Statics and dynamics of skyrmions interacting with disorder and nanostructures

C. Reichhardt, C. J. O. Reichhardt, and M. V. Milošević

Rev. Mod. Phys. 94, 035005 – Published 20 September 2022

Abstract

Magnetic skyrmions are topologically stable nanoscale particlelike objects that were discovered in 2009. Since that time, intense research interest in the field has led to the identification of numerous compounds that support skyrmions over a range of conditions spanning from cryogenic to room temperatures. Skyrmions can be set into motion under various types of driving, and the combination of their size, stability, and dynamics makes them ideal candidates for numerous applications. At the same time, skyrmions represent a new class of system in which the energy scales of the skyrmion-skyrmion interactions, sample disorder, temperature, and drive can compete. A growing body of work indicates that the static and dynamic states of skyrmions can be influenced strongly by pinning or disorder in the sample; thus, an understanding of such effects is essential for the eventual use of skyrmions in applications. The current state of knowledge regarding individual skyrmions and skyrmion assemblies interacting with quenched disorder or pinning is reviewed. The microscopic mechanisms for skyrmion pinning, including the repulsive and attractive interactions that can arise from impurities, grain boundaries, or nanostructures, are outlined. This is followed by descriptions of depinning phenomena, sliding states over disorder, the effect of pinning on the skyrmion Hall angle, the competition between thermal and pinning effects, the control of skyrmion motion using ordered potential landscapes such as one- or two-dimensional periodic asymmetric substrates, the creation of skyrmion diodes, and skyrmion ratchet effects. Highlighted are the distinctions arising from internal modes and the strong gyrotropic or Magnus forces that cause the dynamical states of skyrmions to differ from those of other systems with pinning, such as vortices in type-II superconductors, charge density waves, or colloidal particles. Throughout this review future directions and open questions related to the pinning and dynamics in skyrmion systems are also discussed.

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Received 18 February 2021

DOI:https://doi.org/10.1103/RevModPhys.94.035005

Physics Subject Headings (PhySH)

Dynamical phase transitions Magnetism Skyrmions

Nanostructures

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. Reichhardt ^* and C. J. O. Reichhardt^†

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

M. V. Milošević^‡

NANOlab Center of Excellence, Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

^*reichhardt@lanl.gov
^†cjrx@lanl.gov
^‡milorad.milosevic@uantwerpen.be

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Vol. 94, Iss. 3 — July - September 2022

