We present a framework that explains the strong connection in 2D materials between mechanics and electronic structure, via dislocation theory. Within this framework, moiré patterns created by layered 2D materials may be understood as dislocation arrays, and vice versa. The dislocations are of a unique type that we describe as van der Waals dislocations, for which we present a complete geometrical description, connected to both stretch and twist moiré patterns. A simple computational scheme, which reduces the complexity of the electronic interaction between layers in order to make the problem computationally tractable, is introduced to simulate these dislocation arrays, allowing us to predict and explain all of the observed moiré patterns in 2D material systems within a unique framework. We extend this analysis as well to defects in moiré patterns, which have been reported recently, and which are the result of defects of the same symmetry in the constituent 2D material layers. Finally, we show that linear defects in the moiré space can be viewed as unidimensional topological states, and can be engineered using our framework.

In-plane heterostructures of 2D materials form interface misfit dislocations to relieve lattice mismatch strain, much like heterostructures of 3D materials. Here, using graphene-hexagonal boron nitride (h-BN) as a model system, we consider interface misfit dislocations in 2D lateral heterostructures resting on a flat support layer that prevents out-of-plane deformation. Using an accurate empirical interatomic potential, we carry out a rigorous energetic analysis of the graphene/h-BN interface with 5-7 or 8-6 dislocation cores. We define and extract critical thicknesses for the formation of an interface misfit dislocation in the heterostructure, for the limiting cases when the h-BN or graphene domains are significantly different in size (equivalent to the classic 3D thin film critical thickness problem), and the intermediate case, where the h-BN and graphene domains are of comparable size (equivalent to the classic 3D compliant substrate problem). This makes it possible to compare the alternative dislocation core structures and to determine the resulting dislocation core energy in a continuum analysis. It also reveals a design space where defect-free heterostructures can be grown.

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We investigate the influence of strain and dislocations on band alignment in GaSb/GaAs quantum dot systems. Composition profiles from cross-sectional scanning tunneling microscopy images are interpolated onto a finite element mesh in order to calculate the distribution of local elastic strain, which is converted to a spatially varying band alignment using deformation potential theory. Our calculations predict that dislocation-induced strain relaxation and charging lead to significant local variations in band alignment. Furthermore, misfit strain induces a transition from a nested (type I) to a staggered (type II) band alignment. Although dislocation-induced strain relaxation prevents the type I to type II transition, electrostatic charging at dislocations induces the staggered band alignment once again.