Three-dimensional vortex structures

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Abstract

The review provides a theoretical description of the structures of currently known three-dimensional magnetic vortices in magnets with and without an inversion center. For the case of an isotropic and uniaxial ferromagnet, the following cnoidal and spiral “hedgehogs”, vortex structures of the “inclusion” type, a vortex filament with various two-dimensional topological charges, a vortex circular filament, and a vortex ring domain wall are considered. The structure of magnetic vortices in various nanostructures is described. It is shown how a spin-transfer nanooscillator can be used to create a dissipative magnetic droplet soliton. For magnets without an inversion center, the structure of vortex objects of the following type is considered: a stack of spin spirals, magnetic skyrmion braids and magnetic skyrmion beams. It is shown that the three-dimensional structure of the vortex is the cause of a nontrivial interaction of skyrmions. An experimentally discovered new type of particle-like state in chiral magnets, the chiral bobber, is described and a concept of magnetic solid-state memory is proposed on its basis.

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A. B. Borisov

M.N. Miheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences

Author for correspondence.
Email: borisov@imp.uran.ru
Russian Federation, Ekaterinburg, 620108

E. G. Ekomasov

Ufa University of Science and Technology

Email: borisov@imp.uran.ru
Russian Federation, Ufa, 450076

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Supplementary files

Supplementary Files
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1. JATS XML
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2. Fig. 1. Distribution of the magnetization vector in a three-dimensional “hedgehog” type structure.

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3. Fig. 2. Distribution of cos θ in the “cnoidal hedgehog” in the plane z = 8 (k = 1/3, Q = 1). The inset shows domains with negative (dark areas) or positive (light areas) values of cos θ [10, 18].

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4. Рис. 3. Геликоидальная поверхность постоянных значений n3 компоненты для спирального “ежа” при k =0.125, S = 1, Q = 1 [10, 18].

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5. Fig. 4. Structure of the spiral “hedgehog” (k = 1/3, S = 1, Q = 1) at z = 5 [10, 18].

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6. Fig. 5. Structure of the field Φ2 at z = 0 [25].

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7. Fig. 6. Distribution of the vector Z in the plane z = 1 [25].

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8. Fig. 7. Distribution of the vector Z in the plane z = −1 [25].

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9. Fig. 8. Distribution of magnetization corresponding to solution (1.32) for a vortex ring domain wall. The region of the main change in magnetization in the wall from θ = 0 to θ = π [19] is highlighted.

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10. Fig. 9. Possible implementation of three-dimensional vortex excitation corresponding to solution (1.32) in a magnetic nanodot with fixed magnetization at the ends [19].

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11. Fig. 10. Vortex thread in the Oxy plane of the Cartesian coordinate system (a) and contour Γ in the cylindrical system (b) [32].

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12. Fig. 11. Section of a vortex ring by plane x = 0 at R = 1 [32].

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13. Fig. 12. The energy of the ring E, divided by the radius R (solid gray), and its approximation (1.44) (dashed) [32].

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14. Fig. 13. Calculation of the dependence of the interaction energy of two coaxial rings on the distance between their centers b. The radius of the core a = 0.1R [32].

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15. Fig. 14. Calculation of the dependence of the interaction energy of two rings located in the same plane on the distance between their centers b. The radius of the core a = 0.1R [32].

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16. Fig. 15. Stability diagram of magnetic states in nanodisks of radius R and thickness L (LE is the exchange length). Three stable states of magnetization are possible: a vortex, a single domain in a plane, and a perpendicularly magnetized single domain. The energy of the equilibrium line is shown by a solid line. The dashed line corresponds to the stability boundary of the vortex state. The shaded region is the instability region.

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17. Fig. 16. Schematic representation of a cylindrical STNO.

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18. Fig. 17. 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 either into a conical helix (c) with inclined magnetization and a wave vector along the magnetic field, or into a longitudinal helicoid (d) [107].

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19. Fig. 18. Characteristic dependence of the polar angle (a) and the equilibrium energy density (b) [34].

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20. Fig. 19. Skyrmion tube [11].

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21. Fig. 20. A set of colors indicating the direction of the vector in the section nz = 0 (a) and on the sphere (b).

