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Phase noise analysis and control of VO$_2$-based relaxation type oscillators

T0 review · 0 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read VO2 oscillator linewidths widen at low frequency because thermal noise hits hardest during the slow approach to the insulator-metal switch, and square-wave injection locks them better than sine waves.

desk verdict Solid experimental map of VO2 relaxation-oscillator phase noise: low-Vs linewidth rise is incubation jitter, and square-wave 2f lock beats sine on both noise and Γp. read the letter →

arxiv 2607.27447 v1 pith:E7CGPRBM submitted 2026-07-29 physics.app-ph cond-mat.mes-hall

Artem Litvinenko , Erbin Qiu , Sambit Ghosh , Juan Andres Hofer , Akash Kumar , Jong-Guk Choi , Ivan K. Schuller , Johan Åkerman This is my paper · ORCID
classification physics.app-phcond-mat.mes-hall PACS 85.75.-d05.45.Xt75.40.Gb75.47.-m84.30.Qi
keywords VO2relaxationoscillatorsphasenoiseinsulator-metaltransitionincubationtimeinjectionlockingsquare-wavesynchronizationjitterIsingmachines
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

VO2 relaxation oscillators are simple, CMOS-compatible devices used in neuromorphic circuits and Ising machines, but their free-running signals have broad spectral lines that limit locking and phase encoding. This paper shows that the extra broadening at low supply voltage comes from the incubation interval before the insulator-to-metal transition: the capacitor voltage creeps toward threshold with a shallow slope, so thermal fluctuations produce large timing jitter and a rising white-frequency-noise floor. Time-domain histograms become broad and right-skewed, matching first-passage statistics. Injection locking at twice the free-running frequency suppresses that noise, and a square-wave drive does so more efficiently than a sine wave of the same amplitude because its sharp edges steepen the approach to threshold, shorten the stochastic incubation time, and raise the amplitude relaxation rate. The result gives a physical account of the linewidth and a practical waveform choice for more stable VO2 oscillator networks.

What carries the argument

Stochastic incubation / first-passage timing at the IMT threshold: timing uncertainty scales as δt ∼ δV / (dV/dt)_th, so jitter and white frequency noise grow as the overdrive (Vs − VIMT) shrinks and the approach slope flattens; square-wave edges raise that slope and shorten the noisy interval.

What would settle it

Measure period jitter and phase-noise spectra while independently changing only the metallic-state discharge path (for example by adding a parallel low-resistance shunt or a second switch that dumps the capacitor faster) at fixed insulating charging slope; if metallic discharge is truly negligible, jitter and the 1/f^2 floor should stay unchanged, and square-wave locking should retain its advantage over sine.

Watch

Extended reading notes

Core claim

The dominant source of linewidth broadening in free-running VO2 Pearson-Anson oscillators at low oscillation frequency is heightened susceptibility to thermal fluctuations during the incubation phase preceding the insulator-to-metal transition, where the capacitor voltage approaches VIMT with reduced slew rate. That mechanism elevates the white-frequency (1/f^2) phase-noise contribution; external synchronization at 2f reduces the phase noise, and a square-wave injection signal does so more effectively than a sinusoidal signal of equal amplitude, reaching partial and full lock at lower drive and increasing the amplitude relaxation frequency Γp to about 1–2×10^5 Hz.

Load-bearing premise

The short metallic-state capacitor discharge barely affects the period or phase noise, so nearly all of the voltage-dependent jitter and the square-versus-sine locking difference can be blamed on the slow insulating approach to the insulator-metal threshold.

