Electronic and transport properties of rectangular graphene macromolecules and zigzag carbon nanotubes of finite length

TL;DR

This study analyzes the electronic properties of finite-length graphene ribbons and carbon nanotubes, revealing conditions for metallic behavior, universal dispersion laws, and localized edge states, using the Hückel model.

Contribution

It provides exact energy level calculations for finite graphene structures and identifies universal dispersion features and edge-localized states.

Findings

Metallic ribbons occur when width equals 2+3n.

Universal sin-like dispersion law at Fermi energy.

Localized edge states at zero energy in zigzag structures.

Abstract

We study one dimensional (1D) carbon ribbons with the armchair edges and the zigzag carbon nanotubes and their counterparts with finite length (0D) in the framework of the H\"{u}ckel model. We prove that a 1D carbon ribbon is metallic if its width (the number of carbon rings) is equal to $2+3n$. We show that the dispersion law (electron band energy) of a 1D metallic ribbon or a 1D metallic carbon nanotube has a universal {\it sin-}like dependence at the Fermi energy which is independent of its width. We find that in case of metallic graphene ribbons of finite length (rectangular graphene macromolecules) or nanotubes of finite length the discrete energy spectrum in the vicinity of $\varepsilon=0$ (Fermi energy) can be obtained exactly by selecting levels from the same dispersion law. In case of a semiconducting graphene macromolecule or a semiconducting nanotube of finite length the positions of energy levels around the energy gap can be approximated with a good accuracy. The electron spectrum of 0D carbon structures often include additional states at energy $\varepsilon=0$, which are localized on zigzag edges and do not contribute to the volume conductivity.