Long-wavelength helimagnetic order and skyrmion lattice phase in Cu2OSeO3

Strain-mediated multistate skyrmion for neuron devices

Magnetic skyrmions are potential candidates for neuromorphic computing because of their inherent topological stability, low drive current density and nanoscale size. However, an artificial neuron device based on current-driven skyrmion motion cannot satisfy the requirement of energy efficiency and integration density due to hundreds of millions of interconnected neurons and synapses present in the deep networks. Here, we present a compact and energy efficient skyrmion-based artificial neuron consisting of ferromagnetic/heavy metal/ferroelectric layers which uses strain-mediated voltage manipulation of skyrmion states to mimic the Integrate-and-Fire (IF) function of biological neurons. By implementation of a spiking neural network (SNN) based on the proposed skyrmionic neuronal devices, it can achieve a high accuracy of 95.08% on a modified National Institute of Standards and Technology (MNIST) handwritten digit dataset, as well as a low power consumption of ∼46.8 fJ per epoch per neuron. The present work suggests a novel way to realize energy-efficient and high-density neuromorphic computing.

Spontaneous Skyrmion Bubbles in an Iron-Silicon Alloy with Uniaxial Magnetic Anisotropy

The magnetic skyrmions exhibit intriguing topological behaviors, holding promise for future applications in the realm of spintronic devices. Despite recent advancements, achieving spontaneous magnetic skyrmions and topological transitions in magnets featuring uniaxial magnetic anisotropy, particularly at elevated temperatures (>100 K), remains a challenging endeavor. Here, single-crystal Fe5Si3 nanorods with the central symmetry and uniaxial magnetic anisotropy were successfully synthesized on a mica substrate through chemical vapor deposition, which exhibit a high Curie temperature (TC) of about 372 K. The real-time observation, facilitated by Lorentz transmission electron microscopy, revealed the spontaneous formation of magnetic skyrmions and evolution of domains in focused ion beam-prepared Fe5Si3 thin foils. Moreover, Fe5Si3 device transport measurements expose notable magnetoresistance (MR) effects, enabling the interchange between positive and negative MR across specific temperature settings. These results offer various potential avenues for exploring diverse topological spin textures and their formation mechanisms, indicating inventive applications for iron-silicon alloy in the realm of spintronics.

Regulation of interfacial Dzyaloshinskii-Moriya interaction in ferromagnetic multilayers

Interfacial Dzyaloshinskii-Moriya interaction (i-DMI) exists in the film materials with inversion symmetry breaking, which can stabilize a series of nonlinear spin structures and control their chirality, such as Néel-type domain wall, magnetic skyrmion and spin spiral. In addition, the strength and chirality of i-DMI are directly related to the dynamic behavior of these nonlinear spin structures. Therefore, regulating the strength and chirality of i-DMI not only has an important scientific significance for enriching spintronics and topological physics, but also has a significant practical value for constructing a new generation of memorizer, logic gate, and brain-like devices with low-power. This review summarizes the research progress on the regulation of i-DMI in ferromagnetic films and provides some prospects for future research.

The influence of antiferromagnetic spin cantings on the magnetic helix pitch in cubic helimagnets

In cubic helimagnets MnSi and Cu2OSeO3with their nearly isotropic magnetic properties, the magnetic structure undergoes helical deformation, which is almost completely determined by the helicoid wavenumberk=D/J, where magnetization field stiffnessJis associated with isotropic spin exchange, andDis a pseudoscalar value characterizing the antisymmetric Dzyaloshinskii-Moriya (DM) interaction. Another magnetic feature of these crystals, also caused by the DM interactions, are antiferromagnetic spin cantings, similar to the ferromagnetic cantings responsible for the phenomenon of weak ferromagnetism. Here we show that cantings can strongly influence the helical order through the value of the parameterD. Changing the cantings in a strong magnetic field is predicted to affect the magnon spectrum of the crystals.

High-density, spontaneous magnetic biskyrmions induced by negative thermal expansion in ferrimagnets

Magnetic skyrmions are topologically protected quasiparticles that are promising for applications in spintronics. However, the low stability of most magnetic skyrmions leads to either a narrow temperature range in which they can exist, a low density of skyrmions, or the need for an external magnetic field, which greatly limits their wide application. In this study, high-density, spontaneous magnetic biskyrmions existing within a wide temperature range and without the need for a magnetic field were formed in ferrimagnets owing to the existence of a negative thermal expansion of the lattice. Moreover, a strong connection between the atomic-scale ferrimagnetic structure and nanoscale magnetic domains in Ho(Co,Fe)3 was revealed via in situ neutron powder diffraction and Lorentz transmission electron microscopy measurements. The critical role of the negative thermal expansion in generating biskyrmions in HoCo3 based on the magnetoelastic coupling effect is further demonstrated by comparing the behavior of HoCo2.8Fe0.2 with a positive thermal expansion.

CP2 skyrmions and skyrmion crystals in realistic quantum magnets

Magnetic skyrmions are nanoscale topological textures that have been recently observed in different families of quantum magnets. These objects are called CP¹ skyrmions because they are built from dipoles-the target manifold is the 1D complex projective space, CP¹ ≅ S². Here we report the emergence of magnetic CP² skyrmions in a realistic spin-1 model, which includes both dipole and quadrupole moments. Unlike CP¹ skyrmions, CP² skyrmions can also arise as metastable textures of quantum paramagnets, opening a new road to discover emergent topological solitons in non-magnetic materials. The quantum phase diagram of the spin-1 model also includes magnetic field-induced CP² skyrmion crystals that can be detected with regular momentum- (diffraction) and real-space (Lorentz transmission electron microscopy) experimental techniques.

Evolution of Emergent Monopoles into Magnetic Skyrmion Strings

Topological defects are fundamental concepts in physics, but little is known about the transition between distinct types across different dimensionalities. In topological magnetism, as in field theory, the transition between 1D strings and 0D monopoles is a key process whose observation has remained elusive. Here, we introduce a novel mechanism that allows for the controlled stabilization of emergent monopoles and show that magnetic skyrmion strings can be folded into monopoles. Conversely, they act as seeds out of which the entire string structure can unfold, containing its complete information. In chiral magnets, this process can be observed by resonant elastic X-ray scattering when the objects are in proximity to a polarized ferromagnet, whereby a pure monopole lattice is emerging on the surface. Our experimental proof of the reversible evolution from monopole to string sheds new light on topological defects and establishes the emergent monopole lattice as a new 3D topological phase.

