Three-dimensional Moir\'e crystallography

Abstract

Moir\'e materials, typically confined to stacking atomically thin, two - dimensional (2D) layers such as graphene or transition metal dichalcogenides, have transformed our understanding of strongly correlated and topological quantum phenomena. The lattice mismatch and relative twist angle between 2D layers have shown to result in Moir\'e patterns associated with widely tunable electronic properties, ranging from Mott and Chern insulators to semi- and super-conductors. Extended to three-dimensional (3D) structures, Moir\'e materials unlock an entirely new crystallographic space defined by the elements of the 3D rotation group and translational symmetry of the constituent lattices. 3D Moir\'e crystals exhibit fascinating novel properties, often not found in the individual components, yet the general construction principles of 3D Moir\'e crystals remain largely unknown. Here we establish fundamental mathematical principles of 3D Moir\'e crystallography and propose a general method of 3D Moir\'e crystal construction using Clifford algebras over the field of rational numbers. We illustrate several examples of 3D Moir\'e structures representing realistic chemical frameworks and highlight their potential applications in condensed matter physics and solid-state chemistry.