Curvature induced optical phonon frequency shift in metallic carbon nanotubes

The quantum corrections to the frequencies of the $Gamma$ point longitudinal optical (LO) and transverse optical (TO) phonon modes in carbon nanotubes are investigated theoretically. The frequency shift and broadening of the TO phonon mode strongly depend on the curvature effect due to a special electron-phonon coupling in carbon nanotubes, which is shown by the Fermi energy dependence of the frequency shift for different nanotube chiralities. It is also shown that the TO mode near the $Gamma$ point decouples from electrons due to local gauge symmetry and that a phonon mixing between LO and TO modes is absent due to time-reversal symmetry. Some comparison between theory and experiment is presented.