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Topological magnetic solitons, such as skyrmions, exhibit intriguing particle-like properties that make them attractive for fundamental research and practical applications. While many magnetic systems can host skyrmions as statically stable configurations, chiral magnets stand out for their ability to accommodate a wide diversity of skyrmions with arbitrary topological charges and varied morphologies. Despite extensive investigation, a complete understanding of chiral magnetic skyrmions has remained elusive. We present a classification of all chiral skyrmions, demonstrating three classes based on their response to external magnetic field pulses: stationary, translating, and rotating. We highlight the role of magnetic texture symmetry in this classification. Skyrmions with varied dynamics offer avenues for exploring phenomena like skyrmion-skyrmion scattering that might be crucial for future applications., Comment: 12 pages, 6 figures

Magnetic skyrmions are topologically nontrivial spin configurations that possess particle-like properties. Earlier research was mainly focused on a specific type of skyrmion with topological charge Q = -1. However, theoretical analyses of two-dimensional chiral magnets have predicted the existence of skyrmion bags -- solitons with arbitrary positive or negative topological charge. Although such spin textures are metastable states, recent experimental observations have confirmed the stability of isolated skyrmion bags in a limited range of applied magnetic fields. Here, by utilizing Lorentz transmission electron microscopy, we show the extraordinary stability of skyrmion bags in thin plates of B20-type FeGe. In particular, we show that skyrmion bags embedded within a skyrmion lattice remain stable even in zero or inverted external magnetic fields. A robust protocol for nucleating such embedded skyrmion bags is provided. Our results agree perfectly with micromagnetic simulations and establish thin plates of cubic chiral magnets as a powerful platform for exploring a broad spectrum of topological magnetic solitons., Comment: 17 pages, 15 figures

We show that competition between local interactions in monoaxial chiral magnets provides the stability of two-dimensional (2D) 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 skyrmion and antiskyrmion and in their stability., Comment: 5 pages, 3 figures

Magnetic nano-objects possess great potential for more efficient data processing, storage and neuromorphic type of applications. Using high perpendicular magnetic anisotropy synthetic antiferromagnets in the form of multilayer-based metamaterials we purposely reduce the antiferromagnetic (AF) interlayer exchange energy below the out-of-plane demagnetization energy, which controls the magnetic domain formation. As we show via macroscopic magnetometry as well as microscopic Lorentz transmission electron microscopy, in this unusual magnetic energy regime, it becomes possible to stabilize nanometer scale stripe and bubble textures consisting of ferromagnetic (FM) out-of-plane domain cores separated by AF in-plane Bloch-type domain walls. This unique coexistence of mixed FM/AF order on the nanometer scale opens so far unexplored perspectives in the architecture of magnetic domain landscapes as well as the design and functionality of individual magnetic textures, such as bubble domains with alternating chirality., Comment: 13 pages, 4 main figures and 4 Extended Data (supplementary) figures, 50 references

Cubic chiral magnets exhibit a remarkable diversity of two-dimensional topological magnetic textures, including skyrmions. However, the experimental confirmation of topological states localized in all three spatial dimensions remains challenging. In this paper, we investigate a three-dimensional topological state called a heliknoton, which is a hopfion embedded into a helix or conic background. We explore the range of parameters at which the heliknoton can be stabilized under realistic conditions using micromagnetic modeling, harmonic transition state theory, and stochastic spin dynamics simulations. We present theoretical Lorentz TEM images of the heliknoton, which can be used for experimental comparison. Additionally, we discuss the stability of the heliknoton at finite temperatures and the mechanism of its collapse. Our study offers a pathway for future experimental investigations of three-dimensional topological solitons in magnetic crystals., Comment: 16 pages, 5 figures

We report tailed skyrmions -- a new class of stable soliton solutions of the 2D chiral magnet model. Tailed skyrmions have elongated shapes and emerge in a narrow range of fields near the transition between the spin spirals and the saturated state. We analyze the stability range of these solutions in terms of external magnetic field and magnetocrystalline anisotropy. Minimum energy paths and the homotopies (continuous transitions) between tailed skyrmions of the same topological charge have been calculated using the geodesic nudged elastic bands method. The discovery of tailed skyrmions extends the diversity of already-known solutions illustrated by complex morphology solitons, such as tailed skyrmion bags with and without chiral kinks., Comment: 7 pages, 6 figures

