Dynamics of Skyrmion crystals in metallic thin films

Antiskyrmions in Ferroelectric Barium Titanate

Typical magnetic skyrmion is a string of inverted magnetization within a ferromagnet, protected by a sleeve of a vortexlike spin texture, such that its cross-section carries an integer topological charge. Some magnets form antiskyrmions, the antiparticle strings which carry an opposite topological charge. Here we demonstrate that topologically equivalent but purely electric antiskyrmion can exist in a ferroelectric material as well. In particular, our computer experiments reveal that the archetype ferroelectric, barium titanate, can host antiskyrmions at zero field. The polarization pattern around their cores reminds ring windings of decorative knots rather than the typical magnetic antiskyrmion texture. We show that the antiskyrmion of barium titanate has just 2-3 nm in diameter, a hexagonal cross section, and an exotic topological charge with doubled magnitude and opposite sign when compared to the standard skyrmion string.

Acoustic Skyrmionic Mode Coupling and Transferring in a Chain of Subwavelength Metastructures

Skyrmions, a stable topological vectorial textures characteristic with skyrmionic number, hold promise for advanced applications in information storage and transmission. While the dynamic motion control of skyrmions has been realized with various techniques in magnetics and optics, the manipulation of acoustic skyrmion has not been done. Here, the propagation and control of acoustic skyrmion along a chain of metastructures are shown. In coupled acoustic resonators made with Archimedes spiral channel, the skyrmion hybridization is found giving rise to bonding and antibonding skyrmionic modes. Furthermore, it is experimentally observed that the skyrmionic mode of acoustic velocity field distribution can be robustly transferred covering a long distance and almost no distortion of the skyrmion textures in a chain of metastructures, even if a structure defect is introduced in the travel path. The proposed localized acoustic skyrmionic mode coupling and propagating is expected in future applications for manipulating acoustic information storage and transfer.

Steady motion of 80-nm-size skyrmions in a 100-nm-wide track

The current-driven movement of magnetic skyrmions along a nanostripe is essential for the advancement and functionality of a new category of spintronic devices resembling racetracks. Despite extensive research into skyrmion dynamics, experimental verification of current-induced motion of ultra-small skyrmions within an ultrathin nanostripe is still pending. Here, we unveil the motion of individual 80 nm-size skyrmions in an FeGe track with an ultrathin width of 100 nm. The skyrmions can move steadily along the track over a broad range of current densities by using controlled pulse durations of as low as 2 ns. The potential landscape, arising from the magnetic edge twists in such a geometrically confined system, introduces skyrmion inertia and ensures efficient motion with a vanishing skyrmion Hall angle. Our results showcase the steady motion of skyrmions in an ultrathin track, offering a practical pathway for implementing skyrmion-based spintronic devices.

Topological Spin Textures: Basic Physics and Devices

In the face of escalating modern data storage demands and the constraints of Moore's Law, exploring spintronic solutions, particularly the devices based on magnetic skyrmions, has emerged as a promising frontier in scientific research. Since the first experimental observation of skyrmions, topological spin textures have been extensively studied for their great potential as efficient information carriers in spintronic devices. However, significant challenges have emerged alongside this progress. This review aims to synthesize recent advances in skyrmion research while addressing the major issues encountered in the field. Additionally, current research on promising topological spin structures in addition to skyrmions is summarized. Beyond 2D structures, exploration also extends to 1D magnetic solitons and 3D spin textures. In addition, a diverse array of emerging magnetic materials is introduced, including antiferromagnets and 2D van der Waals magnets, broadening the scope of potential materials hosting topological spin textures. Through a systematic examination of magnetic principles, topological categorization, and the dynamics of spin textures, a comprehensive overview of experimental and theoretical advances in the research of topological magnetism is provided. Finally, both conventional and unconventional applications are summarized based on spin textures proposed thus far. This review provides an outlook on future development in applied spintronics.

Topological Spin Textures in an Insulating van der Waals Ferromagnet

Generation and control of topological spin textures constitutes one of the most exciting challenges of modern spintronics given their potential applications in information storage technologies. Of particular interest are magnetic insulators, which due to low damping, absence of Joule heating and reduced dissipation can provide energy-efficient spin-textures platform. Here, it is demonstrated that the interplay between sample thickness, external magnetic fields, and optical excitations can generate a prolific paramount of spin textures, and their coexistence in insulating CrBr3 van der Waals (vdW) ferromagnets. Using high-resolution magnetic force microscopy and large-scale micromagnetic simulation methods, the existence of a large region in T-B phase diagram is demonstrated where different stripe domains, skyrmion crystals, and magnetic domains exist and can be intrinsically selected or transformed to each-other via a phase-switch mechanism. Lorentz transmission electron microscopy unveils the mixed chirality of the magnetic textures that are of Bloch-type at given conditions but can be further manipulated into Néel-type or hybrid-type via thickness-engineering. The topological phase transformation between the different magnetic objects can be further inspected by standard photoluminescence optical probes resolved by circular polarization indicative of an existence of exciton-skyrmion coupling mechanism. The findings identify vdW magnetic insulators as a promising framework of materials for the manipulation and generation of highly ordered skyrmion lattices relevant for device integration at the atomic level.

Spin disorder control of topological spin texture

Stabilization of topological spin textures in layered magnets has the potential to drive the development of advanced low-dimensional spintronics devices. However, achieving reliable and flexible manipulation of the topological spin textures beyond skyrmion in a two-dimensional magnet system remains challenging. Here, we demonstrate the introduction of magnetic iron atoms between the van der Waals gap of a layered magnet, Fe3GaTe2, to modify local anisotropic magnetic interactions. Consequently, we present direct observations of the order-disorder skyrmion lattices transition. In addition, non-trivial topological solitons, such as skyrmioniums and skyrmion bags, are realized at room temperature. Our work highlights the influence of random spin control of non-trivial topological spin textures.

