First-principles calculations of the radial breathing mode of single-wall carbon nanotubes

The radial breathing mode is a promising candidate for the analysis of conformational and electronic properties of carbon nanotubes. The paper presents calculations based on the local density approximation to determine the radial breathing mode (RBM) frequency of various zigzag and armchair tubes with radii between 3.5 and 8.1 Å. The frequencies are derived from both frozen phonon calculations for nanotubes and from the elastic constants of the flat graphene sheet. The RBM frequency of the (10,10) armchair tube is found to be $175{\mathrm{cm}}^{\mathrm{-}1}$ and $174{\mathrm{cm}}^{\mathrm{-}1}$ for the two techniques, respectively. These values are in very good agreement with one of the strongest components in the structured Raman band of the radial breathing modes observed at $177{\mathrm{cm}}^{\mathrm{-}1}.$ Since the RBM frequency turns out to scale with $1/R$ the calculations also allow one to evaluate the frequency of this mode for tubes with arbitrary chirality.