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arxiv: 2605.08118 · v1 · submitted 2026-04-28 · ⚛️ physics.app-ph · physics.optics

A micromechanical frequency reference with parts-per-trillion holdover stability

Jie Yan , Jintark Kim , Rakibul Islam , Jiawei Yang , Karim Elmeligy , Alkim Bozkurt , Thomas W. Kenny , Pavan K. Hanumolu
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Gaurav Bahl
This is my paper

Pith reviewed 2026-05-12 00:44 UTC · model grok-4.3

classification ⚛️ physics.app-ph physics.optics
keywords MEMS resonatorfrequency stabilitydual-frequency resonance trackingholdover stabilitysilicon clockparts-per-trillionfrequency-locked loopatomic clock alternative
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The pith

A 268 MHz silicon MEMS clock reaches 8 parts-per-trillion fractional frequency stability at 8-hour averages by locking to dual resonances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a compact micromechanical resonator can serve as a high-stability frequency reference for holdover timing. Conventional phase-locked MEMS oscillators drift because temperature and component variations change the gain of the sustaining electronics. The authors replace that architecture with a frequency-locked loop that tracks two resonances at once, so the resonator frequency no longer depends on electronic gain. When this is paired with dual-mode temperature sensing and ratiometric stabilization, the device achieves stability competitive with chip-scale atomic clocks while remaining small and potentially integrable. A sympathetic reader would care because the result suggests a practical route to precise long-term timing without the size or power demands of atomic references.

Core claim

We demonstrate a 268 MHz MEMS clock that achieves record fractional frequency stability of ~8 parts-per-trillion at an averaging time of 8 hours, competitive with chip-scale atomic clocks. This performance is obtained with a single-crystal silicon electrostatic resonator that has no currently known intrinsic drift mechanism and is protected by wafer-level encapsulation. Gain variations in the sustaining electronics are identified as the dominant limitation in conventional phase-locked designs; these are removed by implementing a frequency-locked loop based on dual-frequency resonance tracking that eliminates the specific gain of the supporting electronics as a frequency-determining variable.

What carries the argument

Dual-frequency resonance tracking (DFRT) frequency-locked loop architecture that decouples resonator frequency from the gain of the sustaining electronics

If this is right

  • MEMS clocks can reach holdover stability competitive with chip-scale atomic clocks in a compact form factor.
  • Gain-insensitive DFRT locking becomes a general method for high-stability MEMS frequency references.
  • Combining DFRT with dual-mode tracking and ratiometric temperature stabilization produces dramatic long-term stability gains.
  • Single-crystal silicon resonators under wafer-level encapsulation exhibit no currently known intrinsic drift mechanisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dual-resonance approach could be applied to other MEMS sensors whose output depends on electronic parameters.
  • Portable devices could adopt this level of timing precision without carrying an atomic clock.
  • Longer-term measurements beyond 8 hours would test whether the reported stability continues or saturates at other limits.
  • Similar frequency-tracking techniques might reduce drift in related micromechanical devices such as oscillators or filters.

Load-bearing premise

That the encapsulated single-crystal silicon resonator has no unknown intrinsic long-term drift and that dual-frequency tracking fully removes every contribution from electronic gain and temperature sensitivity.

What would settle it

A controlled test that records frequency drift larger than 8 parts per trillion over 8 hours while temperature and electronics remain stable would show that unaccounted drift sources still limit performance.

Figures

Figures reproduced from arXiv: 2605.08118 by Alkim Bozkurt, Gaurav Bahl, Jiawei Yang, Jie Yan, Jintark Kim, Karim Elmeligy, Pavan K. Hanumolu, Rakibul Islam, Thomas W. Kenny.

