Figure 1

(a) Micrograph of a single drum resonator (similar to the one used in our analysis) superposed with schematic diagrams of the different harmonics ($n$, $m$) depicting their nodal lines (the origin is set at the center of the membrane). (b) Idem for a square membrane resonator. (c) Dissipation $1/Q$ as a function of frequency for the different harmonics of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ drum resonator ($D=14.5\mu \mathrm{m}$, $\mathsf{t}=110\mathrm{nm}$, and $\sigma =0.90\mathrm{GPa}$). (d) Idem for a square membrane resonator ($253.2\mu \mathrm{m}\times 253.2\mu \mathrm{m}\times 0.0125\mu \mathrm{m}$ with $\sigma =0.87\mathrm{GPa}$). Red (medium gray) plot: Measured values with an error of 10% for $1/Q$—we ascribe the splitting of degeneracies observed for the square membrane [cf. (d)] to disorder. Blue (dark gray) plot: Least squares fit of our model to the measured $1/Q$ using as fit parameters an internal dissipation offset ($1/Q_{\mathrm{int}}$) and material properties of the substrate ($E_s=148\mathrm{GPa}$, ${\rho }_s=3.75\mathrm{g}{\mathrm{cm}}^{-3}$, and $1/Q_{\mathrm{int}}=8.5\times {10}^{-7}$ for the square membrane, and $E_s=323\mathrm{GPa}$, $1/Q_{\mathrm{int}}=4.6\times {10}^{-5}$ for the drum). Green (light gray) plot: $1/Q$ without the offset corresponding to the predicted clamping loss—the resulting limits for the $Q$ values of the drum and square membrane are shown, respectively, in (e) and (f). High-$Q$ harmonics $(n,1){|}_{n>0}$ [$(n,n){|}_{n>1}$] of the drum [square] are marked by circles; low-$Q$ harmonics $(n,1){|}_{n>0}$ of the square, by triangles. For the drum [square] all harmonics with frequencies below 130 MHz [9 MHz] are included except (0, 3), (3, 2) [(7, 1)].Reuse & Permissions