Circular dichroism of single-wall carbon nanotubes

Abstract

Circular dichroism (CD) of a single-wall carbon nanotube (SWNT) is calculated as a function of the wavelength of light. Because of the symmetry between the $K$ and ${K}^{\ensuremath{'}}$ points in the hexagonal Brillouin zone, the conventional theory for CD intensity gives a zero value in which the absorption probability near the $K$ point for right-handed circular polarized light and that near the ${K}^{\ensuremath{'}}$ point for left-handed circular polarized light cancel each other. Considering the phase differences of the light for carbon atoms of a nanotube, which are beyond so-called dipole approximation, a formulation of CD for a SWNT is presented. Analytic and numerical calculations show (1) the alternating sign of the CD intensity at ${E}_{ii}$ ($i=1,2,3,...$) van Hove singular energies and (2) opposite sign of the CD values as a function of wavelength of the light for different types and handedness of nanotubes, which reproduce the experimental results. In the metallic SWNTs, we predict the opposite sign of CD values for split ${E}_{ii}^{+}$ and ${E}_{ii}^{\ensuremath{-}}$.