Room-Temperature Magnetic Skyrmions and Large Topological Hall Effect in Chromium Telluride Engineered by Self-Intercalation

Room-temperature magnetic skyrmion materials exhibiting robust topological Hall effect (THE) are crucial for novel nano-spintronic devices. However, such skyrmion-hosting materials are rare in nature. In this study, a self-intercalated transition metal dichalcogenide Cr_{1+ x} Te₂ with a layered crystal structure that hosts room-temperature skyrmions and exhibits large THE is reported. By tuning the self-intercalate concentration, a monotonic control of Curie temperature from 169 to 333 K and a magnetic anisotropy transition from out-of-plane to the in-plane configuration are achieved. Based on the intercalation engineering, room-temperature skyrmions are successfully created in Cr_1.53 Te₂ with a Curie temperature of 295 K and a relatively weak perpendicular magnetic anisotropy. Remarkably, a skyrmion-induced topological Hall resistivity as large as ≈106 nΩ cm is observed at 290 K. Moreover, a sign reversal of THE is also found at low temperatures, which can be ascribed to other topological spin textures having an opposite topological charge to that of the skyrmions. Therefore, chromium telluride can be a new paradigm of the skyrmion material family with promising prospects for future device applications.

Origin of metamagnetism in skyrmion host Cu[Formula: see text]OSeO[Formula: see text]

Skyrmion host chiral Cu[Formula: see text]OSeO[Formula: see text] has attracted researchers due to several intriguing properties. Observation of metamagnetism in low-temperature and low-field makes the magnetic properties of Cu[Formula: see text]OSeO[Formula: see text] more complex. Here, we present an investigation on metamagnetism in Cu[Formula: see text]OSeO[Formula: see text] by analyzing its structural and magnetic properties. Study of magnetic properties reveal spin-flip of one of the Cu[Formula: see text] ions, embedded in square pyramidal CuO[Formula: see text] polyhedra, due to the development of strain in low-temperature and low-field regime. The spin-flip is found to be the main reason for field-induced first-order metamagnetic transition. Magnetic phase diagram of Cu[Formula: see text]OSeO[Formula: see text] has been constructed with the help of magnetization analyses. It is argued that the metamagnetic hysteretic field region may be low-temperature skyrmion phase with additional spiral and tilted-conical phases. A tricritical point has been observed in the phase diagram at which first-order metamagnetic hysteretic field range ceases to exist.

Square skyrmion crystal in centrosymmetric systems with locally inversion-asymmetric layers

We investigate an instability toward a square-lattice formation of magnetic skyrmions in centrosymmetric layered systems. By focusing on a bilayer square-lattice structure with the inversion center at the interlayer bond instead of the atomic site, we numerically examine the stability of the square skyrmion crystal (SkX) based on an effective spin model with the momentum-resolved interaction in the ground state through the simulated annealing. As a result, we find that a layer-dependent staggered Dzyaloshinskii-Moriya (DM) interaction built in the lattice structure becomes the origin of the square SkX in an external magnetic field irrespective of the sign of the interlayer exchange interaction. The obtained square SkX is constituted of the SkXs with different helicities in each layer due to the staggered DM interaction. Furthermore, we show that the interplay between the staggered DM interaction and the interlayer exchange interaction gives rise to a double-Qstate with a uniform component of the scalar chirality in the low-field region. The present results provide another way of stabilizing the square SkX in centrosymmetric magnets, which will be useful to explore further exotic topological spin textures.

Magnetoelectric Classification of Skyrmions

We develop a general theory to classify magnetic skyrmions and related spin textures in terms of their magnetoelectric multipoles. Since magnetic skyrmions are now established in insulating materials, where the magnetoelectric multipoles govern the linear magnetoelectric response, our classification provides a recipe for manipulating the magnetic properties of skyrmions using applied electric fields. We apply our formalism to skyrmions and antiskyrmions of different helicities, as well as to magnetic bimerons, which are topologically, but not geometrically, equivalent to skyrmions. We show that the nonzero components of the magnetoelectric multipole and magnetoelectric response tensors are uniquely determined by the topology, helicity, and geometry of the spin texture. Therefore, we propose straightforward linear magnetoelectric response measurements as an alternative to Lorentz microscopy for characterizing insulating skyrmionic textures.

Topology dependence of skyrmion Seebeck and skyrmion Nernst effect

We explore the dynamics of skyrmions with various topological charges induced by a temperature gradient in an ultra-thin insulating magnetic film. Combining atomistic spin simulations and analytical calculations we find a topology-dependent skyrmion Seebeck effect: while skyrmions and antiskyrmions move to the hot regime, a topologically trivial localized spin structure moves to the cold regime. We further reveal the emergence of a skyrmion Nernst effect, i.e. finite, topology-dependent velocities transverse to the direction of the temperature gradient. These findings are in agreement with accompanying simulations of skyrmionic motion induced by monochromatic magnon currents, allowing us to demonstrate that the magnonic spin Seebeck effect is responsible for both, skyrmion Seebeck and Nernst effect. Furthermore we employ scattering theory together with Thiele's equation to identify linear momentum transfer from the magnons to the skyrmion as the dominant contribution and to demonstrate that the direction of motion depends on the topological magnon Hall effect and the topological charge of the skyrmion.

Screw Dislocations in Chiral Magnets

Helimagnets realize an effective lamellar ordering that supports disclination and dislocation defects. Here, we investigate the micromagnetic structure of screw dislocation lines in cubic chiral magnets using analytical and numerical methods. The far field of these dislocations is universal and classified by an integer strength ν that quantifies its Burgers vector. We demonstrate that a rich variety of dislocation-core structures can be realized even for the same strength ν. In particular, the magnetization at the core can be either smooth or singular. We present a specific example with ν=1 for which the core is composed of a chain of singular Bloch points. In general, screw dislocations carry a noninteger but finite skyrmion charge so that they can be efficiently manipulated by spin currents and should contribute to the topological Hall effect.