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Images

Figure 1
Skyrmion crystal image obtained using Lorentz microscopy on thin-film ${Fe}_{0.5} {Co}_{0.5} Si$ near $T = 25 K$ . (a) Spin structures predicted by simulation. (b) Schematic of the spin configuration in a single skyrmion. (c) Lorentz image of the skyrmion lattice. (d) Magnified view of panel (c). Here the skyrmions are on the order of 90 nm in diameter. From [459].
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Figure 2
Image of skyrmion creation at room temperature by passing current through a constriction. Here the skyrmions are approximately a micron in diameter. The constriction at the center of the image is $3 μ m$ wide and $20 μ m$ long. From [157].
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Figure 3
Different types of skyrmionic textures in real space. (a) Schematic magnetization texture of a square meron lattice. The dashed square is about 100 nm on a side in a typical experiment. From [457]. (b) Image of a polar skyrmion structure. From [67]. (c) Image of a half skyrmion lattice in a liquid-crystal system. The scale bar is $1 μ m$ long. From [287]. (d) Vector representation of the electric field for a Néel-type optical skyrmion that is roughly 500 nm in diameter. From [408].
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Figure 4
Antiskyrmions with noncircular shapes can produce alternative lattice structures. (a),(b) Schematics of the magnetic Mn atom locations in ${Mn}_{1.4} {Pt}_{0.9} {Pd}_{0.1} Sn$ . (c) Lorentz image of a helical stripe state. (d) The corresponding clockwise (CW) and counterclockwise (CCW) magnetization textures from the dashed yellow box in (c). (e) Illustration of the same helical state. (f) Square antiskyrmions forming a square lattice in a Lorentz image under an applied field of 340 mT. (g) The corresponding magnetization texture of the antiskyrmion in the dashed yellow box in (f), where large yellow arrows are Bloch lines. (h) Schematic illustration of the spin texture of the square antiskyrmion. From [307].
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Figure 5
Snapshots at different times of out-of-plane magnetization components from micromagnetic simulations of a skyrmion driven by a spin current applied parallel to a CoPt racetrack sample. (a) In a pure sample, the skyrmion travels to the edge and annihilates. (b) When the sample edge is lined with a repulsive material, the skyrmion is prevented from annihilating. From [191].
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Figure 6
(a) Scanning tunneling microscope image of an ordered superconducting vortex lattice. From [140]. (b) Magneto-optical image of a disordered superconducting vortex lattice. From [118]. (c) Optical microscope image of a colloidal lattice. From [430]. (d) Delaunay triangulation from an optical microscope image of colloidal particle positions in a colloidal glass state. From [309].
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Figure 7
Illustration of the difference between purely overdamped motion and motion with a Magnus force of strength $α_{m}$ for particles with finite damping ( $α_{d} > 0$ ). (a) Initial dense cluster of particles. (b) Trajectories of overdamped particles with $α_{m} = 0$ moving away from the center. (c) Trajectories of particles with a Magnus force $α_{m} > 0$ moving away from the center, showing the emergence of nonconservative rotation.
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Figure 8
The simplest system exhibiting depinning is an overdamped particle (filled circle) in a sinusoidal potential $U (x) = A \cos (k x)$ tilted by a driving force $F_{D}$ . The particle is pinned when $F_{D} < F_{c}$ (upper blue curve), where $F_{c}$ is the critical driving force that must be applied to enable the particle to slide. Steady state motion occurs when $F_{D} > F_{c}$ (lower orange curve).
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Figure 9
(a) Schematic velocity $v$ vs drive $F_{d}$ curves for a system with a finite depinning threshold $F_{c}$ at zero temperature ( $T = 0$ ; lower red curve) and finite temperature ( $T > 0$ ; upper green curve). Finite temperature creep occurs when the velocity remains nonzero for $F_{d} < F_{c}$ . The $T > 0$ velocity-force curve changes shape near $F_{c}$ at the crossover from creep to flow. The dashed line indicates the free-flow limit $v \propto F_{d}$ for a system with no pinning. (b) The same for particles moving over a periodic substrate with a finite depinning threshold $F_{c}$ at zero temperature ( $T = 0$ ). Lower red curve, response in the absence of ac driving; upper green curve, Shapiro steps appear when a finite ac drive is superimposed on the dc driving force.
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Figure 10
Real-space image of skyrmions (red spheres) in a particle-based model driven through randomly arranged pinning sites (blue disks) in a plastic flow phase. The trajectory of a single skyrmion (black line) shows spiraling motions inside the pinning sites. From [330].
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Figure 11
Images from 2D micromagnetic simulations showing (a)–(d) local magnetization $m_{z}$ , (e)–(h) topological charge density $n_{sk}$ , and (i)–(l) energy density $ϵ$ at magnetic fields of $B / B_{c} = 0.44$ , 0.0, $- 0.26$ , and $- 0.32$ (from left to right), where $B_{c}$ is the field at which a uniform ferromagnetic state emerges. These models reveal the size change and shape distortions of the skyrmions as well as the different types of textures that can arise. The size of the skyrmions increases as the lattice transitions from triangular to square. From [390].
