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(color online). 3D skyrmion lattice: (a) Distribution of the magnetization in the skyrmion core demonstrates chiral conical modulations along the cell axis. For clarity, the sizes along the z-axis are magnified; (b) Calculated distribution of the magnetization in the skyrmion cell for top and bottom layers in a film with thickness L/L D = 0.25 and in the applied field H/H D = 0.2.
Source publication
A direct three-dimensional minimization of the standard energy functional
shows that in thin films of cubic helimagnets chiral skyrmions are modulated
along three spatial directions. The structure of such 3D skyrmions can be
thought of as a superposition of conical modulations along the skyrmion axis
and double-twist rotation in the perpendicular p...
Citations
... Corresponding GPA and 6 data are shown in Fig. 4 and Supplementary Information S3. www.nature.com/scientificreports/ three-dimensional magnetic texture that is not purely Bloch-type but shows also Néel-type components near the surfaces, so-called surface twist effect [42][43][44] . Systematic studies as a function of the thickness of the lamella could be carried out to understand the influence of the surfaces on the skyrmion lattice. ...
Magnetic skyrmions are quasi-particles with a swirling spin texture that form two-dimensional lattices. Skyrmion lattices can exhibit defects in response to geometric constraints, variations of temperature or applied magnetic fields. Measuring deformations in skyrmion lattices is important to understand the interplay between the lattice structure and external influences. Geometric phase analysis (GPA) is a Fourier-based image processing method that is used to measure deformation fields in high resolution transmission electron microscopy (TEM) images of crystalline materials. Here, we show that GPA can be applied quantitatively to Lorentz TEM images of two-dimensional skyrmion lattices obtained from a chiral magnet of FeGe. First, GPA is used to map deformation fields around a 5–7 dislocation and the results are compared with the linear theory of elasticity. Second, rotation angles between skyrmion crystal grains are measured and compared with angles calculated from the density of dislocations. Third, an orientational order parameter and the corresponding correlation function are calculated to describe the evolution of the disorder as a function of applied magnetic field. The influence of sources of artifacts such as geometric distortions and large defoci are also discussed.
... The exploration of element doping, using both non-magnetic element Si and magnetic element Co in FeGe, has led to intriguing advancements, such as controlled changes in λ, T C , and stability of SkX as well as the corresponding transport properties (anomalous and topological hall resistivity) [25][26][27][28][29][30][31]. In Si-doped FeGe, both λ and T C decrease with increasing Si composition [27]. ...
... the doping systematics in topological spin textures remains elusive for x < 0.5. Despite extensive research efforts on profound features of topological spin textures, such as spin spiral modulation in various dimensions [29], anisotropic effects [28], and surface twist instability [30,31], there remains significant interest in understanding the stability of topological spin textures in FeGe when elements are doped into the material. ...
... The stability of the SkX is reasonably enhanced with decreasing the dimensionality of helimagnets [9,22,29,42]. ...
Element substitutions with magnetic or non-magnetic atoms are known to significantly impact the magnetic structure and related transport properties of magnets. To clarify the change of magnetic structure of B20-type magnets with element doping, we conduct real-space observations of spin textures and their temperature ( T ) - magnetic field ( H ) phase diagrams of a helimagnet FeGe with partially substituting Fe and Ge with Co and Si, respectively. The helical period ( λ ) changes dramatically by the element doping: λ increases by 147% to 103 nm in 30% Co-doped FeGe, whereas it decreases by around 70% to 49 nm in 30% Si-doped FeGe, compared to the λ = 70 nm in FeGe. Upon applying the magnetic field applying normally to (001), (110), and (111) thin plates of both FeSi 0.3 Ge 0.7 and Fe 0.7 Co 0.3 Ge, the hexagonal skyrmion crystal (SkX) state emerges. The magnetic phase diagrams observed through the real-space imaging reveal that 1) the SkX can extend to a larger T - H window by reducing the sample thickness or by cooling the sample under specific magnetic fields from temperatures above the transition temperature ( T C ); 2) the stability of the SkX phase differs between Si-doped and Co-doped FeGe: the SkX phase is most unstable in the FeSi 0.3 Ge 0.7 (111) thin plate, while it remains robust in the Fe 0.7 Co 0.3 Ge (111) thin plate. These differences indicate distinct anisotropic behavior in FeGe with magnetic (Co) and non-magnetic-element (Si) dopants.
