The curvature of the nanotube sidewall and its effect on the electronic and optical properties of zigzag nanotubes

In this paper, we employ the density functional theory to perform a thorough study of the impact of the nanotube sidewall curvature on the optical properties by investigating several effects arising from this curvature, such as σ–π orbital rehybridization, the carbon–carbon bond length, and the mixing of d and p orbitals. Separating these and studying their effects independently, provides insight into the reasons behind the different values obtained for optical transition energies in experiments and in tight-binding/zonefolding calculations, where the curvature of the nanotube sidewall is neglected. As well, it is useful in the understanding of how and to what extent each of these effects are responsible for the accurate prediction of the electronic and optical properties of carbon nanotubes. Although the σ–π rehybridization is known as the main curvature effect in the literature, we show that the effect of bond length is quite significant as well and needs to be properly taken into account. For example, the first transition energy (E11) of a (4,0) nanotube is decreased by ∼64% and ∼46% due to σ–π rehybridization and bond length effect, respectively. Another important observation is that the overall effect of σ–π rehybridization and bond length does not have a monotonous diameter dependence. Rather, our study suggests a dependence on the nanotube index, n, since zigzag nanotubes with n(mod)3 = 1 show a different behavior compared to the ones with n(mod)3 = 2. We also show that for some of the nanotubes, the effect of including the d orbitals on the optical transition energies can be as significant as the effect of σ–π orbital rehybridization and bond length. Due to the importance of the bond length effect, we also investigate several methods of geometry optimization for nanotubes. It appears that the HSEH1PBE functional might lead to the most accurate prediction of the nanotube geometry compared to the other functionals tested.