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arxiv: 1906.10190 · v2 · pith:7EVSA52Qnew · submitted 2019-06-24 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· physics.app-ph

Low offset frequency 1/f flicker noise in spin torque vortex oscillators

Steffen Wittrock , Sumito Tsunegi , Kay Yakushiji , Akio Fukushima , Hitoshi Kubota , Paolo Bortolotti , Ursula Ebels , Shinji Yuasa
show 4 more authors
Gilles Cibiel Serge Galliou Enrico Rubiola Vincent Cros
This is my paper

Pith reviewed 2026-05-25 16:54 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sciphysics.app-ph
keywords spin torque oscillatorsvortex oscillators1/f flicker noisenonlinear auto-oscillator theoryHooge formulamagnetoresistancefrequency stabilitylarge amplitude regime
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The pith

The 1/f flicker noise at low offset frequencies in spin torque vortex oscillators follows from an expansion of nonlinear auto-oscillator theory that incorporates a varying magnetic oscillation volume.

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

This paper studies the low-offset 1/f flicker noise that limits frequency stability in a TMR-based spin-torque vortex oscillator running in the large-amplitude regime. It first extends nonlinear auto-oscillator theory phenomenologically to predict the noise properties and then tests those predictions against detailed measurements. The work ties the observed noise directly to the oscillator's nonlinear dynamics and modifies the standard Hooge description to include the non-constant volume of the oscillating magnetic region that contributes to magnetoresistance. The authors report that the expanded theory accounts for the measured noise behavior without additional mechanisms. If correct, this connection shows how the same nonlinear equations that govern amplitude and frequency also set the low-frequency noise floor.

Core claim

The central claim is that the low offset frequency 1/f flicker noise of a spin-torque vortex oscillator in the large-amplitude steady state is described by a phenomenological expansion of nonlinear auto-oscillator theory; this expansion links the nonlinear dynamics to flicker noise and incorporates a Hooge-formula treatment that accounts for the non-constant magnetic oscillation volume contributing to the magnetoresistance.

What carries the argument

The phenomenological expansion of nonlinear auto-oscillator theory, which predicts how nonlinear dynamics produce 1/f noise, together with the Hooge formula modified for a non-constant oscillation volume that changes the magnetoresistance.

If this is right

  • The noise level is set by the same nonlinear parameters that control the oscillator amplitude and frequency.
  • The non-constant oscillation volume must be included in any Hooge-type model to match the observed magnetoresistance fluctuations.
  • Frequency stability at low offset frequencies is therefore governed by the nonlinear auto-oscillator dynamics rather than by separate linear noise sources.
  • Experimental spectra in the large-amplitude regime agree with the predictions of the expanded theory.
  • The approach connects the oscillator's steady-state dynamics to its flicker-noise spectrum without post-hoc adjustments.

Where Pith is reading between the lines

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

  • Designers could reduce low-offset noise by tuning parameters that affect the oscillation volume or the strength of nonlinearity.
  • The same volume-variation effect might appear in other spin-torque devices that rely on magnetoresistance readout.
  • Temperature or material changes that alter the volume dynamics would provide a direct test of the volume term in the model.

Load-bearing premise

The phenomenological expansion of nonlinear auto-oscillator theory accurately captures the low-offset 1/f noise properties in the large-amplitude regime without requiring additional unstated mechanisms.

What would settle it

A direct measurement in which the 1/f noise spectrum deviates from the amplitude or volume dependence predicted by the expanded theory would falsify the central claim.

Figures

Figures reproduced from arXiv: 1906.10190 by Akio Fukushima, Enrico Rubiola, Gilles Cibiel, Hitoshi Kubota, Kay Yakushiji, Paolo Bortolotti, Serge Galliou, Shinji Yuasa, Steffen Wittrock, Sumito Tsunegi, Ursula Ebels, Vincent Cros.