Figure 1
Figure 1. Figure 1: Comparison of frequency tracking architectures. (a) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dual-mode wafer level encapsulated resonator and its temperature characteristics. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dual-frequency resonance-tracking (DFRT) principle and clock architecture. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Clock characterization with large temperature deviations applied to the TIAs (20–50 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Long-term operation of the 268 MHz clock in ambient laboratory conditions. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Influence of the amplitude–frequency (A–f) nonlinearity in the FLL based clock. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

Microelectromechanical (MEMS) resonators are widely used in timekeeping applications, and recent advances in fabrication, materials, and encapsulation technology have advanced their potential as high stability frequency references. However, for holdover applications that require the highest levels of long-term frequency stability, compact vapor atomic clocks remain dominant. In this work, we demonstrate a 268 MHz MEMS clock that achieves record fractional frequency stability of ~8 parts-per-trillion at an averaging time of 8 hours, competitive with chip-scale atomic clocks. We achieved this using a single-crystal silicon electrostatic resonator that has no currently known intrinsic drift mechanism and is protected from the environment with a wafer-level encapsulation. We specifically identify gain variations in the sustaining electronics as the dominant limitation in conventional phase-locked oscillator architectures -- originating from temperature sensitivity and drifts in the electronic components -- and overcome this by implementing a frequency-locked loop architecture based on dual-frequency resonance tracking (DFRT). This novel approach removes the specific gain of the supporting electronics as a frequency determining variable in the oscillator. When combined with dual-mode tracking and ratiometric temperature stabilization of the resonator, this approach enables a dramatic enhancement to long-term frequency stability and establishes gain-insensitive DFRT locking as a general paradigm for high-stability MEMS clocks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports an experimental demonstration of a 268 MHz MEMS frequency reference based on a single-crystal silicon electrostatic resonator with wafer-level encapsulation. The authors identify gain variations in sustaining electronics as the dominant prior limitation in conventional oscillators and implement a dual-frequency resonance tracking (DFRT) frequency-locked loop architecture, combined with dual-mode ratiometric temperature stabilization, to achieve a claimed fractional frequency stability of ~8 parts-per-trillion at 8 hours averaging time, competitive with chip-scale atomic clocks. The central innovation is the assertion that DFRT renders the output frequency independent of electronics gain.

Significance. If substantiated with full supporting data, this would represent a notable advance in micromechanical timekeeping by closing much of the stability gap between MEMS resonators and atomic references for holdover applications. The DFRT approach offers a potentially generalizable method to decouple oscillator frequency from electronic component drifts, which could influence design of other high-stability MEMS devices. The work is grounded in direct experimental measurement rather than simulation or parameter fitting.

major comments (2)
  1. [DFRT architecture and frequency-locked loop description] The claim that DFRT fully removes sustaining-electronics gain as a frequency-determining variable (detailed in the architecture and DFRT sections) is load-bearing for attributing the stability improvement to the new locking scheme. This holds only under assumptions of perfect mode orthogonality, infinite DC loop gain, and absence of cross-mode coupling or amplitude-dependent shifts. No explicit experimental test is described, such as deliberate variation of amplifier transconductance while logging frequency output, nor a derivation of the closed-loop transfer function from gain to frequency showing it is identically zero. Without such verification, the removal of gain sensitivity remains an assumption rather than a demonstrated result.
  2. [Results and stability measurements] The headline stability result of ~8 ppt at 8 hours (reported in the results section and associated Allan deviation figure) requires supporting quantitative details to be load-bearing: error bars on the stability metric, full long-term frequency trace data, and a direct comparison table against prior MEMS and chip-scale atomic clock performance metrics. The abstract and results text state the value but the absence of these elements in the presented data weakens the 'record' and 'competitive' assertions.
minor comments (2)
  1. [Introduction and resonator fabrication] The statement that single-crystal silicon 'has no currently known intrinsic drift mechanism' (introduction and resonator description) should be supported by specific citations to long-term drift studies or a brief discussion of why known mechanisms (e.g., stress relaxation, contamination) are ruled out under the encapsulation conditions.
  2. [Figures and captions] Figure captions and axis labels in the stability plots could be clarified to explicitly state averaging times, measurement bandwidth, and whether the data include the full 8-hour trace or only selected segments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough and constructive review. We address each major comment below and will revise the manuscript to strengthen the presentation of the DFRT architecture and the stability results.

read point-by-point responses

  1. Referee: [DFRT architecture and frequency-locked loop description] The claim that DFRT fully removes sustaining-electronics gain as a frequency-determining variable (detailed in the architecture and DFRT sections) is load-bearing for attributing the stability improvement to the new locking scheme. This holds only under assumptions of perfect mode orthogonality, infinite DC loop gain, and absence of cross-mode coupling or amplitude-dependent shifts. No explicit experimental test is described, such as deliberate variation of amplifier transconductance while logging frequency output, nor a derivation of the closed-loop transfer function from gain to frequency showing it is identically zero. Without such verification, the removal of gain sensitivity remains an assumption rather than a demonstrated result.