... Here fortunately, the analysis of a perfect skyrmion lattice showed no significant changes of magnification 159 and rotation across the field-of-view (see Supplementary Information 4). Bloch-type but shows also Néel-type components near the surfaces, so-called surface twist effect [39,40,41]. ...
Magnetic skyrmions are quasi-particles with a swirling spin texture that form two-dimensional lattices. Skyrmion lattices can exhibit defects in response to geometric constraints, variations of temperature or applied magnetic fields. Measuring deformations in skyrmion lattices is important to understand the interplay between the lattice structure and external influences. Geometric phase analysis (GPA) is a Fourier-based image processing method that is used to measure deformation fields in high resolution transmission electron microscopy (TEM) images of crystalline materials. Here, we show that GPA can be applied quantitatively to Lorentz TEM images of two-dimensional skyrmion lattices obtained in a chiral magnet of FeGe. First, GPA is used to map deformation fields around a 5-7 dislocation and the results are compared with linear elastic theory. Second, rotation angles between skyrmion crystal grains are measured and compared with angles calculated from the density of dislocations. Third, an orientational order parameter and the corresponding correlation function are calculated to describe the evolution of the disorder as a function of applied magnetic field. The influence of sources of artifacts such as geometric distortions and large defoci are also discussed.
PACS: 12.39.Dc, 81.40.Lm, 68.37.Lp
... 1(d)-1(h)]. Single skyrmions are stabilized as a form of skyrmion tubes through the samples [1,2,[16][17][18] [ Fig. 1(d)]. In addition, Bloch points (BPs) are pointlike topological defects where the magnetization vanishes in the center and play crucial roles in mediating skyrmion annihilation processes [2,14,[18][19][20]. ...
... Magnetic skyrmions in three dimensions are stabilized in the form of skyrmion tubes [ Fig. 1(d)]. Although the ideal solutions of skyrmions modulated by the bulk DMI are Bloch type, the lack of neighboring spins near the top and bottom surfaces induces distortions of the chiralities along a skyrmion tube [16,17,40,43], i.e., the chirality of the skyrmion profile is π/2 (pure Bloch type) in the middle plane, but gradually increases/decreases with the mixture of a Néeltype profile when moving towards the top/bottom surface. As a result, this "surface twist" effect will induce a spatial inhomogeneity of energy distributions throughout the film, especially the magnetostatic energy [18]. ...
Magnetic skyrmions are nanoscale spin textures whose thermal stability originates from the nontrivial topology in nature. Recently, a plethora of topological spin textures have been theoretically predicted or experimentally observed, enriching the diversity of the skyrmionic family. In this work, we theoretically demonstrate the stabilities of various topological spin textures against homochiral states in chiral magnets, including chiral bobbers, dipole strings, and skyrmion tubes. They can be effectively classified by the associated topological Hall signals. Multiple transition paths are found among these textures, mediated by Bloch-point singularities, and the topological protection property here can be manifested by a finite energy barrier with the saddle point corresponding to the Bloch-point creation/destruction. By carefully modulating the local property of a surface, such as interfacial Dzyaloshinskii-Moriya interaction induced by breaking the structural symmetry, the energy landscape of a magnetic system can be tailored decisively. Significantly, the proposed scenario also enables the manipulation of stabilities and transition barriers of these textures, even accompanied by the discovery of ground-state chiral bobbers. This study may raise great expectations on the coexistence of topological spin textures as spintronics-based information carriers for future applications.
... Due to the topological nature of such a transition, it typically requires elevated temperatures to overcome the energy barrier associated with the nucleation of skyrmions. Due to the effect of chiral surface twist [5,6], which provides an additional energy gain for skyrmions, the thickness of the sample also plays a crucial role. The chiral surface twist represents additional chiral modulations of magnetization in the vicinity of the sample surface. ...
... The chiral surface twist represents additional chiral modulations of magnetization in the vicinity of the sample surface. Besides the energy gain it provides to the skyrmions [5], it is also responsible for the stability of chiral bobbers [7] and stacked spin spirals [8]. Above critical thickness, the conical and stacked spiral phases become energetically more favorable compared to a skyrmion lattice. ...