Topological magnon band structure of emergent Landau levels in a skyrmion lattice

The motion of a spin excitation across topologically nontrivial magnetic order exhibits a deflection that is analogous to the effect of the Lorentz force on an electrically charged particle in an orbital magnetic field. We used polarized inelastic neutron scattering to investigate the propagation of magnons (i.e., bosonic collective spin excitations) in a lattice of skyrmion tubes in manganese silicide. For wave vectors perpendicular to the skyrmion tubes, the magnon spectra are consistent with the formation of finely spaced emergent Landau levels that are characteristic of the fictitious magnetic field used to account for the nontrivial topological winding of the skyrmion lattice. This provides evidence of a topological magnon band structure in reciprocal space, which is borne out of the nontrivial real-space topology of a magnetic order.

Different Critical Exponents on Two Sides of a Transition: Observation of Crossover from Ising to Heisenberg Exchange in Skyrmion Host Cu_{2}OSeO_{3}

We present experimental investigation on critical phenomena in Cu_{2}OSeO_{3} by analyzing the critical behavior of magnetization using a new method. This is necessary as a crossover from 3D Ising to 3D Heisenberg has been observed in Cu_{2}OSeO_{3}. The proposed method is applicable to explore the physics for a wide range of materials showing trivial or nontrivial critical behavior on two sides of the transition. A magnetic phase diagram has been constructed from the critical analysis. Multiple critical points due to multiple phases and transition between them have been observed in the phase diagram of Cu_{2}OSeO_{3}.

Tunable gigahertz dynamics of low-temperature skyrmion lattice in a chiral magnet

Recently, it has been shown that the chiral magnetic insulator Cu₂OSeO₃hosts skyrmions in two separated pockets in temperature and magnetic field phase space. It has also been shown that the predominant stabilization mechanism for the low-temperature skyrmion (LTS) phase is via the crystalline anisotropy, opposed to temperature fluctuations that stabilize the well-established high-temperature skyrmion (HTS) phase. Here, we report on a detailed study of LTS generation by field cycling, probed by GHz spin dynamics in Cu₂OSeO₃. LTSs are populated via a field cycling protocol with the static magnetic field applied parallel to the ⟨100⟩ crystalline direction of plate and cuboid-shaped bulk crystals. By analyzing temperature-dependent broadband spectroscopy data, clear evidence of LTS excitations with clockwise (CW), counterclockwise (CCW), and breathing mode (BR) character at temperatures belowT= 40 K are shown. We find that the mode intensities can be tuned with the number of field-cycles below the saturation field. By tracking the resonance frequencies, we are able to map out the field-cycle-generated LTS phase diagram, from which we conclude that the LTS phase is distinctly separated from the high-temperature counterpart. We also study the mode hybridization between the dark CW and the BR modes as a function of temperature. By using two Cu₂OSeO₃crystals with different shapes and therefore different demagnetization factors, together with numerical calculations, we unambiguously show that the magnetocrystalline anisotropy plays a central role for the mode hybridization.

Chiral Helimagnetism and One-Dimensional Magnetic Solitons in a Cr-Intercalated Transition Metal Dichalcogenide

Chiral magnets endowed with topological spin textures are expected to have promising applications in next-generation magnetic memories. In contrast to the well-studied 2D or 3D magnetic skyrmions, the authors report the discovery of 1D nontrivial magnetic solitons in a transition metal dichalcogenide 2H-TaS₂ via precise intercalation of Cr elements. In the synthetic Cr_1/3 TaS₂ (CTS) single crystal, the coupling of the strong spin-orbit interaction from TaS₂ and the chiral arrangement of the magnetic Cr ions evoke a robust Dzyaloshinskii-Moriya interaction. A magnetic helix having a short spatial period of ≈25 nm is observed in CTS via Lorentz transmission electron microscopy. In a magnetic field perpendicular to the helical axis, the helical spin structure transforms into a chiral soliton lattice (CSL) with the spin structure evolution being consistent with the chiral sine-Gordon theory, which opens promising perspectives for the application of CSL to fast-speed nonvolatile magnetic memories. This work introduces a new paradigm to soliton physics and provides an effective strategy for seeking novel 2D magnets.

Magnetic phase diagram of a spin-1/2 XXZ chain with modulated Dzyaloshinskii-Moriya interaction

We consider the ground-state phase diagram of a one-dimensional spin-1/2 XXZ chain with a spatially modulated Dzyaloshinskii-Moriya interaction in the presence of an alternating magnetic field applied along the z[over ̂] axis. The model is studied using the continuum-limit bosonization approach and the finite system exact numerical technique. In the absence of a magnetic field, the ground-state phase diagram of the model includes, besides the ferromagnetic and gapless Luttinger-liquid phases, two gapped phases: the composite (C1) phase characterized by the coexistence of long-range-ordered (LRO) alternating dimerization and spin chirality patterns, and the composite (C2) phase characterized by, in addition to the coexisting spin dimerization and alternating chirality patterns, the presence of LRO antiferromagnetic order. In the case of mentioned composite gapped phases, and in the case of a uniform magnetic field, the commensurate-incommensurate type quantum phase transitions from a gapful phase into a gapless phase have been identified and described using the bosonization treatment and finite chain exact diagonalization studies. The upper critical magnetic field corresponding to the transition into a fully polarized state has been also determined. It has been shown that the very presence of a staggered component of the magnetic field vapes the composite (C1) in favor of the composite gapped (C2) phase.