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Figure 12
Dynamic phase diagram as a function of current $J$ vs magnetic field $H_{a}$ from 2D micromagnetic simulations. In the absence of a current ( $J = 0$ ), pinned spiral, skyrmion lattice, and ferromagnetic (FM) phases appear. At finite $J$ , a moving skyrmion lattice and a chiral liquid phase form at high drives. This indicates that a drive can be used to nucleate skyrmions from a spiral or ferromagnetic state. From [215].
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Figure 13
Schematic illustrations of possible ways that pinning can arise in skyrmion systems. (a) Surface thickness modulations. (b) Addition of nanodots to the surface. (c) Naturally occurring atomic defects or substitutions in the bulk of the sample. (d) Adatoms on the surface of the sample.
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Figure 14
Micromagnetic simulations of skyrmion pinning using a notch plotted as a function of the applied spin-polarized current $j$ and the notch depth. The geometry appears in the insets, where a dashed white line indicates the skyrmion trajectory. A notch (hatched region) is introduced into a nanotrack (red). The skyrmion (blue circle) either flows past the notch (upper inset and open circles) or becomes pinned near the notch tip (lower inset and filled circles). The current required to prevent pinning increases with increasing notch depth. From [358].
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Figure 15
Shape of the pinning potential produced by a hole in the sample, which has longer-range repulsion and a short-range attraction. From [272].
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Figure 16
The total, exchange (exi), Dzyaloshinskii-Moriya (DMI), anisotropy (ani), and Zeeman (field) energies plotted as a function of distance $ζ$ along the minimum energy path for a skyrmion to escape from the defect for three different types of defects at external fields of (a)–(c) $H = 0 mT$ and (d)–(f) $H = 300 mT$ . Insets: total energy landscape or effective pinning potential fit to an exponential power function. From [383].
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Figure 17
Schematic of atom-by-atom construction of potential landscapes for skyrmions. The leftmost cluster of arrows illustrates the size of a typical skyrmion. Green spheres are atoms placed so as to construct, from left to right, a repulsive, attractive, strongly repulsive, or combined attractive and repulsive pinning potential. From [6].
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Figure 18
Micromagnetic simulation images of the evolution of a skyrmion crystal and the skyrmion distortions for different times under an applied driving current. The times are (a) $t = 1.30 \times 10^{- 8}$ , (b) $2.60 \times 10^{- 8}$ , and (c) $4.87 \times 10^{- 8} s$ . Green dots are the defect sites and red regions are the skyrmion centers. Insets: corresponding structure factor measurements. (d) Magnified view of the distorted skyrmions in (c). From [152].
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Figure 19
The critical depinning force $F_{c}$ vs the ratio $α_{m} / α_{d}$ of the Magnus force to the dissipative term for 2D particle-based simulations of skyrmions moving over pointlike disorder sites for various pinning strengths $F_{p}$ . The depinning threshold decreases with increasing Magnus force. From [330].
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Figure 20
Schematic of a skyrmion (upper red dots) with both Magnus and dissipative terms and a superconducting vortex (lower blue dots) with only a dissipative term interacting with an attractive point pinning site (green dot) to illustrate how the Magnus force decreases the pinning effectiveness. The skyrmion moves in a direction that is the resultant of two velocity components: dissipative (thin brown arrows) and Magnus force induced (thick red arrows). Since the Magnus-force velocity component is perpendicular to the attractive force from the pinning site, the skyrmion deflects around the pinning site. In contrast, the superconducting vortex moves directly toward the potential minimum and is more likely to be trapped by the pinning site.
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Figure 21
(a) Schematic showing the dissipative (thin brown arrows) and Magnus (thick red arrows) velocity components for a skyrmion (upper red dot) or superconducting vortex (lower blue dot) moving toward an attractive extended line defect (green column). The overdamped superconducting vortex moves directly toward the line defect, while the skyrmion is deflected but gradually approaches the defect. Unlike the case for point pinning in Fig. 20, the skyrmion cannot simply move around the line defect but eventually reaches the defect and interacts with it. (b) Schematic of a skyrmion (red dot) located inside a closed grain boundary (green line). The skyrmion may be deflected as it moves toward the grain boundary; however, it must cross the pinning potential minimum in order to pass through the grain boundary.
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Figure 22