Optimizing skyrmionium movement and stability via stray magnetic fields in trilayer nanowire constructs

Skyrmioniums, known for their unique transport and regulatory properties, are emerging as potential cornerstones for future data storage systems. However, the stability of skyrmionium movement faces considerable challenges due to the skyrmion Hall effect, which is induced by deformation. In response, our research introduces an innovative solution: we utilized micro-magnetic simulations to create a sandwiched trilayer nanowire structure augmented with a stray magnetic field. This combination effectively guides the skyrmionium within the ferromagnetic (FM) layer. Our empirical investigations reveal that the use of a stray magnetic field not only reduces the size of the skyrmionium but also amplifies its stability. This dual-effect proficiently mitigates the deformation of skyrmionium movement and boosts their thermal stability. We find these positive outcomes are most pronounced at a particular intensity of the stray magnetic field. Importantly, the required stray magnetic field can be generated using a heavy metal (HM1) layer of suitable thickness, rendering the practical application of this approach plausible in real-world experiments. Additionally, we analyze the functioning mechanism based on the Landau-Lifshitz-Gilbert (LLG) equation and energy variation. We also develop a deep spiking neural network (DSNN), which achieves a remarkable recognition accuracy of 97%. This achievement is realized through supervised learning via the spike timing dependent plasticity rule (STDP), considering the nanostructure as an artificial synapse device that corresponds to the electrical properties of the nanostructure. In conclusion, our study provides invaluable insights for the design of innovative information storage devices utilizing skyrmionium technology. By tackling the issues presented by the skyrmion Hall effect, we outline a feasible route for the practical application of this advanced technology. Our research, therefore, serves as a robust platform for continued investigations in this field.

Field-controlled dynamics of skyrmions and monopoles

Magnetic monopoles, despite their ongoing experimental search as elementary particles, have inspired the discovery of analogous excitations in condensed matter systems. In chiral condensed matter systems, emergent monopoles are responsible for the onset of transitions between topologically distinct states and phases, such as in the case of transitions from helical and conical phase to A-phase comprising periodic arrays of skyrmions. By combining numerical modeling and optical characterizations, we describe how different geometrical configurations of skyrmions terminating at monopoles can be realized in liquid crystals and liquid crystal ferromagnets. We demonstrate how these complex structures can be effectively manipulated by external magnetic and electric fields. Furthermore, we discuss how our findings may hint at similar dynamics in other physical systems and their potential applications.

Strain-Induced Reversible Motion of Skyrmions at Room Temperature

Magnetic skyrmions are topologically protected swirling spin textures with great potential for future spintronic applications. The ability to induce skyrmion motion using mechanical strain not only stimulates the exploration of exotic physics but also affords the opportunity to develop energy-efficient spintronic devices. However, the experimental realization of strain-driven skyrmion motion remains a formidable challenge. Herein, we demonstrate that the inhomogeneous uniaxial compressive strain can induce the movement of isolated skyrmions from regions of high strain to regions of low strain at room temperature, which was directly observed using an in situ Lorentz transmission electron microscope with a specially designed nanoindentation holder. We discover that the uniaxial compressive strain can transform skyrmions into a single domain with in-plane magnetization, resulting in the coexistence of skyrmions with a single domain along the direction of the strain gradient. Through comprehensive micromagnetic simulations, we reveal that the repulsive interactions between skyrmions and the single domain serve as the driving force behind the skyrmion motion. The precise control of skyrmion motion through strain provides exciting opportunities for designing advanced spintronic devices that leverage the intricate interplay between strain and magnetism.

Controlling stable Bloch points with electric currents

The Bloch point is a point singularity in the magnetisation configuration, where the magnetisation vanishes. It can exist as an equilibrium configuration and plays an important role in many magnetisation reversal processes. In the present work, we focus on manipulating Bloch points in a system that can host stable Bloch points-a two-layer FeGe nanostrip with opposite chirality of the two layers. We drive Bloch points using spin-transfer torques and find that Bloch points can move collectively without any Hall effect and report that Bloch points are repelled from the sample boundaries and each other. We study pinning of Bloch points at wedge-shaped constrictions (notches) in the nanostrip and demonstrate that arrays of Bloch points can be moved past a series of notches in a controlled manner by applying consecutive current pulses of different strength. Finally, we simulate a T-shaped geometry and demonstrate that a Bloch point can be moved along different paths by applying current between suitable strip ends.

Ferroelectrically tunable magnetic skyrmions in two-dimensional multiferroics

Magnetic skyrmions are topologically protected entities that are promising for information storage and processing. Currently, an essential challenge for future advances of skyrmionic devices lies in achieving effective control of skyrmion properties. Here, through first-principles and Monte-Carlo simulations, we report the identification of nontrivial topological magnetism in two-dimensional multiferroics of Co2NF2. Because of ferroelectricity, monolayer Co2NF2 exhibits a large Dzyaloshinskii-Moriya interaction. This together with exchange interaction can stabilize magnetic skyrmions with the size of sub-10 nm under a moderate magnetic field. Importantly, arising from the magnetoelectric coupling effect, the chirality of magnetic skyrmions is ferroelectrically tunable, producing the four-fold degenerate skyrmions. When interfacing with monolayer MoSe2, the creation and annihilation of magnetic skyrmions, as well as phase transition between skyrmion and skyrmion lattice, can be realized in a ferroelectrically controllable fashion. A dimensionless parameter κ' is further proposed as the criterion for stabilizing magnetic skyrmions in such multiferroic lattices. Our work greatly enriches the two-dimensional skyrmionics and multiferroics research.

