applications to intercalated transition metals

Abstract

A real-space method using generating integers is used to determine possible commensurate lattice Moiré patterns for a bilayer of two equal hexagonal lattices, which can in principle be extended to lattice mismatched bilayers. These Moiré patterns can be classified by a pair of relatively prime integers (n,m), wherein a rotation θ(n,m) of the top hexagonal lattice maps its lattice vector (n,m) to (m,n) of the bottom lattice. Within this formulation, the area of the commensurate supercell is proportional to (n2+m2+nm) and the number of coincident lattice sites per supercell is given by (n−m)2. Taking bilayer boron nitride (BN) as an example, we present how to systematically generate Moiré patterns and explore the differences in local chemistry in the interstitial region by impurity intercalation. Systematic calculations of the properties of intercalated 3d transition metals were performed in an h-BN (4,3) bilayer, corresponding to a rotation of 9.43 degrees. These calculations reveal that local symmetry in the intercalated region significantly affect the energetics and magnetization of the intercalated species. These results highlight that Moiré pattern physics is not limited to optoelectronic/electronic phenomena, such as interfacial exciton formation or magic angle superconductivity, but it also produces chemical and magnetic atomic site selectivity, which may play important roles in adsorption, catalysis, or quantum information.