Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model

TL;DR

This paper extends a quantum graph model to bilayer and trilayer Bernal-stacked graphene, showing Dirac cones only in trilayer graphene and aligning with existing physics models.

Contribution

It provides a rigorous quantum graph framework for multilayer graphene, explicitly characterizing spectra and dispersion relations, and confirms the presence of Dirac cones solely in trilayer graphene.

Findings

Dirac cones are present only in trilayer graphene.

Bilayer graphene has a gapless parabolic band component.

Model aligns with tight-binding and continuum models.

Abstract

A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac cones are then proven to be present only for trilayer graphene, although the bilayer has a gapless parabolic band component. Our model rigorously exhibits basic facts from tight-binding calculations, effective two-dimensional models and a $\pi$-orbital continuum model with nearest-neighbour tunneling that have been discussed in the physics literature.