Instability of Magnetic Skyrmion Strings Induced by Longitudinal Spin Currents

It is well established that spin-transfer torques exerted by in-plane spin currents give rise to a motion of magnetic skyrmions resulting in a skyrmion Hall effect. In films of finite thickness or in three-dimensional bulk samples the skyrmions extend in the third direction forming a string. We demonstrate that a spin current flowing longitudinally along the skyrmion string instead induces a Goldstone spin wave instability. Our analytical results are confirmed by micromagnetic simulations of both a single string as well as string lattices, suggesting that the instability eventually breaks the strings. A longitudinal current is thus able to melt the skyrmion string lattice via a nonequilibrium phase transition. For films of finite thickness or in the presence of disorder a threshold current will be required, and we estimate the latter assuming weak collective pinning.

Two methods of forming flat magnetic structures in magnetic films with topological features

The paper investigates vortex-like structures observed in ferromagnetic films with strong uniaxial easy-plane anisotropy in the presence of topological features in them. Two approaches to the creation of such features are considered, namely, perforation of the sample and the inclusion of artificial defects in it, and a theorem on their equivalence is proved, according to which the structure of magnetic inhomogeneities arising in the film itself turns out to be the same for both approaches. In the second case, the properties of magnetic vortices formed on defects are also studied, and for cylindrical defects explicit analytical expressions for the energy and configuration of vortices are obtained, which are applicable in a wide range of values of material parameters.

Control and regulation of skyrmionic topological charge in a novel synthetic antiferromagnetic nanostructure

Skyrmionium is a combination of a skyrmion with a topological charge (Q is +1 or -1), resulting in a magnetic configuration with a total topological charge of Q = 0. Skyrmionium has distinctive characteristics, including a slightly higher velocity, motion restricted to the middle of the track without the skyrmion Hall effect (SkHE), and absence of an acceleration phase. However, there is little stray field due to the zero net magnetization, the topological charge Q is zero due to the magnetic configuration, and detecting skyrmionium is still challenging. In the present work, we propose a novel nanostructure composed of triple nanowires with a narrow channel. It was found that the skyrmionium is converted into a DW pair or skyrmion by the concave channel. It was also found that the topological charge Q can be regulated by Ruderman-Kittel-Kasuya-Yosida (RKKY) antiferromagnetic (AFM) exchange coupling. Moreover, we analyzed the mechanism of the function based on the Landau-Lifshitz-Gilbert (LLG) equation and energy variation and constructed a deep spiking neural network (DSNN) with a recognition accuracy of 98.6% with supervised learning via the spike timing dependent plasticity rule (STDP) by considering the nanostructure as an artificial synapse device corresponding to the electrical properties of the nanostructure. These results provide the means for skyrmion-skyrmionium hybrid application and neuromorphic computing applications.

Mechanics of a ferromagnetic domain wall

This paper gives a pedagogical introduction to the mechanics of ferromagnetic solitons. We start with the dynamics of a single spin and develop all the tools required for the description of the dynamics of solitons in a ferromagnet.

Emergent Magnetomultipoles and Nonlinear Responses of a Magnetic Hopfion

The three-dimensional emergent magnetic field B^{e} of a magnetic hopfion gives rise to emergent magnetomultipoles in a similar manner to the multipoles of classical electromagnetic field. Here, we show that the nonlinear responses of a hopfion are characterized by its emergent magnetic toroidal moment T_{z}^{e}=1/2∫(r×B^{e})_{z}dV and emergent magnetic octupole component Γ^{e}=∫[(x^{2}+y^{2})B_{z}^{e}-xzB_{x}^{e}-yzB_{y}^{e}]dV. The hopfion exhibits nonreciprocal dynamics (nonlinear hopfion Hall effect) under an ac driving current applied along (perpendicular to) the direction of T_{z}^{e}. The sign of nonreciprocity and nonlinear Hall angle is determined by the polarity and chirality of hopfion. The nonlinear electrical transport induced by a magnetic hopfion is also discussed. This Letter reveals the vital roles of emergent magnetomultipoles in nonlinear hopfion dynamics and could stimulate further investigations on the dynamical responses of topological spin textures induced by emergent electromagnetic multipoles.

Magnus induced diode effect for skyrmions in channels with periodic potentials

Using a particle based model, we investigate the skyrmion dynamical behavior in a channel where the upper wall contains divots of one depth and the lower wall contains divots of a different depth. Under an applied driving force, skyrmions in the channels move with a finite skyrmion Hall angle that deflects them toward the upper wall for -xdirection driving and the lower wall for +xdirection driving. When the upper divots have zero height, the skyrmions are deflected against the flat upper wall for -xdirection driving and the skyrmion velocity depends linearly on the drive. For +xdirection driving, the skyrmions are pushed against the lower divots and become trapped, giving reduced velocities and a nonlinear velocity-force response. When there are shallow divots on the upper wall and deep divots on the lower wall, skyrmions get trapped for both driving directions; however, due to the divot depth difference, skyrmions move more easily under -xdirection driving, and become strongly trapped for +xdirection driving. The preferred -xdirection motion produces what we call a Magnus diode effect since it vanishes in the limit of zero Magnus force, unlike the diode effects observed for asymmetric sawtooth potentials. We show that the transport curves can exhibit a series of jumps or dips, negative differential conductivity, and reentrant pinning due to collective trapping events. We also discuss how our results relate to recent continuum modeling on a similar skyrmion diode system.

