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Skyrmion Gas Manipulation for Probabilistic Computing
Daniele Pinna, Flavio Abreu Araujo, Joo-Von Kim, Vincent Cros, Damien Querlioz, Perre Bessiere, Jacques Droulez, Julie Grollier
TL;DR
Stochastic computing needs signal reshuffling because correlations can undermine cascaded operations, while efficient hardware reshufflers are lacking. This paper models skyrmion ensembles with a collective-coordinate framework including skyrmion interactions and proposes a thermally driven reshuffler. The device produces an output with the input’s statistical properties while reducing correlations, with a stated low-power and compact-device scope.
Problem
Correlations rapidly propagate through cascaded stochastic gates, while efficient hardware reshufflers are not known and existing simulation methods are impractical for large structures and long timescales.
Method
The paper develops a collective-coordinate N-body model of skyrmion ensembles that phenomenologically accounts for skyrmion-skyrmion and skyrmion-boundary interactions.
Results
The Skyrmion Reshuffler thermally reshuffles skyrmion order to reconstruct an output signal with the input’s statistical properties and negligible correlations to the input.
Takeaways & Limitations
The results support a compact probabilistic-computing building block based on stable, thermally diffusing skyrmions, with operation expected at approximately µW energy.
Takeaways & Limitations
Higher driving currents speed signal generation but produce highly correlated outputs, reaching 64% correlation at 3.8 · 10^11 A/m^2 in the sampled case.
Abstract
from arXiv · showhide
The topologically protected magnetic spin configurations known as skyrmions offer promising applications due to their stability, mobility and localization. In this work, we emphasize how to leverage the thermally driven dynamics of an ensemble of such particles to perform computing tasks. We propose a device employing a skyrmion gas to reshuffle a random signal into an uncorrelated copy of itself. This is demonstrated by modelling the ensemble dynamics in a collective coordinate approach where skyrmion-skyrmion and skyrmion-boundary interactions are accounted for phenomenologically. Our numerical results are used to develop a proof-of-concept for an energy efficient ($\simμ\mathrm{W}$) device with a low area imprint ($\simμ\mathrm{m}^2$). Whereas its immediate application to stochastic computing circuit designs will be made apparent, we argue that its basic functionality, reminiscent of an integrate-and-fire neuron, qualifies it as a novel bio-inspired building block.
1 Introduction
Skyrmions offer stable, mobile information carriers, but existing applications largely constrain their motion to one-dimensional tracks. The paper proposes using two-dimensional ensemble dynamics and a particle-based model to develop compact probabilistic computing devices.
- 1 Introduction: Skyrmions are stable spin textures that can be manipulated at small current densities and resist pinning onto defects.Their topological resilience also supports particle-number conservation under many operating conditions.
- 1 Introduction: Existing skyrmion applications typically constrain motion to one-dimensional tracks, leaving two-dimensional freedom underused.The paper identifies this underuse as both a functional opportunity and a route toward reducing device area.
- 1 Introduction: Micromagnetic simulations become impractical for large structures and long timescales because execution time scales rapidly with structure volume.The paper therefore motivates simulation methods whose cost scales with skyrmion number.
- 1 Introduction: Probabilistic skyrmion devices are proposed to leverage particle-number conservation, thermal susceptibility, and low-power transport.The stated goal is to implement basic building blocks for probabilistic computing.
- 1 Introduction: A collective-coordinate N-body framework models skyrmion ensembles inside device geometries and supports their use in probabilistic computing.The framework starts from Thiele’s dynamical theory and incorporates ensemble behavior for device design.
2 Motivation: Stochastic Computing
Stochastic computing encodes values as probabilities in random bitstreams, but correlations accumulate across cascaded operations and make reshuffling necessary. The proposed skyrmion reshuffler uses thermal diffusion and current-driven transport to produce an output with the same statistical properties as the input while removing correlations.
