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Efficient Synchronization of Dipolarly Coupled Vortex-Based Spin Transfer Nano-Oscillators

N. Locatelli, A. Hamadeh, F. Abreu Araujo, A. D. Belanovsky, P. N. Skirdkov, R. Lebrun, V. V. Naletov, K. A. Zvezdin, M. Munoz, J. Grollier, O. Klein, V. Cros, G. de Loubens

arXiv:1506.03603v2cond-mat.mes-hall

TL;DR

STNOs’ nonlinear frequency adaptation makes synchronization a useful subject of study, and prior mutual synchronization has used spin-wave or exchange coupling. This paper studies dipolar coupling between adjacent vortex-based oscillators and finds stronger effective coupling when their vortices gyrate in opposite directions.

  • Problem

    STNO nonlinear frequency adaptation makes synchronization a relevant problem, while mutual synchronization had been demonstrated through spin-wave and exchange coupling.

    Source: Introduction · pp. 1–2

  • Method

    The study combines experiments and analytical modeling of dipolar coupling between adjacent vortex-based STNOs, alongside numerical investigation.

    Source: Introduction · p. 2, Experimental results · p. 2

  • Results

    Effective dipolar coupling energy and the critical frequency mismatch for synchronization are stronger when the vortices gyrate in opposite directions than in the same direction.

    Source: Discussion · p. 8

  • Takeaways & Limitations

    Synchronization was observed at frequency mismatches of approximately 10% and 6% of the oscillators’ mean frequency in the optimal configuration.

    Source: Discussion · p. 9

    The model neglects fast-varying coupling-energy terms that may increase phase fluctuations and synchronized-signal linewidth, potentially limiting coherence gains.

    Source: Analytical model for synchronization · p. 7

Abstract

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Due to their nonlinear properties, spin transfer nano-oscillators can easily adapt their frequency to external stimuli. This makes them interesting model systems to study the effects of synchronization and brings some opportunities to improve their microwave characteristics in view of their applications in information and communication technologies and to design innovative computing architectures. So far, mutual synchronization of spin transfer nano-oscillators through propagating spin-waves and exchange coupling in a common magnetic layer has been demonstrated. Here we show that the dipolar interaction is also an efficient mechanism to synchronize neighbouring oscillators. We experimentally study a pair of vortex-based spin-transfer nano-oscillators, in which mutual synchronization can be achieved despite a significant frequency mismatch between oscillators. Importantly, the coupling efficiency is controlled by the magnetic configuration of the vortices, as confirmed by an analytical model highlighting the physics at play in the synchronization process as well as by micromagnetic simulations.

Introduction

STNOs’ nonlinear frequency adaptation makes them useful systems for studying synchronization, though it can broaden their spectral linewidth. This section introduces dipolar coupling as a route to mutual synchronization and outlines the study of adjacent vortex-based oscillators.

Introduction

  • STNOs adapt their frequency to environmental changes, but this nonlinearity can broaden spectral linewidth and reduce quality factor.Their nonlinear dynamics also make them model nanoscale systems for studying synchronization and possible chaotic regimes under spin-transfer forces.Source: Introduction · p. 1
  • Synchronization is pursued to stabilize self-sustained oscillations for radio-frequency applications and to support oscillator-based associative memories that encode information in oscillator phases [8–12].Source: Introduction · pp. 1–2
  • Earlier mutual synchronization studies investigated coupling through propagating spin-waves [24–26] and exchange interactions [27].Prior work also examined phase locking to external rf fields [4, 13–15] and rf currents [16–23].Source: Introduction · p. 2
  • Coupling approaches may be difficult to scale to large STNO arrays, motivating study of electrical self-modulation and dipolar coupling between closely packed oscillators [28–32].The paper combines experiments, numerical work, and theory to study dipolar synchronization in two adjacent vortex-based STNOs.Source: Introduction · p. 2

1 Synchronization of two neighbouring vortex-based STNOs

Experiments establish reversible mutual synchronization of neighbouring vortex-based STNOs through dipolar coupling despite substantial frequency mismatch. Measurements and models both find coupling more effective for opposite core polarities.

