Electronic States of Graphene Nanoribbons

TL;DR

This paper analyzes the electronic states of graphene nanoribbons with zigzag and armchair edges, demonstrating how boundary conditions influence their band structures and surface states, with results matching tight-binding calculations.

Contribution

It provides a quantitative understanding of the electronic states in graphene nanoribbons using Dirac equations with boundary conditions, clarifying edge effects and band structure variations.

Findings

Zigzag nanoribbons have surface states that evolve into zero-energy states as width increases.

Armchair nanoribbons exhibit metallic or insulating behavior depending on width divisible by 3.

Dirac equation solutions agree with tight-binding results except for very narrow ribbons.

Abstract

We study the electronic states of narrow graphene ribbons (``nanoribbons'') with zigzag and armchair edges. The finite width of these systems breaks the spectrum into an infinite set of bands, which we demonstrate can be quantitatively understood using the Dirac equation with appropriate boundary conditions. For the zigzag nanoribbon we demonstrate that the boundary condition allows a particle- and a hole-like band with evanescent wavefunctions confined to the surfaces, which continuously turn into the well-known zero energy surface states as the width gets large. For armchair edges, we show that the boundary condition leads to admixing of valley states, and the band structure is metallic when the width of the sample in lattice constant units is divisible by 3, and insulating otherwise. A comparison of the wavefunctions and energies from tight-binding calculations and solutions of the Dirac equations yields quantitative agreement for all but the narrowest ribbons.