    Authors: We agree that explicit verification strengthens the central claim. In the revised manuscript we will add a derivation of the closed-loop transfer function showing that the locked output frequency is independent of sustaining-amplifier gain to first order under the DFRT architecture (assuming finite but high loop gain and the measured mode orthogonality). We will also include new experimental data in which amplifier transconductance is deliberately varied while the DFRT loop is closed, demonstrating that the output frequency remains unchanged within the measurement noise floor. These additions will convert the theoretical independence into a demonstrated result. revision: yes

  2. Referee: [Results and stability measurements] The headline stability result of ~8 ppt at 8 hours (reported in the results section and associated Allan deviation figure) requires supporting quantitative details to be load-bearing: error bars on the stability metric, full long-term frequency trace data, and a direct comparison table against prior MEMS and chip-scale atomic clock performance metrics. The abstract and results text state the value but the absence of these elements in the presented data weakens the 'record' and 'competitive' assertions.

    Authors: We accept that these supporting elements are needed for the result to be fully load-bearing. In the revision we will add statistical error bars to the Allan deviation plot, include the complete long-term frequency time series as supplementary material, and insert a comparison table that places the 8 ppt / 8 h result against representative prior MEMS resonators and chip-scale atomic clocks using the same metrics (Allan deviation at 8 h, power, volume). These changes will allow readers to evaluate the claimed performance directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; experimental demonstration with independent measurements

full rationale

The paper presents an experimental MEMS clock demonstration achieving measured fractional frequency stability of ~8 ppt at 8 hours. The central claim rests on physical fabrication, wafer-level encapsulation, dual-frequency resonance tracking (DFRT) implementation, and direct Allan deviation measurements rather than any derivation that reduces to fitted parameters or self-referential equations. No load-bearing steps invoke self-citation chains, ansatzes smuggled via prior work, or renaming of known results as new predictions. The DFRT architecture is described as removing electronics gain as a variable through mode tracking and ratiometric stabilization, but this is presented as an implemented technique whose efficacy is shown by the reported stability data, not derived by construction from the inputs. The result is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that single-crystal silicon resonators lack intrinsic long-term drift mechanisms and that DFRT eliminates electronic gain as a frequency variable. No free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Single-crystal silicon electrostatic resonator has no currently known intrinsic drift mechanism
    Explicitly stated as the basis for long-term stability in the abstract.

pith-pipeline@v0.9.0 · 5558 in / 1186 out tokens · 35408 ms · 2026-05-12T00:44:36.681033+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Nguyen, C. T.-C. MEMS technology for timing and frequency control.IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control54, 251–270 (2007)

    work page 2007

  2. [2]

    & Puers, R

    Van Beek, J. & Puers, R. A review of MEMS oscillators for frequency reference and timing applications.Journal of Micromechanics and Microengineering22, 013001 (2011)

    work page 2011

  3. [3]

    Wu, G., Xu, J., Ng, E. J. & Chen, W. MEMS resonators for frequency reference and timing applications.Journal of Microelectromechanical Systems29, 1137–1166 (2020)

    work page 2020

  4. [4]

    C.et al.Low-power dual mode MEMS resonators with ppb stability over temperature.Journal of Microelectromechanical Systems29, 190–201 (2020)

    Ortiz, L. C.et al.Low-power dual mode MEMS resonators with ppb stability over temperature.Journal of Microelectromechanical Systems29, 190–201 (2020)

    work page 2020

  5. [5]

    Engineering36, 124–131 (2024)

    Wei, X.et al.MEMS huygens clock based on synchronized micromechanical resonators. Engineering36, 124–131 (2024)

    work page 2024

  6. [6]

    In2025 IEEE 38th International Conference on Micro Electro Mechanical Systems (MEMS), 205–208 (IEEE, 2025)

    Kim, J.et al.Ultra-stable MEMS clock with 53 parts-per-trillion fractional frequency stability at 8 hours. In2025 IEEE 38th International Conference on Micro Electro Mechanical Systems (MEMS), 205–208 (IEEE, 2025)

    work page 2025

  7. [7]

