Evidence of surface loss as ubiquitous limiting damping mechanism in SiN micro- and nanomechanical resonators

Silicon nitride (SiN) micro- and nanomechanical resonators have attracted a lot of attention in various research fields due to their exceptionally high quality factors ($Q$s). Despite their popularity, the origin of the limiting loss mechanisms in these structures has remained controversial. In this paper we propose an analytical model combining acoustic radiation loss with intrinsic loss. The model accurately predicts the resulting mode-dependent $Q$s of a low-stress silicon-rich and a high-stress stoichiometric SiN membrane. The large acoustic mismatch of the low-stress membrane to the substrate seems to minimize radiation loss and $Q$s of higher modes ($n \wedge m \geq 3$) are limited by intrinsic losses. The study of these intrinsic losses in low-stress membranes with varying lengths $L$ and thicknesses $h$ reveals an inverse linear dependence of the intrinsic loss with $h$ for thin resonators independent of $L$. This finding was confirmed by comparing the intrinsic dissipation of arbitrary (membranes, strings, and cantilevers) SiN resonators extracted from literature, suggesting surface loss as ubiquitous damping mechanism in thin SiN resonators with $Q_{surf} = \beta \cdot h$ and $\beta = 6\times10^{10} \pm4\times 10^{10}$~m$^{-1}$. Based on the intrinsic loss the maximal achievable $Q$s and $Q\cdot f$ products for SiN membranes and strings are outlined.