Electronic Observables for Relaxed Bilayer Two-Dimensional Heterostructures in Momentum Space

Momentum space transformations for incommensurate two-dimensional 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 [Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 12233-12237]. 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. First, 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 straightforward algorithm for writing the complex momentum space model from the original real space model. Second, we implement this for twisted bilayer graphene with mechanical relaxation effects 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.