Two-dimensional particle-based simulations of a skyrmion interacting with a 1D defect line showing the critical depinning force for driving applied parallel ( $F_{| |}^{c}$ ; blue squares) and perpendicular ( $F_{⊥}^{c}$ ; red circles) to the line vs the relative strength $α_{m} / α_{d}$ of the Magnus force. $F_{| |}^{c}$ is insensitive to the Magnus force, while $F_{⊥}^{c}$ decreases with increasing Magnus force. From [328].
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Figure 23
Illustration of skyrmion motion through a heterochiral interface. The initial skyrmion position is on the left side of the interface. As the relative DMI strengths $D_{1}$ and $D_{2}$ are varied with respect to each other, the skyrmion trajectory is deflected by a distance $δ y$ in the positive (upper black arrows) or negative (lower blue arrows) $y$ direction. From [254].
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Figure 24
Schematic of possible 3D defects that could be created for bulk skyrmions. (a) Columnar defect tracks, which could induce the formation of a skyrmion Bose glass. (b) Splayed columnar defects, which could create a splayed skyrmion glass or promote skyrmion entanglement. (c) Random point defects, which could generate a skyrmion glass. (d) 3D planar defects.
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Figure 25
Magnetic force microscope image of disordered skyrmions in $Ir / Fe / Co / Pt$ multilayers. Black (blue) areas indicate low magnetic field and yellow (white) areas indicate high magnetic field. The scale bar is $0.5 μ m$ long. From [381].
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Figure 26
Examples of Delaunay triangulations of skyrmion lattices. (a) Image of a lattice defect consisting of sevenfold-coordinated (left black) and fivefold-coordinated (right red) skyrmions adjacent to each other. (b) Map of the local spatial angle superimposed on top of the Delaunay triangulation with a defect at the center. (c) Enlarged view of the region marked with a square in (b) showing the presence of a dislocation line at the domain boundary. From [320].
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Figure 27
(a) Schematic phase diagram as a function of quenched disorder $δ$ vs temperature $T / T_{m}$ for a 2D system, where $T_{m}$ is the melting temperature. The solid line indicates the predicted transition from a crystal to a disordered noncrystalline state ([280]). The disordered state becomes reentrant when the temperature overpowers the quenched disorder before the crystal lattice melts. The dashed red line is from the modified phase diagram proposed by [49], where the system is ordered at $T = 0$ and a low temperature disordered state does not appear until a critical amount of disorder $δ_{c}$ has been added. (b) The same for 2D colloidal experiments ([73]), where an intermediate hexatic phase appears between the crystal and disordered phases.
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Figure 28
Schematic illustrations of scenarios leading to disordered skyrmion structures without quenched disorder or temperature. (a),(b) Disordering induced by size polydispersity. (c),(d) Jamming mechanism for skyrmions with short-range contact forces. (c) Below a critical density $ρ < ρ_{c}$ , the system is liquidlike, while (d) for $ρ > ρ_{c}$ the skyrmions are in contact and form a solid.
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Figure 29
Image of a superconducting vortex lattice with intermediate disorder showing regions of crystalline sixfold-coordinated vortices (hollow green circles) and grain boundaries composed of fivefold- and sevenfold-coordinated vortices (filled red and blue circles). Under a driving current, the trajectories (black lines) indicate that depinning occurs first along the grain boundaries. From [267].
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Figure 30
Images from Monte Carlo simulations of spiral (left column) and skyrmion (right column) states with increasing magnetic spin vacancy densities $ρ$ . At $ρ = 0$ , an ordered spiral or triangular skyrmion lattice state forms. As $ρ$ increases, skyrmions nucleate in the spiral state, the skyrmion lattice becomes disordered and bimerons appear. From [375].
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Figure 31
Schematic of a possible phase diagram as a function of disorder strength $δ$ vs magnetic field $H$ for a skyrmion system. A helical state (Hl) forms at low fields. At low $δ$ and high skyrmion density there is a skyrmion Bragg glass (Sk-BrG), while at low skyrmion densities and intermediate $δ$ a skyrmion glass (Sk-Gl) appears. For large $δ$ , a mixed skyrmion-meron state with skyrmion breaking (disordered) emerges. A ferromagnetic state (ferro) appears at the highest fields.
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Figure 32
Construction of skyrmion velocity-current curves based on measurements of the topological Hall effect $ρ_{x y}^{T}$ . The results are from two different devices, one smaller (upper row) and one larger (lower row). (a),(d) $ρ_{x y}^{T}$ vs magnetic field $B$ for different applied current densities $j$ . (b),(e) Average value of $ρ_{x y}^{T}$ over the range $B = 0.2$ to 0.4 T vs $j$ at different temperatures. (c),(f) Estimated skyrmion drift velocity $v_{d}$ vs $j$ . From [207].
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Figure 33
Direct imaging measurements of skyrmion velocity. Top panel: skyrmion velocity $V_{skyrmion}$ vs current $j_{e}$ for room-temperature skyrmions. From [157]. Bottom panel: skyrmion velocity $⟨ s ⟩$ vs current $J$ for $Pt / Co / Os / Pt$ thin films showing a linear fit. From [400].
