Optical transitions in semiconducting zigzag carbon nanotubes with small diameters: A first-principles broad-range study

Abstract

We study all possible band-to-band transitions between 12 valence and 16 conduction bands of (8,0), (10,0), and (7,0) nanotubes and calculate the corresponding dipole moments using first-principles methods in a wide ultraviolet-visible-infrared range of photon energies. The goal is to investigate the optical transitions and the selection rules for nanotubes with small diameters in a broad range of energies, taking into account the curvature and rehybridization effects on the dipole moment. Our calculations show the conservation of a modified quantum number, $\stackrel{\fontencoding{LECO}\selectfont\char177{}}{m}$, for all the transitions and reversal of the horizontal parity. Besides the $\ensuremath{\pi}\text{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{\ast}}$ transitions, we observe the possibility of $\ensuremath{\pi}\text{\ensuremath{-}}{\ensuremath{\sigma}}^{\ensuremath{\ast}}$, $\ensuremath{\sigma}\text{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{\ast}}$, and $\ensuremath{\sigma}\text{\ensuremath{-}}{\ensuremath{\sigma}}^{\ensuremath{\ast}}$ transitions. In fact, the maximum dipole moment for the three nanotubes corresponds to the transition between valence and conduction bands that are both of $\ensuremath{\sigma}$ nature. The maximum transition rate for an (8,0) nanotube happens at approximately 1.42 eV, corresponding to the transition between the first valence and fourth conduction bands and mapping to the infrared region. The maximum of transition rate for (10,0) and (7,0) nanotubes happens at 0.79 eV and 3.03 eV, respectively. There exist high absorption probabilities in the infrared and visible region in all three nanotubes. Also, high absorption probability in the low ultraviolet region is demonstrated, mostly as a result of $\ensuremath{\sigma}\text{\ensuremath{-}}{\ensuremath{\sigma}}^{\ensuremath{\ast}}$ transitions.