Energy gaps of semiconducting nanotubules.

TLDR: An approximate expression for E, which can be applicable even to small-radius semiconducting tubules, is presented and it is shown that the E is well correlated with a fraction of the \ensuremath{\sigma} component in the wave functions for the conduction-band bottom and valence-band top.

Abstract: The effect of the mixing between \ensuremath{\pi} and \ensuremath{\sigma} orbitals upon the energy gaps (${\mathit{E}}_{\mathit{g}}$'s) of semiconducting carbon nanotubules has been theoretically studied within the tight-binding approximation. The magnitude of a chirality-dependent energy gap (${\mathit{E}}_{\mathrm{\ensuremath{\theta}}}$) has been estimated. This term cannot be neglected for tubules with a small radius. The relation of ${\mathit{E}}_{\mathit{g}}$ to R, ${\mathit{E}}_{\mathit{g}}$\ensuremath{\propto}${\mathit{R}}^{\mathrm{\ensuremath{-}}1}$, no longer holds for small-radius tubules. An approximate expression for ${\mathit{E}}_{\mathit{g}}$, which can be applicable even to small-radius semiconducting tubules, is presented. In addition, it is shown that the ${\mathit{E}}_{\mathrm{\ensuremath{\theta}}}$ is well correlated with a fraction of the \ensuremath{\sigma} component in the wave functions for the conduction-band bottom and valence-band top.