    In2025 23rd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), 184–187 (IEEE, 2025)

    Yang, J.et al.Precision MEMS timing: Microheater and machine learning compensation for 21 ppt stability at 7.5 hours. In2025 23rd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), 184–187 (IEEE, 2025)

    work page 2025

  8. [8]

    Rubiola, E.Phase noise and frequency stability in oscillators(Cambridge University Press, 2008). 12

    work page 2008

  9. [9]

    Leeson, D. B. A simple model of feedback oscillator noise spectrum.Proceedings of the IEEE54, 329–330 (1966)

    work page 1966

  10. [10]

    J.et al.Temperature dependence of the elastic constants of doped silicon.Journal of Microelectromechanical Systems24, 730–741 (2014)

    Ng, E. J.et al.Temperature dependence of the elastic constants of doped silicon.Journal of Microelectromechanical Systems24, 730–741 (2014)

    work page 2014

  11. [11]

    & Pekko, P

    Jaakkola, A., Prunnila, M., Pensala, T., Dekker, J. & Pekko, P. Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators.IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control61, 1063–1074 (2014)

    work page 2014

  12. [12]

    Samarao, A. K. & Ayazi, F. Temperature compensation of silicon resonators via degenerate doping.IEEE Transactions on Electron Devices59, 87–93 (2011)

    work page 2011

  13. [13]

    & Pourkamali, S

    Hajjam, A., Logan, A. & Pourkamali, S. Doping-induced temperature compensation of thermally actuated high-frequency silicon micromechanical resonators.Journal of Microelectromechanical systems21, 681–687 (2012)

    work page 2012

  14. [14]

    Journal of Microelectromechanical Systems18, 1409–1419 (2009)

    Melamud, R.et al.Temperature-insensitive composite micromechanical resonators. Journal of Microelectromechanical Systems18, 1409–1419 (2009)

    work page 2009

  15. [15]

    Shahmohammadi, M., Harrington, B. P. & Abdolvand, R. Turnover temperature point in extensional-mode highly doped silicon microresonators.IEEE Transactions on Electron Devices60, 1213–1220 (2013)

    work page 2013

  16. [16]

    & Nguyen, C.-C

    Hsu, W.-T. & Nguyen, C.-C. Stiffness-compensated temperature-insensitive microme- chanical resonators. InTechnical Digest. MEMS 2002 IEEE International Conference. Fifteenth IEEE International Conference on Micro Electro Mechanical Systems (Cat. No. 02CH37266), 731–734 (IEEE, 2002)

    work page 2002

  17. [17]

    & Li, W.-C

    Liu, J.-R. & Li, W.-C. Temperature-compensated CMOS-MEMS resonators via electrical stiffness frequency pulling.Journal of Micromechanics and Microengineering30, 014002 (2020)

    work page 2020

  18. [18]

    & Segura, J

    Perell´ o-Roig, R., Barcel´ o, S., Verd, J., Bota, S. & Segura, J. Geometrically engineered mass sensing CMOS-MEMS resonators for temperature compensation via folded anchors. Sensors and Actuators A: Physical373, 115435 (2024)

    work page 2024

  19. [19]

    Jha, C.et al.High resolution microresonator-based digital temperature sensor.Applied Physics Letters91, 074101 (2007)

    work page 2007

  20. [20]

    C., Melamud, R., Chandorkar, S

    Salvia, J. C., Melamud, R., Chandorkar, S. A., Lord, S. F. & Kenny, T. W. Real-time temperature compensation of MEMS oscillators using an integrated micro-oven and a phase-locked loop.Journal of Microelectromechanical Systems19, 192–201 (2009)

    work page 2009

  21. [21]

    Applied Physics Letters126, 133503 (2025)

    Dabas, S.et al.A temperature-insensitive nonlinear silicon bulk acoustic oscillator. Applied Physics Letters126, 133503 (2025)

    work page 2025

  22. [22]

    H.et al.Dual-MEMS-resonator temperature-to-digital converter with 40 µK resolution and FOM of 0.12 pJK 2