Editorial extensions

If this is right

  • Operating VO2 oscillators well above the IMT threshold keeps the incubation slope steep and holds white-frequency noise and period jitter near their high-bias floor (~few ns).
  • Square-wave injection at 2f reaches usable phase-noise plateaus at lower drive amplitude than sine injection and raises Γp, which shortens network convergence time in oscillator Ising machines.
  • Phase-slip rates and random spin flips in VO2 Ising machines should drop when the free-running incubation noise is suppressed by waveform-shaped locking.
  • Time-delay and phase encoding in VO2 spiking or oscillatory neural nets become more reliable once the shallow-drift jitter source is identified and controlled.
  • Device and circuit design can target slew rate at threshold (overdrive and edge shape) as the primary knob for spectral purity rather than only raw thermal noise power.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same slew-rate argument should apply to other thermally triggered Mott or phase-change relaxation oscillators, so square-edge injection may be a general fix beyond VO2.
  • If Γp sets time-to-solution in large Ising arrays, quoting lock bandwidth under square-wave drive becomes as important a figure of merit as free-running linewidth.
  • Asymmetric first-passage histograms at low overdrive could be turned into a diagnostic of local thermal noise strength or threshold variance across a fabricated array.
  • Combining moderate overdrive with modest square-wave lock may give a better power-stability tradeoff than pure free-running high-bias operation in dense neuromorphic chips.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. The manuscript presents an experimental study of phase noise in VO2-based Pearson–Anson relaxation oscillators. It attributes the strong linewidth broadening at low supply voltage (low frequency) to increased thermal susceptibility during the slow capacitor-voltage approach to the insulator-to-metal transition (incubation phase), supported by phase-noise spectrograms/PSDs (flicker-frequency 1/f^3 and white-frequency 1/f^2 regimes), intensity-graded time traces, period histograms with FWHM jitter, and an inverse-overdrive fit extracting V_IMT ≈ 1.79 V with a ~3.5 ns high-bias floor. It further shows that external injection locking at 2f with a square-wave drive suppresses low-offset phase noise more efficiently than a sine drive of equal amplitude, reaches partial/full lock at lower V_sync, and raises the amplitude relaxation frequency Γ_p to ~1–2×10^5 Hz, which the authors link to steeper threshold approach and shortened stochastic incubation time.

Significance. VO2 relaxation oscillators are actively pursued for neuromorphic networks and oscillator Ising machines, where broad linewidth and phase slips limit locking fidelity and time-to-solution. The work supplies a coherent, multi-modal experimental account of the dominant free-running noise mechanism and a practical, waveform-level control knob (square vs. sine at 2f) that improves both phase stability and Γ_p. The Hilbert-transform S_φ extraction, voltage-dependent jitter statistics (including right-skewed first-passage-time histograms), and the inverse-overdrive fit are concrete and falsifiable contributions that strengthen the applied-physics case for using these devices in phase-binarized computing.

minor comments (7)
  1. [Full text] Throughout the main text there are numerous missing spaces and run-on words (e.g., “broadlinewidthinoscillatoryneuralnetworks”, “Italsodisturbs”, “relaxationoscillatorschematic”, “V O2” in figure captions). A full copy-edit pass is needed before production.
  2. [§II, Eqs. (1)–(4)] §II after Eq. (1) and the frequency approximation Eq. (4) assert that metallic-state discharge is short and contributes negligibly to period and phase noise. A brief quantitative bound (measured or estimated τ_metal vs. τ_ins, or fraction of the period) would make this standard assumption explicit for non-specialist readers.
  3. [§III, Fig. 2; Appendix B] Fig. 2b and Fig. 4: state explicitly how S_φ is normalized/referenced (rad^2/Hz) and the carrier-tracking or unwrapping procedure used when the free-running linewidth is very broad (Vs = 1.8 V), so that the reported 1/f^2 rise is unambiguously distinguished from amplitude or digitization artifacts.
  4. [§V, Fig. 4] Fig. 4 and the definition of Γ_p: the text identifies Γ_p as the corner of the synchronization plateau (−3 dB from the low-frequency floor). Mark this corner on the plotted curves or give a short table of extracted Γ_p(V_sync) for sine and square drives to make the claimed 1–2×10^5 Hz increase unambiguous.
  5. [§III–V; Appendix A] Clarify whether all free-running and locked data are from a single device or representative of multiple devices; if single-device, a short statement on device-to-device spread (or lack of it) would help assess generality for network applications.
  6. [§IV, Fig. 3; Supp. Fig. S1] Supplementary Fig. S1: the 10 ns sampling-grid ripple artifact is well explained; consider noting it once in the main-text jitter caption (Fig. 3) so readers do not misread the oscillations as physical.
  7. [Front matter] PACS line and author affiliations contain minor formatting inconsistencies; arXiv stamp and “July 31, 2026” date should be cleaned for the journal version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: experimental phase-noise and jitter results are read from time traces, not forced by construction.