... Taking into account the above invariance of the skyrmion energy with respect to arbitrary rotation about the principal axis (4), we consider the skyrmion dynamics under circulating magnetic field [26], ϕ h (t ) = 2πωt, ϑ h = const. First, we parametrize magnetization n in a frame x , y , z related to the frame x, y, z by rotation about the x axis by angle ϕ h (t ): n = ⎛ ⎜ ⎝ sin cos sin sin cos ϕ h (t ) − cos sin ϕ h (t ) cos cos ϕ h (t ) + sin sin sin ϕ h (t ) ⎞ ⎟ ⎠, (5) where and are the standard spherical angles in the x, y, z frame. Using (5), the Landau-Lifshitz-Gilbert (LLG) equation [27] can be written as ...
We show that competition between local interactions in monoaxial chiral magnets provides the stability of two-dimensional solitons with identical energies but opposite topological charges. These skyrmions and antiskyrmions represent metastable states in a wide range of parameters above the transition into the saturated ferromagnetic phase. The symmetry of the underlying micromagnetic functional gives rise to soliton zero modes allowing efficient control of their translational movement by the frequency of the circulating external magnetic field. We also discuss the role of demagnetizing fields in the energy balance between skyrmions and antiskyrmions and in their stability.
... The enhanced stability skyrmions' gain in thin layers of chiral magnets rests on additional surface twists [21,22] of the magnetization near the confining surfaces. Indeed, in bulk helimagnets, only the Lifshitz invariants L (x,y) ...
... x,y leads to the gradual variation in the skyrmion helicity (γ = π/2 ± δ(z)) towards upper and lower surfaces [21,22]. This effect accumulates additional negative energy as compared with the cones and leads to the SkL stability [22,23]. ...
... For low field values, such spirals have lower energy compared to the conical state, which is readily explained by the additional surface twists for any thickness of the film (mechanism described in the introduction). This effect accumulates additional negative energy as opposed to the cones [21,22], which are not decorated by the surface twists [21,22]. Cones in this geometry (schematically shown in Figure 1c) are oriented along the field and stabilize in the green-shaded region of the phase diagram ( Figure 1a). ...
The skyrmion Hall effect, which is regarded as a significant hurdle for skyrmion implementation in thin-film racetrack devices, is theoretically shown to be suppressed in wedge-shaped nanostructures of cubic helimagnets. Under an applied electric current, ordinary isolated skyrmions with the topological charge 1 were found to move along the straight trajectories parallel to the wedge boundaries. Depending on the current density, such skyrmion tracks are located at different thicknesses uphill along the wedge. Numerical simulations show that such an equilibrium is achieved due to the balance between the Magnus force, which instigates skyrmion shift towards the wedge elevation, and the force, which restores the skyrmion position near the sharp wedge boundary due to the minimum of the edge–skyrmion interaction potential. Current-driven dynamics is found to be highly non-linear and to rest on the internal properties of isolated skyrmions in wedge geometries; both the skyrmion size and the helicity are modified in a non-trivial way with an increasing sample thickness. In addition, we supplement the well-known theoretical phase diagram of states in thin layers of chiral magnets with new characteristic lines; in particular, we demonstrate the second-order phase transition between the helical and conical phases with mutually perpendicular wave vectors. Our results are useful from both the fundamental point of view, since they systematize the internal properties of isolated skyrmions, and from the point of view of applications, since they point to the parameter region, where the skyrmion dynamics could be utilized.
... In fact, such rope-like twisted patterns of skyrmion strings were only confirmed recently [35]. On the other hand, also the emergence of 3D localized hybrid magnetization textures consisting of a smooth skyrmion (tube) field terminated by one, as in the case of the chiral bobber [36,37], or two monopoles or Bloch points, respectively, as in the case of the chiral magnetic globule [38] (also coined toron FIG. 1. Illustration of the geometry of an isotropic toroidal hopfion with unit Hopf invariant. The toroid is formed by the n z = 0 isosurface of the spin direction n(r). ...
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.
... Magnetic skyrmions are topologically ordered, swirling spin configurations, which extend into strings in threedimensional (3D) space [1]. The study of skyrmion stabilization and evolution mechanisms has been a long-standing task [2][3][4][5][6][7][8][9][10][11][12][13][14][15], which is intimately related to singularity dynamics and topology-driven phase transitions [4,[16][17][18][19][20][21][22][23][24][25]. Recently, it has been recognized that skyrmion order is able to energetically dominate over other competing phases by adjusting its 3D string configurations [9,10,17,19,22,[26][27][28][29][30][31][32][33][34][35][36][37][38][39], providing a new perspective for studying their nucleation process. ...
