Optical phonon interacting with electrons in carbon nanotubes

Effects of interactions with electrons on optical phonons are studied in an effective-mass approximation. The longitudinal mode with displacement in the axis direction is lowered in its frequency, while the transverse mode with displacement in the circumference direction is raised, in metallic nanotubes. The shifts are opposite but their absolute values are smaller in semiconducting nanotubes. Only the longitudinal mode has a considerable broadening in metallic nanotubes. In the presence of an Aharonov-Bohm magnetic flux, the broadening appears for the transverse mode and diverges when the induced gap becomes the same as the frequency of the optical phonon.