Optical phonon tuned by fermi level in carbon nanotubes

The dependence of the frequency and broadening of optical phonons on carrier concentration is studied in carbon nanotubes within an effective-mass approximation. In metallic nanotubes, the frequency shift exhibits a logarithmic divergence and the broadening vanishes discontinuously when the Fermi level reaches the half of the optical-phonon frequency for the longitudinal mode with displacement in the axis direction, while the transverse mode is not affected. In semiconducting nanotubes, the frequency is raised for both transverse and longitudinal modes, exhibiting a behavior similar to level crossing.