  We calculate the current-voltage characteristic of metallic nanotubes at high
bias voltage showing that a bottleneck exists for short nanotubes in contrast
to large ones. We attribute this to a redistribution of lower-lying acoustic
phonons caused by phonon-phonon scattering with hot optical phonons. The
current-voltage characteristic and the electron and phonon distribution
functions are derived analytically, and serve to obtain in a self-contained way
the frequency shift and line broadening of the zone center optical phonons due
to the electron-phonon coupling at high bias. We obtain a positive frequency
shift from the zero bias shift and no broadening of the optical phonon mode at
very high voltages, in agreement with recent experiments.