    Roshan, M. H.et al.Dual-MEMS-resonator temperature-to-digital converter with 40 µK resolution and FOM of 0.12 pJK 2. In2016 IEEE International Solid-State Circuits Conference (ISSCC), 200–201 (IEEE, 2016)

    work page 2016

  23. [23]

    In2026 IEEE 39th International Conference on Micro Electro Mechanical Systems (MEMS), 1371–1374 (IEEE, 2026)

    Kim, J.et al.Non-ovenized MEMS clock with ±10 ppb dynamic temperature stability and sub-ppb long-term stability. In2026 IEEE 39th International Conference on Micro Electro Mechanical Systems (MEMS), 1371–1374 (IEEE, 2026)

    work page 2026

  24. [24]

    In2025 23rd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), 772–774 (IEEE, 2025)

    Yan, J.et al.Electronics insensitive tracking for ultra-stable MEMS frequency refer- ences. In2025 23rd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), 772–774 (IEEE, 2025). 13

    work page 2025

  25. [25]

    J., Callahan, C., Kalinin, S

    Rodriguez, B. J., Callahan, C., Kalinin, S. V. & Proksch, R. Dual-frequency resonance- tracking atomic force microscopy.Nanotechnology18, 475504 (2007)

    work page 2007

  26. [26]

    Sullivan, D.et al.Primary atomic frequency standards at NIST.Journal of Research of the National Institute of Standards and Technology106, 47 (2001)

    work page 2001

  27. [27]

    D.et al.A chip-scale atomic beam clock.Nature Communications14, 3501 (2023)

    Martinez, G. D.et al.A chip-scale atomic beam clock.Nature Communications14, 3501 (2023)

    work page 2023

  28. [28]

    J., Yang, Y., Chen, Y

    Ng, E. J., Yang, Y., Chen, Y. & Kenny, T. W. An etch hole-free process for temperature- compensated, high Q, encapsulated resonators. InProc. Solid-State Sens., Actuators, Microsyst. Workshop, 99–100 (2014)

    work page 2014

  29. [29]

    Kim, B.et al.Frequency stability of wafer-scale film encapsulated silicon based MEMS resonators.Sensors and Actuators A: Physical136, 125–131 (2007)

    work page 2007

  30. [30]

    & Bernier, L.-G

    Vanier, J. & Bernier, L.-G. On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards.IEEE Transactions on Instrumentation and Measurement 277–282 (2012)

    work page 2012

  31. [31]

    URL https://ww1.microchip.com/downloads/en/DeviceDoc/00002985.pdf

    Microchip Technology Inc.SA.45s Chip-Scale Atomic Clock (CSAC) Datasheet(2021). URL https://ww1.microchip.com/downloads/en/DeviceDoc/00002985.pdf. Document No. 900-00744-000 Rev G

    work page arXiv 2021

  32. [32]

    & Zhao, J

    Zhang, Y., Yang, W., Zhang, S. & Zhao, J. Rubidium chip-scale atomic clock with improved long-term stability through light intensity optimization and compensation for laser frequency detuning.Journal of the Optical Society of America B33, 1756–1763 (2016)

    work page 2016

  33. [33]

    M.et al.Microcell atomic clock using laser current-actuated power modulation with 10 −12 range stability at 1 day.Scientific Reports15, 36000 (2025)

    Rivera-Aguilar, C. M.et al.Microcell atomic clock using laser current-actuated power modulation with 10 −12 range stability at 1 day.Scientific Reports15, 36000 (2025)

    work page 2025

  34. [34]

    M., Axelrad, P., Seubert, J

    Rybak, M. M., Axelrad, P., Seubert, J. & Ely, T. Chip scale atomic clock–driven one-way radiometric tracking for low-earth-orbit CubeSat navigation.Journal of Spacecraft and Rockets58, 200–209 (2021)

    work page 2021

  35. [35]

    & Cross, M

    Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators.Reviews of Nonlinear Dynamics and Complexity1, 1–52 (2008)

    work page 2008

  36. [36]

    Kaajakari, V., Koskinen, J. K. & Mattila, T. Phase noise in capacitively coupled micromechanical oscillators.IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control52, 2322–2331 (2005). Acknowledgments The authors express their continued gratitude to Dr. Gabrielle Vukasin and Dr. Hyun-Keun Kwon for their work in fabricating the MEMS resona...

    work page 2005