Edge States in a Relativistic Description of Carbon Nanotubes

Graphene based systems admit a relativistic description in their low energy sector. This is due to the linear dispersion around the systems Dirac points. At such low energies where the dispersion is linear, support for bulk states at the edges of zigzag carbon nanotubes has been theoretically demonstrated. This is due to the wavefunctions of the two triangular sub-lattices being out of phase for certain system configurations, and the less stringent boundary conditions applied to relativistic systems. We demonstrate that no such theoretical support for bulk states at the edge is found for armchair carbon nanotubes. Instead, it is shown that the armchair carbon nanotube exhibits non-relativistic results where the charge density necessarily goes to zero at the edge, as would be expected in a system that admitted a Schrodinger description. These results are explained in terms of the shape of the edges and the resultant boundary conditions that we use when solving the Dirac equation for both zigzag and armchair systems in their low energy limit. The shape of the armchair edge requires that the wavefunctions of both sub-lattices equal zero at the edge and therefore, the density must be zero at the edge as well.