A Discrete-to-Continuum Model of Weakly Interacting Incommensurate Two-Dimensional Lattices

TL;DR

This paper develops a continuum variational model for two-dimensional incommensurate lattices with slight lattice mismatch and rotation, capturing the formation of domain walls and deformation patterns observed in discrete atomic models.

Contribution

It introduces a novel continuum model that accurately describes the behavior of weakly interacting incommensurate 2D lattices, including domain wall formation and lattice mismatch effects.

Findings

Model reproduces discrete lattice behavior qualitatively and quantitatively.

Predicts network of domain walls with large deformation.

Describes lattice mismatch using integer-valued parameters.

Abstract

In this paper we propose a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The two lattices have slightly different lattice parameters and there is a small relative rotation between them. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching, and bending deformation that accommodate the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters that can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the detail of deformation associated with the domain walls.