Static Hopf Solitons and Knotted Emergent Fields in Solid-State Noncentrosymmetric Magnetic Nanostructures

Jung-Shen B. Tai (戴榮身) and Ivan I. Smalyukh

Phys. Rev. Lett. 121, 187201 – Published 30 October 2018

Abstract

Two-dimensional topological solitons, commonly called Skyrmions, are extensively studied in solid-state magnetic nanostructures and promise many spintronics applications. However, three-dimensional topological solitons dubbed hopfions have not been demonstrated as stable spatially localized structures in solid-state magnetic materials. Here we model the existence of such static solitons with different Hopf index values in noncentrosymmetric solid magnetic nanostructures with a perpendicular interfacial magnetic anisotropy. We show how this surface anisotropy, along with the Dzyaloshinskii-Moriya interactions and the geometry of nanostructures, stabilize hopfions. We demonstrate knots in emergent field lines and computer simulate Lorentz transmission electron microscopy images of such solitonic configurations to guide their experimental discovery in magnetic solids.

Received 30 May 2018
Revised 4 September 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.187201

Physics Subject Headings (PhySH)

Magnetism

Chiral magnets

Solitons Topology

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jung-Shen B. Tai (戴榮身)¹ and Ivan I. Smalyukh^1,2,3,*

¹Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
²Materials Science and Engineering Program, Soft Materials Research Center and Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA
³Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA

^*Corresponding author. ivan.smalyukh@colorado.edu

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Vol. 121, Iss. 18 — 2 November 2018

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Images

Figure 1
3D topological soltion–hopfion. (a) Midplane cross sections of a hopfion in the plane perpendicular to $m_{0}$ (upper) and in the vertical plane containing $m_{0}$ (lower). The magnetization fields are shown with cones colored according to the corresponding points on $S^{2}$ (lower-left insets). In the $x − z$ cross section, the black stripes at the top and bottom indicate fixed boundary conditions that can be achieved, for example, using thin films of a different material (e.g., oxide or lattice mismatch layer). (b) The 3D preimages of points on $S^{2}$ indicated as cones in the upper-right inset. The linking number of preimages yields $Q = 1$ . (c) Streamlines of $B_{em}$ form the Hopf fibration. Subsets of streamlines originating from points on a horizontal black line are illustrated by the blue lines, with some highlighted by red tubes to show interlinking of the ensuing closed loops, with the linking number 1. (d) Visualization of $B_{em}$ by the isosurfaces of constant magnitude and streamlines with cones indicating directions.
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Figure 2
Stable hopfions in circular nanodisks. (a),(b),(c) Horizontal midplane cross sections (upper) and vertical midplane cross sections (lower) of hopfions with $Q = 1$ , $- 1$ , and 2, respectively, shown along with preimages of points on $S^{2}$ (corresponding to cones in the upper-right insets). (d) Computer-simulated Lorentz TEM images of a $Q = 1$ hopfion shown in (a) for viewing directions along $\hat{z}$ (upper) and perpendicular to it (lower). (e) Ground-state stability diagram of solitonic structures in nanodisks. The parameter space of stable hopfions, Skyrmions, and helical states are shown in red, blue, and green, respectively, and that for the conical state is left blank. (f) A half-hopfion with PMA only on the bottom interface for $d = 0.9 λ$ and $D = 4 λ$ . (g) Visualization of half-hopfion’s $B_{em}$ derived from (f) by the isosurfaces colored by magnitude and streamlines with cones indicating directions.
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Figure 3
Hexagonal hopfion crystal in a thin film of a chiral magnet. (a) Midplane cross sections of a unit cell of the hopfion crystal in a plane perpendicular to $m_{0}$ (upper) and in the vertical plane (indicated by a black line in the upper panel) containing $m_{0}$ (lower). (b) 3D preimages of points on $S^{2}$ indicated by cones in the lower-right inset. (c) Diagram of metastability of a hopfion crystal (shown in red) vs $ξ / λ$ , $d / λ$ and $H / H_{D}$ . (d) Dependence of $θ_{c}$ on $ξ / λ$ at $d / λ = 0.9$ . Shown in the inset is the target $S^{2}$ with the boundary at $θ = 45 °$ for $ξ / λ = 0.056$ . (e) A unit cell of the hopfion crystal confined in the space $Ω \cup Ω^{'}$ extended by $ξ$ . (f) A Lorentz TEM image of a 2D hopfion crystal.
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Figure 4
Hopfions in magnetic channels shown for three unit cells. (a) Midplane cross sections of a channel of hopfions in the plane perpendicular to $m_{0}$ (upper) and in the vertical plane containing $m_{0}$ (lower). (b) Computer-simulated Lorentz TEM images of hopfions in a channel when viewed along $\hat{z}$ (upper) and orthogonally to it (lower). (c) 3D preimages of points on $S^{2}$ indicated as cones in the lower-right inset.
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