Topological spin crystals by itinerant frustration

Spin textures with nontrivial topology, such as vortices and skyrmions, have attracted attention as a source of unconventional magnetic, transport, and optical phenomena. Recently, a new generation of topological spin textures has been extensively studied in itinerant magnets; in contrast to the conventional ones induced, e.g., by the Dzyaloshinskii-Moriya interaction in noncentrosymmetric systems, they are characterized by extremely short magnetic periods and stable even in centrosymmetric systems. Here we review such new types of topological spin textures with particular emphasis on their stabilization mechanism. Focusing on the interplay between charge and spin degrees of freedom in itinerant electron systems, we show that itinerant frustration, which is the competition among electron-mediated interactions, plays a central role in stabilizing a variety of topological spin crystals including a skyrmion crystal with unconventional high skyrmion number, meron crystals, and hedgehog crystals. We also show that the essential ingredients in the itinerant frustration are represented by bilinear and biquadratic spin interactions in momentum space. This perspective not only provides a unified understanding of the unconventional topological spin crystals but also stimulates further exploration of exotic topological phenomena in itinerant magnets.

Magnetic Skyrmion Materials

Skyrmion, a concept originally proposed in particle physics half a century ago, can now find the most fertile field for its applicability, that is, the magnetic skyrmion realized in helimagnetic materials. The spin swirling vortex-like texture of the magnetic skyrmion can define the particle nature by topology; that is, all the constituent spin moments within the two-dimensional sheet wrap the sphere just one time. Such a topological nature of the magnetic skyrmion can lead to extraordinary metastability via topological protection and the driven motion with low electric-current excitation, which may promise future application to spintronics. The skyrmions in the magnetic materials frequently show up as the crystal lattice form, e.g., hexagonal lattice, but sometimes as isolated or independent particles. These skyrmions in magnets were initially found in acentric magnets, such as chiral, polar, and bilayered magnets endowed with antisymmetric spin exchange interaction, while the skyrmion host materials have been explored in a broader family of compounds including centrosymmetric magnets. This review describes the materials science and materials chemistry of magnetic skyrmions using the classification scheme of the skyrmion forming microscopic mechanisms. The emergent phenomena and functions mediated by skyrmions are described, including the generation of emergent magnetic and electric field by statics and dynamics of skrymions and the inherent magnetoelectric effect. The other important magnetic topological defects in two or three dimensions, such as biskyrmions, antiskyrmions, merons, and hedgehogs, are also reviewed in light of their interplay with the skyrmions.

Nonreciprocity of spin waves in the conical helix state

Nonreciprocity emerges in nature and in artificial objects from various physical origins, being widely utilized in contemporary technologies as exemplified by diode elements in electronics. While most of the nonreciprocal phenomena are realized by employing interfaces where the inversion symmetry is trivially lifted, nonreciprocal transport of photons, electrons, magnons, and possibly phonons also emerge in bulk crystals with broken space inversion and time reversal symmetries. Among them, directional propagation of bulk magnons (i.e., quanta of spin wave excitation) is attracting much attention nowadays for its potentially large nonreciprocity suitable for spintronic and spin-caloritronic applications. Here, we demonstrate nonreciprocal propagation of spin waves for the conical spin helix state in Cu₂OSeO₃ due to a combination of dipole and Dzyaloshinskii-Moriya interactions. The observed nonreciprocal spin dispersion smoothly connects to the hitherto known magnetochiral nonreciprocity in the field-induced collinear spin state; thus, all the spin phases show diode characteristics in this chiral insulator.

Microwave Spectroscopy of the Low-Temperature Skyrmion State in Cu_{2}OSeO_{3}

In the cubic chiral magnet Cu_{2}OSeO_{3} a low-temperature skyrmion state (LTS) and a concomitant tilted conical state are observed for magnetic fields parallel to ⟨100⟩. Here, we report on the dynamic resonances of these novel magnetic states. After promoting the nucleation of the LTS by means of field cycling, we apply broadband microwave spectroscopy in two experimental geometries that provide either predominantly in-plane or out-of-plane excitation. By comparing the results to linear spin-wave theory, we clearly identify resonant modes associated with the tilted conical state, the gyrational and breathing modes associated with the LTS, as well as the hybridization of the breathing mode with a dark octupole gyration mode mediated by the magnetocrystalline anisotropies. Most intriguingly, our findings suggest that under decreasing fields the hexagonal skyrmion lattice becomes unstable with respect to an oblique deformation, reflected in the formation of elongated skyrmions.

Particle-size dependent structural transformation of skyrmion lattice

Magnetic skyrmion is a topologically protected particle-like object in magnetic materials, appearing as a nanometric swirling spin texture. The size and shape of skyrmion particles can be flexibly controlled by external stimuli, which suggests unique features of their crystallization and lattice transformation process. Here, we investigated the detailed mechanism of structural transition of skyrmion lattice (SkL) in a prototype chiral cubic magnet Cu₂OSeO₃, by combining resonant soft X-ray scattering (RSXS) experiment and micromagnetic simulation. This compound is found to undergo a triangular-to-square lattice transformation of metastable skyrmions by sweeping magnetic field (B). Our simulation suggests that the symmetry change of metastable SkL is mainly triggered by the B-induced modification of skyrmion core diameter and associated energy cost at the skyrmion-skyrmion interface region. Such internal deformation of skyrmion particle has further been confirmed by probing the higher harmonics in the RSXS pattern. These results demonstrate that the size/shape degree of freedom of skyrmion particle is an important factor to determine their stable lattice form, revealing the exotic manner of phase transition process for topological soliton ensembles in the non-equilibrium condition.

Skyrmions are topological spin textures and are of great interest due to their impressive stability. Here, by sweeping an applied magnetic field, the authors observe a change in the skyrmion lattice structure, shedding light on the relation between skyrmion size and stability.

Effect of chemical substitution on the skyrmion phase in Cu2OSeO3

Magnetic skyrmions have been the focus of intense research due to their unique qualities which result from their topological protections. Previous work on Cu2OSeO3, the only known insulating multiferroic skyrmion material, has shown that chemical substitution alters the skyrmion phase. We chemically substitute Zn, Ag, and S into powdered Cu2OSeO3 to study the effect on the magnetic phase diagram. In both the Ag and the S substitutions, we find that the skyrmion phase is stabilized over a larger temperature range, as determined via magnetometry and small-angle neutron scattering (SANS). Meanwhile, while previous magnetometry characterization suggests two high temperature skyrmion phases in the Zn-substituted sample, SANS reveals the high temperature phase to be skyrmionic while we are unable to distinguish the other from helical order. Overall, chemical substitution weakens helical and skyrmion order as inferred from neutron scattering of the q ≈ 0.01 Å - 1 magnetic peak.

