Energy gaps, electronic structures, and x-ray spectroscopies of finite semiconductor single-walled carbon nanotubes

We report hybrid density functional theory calculations for electronic structures of hydrogen-terminated finite single-walled carbon nanotubes (6,5) and (8,3) up to 100 nm in length. Gap states that are mainly arisen from the hydrogen-terminated edges have been found in (8,3) tubes, but their contributions to the density of states become invisible when the tube is longer than 10 nm. The electronic structures of (6,5) and (8,3) tubes are found to be converged around 20 nm. The calculated band-gap energies of 100 nm long nanotubes are in good agreement with experimental results. The valence band structures of (6,5), (8,3), as well as (5,5) tubes are also investigated by means of ultraviolet photoelectron spectra (UPS), x-ray emission spectroscopy (XES), and the resonant inelastic x-ray scattering (RIXS) spectra theoretically. The UPS, XES and RIXS spectra become converged already at 10 nm. The length-dependent oscillation behavior is found in the RIXS spectra of (5,5) tubes, indicating that the RIXS spectra may be used to determine the size and length of metallic nanotubes. Furthermore, the chiral dependence observed in the simulated RIXS spectra suggests that RIXS spectra could be a useful technique for the determination of chirality of carbon nanotubes.