  Free-standing silicon nitride (SiN) mechanical resonators are of central
interests in applications such as temperature and mass sensing, and for
fundamental optomechanical reasearch. Understanding thermal coupling between a
membrane resonator and its environment is required for predicting thermal
noise, frequency noise, as well as sensors responses to temperature changes. In
this work, we provide closed-form derivations of intrinsic thermal coupling
quantities in free-standing thin films, namely total thermal conductance with
the surroundings, thermal response time, and the relative contribution of
thermal radiation. Our model is valid for any free-standing thin film anchored
on all sides, although we particularly emphasize the specific case of SiN for
which spectral emissivity is thoroughly investigated as a function of thickness
and temperature. We find that radiative heat exchanges can play a
non-negligible role, and even dominate thermal coupling for membranes of sizes
commonly employed in optomechanics experiments. We experimentally confirm the
validity of our model by measuring radiative thermal coupling between a SiN
mechanical resonator and a ceramic heater in high vacuum.
