Dispersion of "Dispersionless Zero Mode": Comments on L. Brey and H.A. Fertig paper Electronic States of Graphene Nanoribbons Studied With The Dirac Equation

TL;DR

This paper derives an exact analytical dispersion relation for edge states in graphene zigzag ribbons, challenging previous explanations by showing their limited validity across the wave vector spectrum.

Contribution

It provides a precise analytical expression for the dispersion of edge states, clarifying the limitations of earlier models for graphene nanoribbons.

Findings

Exact dispersion relation for edge states derived

Previous models only valid in narrow wave vector range

Revised understanding of zero mode behavior

Abstract

As a particular application of the earlier proposed model of graphene as a macromolecule, we found the exact analytical expression of dispersion relation for the band of edge states in graphene zigzag ribbons. This band is often referred to as "dispersionless band" or "zero mode". The obtained result contrasts description of edge states given in the referenced paper, showing that the earlier given explanation is valid only for a very narrow region of values of the electron/hole wave vector, but for the rest, it is not correct.