- 2.1 Motivation: Stochastic computing represents numerical values by the probability of 1s or 0s in random bitstreams and performs arithmetic on their p-values.An AND-gate multiplies p-values when its input streams are random and uncorrelated.
- 2.1 Motivation: Correlations between input streams cause cascaded stochastic gates to produce incorrect results, creating a need for regular signal reshuffling.The reshuffler should preserve the input p-value while generating an uncorrelated copy.
- 2.1 Motivation: The proposed Skyrmion Reshuffler addresses the absence of efficient hardware reshufflers with a compact, low-energy design.Existing CMOS approaches require long-term memories, large area imprints, and substantial energy.
- 2.2 Skyrmion Reshuffler: The device converts an input bitstream into skyrmions whose order is thermally reshuffled through diffusive two-dimensional motion.Reading the reordered skyrmions produces an output signal with the input’s p-value and reduced correlation.
- 2.2 Skyrmion Reshuffler: Input bit states select one of two chambers, where current-driven drift and thermal diffusion determine the exit order.The two chamber populations encode the input p-value, and outgoing skyrmions reconstruct the output signal.
- 2.2 Skyrmion Reshuffler: The reshuffled output retains the input’s statistical properties while being uncorrelated with the original signal.This behavior is the device’s intended stochastic-computing function.
3 Modelling the Dynamics of Interacting Skyrmions
The paper models interacting skyrmions with collective-coordinate Thiele dynamics, incorporating current-driven motion, thermal noise, skyrmion interactions, and boundary forces. Phenomenological interactions fitted to micromagnetic simulations reproduce ensemble dynamics, while the model is limited near separations below the skyrmion diameter.
- 3.1 Isolated Skyrmion Dynamics: The collective-coordinate model describes skyrmion centers using Thiele dynamics with current-induced, thermal, interskyrmion, and boundary-force contributions.The current-driven term depends on spin-drift velocity and torque parameters, while thermal forces enter as additive homogeneous zero-mean noise.
- 3.1 Isolated Skyrmion Dynamics: The simulated isolated skyrmions have a typical diameter of ∼60nm under the selected material parameters.The chosen parameters correspond to interfacially stabilized skyrmions, and the model assumes size-independent force terms over the relevant range.
- 3.2 Skyrmion Interactions: The fitted skyrmion-skyrmion repulsion is Gaussian-like below the skyrmion diameter and exponentially decaying at larger separations.The interaction form was fitted using two-skyrmion micromagnetic relaxation data and captures exchange and dipolar effects.
- 3.2 Skyrmion Interactions: Two interacting skyrmions spiral away from each other, consistent with gyrotropic terms in the Thiele model.The gyrotropic matrix stabilizes rapidly, with G = −4π reported as expected from the continuum-limit topological charge.
- 3.3 Boundary Effects: The interaction model is not valid below the skyrmion diameter, where magnetic textures are radically altered and annihilation can occur.This regime is inconsistent with both the Thiele dynamical model and the particle picture of topologically protected skyrmions.
- 3.3 Boundary Effects: Boundary forces produce drift both along and away from a circular boundary and require a distinct phenomenological scaling from pair repulsion.The boundary interaction is important near the boundary and substantially improves force fits for nearby particles, while having virtually no effect near the nanodot center.
4 Skyrmion Reshuffler
The Skyrmion Reshuffler uses thermally diffusing skyrmions in two chambers to scramble injected signal sequences, producing output signals with reduced input correlation. Correlation can be tuned through current intensity and chamber size, balancing speed against decorrelation.
- Device operation: The device injects skyrmions into two 1024 nm chambers according to a telegraph-noise input, then reads their reordered output sequence.Skyrmions are injected every 5 ns through 100 nm-wide conduits and read on arrival at the output track.
- Decorrelation: Strong decorrelation is achieved at current intensities of ∼10^10 A/m2 because thermal diffusion scrambles the injected skyrmion sequence.Larger chambers allow more diffusion before particles exit, improving decorrelation.
