Symmetry of boundary conditions of the Dirac equation for electrons in carbon nanotubes

TL;DR

This paper classifies boundary conditions for the Dirac equation in carbon nanotubes, showing how different conditions affect valley degeneracy and electronic properties.

Contribution

It derives and classifies all energy-independent boundary conditions for the Dirac equation in metallic nanotubes based on symmetry considerations.

Findings

Different boundary conditions lead to varying valley degeneracy breaking.

Classification of boundary conditions based on symmetry properties.

Implications for electronic spectrum tuning in nanotubes.

Abstract

We consider the effective mass model of spinless electrons in single wall carbon nanotubes that is equivalent to the Dirac equation for massless fermions. Within this framework we derive all possible energy independent hard wall boundary conditions that are applicable to metallic tubes. The boundary conditions are classified in terms of their symmetry properties and we demonstrate that the use of different boundary conditions will result in varying degrees of valley degeneracy breaking of the single particle energy spectrum.