Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

TL;DR

This paper develops a rigorous framework for momentum space transformations in incommensurate 2D heterostructures, including mechanical relaxation effects, and demonstrates their impact on electronic coupling ranges in twisted bilayer graphene.

Contribution

It provides a clear algorithm for transforming between key spaces and extends the Bistritzer-MacDonald model to include relaxation effects, analyzing their influence on electronic properties.

Findings

Transformation algorithms for complex 2D heterostructures

Inclusion of mechanical relaxation in momentum space models

Coupling range increases for smaller twist angles

Abstract

Momentum space transformations for incommensurate 2D electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer-MacDonald model. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. Firstly, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straight-forward algorithm for writing the complex momentum space model from the original real space model. Secondly, we implement this for twisted bilayer graphene with mechanical relaxation affects included. We also analyze the convergence rates of the approximations, and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer-MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.