🔬 applied physics

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

This paper investigates phase noise in VO $_2$ -based relaxation oscillators, identifying thermal fluctuations during the incubation phase as the primary cause of spectral linewidth broadening at low frequencies and demonstrating that synchronization with an external square-wave signal offers superior phase noise suppression compared to sinusoidal injection locking.

Original authors: Artem Litvinenko, Erbin Qiu, Sambit Ghosh, Juan Andres Hofer, Akash Kumar, Jong-Guk Choi, Ivan K. Schuller, Johan Åkerman

Published 2026-07-31

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CC BY 4.0

Original authors: Artem Litvinenko, Erbin Qiu, Sambit Ghosh, Juan Andres Hofer, Akash Kumar, Jong-Guk Choi, Ivan K. Schuller, Johan Åkerman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where computers don't just crunch numbers like a calculator, but think more like a brain, using tiny pulses of electricity to solve complex puzzles. To make this happen, scientists are building special "neurons" out of materials that can switch states instantly, acting like the firing neurons in our heads. One of the most promising materials for this job is Vanadium Dioxide (VO2). Think of VO2 as a tiny, super-fast light switch that flips between being an insulator (blocking electricity) and a metal (letting electricity flow) based purely on heat. When you heat it up, it snaps to metal; when it cools, it snaps back.

The problem is that these switches aren't perfect. They are a bit jittery, like a nervous singer hitting a note that wavers slightly off-key. In the world of electronics, this wavering is called "phase noise," and it makes the signal fuzzy. If you are trying to build a super-fast computer or a machine that solves math problems by mimicking the way spins in a magnet align (called an Ising machine), this fuzziness is a disaster. It causes the machine to make random mistakes or get stuck. So, the big question for engineers is: How do we stop this jittery switch from wobbling so much that it ruins the whole show?

This is exactly what a team of researchers set out to investigate. They studied these VO2-based oscillators, which are essentially circuits that use the material's heat-driven switching to create a rhythmic pulse. They discovered that the main reason these pulses get so messy, especially when the circuit runs slowly, is a "hesitation phase." Before the switch flips, the voltage has to climb a hill to reach a critical tipping point. If the climb is too slow, tiny, random heat fluctuations in the material act like little bumps on the road, pushing the switch over the edge at random times. This creates a lot of noise.

The researchers then tested a clever trick to fix this: they tried to force the oscillator to sync up with an outside signal, kind of like a conductor telling a jittery orchestra when to play. They compared two types of signals: a smooth, rolling wave (a sine wave) and a sharp, blocky signal (a square wave). They found that the smooth wave helped a little, but the sharp, square wave was a game-changer. Because the square wave rises to the target voltage much more steeply, it gives the switch a decisive push that gets it over the threshold quickly, shortening the time it spends in that vulnerable state where random heat bumps can mess things up.

The results showed that using this square-wave signal didn't just clean up the noise; it made the whole system snap into a stable rhythm much faster and at lower power levels than the smooth wave could. The team measured that this method significantly improved stability, reducing timing errors (jitter) from a messy 280 nanoseconds down to a high-bias floor of approximately 3.5 nanoseconds. They also found that this sharp signal made the oscillator respond to external changes much more rapidly, increasing its response speed to up to 100,000 times per second. Essentially, they proved that if you want these brain-like computers to work reliably, you don't just need a good switch; you need to push that switch with a sharp, decisive hand rather than a gentle, gradual one. This discovery gives engineers a clear recipe for building more stable, faster, and smarter computing devices in the future.

Technical Summary: Phase Noise Analysis and Control of VO2-Based Relaxation Oscillators

Problem Statement
Vanadium dioxide (VO2)-based relaxation oscillators are emerging as promising components for neuromorphic computing, Ising machines, and unconventional signal processing due to their CMOS compatibility, nanoscale dimensions, and robust synchronization properties. However, their practical utility is currently limited by a relatively broad spectral linewidth, often on the order of tens of kHz. This broad linewidth is particularly detrimental for applications such as oscillator-based Ising machines, where it disrupts injection locking and induces random phase slips (analogous to random spin flips). Similarly, in oscillatory neural networks, increased phase noise degrades collective synchronization patterns, reduces signal-to-noise ratios, and disturbs time-delay encoding mechanisms essential for associative memory and frequency-locked computing. Unlike conventional CMOS oscillators, VO2 oscillators rely on thermally induced phase transitions between insulating and metallic states, making them inherently susceptible to thermal fluctuations that increase jitter and phase noise.