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Figure 34
Micromagnetic simulation measurements of the current-induced longitudinal velocities $v_{| |}$ of the helical (HL) and skyrmion crystal (SkX) phases vs current density $j$ in the clean (impurity-free) and dirty limits for different values of the nonadiabatic term $β$ . Center blue lines, skyrmion phases; outer red and magenta lines, helical phases. The skyrmions are much more weakly pinned than the helical phases and show an elastic depinning transition. (b) Magnification of (a) in the region of low current density. From [152].
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Figure 35
Dynamic phase diagram as a function of the driving force $F_{D}$ vs the pinning strength $F_{p}$ from particle-based simulations, showing a pinned crystal to pinned glass transition. The pinned crystal depins elastically into a moving crystal, and the pinned glass depins plastically into a plastic flow regime that transitions into a moving liquid. At high drives, a moving crystal appears. The pinned to moving crystal transition follows $F_{c} \propto F_{p}^{2}$ , while the pinned glass to plastic flow transition obeys $F_{c} \propto F_{p}$ . From [336].
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Figure 36
Plastic flow phase just above depinning from particle-based simulations of skyrmions in strong random pinning. Skyrmion positions (dots) and trajectories (lines) are obtained for a fixed time interval, and the drive is in the $+ x$ direction. (a) Near depinning, channels of flow coexist with pinned skyrmions. (b) The number of pinned skyrmions decreases with increasing drive. (c) At higher drives, plastic flow persists and the direction of motion rotates away from the driving direction. (d) Trajectories obtained over a shorter time period in a high drive dynamically ordered state where the skyrmions move at an angle of $- 79.8 °$ to the drive. From [333].
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Figure 37
Images showing current-induced plastic motion of dipole skyrmions from room-temperature experiments on Ta $(5 nm) / [Fe (0.34 nm) / Gd (0.4 nm)] \times 100 / Pt (3 nm)$ . (a) Original soft x-ray microscopy image of a close-packed skyrmion lattice. (b) Postprocessed binary image of (a) where the background has been subtracted. (c)–(f) Skyrmion dynamics obtained by summing images of the domain morphology before and after a current pulse is injected, where purple indicates places where the domain morphology has changed. From [264].
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Figure 38
Static structure factor $S (q)$ from particle-based skyrmion simulations. The driving force increases from left to right in each row. An overdamped system with an intrinsic Hall angle of $θ_{SkH}^{int} = 0$ is displayed in (a) the plastic flow state, (b) the moving smectic state, and (c) the moving anisotropic crystal state. A system with $θ_{SkH}^{int} = 45 °$ is shown in (d) the moving liquid state, (e) a slightly anisotropic moving crystal state, and (f) the moving crystal state. A system with $θ_{SkH}^{int} = 70 °$ is displayed in (g) the moving liquid state, (h) a slightly anisotropic moving crystal state, and (i) the moving crystal state. From [75].
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Figure 39
Spectral density plots from particle-based skyrmion simulations showing $S_{| |} (ω)$ (upper blue lines) and $S_{⊥} (ω)$ (lower red lines) for velocity fluctuations parallel and perpendicular, respectively, to $θ_{SkH}$ . The driving force increases from left to right. An overdamped sample with $θ_{SkH}^{int} = 0 °$ is displayed in (a) the disordered flow state and (b),(c) two drives in the moving smectic phase. A sample with $θ_{SkH}^{int} = 45 °$ is shown in (d) the disordered flow state, (e) the moving liquid phase, and (f) the moving crystal phase. A sample with $θ_{SkH}^{int} = 70 °$ is displayed in (g) the disordered flow state, (h) the moving liquid phase, and (i) the moving crystal phase. From [75].
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Figure 40
(a) Spectral voltage noise power vs applied current density obtained from micrometer-size MnSi samples in the skyrmion lattice phase. Representative power spectra as a function of $S_{v}$ vs frequency are displayed in (b) the white noise regime (left white region), (c) the broadband noise regime (center yellow area), and (d) the narrowband noise regime (right blue region). From [360].
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Figure 41
Images from particle-based simulations of skyrmion avalanches where skyrmions are added slowly to the pin-free region (left gray regions) and move into the pinned region (right white areas) under their own gradient-induced repulsion. In each avalanche event, light red dots indicate skyrmions that translated a distance greater than a pinning site radius, dark blue dots are stationary skyrmions, and lines show the skyrmion trajectories. Here (a) $θ_{SkH}^{int} = 0 °$ , (b) 30°, (c) 60°, and (d) 80°. The avalanche motion curves increasingly into the $+ y$ direction as the magnitude of the Magnus term increases. From [74].
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Figure 42
Dynamic phase diagrams as a function of applied current $j$ vs impurity strength $K_{imp}$ from continuum simulations. The applied magnetic field is (a) $h = 0.025$ and (b) 0.04. The dynamic phases include the skyrmion glass (SkG) and moving skyrmion crystal (SkX) states. From [183].