Low-Energy Excitations of Skyrmion Crystals in a Centrosymmetric Kondo-Lattice Magnet: Decoupled Spin-Charge Excitations and Nonreciprocity

We theoretically study spin and charge excitations of skyrmion crystals stabilized by conduction-electron-mediated magnetic interactions via spin-charge coupling in a centrosymmetric Kondo-lattice model by large-scale spin-dynamics simulations combined with the kernel polynomial method. We reveal clear segregation of spin and charge excitation channels and nonreciprocal nature of the spin excitations governed by the Fermi-surface geometry, which are unique to the skyrmion crystals in centrosymmetric itinerant hosts and can be a source of novel physical phenomena.

Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2

Novel magnetic ground states have been stabilized in two-dimensional (2D) magnets such as skyrmions, with the potential next-generation information technology. Here, we report the experimental observation of a Néel-type skyrmion lattice at room temperature in a single-phase, layered 2D magnet, specifically a 50% Co-doped Fe₅GeTe₂ (FCGT) system. The thickness-dependent magnetic domain size follows Kittel's law. The static spin textures and spin dynamics in FCGT nanoflakes were studied by Lorentz electron microscopy, variable-temperature magnetic force microscopy, micromagnetic simulations, and magnetotransport measurements. Current-induced skyrmion lattice motion was observed at room temperature, with a threshold current density, j_th = 1 × 10⁶ A/cm². This discovery of a skyrmion lattice at room temperature in a noncentrosymmetric material opens the way for layered device applications and provides an ideal platform for studies of topological and quantum effects in 2D.

Electrical manipulation of skyrmions in a chiral magnet

Writing, erasing and computing are three fundamental operations required by any working electronic device. Magnetic skyrmions could be essential bits in promising in emerging topological spintronic devices. In particular, skyrmions in chiral magnets have outstanding properties like compact texture, uniform size, and high mobility. However, creating, deleting, and driving isolated skyrmions, as prototypes of aforementioned basic operations, have been a grand challenge in chiral magnets ever since the discovery of skyrmions, and achieving all these three operations in a single device is even more challenging. Here, by engineering chiral magnet Co₈Zn₁₀Mn₂ into the customized micro-devices for in-situ Lorentz transmission electron microscopy observations, we implement these three operations of skyrmions using nanosecond current pulses with a low current density of about 10¹⁰ A·m⁻² at room temperature. A notched structure can create or delete magnetic skyrmions depending on the direction and magnitude of current pulses. We further show that the magnetic skyrmions can be deterministically shifted step-by-step by current pulses, allowing the establishment of the universal current-velocity relationship. These experimental results have immediate significance towards the skyrmion-based memory or logic devices.

There has been much interest in using skyrmions for new approaches to compution, however, creating, deleting and driving skyrmions remains a challenge. Here, Wang et al demonstrate all three operations for skyrmions in tailored Co8Zn10Mn2 nanodevices using tailored current pulses.

Behavior of Vortex-Like Inhomogeneities Originating in Magnetic Films with Modulated Uniaxial Anisotropy in a Planar Magnetic Field

This paper investigates the processes of magnetization reversal of a uniaxial ferromagnetic disk containing a columnar defect of the potential well type in perpendicular and planar magnetic fields. The characteristic stages of magnetization reversal of the domain structure of the disk and vortex-like inhomogeneities forming on the defect are determined. The critical fields of their existence are found and an explanation is given for the presence of a significant difference in their values for the perpendicular and planar fields of the defect magnetization reversal. The role of chirality in the behavior of a Bloch-type magnetic skyrmion during the magnetization reversal of a defect in a planar field is shown.

Controlled Switching of the Number of Skyrmions in a Magnetic Nanodot by Electric Fields

Magnetic skyrmions are topological swirling spin configurations that hold promise for building future magnetic memories and logic circuits. Skyrmionic devices typically rely on the electrical manipulation of a single skyrmion, but controllably manipulating a group of skyrmions can lead to more compact and memory-efficient devices. Here, an electric-field-driven cascading transition of skyrmion clusters in a nanostructured ferromagnetic/ferroelectric multiferroic heterostructure is reported, which allows a continuous multilevel transition of the number of skyrmions in a one-by-one manner. Most notably, the transition is non-volatile and reversible, which is crucial for multi-bit memory applications. Combined experiments and theoretical simulations reveal that the switching of skyrmion clusters is induced by the strain-mediated modification of both the interfacial Dzyaloshinskii-Moriya interaction and effective uniaxial anisotropy. The results not only open up a new direction for constructing low-power-consuming, non-volatile, and multi-bit skyrmionic devices, but also offer valuable insights into the fundamental physics underlying the voltage manipulation of skyrmion clusters.

Unveiling the three-dimensional magnetic texture of skyrmion tubes

Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. Their nanoscopic size and the low electric currents required to control their motion has opened a new field of research, skyrmionics, that aims for the usage of skyrmions as information carriers. Further advances in skyrmionics call for a thorough understanding of their three-dimensional (3D) spin texture, skyrmion-skyrmion interactions and the coupling to surfaces and interfaces, which crucially affect skyrmion stability and mobility. Here, we quantitatively reconstruct the 3D magnetic texture of Bloch skyrmions with sub-10-nanometre resolution using holographic vector-field electron tomography. The reconstructed textures reveal local deviations from a homogeneous Bloch character within the skyrmion tubes, details of the collapse of the skyrmion texture at surfaces and a correlated modulation of the skyrmion tubes in FeGe along their tube axes. Additionally, we confirm the fundamental principles of skyrmion formation through an evaluation of the 3D magnetic energy density across these magnetic solitons.

