  Silicon nitride (SiN) drumhead resonators offer a promising platform for
thermal sensing due to their high mechanical quality factor and the high
temperature sensitivity of their resonance frequency. As such, gaining an
understanding of heat transport in SiN resonators as well as their sensing
noise limitations is of interest, both of which are goals of the present work.
We first present new experimental results on radiative heat transport in SiN
membrane, which we use for benchmarking two recently proposed theoretical
models. We measure the characteristic thermal response time of square SiN
membranes with a thickness of 90 $\pm$ 1.7 nm and side lengths from 1.5 to 12
mm. A clear transition between radiation and conduction dominated heat
transport is measured, in close correspondence with theory. In the second
portion of this work, we use our experimentally validated heat transport model
to provide a closed-form expression for thermal fluctuation-induced frequency
noise in SiN membrane resonators. We find that, for large area SiN membranes,
thermal fluctuations can be greater than thermomechanical contributions to
frequency noise. For the specific case of thermal radiation sensing
applications, we also derive the noise equivalent power resulting from thermal
fluctuation-induced frequency noise, and we show in which conditions it reduces
to the classical detectivity limit of thermal radiation sensors. Our work
therefore provides a path towards achieving thermal radiation sensors operating
at the never attained fundamental detectivity limit of bolometric sensing. We
also identify questions that remain when attempting to push the limits of
radiation sensing, in particular, the effect of thermal fluctuation noise in
closed-loop frequency tracking schemes remains to be clarified.
