Theory for constructing effective models for electrons in generic bilayer graphene

TL;DR

This paper develops practical techniques for formulating effective low-energy electron models in bilayer graphene, enabling better understanding and simulation of their electronic properties.

Contribution

It introduces a systematic approach to derive effective models from tight-binding descriptions, including parameterization schemes and applicability to small twist angle configurations.

Findings

Effective Hamiltonians can be constructed with kinetic and potential terms.

The approach is applicable to small twist angle bilayer graphene.

Numerical validation of model accuracy and applicability.

Abstract

We present and discuss in detail practical techniques in formulating effective models to describe the dynamics of low-energy electrons in generic bilayer graphene. Starting from a tight-binding model using the $p_z$ orbital of carbon atoms as a representation basis set, we reformulate it into the problem of coupling between Bloch states defined in each graphene layer. This approach allows transferring the original problem into the determination of Bloch states in two independent material layers and coupling rules of such states. We show two schemes to parameterize coupled Bloch state vectors. For the bilayer graphene configurations of small twist angle in which the long wavelength approximation is applicable, we show that an effective Hamiltonian can be written in the canonical form of a kinetic term defined by the momentum operator and a potential term defined by the position operator. The validity of effective models of different sophistication levels and their potential application in treating various physical aspects are numerically discussed.