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 comparisons between theory and experiment are presented.