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22. Fig. 21. Distribution of magnetization in the core of one skyrmion (a) in the upper (b) and lower (c) layers of a film with thickness L/LD = 0.25 in a field H/HD = 0.2 [121].

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23. Fig. 22. Dependence of the equilibrium energy of isolated 3D skyrmions (2.10) (solid line) and 2D skyrmions uniform along the z axis (dashed line) on the field H/HD. Film thickness L/LD = 0.25. The inset shows the dependence of the equilibrium energy of a skyrmion on the film thickness in a fixed field H/HD = 0.4 [121].

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24. Fig. 23. Dependence of the energy difference ∆e of the skyrmion lattice and the cone phase (solid blue line), as well as the helical and cone phases (dashed green line) on the applied field. The ∆e functions are plotted for two values of the normalized film thickness L/LD = 0.25 and 0.5. For comparison, the corresponding results for the 1D helicoid [100] and the hexagonal 2D skyrmion lattice [105] in bulk samples [121] are shown.

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25. Fig. 24. Schematic representations of skyrmion tubes. Vector fields and isosurfaces nz = 0 for a skyrmion tube with uniform magnetization along the axis of symmetry (a). Skyrmion tube with non-uniform magnetization in three directions and rotations of magnetic moments induced by the film surface (b). Note that nz > 0, nx > 0 in the sections z = const [126].

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26. Fig. 25. (a) Phase diagram of the magnetic states of an isotropic chiral magnet film in units of L/LD and the field H/HD directed normal to the film surface. In the central panel, the film thickness varies in the range from 0 to 50 LD (LD is the helicoid period at H = 0). The inset shows details of the phase diagram for films with thickness 0 < L ≤ 10LD. In the small shaded region of the conical phase, isolated metastable skyrmions have the lowest energy. The left panel corresponds to a monolayer (two-dimensional helimagnet). The right panel, to the case of an extremely thick film (bulk crystal). Small open circles I and II with LD and H/HD coordinates given in brackets correspond to triple points. (b) Average energy density of different phases as a function of the applied field for a film of thickness L = 6L [126].

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27. Fig. 26. Stacks of spin helices. Local directions of magnetization in the film are indicated by color in accordance with the scale of Fig. 20 [126].

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28. Fig. 27. Skyrmion braid in chiral magnets [132].

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29. Fig. 28. Energy gain of a skyrmion braid in a bulk sample: (a) a cluster of six straight skyrmion strings (left) and the corresponding skyrmion braid (right) in a bulk sample with periodic boundary conditions in all three dimensions; (b) the dependence of the energetically optimal twist per unit length on the applied magnetic field calculated without taking into account the dipole-dipole interaction; (c) the difference in the energy components between a cluster of straight skyrmion strings and a skyrmion braid at Bext = 0.392Bc for a z period of 25LD, with ∆ϕ ≈ 14 deg/LD [132].

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30. Fig. 29. Schematic of magnetic skyrmion beams. Typical two-dimensional skyrmion textures: (a) skyrmion with Q = 1; (b) skyrmion with Q = 0; (c) skyrmion bag with Q = 5; (d) schematic of Q = 5 skyrmion in a thin FeGe plate of 150 nm thickness. Chiral magnetic vortex structures with skyrmion bags inside appear around the two surfaces. The inner skyrmion tubes are shown as isosurfaces of normal magnetization nz = 0.7. The colors in Fig. (a–d) represent the magnetization orientation according to the color scheme in Fig. (b). The contrast of dark and white represents downward and upward magnetization, respectively; (e) average mapping of the magnetization in the plane at the depth of the skyrmion beam obtained by TEM measurements. Dark contrasts indicate that the in-plane magnetization is zero [135].

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31. Fig. 30. Schematic representation of the vector field n for the chiral bobber in the conical phase (left) and the Bloch point (right) [141].

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32. Fig. 31. Dependence of the energy of an isolated CS (light circles) and CB (solid red circles) on the magnetic field at a fixed film thickness (a, b) and on the film thickness at a fixed field (c, d), as well as the dependence of the energies of the CS and CB on the parameters of cubic (d) and uniaxial anisotropy (e) at fixed values of the film thickness and field [141].

