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Magnetic skyrmions and hopfions are topological solitons 1 —well-localized field configurations that have gained considerable attention over the past decade owing to their unique particle-like properties, which make them promising objects for spintronic applications. Skyrmions 2 , 3 are two-dimensional solitons resembling vortex-like string structures that can penetrate an entire sample. Hopfions 4 – 9 are three-dimensional solitons confined within a magnetic sample volume and can be considered as closed twisted skyrmion strings that take the shape of a ring in the simplest case. Despite extensive research on magnetic skyrmions, the direct observation of magnetic hopfions is challenging 10 and has only been reported in a synthetic material 11 . Here we present direct observations of hopfions in crystals. In our experiment, we use transmission electron microscopy to observe hopfions forming coupled states with skyrmion strings in B20-type FeGe plates. We provide a protocol for nucleating such hopfion rings, which we verify using Lorentz imaging and electron holography. Our results are highly reproducible and in full agreement with micromagnetic simulations. We provide a unified skyrmion–hopfion homotopy classification and offer insight into the diversity of topological solitons in three-dimensional chiral magnets. Transmission electron microscopy is used to observe three-dimensional topological solitons known as hopfions that in a chiral magnet are found to form rings around skyrmion strings, and a nucleation protocol for these rings is provided.

Chiral magnets are an emerging class of topological matter harboring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as an entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets can not only host skyrmions but also antiskyrmions as least energy configurations over all non-trivial homotopy classes. We derive practical criteria for their occurrence and coexistence with skyrmions that can be fulfilled by (110)-oriented interfaces depending on the electronic structure. Relating the electronic structure to an atomistic spin-lattice model by means of density functional calculations and minimizing the energy on a mesoscopic scale by applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultra-thin magnetic films grown on semiconductor or heavy metal substrates with C 2 v symmetry are prototype classes of materials hosting magnetic antiskyrmions. Skyrmions, localized defects in the magnetization, can be stabilised in materials by the Dzyaloshinskii-Moriya interaction (DMI). Hoffmann et al. predict that, when the DMI is anisotropic, antiskyrmions can be formed and coexist with skyrmions, enabling studies and exploitation of their interactions.

Skyrmions are vortex-like spin textures that form strings in magnetic crystals. Due to the analogy to elastic strings, skyrmion strings are naturally expected to braid and form complex three-dimensional patterns, but this phenomenon has not been explored yet. We found that skyrmion strings can form braids in cubic crystals of chiral magnets. This finding is confirmed by direct observations of skyrmion braids in B20-type FeGe using transmission electron microscopy. The theoretical analysis predicts that the discovered phenomenon is general for a wide family of chiral magnets. These findings have important implications for skyrmionics and propose a solid-state framework for applications of the mathematical theory of braids. Skyrmions are topological two-dimensional spin textures that in three-dimensional systems resemble strings or tubes. Here, using transmission electron microscopy Zheng et al observe the braiding of skyrmion strings in FeGe and predict this phenomenon for a large family of magnets.

Chiral magnetic skyrmions1,2 are nanoscale vortex-like spin textures that form in the presence of an applied magnetic field in ferromagnets that support the Dzyaloshinskii–Moriya interaction (DMI) because of strong spin–orbit coupling and broken inversion symmetry of the crystal3,4. In sharp contrast to other systems5,6 that allow for the formation of a variety of two-dimensional (2D) skyrmions, in chiral magnets the presence of the DMI commonly prevents the stability and coexistence of topological excitations of different types7. Recently, a new type of localized particle-like object—the chiral bobber (ChB)—was predicted theoretically in such materials8. However, its existence has not yet been verified experimentally. Here, we report the direct observation of ChBs in thin films of B20-type FeGe by means of quantitative off-axis electron holography (EH). We identify the part of the temperature–magnetic field phase diagram in which ChBs exist and distinguish two mechanisms for their nucleation. Furthermore, we show that ChBs are able to coexist with skyrmions over a wide range of parameters, which suggests their possible practical applications in novel magnetic solid-state memory devices, in which a stream of binary data bits can be encoded by a sequence of skyrmions and bobbers.

