Two-dimensional magnetic vortices

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In the proposed review, the structure of peculiar topological excitations of magnetically ordered media, the so-called two-dimensional magnetic vortices, is described as completely and in detail as possible. Magnetic vortices represent a distinct category of defects within the field of condensed matter physics. Accordingly, the structure of vortices in hydrodynamics and superfluids, as well as dislocations in solid-state physics, is presented at the beginning of the review. A specific section of the review is dedicated to elucidating the structural characteristics of plane vortices, instantons, spiral vortices, magnetic “targets,” vortex stripes, and their interactions employing analytical methods. A general solution of a two-dimensional isotropic ferromagnetic system is presented using methods of differential geometry. The discussion encompasses twodimensional vortices with anisotropic exchange interactions. A substantial portion of the review is devoted to helicoidal structures and vortices (skyrmions) in chiral magnets, encompassing their theoretical characterization based on a functional incorporating the DMI, as well as the outcomes of the early experiments on the detection of one-dimensional helical structures. A theoretical description of skyrmions and two-dimensional skyrmion lattices in bulk crystals is provided. It is observed that the DMI significantly alters the morphology of skyrmions with an arbitrary topological charge. Such structures can be represented as a “sack” with the shell comprised of kπ-skyrmions. The observed Archimedean spiral vortices are described, and a hexagonal lattice of Archimedean spiral is predicted to represent a new equilibrium phase.

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Sobre autores

A. Borisov

Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences

Autor responsável pela correspondência
Email: borisov@imp.uran.ru
Rússia, Ekaterinburg

Bibliografia

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2. Fig. 1. Distribution of the normalized velocity of the drain (a), vortex (b), vortex source (c).

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3. Fig. 2. (a) An edge dislocation is formed by the presence of an unfinished half-plane of atoms; (b) a screw dislocation is a complete shift of a section of the lattice.

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4. Fig. 3. Schematic representation of a vortex in a type II superconductor. The vortex is parallel to the external magnetic field. The field lines outside the conductor and in the center of the vortex are indicated by straight arrows, and the eddy currents are indicated by closed circular arrows.

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5. Fig. 4. Triangular lattice of vortices, if you look in the direction of the magnetic field. Each circle with an arrow symbolically depicts an eddy current, and the dot in the middle of the circle is the magnetic field line directed towards us. Every three adjacent vortices form a regular triangle.

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6. Fig. 5. Distribution of the vector n in the plane for a plane vortex at Q = 1 (a, b, c) and Q = −1 (d).

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7. Fig. 6. Distribution of the vector n in a two-vortex structure at Q1 = Q2 = 1.

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8. Fig. 7. Stereographic projection of a sphere onto a plane.

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9. Fig. 8. Distribution of magnetization in a two-dimensional vortex (intanton) with N = 1.

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10. Fig. 9. The structure of the core (surface n3 = n3 (x, y)), corresponding to a single-turn spiral (N = 1, Q = 1, r0 = 1, k = 1/2). At the bottom are shown regions on the xOy plane with positive (white) and negative (black) values ​​of the magnetization component n3. The energy of the spiral vortex, as for the flat vortex, is proportional to InR / l.

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11. Fig. 10. Structure of the magnetic “target” type.

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12. Fig. 11. Graph of the function F2(θ) for A = −1, U = 1.

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13. Fig. 12. Distribution of the vector field n in a circular strip at A = −1, U = 1.

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14. Fig. 13. Domains of definition of a two-vortex structure at b = 0 (a), (b), (c).

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15. Fig. 14. Two-vortex structure at b = 0 (a), (b).

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16. Fig. 15. Graphs of the functions θ(r) for regular solutions. The solid line corresponds to a vortex in the absence of anisotropic exchange.

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17. Fig. 16. Dependence of the parameter B(λ) on the anisotropic exchange constant.

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18. Fig. 17. Crystal structure of the right (a) and left (b) modifications of MnSi.

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19. Fig. 18. Schematic representation of various modulated states in chiral magnets. (a) Spin helix in the absence of a magnetic field with a wave vector k along the Oz axis; (b) the location of the helix in planes. Under the influence of a magnetic field, the helix (a) is transformed into a conical helix; (c) with inclined magnetization and a wave vector along the magnetic field or into a longitudinal helicoid (d).

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20. Fig. 19. Topological spin textures: a — Neel-type skyrmion N = 1; b — Bloch-type skyrmion N = 1; c — anti-skyrmion N = −1.

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21. Fig. 20. Topological spin textures in the Fe0.5Co0.5Si film [66]. Helical (a) and skyrmion (b) structures predicted by Monte Carlo simulation; (c) scheme of the skyrmion spin configuration. Experimentally observed in real space images of spin textures (d–f), according to TEM data: (d) helical structure in the absence of a magnetic field, (d) structure of a skyrmion crystal in the presence of a weak magnetic field (50 mT) directed normal to the plate, (e) enlarged view of an individual skyrmion. The color map and white arrows indicate the direction of magnetization at each point of the film.

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22. Fig. 21. Effect of magnetic field and temperature on the change in spin texture in Fe0.5Co0.5Si: (a–g) — TEM images of the dependence of texture on magnetic field; (e–h) — fast Fourier analysis processing of TEM images (a–g); (i–m) — temperature dependence of spin texture in a magnetic field of 50 mT. The magnetic field is directed along the normal (001) to the film surface. The color wheel determines the direction of magnetization at each point.

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23. Fig. 22. Experimental phase diagram of the magnetic structure in a thin Fe0.5Co0.5Si film in the variables “magnetic induction – temperature” (B – T).

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24. Fig. 23. Plot of θ(r) for 1π- and 2π-skyrmions.

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25. Fig. 24. Color map for Sz of the first kπ-skyrmions.

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26. Fig. 25. Morphology of stable chiral skyrmions with topological charges N = −3, −2, ..., 2. The upper row of images (a) corresponds to zero magnetocrystalline anisotropy (u = 0) in an external magnetic field applied perpendicular to the plane, h = 0.65. The lower row of images (b) corresponds to the case of uniaxial anisotropy, u = 1.3 and zero external field, h = 0. The colors reflect the direction of n vectors according to the scheme: black and white denote spins up and down, respectively, and red-green-blue reflect the azimuthal angle relative to the Ox axis.

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27. Fig. 26. Vortex-like stripe spin patterns.

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28. Fig. 27. Calculated AS surrounded by different structures: a, b — AS in a labyrinth structure in the absence of a magnetic field; c — in a skyrmion lattice in the absence of a field; d — in a field h = 0.5.

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29. Fig. 28. The structure of the RAS with five turns in the absence of a magnetic field (a) and three turns in a field h = 0.15 (b).

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