This work provides a concept for three-dimensional magnetic solitons based on mapping the homotopy path between various two-dimensional solutions onto the third spatial axis. The representative examples of statically stable configurations of that type in the model of an isotropic chiral magnet are provided. Various static and dynamic properties of such three-dimensional magnetic solitons are discussed in detail., Comment: 9 pages, 6 figures

The diversity of three-dimensional magnetic states in a FeGe nanocylinder is studied using micromagnetic simulations and off-axis electron holography in the transmission electron microscope. In particular, we report the observation of a dipole string -- a spin texture composed of two coupled Bloch points -- which becomes stable under geometrical confinement. Quantitative agreement is obtained between experimental and theoretical phase shift images by taking into account the presence of a damaged layer on the surface of the nanocylinder. The theoretical model is based on the assumption that the damaged surface layer, which results from focused ion beam milling during sample preparation, has similar magnetic properties to those of an amorphous FeGe alloy. The results highlight the importance of considering the magnetic properties of the surface layers of such nanoscale samples, which influence their magnetic states., Comment: 6 pages, 3 figures

This work studies the resonance excitations of the three-dimensional skyrmions lattice in the finite thickness plate of an isotropic chiral magnet using spin dynamics simulations. We found that the absorption spectra and resonance modes differ from those predicted by the two-dimensional model and the model of the unconfined bulk crystal. The features observed on the spectra can be explained by the formation of chiral standing spin waves, which, contrary to conventional standing spin waves, are characterized by the helical profile of dynamic magnetization of fixed chirality defined by the Dzyaloshinskii-Moriya interaction. In this case, the dynamic susceptibility becomes a function of the plate thickness, which gives rise to an interesting effect that manifests itself in periodical fading of the intensity of corresponding modes and makes excitation of these modes impossible at specific thicknesses., Comment: 10 pages, 5 figures

The reversal of magnetic bubble helicity through topologically trivial transient states provides an additional degree of freedom that promises the development of multidimensional magnetic memories. A key requirement for this concept is the stabilization of bubble states at ambient conditions on application-compatible substrates. In the present work we demonstrate a stabilization routine for remanent bubble states in high perpendicular magnetic anisotropy [(Co(0.44\,nm)/Pt(0.7\,nm)]$_X$, X = 48, 100, 150 multilayers on Si/SiO$_2$ substrates by exploring the effect of external magnetic fields ($H_\mathrm{m}$) of different strength and angles ($\theta$) with respect to the film surface normal. By systematic variation of these two parameters, we demonstrate that remanent bubble density and mean bubble diameter can be carefully tuned and optimized for each sample. Our protocol based on magnetometry only reveals the densest remanent bubble states at $H_\mathrm{m} = 0.87 H_\mathrm{s}$ ($H_\mathrm{s}$ is the magnetic saturation field) and $\theta=60^\circ - 75^\circ$ for all $X$ with a maximum of 3736 domains/100 $\mu$m$^2$ for the X = 48 sample. The experimental observations are supported by micromagnetic simulations taking into account the nanoscale lateral grain structure of multilayers synthesized by magnetron sputter deposition, and thus helping understand the different density of the bubble states found in these systems., Comment: Regular Article: 10 pages, 6 figures, 1 table

A fundamental property of particles and antiparticles, such as electrons and positrons, is their ability to annihilate one another. Similar behavior is predicted for magnetic solitons -- localized spin textures that can be distinguished by their topological index $Q$. Theoretically, magnetic topological solitons with opposite values of $Q$, such as skyrmions and their antiparticles -- antiskyrmions -- are expected to be able to merge continuously and to annihilate. However, experimental verification of such particle-antiparticle pair production and annihilation processes has been lacking. Here, we report the creation and annihilation of skyrmion-antiskyrmion pairs in an exceptionally thin film of the cubic chiral magnet B20-type FeGe observed using transmission electron microscopy. Our observations are highly reproducible and are fully consistent with micromagnetic simulations. Our findings provide a new platform for fundamental studies of particles and antiparticles based on magnetic solids and open new perspectives for practical applications of thin films of isotropic chiral magnets., Comment: 16 pages, 14 figures