... The study of skyrmion stabilization and evolution mechanisms has been a long-standing task [2][3][4][5][6][7][8][9][10][11][12][13][14][15], which is intimately related to singularity dynamics and topology-driven phase transitions [4,[16][17][18][19][20][21][22][23][24][25]. Recently, it has been recognized that skyrmion order is able to energetically dominate over other competing phases by adjusting its 3D string configurations [9,10,17,19,22,[26][27][28][29][30][31][32][33][34][35][36][37][38][39], providing a new perspective for studying their nucleation process. ...
... For thin plates, the breaking of translational symmetry at the two surfaces becomes energetically dominant [43,44], providing an exotic stabilization mechanism for skyrmions [9]. The surface state is represented by a unique 3D skyrmion structure with modulated skyrmion helicity angle along the * These authors contributed equally to this work. ...
We identify a three-dimensional skyrmion side-face state in chiral magnets that consists of a thin layer of modulated surface spirals and an array of phase-locked skyrmion screws. Such chiral spin structures lead to a characteristic X-shaped magnetic diffraction pattern in resonant elastic x-ray scattering, reminiscent of Photo 51 of the DNA double-helix diffraction. By measuring both thin plates and bulk Cu2OSeO3 crystals in the field-in-plane geometry, we unambiguously identify the modulated skyrmion strings by retrieving their chirality and helix angle. The breaking of the translational symmetry along the side faces suppresses the bulk-favored conical state, providing a stabilization mechanism for the skyrmion lattice phase that has been overlooked so far.
... Skyrmions are normally stabilized by the competition of direct exchange, the Zeeman interaction, thermal fluctuations, and the Dzyaloshinskii-Moriya interaction (DMI), which requires a noncentrosymmetric crystal structure as present in conventional helimagnetic systems, such as MnSi [3], FeGe [6], and the insulating ferrimagnet Cu 2 OSeO 3 [7]. Characterized by an integer winding number, skyrmions arrange in a periodic hexagonal crystal in two dimensions with a tubelike nature in the third dimension [8,9]. Their structure, size, and dynamical properties indicate promise for skyrmions as elements in complex computing devices, such as skyrmion racetrack memory [10,11], logic gates [12], Boolean processors [13], skyrmion transistors [14], as well as neuromorphic [15], stochastic [16], and reservoir computing [17]. ...
Mechanical straining of skyrmion hosting materials has previously demonstrated increased phase stability through the expansion of the skyrmion equilibrium pocket. Additionally, metastable skyrmions can be generated via rapid field cooling to form significant skyrmion populations at low temperatures. Using small-angle x-ray scattering and x-ray holographic imaging on a thermally strained 200-nm-thick FeGe lamella, we observe temperature-induced strain effects on the structure and metastability of the skyrmion lattice. We find that in this sample orientation (H ∥ [1¯10]) with no strain, metastable skyrmions produced by field cooling through the equilibrium skyrmion pocket vanish from the sample upon dropping below the well-known helical reorientation temperature. However, when strain is applied along the [110] axis, and this procedure is repeated, a substantial volume fraction of metastable skyrmions persist upon cooling below this temperature down to 100 K. Additionally, we observe a large number of skyrmions retained after a complete magnetic field polarity reversal, implying that the metastable energy barrier protecting skyrmions from decay is enhanced.
... In fact, such rope-like twisted patterns of skyrmion strings were only confirmed recently [35]. On the other hand, also the emergence of 3D localized hybrid magnetization textures consisting of a smooth skyrmion (tube) field terminated by one, as in the case of the chiral bobber [36,37], or two monopoles or Bloch points, respectively, as in the case of the chiral magnetic globule [38] (also coined toron as in Ref. [39]) are theoretically conceivable and have lately been confirmed experimentally [40]. The hybrid nature of these localized particles consisting of a smooth and a singular magnetic texture leads to electric transport properties very different from the smooth skyrmion textures [41]. ...
Magnetic hopfions are string-like three-dimensional topological solitons, characterised by the Hopf invariant. They serve as a fundamental prototype for three-dimensional magnetic quasi-particles 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 $Q_\text{H}=1$ 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.