Formation Mechanism of the Helical Q Structure in Gd-Based Skyrmion Materials

Using the ab initio local force method, we investigate the formation mechanism of the helical spin structure in GdRu_{2}Si_{2} and Gd_{2}PdSi_{3}. We calculate the paramagnetic spin susceptibility and find that the Fermi surface nesting is not the origin of the incommensurate modulation, in contrast to the naive scenario based on the Ruderman-Kittel-Kasuya-Yosida mechanism. We then decompose the exchange interactions between the Gd spins into each orbital component, and show that spin-density-wave type interaction between the Gd-5d orbitals is ferromagnetic, but the interaction between the Gd-4f orbitals is antiferromagnetic. We conclude that the competition of these two interactions, namely, the interorbital frustration, stabilizes the finite-Q structure.

Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase

The phase transition most commonly observed is probably melting, a transition from ordered crystalline solids to disordered isotropic liquids. In three dimensions, melting is a single, first-order phase transition. In two-dimensional systems, however, theory predicts a general scenario of two continuous phase transitions separated by an intermediate, oriented liquid state, the so-called hexatic phase with short-range translational and quasi-long-range orientational orders. Such hexatic phases occur in colloidal systems, Wigner solids and liquid crystals, all composed of real-matter particles. In contrast, skyrmions are countable soliton configurations with non-trivial topology and these quasi-particles can form two-dimensional lattices. Here we show, by direct imaging with cryo-Lorentz transmission electron microscopy, that magnetic field variations can tune the phase of the skyrmion ensembles in Cu₂OSeO₃ from a two-dimensional solid through the long-speculated skyrmion hexatic phase to a liquid. The local spin order persists throughout the process. Remarkably, our quantitative analysis demonstrates that the aforementioned topological-defect-induced crystal melting scenario well describes the observed phase transitions.

Giant Topological Hall Effect and Superstable Spontaneous Skyrmions below 330 K in a Centrosymmetric Complex Noncollinear Ferromagnet NdMn2Ge2

Skyrmions with topologically nontrivial spin textures are promising information carriers in next-generation ultralow power consumption and high-density spintronic devices. To promote their further development and utilization, the search for new room temperature skyrmion-hosting materials is crucial. Considering that most of the previous skyrmion-hosting materials are noncollinear magnets, here, the detection of the topological Hall effect (THE) and the discovery of skyrmions at room temperature are first reported in a centrosymmetric complex noncollinear ferromagnet NdMn₂Ge₂. Below 330 K, the compound can host stable Bloch-type skyrmions with about 75 nm diameter in a wide window of magnetic field and temperature, including zero magnetic field and room temperature. Moreover, the skyrmions can induce a giant topological Hall effect in a wide temperature range with a maximum value of -2.05 μΩ cm. These features make the compound attractive for both fundamental research and potential application in novel spintronic devices.

Skyrmion Crystal from RKKY Interaction Mediated by 2D Electron Gas

We consider a C_{6} invariant lattice of magnetic moments coupled via a Kondo exchange J with a 2D electron gas (2DEG). The effective Ruderman-Kittel-Kasuya-Yosida interaction between the moments stabilizes a magnetic skyrmion crystal in the presence of magnetic field and easy-axis anisotropy. An attractive aspect of this mechanism is that the magnitude of the magnetic ordering wave vectors, Q_{ν} (ν=1, 2, 3), is dictated by the Fermi wave number k_{F}: |Q_{ν}|=2k_{F}. Consequently, the topological contribution to the Hall conductivity of the 2DEG becomes of the order of the quantized value, e^{2}/h, when J is comparable to the Fermi energy ε_{F}.

Room-temperature skyrmion phase in bulk Cu2OSeO3 under high pressures

A skyrmion state in a noncentrosymmetric helimagnet displays topologically protected spin textures with profound technological implications for high-density information storage, ultrafast spintronics, and effective microwave devices. Usually, its equilibrium state in a bulk helimagnet occurs only over a very restricted magnetic field-temperature phase space and often in the low-temperature region near the magnetic transition temperature T_c We have expanded and enhanced the skyrmion phase region from the small range of 55 to 58.5 K to 5 to 300 K in single-crystalline Cu₂OSeO₃ by pressures up to 42.1 GPa through a series of phase transitions from the cubic P2₁3, through orthorhombic P2₁2₁2₁ and monoclinic P2₁, and finally to the triclinic P1 phase, using our newly developed ultrasensitive high-pressure magnetization technique. The results are in agreement with our Ginzburg-Landau free energy analyses, showing that pressures tend to stabilize the skyrmion states and at higher temperatures. The observations also indicate that the skyrmion state can be achieved at higher temperatures in various crystal symmetries, suggesting the insensitivity of skyrmions to the underlying crystal lattices and thus the possible more ubiquitous presence of skyrmions in helimagnets.

Robust Perpendicular Skyrmions and Their Surface Confinement

Magnetic skyrmions are two-dimensional magnetization swirls that stack in the form of tubes in the third dimension and which are proposed as prospective information carriers for nonvolatile memory devices due to their unique topological properties. From resonant elastic X-ray scattering measurements on Cu₂OSeO₃ with an in-plane magnetic field, we find that a state of perpendicularly ordered skyrmions forms, in stark contrast to the well-studied bulk state. The surface state is stable over a wide temperature range, unlike the bulk state in out-of-plane fields which is confined to a narrow region of the temperature-field phase diagram. In contrast to ordinary skyrmions found in the bulk, the surface state skyrmions result from the presence of magnetic interactions unique to the surface which stabilize them against external perturbations. The surface guiding makes the robust state particular interesting for racetracklike devices, ultimately allowing for much higher storage densities due to the smaller lateral footprint of the perpendicular skyrmions.