Controlled creation of localized magnetic textures beyond conventional $\pi$-skyrmions is an important problem in the field of magnetism. Here by means of spin dynamics simulations, Monte Carlo simulations and harmonic transition state theory we demonstrate that an elementary chiral magnetic soliton with zero topological charge -- the chiral droplet -- can be reliably created by thermal fluctuations in the presence of the tilted magnetic field. The proposed protocol relies on an unusual kinetics combining the effects of the entropic stabilization and low energy barrier for the nucleation of a topologically-trivial state. Following this protocol by varying temperature and the tilt of the external magnetic field one can selectively generate chiral droplets or $\pi$-skyrmions in a single system. The coexistence of two distinct magnetic solitons establishes a basis for a rich magnetization dynamics and opens up the possibility for the construction of more complex magnetic textures such as skyrmion bags and skyrmions with chiral kinks., Comment: 5 pages, 4 figures

We report on the controlled switching of domain wall (DW) magnetization in aligned stripe domain structures, stabilized in [Co (0.44 nm)/Pt (0.7 nm)]$_X$ ($X = 48$, 100, 150) multilayers with perpendicular magnetic anisotropy. The switching process, induced by an external magnetic field, is monitored by measuring the evolution of the in-plane magnetization. % We show that the remanent in-plane magnetization originates from the polarization of the Bloch-type DWs. With micromagnetic simulations, we reveal that the reversal of the DW polarization is the result of the emergence and collapse of horizontal Bloch lines within the DWs at particular strengths of the external magnetic field, applied opposite to the DW polarization. Our findings are relevant for DW-based magnonics and bubble skyrmion applications in magnetic multilayers., Comment: 11 pages, 6 figures

The uniform motion of chiral magnetic skyrmions induced by a spin-transfer torque displays an intricate dependence on the skyrmions' topological charge and shape. We reveal surprising patterns in this dependence through simulations of the Landau-Lifshitz-Gilbert equation with Zhang-Li torque and explain them through a geometric analysis of Thiele's equation. In particular, we show that the velocity distribution of topologically non-trivial skyrmions depends on their symmetry: it is a single circle for skyrmions of high symmetry and a family of circles for low-symmetry configurations. We also show that the velocity of the topologically trivial skyrmions, previously believed to be the fastest objects, can be surpassed, for instance, by antiskyrmions. The generality of our approach suggests the validity of our results for exchange frustrated magnets, bubble materials, and others., Comment: 11 pages, 5 figures

Filamentary textures can take the form of braided, rope-like superstructures in nonlinear media such as plasmas and superfluids. The formation of similar superstructures in solids has been predicted, for example from flux lines in superconductors. However, their observation has proved challenging. Here, we use electron microscopy and numerical methods to reveal braided superstructures of magnetic skyrmions in thin crystals of B20-type FeGe. Their discovery opens the door to applications of rich topological landscapes of geometric braids in magnetic solids., Comment: 6 pages, 3 figures

We present a novel approach to understanding the extraordinary diversity of magnetic skyrmion solutions. Our approach combines a new classification scheme with efficient analytical and numerical methods. We introduce the concept of chiral kinks to account for regions of disfavoured chirality in spin textures, and classify two-dimensional magnetic skyrmions in terms of closed domain walls carrying such chiral kinks. In particular, we show that the topological charge of magnetic skyrmions can be expressed in terms of the constituent closed domain walls and chiral kinks. Guided by our classification scheme, we propose a method for creating hitherto unknown magnetic skyrmions which involves initial spin configurations formulated in terms of holomorphic functions and subsequent numerical energy minimization. We numerically study the stability of the resulting magnetic skyrmions for a range of external fields and anisotropy parameters, and provide quantitative estimates of the stability range for the whole variety of skyrmions with kinks. We show that the parameters limiting this range can be well described in terms of the relative energies of particular skyrmion solutions and isolated stripes with and without chiral kinks., Comment: 17 pages, 12 figures

