The effect of the mixing between π and σ 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 (${\mathit{E}}_{\mathrm{\theta }}$) has been estimated. This term cannot be neglected for tubules with a small radius. The relation of ${\mathit{E}}_{\mathit{g}}$ to R, ${\mathit{E}}_{\mathit{g}}$∝${\mathit{R}}^{\mathrm{-}1}$, 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 ${\mathit{E}}_{\mathrm{\theta }}$ is well correlated with a fraction of the σ component in the wave functions for the conduction-band bottom and valence-band top.

• Received 22 March 1995

DOI:https://doi.org/10.1103/PhysRevB.52.2723