Electronic band structure of isolated and bundled carbon nanotubes

We study the electronic dispersion in chiral and achiral isolated nanotubes as well as in carbon nanotube bundles. The curvature of the nanotube wall is found not only to reduce the band gap of the tubes by hybridization, but also to alter the energies of the electronic states responsible for transitions in the visible energy range. Even for nanotubes with larger diameters (1–1.5 nm) a shift of the energy levels of $\approx 100\hspace{0.5em}\mathrm{meV}$ is obtained in our ab initio calculations. Bundling of the tubes to ropes results in a further decrease of the energy gap in semiconducting nanotubes; the bundle of (10,0) nanotubes is even found to be metallic. The intratube dispersion, which is on the order of 100 meV, is expected to significantly broaden the density of states and the optical absorption bands in bundled tubes. We compare our results to scanning tunneling microscopy and Raman experiments, and discuss the limits of the tight-binding model including only $\pi$ orbitals of graphene.