Hopfion rings in a cubic chiral magnet.

Magnetic skyrmions and hopfions are topological solitons1-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. Skyrmions2,3 are two-dimensional solitons resembling vortex-like string structures that can penetrate an entire sample. Hopfions4-9 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 challenging10 and has only been reported in a synthetic material11. 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.

Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method.

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.

Interaction of Individual Skyrmions in a Nanostructured Cubic Chiral Magnet.

We report direct evidence of the 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 an external magnetic field the character of such long-range skyrmion interactions changes from attraction to repulsion. Experimentally measured equilibrium inter-skyrmion and skyrmion-edge distances as a function of the applied magnetic field shows quantitative agreement with the results of micromagnetic simulations. The important role of demagnetizing fields and the internal symmetry of three-dimensional magnetic skyrmions are discussed in detail.

Interlayer Exchange Coupling: A General Scheme Turning Chiral Magnets into Magnetic Multilayers Carrying Atomic-Scale Skyrmions.

We report on a general principle using interlayer exchange coupling to extend the regime of chiral magnetic films in which stable or metastable magnetic Skyrmions can appear at a zero magnetic field. We verify this concept on the basis of a first-principles model for a Mn monolayer on a W(001) substrate, a prototype chiral magnet for which the atomic-scale magnetic texture is determined by the frustration of exchange interactions, impossible to unwind by laboratory magnetic fields. By means of ab initio calculations for the Mn/W_{m}/Co_{n}/Pt/W(001) multilayer system we show that for certain thicknesses m of the W spacer and n of the Co reference layer, the effective field of the reference layer fully substitutes the required magnetic field for Skyrmion formation.

New type of stable particlelike states in chiral magnets.

We present a new type of thermodynamically stable magnetic state at interfaces and surfaces of chiral magnets. The state is a soliton solution of micromagnetic equations localized in all three dimensions near a boundary, and it contains a singularity but nevertheless has finite energy. Both features combine to form a quasiparticle state for which we expect unusual transport and dynamical properties. It exhibits high thermal stability and thereby can be considered as a promising object for fundamental research and practical applications in spintronic devices. We identified the range of existence of such particlelike states in the thickness dependent magnetic phase diagram for helimagnet films and analyzed its stability in comparison with the isolated skyrmion within the conical phase. We provide arguments that such a state can be found in different B20-type alloys, e.g., Mn_{1-x}Fe_{x}Ge, Mn_{1-x}Fe_{x}Si, Fe_{1-x}Co_{x}Si.

Geometric phase analysis of magnetic skyrmion lattices in Lorentz transmission electron microscopy images.

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.

Dipolar skyrmions and antiskyrmions of arbitrary topological charge at room temperature.

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.

Magnetic skyrmion braids.

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.

Lifetime of racetrack skyrmions.

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.

Experimental observation of chiral magnetic bobbers in B20-type FeGe.

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 types 7 . Recently, a new type of localized particle-like object-the chiral bobber (ChB)-was predicted theoretically in such materials 8 . 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.

Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions.

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 2v 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.

Control of morphology and formation of highly geometrically confined magnetic skyrmions.

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.

Switching of chiral magnetic skyrmions by picosecond magnetic field pulses via transient topological states.

Magnetic chiral skyrmions are vortex like spin structures that appear as stable or meta-stable states in magnetic materials due to the interplay between the symmetric and antisymmetric exchange interactions, applied magnetic field and/or uniaxial anisotropy. Their small size and internal stability make them prospective objects for data storage but for this, the controlled switching between skyrmion states of opposite polarity and topological charge is essential. Here we present a study of magnetic skyrmion switching by an applied magnetic field pulse based on a discrete model of classical spins and atomistic spin dynamics. We found a finite range of coupling parameters corresponding to the coexistence of two degenerate isolated skyrmions characterized by mutually inverted spin structures with opposite polarity and topological charge. We demonstrate how for a wide range of material parameters a short inclined magnetic field pulse can initiate the reliable switching between these states at GHz rates. Detailed analysis of the switching mechanism revealed the complex path of the system accompanied with the excitation of a chiral-achiral meron pair and the formation of an achiral skyrmion.