Intrinsic topological magnons in arrays of magnetic dipoles

We study a simple magnetic system composed of periodically modulated magnetic dipoles with an easy axis. Upon adjusting the geometric modulation amplitude alone, chains and two-dimensional stacked chains exhibit a rich magnon spectrum where frequency gaps and magnon speeds are easily manipulable. The blend of anisotropy due to dipolar interactions between magnets and geometrical modulation induces a magnetic phase with fractional Zak number in infinite chains and end states in open one-dimensional systems. In two dimensions it gives rise to topological modes at the edges of stripes. Tuning the amplitude in two-dimensional lattices causes a band touching, which triggers the exchange of the Chern numbers of the volume bands and switches the sign of the thermal conductivity.

Magnetic skyrmion bundles and their current-driven dynamics

Topological charge Q classifies non-trivial spin textures and determines many of their characteristics. Most abundant are topological textures with |Q| ≤ 1, such as (anti)skyrmions, (anti)merons or (anti)vortices. In this study we created and imaged in real space magnetic skyrmion bundles, that is, multi-Q three-dimensional skyrmionic textures. These textures consist of a circular spin spiral that ties together a discrete number of skyrmion tubes. We observed skyrmion bundles with integer Q values up to 55. We show here that electric currents drive the collective motion of these particle-like textures similar to skyrmions. Bundles with Q ≠ 0 exhibit a skyrmion Hall effect with a Hall angle of ~62°, whereas Q = 0 bundles, the so-called skyrmioniums, propagate collinearly with respect to the current flow, that is, with a skyrmion Hall angle of ~0°. The experimental observation of multi-Q spin textures adds another member to the family of magnetic topological textures, which may serve in future spintronic devices.

Current-Controlled Topological Magnetic Transformations in a Nanostructured Kagome Magnet

Topological magnetic charge Q is a fundamental parameter that describes the magnetic domains and determines their intriguing electromagnetic properties. The ability to switch Q in a controlled way by electrical methods allows for flexible manipulation of electromagnetic behavior in future spintronic devices. Here, the room-temperature current-controlled topological magnetic transformations between Q = -1 skyrmions and Q = 0 stripes or type-II bubbles in a kagome crystal Fe₃ Sn₂ are reported. It is shown that reproducible and reversible skyrmion-bubble and skyrmion-stripe transformations can be achieved by tuning the density of nanosecond pulsed current of the order of ≈10¹⁰ A m^-2 . Further numerical simulations suggest that spin-transfer torque combined with Joule thermal heating effects determine the current-induced topological magnetic transformations.

Transcription and logic operations of magnetic skyrmions in bilayer cross structures

Magnetic skyrmions are potential building blocks for future information storage and computing devices. Here, we computationally study the skyrmion dynamics in a cross structure made of two ferromagnetic nanotracks. We show that by controlling the skyrmion motion in the cross structure using spin currents, it is possible to realize the transcription of skyrmion at the intersection of the cross structure at certain conditions. Based on the transcription of skyrmion, we computationally demonstrate the AND, OR and NOT logical gates using the cross structures with modified geometries and appropriate magnetic parameters. Our results may provide guidelines to design future three-dimensional spintronics devices based on magnetic skyrmions.

Motion-induced inertial effects and topological phase transitions in skyrmion transport

When the skyrmion dynamics beyond the particle-like description is considered, this topological structure can deform due to a self-induced field. In this work, we perform Monte Carlo simulations to characterize the skyrmion deformation during its steady movement. In the low-velocity regime, the deformation in the skyrmion shape is quantified by an effective inertial mass, which is related to the dissipative force. When skyrmions move faster, the large self-induced deformation triggers topological transitions. These transitions are characterized by the proliferation of skyrmions and a different total topological charge, which is obtained as a function of the skyrmion velocity. Our findings provide an alternative way to describe the dynamics of a skyrmion that accounts for the deformations of its structure. Furthermore, such motion-induced topological phase transitions make it possible to control the number of ferromagnetic skyrmions through velocity effects.

Skyrmion crystals in centrosymmetric itinerant magnets without horizontal mirror plane

We theoretically investigate a new stabilization mechanism of a skyrmion crystal (SkX) in centrosymmetric itinerant magnets with magnetic anisotropy. By considering a trigonal crystal system without the horizontal mirror plane, we derive an effective spin model with an anisotropic Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction for a multi-band periodic Anderson model. We find that the anisotropic RKKY interaction gives rise to two distinct SkXs with different skyrmion numbers of one and two depending on a magnetic field. We also clarify that a phase arising from the multiple-Q spin density waves becomes a control parameter for a field-induced topological phase transition between the SkXs. The mechanism will be useful not only for understanding the SkXs, such as that in Gd[Formula: see text]PdSi[Formula: see text], but also for exploring further skyrmion-hosting materials in trigonal itinerant magnets.

Confinement and Protection of Skyrmions by Patterns of Modified Magnetic Properties

Magnetic skyrmions are versatile topological excitations that can be used as nonvolatile information carriers. The confinement of skyrmions in channels is fundamental for any application based on the accumulation and transport of skyrmions. Here, we report a method that allows effective position control of skyrmions in designed channels by engineered energy barriers and wells, which is realized in a magnetic multilayer film by harnessing the boundaries of patterns with modified magnetic properties. We experimentally and computationally demonstrate that skyrmions can be attracted or repelled by the boundaries of areas with modified perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. By fabricating square and stripe patterns with modified magnetic properties, we show the possibility of building reliable channels for confinement, accumulation, and transport of skyrmions, which effectively protect skyrmions from being destroyed at the device edges. Our results are useful for the design of spintronic applications using either static or dynamic skyrmions.

