Measurement of electronic heat dissipation in highly disordered graphene

We have measured the electronic heat dissipation of hot electrons in highly disordered millimeter scale graphene at temperatures $T=0.3\text{--}3$ K. Disorder was introduced by hydrogenation of graphene, bringing low-temperature electron conduction below the Ioffe-Regel criterion for metallic conduction. Resistive thermometry was employed to determine the dependence of electron temperature on applied electrical power. The relation between heat flow and electron temperature was found to be well described by a power law with an exponent $\ensuremath{\beta}\ensuremath{\sim}3.7\text{--}3.9$ and a coupling coefficient $\mathrm{\ensuremath{\Sigma}}\ensuremath{\sim}1\phantom{\rule{0.28em}{0ex}}\mathrm{mW}/{\mathrm{m}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{\beta}}$. Our observations are similar to electronic heat dissipation of a two-dimensional electron gas in the hydrodynamic limit of electron-phonon coupling, corresponding to acoustic phonon emission into the substrate.