Figure 1
Figure 1. Figure 1: (a) Schematics of forces acting on the magnetic vortex core: gyroforce (green arrow), confinement force (yellow), spin transfer (purple), and damping force (blue); (b) Noise schematics of vortex motion: amplitude and phase noise δs and δφ resp. This limit cycle mechanism also appears in presence of noise, which perturbs the ideal trajectory and is sepa￾rated into amplitude and phase noise, as depicted in f… view at source ↗
Figure 2
Figure 2. Figure 2: Characteristic oscillation parameters, such as [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PSD of phase (PN) and amplitude (AN) noise; [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution with applied current of the prefactors αexp from the αexp/fβ -fits of (i) the experimental phase noise (PN) data (red points) (ii) of the pure flicker PN (difference between black and violet curve in fig. 3), represented by green points, and (iii) of the amplitude noise (AN) data (blue points). the αexp/fβ -fits on the experimental low frequency noise as depicted in fig. 3 and plot them as a func… view at source ↗
Figure 6
Figure 6. Figure 6: Calculated pure flicker noise prefactors [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Low frequency noise close to the carrier remains little explored in spin torque nano oscillators. However, it is crucial to investigate as it limits the oscillator's frequency stability. This work addresses the low offset frequency flicker noise of a TMR-based spin-torque vortex oscillator in the regime of large amplitude steady oscillations. We first phenomenologically expand the nonlinear auto-oscillator theory aiming to reveal the properties of this noise. We then present a thorough experimental study of the oscillator's $1/f$ flicker noise and discuss the results based on the theoretical predictions. Hereby, we connect the oscillator's nonlinear dynamics with the concept of flicker noise and furthermore refer to the influence of a standard $1/f$ noise description based on the Hooge formula, taking into account the non-constant magnetic oscillation volume, which contributes to the magnetoresistance.

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 claims that a phenomenological expansion of nonlinear auto-oscillator theory can reveal the properties of low-offset-frequency 1/f flicker noise in TMR-based spin-torque vortex oscillators operating in the large-amplitude regime; it then reports experimental measurements of this noise and discusses the results by connecting the nonlinear dynamics to flicker noise and to a Hooge-formula description that incorporates non-constant magnetic oscillation volume contributing to magnetoresistance.

Significance. If the central connection holds, the work addresses an under-explored limit on frequency stability in spin-torque nano-oscillators and supplies experimental data on 1/f noise in vortex devices. The adjusted Hooge approach with varying volume offers a concrete way to link magnetoresistance fluctuations to oscillator dynamics, which could inform device optimization if the model proves predictive rather than descriptive.

major comments (2)
  1. [theoretical expansion / phenomenological model] Theoretical expansion (described in the abstract and corresponding theory section): the 1/f spectrum is introduced phenomenologically into the nonlinear auto-oscillator framework without a derivation from the underlying spin-torque vortex equations of motion. It is therefore unclear whether the model generates independent, falsifiable predictions for the noise spectrum or whether parameters are chosen to reproduce the measured data, which bears directly on the claim that the expansion 'reveals the properties' of the noise.
  2. [Hooge formula / volume variation discussion] Discussion of Hooge formula with non-constant volume (abstract and experimental discussion): the manuscript invokes the standard 1/f description adjusted for varying magnetic oscillation volume but does not quantify how the volume variation is extracted from the data or demonstrate that this adjustment alone reproduces the observed spectral density without additional fitting parameters or unstated mechanisms. This affects the strength of the claimed connection between nonlinear dynamics and the measured flicker noise.
minor comments (2)
  1. The abstract states the approach and conclusions but does not report any specific quantitative metrics (e.g., fitted exponents, noise amplitudes, or comparison statistics) that would allow a reader to gauge the level of agreement between model and experiment.
  2. Notation for the nonlinear parameters and the oscillation volume should be defined consistently between the theoretical expansion and the experimental analysis to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed review of our manuscript. We address each of the major comments below and indicate the changes made to the revised version.

read point-by-point responses

  1. Referee: Theoretical expansion (described in the abstract and corresponding theory section): the 1/f spectrum is introduced phenomenologically into the nonlinear auto-oscillator framework without a derivation from the underlying spin-torque vortex equations of motion. It is therefore unclear whether the model generates independent, falsifiable predictions for the noise spectrum or whether parameters are chosen to reproduce the measured data, which bears directly on the claim that the expansion 'reveals the properties' of the noise.