Dynamical magnetoelectric phenomena of skyrmions in multiferroics

Magnetic skyrmions, nanoscopic spin vortices carrying a quantized topological number in chiral-lattice magnets, are recently attracting great research interest. Although magnetic skyrmions had been observed only in metallic chiral-lattice magnets such as B20 alloys in the early stage of the research, their realization was discovered in 2012 also in an insulating chiral-lattice magnet Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . A characteristic of the insulating skyrmions is that they can host multiferroicity, that is, the noncollinear magnetization alignment of skyrmion induces electric polarizations in insulators with a help of the relativistic spin-orbit interaction. It was experimentally confirmed that the skyrmion phase in Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ is indeed accompanied by the spin-induced ferroelectricity. The resulting strong magnetoelectric coupling between magnetizations and electric polarizations can provide us with a means to manipulate and activate magnetic skyrmions by application of electric fields. This is in sharp contrast to skyrmions in metallic systems, which are driven through injection of electric currents. The magnetoelectric phenomena specific to the skyrmion-based multiferroics are attracting intensive research interest, and, in particular, those in dynamical regime are widely recognized as an issue of vital importance because their understanding is crucial both for fundamental science and for technical applications. In this article, we review recent studies on multiferroic properties and dynamical magnetoelectric phenomena of magnetic skyrmions in insulating chiral-lattice magnet Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . It is argued that the multiferroic skyrmions show unique resonant excitation modes of coupled magnetizations and polarizations, so-called electromagnon excitations, which can be activated both magnetically with a microwave magnetic field and electrically with a microwave electric field. The interference between these two activation processes gives rise to peculiar phenomena in the gigahertz regime. As its representative example, we discuss a recent theoretical prediction of unprecedentedly large nonreciprocal directional dichroism of microwaves in the skyrmion phase of Cu 2 OSeO 3 $\textrm{Cu}_2\textrm{OSeO}_3$ . This phenomenon can be regarded as a one-way window effect on microwaves, that is, the extent of microwave absorption changes significantly when its incident direction is reversed. This dramatic effect was indeed observed by subsequent experiments. These studies demonstrated that the multiferroic skyrmions can be a promising building block for microwave devices.

Propagation dynamics of spin excitations along skyrmion strings

Magnetic skyrmions, topological solitons characterized by a two-dimensional swirling spin texture, have recently attracted attention as stable particle-like objects. In a three-dimensional system, a skyrmion can extend in the third dimension forming a robust and flexible string structure, whose unique topology and symmetry are anticipated to host nontrivial functional responses. Here we experimentally demonstrate the coherent propagation of spin excitations along skyrmion strings for the chiral-lattice magnet Cu₂OSeO₃. We find that this propagation is directionally non-reciprocal and the degree of non-reciprocity, as well as group velocity and decay length, are strongly dependent on the character of the excitation modes. These spin excitations can propagate over a distance exceeding 50 μm, demonstrating the excellent long-range ordered nature of the skyrmion-string structure. Our combined experimental and theoretical analyses offer a comprehensive account of the propagation dynamics of skyrmion-string excitations and suggest the possibility of unidirectional information transfer along such topologically protected strings.

Low-energy magnons in the chiral ferrimagnet Cu2OSeO3: A coarse-grained approach

We report a comprehensive neutron scattering study of low energy magnetic excitations in the breathing pyrochlore helimagnetic Cu₂OSeO₃. Fully documenting the four lowest energy magnetic modes that leave the ferrimagnetic configuration of the "strong tetrahedra" intact ( | ℏ ω | < 13 meV), we find gapless quadratic dispersion at the point for energies above 0.2 meV, two doublets separated by 1.6(2) meV at the R point, and a bounded continuum at the X point. Our constrained rigid spin cluster model relates these features to Dzyaloshinskii-Moriya (DM) interactions and the incommensurate helical ground state. Combining conventional spin wave theory with a spin cluster form factor accurately reproduces the measured equal time structure factor through multiple Brillouin zones. An effective spin Hamiltonian describing complex anisotropic intercluster interactions is obtained.

Ferromagnetic Resonance with Magnetic Phase Selectivity by Means of Resonant Elastic X-Ray Scattering on a Chiral Magnet

Cubic chiral magnets, such as Cu_{2}OSeO_{3}, exhibit a variety of noncollinear spin textures, including a trigonal lattice of spin whirls, the so-called skyrmions. Using magnetic resonant elastic x-ray scattering (REXS) on a crystalline Bragg peak and its magnetic satellites while exciting the sample with magnetic fields at gigahertz frequencies, we probe the ferromagnetic resonance (FMR) modes of these spin textures by means of the scattered intensity. Most notably, the three eigenmodes of the skyrmion lattice are detected with large sensitivity. As this novel technique, which we label REXS FMR, is carried out at distinct positions in reciprocal space, it allows us to distinguish contributions originating from different magnetic states, providing information on the precise character, weight, and mode mixing as a prerequisite of tailored excitations for applications.

Spiral spin structures and skyrmions in multiferroics

In this article, we focus on (1) type-II multiferroics driven by spiral spin orderings and (2) magnetoelectric couplings in multiferroic skyrmion-hosting materials. We present both phenomenological understanding and microscopic mechanisms for spiral spin state, which is one of the essential starting points for type-II multiferroics and magnetic skyrmions. Two distinct mechanisms of spiral spin states (frustration and Dzyaloshinskii–Moriya [DM] interaction) are discussed in the context of the lattice symmetry. We also discuss the spin-induced ferroelectricity on the basis of the symmetry and microscopic atomic configurations. We compare two well-known microscopic models: the generalized inverse DM mechanism and the metal-ligand d-p hybridization mechanism. As a test for these models, we summarize the multiferroic properties of a family of triangular-lattice antiferromagnets. We also give a brief review of the magnetic skyrmions. Three types of known skyrmion-hosting materials with multiferroicity are discussed from the view point of crystal structure, magnetism, and origins of the magnetoelectric couplings. For exploration of new skyrmion-hosting materials, we also discuss the theoretical models for stabilizing skyrmions by magnetic frustration in centrosymmetric system. Several basic ideas for material design are given, which are successfully demonstrated by the recent experimental evidences for the skyrmion formation in centrosymmetric frustrated magnets.

