Band structures of periodic carbon nanotube junctions and their symmetriesanalyzed by the effective mass approximation

The hand structures of the periodic nanotube junctions are investigated by the effective mass theory and the tight binding model. The periodic junctio ns are constructed by introducing pairs of a pentagonal defect and a heptag onal defect periodically in the carbon nanotube. We treat the periodic junc tions composed by two kinds of metallic nanotubes with almost same radii, t he ratio of which is between 0.7 and 1. The discussed energy region is near the undoped Fermi level. The energy bands are expressed with closed analyt ical forms by the effective mass theory. They are similar to the dispersion relation of Kronig-Penny model and coincide well with the numerical result s by the tight binding model. The width of the gap and the hand are in inve rse proportion to the length of the unit cell. The degeneracy and repulsion between the two bands are determined only from symmetries.