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Paths to annihilation of first- and second-order (anti)skyrmions via (anti)meron nucleation on the frustrated square lattice

L. Desplat, J.-V. Kim, and R. L. Stamps
Phys. Rev. B 99, 174409 – Published 13 May 2019
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  • INTRODUCTION
  • MODEL
  • RESULTS
  • Langevin simulations
  • Annihilation rates
  • DISCUSSION AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study annihilation mechanisms of small first- and second-order skyrmions and antiskyrmions on the frustrated J1−J2−J3 square lattice with broken inversion symmetry (Dzyaloshinskii-Moriya interaction). We find that annihilation happens via the injection of the opposite topological charge in the form of meron or antimeron nucleation. Overall, the exchange frustration generates a complex energy landscape with not only many (meta)stable and unstable local energy solutions but also many possible paths connecting them. Whenever possible, we compute the activation energy and attempt frequency for the annihilation of isolated topological defects. In particular, we compare the average lifetime of the antiskyrmion calculated with transition state theory with direct Langevin simulations, for which excellent agreement is obtained.

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    • Received 25 February 2019
    • Revised 29 March 2019

    DOI:https://doi.org/10.1103/PhysRevB.99.174409

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    MagnetismMicromagnetismSkyrmionsStochastic processes
    Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    L. Desplat1,2,*, J.-V. Kim1, and R. L. Stamps2,3

    • 1Centre for Nanoscience and Nanotechnology, CNRS, Université Paris-Sud, Université Paris-Saclay, 91120 Palaiseau, France
    • 2SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
    • 3Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

    • *l.desplat.1@research.gla.ac.uk

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    Vol. 99, Iss. 17 — 1 May 2019

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    Images

    • Figure 1
      Figure 1

      Different isolated topological defects stabilized in a frustrated magnet with inversion symmetry (d=0) and reduced parameters (b,k)=(0.1,0.1): (a) antiskyrmion (n=−1), (b) skyrmion (n=1; Bloch), (c) skyrmion (n=1; Néel), (d) second-order skyrmion (n=2), (e) second-order antiskyrmion (n=−2).

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    • Figure 2
      Figure 2

      For k=0.1 and different values of the reduced DMI constant d and the reduced applied field b, the existence of (a) an isolated skyrmion solution, (b) an isolated antiskyrmion solution, (c) an isolated second-order skyrmion solution, (d) an isolated second-order antiskyrmion solution. Only metastable solutions of interest corresponding to excitations of the FM ground state are marked with an orange dot. The spin maps are zoomed in to show the topological defects.

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    • Figure 3
      Figure 3

      Spin maps (zoomed) and corresponding topological charge density along the transition path for skyrmion annihilation. The parameters are (a) (b,d)=(0.2,0.03) (metastable antiskyrmion solutions exist), (b) (b,d)=(0.3,0.07) (close to the existence of antiskyrmion solutions), and (c) (b,d)=(0.7,0.2) (antiskyrmion solutions do not exist). The image index is given in the top left corner.

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    • Figure 4
      Figure 4

      For skyrmion annihilation at different values of the reduced applied field b and reduced DMI constant d, we show the evolution along the transition path of (a) the internal energy barrier in units of J1 and (b) the topological charge. The reaction coordinate is normalized by the largest path length. The insets show the spin configuration and the topological charge density at the SP.

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    • Figure 5
      Figure 5

      Spin maps (zoomed) and corresponding topological charge density along the transition path for antiskyrmion annihilation with (b,d)=(0.2,0.03), where (a) shows the path over SP1 and (b) shows the path over SP2. The image index is given in the top left corner.

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    • Figure 6
      Figure 6

      Internal energy barrier in units of J1 (solid blue dots) and topological charge Ns (open red squares) along the transition path for an antiskyrmion collapse with (b,d)=(0.2,0.03) for (a) the path over SP1 and (b) the path over SP2. The inset shows the spin configuration and the topological charge density at the SP.

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    • Figure 7
      Figure 7

      Spin maps (zoomed) and corresponding topological charge density along the transition path for (a) the decay of a second-order skyrmion into a first-order skyrmion with (b,d)=(0.14,0.005) and (b) the division of a second-order skyrmion into a bound skyrmion pair with (b,d)=(0.1,0.005). The image index is given in the top left corner.

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    • Figure 8
      Figure 8

      Internal energy barrier in units of J1 (solid blue dots) and topological charge Ns (open red squares) along the transition path for (a) the decay of a second-order skyrmion into a first-order skyrmion with (b,d)=(0.14,0.005) and (b) the division of a second-order skyrmion into a bound skyrmion pair with (b,d)=(0.1,0.005). The inset shows the spin configuration and the topological charge density at the SP.

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    • Figure 9
      Figure 9

      Langevin simulation snapshots (zoomed) of (a) the annihilation of an antiskyrmion at T=80K, (b,d)=(0.2,0.03) and, at T=50K, (b,d)=(0.14,0.005), (b) and (c) the division of a second-order skyrmion into a bound skyrmion pair, (d) the decay of a second-order skyrmion into a first-order skyrmion, and (e) the division of a second-order antiskyrmion into a bound antiskyrmion pair.

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    • Figure 10
      Figure 10

      Mapping of the spin configurations along the MEP onto the unit sphere. Vertices correspond to the tip of the magnetic vectors with their origin in the center of the sphere, and the edges represent the exchange coupling between first neighbors. The solid sphere represents the “ball” that gets extracted from the “net.” The image index is given in the top left corner. We show (a) skyrmion annihilation via antimeron nucleation [Fig. 3], (b) skyrmion annihilation via isotropic collapse [Fig. 3], (c) antiskyrmion annihilation via meron nucleation [Fig. 5], and (d) decay of a second-order skyrmion into a first-order skyrmion [Fig. 7].

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