Suspended Quartz Phononic Crystal Resonators

• The paper demonstrates that suspended quartz resonators use periodic phononic shields to localize defect modes and achieve high quality factors (e.g., Q ≈ 1.6×10^5 at 500 MHz).
• The resonators are fabricated as released thin-film quartz bridges or double‐clamped Z-cut devices, employing XeF2 or KOH undercut processes to suspend the structure and minimize substrate leakage.
• The systems integrate piezoelectric transduction with electromechanical coupling, enabling quantum acoustics and hybrid superconducting circuits with low dissipation.

Suspended quartz phononic crystal resonators are piezoelectric acoustic defect cavities formed in released quartz structures whose surrounding periodic sections create phononic bandgaps and suppress elastic leakage into the supports. In the current literature, the topic is represented most directly by one-dimensional suspended thin-film quartz bridges at approximately $500$ MHz and by suspended Z-cut quartz phononic crystal resonators near $100$ MHz, with the central technical themes being defect-mode localization, anchor-loss suppression, low-power loss from two-level systems (TLS), and electromechanical interfacing that preserves the low-loss character of quartz (Emser et al., 2024, Hu et al., 9 Sep 2025).

1. Structural archetypes and material platforms

The most direct suspended quartz implementations reported to date use released one-dimensional phononic crystal geometries. In one case, the device is a one-dimensional suspended phononic crystal resonator fabricated from a $1~\mu\text{m}$-thick quartz film on silicon, patterned into a released quartz bridge containing a central defect cavity between periodic shield sections, and suspended by $\mathrm{XeF_2}$ undercut of the silicon beneath the bridge (Emser et al., 2024). In another, the devices are approximately 3.5 $\mu$m thick, double-clamped, suspended, 1-D quartz phononic crystal resonators fabricated from Z-cut $\alpha$-quartz bonded to high-resistivity silicon, with final suspension obtained by KOH undercutting of silicon (Hu et al., 9 Sep 2025).

The two direct quartz platforms occupy different frequency regimes and design scales. The $500$ MHz class uses a defect cell flanked by 7 phononic shield cells on each side and was designed across an array of 20 resonators spanning defect-mode frequencies from 480 to 520 MHz (Emser et al., 2024). The $\sim 100$ MHz class uses periodic Bragg mirrors with 3M and 5M arrays, where the central rectangular defect block localizes a width-extension mode and the defect width was swept from 23.1 $\mu$m to 28.6 $\mu$m (Hu et al., 9 Sep 2025).

Platform	Suspended quartz structure	Representative reported performance
(Emser et al., 2024)	$100$0-thick quartz bridge with central defect and 7 shield cells per side	$100$1 at $100$2 MHz and $100$3
(Hu et al., 9 Sep 2025)	$100$4-thick double-clamped Z-cut quartz 1-D PnC near $100$5 MHz	$100$6 and $100$7 ms at about $100$8 K

These structures are explicitly suspended rather than substrate-bound. That distinction is not merely geometric. Suspension removes direct leakage pathways into the underlying substrate and allows the periodic mirror sections to act as the dominant acoustic isolation mechanism. In both quartz platforms, the resonator is therefore best understood as a defect mode hosted inside a released phononic shield rather than as a conventional bulk quartz resonator with incidental patterning (Emser et al., 2024, Hu et al., 9 Sep 2025).

2. Bandgap formation, defect localization, and mode engineering

The confinement physics is standard defect-in-bandgap phononic-crystal physics, but implemented in quartz with cut- and mode-specific design choices. In the $100$9 MHz thin-film quartz bridge, the periodic shield is designed so that the unit-cell length is chosen near $1~\mu\text{m}$0 to place the bandgap near $1~\mu\text{m}$1 MHz, with the bandgap center estimated by $1~\mu\text{m}$2 Finite-element simulation predicts a bandgap center near $1~\mu\text{m}$3 MHz, a bandgap width $1~\mu\text{m}$4 MHz, and a complete bandgap roughly 421–598 MHz (Emser et al., 2024). In the $1~\mu\text{m}$5 MHz Z-cut devices, the periodic mirror cells are designed to produce a complete acoustic bandgap of about 20 MHz centered near 100 MHz, and measured $1~\mu\text{m}$6 rises strongly when the defect mode enters this gap (Hu et al., 9 Sep 2025).

