  Several data sets of electrical breakdown in air of single-wall carbon
nanotubes (SWNTs) on insulating substrates are collected and analyzed. A
universal scaling of the Joule breakdown power with nanotube length is found,
which appears independent of the insulating substrates used or their thickness.
This suggests the thermal resistances at the interface between SWNT and
insulator, and between SWNT and electrodes, govern heat sinking from the
nanotube. Analytical models for the breakdown power scaling are presented,
providing an intuitive, physical understanding of the breakdown process. The
electrical and thermal resistance at the electrode contacts limit the breakdown
behavior for sub-micron SWNTs, the breakdown power scales linearly with length
for microns-long tubes, and a minimum breakdown power (~ 0.05 uW) is observed
for the intermediate (~ 0.5 um) length range.
