  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.
