Tight-binding calculation with up to the 3rd nearest neighbor coupling for small-radius carbon nanotubes

We present the empirical tight-binding calculation of band structures of small-radius single-walled carbon nanotubes (SWCNTs). It incorporates the curvature effect as well as the electronic hopping up to the 3rd nearest neighbor and, as a result, closes the discrepancy between previous tight-binding calculations and first-principle studies. The calculation produces a non-monotonous variation of band gaps, where the gap increases initially and then decreases, when the tube radius decreases. In particular, in the zigzag (6, 0) case, it yields a vanishing band gap. We conclude that the inclusion of the 3rd nearest neighbor hopping is of paramount importance in the tight-binding calculation of small-radius SWCNTs.