Nanoscale thermal imaging of VO$_2$ via Poole-Frenkel conduction

We present a novel method for nanoscale thermal imaging of insulating thin films. We demonstrate this method on VO$_2$, which undergoes a sharp insulator-to-metal transition at 340 K. We sweep the voltage applied to a conducting atomic force microscope tip in contact mode at room temperature and measure the resultant current through a VO$_2$ film. The Poole-Frenkel (PF) conduction mechanism, which dominates in the insulating state of VO$_2$, is fit to extract the local temperature of the film using fundamental constants and known film properties. We measure the local electric field and temperature immediately preceding the insulator-to-metal transition in VO$_2$ to determine whether the transition can be triggered by an applied electric field alone. We calculate an average temperature of 334 $\pm$ 5 K, implying that Joule heating has locally warmed the sample very close to the transition temperature. Our thermometry technique opens up the possibility to measure the local temperature of any film dominated by the PF conduction mechanism, and presents the opportunity to extend our technique to other conduction mechanisms.