Review of the tight-binding method applicable to the properties of moir\'e superlattices

TL;DR

This review comprehensively discusses tight-binding methods for modeling the electronic properties of moiré superlattices, addressing their complexity and connection to continuum models, with practical guidance for researchers.

Contribution

It provides an extensive overview of atomistic tight-binding Hamiltonians, numerical techniques, and their application to various moiré materials, highlighting advantages and practical considerations.

Findings

Detailed description of atomistic TB Hamiltonians for moiré systems

Comparison between TB models and continuum approaches

Practical examples demonstrating TB method advantages

Abstract

Moir\'e superlattices have emerged as a versatile platform for exploring a wide range of ex- otic quantum phenomena. Unlike angstrom-scale materials, the moir\'e length-scale system contains a large number of atoms, and its electronic structure is significantly modulated by the lattice relaxation. These features pose a huge theoretical challenge. Among the available theoretical approaches, tight-binding (TB) methods are widely employed to predict the electronic, transport, and optical properties of systems such as twisted graphene, twisted transition-metal dichalcogenides (TMDs), and related moir\'e materials. In this review, we pro- vide a comprehensive overview of atomistic TB Hamiltonians and the numerical techniques commonly used to model graphene-based, TMD-based and hBN-based moir\'e superlattices. We also discuss the connection between atomistic TB descriptions and effective low-energy continuum models. Two examples of different moir\'e materials and geometries are provided to emphasize the advantages of the TB methods. This review is intended to serve as a theoretical and practical guide for those seeking to apply TB methods to the study of various properties of moir\'e superlattices.