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Stability of hopfions in bulk magnets with competing exchange interactions

Moritz Sallermann, Hannes Jónsson, and Stefan Blügel
Phys. Rev. B 107, 104404 – Published 6 March 2023
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  • INTRODUCTION
  • MODELS
  • ATOMISTIC SPIN-SIMULATIONS…
  • RESULTS AND DISCUSSIONS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Magnetic hopfions are string-like three-dimensional topological solitons, characterised by the Hopf number. They serve as a fundamental prototype for three-dimensional magnetic quasiparticles and are an inspiration for novel device concepts in the field of spintronics. Based on a micromagnetic model and without considering temperature, the existence of such hopfions has been predicted in certain magnets with competing exchange interactions. However, physical realisation of freely moving hopfions in bulk magnets have so far been elusive. Here, we consider an effective Heisenberg model with competing exchange interactions and study the stability of small toroidal hopfions with Hopf number QH=1 by finding first-order saddle points on the energy surface representing the transition state for the decay of hopfions via the formation of two coupled Bloch points. We combine the geodesic nudged elastic band method and an adapted implementation of the dimer method to resolve the sharp energy profile of the reaction path near the saddle point. Our analysis reveals that the energy barrier can reach substantial height and is largely determined by the size of the hopfion relative to the lattice constant.

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    • Received 7 December 2022
    • Revised 23 January 2023
    • Accepted 22 February 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.104404

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetic textureMicromagnetism
    1. Physical Systems
    Magnetic systemsTopological materials
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Moritz Sallermann1,2,3,*, Hannes Jónsson2, and Stefan Blügel1

    • 1Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
    • 2Science Institute and Faculty of Physical Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland
    • 3Department of Physics, RWTH Aachen University, 52056 Aachen, Germany

    • *m.sallermann@fz-juelich.de

    See Also

    Lifetime, collapse, and escape paths for hopfions in bulk magnets with competing exchange interactions

    I. S. Lobanov and V. M. Uzdin
    Phys. Rev. B 107, 104405 (2023)

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    Vol. 107, Iss. 10 — 1 March 2023

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    Images

    • Figure 1
      Figure 1

      Illustration of the geometry of an isotropic toroidal hopfion with unit Hopf invariant. The toroid is formed by the nz=0 isosurface of the spin direction n(r). Further indicated are the equatorial plane and one of its normals passing through the center point of the toroid. The latter is called the hopfion normal. The colored curves (and cones) are depictions of pre-images, spatial curves along which the spin direction is constant and points into the direction of the cone tip. The color is chosen according to the spin orientation as indicated in the inset. Since this hopfion has a Hopf invariant of QH=1, each pair of pre-images is linked exactly once.

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    • Figure 2
      Figure 2

      (a) The investigated parameter space (γ, r0). The characteristic energy E0 has been kept constant at 1meV, while γ has been varied in steps of 1/7 and r0 in steps of 0.5a, where a is the lattice constant of the simple cubic atomic lattice. The markers show the discrete pairs of (γ, r0) for which the energy barrier was investigated. A blue circle signifies a stabilized hopfion and a red cross signifies a point where a hopfion could not be stabilized and the system would decay into the ferromagnetic ground state. The blue (red) shaded background illustrates the region where the criterion Eq. (8), which is derived from micromagnetic models, predicts hopfion stability (instability). The images of the magnetization textures at the top and right side show nz=0 isosurfaces of the magnetization texture, for different values of γ and r0. The color tone of the surface depends on the azimuthal angle of the spin direction; additionally, a detailed depiction of the colorcode can be found in Appendix pp2. Small white dots on top of the blue circles mark (γ, r0) points for which isosurfaces are shown. (b) Cross section of the isotropic (γ, r0) = (6/7, 5a) hopfion in a plane containing the central normal of the toroid. The bottom inset illustrates the cross section plane in relation to the nz=0 isosurface. (c) Cross section of the same hopfion in the equatorial plane of the toroid. Again, the bottom inset illustrates the cross section plane in relation to the nz=0 isosurface.

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    • Figure 3
      Figure 3

      (Left) Minimum energy path for (γ,r0)=(6/7,5a). The inset shows a zoomed subsection of the path, where many intermediate shallow minima form. Beyond the saddle point A, a pair of Bloch points emerges. The orange shading marks the interval in which this pair exists. (Top) The images show renderings of the nz=0 isosurface at different points of the path—marked by capital letters. The surfaces are colored according to the azimuthal angles of the spins, within the xy plane, and show the evolution of the spin texture from the emergence of the pair of Bloch points at “A” to its collapse at “F”. For details of the colorcode see Appendix pp2. (Right) The distance between the pair of Bloch points as a function of the reaction coordinate. The gray shade in the background represents the energy and is intended as a visual guide.

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    • Figure 4
      Figure 4

      Comparison of several hopfions to their saddle points. The top row contains renderings of the nz=0 isosurface, while the bottom row shows cross sections in a plane orthogonal to the equatorial plane of the hopfion. The grey shape in the background of the cross sections is a projection of the nz=0 isosurface and intended as a visual guide. All of the saddle points are a result of the formation of two Bloch points along the hopfion normal.

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    • Figure 5
      Figure 5

      Energy barriers as functions of the characteristic length scale r0 (top panel) and as a function of RH (bottom panel). The radius RH is related to the overall size of the Hopfion and defined as the average distance of points in the nz=0 isosurface from the toroid center point parallel-projected to the equatorial plane, (bottom panel). The inset in the bottom panel illustrates the definition of RH.

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    • Figure 6
      Figure 6

      The top (bottom) panel shows the energy difference between the hopfion (saddle point) and the ferromagnetic ground state. Notice the relation between the shade of the points and the value of the symmetry parameter γ.

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    • Figure 7
      Figure 7

      Color code that maps the direction of the spin unit vector to hue and saturation. The azimuthal angle of the spin direction corresponds to the hue, while the polar angle determines the saturation. The spherical shape at the center of the figure is colored according to the radial vector of points on the surface. The surrounding rings show colors according to spins oriented in the xy, yz, or xz plane respectively.

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    • Figure 8
      Figure 8

      Several depictions of the first globule state encountered in the collapse of the isotropic hopfion with γ=6/7 and r0=5a. (a) The nz=0 isosurface and arrows for any spin with nz<0 (b) Cut through the center of the system orthogonal to the orientation of the core spins. (c) The uncolored nz=0 isosurface and a cut through the spin system. (d) Cut through the center of the system parallel to the orientation of the core spins.

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