Chirality dependence of exciton effects in single-wall carbon nanotubes: Tight-binding model

We have studied the exciton properties of single-wall carbon nanotubes by solving the Bethe-Salpeter equation within tight-binding models. The screening effect of the $\pi$ electrons in carbon nanotubes is treated within the random phase and static screened approximations. The exciton wave functions along the tube axis and circumference are discussed as a function of $(n,m)$. A $2n+m=\mathrm{const}$ family behavior is found in the exciton wave function length, excitation energy, binding energy, and environmental shift. This family behavior is understood in terms of the trigonal warping effect around the $K$ point of a graphene layer and curvature effects. The large family spread in the excitation energy of the Kataura plot is found to come from the single-particle energy.