A fundamental property of particles and antiparticles (such as electrons and positrons, respectively) is their ability to annihilate one another. A similar behaviour is predicted for magnetic solitons 1 —localized spin textures that can be distinguished by their topological index Q . Theoretically, magnetic topological solitons with opposite values of Q , such as skyrmions 2 and their antiparticles (namely, antiskyrmions), are expected to be able to continuously merge and annihilate 3 . 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 of 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 the fundamental studies of particles and antiparticles based on magnetic solids and open new perspectives for practical applications of thin films of isotropic chiral magnets. Magnetic skyrmions—a type of localized spin texture—have been theoretically predicted to annihilate with counterparts known as antiskyrmions. By means of electron microscopy, such annihilation has now been observed in a cubic chiral magnet.

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 realising 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 skyrmion lifetime is found to be consistent with reported experimental measurements. Our simulations also show that the Arrhenius pre-exponential factor of escape depends only weakly on the external magnetic field, whereas the pre-exponential factor for collapse is strongly field dependent. Our results open the door for predictive simulations, free from empirical parameters, to aid the design of skyrmion-based information technology.

Magnetic skyrmions are localized, stable topological magnetic textures that can move and interact with each other like ordinary particles when an external stimulus is applied. The efficient control of the motion of spin textures using spin-polarized currents opened an opportunity for skyrmionic devices such as racetrack memory and neuromorphic or reservoir computing. The coexistence of skyrmions with high topological charge in the same system promises further possibilities for efficient technological applications. In this work, we directly observe dipolar skyrmions and antiskyrmions with arbitrary topological charge in Co/Ni multilayers at room temperature. We explore the dipolar-stabilized spin objects with topological charges of up to 10 and characterize their nucleation process, their energy dependence on the topological charge and the effect of the material parameters on their stability. Furthermore, our micromagnetic simulations demonstrate spin-transfer-induced motion of these spin objects, which is important for their potential device application. Control over magnetic skyrmions at room temperature has important applications in technology. Now the observation of skyrmions with high topological charge widens the potential for them to be used in unconventional computing techniques.

We present the phase diagram of magnetic states for films of isotropic chiral magnets (ChMs) calculated as function of applied magnetic field and thickness of the film. We have found a novel magnetic state driven by the natural confinement of the crystal, localized at the surface and stacked on top of the conical bulk phase. This magnetic surface state has a three-dimensional (3D) chiral spin-texture described by the superposition of helical and cycloidal spin spirals. This surface state exists for a large range of applied magnetic fields and for any film thickness beyond a critical one. We also identified the whole thickness and field range for which the skyrmion lattice becomes the ground state of the system. Below a certain critical thickness the surface state and bulk conical phase are suppressed in favor of the skyrmion lattice. Unraveling of those phases and the construction of the phase diagram became possible using advanced computational techniques for direct energy minimization applied to a basic 3D model for ChMs. Presented results provide a comprehensive theoretical description for those effects already observed in experiments on thin films of ChMs, predict new effects important for applications and open perspectives for experimental studies of such systems.

The ability to controllably manipulate magnetic skyrmions, small magnetic whirls with particle-like properties, in nanostructured elements is a prerequisite for incorporating them into spintronic devices. Here, we use state-of-the-art electron holographic imaging to directly visualize the morphology and nucleation of magnetic skyrmions in a wedge-shaped FeGe nanostripe that has a width in the range of 45–150 nm. We find that geometrically-confined skyrmions are able to adopt a wide range of sizes and ellipticities in a nanostripe that are absent in both thin films and bulk materials and can be created from a helical magnetic state with a distorted edge twist in a simple and efficient manner. We perform a theoretical analysis based on a three-dimensional general model of isotropic chiral magnets to confirm our experimental results. The flexibility and ease of formation of geometrically confined magnetic skyrmions may help to optimize the design of skyrmion-based memory devices. Proposals for skyrmion-based high-density memory devices require an understanding of the formation and shape of skyrmions in confined geometries. Here, the authors use electron holography to image magnetic textures in FeGe nanostripes and explore the parameters governing skyrmion morphology.

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.