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arxiv: 2604.08073 · v1 · submitted 2026-04-09 · ❄️ cond-mat.mtrl-sci

Emergence of Lissajous trajectories in skyrmion oscillator

Tamali Mukherjee , V Satya Narayana Murthy This is my paper

Pith reviewed 2026-05-10 17:07 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords skyrmion dynamicsLissajous trajectoriesAC current driveskyrmion oscillatortemperature effectsCo/Pt thin filmHall angleforced oscillator
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The pith

Skyrmions trace Lissajous figures when driven by two perpendicular AC current pulses in a magnetic film.

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

The paper demonstrates that a skyrmion in a Co/Pt thin film follows sinusoidal paths and draws out the same closed curves seen in classical mechanics when subjected to independent AC currents along the x and y axes. In the stated ranges of current density and frequency the skyrmion behaves as a forced oscillator whose trajectory is set by the amplitudes, frequencies, and relative phase of the two drives. Finite temperature introduces deviations through the skyrmion Hall angle and random thermal forces, turning ideal figures into recognizably deformed versions. If this mapping holds, simple electrical signals could be used to steer skyrmions along prescribed paths without additional magnetic fields.

Core claim

For a current pulse of the form (A₁ sin ω₁ t, A₂ sin(ω₂ t + ϕ), 0) the skyrmion position in the x-y plane traces a Lissajous figure identical to the trajectory of two perpendicular harmonic oscillators. The motion remains sinusoidal and oscillator-like across current amplitudes from 1×10¹¹ A/m² to 1×10¹² A/m² and frequencies from 5×10⁸ Hz to 1×10¹⁰ Hz. At T = 0 K the paths match the classical expectation exactly; at T > 0 K the temperature-dependent Hall angle and stochastic thermal force cause progressive distortion while the overall Lissajous character is retained.

What carries the argument

The orthogonal pair of sinusoidal current components that drive the skyrmion velocity response, incorporating the temperature-dependent skyrmion Hall angle and stochastic thermal force term.

If this is right

  • Skyrmion position can be controlled as a tunable harmonic oscillator whose closed curves are set directly by drive parameters.
  • Temperature provides a continuous knob that converts ideal Lissajous figures into progressively distorted but still identifiable patterns.
  • The same driving scheme supplies a direct experimental test of whether skyrmion dynamics remain linear over the quoted amplitude and frequency window.
  • Lissajous patterns offer a simple visual diagnostic for verifying the combined effects of Hall angle and thermal noise in device-scale simulations.

Where Pith is reading between the lines

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

  • If the trajectories remain stable under realistic pinning, the Lissajous control could support phase- or frequency-based encoding of information in skyrmion circuits.
  • The same orthogonal-drive method might be applied to other magnetic textures or multilayer stacks to test how material-specific damping changes the observed figures.
  • Room-temperature distortions could still preserve enough symmetry information to allow readout of input frequency ratios or phase shifts.

Load-bearing premise

Skyrmion motion is fully determined by the applied current, the temperature-dependent Hall angle, and random thermal kicks, with no significant pinning or material defects altering the path.

What would settle it

Time-resolved imaging of the skyrmion trajectory under the stated dual-frequency current pulse in a real Co/Pt sample; if the observed path deviates markedly from the calculated Lissajous figure after the measured Hall angle and temperature are accounted for, the claim is falsified.

Figures

Figures reproduced from arXiv: 2604.08073 by Tamali Mukherjee, V Satya Narayana Murthy.

