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Nanoscale Thermal Imaging of VO$_2$ via Poole-Frenkel Conduction

We present a method for nanoscale thermal imaging of insulating thin films using atomic force microscopy (AFM), and we demonstrate its utility on VO$_2$. We sweep the applied voltage $V$ to a conducting AFM tip in contact mode and measure the local current $I$ through the film. By fitting the $IV$ curves to a Poole-Frenkel conduction model at low $V$, we calculate the local temperature with spatial resolution better than 50 nm using only fundamental constants and known film properties. Our thermometry technique enables local temperature measurement of \textit{any} insulating film dominated by the Poole-Frenkel conduction mechanism, and can be extended to insulators that display other conduction mechanisms.