Spectrum of $\pi$-electrons in Graphene As a Macromolecule

TL;DR

This paper provides an exact analytical solution for the spectral properties of π-electrons in a graphene sheet with specific boundary shapes, revealing how symmetry influences electronic band structure and edge states.

Contribution

It presents the first exact solution for the spectral problem of π-electrons in graphene with armchair and zigzag boundaries, highlighting symmetry effects on electronic properties.

Findings

Closed-form expression for edge-state energies

Lower symmetry significantly affects band structure

Implications for graphene electronic applications

Abstract

We report the exact solution of spectral problem for a graphene sheet framed by two armchair- and two zigzag-shaped boundaries. The solution is found for the $\pi$ electron Hamiltonian and gives, in particular, a closed analytic expression of edge-state energies in graphene. It is shown that the lower symmetry of graphene, in comparison with $C_{6h}$ of 2D graphite, has a profound effect on the graphene band structure. This and other obtained results have far going implications for the understanding of graphene electronics. Some of them are briefly discussed.