Topological nature of dislocation networks in two-dimensional moiré materials

Moiré superlattice patterns at the interface of two-dimensional (2D) van der Waals (vdW) materials, arising from a small displacement between similar lattices, have been extensively studied over the past decade for their dramatic ability to tune material properties. However, previous work to understand the structure of these interfaces has largely focused on some special cases, particularly honeycomb lattices displaced by twist or isotropic scaling. In this work, we develop practical and analytical tools for understanding the moiré structure that can be generalized to other lattice distortions and lattice types. At large enough moiré lengths, all moiré systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortexlike topological defects. However, we find these topological defects to exist on a punctured torus, requiring different mathematical formalism than the topological defects in a superconductor or planar ferromagnet. In the case of twisted bilayer graphene, the defects are characterized by the free group on two generators. We find that antivortices occur in the presence of anisotropic heterostrain, such as a shear or anisotropic displacement, while arrays of vortices appear under a twist or isotropic displacement between vdW materials. Utilizing the dark field imaging capability of transmission electron microscopy (TEM), we experimentally demonstrate the existence of vortex and antivortex pair formation in a moiré system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moiré structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moiré systems.