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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.

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.

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

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.

We present the phase diagram of magnetic states for films of isotropic chiral magnets 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 chiral magnets. Presented results provide a comprehensive theoretical description for those effects already observed in experiments on thin films of chiral magnets, predict new effects important for applications and open perspectives for experimental studies of such systems.

We present a new type of a 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 contains a singularity, but nevertheless has a finite energy. Both features combine to a quasi-particle state for which we expect unusual transport and dynamical properties. It exhibits high thermal stability and thereby can be considered as promising object for fundamental research and practical applications in spintronic devices. 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., Comment: accepted in PRL

Axisymmetric magnetic lines of nanometer sizes (chiral vortices or skyrmions) have been predicted to exist in a large group of noncentrosymmetric crystals more than two decades ago. Recently these magnetic textures have been directly observed in nanolayers of cubic helimagnets and monolayers of magnetic metals. We develop a micromagnetic theory of chiral skyrmions in thin magnetic layers for magnetic materials with intrinsic and induced chirality. Such particle-like and stable micromagnetic objects can exist in broad ranges of applied magnetic fields including zero field. Chiral skyrmions can be used as a new type of highly mobile nanoscale data carriers.

In antiferromagnetically coupled multilayers with perpendicular anisotropy unusual multidomain textures can be stabilized due to a close competition between long-range demagnetization fields and short-range interlayer exchange coupling. In particular, the formation and evolution of specific topologically stable planar defects within the antiferromagnetic ground state, i.e. wall-like structures with a ferromagnetic configuration extended over a finite width, explain configurational hysteresis phenomena recently observed in [Co/Pt(Pd)]/Ru and [Co/Pt]/NiO multilayers. Within a phenomenological theory, we have analytically derived the equilibrium sizes of these "ferroband" defects as functions of the antiferromagnetic exchange, a bias magnetic field, and geometrical parameters of the multilayers. In the magnetic phase diagram, the existence region of the ferrobands mediates between the regions of patterns with sharp antiferromagnetic domain walls and regular arrays of ferromagnetic stripes. The theoretical results are supported by magnetic force microscopy images of the remanent states observed in [Co/Pt]/Ru., Comment: Paper submitted by the Joint European Magnetics Symposia 2008, Dublin (4 pages, 3 figures)

A rich variety of specific multidomain textures recently observed in antiferromagnetically coupled multilayers with perpendicular anisotropy include regular (equilibrium) multidomain states as well as different types of topological magnetic defects. Within a phenomenological theory we have classified and analyzed the possible magnetic defects in the antiferromagnetic ground state and determine their structures. We have derived the optimal sizes of the defects as functions of the antiferromagnetic exchange, the applied magnetic field, and geometrical parameters of the multilayer. The calculated magnetic phase diagrams show the existence regions for all types of magnetic defects. Experimental investigations of the remanent states (observed after different magnetic pre-history) in [Co/Pt]/Ru multilayers with wedged Co layers reveal a corresponding succession of different magnetic defect domain types., Comment: 3 pages, 4 figures

The evolution of multivariant patterns in thin plates of magnetic shape memory materials with an applied magnetic field was studied theoretically. A geometrical domain-model is considered composed of straight stripe-like martensite variants with constant internal magnetization (high anisotropy limit) and magnetic domain wall orientation fixed by the twin boundaries. Through integral transforms of the demagnetization energy, the micromagnetic energy is cast into a form convenient for direct numerical evaluation and analytical calculations. The equilibrium geometrical parameters of multivariant patterns with straight and oblique twin boundaries have been derived as functions of the applied field and the material parameters of a plate. It is shown that the oblique multivariant states exist only in plates with thicknesses L larger than a certain critical value L_0. In samples with L < L_0 a magnetic-field-driven transformation occurs directly between single variant states., Comment: Materials for E-MRS Fall Meeting, Warsaw 2007. Corrected figures

Recently synthesized magnetic multilayers with strong perpendicular anisotropy exhibit unique magnetic properties including the formation of specific multidomain states. In particular, antiferromagnetically coupled multilayers own rich phase diagrams that include various multidomain ground states. Analytical equations have been derived for the stray-field components of these multidomain states in perpendicular multilayer systems. In particular, closed expressions for stray fields in the case of ferromagnetic and antiferromagnetic stripes are presented. The theoretical approach provides a basis for the analysis of magnetic force microscopy (MFM) images from this novel class of nanomagnetic systems. Peculiarities of the MFM contrast have been calculated for realistic tip models. These characteristic features in the MFM signals can be employed for the investigations of the different multidomain modes. The obtained results are applied for the analysis of multidomain modes that have been reported earlier in the literature from experiments on [Co/Cr]Ru superlattices., Comment: 7 pages, 6 figures

Antiferromagnetically coupled multilayers with perpendicular anisotropy, as [CoPt]/Ru, Co/Ir, Fe/Au, display ferromagnetic stripe phases as the ground states. It is theoretically shown that the antiferromagnetic interlayer exchange causes a relative shift of domains in adjacent layers. This ``exchange shift'' is responsible for several recently observed effects: an anomalous broadening of domain walls, the formation of so-called ``tiger-tail'' patterns, and a ``mixed state'' of antiferromagnetic and ferromagnetic domains in [CoPt]/Ru multilayers. The derived analitical relations between the values of the shift and the strength of antiferromagnetic coupling provide an effective method for a quantitative determination of the interlayer exchange interactions., Comment: 4 pages, 3 figures