Trigonal warping effect of carbon nanotubes

Chirality-dependent van Hove singularities (vHs) of the one-dimensional electronic density of states (DOS) are discussed in connection with resonant Raman spectroscopy. The effect of trigonal warping on the energy dispersion relations near the Fermi energy splits the peaks of the density of states for metallic nanotubes, and the magnitude of this effect depends on the chiral angle of the carbon nanotube. The width of the peak splitting has a maximum for metallic zigzag nanotubes, and no splitting is obtained for armchair nanotubes or semiconducting nanotubes. We also find an additional logarithmic singularity in the electronic DOS for carbon nanotubes that is related to a two-dimensional singularity, which does not depend on either the diameter or the chirality.