Mode selection is equally specific. The $1~\mu\text{m}$7 MHz devices target a fundamental extensional mode with primary displacement along the quartz $1~\mu\text{m}$8 axis, explicitly chosen to maximize $1~\mu\text{m}$9 and exploit the largest available piezoelectric tensor component in the adopted ST-cut coordinates, $\mathrm{XeF_2}$0 (Emser et al., 2024). The $\mathrm{XeF_2}$1 MHz devices instead localize a width-extension mode whose dominant strain is along the crystal Y-axis, and finite-element simulation shows both the mechanical displacement and the induced electric potential needed for piezoelectric coupling to nearby electrodes (Hu et al., 9 Sep 2025).

A useful conceptual boundary is provided by quartz phononic structures that are not suspended. The Love-wave device in "Highly confined Love waves modes by defect states in a holey SiO$\mathrm{XeF_2}$2 /quartz phononic crystal" is explicitly substrate-supported, with a $\mathrm{XeF_2}$3 quartz substrate, a square lattice of cylindrical air holes etched only into the SiO$\mathrm{XeF_2}$4 guiding layer, and defect localization produced by removing lines of holes. It demonstrates the same periodicity-plus-defect logic, including nearly flat in-gap defect branches and cavity-like transmission peaks, but its confinement is due to Love-wave guidance in a slow SiO$\mathrm{XeF_2}$5 film on a faster quartz substrate rather than suspension of the quartz resonator itself (Liu et al., 2018).

This distinction is technically important. A suspended quartz phononic crystal resonator relies on release geometry and phononic shielding to reduce anchor loss, whereas the Love-wave structure relies on surface-wave guidance in a substrate-supported layered system. The shared vocabulary of defect states, bandgaps, and cavity branches should not obscure the different leakage channels and mode taxonomies (Liu et al., 2018).

3. Electromechanical coupling and circuit descriptions

Suspended quartz phononic crystal resonators are piezoelectric rather than purely mechanical objects, and their practical significance depends on how electrical access is achieved without reintroducing loss. In the $\mathrm{XeF_2}$6 MHz thin-film quartz bridge, patterned aluminum electrodes on the defect region connect to a microwave coplanar waveguide (CPW), and the piezoelectric interaction is written as $\mathrm{XeF_2}$7 For the chosen ST-cut quartz mode, the coupling is approximated as $\mathrm{XeF_2}$8 Around resonance, the measured reflection response is fit using $\mathrm{XeF_2}$9 with $\mu$0 The on-resonance intracavity phonon occupancy is taken as $\mu$1 These expressions are central to the single-phonon characterization of the device (Emser et al., 2024).

The $\mu$2 MHz quartz memory platform adopts a different strategy: contactless electrodes on a separate superconducting chip. Instead of depositing electrodes on the quartz resonator, the authors place electrodes 1 $\mu$3m above the resonator across an air gap, preserving the quartz surface and suppressing electrode-induced dissipation (Hu et al., 9 Sep 2025). The electromechanical mode is then described with a Butterworth-Van-Dyke equivalent circuit. For the maximum-$\mu$4 device, finite-element simulation gives $\mu$5 The coupling to a superconducting circuit mode is estimated as $\mu$6 and when $\mu$7, approximately $\mu$8 (Hu et al., 9 Sep 2025)

This electromechanical framework is already developed to the level of qubit integration. For fluxonium, the paper considers direct resonant coupling with $\mu$9 yielding $\alpha$0 For transmons, it proposes parametric coupling via a SNAIL three-wave mixer, with static Hamiltonian $\alpha$1 and effective beam-splitter interaction $\alpha$2 The effective coupling is written as $\alpha$3 with an estimated $\alpha$4 (Hu et al., 9 Sep 2025)

