Radiative Heat Transfer in Freestanding Silicon Nitride Membranes
Chang Zhang, Mathieu Giroux, Thea Abdul Nour, Raphael St-Gelais
Abstract
Freestanding silicon nitride ($\mathrm{Si}\mathrm{N}$) mechanical resonators are of central interest in applications such as temperature and mass sensing, and for fundamental optomechanical research. Understanding thermal coupling between a membrane resonator and its environment is required to predict thermal noise, frequency noise, and sensor responses to temperature changes. In this work, we provide closed-form derivations of intrinsic thermal-coupling quantities in freestanding 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 freestanding thin film anchored on all sides when small temperature differences between the film and its environment are considered. We particularly emphasize the specific case of $\mathrm{Si}\mathrm{N}$, 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 can even dominate thermal coupling for membranes of sizes commonly used in optomechanics experiments. We find that our model is in agreement with preliminary experimental results on radiative heat transfer between a ceramic heater and a $3\ifmmode\times\else\texttimes\fi{}3\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{2}$ membrane in a high vacuum.