Elastic plate basis for the deformation and electron diffraction of twisted bilayer graphene on a substrate

TL;DR

This paper introduces an elastic plate basis derived from plate theory to model deformation and electron diffraction in twisted bilayer graphene, validated against experiments and atomistic simulations.

Contribution

It presents a novel elastic plate basis that efficiently captures TBG deformation and diffraction patterns, bridging atomistic simulations and experimental data.

Findings

Few basis terms accurately model relaxed TBG structures.

Elastic plate models reproduce low-energy phonon modes.

Code provided for extracting basis coefficients from diffraction images.

Abstract

A basis is derived from elastic plate theory that quantifies equilibrium and dynamic deformation and electron diffraction patterns of twisted bilayer graphene (TBG). The basis is derived by solving in-plane and out-of-plane normal modes of an unforced parallelogram elastic plate. We show that a combination of only a few basis terms successfully captures the relaxed TBG structure with and without an underlying substrate computed using atomistic simulations. The results are validated by comparison with electron diffraction experiments. A code for extracting the elastic plate basis coefficients from an experimental electron diffraction image accompanies this paper. TBG dynamics are also studied by computing the phonon band structure from atomistic simulations. Low-energy phonons at the $\Gamma$ point are examined in terms of the mode shape and frequency. These modes are captured by simple elastic plate models with uniformly distributed springs for inter-layer and substrate interactions.