4. Dissipation, coherence, and loss budgeting

The dominant technical question in suspended quartz phononic crystal resonators is not whether phononic mirrors create a bandgap; it is which loss channel remains once radiation leakage has been sufficiently reduced. In the thin-film quartz bridge work, the low-power response is modeled as $\alpha$5 with resonant TLS loss $\alpha$6 relaxation TLS loss $\alpha$7 and TLS-induced fractional frequency shift $\alpha$8 For the $\alpha$9 MHz resonance, the fit gives $500$0 and the reactive TLS analysis yields $500$1 The central materials conclusion is therefore that quartz is substantially cleaner than the aluminum-associated loss channel, and that a significant portion of the remaining low-power loss is associated with the aluminum electrodes and/or associated aluminum oxide (Emser et al., 2024).

The $500$2 MHz memory platform frames the same problem differently, by eliminating direct electrodes on the resonator and identifying a total loss budget containing Gas damping, Electrode-induced loss, Anchor loss, Phonon-phonon scattering, Thermoelastic damping (TED), Impurity-related relaxation, and Two-level systems (TLS) (Hu et al., 9 Sep 2025). Its temperature dependence is interpreted using the Zener-type relaxation form $500$3 with a Landau-Rumer regime at low temperature characterized by approximately $500$4 (Hu et al., 9 Sep 2025)

A recurring misconception is that phononic mirrors alone determine the measured $500$5. The quartz literature does not support that simplification. In the $500$6 MHz bridge, radiation-limited loss was modeled as $500$7 and for the main device with 7 mirror cells per side, the inferred radiation-limited quality factor was about $500$8 far above the measured low-power $500$9 (Emser et al., 2024). In the $\sim 100$0 MHz resonators, the simulated maximum radiative quality factor is about $\sim 100$1 whereas the measured maximum is $\sim 100$2 In both cases, anchor loss is strongly suppressed but does not set the observed low-power limit (Hu et al., 9 Sep 2025).

This convergence of independent quartz studies suggests a stable hierarchy of loss channels. Suspended quartz phononic crystal resonators can suppress radiation loss enough that internal mechanisms—TLS, phonon-phonon scattering, and electrode-associated dissipation—become the principal constraints. A plausible implication is that further progress depends less on stronger mirrors than on surface, interface, and transduction engineering (Emser et al., 2024, Hu et al., 9 Sep 2025).

5. Measured performance and operating regimes

The present experimental record already spans two distinct operating regimes. In the $\sim 100$3 MHz thin-film quartz bridge, the headline low-power result is $\sim 100$4 for the $\sim 100$5 MHz mode at $\sim 100$6 At higher power, ringdown yields $\sim 100$7 corresponding to $\sim 100$8 These measurements were made in an Oxford Triton dilution refrigerator, with lowest operating temperature around 8 mK, and reflection measurements were performed with a Keysight E5071C VNA, while ringdown used a Zurich Instruments SHFQC (Emser et al., 2024).

The $\sim 100$9 MHz suspended quartz resonators target a different niche: long-lived mechanical memory modes at elevated cryogenic temperature. The central result is $\mu$0 and $\mu$1 with ringdown on the maximum-$\mu$2 mode at 13 K giving $\mu$3 At room temperature in vacuum, once the mode lies well inside the bandgap, the quality factor plateaus around $\mu$4 while at room temperature and atmospheric pressure it is limited to about $\mu$5 Cooling below 10 K raises the plateau to above $\mu$6 (Hu et al., 9 Sep 2025)

The two regimes are complementary rather than redundant. The $\mu$7 MHz device demonstrates single-phonon and millikelvin performance in a piezoelectric thin-film PCR. The $\mu$8 MHz device demonstrates millisecond lifetimes at $\mu$9 K in a geometry explicitly optimized for hybrid acoustic quantum memories. The former emphasizes quantum-regime coherence benchmarking; the latter emphasizes compatibility with contactless superconducting interfaces and long-lived storage at larger physical scale (Emser et al., 2024, Hu et al., 9 Sep 2025).