Spin-Hall Topological Hall Effect in Highly Tunable Pt/Ferrimagnetic-Insulator Bilayers

Electrical detection of topological magnetic textures such as skyrmions is currently limited to conducting materials. Although magnetic insulators offer key advantages for skyrmion technologies with high speed and low loss, they have not yet been explored electrically. Here, we report a prominent topological Hall effect in Pt/Tm₃Fe₅O₁₂ bilayers, where the pristine Tm₃Fe₅O₁₂ epitaxial films down to 1.25 unit cell thickness allow for tuning of topological Hall stability over a broad range from 200 to 465 K through atomic-scale thickness control. Although Tm₃Fe₅O₁₂ is insulating, we demonstrate the detection of topological magnetic textures through a novel phenomenon: "spin-Hall topological Hall effect" (SH-THE), where the interfacial spin-orbit torques allow spin-Hall-effect generated spins in Pt to experience the unique topology of the underlying skyrmions in Tm₃Fe₅O₁₂. This novel electrical detection phenomenon paves a new path for utilizing a large family of magnetic insulators in future skyrmion technologies.

Strong hopping induced Dzyaloshinskii–Moriya interaction and skyrmions in elemental cobalt

The Dzyaloshinskii–Moriya interaction (DMI) is well known to favor a chiral rotation of the magnetic moments, which accounts for the emergence of the skyrmions. The DMI is a combined effect of spin–orbit coupling with broken inversion symmetry in magnets. Most of the noncentrosymmetric magnetic materials that bear skyrmions involve nonmagnetic elements. This work shows that strong DMIs exist in elemental cobalt with a β-Mn-type metastable structure. The variation of DMI among different cobalt pairs largely follows the variation of hopping magnitude in which p electrons play an important role. Although the DMIs between different atomic pairs partly cancels with each other, the net interaction is sufficient to result in a left-handed Bloch-type spiral. Spin dynamics simulation shows that a critical magnetic field of 2.9 T stabilizes skyrmions at 0 K.

Optically Driven Collective Spin Excitations and Magnetization Dynamics in the Néel-type Skyrmion Host GaV_{4}S_{8}

GaV_{4}S_{8} is a multiferroic semiconductor hosting magnetic cycloid (Cyc) and Néel-type skyrmion lattice (SkL) phases with a broad region of thermal and magnetic stability. Here, we use time-resolved magneto-optical Kerr spectroscopy to show the coherent generation of collective spin excitations in the Cyc and SkL phases. Our micromagnetic simulations reveal that these are driven by an optically induced modulation of uniaxial anisotropy. Our results shed light on spin dynamics in anisotropic materials hosting skyrmions and pave a new pathway for the optical manipulation of their magnetic order.

Response of the Skyrmion Lattice in MnSi to Cubic Magnetocrystalline Anisotropies

We report high-precision small-angle neutron scattering of the orientation of the Skyrmion lattice in a spherical sample of MnSi under systematic changes of the magnetic field direction. For all field directions the Skyrmion lattice may be accurately described as a triple-Q[over →] state, where the modulus |Q[over →]| is constant and the wave vectors enclose rigid angles of 120°. Along a great circle across ⟨100⟩, ⟨110⟩, and ⟨111⟩ the normal to the Skyrmion-lattice plane varies systematically by ±3° with respect to the field direction, while the in-plane alignment displays a reorientation by 15° for magnetic field along ⟨100⟩. Our observations are qualitatively and quantitatively in excellent agreement with an effective potential, which is determined by the symmetries of the tetrahedral point group T and includes contributions up to sixth order in spin-orbit coupling, providing a full account of the effect of cubic magnetocrystalline anisotropies on the Skyrmion lattice in MnSi.

Stabilization and Reversal of Skyrmion Lattice in Ta/CoFeB/MgO Multilayers

Recently, magnetic skyrmion has attracted much attention due to its potential application in racetrack memory and other nanodevices. In bulk chiral magnets with non-centrosymmetric crystal structures, skyrmion lattice phase has been extensively observed. However, in film or multilayers with interfacial Dzyaloshinskii-Moriya interaction, individual skyrmion is often observed. Here, we report a short-ordered skyrmion lattice observed in [Ta(5.0 nm)/CoFeB(1.5 nm)/MgO(1.0 nm)]₁₅ multilayer in a remnant state. The structure, stabilization, and reversal of these skyrmions are discussed. Applying a slightly tilted in-plane magnetic field caused reversal of the skyrmion lattice. This reversal came from disappearance of skyrmions and nucleation of new skyrmions in the interstitial regions of the lattice. Also, we investigated how the skyrmion lattice depended on the CoFeB thickness. Our findings provide a pathway to stabilize and reverse the skyrmions in multilayers films.

Direct electric field control of the skyrmion phase in a magnetoelectric insulator

Magnetic skyrmions are topologically protected spin-whirls currently considered as promising for use in ultra-dense memory devices. Towards achieving this goal, exploration of the skyrmion phase response and under external stimuli is urgently required. Here we show experimentally, and explain theoretically, that in the magnetoelectric insulator Cu2OSeO3 the skyrmion phase can expand and shrink significantly depending on the polarity of a moderate applied electric field (few V/μm). The theory we develop incorporates fluctuations around the mean-field that clarifies precisely how the electric field provides direct control over the free energy difference between the skyrmion and the surrounding conical phase. The quantitative agreement between theory and experiment provides a solid foundation for the development of skyrmionic applications based on magnetoelectric coupling.

Reciprocal space tomography of 3D skyrmion lattice order in a chiral magnet

It is commonly assumed that surfaces modify the properties of stable materials within the top few atomic layers of a bulk specimen only. Exploiting the polarization dependence of resonant elastic X-ray scattering to go beyond conventional diffraction and imaging techniques, we have determined the depth dependence of the full 3D spin structure of skyrmions-that is, topologically nontrivial whirls of the magnetization-below the surface of a bulk sample of Cu₂OSeO₃ We found that the skyrmions change exponentially from pure Néel- to pure Bloch-twisting over a distance of several hundred nanometers between the surface and the bulk, respectively. Though qualitatively consistent with theory, the strength of the Néel-twisting at the surface and the length scale of the variation observed experimentally exceed material-specific modeling substantially. In view of the exceptionally complete quantitative theoretical account of the magnetic rigidities and associated static and dynamic properties of skyrmions in Cu₂OSeO₃ and related materials, we conclude that subtle changes of the materials properties must exist at distances up to several hundred atomic layers into the bulk, which originate in the presence of the surface. This has far-reaching implications for the creation of skyrmions in surface-dominated systems and identifies, more generally, surface-induced gradual variations deep within a bulk material and their impact on tailored functionalities as an unchartered scientific territory.