- Speed–decorrelation trade-off: 64% correlation remains at the highest sampled driving current, 3.8 · 10^11 A/m2, because particles have insufficient time to diffuse and interact.Higher current increases speed but reduces decorrelation efficiency.
- Robustness: Accidental annihilation preserves the output p-value when its rate is proportional to chamber population and both chambers are similarly constructed.Such annihilation may nevertheless reduce the output signal frequency.
- Device implications: The device is expected to operate at ∼µW energy, while higher temperature could increase diffusion at the cost of greater energy consumption and possible skyrmion annihilation.The paper also identifies antiferromagnetic skyrmions as a theoretical improvement because their vanishing gyrovector increases diffusion and can reduce correlations.
5 Skyrmion Neuron
The Skyrmion Neuron adapts the reshuffler into a leaky integrate-and-fire device: skyrmions accumulate behind a voltage gate, and density-triggered release produces output activity. Chamber capacity and skyrmion stability constrain operation.
- Device operation: The device injects skyrmions in response to input telegraph-noise signals while an ON voltage gate blocks output and causes particle accumulation.A chamber readout estimates skyrmion density and releases the gate when a critical density is reached.
- Operating limits: Skyrmion capacity is limited because compression, dipolar strain, thermal effects, and defects can promote annihilation or nucleation.Topological stability is guaranteed only under continuous deformations, while sufficiently large thermal fields can induce collapse.
- Operating limits: At 300 K, simulations estimate the maximum number of approximately 39 nm skyrmions stabilized in chambers of varying diameters.Skyrmions were randomly nucleated, stabilized for 2 ns, counted, and averaged over ten repetitions for each chamber size.
- Neuron analogy: The chamber stores input history through skyrmion conservation while annihilations provide memory loss, making the device analogous to a leaky memory.This physical behavior motivates the leaky integrate-and-fire interpretation.
- Neuron analogy: The device emulates a leaky integrate-and-fire neuron by collecting skyrmions and successively dumping them into the output conduit.Skyrmion accumulation substitutes for voltage accumulation, while release corresponds to firing.
6 Conclusion
The paper uses a phenomenological n-body Thiele model to design skyrmion devices that are difficult to study with standard micromagnetic simulations. It proposes a thermally driven reshuffler for decorrelating stochastic signals and a gated neuron-like device, targeting compact, efficient probabilistic computing.
- Modeling approach: A phenomenologically tuned Thiele particle model accounts for skyrmion interactions and enables simulations of devices computationally intractable with standard micromagnetic techniques.The approach reduces solitonic spin-texture dynamics to classical n-body problems under suitable simplifying assumptions.
- Skyrmion Reshuffler: The Skyrmion Reshuffler uses thermal diffusion in two chambers to scramble injected skyrmion order and produce an output with negligible input correlations.Scaling analyses examine how system parameters balance reshuffler speed against effectiveness.
- Skyrmion Reshuffler: The reshuffler addresses stochastic computing requirements by producing an uncorrelated copy of a random signal while retaining an attractive area imprint and energy cost.Its operation leverages skyrmion stability together with thermal diffusivity.
- Skyrmion Neuron: The Skyrmion Neuron adds a voltage gate and chamber readout so density accumulation reaches a threshold before skyrmions are released.The design emulates integrate-and-fire behavior and is described as compatible with standard CMOS.
- Implications: Sensitivity to small currents and thermal noise, combined with nanoscale particle size, supports proposed probabilistic devices that are energy efficient and scalable.The paper presents these properties as opportunities for future spintronic applications.
A Methods
The methods combine three-dimensional micromagnetic simulations with particle tracking and a GPU-accelerated stochastic n-body Thiele solver. The simulations establish skyrmion properties and interaction parameters used in the modeled devices.