Figure 1
Figure 1. Figure 1: FIG. 1: Co/Pt multilayer of (200 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Skyrmion motion under application of an ac pulse: (a) the displacement of the skyrmion from the centre of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Skyrmion oscillation under application of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Skyrmion is subjected to motion under (A [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Skyrmion motion at finite temperature: (a) the displacement of the skyrmion for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: confirms that Asky, and consequently ω resonance sky depends on A, demonstrating the nonlinearity of the skyrmion oscillator. In summary, we notice that decreas￾ing A from 5 × 1011 A/m2 to 1 × 1011 A/m2 , results in a reduction of ω resonance sky from 8 × 108 Hz to 5 × 108 Hz. B. Motion of skyrmion for α ̸= β at T = 0 K We explore the motion of the skyrmion while consider￾ing β = 0.2 which is not equal to … view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The skyrmions of radii (a) 5.1 nm (Q = -0.95) [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

Understanding the dynamics of current-driven skyrmion is essential for their practical applications. In this study, we apply an AC current pulse (a) in x-- direction, and (b) in both x-- and y-- directions through the free layer of a Co/Pt thin film and investigate the motion of the skyrmion. We show that the skyrmion follows the sinusoidal current pulse and behaves like a forced oscillator in the range of current amplitude 1 $\times$ 10$^{11}$ A/m$^2$ to 1 $\times$ 10$^{12}$ A/m$^2$ and frequency 5 $\times$ 10$^{8}$ Hz to 1 $\times$ 10$^{10}$ Hz. For current pulse of (A$_1$sin$\omega_1$t, A$_2$sin($\omega_2$t+$\phi$), 0), the skyrmion forms Lissajous figures in the x-y plane, same as observed in classical mechanics. The results are compared at T = 0 K and T $>$ 0 K to analyze the effect of temperature. As the skyrmion Hall angle ($\theta_{SkH}$) and stochastic thermal fluctuation ($\textbf{F}^{Th}$) are functions of temperature, the skyrmion starts deviating from its path at T = 0 K with increasing temperature and eventually generates somewhat deformed Lissajous figures from ideal.

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 uses micromagnetic simulations to study current-driven skyrmion motion in a Co/Pt thin film. It reports that skyrmions follow the applied sinusoidal AC current pulses (in x-direction or both x- and y-directions) and act as forced oscillators for current densities 1e11–1e12 A/m² and frequencies 5e8–1e10 Hz. For biaxial driving with frequencies ω1 and ω2, the skyrmion traces closed Lissajous figures in the x-y plane, stated to be identical to those in classical mechanics. Temperature effects are compared at T=0 K (ideal figures) versus T>0 K, where the temperature-dependent skyrmion Hall angle and stochastic thermal force produce deformed trajectories.

Significance. If the reported trajectories are shown to be free of frequency mixing and to match classical Lissajous curves exactly, the work would establish a useful mechanical analogy for skyrmion oscillators and could inform device design. The simulations at finite temperature also illustrate practical deviations, which is relevant for applications. However, the central analogy rests on whether the gyroscopic coupling inherent to skyrmion dynamics is overcome or absent in the reported regime; without that demonstration the significance is reduced.

major comments (2)
  1. [Abstract] Abstract: The claim that the skyrmion 'forms Lissajous figures in the x-y plane, same as observed in classical mechanics' for the drive (A₁ sin ω₁t, A₂ sin(ω₂t + ϕ), 0) is not supported by the governing dynamics. The Thiele equation G × v + α D v = F(j(t)) contains a nonzero gyroscopic term that produces a finite skyrmion Hall angle; this linearly mixes the two drive frequencies into both velocity components, so that both ω₁ and ω₂ appear in x(t) and in y(t). Classical Lissajous curves require each coordinate to contain only a single frequency. The manuscript must either derive the integrated trajectories analytically or demonstrate (via explicit comparison to the integrated Thiele solution) that the reported numerical paths are free of this mixing at T = 0 K.
  2. [Results] Results section (micromagnetic simulations): No validation against the analytic Thiele equation, no error bars on trajectory closure, and no parameter-sensitivity study are described for the claimed frequency and amplitude ranges. Because the Hall-angle mixing is a load-bearing issue for the 'same as classical mechanics' statement, the absence of this cross-check undermines the central claim.
minor comments (2)
  1. [Methods] The ranges of current density and frequency are stated only in the abstract; the main text should tabulate the exact simulation parameters (mesh size, damping, saturation magnetization, etc.) used to obtain the reported trajectories.
  2. [Figures] Figure captions should explicitly state whether the plotted paths are at T = 0 K or finite T and should include the precise values of ω₁, ω₂, ϕ, and A₁/A₂ for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concerns point by point below, providing the strongest honest defense of our work while committing to revisions that directly respond to the raised issues.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the skyrmion 'forms Lissajous figures in the x-y plane, same as observed in classical mechanics' for the drive (A₁ sin ω₁t, A₂ sin(ω₂t + ϕ), 0) is not supported by the governing dynamics. The Thiele equation G × v + α D v = F(j(t)) contains a nonzero gyroscopic term that produces a finite skyrmion Hall angle; this linearly mixes the two drive frequencies into both velocity components, so that both ω₁ and ω₂ appear in x(t) and in y(t). Classical Lissajous curves require each coordinate to contain only a single frequency. The manuscript must either derive the integrated trajectories analytically or demonstrate (via explicit comparison to the integrated Thiele solution) that the reported numerical paths are free of this mixing at T = 0 K.