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Figure 43
Results from Hall measurements of 3D skyrmions in MnSi thin-plate samples. (a) Sample temperature $T$ and (b) the real part of the second-harmonic Hall resistivity $Re ρ_{z x}^{2 f}$ vs the driving current density measured at a frequency of $f = 13 Hz$ . (c) Dynamic phase diagram as a function of the current density vs the temperature $T$ showing regions where the skyrmions are pinned (left green band), bending (center blue area), and straight (right white region). From [449].
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Figure 44
Dynamic phase diagram from numerical simulations of skyrmion strings as a function of $L_{Z}$ , the thickness of the 3D system, vs $j_{s}$ , the applied current density. From [184].
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Figure 45
Particle-based simulation measurements of the behavior of the skyrmion Hall angle $θ_{sk}$ for skyrmions driven over random disorder. (a) Skyrmion velocities in the directions parallel ( $| V_{| |} |$ , lower blue line) and perpendicular ( $| V_{⊥} |$ , upper red line) to the driving force vs $F_{D}$ . Inset: enlargement of the main panel in the region just above depinning where there is a crossing of the velocity-force curves. (b) Corresponding $R = | V_{⊥} / V_{| |} |$ vs $F_{D}$ . The solid straight line is a linear fit and the dashed line is the clean limit value of $R \approx 6.0$ . Inset: $θ_{sk} = \tan^{- 1} (R)$ vs $F_{D}$ . The dashed line is the clean limit value of $θ_{sk}$ . From [333].
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Figure 46
Schematic illustration of how pinning changes the effective skyrmion Hall angle. The left blue dot is the skyrmion and the right red circle is the pinning site, while $J$ is the direction of the applied current and $θ_{sk}$ is the intrinsic skyrmion Hall angle. (a) At low drives, the skyrmion executes a Magnus-induced orbital motion as it traverses the pinning site, leading to a side jump in the direction of the current that reduces the effective skyrmion Hall angle. (b) At higher drives, the skyrmion moves rapidly through the pinning site and the magnitude of the side jump is strongly reduced.
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Figure 47
Multiscale simulations of the trajectories of skyrmions scattering from a defect site (black dot) consisting of a single atom. (a) Skyrmions are pinned at low currents. (b) For higher currents, the skyrmions escape but the trajectories are deflected by an amount that decreases with increasing current. From [97].
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Figure 48
Skyrmion velocity and skyrmion Hall angle obtained from direct imaging of the skyrmion motion. (a) Average skyrmion velocity vs current density $j_{e}$ showing a pinned regime (left blue band) and a $θ_{SkH} = 0 °$ region (center orange band). (b) The corresponding $θ_{SkH}$ vs $j_{e}$ . (c) $θ_{SkH}$ for positive and negative driving currents $j_{e}$ under positive and negative applied magnetic fields. In each case, $θ_{SkH}$ saturates for sufficiently large magnitudes of $j_{e}$ . From [158].
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Figure 49
Image-based experimental measurements of $θ_{SkH}$ vs skyrmion velocity $v$ showing a linear dependence. From [219].
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Figure 50
Composite magneto-optical polar Kerr effect images showing the current-induced motion (yellow arrows) of (a) synthetic antiferromagnetic skyrmion bubbles and (b) ferromagnetic skyrmion bubbles under a pulsed current. (c) Schematic of the experiment in which each bubble moves during the current pulse. (d) Skyrmion Hall angle as a function of skyrmion velocity in the two systems indicating that the skyrmion Hall angle in the ferromagnet is more sensitive to skyrmion velocity than that of the synthetic antiferromagnet. From [80].
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Figure 51
Continuum simulations of skyrmion motion through a disordered landscape composed of grains of different sizes $g$ . (a) Mean skyrmion velocity $v$ vs the driving current $J$ showing a finite depinning threshold. (b) Skyrmion Hall angle vs $J$ showing that the angle increases with an increasing $J$ from a value near zero at zero current. From [195].
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Figure 52
Images of skyrmion motion in a multilayer system at varied magnetic fields. In each case, $θ_{SkH} = 10 °$ . The skyrmion size changes as the field varies, so this result indicates that $θ_{SkH}$ is independent of the skyrmion diameter. From [470].
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Figure 53
Theoretical predictions for skyrmion velocity response at different temperatures, showing a creep regime below the zero-temperature depinning threshold. Top panel: longitudinal velocity $u_{x}$ vs driving current $j$ . Bottom panel: transverse velocity $u_{y}$ vs $j$ . From [407].
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Figure 54
Particle-based simulations of skyrmion motion with finite thermal fluctuations. (a) Skyrmion Hall angle $θ_{sk}$ vs the driving force $F_{D}$ . (b) Corresponding skyrmion velocity parallel $⟨ V_{| |} ⟩$ (blue squares) and perpendicular $⟨ V_{⊥} ⟩$ (red circles) to the drive vs $F_{D}$ . There is a pinned phase (left yellow band), a creep phase with $θ_{sk} \approx 0 °$ (center green band), and a flowing phase. From [337].
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Figure 55
