Heat Transport in Silicon Nitride Drum Resonators and its Influence on Thermal Fluctuation-Induced Frequency Noise

Abstract

Silicon nitride ($\mathrm{Si}\mathrm{N}$) drumhead resonators offer a promising platform for thermal sensing owing to their high mechanical quality factor and the high temperature sensitivity of their resonance frequency. As such, gaining an understanding of heat transport in $\mathrm{Si}\mathrm{N}$ resonators as well as their noise limitations are of interest, both of which are goals of the present work. We first present measurements of radiative heat transport in $\mathrm{Si}\mathrm{N}$ membranes, which we use for benchmarking two recently proposed theoretical models. We measure the characteristic thermal response time of square $\mathrm{Si}\mathrm{N}$ membranes with a thickness of 90 \ifmmode\pm\else\textpm\fi{} 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 $\mathrm{Si}\mathrm{N}$ membrane resonators. We find that, for large-area $\mathrm{Si}\mathrm{N}$ 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 unattained 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.