Moir\'e of Moir\'e: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

TL;DR

This paper develops a multi-scale model to analyze the complex mechanical relaxation patterns in incommensurate twisted trilayer van der Waals heterostructures, revealing a 'moiré of moiré' phenomenon and symmetry-breaking effects.

Contribution

It introduces a configuration space-based continuum model for incommensurate trilayer systems with two twist angles, enabling relaxation pattern prediction without supercell approximations.

Findings

Relaxation patterns form a 'moiré of moiré' due to incommensurate coupling.

Trilayer relaxation can break in-plane symmetry, unlike bilayers.

Model accurately captures relaxation in twisted graphene and WSe₂.

Abstract

The incommensurate stacking of multi-layered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. Here we present a multi-scale model to obtain the mechanical relaxation pattern of twisted trilayer van der Waals (vdW) heterostructures with two independent twist angles, a generally incommensurate system without a supercell description. We adopt the configuration space as a natural description of such incommensurate layered materials, based on the local environment of atomic positions, bypassing the need for commensurate approximations. To obtain the relaxation pattern, we perform energy minimization with respect to the relaxation displacement vectors. We use a continuum model in combination with the Generalized Stacking Fault energy to describe the interlayer coupling, obtained from first-principles calculations based on Density Functional Theory. We show that the relaxation patterns of twisted trilayer graphene and $\mathrm{WSe_2}$ are "moir\'e of moir\'e", as a result of the incommensurate coupling two bilayer moir\'e patterns. We also show that, in contrast to the symmetry-preserving in-plane relaxation in twisted bilayers, trilayer relaxation can break the two-fold rotational symmetry about the xy-plane when the two twist angles are equal.