Manipulation of skyrmion motion by magnetic field gradients

Magnetic skyrmions are particle-like, topologically protected magnetisation entities that are promising candidates as information carriers in racetrack memory. The transport of skyrmions in a shift-register-like fashion is crucial for their embodiment in practical devices. Here, we demonstrate that chiral skyrmions in Cu2OSeO3 can be effectively manipulated under the influence of a magnetic field gradient. In a radial field gradient, skyrmions were found to rotate collectively, following a given velocity-radius relationship. As a result of this relationship, and in competition with the elastic properties of the skyrmion lattice, the rotating ensemble disintegrates into a shell-like structure of discrete circular racetracks. Upon reversing the field direction, the rotation sense reverses. Field gradients therefore offer an effective handle for the fine control of skyrmion motion, which is inherently driven by magnon currents. In this scheme, no local electric currents are needed, thus presenting a different approach to shift-register-type operations based on spin transfer torque.

Helimagnon Resonances in an Intrinsic Chiral Magnonic Crystal

We experimentally study magnetic resonances in the helical and conical magnetic phases of the chiral magnetic insulator Cu_{2}OSeO_{3} at the temperature T=5  K. Using a broadband microwave spectroscopy technique based on vector network analysis, we identify three distinct sets of helimagnon resonances in the frequency range 2  GHz≤f≤20  GHz with low magnetic damping α≤0.003. The extracted resonance frequencies are in accordance with calculations of the helimagnon band structure found in an intrinsic chiral magnonic crystal. The periodic modulation of the equilibrium spin direction that leads to the formation of the magnonic crystal is a direct consequence of the chiral magnetic ordering caused by the Dzyaloshinskii-Moriya interaction. The mode coupling in the magnonic crystal allows excitation of helimagnons with wave vectors that are multiples of the spiral wave vector.

Magnetic Skyrmions and Skyrmion Clusters in the Helical Phase of Cu_{2}OSeO_{3}

Skyrmions are nanometric spin whirls that can be stabilized in magnets lacking inversion symmetry. The properties of isolated Skyrmions embedded in a ferromagnetic background have been intensively studied. We show that single Skyrmions and clusters of Skyrmions can also form in the helical phase and investigate theoretically their energetics and dynamics. The helical background provides natural one-dimensional channels along which a Skyrmion can move rapidly. In contrast to Skyrmions in ferromagnets, the Skyrmion-Skyrmion interaction has a strong attractive component and thus Skyrmions tend to form clusters with characteristic shapes. These clusters are directly observed in transmission electron microscopy measurements in thin films of Cu_{2}OSeO_{3}. Topological quantization, high mobility, and the confinement of Skyrmions in channels provided by the helical background may be useful for future spintronics devices.

Theory of magnons in spin systems with Dzyaloshinskii-Moriya interaction

We study in this paper magnetic properties of a system of quantum Heisenberg spins interacting with each other via a ferromagnetic exchange interaction J and an in-plane Dzyaloshinskii-Moriya interaction D. The non-collinear ground state due to the competition between J and D is determined. We employ a self-consistent Green'function theory to calculate the spin-wave spectrum and the layer magnetizations at finite T in two and three dimensions as well as in a thin film with surface effects. Analytical details and the validity of the method are shown and discussed. Numerical solutions are shown for realistic physical interaction parameters. Discussion on possible experimental verifications is given.

Linearly polarized GHz magnetization dynamics of spin helix modes in the ferrimagnetic insulator Cu2OSeO3

Linear dichroism - the polarization dependent absorption of electromagnetic waves- is routinely exploited in applications as diverse as structure determination of DNA or polarization filters in optical technologies. Here filamentary absorbers with a large length-to-width ratio are a prerequisite. For magnetization dynamics in the few GHz frequency regime strictly linear dichroism was not observed for more than eight decades. Here, we show that the bulk chiral magnet Cu₂OSeO₃ exhibits linearly polarized magnetization dynamics at an unexpectedly small frequency of about 2 GHz at zero magnetic field. Unlike optical filters that are assembled from filamentary absorbers, the magnet is shown to provide linear polarization as a bulk material for an extremely wide range of length-to-width ratios. In addition, the polarization plane of a given mode can be switched by 90° via a small variation in width. Our findings shed a new light on magnetization dynamics in that ferrimagnetic ordering combined with antisymmetric exchange interaction offers strictly linear polarization and cross-polarized modes for a broad spectrum of sample shapes at zero field. The discovery allows for novel design rules and optimization of microwave-to-magnon transduction in emerging microwave technologies.

Observation of Various and Spontaneous Magnetic Skyrmionic Bubbles at Room Temperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy

The quest for materials hosting topologically protected skyrmionic spin textures continues to be fueled by the promise of novel devices. Although many materials have demonstrated the existence of such spin textures, major challenges remain to be addressed before devices based on magnetic skyrmions can be realized. For example, being able to create and manipulate skyrmionic spin textures at room temperature is of great importance for further technological applications because they can adapt to various external stimuli acting as information carriers in spintronic devices. Here, the first observation of skyrmionic magnetic bubbles with variable topological spin textures formed at room temperature in a frustrated kagome Fe₃ Sn₂ magnet with uniaxial magnetic anisotropy is reported. The magnetization dynamics are investigated using in situ Lorentz transmission electron microscopy, revealing that the transformation between different magnetic bubbles and domains is via the motion of Bloch lines driven by an applied external magnetic field. These results demonstrate that Fe₃ Sn₂ facilitates a unique magnetic control of topological spin textures at room temperature, making it a promising candidate for further skyrmion-based spintronic devices.