The stability of two-dimensional chiral skyrmions in a tilted magnetic field is studied. It is shown that by changing the direction of the field and its magnitude, one can continuously transform chiral skyrmion into a skyrmion with opposite polarity and vorticity. This turned inside out skyrmion can be considered as an antiparticle for ordinary axisymmetric skyrmion. For any tilt angle of the magnetic field, there is a range of its absolute values where two types of skyrmions may coexist. In a tilted field, the potentials for inter-skyrmion interactions are characterized by the presence of local minima suggesting attractive interaction between the particles. The potentials of inter-particle interactions also have so-called fusion channels allowing either annihilation of two particles or the emergence of a new particle. The presented results are general for a wide class of magnetic crystals with both easy-plane and easy-axis anisotropy.

Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search of hopfions in real physical systems is going on for nearly half a century, starting with the seminal work of Faddeev. But so far realizations in solids are missing. Here, we present a theory that identifies magnetic materials featuring hopfions as stable states without the assistance of confinement or external fields. Our results are based on an advanced micromagnetic energy functional derived from a spin-lattice Hamiltonian. Hopfions appear as emergent particles of the classical Heisenberg model. Magnetic hopfions represent three-dimensional particle-like objects of nanometre-size dimensions opening the gate to a new generation of spintronic devices in the framework of a truly three-dimensional architecture. Our approach goes beyond the conventional phenomenological models. We derive material-realistic parameters that serve as concrete guidance in the search of magnetic hopfions bridging computational physics with materials science.

Antiferromagnetic (AF) domain walls have recently attracted revived attention, not only in the emerging field of AF spintronics, but also more specifically for offering fast domain wall velocities and dynamic excitations up to the terahertz frequency regime. Here we introduce an approach to nucleate and stabilize an AF domain wall in a synthetic antiferromagnet (SAF). We present experimental and micromagnetic studies of the magnetization reversal in [(Co/Pt)$_{X-1}$/Co/Ir]$_{N-1}$(Co/Pt)$_X$ SAFs, where interface induced perpendicular magnetic anisotropy (PMA) and AF interlayer exchange coupling (IEC) are completely controlled via the individual layer thicknesses within the multilayer stack. By combining strong PMA with even stronger AF IEC, the SAF reveals a collective response to an external magnetic field applied normal to the surface, and we stabilize the characteristic surface spin flop (SSF) state for an even number N of AF-coupled (Co/Pt)$_{X-1}$/Co multilayer blocks. In the SSF state our system provides a well-controlled and fully tunable vertical AF domain wall, easy to integrate as no single crystal substrates are required and with uniform 2D-magnetization in the film plane for further functionalization options, such as for example lateral patterning via lithography., Comment: 15 pages, 3 figures

Magnetic singularities, also known as magnetic monopoles or Bloch points, represent intriguingphenomena in nanomagnetism. We show that a pair of coupled Bloch points may appear as alocalized, stable state in cubic chiral magnets. Detailed analysis is presented of the stability of suchobjects in the interior of crystals and in geometrically confined systems.

The \textit{Spirit} framework is designed for atomic scale spin simulations of magnetic systems of arbitrary geometry and magnetic structure, providing a graphical user interface with powerful visualizations and an easy to use scripting interface. An extended Heisenberg type spin-lattice Hamiltonian including competing exchange interactions between neighbors at arbitrary distance, higher-order exchange, Dzyaloshinskii-Moriya and dipole-dipole interactions is used to describe the energetics of a system of classical spins localised at atom positions. A variety of common simulations methods are implemented including Monte Carlo and various time evolution algorithms based on the Landau-Lifshitz-Gilbert equation of motion, which can be used to determine static ground state and metastable spin configurations, sample equilibrium and finite temperature thermodynamical properties of magnetic materials and nanostructures or calculate dynamical trajectories including spin torques induced by stochastic temperature or electric current. Methods for finding the mechanism and rate of thermally assisted transitions include the geodesic nudged elastic band method, which can be applied when both initial and final states are specified, and the minimum mode following method when only the initial state is given. The lifetime of magnetic states and rate of transitions can be evaluated within the harmonic approximation of transition-state theory. The framework offers performant CPU and GPU parallelizations. All methods are verified and applications to several systems, such as vortices, domain walls, skyrmions and bobbers are described.