    Authors: We acknowledge the referee's point that the gyroscopic term in the Thiele equation can in principle introduce frequency mixing via the skyrmion Hall angle. However, our micromagnetic simulations are performed in a regime where the damping term dominates over the gyroscopic coupling for the chosen current densities and frequencies, resulting in trajectories that closely match classical Lissajous figures with negligible observable mixing. To substantiate this, we will add to the revised manuscript an explicit numerical comparison of the micromagnetic trajectories at T=0 K against solutions obtained by direct integration of the Thiele equation for identical driving conditions. This will quantify any residual mixing and delineate the parameter range where the classical analogy holds. revision: yes

  2. Referee: [Results] Results section (micromagnetic simulations): No validation against the analytic Thiele equation, no error bars on trajectory closure, and no parameter-sensitivity study are described for the claimed frequency and amplitude ranges. Because the Hall-angle mixing is a load-bearing issue for the 'same as classical mechanics' statement, the absence of this cross-check undermines the central claim.

    Authors: We agree that the absence of a direct cross-check against the Thiele equation and supporting quantitative measures weakens the central claim. In the revised manuscript we will include a side-by-side comparison of the micromagnetic results with numerically integrated Thiele trajectories over the reported ranges of current density (1e11–1e12 A/m²) and frequency (5e8–1e10 Hz). We will also add error bars quantifying the closure of the Lissajous trajectories and a brief parameter-sensitivity study confirming that the reported behavior remains robust within the stated intervals. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical outcomes of standard Thiele-equation integration under AC drive

full rationale

The paper reports micromagnetic or Thiele-equation simulations of skyrmion motion under specified AC current pulses, comparing trajectories at T=0 K and finite T. The Lissajous-figure claim is presented as an observed numerical result, not derived from a fitted parameter, self-referential definition, or load-bearing self-citation. No step reduces the output trajectory to an input by algebraic construction or renaming; the temperature dependence of θ_SkH and F^Th is an explicit model input whose effect is shown by direct integration. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on standard micromagnetic modeling assumptions for current-driven skyrmion motion and temperature effects; no new entities are introduced.

free parameters (2)
  • Current density amplitude
    Ranges 1e11 to 1e12 A/m² selected to observe forced-oscillator behavior.
  • Driving frequency
    Range 5e8 to 1e10 Hz chosen to ensure skyrmion follows sinusoidal pulse.
axioms (2)
  • domain assumption Skyrmion dynamics governed by Landau-Lifshitz-Gilbert equation augmented with spin-transfer torque and thermal stochastic field
    Implicit foundation for all such current-driven skyrmion simulations.
  • domain assumption Skyrmion Hall angle and thermal fluctuation force are the dominant temperature-dependent quantities
    Explicitly invoked in the abstract to explain path deviations.

pith-pipeline@v0.9.0 · 5564 in / 1358 out tokens · 60134 ms · 2026-05-10T17:07:23.118041+00:00 · methodology

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