Measurement methodology also differs accordingly. The $\mu$0 MHz work is primarily microwave reflection and ringdown metrology. The $\mu$1 MHz work uses microwave drive together with optical photoelastic readout and spatially resolved photoelastic imaging, which directly visualizes exponential decay into the mirror region and thereby confirms defect localization (Hu et al., 9 Sep 2025). This combination of spectral and spatial evidence parallels the broader suspended phononic-crystal literature, where mode-resolved real-space mapping has been used to identify cavity modes, waveguide branches, and loading effects in suspended membranes (Hatanaka et al., 2019).

6. Comparative context, design transfers, and limits of generalization

Suspended quartz phononic crystal resonators belong to a wider family of phononic devices, but that family contains several architectures with distinct confinement physics. Substrate-supported quartz-based phononic crystals can localize Love waves by periodic holes in a SiO$\mu$2 guiding layer on ST-cut quartz, producing bandgaps and nearly flat defect branches without suspension (Liu et al., 2018). Suspended GaAs membranes demonstrate complete hypersonic bandgaps, line-defect cavities, line-defect waveguides, and side-coupled cavity-waveguide systems, with wavelength-scale mode volumes and cavity $\mu$3 up to 4200 under atmospheric conditions (Hatanaka et al., 2019). Phononic bandgap shields in SiN show that a periodic support can suppress support-mode density and external drive coupling by up to 30 dB, establishing the general principle that a resonator plus immediate frame can be treated as a defect inside a shield (Yu et al., 2013). Density-engineered suspended membranes show that periodic modulation of effective areal mass density can realize defect localization and soft clamping without perforating the membrane into a conventional holey crystal (Høj et al., 2022).

Quartz-specific comparison is equally instructive at the non-phononic-crystal boundary. Chip-scale confocal bulk acoustic wave resonators in $\mu$4-cut quartz reach an acoustic $\mu$5-factor of 28 million at 12.7 GHz at cryogenic temperatures, but they are bulk acoustic cavities rather than suspended phononic crystals and use Gaussian-like longitudinal trapping in a plano-convex geometry rather than bandgap confinement (Kharel et al., 2018). Suspended ScAlN-on-SOI XBAR devices with a spherical acoustic lens achieve an approximately 4× improvement in Q by suppressing lateral leakage of a shear-horizontal bulk overtone mode, again without a phononic crystal (Shokati et al., 9 Jan 2026). These neighboring results clarify that high-$\mu$6 confinement in quartz or piezoelectric membranes can arise from phononic bandgaps, geometric wavefront refocusing, or bulk acoustic Gaussian trapping, but suspended quartz phononic crystal resonators are distinguished by defect-in-bandgap localization in a released piezoelectric crystal.

Three design lessons recur across this comparative landscape. First, suspension by itself is insufficient; the decisive step is engineering the leakage channels, whether by periodic mirrors, full shields, or alternative stable-cavity confinement. Second, the widest bandgap or strongest nominal mirror is not automatically the route to the best measured resonance, because the localized mode, the transduction scheme, and the internal loss participation must be co-optimized. Third, quartz changes the balance relative to more strongly piezoelectric platforms: it offers weaker raw electromechanical coupling than materials such as LiNbO$\mu$7, but its low intrinsic loss makes it attractive precisely when long-lived, spectrally isolated, on-chip acoustic modes are required (Liu et al., 2018, Hatanaka et al., 2019, Yu et al., 2013, Høj et al., 2022, Kharel et al., 2018, Shokati et al., 9 Jan 2026).

Within that design space, the most distinctive current result is that suspended quartz phononic crystal resonators are no longer merely a conceptual transfer from GaAs or SiN platforms. They now exist as released thin-film quartz bridges and suspended Z-cut quartz defect chains, with experimentally demonstrated high-$\mu$8 operation, quantitative loss modeling, and explicit circuit-quantum-acoustodynamics interface proposals (Emser et al., 2024, Hu et al., 9 Sep 2025).