full rationale

This is a measurement paper. Free-running Sφ spectra (Fig. 2), intensity-graded traces and period histograms (Fig. 3), and injection-locking plateaus under sine vs square 2f drive (Fig. 4) are obtained directly from digitized waveforms via Hilbert-transform phase extraction and threshold-crossing statistics. The frequency formula (Eqs. 1–4) is the standard Pearson–Anson charging expression used only to frame the operating point; it does not define the measured noise slopes or lock bandwidths. The Supp. Fig. S2 inverse-overdrive fit σ∝(Vs−VIMT)−1 extracts VIMT≈1.79 V from the same jitter-vs-Vs data it illustrates—a routine parameter extraction, not a claimed independent prediction. Method citations (Hilbert phase noise, prior STNO work) are procedural, not load-bearing uniqueness theorems. No central claim reduces to its inputs by definition.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

Load-bearing content is experimental. Background domain assumptions are standard RC relaxation-oscillator and IMT physics from the cited literature. One fitted threshold and one asymptotic jitter floor are extracted from the authors’ own jitter curve. No new particles, fields, or conserved quantities are postulated.

free parameters (2)
  • V_IMT (extracted threshold) = ≈1.79 V
    Critical insulator-to-metal threshold used in the inverse-overdrive jitter model σ∝(Vs−VIMT)^−1; fitted to the measured jitter-vs-Vs curve rather than taken from an independent DC sweep in the main claim chain.
  • high-bias jitter floor = ≈3.5 ns
    Asymptotic timing jitter at large overdrive in Supp. Fig. S2; empirical floor attributed to residual noise/instrumentation, not derived from a microscopic model.
assumptions (4)
  • domain assumption Oscillation period is dominated by insulating-state capacitor charging; metallic discharge time is negligible for frequency and phase noise.
    Stated in §II after Eq. 1 and used to reduce dynamics to Eqs. 1–4 and to focus all jitter/locking analysis on the IMT incubation approach.
  • domain assumption Timing jitter at threshold obeys δt∼δV/(dV/dt)_th with slew rate proportional to overdrive (Vs−VIMT).
    Invoked in §V and Supp. S1 to explain both the low-Vs white-frequency-noise rise and why sharp square edges outperform sine injection.
  • domain assumption Instantaneous phase from Hilbert transform of the AC-coupled output yields a valid Sφ for this strongly nonsinusoidal, moderate-SNR waveform.
    Appendix B method, citing prior STNO phase-noise work; underpins all phase-noise figures.
  • standard math Standard classification of phase-noise PSD slopes (f^−3 flicker-frequency, f^−2 white-frequency, white phase floor).
    Used in §III / Fig. 2 following conventional oscillator noise taxonomy (e.g. Rubiola references).

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Pith. "Pith review of Phase noise analysis and control of VO$_2$-based relaxation type oscillators." pith.science (2026). https://pith.science/paper/E7CGPRBM

@misc{pith2026260727447,
  author       = {Pith},
  title        = {Pith review of: Phase noise analysis and control of VO$_2$-based relaxation type oscillators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7CGPRBM}},
  note         = {Machine review of arXiv:2607.27447}
}
abstract

VO$_2$-based relaxation oscillators form a rapidly developing field that finds applications in neuromorphic computing, Ising machines, and numerous signal processing concepts. These oscillators operate in a deeply nonlinear relaxation regime based on rapid phase transitions between insulating and metallic states in the VO$_2$ material. This process is governed by thermal effects, which lead to additional voltage fluctuations and contribute to a considerably wide spectral linewidth in the VO$_2$-based oscillator signal. In this work, we thoroughly study the phase noise in VO$_2$-based relaxation oscillators and demonstrate that the broadening of the generation spectrum linewidth at low oscillation frequencies is caused by an increased susceptibility to thermal fluctuations during the incubation phase. We explore the types of noise affecting oscillator stability and show that synchronization with an external square-wave signal improves the phase noise more effectively than a sinusoidal-shape injection locking signal.

Figures

Figures reproduced from arXiv: 2607.27447 by the authors.

Figure 1
Figure 1. FIG. 1. Basic characterization of the VO [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase noise characterization. (a) Phase noise spectrogram as a function of applied voltage [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Time-domain traces and statistical jitter analysis of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Phase noise suppression and synchronization of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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