Anisotropic tight-binding model applied to zigzag ultra-small nanotubes

A single-wall carbon nanotube (SWCNT) can be visualized as a graphene rolled into a cylinder. Tight-binding band structure calculations, with hopping between nearest-neighbor pi orbitals only (NNTB), established rules by which both the mode in which the graphene is rolled up and the diameter determine whether the SWCNT is a metal or a semiconductor. However, when the diameter of the SWCNT is ultra-small its large curvature results in the breakage of these rules. In this work, we studied zigzag (n, 0) SWCNTs with diameters smaller than 0.7 nm using a pi orbital-only tight-binding model including anisotropy in the hopping between next-nearest-neighbor sites (ANNNTB). Band overlaps were found in the electronic band structures of the zigzag SWCNTs for n=3, 4, 5, and 6, indicating that they are metals. The reason why the band structures of armchair and chiral SWCNTs are less affected by curvature effects becomes clear with the ANNNTB model, as does the reason why non-degenerate states cause band overlaps of the zigzag SWCNTs for n=3, 4, 5, and 6. Our results show that a pi orbital-only tight-binding model is able to describe both the band overlaps and gaps obtained by ab initio calculations for zigzag SWCNTs.