Continuum simulations of $θ_{SkH}$ vs skyrmion velocity $v$ in a sample with no thermal disorder (black squares) and at two different finite temperatures (open symbols). $θ_{SkH}$ is nearly independent of velocity in the absence of temperature, but when thermal fluctuations are present $θ_{SkH}$ increases with increasing velocity. Insets: display of change in skyrmion shape from nearly circular at low velocities to strongly distorted at high velocities. From [220].
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Figure 56
Examples of skyrmions interacting with nanostructured pinning. Left panels: 2D periodic pinning, where commensuration can occur between the number of skyrmions and the number of pinning sites. Center panels: asymmetric 2D periodic pinning capable of generating ratchet and diode effects. Right panels: 1D periodic pinning. The lower panels show schematic transport curves that could be observed with each pinning geometry.
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Figure 57
An example of a periodic quasi-one-dimensional substrate for skyrmions. The substrate is sinusoidal along the $x$ direction with regular minima (white bands) and maxima (green bands). The skyrmions (red dots) are driven parallel to the substrate periodicity by $F_{| |}^{D}$ (blue arrow), or perpendicular by $F_{⊥}^{D}$ (red arrow). From [328].
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Figure 58
Illustration of the speedup effect from particle-based simulations of skyrmion velocities parallel ( $⟨ V_{| |} ⟩$ ; lower blue lines) and perpendicular ( $⟨ V_{⊥} ⟩$ ; upper red lines) to the substrate periodicity direction for perpendicular driving $F_{⊥}^{D}$ in the quasi-1D potential illustrated in Fig. 57. (a) At $θ_{SkH}^{int} = 30 °$ , the initial skyrmion motion is locked in the perpendicular direction. There is a drop in $⟨ V_{⊥} ⟩$ at the critical drive $F_{c}^{| |}$ for the onset of motion in the parallel direction. At (b) $θ_{SkH}^{int} = 64 °$ and (c) $θ_{SkH}^{int} = 84.3 °$ , $F_{c}^{| |}$ shifts to lower drives and the drop in $⟨ V_{⊥} ⟩$ becomes more pronounced. (d) Total velocity $⟨ V_{tot} ⟩$ vs $F_{⊥}^{D}$ at $θ_{SkH}^{int} = 84.3 °$ . The dashed line indicates the response $⟨ V_{0} ⟩$ expected in a system with no substrate. In the speedup effect, $⟨ V_{tot} ⟩ > ⟨ V_{0} ⟩$ . From [328].
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Figure 59
Results from continuum simulations of a skyrmion interacting with a line along which the DMI has been changed by an amount $δ$ compared to the rest of the sample. (a) Phase diagram as a function of the product of the skyrmion Hall angle $θ_{H}$ and current $J_{H M}$ vs $δ$ . (b) Illustration of motion in the six different regimes. The green vertical line is the defect and the curved gray line is the skyrmion trajectory. From [47].
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Figure 60
Phase locking and Shapiro steps for current-driven skyrmions in MnSi under combined dc and ac driving. (a) Magnitude of the narrowband noise $f_{NBN}$ as a function of dc driving current $j_{dc}$ for different values of the ac current $j_{ac}$ , showing the emergence of a locking step when $j_{ac} = 1.95 \times 10^{8} A / m^{2}$ . (b) Dependence of the locking step width $Δ j_{dc}$ on ac current amplitude $j_{ac}$ showing Bessel function oscillations consistent with Shapiro steps. From [359].
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Figure 61
Particle-based simulations of skyrmions moving over a square array of pinning sites showing the quantization of $θ_{SkH}$ . (a) The velocity parallel ( $⟨ V_{| |} ⟩$ ; upper blue line) and perpendicular ( $⟨ V_{⊥} ⟩$ ; lower red line) to the driving direction vs the dc drive amplitude $F_{D}$ at a Magnus-to-damping ratio of $α_{m} / α_{d} = 4.925$ . (b) $⟨ V_{| |} ⟩$ vs $F_{D}$ for a larger ratio $α_{m} / α_{d} = 9.962$ . (c) The ratio $R = ⟨ V_{⊥} ⟩ / ⟨ V_{| |} ⟩ = \tan (θ_{SkH})$ vs $F_{D}$ for the sample in (a), where steps appear at rational fractions. (d) $R$ vs $F_{D}$ for the sample in (b) also exhibits a series of steps. From [331].
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Figure 62
Skyrmion trajectories (lines) for particle-based simulations of the system in Fig. 61 with a skyrmion (red circle) moving through a periodic array of pinning sites (black dots) at (a) $| R | = 1$ , (b) $| R | = 5 / 3$ , (c) $| R | = 2$ , and (d) $| R | = 3$ . From [331].
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Figure 63
Combined micromagnetic and analytic calculations of skyrmion trajectories (red lines) in a square array of magnetic dots for different values of the damping coefficient $α$ . By varying the direction of the applied current pulse, the skyrmion can be steered to any position in the array. From [94].
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Figure 64
Continuum simulations of chiral liquid-crystal skyrmions (blue rings) interacting with a periodic array of obstacles (black circles). (a) Filling ratio of $f = 1$ , where the skyrmions form a square lattice. (b) Alternating dimer ordering for $f = 2$ . (c) Trimer arrangement at $f = 3$ . (d) Ordered quadrimer state at $f = 4$ . From [85].
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Figure 65
