Band structures of periodic carbon nanotube junctions and their   symmetries analyzed by the effective mass approximation
  The band structures of the periodic nanotube junctions are investigated by the effective mass theory and the tight binding model.   The periodic junctions are constructed by introducing pairs of a pentagonal defect and a heptagonal defect periodically in the carbon nanotube.   We treat the periodic junctions whose unit cell is composed by two kinds of metallic nanotubes with almost same radii, the ratio of which is between 0.7 and 1 .   The discussed energy region is near the undoped Fermi level where the channel number is kept to two, so there are two bands.   The energy bands are expressed with closed analytical forms by the effective mass theory with some assumptions, and they coincide well with the numerical results by the tight binding model. Differences between the two methods are also discussed. Origin of correspondence between the band structures and the phason pattern discussed in Phys. Rev. B {\bf 53}, 2114, is clarified. The width of the gap and the band are in inverse proportion to the length of the unit cell, which is the sum of the lengths measured along the tube axis in each tube part and along 'radial' direction in the junction part. The degeneracy and repulsion between the two bands are determined only from symmetries. 
Ryo Tamura and Masaru Tsukada (University of Tokyo)