Dipolar interaction between the oscillators

  • The oscillating in-plane stray fields couple the vortices; the model represents them as rotating macro-dipoles and neglects out-of-plane coupling because the cores are small.Polarity changes the variation rates of the two coupling terms through sums or differences of the vortex frequencies.Source: Experimental results · p. 3, Dipolar interaction between the… · pp. 4–5

Experimental results

  • The measured linewidth remained about 500kHz before and after synchronization, while the model warns that fast coupling-energy oscillations may increase phase fluctuations and linewidth.The synchronized frequency was generally close to one oscillator’s frequency, which the authors identify as the leader.Source: Experimental results · pp. 3–4, Analytical model for synchronization · p. 7
  • At a critical mismatch of ∆fAP ≃80MHz, the two spectral peaks give way to one, reversible peak with frequency pulling, indicating synchronization.The transition occurs as perpendicular field is swept and is not attributed to vortex-core reversal.Source: Experimental results · p. 3, Figure 2 · p. 4
  • At Idc = +35mA, the maximum mismatch is ∆fAP = 48±2MHz for opposite polarities versus ∆fP = 20±2MHz for identical polarities.The opposite-polarity vortices gyrate in opposite directions; identical-polarity vortices gyrate in the same direction.Source: Experimental results · pp. 3–4, Figure 3 · p. 5

Analytical model for synchronization

  • The synchronization model identifies direct dipolar phase forcing and dipole-driven orbit changes that shift gyration frequency; the latter mechanism dominates for these STNOs.The authors estimate αη ≃0.01 and ν ≃1.Source: Analytical model for synchronization · p. 6
  • The model predicts an APc-to-Pc synchronization mismatch ratio of 3, compared with the measured 48MHz/20MHz ≃2.4 on the second sample.The authors interpret this agreement as evidence of stronger effective coupling for opposite-polarity vortices.Source: Analytical model for synchronization · p. 7
  • After locking, the predicted phase relation is Ψ = ϕ1 − ϕ2 ≈ 0 for identical polarities and Ψ = ϕ1 + ϕ2 ≈ π for opposite polarities.Reversing the relative chiralities reverses this phase relation without changing coupling amplitude or synchronization features.Source: Analytical model for synchronization · p. 7

Study of the dipolar energy versus relative polarities

  • At D12 = 300nm, simulations give μeff(APc)/μeff(Pc) ∼3, and predict that the ratio decreases as interpillar distance shrinks.Opposite-polarity gyration maintains favorable magnetization alignment and a mean interaction energy three times larger than for identical polarities.Source: Figure 5 · p. 8, Study of the dipolar energy versus… · p. 8

Discussion

Experiments and analytical modelling show that dipolar coupling and the critical frequency mismatch for synchronization are stronger when interacting vortices have opposite polarities than when they have identical polarities. Synchronization was observed at mismatches of approximately 10% and 6% of the oscillators’ mean frequency in the optimal configuration.

Discussion

  • Opposite-polarity vortices produce stronger effective dipolar coupling and a larger critical synchronization mismatch than identical-polarity vortices.Source: Discussion · p. 8
  • Synchronization was observed with frequency mismatches of approximately 10% and 6% of the oscillators’ mean frequency in the optimal configuration.Source: Discussion · p. 9
  • The results suggest potential synchronization in larger oscillator networks, where controlled coupling could support information coding in STNO-based associative memories.The paper also says these results can be extended to other spin-transfer oscillator architectures, including magnetic tunnel junctions.Source: Discussion · p. 9

Methods

The study uses paired spin-valve nanopillars and controls vortex polarity with a perpendicular magnetic field. The pillars share a parallel electrical supply, so individual magnetoresistive signals and current amplitudes cannot be measured or controlled independently.

Methods

  • Each sample contains two 200 nm-diameter circular spin-valve nanopillars separated by 100 nm, patterned from a Cu/NiFe/Cu/NiFe/Au stack.The stack layers are Cu(60 nm)/NiFe(15 nm)/Cu(10 nm)/NiFe(4 nm)/Au(25 nm).Source: Methods · p. 9
  • Positive current above 10 mA sets identical positive vortex chiralities, while a perpendicular magnetic field controls vortex polarity.Source: Methods · p. 9
  • The pillars are supplied in parallel, requiring an assumption of equal current and allowing measurement only of their combined voltage, not separate magnetoresistive signals.Neighboring-pillar Oersted fields of 100–200 Oe at 20 mA shift core positions by a few nanometers but are not sufficient to affect vortex stability.Source: Methods · p. 9
  • The ac voltage is isolated from the bias circuit by a bias-T and measured on a spectrum analyzer after 30 dB amplification.Source: Methods · p. 9

Author contributions statement

The authors divide project responsibilities across coordination, experiments, sample fabrication, simulations, analytical modelling, and manuscript preparation.

Author contributions statement

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