Numerical model for the motion of a skyrmion over a moiré pattern formed by a van der Waals 2D magnet. (a) The localized region (red) indicates the location of the skyrmion as a function of time. (b) The time profile of the applied current $j$ (green profile) and the energy of the skyrmion $E / I$ during the motion illustrated in (a). (c) Schematic of the use of a spin-polarized scanning tunneling microscopy tip (upper left gray area) to write a skyrmion, which is then moved from one substrate minimum to another with a current pulse (green profile). From [403].
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Figure 66
Artificial ice geometry for skyrmions. (a) Geometry constructed using elliptical blind holes with opposing magnetization directions inside and outside the holes. (b) Perpendicular or $z$ component of the resulting stray field. (c) Images of the spin configuration (left panel) and the topological density distribution (right panel) of an isolated individual skyrmion. (d) Large skyrmions sit at the center of each blind hole to form a nonfrustrated configuration. (e) Small skyrmions sit at one end of each blind hole and form a frustrated state. From [234].
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Figure 67
Micromagnetic simulations of skyrmion localization in a sample with blind holes etched on the top and bottom faces. In (a)–(e), each blind hole is able to capture a single skyrmion, but if the spacing between etched regions or the distance to the sample edge becomes too small, only some blind holes capture a skyrmion, as shown in (f). From [306].
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Figure 68
(a) Schematic of skyrmions in a nanowire interacting with a 1D periodic substrate at a filling just above $1 ∶ 1$ matching. The additional skyrmion forms a mobile kink that moves in the driving direction. Every time the kink moves through the system, the skyrmion translates by one lattice constant. (b) The same for an antikink just below $1 ∶ 1$ matching that moves in the opposite direction. (c) Two coupled wires with different skyrmion species that could bind together into skyrmion excitons.
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Figure 69
Schematic of possible orderings on square and triangular pinning arrays (large blue circles) at half filling. Left side: skyrmion (small red circles) orderings. Right side: skyrmions elongate into meron pairs (red lozenges) to create a $1 ∶ 1$ filling for the square pinning array but still leave unoccupied pins in the triangular pinning array.
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Figure 70
Schematic of a quasi-one-dimensional asymmetric ratchet potential. A skyrmion (red circle) can be driven by an ac current applied parallel ( $F_{| |}^{ac}$ ; left green arrow) or perpendicular ( $F_{⊥}^{ac}$ ; right blue arrow) to the substrate periodicity direction. An overdamped particle would exhibit no ratcheting effect under $F_{⊥}^{ac}$ , but due to the Magnus effect a skyrmion can undergo ratcheting motion under either ac driving direction. From [329].
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Figure 71
Particle-based simulations of skyrmion ratchet motion under perpendicular driving $F_{⊥}^{ac}$ on the asymmetric substrate illustrated in Fig. 70. (a)–(c) Velocities parallel ( $⟨ V_{| |} ⟩$ ; upper red area) and perpendicular ( $⟨ V_{⊥} ⟩$ ; lower blue region) to the substrate asymmetry as a function of the ac driving force magnitude $F_{⊥}^{ac}$ for different values of the Magnus-force-to-damping-force ratio $α_{m} / α_{d}$ . Ratcheting with quantized velocity values occurs in both the parallel and perpendicular directions above a threshold value of $F_{⊥}^{ac}$ , and there can be drive windows in which no ratcheting motion occurs. Inset of (a): In an overdamped system, no ratcheting occurs in either direction at any value of $F_{⊥}^{ac}$ . Inset of (b): Illustration of the skyrmion trajectory on the $n = 2$ ratcheting step from the main panel. Inset of (c): enlargement of (c) highlighting the presence of fractional velocity steps. From [329].
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Figure 72
Thiele-based simulations showing the operation of a ratchet mechanism in a skyrmion racetrack. (a) An asymmetry in the racetrack edge combining with the Magnus force to produce a 2D orbit that translates over time. (b) Plot of the skyrmion position vs time showing deterministic ratcheting motion in the $+ x$ direction. (c) Shape of the skyrmion orbit as a function of the $x$ direction velocity $v_{x}$ vs the relative displacement in $x$ from the average position. From [120].
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Figure 73
Schematic of the coupling between a chiral ferromagnet (CM) (lower tan region) containing a skyrmion crystal (lower purple circles) and a superconducting film (SC) (upper blue area). The materials are separated by a thin insulating barrier (center gray section) to ensure that only the magnetic fields from the skyrmion lattice pass into the superconductor. The attractive interaction between vortices and skyrmions generates vortices (upper orange circles) in the superconductor. From [64].
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Figure 74
Micromagnetic calculations (arrows) and Thiele equation calculations (thin lines) of skyrmion trajectories in the moving frame produced by interactions with a moving superconducting vortex. The background coloring represents the $z$ component of the magnetization from the vortex that would appear in the absence of the skyrmion. Open dots represent fixed saddle points and filled dots indicate stable spiral points. From [255].
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