The Effects of Finite Length on the Electronic Structure of Carbon   Nanotubes
  The electronic structure of finite-length armchair carbon nanotubes has been studied using several ab-initio and semi-empirical quantum computational techniques. The additional confinement of the electrons along the tube axis leads to the opening of a band-gap in short armchair tubes. The value of the band-gap decreases with increasing tube length, however, the decrease is not monotonic but shows a well defined oscillation in short tubes. This oscillation can be explained in terms of periodic changes in the bonding characteristics of the HOMO and LUMO orbitals of the tubes. Finite size graphene sheets are also found to have a finite band-gap, but no clear oscillation is observed. As the length of the tube increases the density of states (DOS) spectrum evolves from that characteristic of a zero-dimensional (0-D) system to that characteristic of a delocalized one-dimensional (1-D) system. This transformation appears to be complete already for tubes 5-10 nm long. The chemical stability of the nanotubes, expressed by the binding energy of a carbon atom, increases in a similar manner. 
Alain Rochefort, Dennis R. Salahub, Phaedon Avouris