Deriving the skyrmion Hall angle from skyrmion lattice dynamics

Magnetic skyrmions are topologically non-trivial, swirling magnetization textures that form lattices in helimagnetic materials. These magnetic nanoparticles show promise as high efficiency next-generation information carriers, with dynamics that are governed by their topology. Among the many unusual properties of skyrmions is the tendency of their direction of motion to deviate from that of a driving force; the angle by which they diverge is a materials constant, known as the skyrmion Hall angle. In magnetic multilayer systems, where skyrmions often appear individually, not arranging themselves in a lattice, this deflection angle can be easily measured by tracing the real space motion of individual skyrmions. Here we describe a reciprocal space technique which can be used to determine the skyrmion Hall angle in the skyrmion lattice state, leveraging the properties of the skyrmion lattice under a shear drive. We demonstrate this procedure to yield a quantitative measurement of the skyrmion Hall angle in the room-temperature skyrmion system FeGe, shearing the skyrmion lattice with the magnetic field gradient generated by a single turn Oersted wire.

Skyrmions, when driven by any applied force, experience an addition sideways motion known as the skyrmion hall effect. Here, Brearton et al. present a reciprocal space method for determining the strength of the skyrmion hall effect, making measurement possible for skyrmion lattices.

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.

Skyrmion ratchet propagation: utilizing the skyrmion Hall effect in AC racetrack storage devices

Magnetic skyrmions are whirl-like nano-objects with topological protection. When driven by direct currents, skyrmions move but experience a transverse deflection. This so-called skyrmion Hall effect is often regarded a drawback for memory applications. Herein, we show that this unique effect can also be favorable for spintronic applications: We show that in a racetrack with a broken inversion symmetry, the skyrmion Hall effect allows to translate an alternating current into a directed motion along the track, like in a ratchet. We analyze several modes of the ratchet mechanism and show that it is unique for topological magnetic whirls. We elaborate on the fundamental differences compared to the motion of topologically trivial magnetic objects, as well as classical particles driven by periodic forces. Depending on the exact racetrack geometry, the ratchet mechanism can be soft or strict. In the latter case, the skyrmion propagates close to the efficiency maximum.

Recent Progress on Topological Structures in Ferroic Thin Films and Heterostructures

Topological spin/polarization structures in ferroic materials continue to draw great attention as a result of their fascinating physical behaviors and promising applications in the field of high-density nonvolatile memories as well as future energy-efficient nanoelectronic and spintronic devices. Such developments have been made, in part, based on recent advances in theoretical calculations, the synthesis of high-quality thin films, and the characterization of their emergent phenomena and exotic phases. Herein, progress over the last decade in the study of topological structures in ferroic thin films and heterostructures is explored, including the observation of topological structures and control of their structures and emergent physical phenomena through epitaxial strain, layer thickness, electric, magnetic fields, etc. First, the evolution of topological spin structures (e.g., magnetic skyrmions) and associated functionalities (e.g., topological Hall effect) in magnetic thin films and heterostructures is discussed. Then, the exotic polar topologies (e.g., domain walls, closure domains, polar vortices, bubble domains, and polar skyrmions) and their emergent physical properties in ferroelectric oxide films and heterostructures are explored. Finally, a brief overview and prospectus of how the field may evolve in the coming years is provided.

Chiral-Bubble-Induced Topological Hall Effect in Ferromagnetic Topological Insulator Heterostructures

We report compelling evidence of an emergent topological Hall effect (THE) from chiral bubbles in a two-dimensional uniaxial ferromagnet, V-doped Sb₂Te₃ heterostructure. The sign of THE signal is determined by the net curvature of domain walls in different domain configurations, and the strength of THE signal is correlated with the density of nucleation or pinned bubble domains. The experimental results are in good agreement with the integrated linear transport and Monte Carlo simulations, corroborating the emergent gauge field at chiral magnetic bubbles. Our findings not only reveal a general mechanism of THE in two-dimensional ferromagnets but also pave the way for the creation and manipulation of topological spin textures for spintronic applications.

Emergent Topological Hall Effect at a Charge-Transfer Interface

Exploring exotic interface magnetism due to charge transfer and strong spin-orbit coupling has profound application in the future development of spintronic memory. Here, the emergence and tuning of topological Hall effect (THE) from a CaMnO₃ /CaIrO₃ /CaMnO₃ trilayer structure are studied in detail, which suggests the presence of magnetic Skyrmion-like bubbles. First, by tilting the magnetic field direction, the evolution of the Hall signal suggests a transformation of Skyrmions into topologically-trivial stripe domains, consistent with behaviors predicted by micromagnetic simulations. Second, by varying the thickness of CaMnO₃ , the optimal thicknesses for the THE signal emergence are found, which allow identification of the source of Dzyaloshinskii-Moriya interaction (DMI) and its competition with antiferromagnetic superexchange. Employing high-resolution transmission electron microscopy, randomly distributed stacking faults are identified only at the bottom interface and may avoid mutual cancellation of DMI. Last, a spin-transfer torque experiment also reveals a low threshold current density of ≈10⁹ A m^-2 for initiating the bubbles' motion. This discovery sheds light on a possible strategy for integrating Skyrmions with antiferromagnetic spintronics.

