Evidence of Joule heat and carriers injection from temperature-dependent IV characteristics on VO$_2$ sub-micron devices

Temperature dependence of the current-induced Metal-Insulator Transition (MIT) of thin VO$_2$ films was measured for devices ranging from macroscopic size down to 1 micron length. While the V vs. I characteristics of macroscopic VO$_2$ device showed smooth transition, due to Joule heating, the small, 1 $\mu$m, device exhibit a completely different behavior. We find an V vs. I exhibiting negative differential resistance with many discrete voltage jumps. The jumps range from a few millivolts to a few volts, similar to resistance jumps we observed in the temperature driven transition of nanoscale VO$_2$ devices. From the temperature dependence of these jumps we can differentiate between MIT induced by Joule heating or due to carrier injection. These results are consistent with previous works indicating a phase separation across MIT in VO$_2$ thin films and could shed light on the VO$_2$ MIT mechanism.