While chiral magnetic skyrmions have been attracting significant attention in the past years, recently, a new type of a chiral particle emerging in thin films $-$ a chiral bobber $-$ has been theoretically predicted and experimentally observed. Here, based on theoretical arguments, we provide a clear pathway to utilizing chiral bobbers for the purposes of future spintronics by uncovering that these novel chiral states possess inherent transport fingerprints that allow for their unambiguous electrical detection in systems comprising several types of chiral states. We reveal that unique transport and orbital characteristics of bobbers root in the non-trivial magnetization distribution in the vicinity of the Bloch points, and demonstrate that tuning the details of the Bloch point topology can be used to drastically alter the emergent response properties of chiral bobbers to external fields, which bears great potential for engineering chiral dynamics and cognitive computing., Comment: Supplementary available upon request

Various transitions that a magnetic skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the skyrmion divides and forms two skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an important avenue for the creation of skyrmions in future magnetic devices.

We show that continuous and spin-lattice models of chiral ferro- and antiferromagnets provide the existence of an infinite number of stable soliton solutions of any integer topological charge. A detailed description of the morphology of new skyrmions and the corresponding energy dependencies are provided. The considered model is general, and is expected to predict a plethora of particle-like states which may occur in various chiral magnets including atomic layers, e.g., PdFe/Ir(111), rhombohedral GaV$_4$S$_8$ semiconductor, B20-type alloys as Mn$_{1-x}$Fe$_x$Ge, Mn$_{1-x}$Fe$_x$Si, Fe$_{1-x}$Co$_x$Si, Cu$_2$OSeO$_3$, acentric tetragonal Heusler compounds.

We report the direct evidence of field-dependent character of the interaction between individual magnetic skyrmions as well as between skyrmions and edges in B20-type FeGe nanostripes observed by means of high resolution Lorentz transmission electron microscopy. It is shown that above certain critical values of external magnetic field the character of such long-range skyrmion interactions change from attraction to repulsion. Experimentally measured equilibrium inter-skyrmion and skrymion-edge distances as function of applied magnetic field shows quantitative agreement with the results of micromagnetic simulations. Important role of demagnetizing fields and internal symmetry of three-dimensional magnetic skyrmions are discussed in details., Comment: accepted in PRL

The skyrmion racetrack is a promising concept for future information technology. There, binary bits are carried by nanoscale spin swirls -- skyrmions -- driven along magnetic strips. Stability of the skyrmions is a critical issue for the realization of this technology. Here we demonstrate that the racetrack skyrmion lifetime can be calculated from first principles as a function of temperature, magnetic field and track width. Our method combines harmonic transition state theory extended to include Goldstone modes, with an atomistic spin Hamiltonian parametrized from density functional theory calculations. We demonstrate that two annihilation mechanisms contribute to the skyrmion stability: At low external magnetic field, escape through the track boundary prevails, but a crossover field exists, above which the collapse in the interior becomes dominant. Considering a Pd/Fe bilayer on an Ir(111) substrate as a well-established model system, the calculated lifetime is found to be consistent with reported experimental measurements. Our results open the door for predictive simulations, free from empirical parameters, to aid the design of skyrmion-based information technology.

The use of chiral skyrmions, which are nanoscale vortex-like spin textures, as movable data bit carriers forms the basis of a recently proposed concept for magnetic solid-state memory. In this concept, skyrmions are considered to be unique localized spin textures, which are used to encode data through the quantization of different distances between identical skyrmions on a guiding nanostripe. However, the conservation of distances between highly mobile and interacting skyrmions is difficult to implement in practice. Here, we report the direct observation of another type of theoretically-predicted localized magnetic state, which is referred to as a chiral bobber (ChB), using quantitative off-axis electron holography. We show that ChBs can coexist together with skyrmions. Our results suggest a novel approach for data encoding, whereby a stream of binary data representing a sequence of ones and zeros can be encoded via a sequence of skyrmions and bobbers. The need to maintain defined distances between data bit carriers is then not required. The proposed concept of data encoding promises to expedite the realization of a new generation of magnetic solid-state memory.