Controlling bimerons as skyrmion analogues by ferroelectric polarization in 2D van der Waals multiferroic heterostructures

Atom-thick van der Waals heterostructures with nontrivial physical properties tunable via the magnetoelectric coupling effect are highly desirable for the future advance of multiferroic devices. In this work on LaCl/In₂Se₃ heterostructure consisting of a 2D ferromagnetic layer and a 2D ferroelectric layer, reversible switch of the easy axis and the Curie temperature of the magnetic LaCl layer has been enabled by switching of ferroelectric polarization in In₂Se₃. More importantly, magnetic skyrmions in the bimerons form have been discovered in the LaCl/In₂Se₃ heterostructure and can be driven by an electric current. The creation and annihilation of bimerons in LaCl magnetic nanodisks were achieved by polarization switching. It thus proves to be a feasible approach to achieve purely electric control of skyrmions in 2D van der Waals heterostructures. Such nonvolatile and tunable magnetic skyrmions are promising candidates for information carriers in future data storage and logic devices operated under small electrical currents.

Bulk and surface topological indices for a skyrmion string: current-driven dynamics of skyrmion string in stepped samples

The magnetic skyrmion is a topological magnetic vortex, and its topological nature is characterized by an index called skyrmion number which is a mapping of the magnetic moments defined on a two-dimensional space to a unit sphere. In three-dimensions, a skyrmion, i.e., a vortex penetrating though the magnet naturally forms a string, which terminates at the surfaces of the magnet or in the bulk. For such a string, the topological indices, which control its topological stability are less trivial. Here, we study theoretically, in terms of numerical simulation, the dynamics of current-driven motion of a skyrmion string in a film sample with the step edges on the surface. In particular, skyrmion–antiskyrmion pair is generated by driving a skyrmion string through the side step with an enough height. We find that the topological indices relevant to the stability are the followings; (1) skyrmion number along the developed surface, and (2) the monopole charge in the bulk defined as the integral over the surface enclosing a singular magnetic configuration. As long as the magnetic configuration is slowly varying, the former is conserved while its changes is associated with nonzero monopole charge. The skyrmion number and the monoplole charge offer a coherent understanding of the stability of the topological magnetic texture and the nontrivial dynamics of skyrmion strings.

Spin waves in skyrmionic structures with various topological charges

Equilibrium properties and localized magnon excitations are investigated in topologically distinct skyrmionic textures. The observed shape of the structures and their orientation on the lattice is explained based on their vorticities and the symmetry of the crystal. The transformation between different textures and their annihilation as a function of magnetic field is understood based on the energy differences between them. The angular momentum spin-wave eigenmodes characteristic of cylindrically symmetric structures are combined in the distorted spin configurations, leading to avoided crossings in the magnon spectrum. The susceptibility of the skyrmionic textures to homogeneous external fields is calculated, revealing that a high number of modes become detectable due to the hybridization between the angular momentum eigenmodes. These findings should contribute to the observation of spin waves in distorted skyrmionic structures via experiments and numerical simulations, widening the range of their possible applications in magnonic devices.

Electron Beam Lithography of Magnetic Skyrmions

The emergence of magnetic skyrmions, topological spin textures, has aroused tremendous interest in studying the rich physics related to their topology. While skyrmions promise high-density and energy-efficient magnetic memory devices for information technology, the manifestation of their nontrivial topology through single skyrmions and ordered and disordered skyrmion lattices could also give rise to many fascinating physical phenomena, such as chiral magnon and skyrmion glass states. Therefore, generating skyrmions at designated locations on a large scale, while controlling the skyrmion patterns, is the key to advancing topological magnetism. Here, a new, yet general, approach to the "printing" of skyrmions with zero-field stability in arbitrary patterns on a massive scale in exchange-biased magnetic multilayers is presented. By exploiting the fact that the antiferromagnetic order can be reconfigured by local thermal excitations, a focused electron beam with a graphic pattern generator to "print" skyrmions is used, which is referred to as skyrmion lithography. This work provides a route to design arbitrary skyrmion patterns, thereby establishing the foundation for further exploration of topological magnetism.

Electric-field-driven non-volatile multi-state switching of individual skyrmions in a multiferroic heterostructure

Electrical manipulation of skyrmions attracts considerable attention for its rich physics and promising applications. To date, such a manipulation is realized mainly via spin-polarized current based on spin-transfer torque or spin-orbital torque effect. However, this scheme is energy consuming and may produce massive Joule heating. To reduce energy dissipation and risk of heightened temperatures of skyrmion-based devices, an effective solution is to use electric field instead of current as stimulus. Here, we realize an electric-field manipulation of skyrmions in a nanostructured ferromagnetic/ferroelectrical heterostructure at room temperature via an inverse magneto-mechanical effect. Intriguingly, such a manipulation is non-volatile and exhibits a multistate feature. Numerical simulations indicate that the electric-field manipulation of skyrmions originates from strain-mediated modification of effective magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Our results open a direction for constructing low-energy-dissipation, non-volatile, and multistate skyrmion-based spintronic devices.

Topology-Dependent Brownian Gyromotion of a Single Skyrmion

Noninteracting particles exhibiting Brownian motion have been observed in many occasions of sciences, such as molecules suspended in liquids, optically trapped microbeads, and spin textures in magnetic materials. In particular, a detailed examination of Brownian motion of spin textures is important for designing thermally stable spintronic devices, which motivates the present study. In this Letter, through using temporally and spatially resolved polar magneto-optic Kerr effect microscopy, we have experimentally observed the thermal fluctuation-induced random walk of a single isolated Néel-type magnetic skyrmion in an interfacially asymmetric Ta/CoFeB/TaO_{x} multilayer. An intriguing topology-dependent Brownian gyromotion behavior of skyrmions has been identified. The onset of Brownian gyromotion of a single skyrmion induced by thermal effects, including a nonlinear temperature-dependent diffusion coefficient and topology-dependent gyromotion are further formulated based on the stochastic Thiele equation. The experimental and numerical demonstration of topology-dependent Brownian gyromotion of skyrmions can be useful for understanding the nonequilibrium magnetization dynamics and implementing spintronic devices.

