  Graphene is a material with remarkable electronic properties and exceptional
thermal transport properties near room temperature, which have been well
examined and understood. However at very low temperatures the thermodynamic and
thermal transport properties are much less well explored and somewhat
surprisingly, is expected to exhibit extreme thermal isolation. Here we
demonstrate an ultra-sensitive, wide-bandwidth measurement scheme to probe the
thermal transport and thermodynamic properties of the electron gas of graphene.
We employ Johnson noise thermometry at microwave frequency to sensitively
measure the temperature of the electron gas with resolution of $4 mK/\sqrt{Hz}$
and a bandwidth of 80 MHz. We have measured the electron-phonon coupling from
2-30 K at a charge density of $2\cdot 10^{11} cm^{-2}$. Utilizing bolometric
mixing, we have sensed temperature oscillations with period of 430 ps and have
determined the heat capacity of the electron gas to be $2\cdot 10^{-21}
J/(K\cdot \mu m^2)$ at 5 K which is consistent with that of a two dimensional,
Dirac electron gas. These measurements suggest that graphene-based devices
together with wide bandwidth noise thermometry can generate substantial
advances in the areas of ultra-sensitive bolometry, calorimetry, microwave and
terahertz photo-detection, and bolometric mixing for applications in areas such
as observational astronomy and quantum information and measurement.
