Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortex, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of crystalline order composed of topologically stable spin states.

Skyrmion Lattice in a Chiral Magnet

Magnetic skyrmions are particle-like nanometre-sized spin textures of topological origin found in several magnetic materials, and are characterized by a long lifetime. Skyrmions have been observed both by means of neutron scattering in momentum space and microscopy techniques in real space, and their properties include novel Hall effects, current-driven motion with ultralow current density and multiferroic behaviour. These properties can be understood from a unified viewpoint, namely the emergent electromagnetism associated with the non-coplanar spin structure of skyrmions. From this description, potential applications of skyrmions as information carriers in magnetic information storage and processing devices are envisaged.

Topological properties and dynamics of magnetic skyrmions

Crystal order is not restricted to the periodic atomic array, but can also be found in electronic systems such as the Wigner crystal or in the form of orbital order, stripe order and magnetic order. In the case of magnetic order, spins align parallel to each other in ferromagnets and antiparallel in antiferromagnets. In other, less conventional, cases, spins can sometimes form highly nontrivial structures called spin textures. Among them is the unusual, topologically stable skyrmion spin texture, in which the spins point in all the directions wrapping a sphere. The skyrmion configuration in a magnetic solid is anticipated to produce unconventional spin-electronic phenomena such as the topological Hall effect. The crystallization of skyrmions as driven by thermal fluctuations has recently been confirmed in a narrow region of the temperature/magnetic field (T-B) phase diagram in neutron scattering studies of the three-dimensional helical magnets MnSi (ref. 17) and Fe(1-x)Co(x)Si (ref. 22). Here we report real-space imaging of a two-dimensional skyrmion lattice in a thin film of Fe(0.5)Co(0.5)Si using Lorentz transmission electron microscopy. With a magnetic field of 50-70 mT applied normal to the film, we observe skyrmions in the form of a hexagonal arrangement of swirling spin textures, with a lattice spacing of 90 nm. The related T-B phase diagram is found to be in good agreement with Monte Carlo simulations. In this two-dimensional case, the skyrmion crystal seems very stable and appears over a wide range of the phase diagram, including near zero temperature. Such a controlled nanometre-scale spin topology in a thin film may be useful in observing unconventional magneto-transport effects.

Real-space observation of a two-dimensional skyrmion crystal

Magnetic skyrmions are nanoscale spin configurations that hold promise as information carriers in ultradense memory and logic devices owing to the extremely low spin-polarized currents needed to move them.

Skyrmions on the track

We report on the design, verification and performance of mumax3, an open-source GPU-accelerated micromagnetic simulation program. This software solves the time- and space dependent magnetization evolution in nano- to micro scale magnets using a finite-difference discretization. Its high performance and low memory requirements allow for large-scale simulations to be performed in limited time and on inexpensive hardware. We verified each part of the software by comparing results to analytical values where available and to micromagnetic standard problems. mumax3 also offers specific extensions like MFM image generation, moving simulation window, edge charge removal and material grains.

The design and verification of Mumax3

Since the 1950s, Heisenberg and others have addressed the problem of how to explain the appearance of countable particles in continuous fields. Stable localized field configurations were searched for an ingredient for a general field theory of elementary particles, but the majority of nonlinear field models were unable to predict them. As an exception, Skyrme succeeded in describing nuclear particles as localized states, so-called 'skyrmions'. Skyrmions are a characteristic of nonlinear continuum models ranging from microscopic to cosmological scales. Skyrmionic states have been found under non-equilibrium conditions, or when stabilized by external fields or the proliferation of topological defects. Examples are Turing patterns in classical liquids, spin textures in quantum Hall magnets, or the blue phases in liquid crystals. However, it has generally been assumed that skyrmions cannot form spontaneous ground states, such as ferromagnetic or antiferromagnetic order, in magnetic materials. Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed-matter systems with chiral interactions without the assistance of external fields or the proliferation of defects. We show this within a phenomenological continuum model based on a few material-specific parameters that can be determined experimentally. Our model has a condition not considered before: we allow for softened amplitude variations of the magnetization, characteristic of, for instance, metallic magnets. Our model implies that spontaneous skyrmion lattice ground states may exist generally in a large number of materials, notably at surfaces and in thin films, as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.

Spontaneous skyrmion ground states in magnetic metals

Thermodynamically stable magnetic vortex states in magnetic crystals

Thermodynamically stable "vortices" in magnetically ordered crystals. The mixed state of magnets

A phenomenological theory of chiral symmetry breaking in magnetic nanostructures is developed considering induced, inhomogeneous chiral interactions (Dzyaloshinsky-Moriya--type). Application of the theory to films and multilayers with in-plane and out-of-plane magnetization predicts modulated and two-dimensional localized patterns (vortices). These new classes of magnetic patterns are intrinsically stable and localized on nanometer scale. Various experimental observations agree qualitatively with structures derived from this theory.

Chiral Symmetry Breaking in Magnetic Thin Films and Multilayers

Axisymmetric magnetic string-like objects of nanometre 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.

https://iopscience.iop.org/article/10.1088/0022-3727/44/39/392001/pdf

Alexei N. Bogdanov

Papers

Spontaneous skyrmion ground states in magnetic metals

Thermodynamically stable magnetic vortex states in magnetic crystals

Thermodynamically stable "vortices" in magnetically ordered crystals. The mixed state of magnets

Chiral Symmetry Breaking in Magnetic Thin Films and Multilayers

Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies?