Formal justification of a continuum relaxation model for one-dimensional moir\'e materials

TL;DR

This paper rigorously derives a one-dimensional continuum relaxation model for moiré materials from an atomistic perspective, clarifying the conditions under which the continuum approximation is valid.

Contribution

It provides a formal derivation of the continuum model from atomistic models, establishing the limit process and parameter relations.

Findings

The continuum model emerges as the limit when ownarrow 0 and ownarrow 0 with ta fixed.

Minimizers of the 1D model are computed and analyzed.

The derivation clarifies the connection between atomistic and continuum descriptions.

Abstract

Mechanical relaxation in moir\'e materials is often modeled by a continuum model where linear elasticity is coupled to a stacking penalty known as the Generalized Stacking Fault Energy (GSFE). We review and compute minimizers of a one-dimensional version of this model, and then show how it can be formally derived from a natural atomistic model. Specifically, we show that the continuum model emerges in the limit $\epsilon \downarrow 0$ and $\delta \downarrow 0$ while holding the ratio $\eta := \frac{\epsilon^2}{\delta}$ fixed, where $\epsilon$ is the ratio of the monolayer lattice constant to the moir\'e lattice constant and $\delta$ is the ratio of the typical stacking energy to the monolayer stiffness.