Energy Spectrum and Optical Properties of Single-Wall Carbon Nanotubes

Abstract

The energy spectrum and the optical absorption spectra of single-wall carbon nanotubes with chiralities (5,5), (10,0), (9,0), (12,0), and (15,0) is calculated within the framework of the Shubin-Wonsowskii-Hubbard model taking into account distant (adjacent to the nearest) interstitial electron transfer. It is demonstrated that all of them, irrespective of the chirality, are narrow-gap semiconductors with a gap of ~0.1 eV that coincides with the available experimental data. The optical absorption spectra are also in good agreement with the experimentally measured spectra. orbitals of deeply lying bands forming the skeleton of the structure and the nonhybrid p-orbital forming the so-called wandering π -bonds. The states of these electrons are partially localized, and they can be considered in the Wannier representation. It then follows that the electrical conductivity of the CNT must be completely determined by the π -electron subsystem. At present there is a widespread opinion that the electrical conductivity and other properties of the CNT depend critically on the chirality indices. Thus it is believed that the CNT with chiralities (n, m) are metals if the difference between the chiralities n - m is a multiple of three; otherwise, they are semiconductors or dielectrics. This opinion is based on the energy spectrum of the π -electron subsystem obtained by Dresselhaus et al. (2-4) using the results of calculations carried out by Wallace (5) as far back as 1947. In (5) only the electron transfer from site to the neighboring site was taken into account. However, the experimental material accumulated by the present time demonstrates that there is no unambiguous correspondence between the chirality and the electrical conductivity of the CNT (6). This is most likely due to the fact that the Coulomb interaction in the π -electron subsystem whose energy, according to (7), can reach 10 eV and higher, is not taken into account. Results of calculations carried out in (8, 9) in which the Coulomb interaction in the same site was considered demonstrated that all CNT, irrespective of chiralities, were semiconductors with a gap of ~1 eV. However, experimental data on the electrical conductivity (10-12) and direct measurement (13) of the density of states for the CNT with chiralities (15,0), (12,0), and (9,0) demonstrated that the CNT, irrespective of the chiralities, are narrow-gap semiconductors whose gap was in the range ~0.01-0.1 eV. This circumstance calls for a more detailed study of the examined problem.