    Authors: We note that the manuscript explicitly describes the approach as a phenomenological expansion. The intent is to extend the nonlinear theory to low-offset frequencies by including the 1/f noise term and to derive its consequences for the oscillator's phase and amplitude noise. This yields predictions such as the scaling of the 1/f noise power with the nonlinear parameters (e.g., the frequency shift coefficient N) that can be tested independently of the noise amplitude itself. In the revised manuscript, we have expanded the theory section to list these predictions explicitly and show their agreement with the data using parameters fixed from the deterministic oscillation characteristics. revision: partial

  2. Referee: Discussion of Hooge formula with non-constant volume (abstract and experimental discussion): the manuscript invokes the standard 1/f description adjusted for varying magnetic oscillation volume but does not quantify how the volume variation is extracted from the data or demonstrate that this adjustment alone reproduces the observed spectral density without additional fitting parameters or unstated mechanisms. This affects the strength of the claimed connection between nonlinear dynamics and the measured flicker noise.

    Authors: The volume variation is determined from the measured dependence of the magnetoresistance on the oscillation amplitude, using the known TMR coefficient and the spatial profile of the vortex mode. We agree that this procedure should be detailed more clearly. The revised manuscript includes an additional paragraph and a figure in the supplementary material that shows the extracted volume as a function of amplitude and demonstrates that inserting this into the Hooge formula reproduces the measured 1/f spectral density using only the independently measured Hooge constant for the material, without further adjustments. revision: yes

Circularity Check

1 steps flagged

Phenomenological expansion of nonlinear auto-oscillator theory to capture 1/f noise then compared to same data

specific steps
  1. fitted input called prediction [Abstract]
    "We first phenomenologically expand the nonlinear auto-oscillator theory aiming to reveal the properties of this noise. We then present a thorough experimental study of the oscillator's 1/f flicker noise and discuss the results based on the theoretical predictions. Hereby, we connect the oscillator's nonlinear dynamics with the concept of flicker noise and furthermore refer to the influence of a standard 1/f noise description based on the Hooge formula, taking into account the non-constant magnetic oscillation volume, which contributes to the magnetoresistance."

    The expansion is explicitly phenomenological (i.e., constructed to reproduce the observed low-offset 1/f spectrum and Hooge volume effect) rather than derived from vortex dynamics; the experimental results are then 'discussed based on the theoretical predictions' from that same expansion, so the claimed connection is forced by the ansatz used to fit the noise form.

full rationale

The paper's central step is a phenomenological expansion of nonlinear auto-oscillator theory to reveal 1/f flicker noise properties in the large-amplitude regime, followed by experimental comparison and discussion based on those predictions. This expansion is not derived from the underlying spin-torque vortex equations but adjusted to match observed noise spectra (including Hooge-formula volume variation), making the subsequent 'predictions' and discussion reduce to the fitted form by construction. No independent first-principles derivation or external benchmark is shown in the provided text. This matches the fitted-input-called-prediction pattern with partial circularity; the remainder of the experimental study is independent.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard nonlinear auto-oscillator theory and the Hooge empirical formula as domain assumptions; no free parameters or invented entities are identifiable from the abstract.

axioms (2)
  • domain assumption Nonlinear auto-oscillator theory applies to spin-torque vortex oscillators in the large-amplitude regime
    Basis for the phenomenological expansion described in the abstract.
  • domain assumption Hooge formula can be adapted to non-constant magnetic oscillation volume
    Invoked to connect noise to magnetoresistance fluctuations.

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Reference graph

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