  We report electrical transport experiments using the phenomenon of electrical
breakdown to perform thermometry that probe the thermal properties of
individual multi-walled nanotubes. Our results show that nanotubes can readily
conduct heat by ballistic phonon propagation, reaching a quantum-mechanical
limit to thermal conductance. We determine the thermal conductance quantum, the
ultimate limit to thermal conductance for a single phonon channel, and find
good agreement with theoretical calculations. Moreover, our results suggest a
breakdown mechanism of thermally activated C-C bond breaking coupled with the
electrical stress of carrying ~10^12 A/m^2. We also demonstrate a
current-driven self-heating technique to improve the conductance of nanotube
devices dramatically.