- Micromagnetic simulations: Three-dimensional micromagnetic simulations use MuMax3 and include exchange, anisotropy, Zeeman, magnetostatic, and DMI energy terms.The simulated nanotracks are 2 nm thick, with 50 nm input/output conduits and 100 nm conduit width.
- Micromagnetic simulations: The simulations discretize models into 2 × 2 × 2 nm^3 tetragonal cells, smaller than the stated exchange-DMI length scale and domain-wall width.This discretization supports the reported micromagnetic calculations.
- Micromagnetic simulations: The models stabilize skyrmions averaging approximately 39 nm in diameter at both 0 K and 300 K.Integration time steps are 5 · 10^-12 s at 0 K and 5 · 10^-14 s at 300 K; finite-temperature breathing modes occur at GHz frequencies.
- Particle tracking: Skyrmions are tracked from dynamical snapshots using a DLIB HOG detector trained on 200 stabilized skyrmion profiles.The detector identifies regions containing individual skyrmions from the total magnetization profile.
- Particle tracking: Skyrmion centers are obtained by cubic-spline interpolation followed by locating the maximum z-component of magnetization, yielding approximately pm resolution.Gyrotropic parameters are computed from surface integrals over the interpolated profiles.
- Particle dynamics: The stochastic n-body Thiele equations are solved with a CUDA/C++ GPU solver using a Heun scheme for convergence to the Stratonovich solution.Current-density distributions are first obtained with COMSOL, while breathing modes are stated not to affect net dynamics.
- Particle dynamics: For the modeled material parameters, fitted interaction coefficients specify skyrmion-skyrmion and boundary repulsions in the Thiele framework.The reported coefficient sets are a ≃ [2.709, −5.643, 0.964] and b ≃ [0.001419, −0.02631, 0.24135, −1.1609, 3.3547, −2.1786].
B Qualitative influence of dipole effects on skyrmion repulsion in the ultrathin film limit.
The appendix derives repulsion between magnetic domain walls in the ultrathin-film limit and uses this result to motivate the qualitative behavior of skyrmion repulsion. The interaction force decays exponentially with wall separation.
- Energy model: The reduced magnetic energy contains exchange, anisotropy, magnetostatic, and surface-DMI contributions with distinct preferred magnetization configurations.Exchange favors constant magnetization, anisotropy favors out-of-plane magnetization, magnetostatics favors divergence-free configurations, and DMI favors chiral symmetry breaking.
- Ultrathin-film reduction: In the ultrathin-film limit, the magnetic free energy can be reduced by expressing stray-field energy as a local shape-anisotropy term.This reduction permits treatment of a simplified one-dimensional magnetic wire with DMI.
- Domain-wall model: The magnetization is parametrized by θ, and the quality factor Q combines crystalline anisotropy and stray-field contributions in the ultrathin-film model.The energy is minimized by domain-wall solutions expressed using this parametrization.
- Domain-wall interaction: For two domain walls separated by δ, η = exp(−Qδ/2) enters the total energy and the interaction potential.The repulsive force follows from differentiating the interaction potential with respect to separation.
- Domain-wall interaction: The resulting repulsive force decays exponentially with the distance δ between domain walls.This behavior follows from powers of η in the force prefactors and is attributed to the ultrathin-film treatment of the stray field.
C Skyrmion properties under different material conditions.
The section lists material conditions explored in micromagnetic simulations and reports the observables used to characterize skyrmions and their repulsive interactions.
- Material conditions: Seven material-parameter sets, named S1–S7, were explored in micromagnetic simulations.The simulations required magnetic textures to relax to stable values on sufficiently short timescales for subsequent dynamics to be considered physically relevant.
- Material conditions: The material-parameter table organizes values for magnetization, exchange coupling, D, anisotropy, magnetic field, and skyrmion radius.The listed quantities are M_S, J_ex, D, K, B, and R_skx.
- Skyrmion interactions: A second table reports skyrmion–skyrmion repulsion as a function of n.