Engineering the Dissipation of Crystalline Micromechanical Resonators

High-quality micro- and nanomechanical resonators are widely used in sensing, communications, and timing, and have future applications in quantum technologies and fundamental studies of quantum physics. Crystalline thin films are particularly attractive for such resonators due to their prospects for high quality, high intrinsic stress, high yield strength, and low dissipation. However, when such films are grown on a silicon substrate, interfacial defects arising from lattice mismatch with the substrate have been postulated to introduce additional dissipation. Here, we develop a back-side etching process for single-crystal silicon carbide microresonators that allows us to quantitatively verify this prediction. By engineering the geometry of the resonators and removing the defective interfacial layer, we achieve quality factors exceeding a million in silicon carbide trampoline resonators at room temperature, a factor of five higher than those achieved without removal of the interfacial defect layer. We predict that similar devices fabricated from ultrahigh-purity silicon carbide, leveraging its high yield strength, could enable room-temperature quality factors as high as $6\times {10}^9$.

Figure 1

(a) Mode shape of the fundamental vibrational out-of-plane mode of a trampoline with lateral length $L=700\mu \mathrm{m}$ and clamping points rounded with a radius $R=30\mu \mathrm{m}$ (inset). (b) Fabrication steps for single-crystal $\mathrm{Si}\mathrm{C}$ trampoline resonators. (c) Image of a $\mathrm{Si}\mathrm{C}$-on-$\mathrm{Si}$ trampoline chip sitting on an aluminum holder. The different colors of the four different chip sections are produced by different thicknesses of the $\mathrm{Si}\mathrm{C}$. Each device is released on a fully etched silicon window, which can be directly observed in the image. (d) SEM image of a $\mathrm{Si}\mathrm{C}$ trampoline resonator. The trampoline is suspended above a fully back-side-etched hole in a 0.5-mm-thick silicon substrate. (e) Resonance frequency of a trampoline calculated numerically using finite-element modeling (blue dots), compared with the resonance frequency of a string of length $L_s=\sqrt{2}L$ calculated analytically (red line) using Eq. (2) for a thickness $h=337\mathrm{nm}$.

Figure 2

(a) Normalized ringdown measurement of the fundamental mode of a trampoline resonator with resonance frequency ${\omega }_m/2\pi \sim 211\mathrm{kHz}$ and mechanical quality factor $Q=1.7\times {10}^6$. Ringdown fitting (black dashed line) using linear regression of the free-ringdown signal captured in a single shot. PSD, power spectral density. (b) Measured resonance frequencies ${\omega }_m/2\pi$ and quality factors $Q$ of various single-crystal $\mathrm{Si}\mathrm{C}$ trampoline resonators. The blue color code is for front-side-etched devices. Each point represents an individual device, and different shapes correspond to different thicknesses: circles, $h=337\mathrm{nm}$; squares, $h=293\mathrm{nm}$; rhombuses, $h=221\mathrm{nm}$; triangles pointing up, $h=140\mathrm{nm}$; and triangles pointing down, $h=75\mathrm{nm}$. The red triangles represent the devices that are back-side etched, with a total thickness $h=260\mathrm{nm}$.

Figure 3

Mean values of the measured $Q$ of $\mathrm{Si}\mathrm{C}$ trampolines with different thicknesses $h$, obtained from the raw data presented in Fig. 2. The blue squares (red triangles) are for the front-side- (back-side)-thinned resonators, and the error bars show the standard error. The blue line shows a theoretical fit for $Q$ using Eq. (1) for front-side-etched resonators, and the shaded region shows the uncertainty of the fit. The mean $Q$ for the back-side-etched resonators is shown by a red triangle. The green (orange) dashed line shows $\mathcal{D}(h)\times Q_{\mathrm{def}}$ ($\mathcal{D}(h)\times Q_{\mathrm{hq}}$), the limit for trampoline resonators made solely from the defect-rich (high-quality) $\mathrm{Si}\mathrm{C}$ layer. The purple dashed line shows $Q_{\mathrm{clamp}}$ calculated from Eq. (4) for a $\mathrm{Si}\mathrm{C}$ trampoline resonator with intrinsic stress $\sigma (h)$. The black dashed line shows the diluted intrinsic quality factor $\mathcal{D}(h)\times Q_{\mathrm{int}}$ when clamping losses are neglected.

Figure 4

(a) Mean stress $\sigma (h)$ as a function of the total thickness of the $\mathrm{Si}\mathrm{C}$ thin film for front-side-etched resonators. The experimental data are the mean values obtained from the measured resonance frequency using Eq. (3). The error bars represent the standard error. The black line shows the theoretical fit described by Eq. (3), and the gray band shows the associated uncertainty of the fit. (b) Dilution factor $\mathcal{D}(h)$ for a thin-film trampoline with mean stress $\sigma (h)$. The blue circles are determined from the experimental data shown in Fig. 4. The light blue band shows the uncertainty of the theoretical fit.

Figure 5

(a) FEM simulation of the acoustic power $P_{\mathrm{acou}}$ propagating through the $\mathrm{Si}$ substrate, with a perfectly matched layer at the surface $S$, calculated in a quarter of the total domain using symmetries, with enlargement of the quarter of the trampoline used in the calculation. (b) Top view of the trampoline design, where the blue-shaded region represents the domain used in the FEM simulation. The clamped boundaries, shown by colored lines, are mapped into the substrate for reference. (c) Schematic lateral view of the $\mathrm{Si}\mathrm{C}$ single crystal atop $\mathrm{Si}$, based on TEM studies and images [33, 49]. The region labeled $h_0$ represents the thickness of the defect-rich layer localized near the interface. The lines near the interface illustrate stacking defects and dislocations. (d) Thickness-dependent fitting parameters for the bilayer system: the intrinsic parameter $Q_{\mathrm{def}}$ of the defect-rich layer (green line) and that of the high-quality single crystal, $Q_{\mathrm{hq}}$ (orange line), and the thickness-dependent surface loss $Q^{\mathrm{surf}}(h)$ (purple dashed line) of the bilayer system. Each layer is limited by its respective volume loss $Q_{\mathrm{vol}}^{\mathrm{hq},\mathrm{def}}$ (gray dashed lines). A theoretical fit of $Q_{\mathrm{int}}$ obtained from Eq. (5) for the front-side-etched resonators is compared with the experimental data, with the blue-shaded region being the uncertainty of the fit.