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33. Fig. 32. Micromagnetic simulation of EG skyrmions and bobbers: (a) — results of micromagnetic simulation for a chain of SC and CB structures in a FeGe nanotrack in an external magnetic field applied along the z-axis; (b) — phase shift of the electron beam calculated for the magnetization distribution shown in (a); (c) — profile of the phase shift map in (b), averaged over the bandwidth. Blue, red and black dotted lines correspond to the averaged signal of SC, in CB and the nearly uniform state between them, respectively. Red arrows point to the peaks corresponding to CB; ∆φ is the difference height of the peaks corresponding to SC and CB [148].

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34. Fig. 33. Experimental evidence of magnetic bobber nucleation under conditions of sample annealing in a magnetic field. Dimensions and geometry of wedge-shaped (a) and rectangular (b) FeGe stripes. (c, d) — phase shift images after cooling samples (a) and (b), respectively, in a magnetic field of 300 mT. Image (c) corresponds to the part of the sample highlighted by the dashed line in Fig. (a). Red arrows point to objects with low contrast, which are identified with CBs; (d, e) — positions of phase shift profiles along stripes of the corresponding colors in images (c) and (d) [148].

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35. Fig. 34. Stability of chiral magnetic bobbers for different fields and temperatures: (a–g) — images of the phase shift from the CS and CB are located from left to right as the field increases. Images (a), (d), (g) were obtained in one experiment, and the others — in different ones. These images guarantee the reproducibility of the effect. All images were obtained at a temperature of 95 K. The position of the CB is marked with a red arrow. (h) — phase shift calculated for magnetic configurations by micromagnetic simulation. Two brighter spots correspond to the skyrmion, the gray one — the CB; (i) — phase shift along the rectangular stripes in Figs. (d) and (h). Red and blue lines — theoretical calculation and experimental data; (j) — difference in phase shift ∆φ between the signal from the CS and CB depending on the applied field. Blue squares represent experimental data at T = 95 K, red circles — theoretical calculations. The vertical dashed lines are the boundaries of the region of existence of the KB at T = 95 K. The green dashed line corresponds to the elliptical instability, and the red one to the collapse of the KB; (l) is the magnetic phase diagram of the experimentally established regions of existence of the KS (gray region) and KB (marked with double hatching) in the S2 sample with a thickness of 175 nm. The blue line gives the dependence of the skyrmion nucleation field on the temperature. The green line represents the dependence of the elliptical instability field of the KS and KB on the temperature. The red and black lines give the temperature dependences of the collapse fields of the KB and KS [148].

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36. Fig. 35. An alternative concept of magnetic solid-state memory based on encoding a data stream in a nanostrip. The strip has the form of a closed track and contains a chain of alternating magnetic skyrmions and chiral bobbers, which play the role of bits “1” and “0”. The actions of writing, reading and deleting information are performed on different sections of the guide track. The spin structures for the CS and KB are schematically represented by their isosurfaces.

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37. Fig. 36. Evolution of a pair of skyrmions with a change in the magnetic field in a FeGe plate. The directions of magnetization in the plane are indicated by color. The attraction of skyrmions to each other and the attraction of scorpions to the edges of the sample is observed in a weak magnetic field; in a strong field, the nature of the interactions becomes repulsive. The reversibility of states with a very weak hysteresis effect is illustrated by an increase (a–e) and subsequent decrease (g–m) in the magnetic field [155].

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38. Fig. 37. Micromagnetic modeling of the structure and interaction of skyrmions in the cone phase: (a–b) — isosurfaces θ = 90° and θ = 5° for an individual skyrmion and a coupled pair of CSs. For films of finite thickness, the ends of the tubular isosurfaces are distorted by a twist of magnetization near the film boundaries. The standard color map (Fig. 20) determines the directions of magnetic moments on a unit sphere, in particular, the black and white areas correspond to the values nz = 1 (θ = 0) and nz = −1 (θ = 180°); (c–d) — magnetization distributions in the z = const plane for skyrmions. The isolines of the angle θ are shown as closed curves. Dependences of the skyrmion-edge (d) and skyrmion-skyrmion (e) distances on the magnetic field. Red and blue circles represent the results of numerical simulation; (g) — field dependences of dss and dse calculated neglecting magnetostatic fields [155].

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