Fermi energy in the non-parabolic band structure of a carbon nanotube

The band structure of a carbon nanotube is obtained from the graphene band structure. The band energy of CNT near to the minimum band energy is parabolic. However it is not parabolic in other regions. In the non-parabolic regime the density of states is proportional to a numerical integral that is different from Fermi integrals. The carrier statistics to all degeneracy levels are presented and limits are obtained. In the nondegenrate regime the results reproduce what is expected from, the Boltzmann statistics. However, the results differ in the degenerate regime. The Fermi energy with respect to the transformed conduction band edge is a function of carrier concentration. Given the density of states, the carrier concentration is obtained based on a numerical method as an alternative to the usual Fermi-Dirac integrals and we call it a carbon nanotube Fermi integral. The Fermi energy with respect to the band edge is a function of temperature and is independent of the carrier concentration in the nondegenrate regime. In the other strongly degenerate regime, the Fermi energy is a function of carrier concentration appropriate for given dimensionality, but is independent of temperature