Methodology
The authors conducted a comprehensive experimental analysis of phase noise in VO2-based relaxation oscillators operating in a deeply nonlinear regime. The study utilized a standard Pearson-Anson oscillator schematic, consisting of an external RC circuit and a VO2 film acting as a nonlinear switching element.

Characterization: The team measured the phase noise spectral density ( $S_\phi$ ) across various supply voltages ( $V_s$ ) ranging from 1.8 V to 3.5 V.
Time-Domain Analysis: To correlate spectral broadening with temporal instability, high-resolution time-domain traces were captured using AC coupling to remove DC offsets. The authors analyzed cycle-to-cycle timing jitter using intensity-graded persistence maps and statistical histograms of period lengths.
Synchronization Experiments: To mitigate thermal fluctuations, the oscillators were synchronized with external signals at twice the fundamental frequency ( $2f$ ). The study compared the efficacy of two injection signal shapes: sinusoidal and square-wave. The synchronization amplitude ( $V_{sync}$ ) was varied to observe the transition from free-running to partially and fully locked states.

Key Results

Origin of Linewidth Broadening: The analysis reveals that the broadening of the generation spectrum at low oscillation frequencies is primarily caused by an increased susceptibility to thermal fluctuations during the "incubation phase." As the supply voltage approaches the insulator-to-metal transition (IMT) threshold, the voltage approach to the switching threshold becomes shallow. In this regime, the system relies on rare, high-amplitude thermal and voltage fluctuations to trigger the transition, leading to a stochastic incubation time.
Noise Composition: At higher supply voltages (e.g., 3.5 V), the phase noise spectrum exhibits distinct regions: a $1/f^3$ slope (flicker frequency noise) at low offsets, a $1/f^2$ slope (white frequency noise) in the mid-range, and a white noise floor at high offsets. As the supply voltage decreases, the white frequency noise component ( $1/f^2$ ) rises significantly, eventually dominating the spectrum and masking the flicker noise.
Jitter and Statistics: Time-domain measurements confirm that reducing the supply voltage dramatically increases temporal jitter (from ~9.8 ns at 3.4 V to ~280.2 ns at 1.81 V). The period distribution shifts from narrow and symmetric to broad and right-skewed, consistent with first-passage-time statistics governing threshold crossing in a shallow-drift regime.
Impact of Synchronization Waveform: External synchronization effectively suppresses phase noise, but the waveform shape is critical.
- Square-Wave Superiority: Square-wave synchronization signals achieve partial and full injection locking at significantly lower amplitudes (5–25 mV) compared to sinusoidal signals (requiring ~40 mV for full locking).
- Mechanism: The sharp edges of the square wave steepen the voltage approach to the switching threshold, thereby shortening the stochastic incubation time and reducing the window for thermal fluctuations to induce timing errors.
- Amplitude Relaxation Frequency ( $\Gamma_p$ ): Square-wave synchronization increases the amplitude relaxation frequency $\Gamma_p$ up to $1-2 \times 10^5$ Hz, indicating a faster oscillator response to external signals compared to sinusoidal driving.

Significance and Claims
The paper establishes that the physical origin of phase noise in VO2 oscillators is the stochastic nature of the incubation period preceding the IMT, which is highly sensitive to thermal noise when the driving voltage is low. The authors claim that optimizing the synchronization waveform is as critical as the signal amplitude for controlling phase stability. Specifically, they demonstrate that square-wave injection locking is a more efficient control mechanism than sinusoidal locking because it directly suppresses the stochastic threshold-crossing process. These findings provide a practical route for improving the phase stability of VO2-based networks, which is essential for enhancing the convergence speed and reliability of Ising machines and neuromorphic computing systems. The work underscores that for these thermally driven oscillators, the shape of the synchronization signal is a fundamental parameter for robust operation.

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