Room Temperature Skyrmions in Strain-Engineered FeGe thin films

Skyrmions hold great promise for low-energy consumption and stable high density information storage, and stabilization of the skyrmion lattice (SkX) phase at or above room temperature is greatly desired for practical use. The topological Hall effect can be used to identify candidate systems above room temperature, a challenging regime for direct observation by Lorentz electron microscopy. Atomically ordered FeGe thin films are grown epitaxially on Ge(111) substrates with ~ 4 % tensile strain. Magnetic characterization reveals enhancement of Curie temperature to 350 K due to strain, well above the bulk value of 278 K. Strong topological Hall effect was observed between 10 K and 330 K, with a significant increase in magnitude observed at 330 K. The increase in magnitude occurs just below the Curie temperature, a similar relative temperature position as the onset of Skx phase in bulk FeGe. The results suggest that strained FeGe films may host a SkX phase above room temperature when significant tensile strain is applied.

Skyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applications

The field of magnetic skyrmions has been actively investigated across a wide range of topics during the last decades. In this topical review, we mainly review and discuss key results and findings in skyrmion research since the first experimental observation of magnetic skyrmions in 2009. We particularly focus on the theoretical, computational and experimental findings and advances that are directly relevant to the spintronic applications based on magnetic skyrmions, i.e. their writing, deleting, reading and processing driven by magnetic field, electric current and thermal energy. We then review several potential applications including information storage, logic computing gates and non-conventional devices such as neuromorphic computing devices. Finally, we discuss possible future research directions on magnetic skyrmions, which also cover rich topics on other topological textures such as antiskyrmions and bimerons in antiferromagnets and frustrated magnets.

Dynamic tailoring of an optical skyrmion lattice in surface plasmon polaritons

A skyrmion is a topologically protected soliton with a spin structure on the micro/nano scale that has promising applications in magnetic information storage and spintronics devices. This study focuses on the optical skyrmion lattice structures created in the surface plasmon polaritons (SPPs) field. Both the Néel-type optical skyrmion lattice formed by the electric field vector and Bloch-type optical skyrmion lattice formed by the magnetic field vector are generated via exciting a hexagonal grating structure on the metal surface with six Gaussian optical spots. Such a multiple-spot excitation can be realized through tightly focusing a specially designed complex field with a high NA lens. Through introducing the phase difference of the excitation beams to shift the SPP standing waves, the shape and position of the optical skyrmion lattice can be dynamically controlled. Both the electric field vector and magnetic field vector are evaluated quantitatively based on the electric and magnetic field obtained by finite difference time domain (FDTD) simulation to demonstrate the validity and capability of the proposed technique.

Three-Dimensional Dynamics of a Magnetic Hopfion Driven by Spin Transfer Torque

Magnetic hopfion is a three-dimensional (3D) topological soliton with novel spin structure that would enable exotic dynamics. Here, we study the current-driven 3D dynamics of a magnetic hopfion with a unit Hopf index in a frustrated magnet. Attributed to the spin Berry phase and symmetry of the hopfion, the phase space entangles multiple collective coordinates, thus the hopfion exhibits rich dynamics including longitudinal motion along the current direction, transverse motion perpendicular to the current direction, rotational motion, and dilation. Furthermore, the characteristics of hopfion dynamics is determined by the ratio between the nonadiabatic spin transfer torque parameter and the damping parameter. Such peculiar 3D dynamics of magnetic hopfion could shed light on understanding the universal physics of hopfions in different systems and boost the prosperous development of 3D spintronics.

Inhomogeneous-strain-induced magnetic vortex cluster in one-dimensional manganite wire

Mixed-valance manganites with strong electron correlation exhibit strong potential for spintronics, where emergent magnetic behaviors, such as propagation of high-frequency spin waves and giant topological Hall Effects can be driven by their mesoscale spin textures. Here, we create magnetic vortex clusters with flux closure spin configurations in single-crystal La_0.67Sr_0.33MnO₃ wire. A distinctive transformation from out-of-plane domains to a vortex state is directly visualized using magnetic force microscopy at 4 K in wires when the width is below 1.0 μm. The phase-field modeling indicates that the inhomogeneous strain, accompanying with shape anisotropy, plays a key role for stabilizing the flux-closure spin structure. This work offers a new perspective for understanding and manipulating the non-trivial spin textures in strongly correlated systems.

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.

Diameter-independent skyrmion Hall angle observed in chiral magnetic multilayers

Magnetic skyrmions are topologically non-trivial nanoscale objects. Their topology, which originates in their chiral domain wall winding, governs their unique response to a motion-inducing force. When subjected to an electrical current, the chiral winding of the spin texture leads to a deflection of the skyrmion trajectory, characterised by an angle with respect to the applied force direction. This skyrmion Hall angle is predicted to be skyrmion diameter-dependent. In contrast, our experimental study finds that the skyrmion Hall angle is diameter-independent for skyrmions with diameters ranging from 35 to 825 nm. At an average velocity of 6 ± 1 ms⁻¹, the average skyrmion Hall angle was measured to be 9° ± 2°. In fact, the skyrmion dynamics is dominated by the local energy landscape such as materials defects and the local magnetic configuration.

Magnetic skyrmions are promising objects for future spintronic devices. However, a better understanding of their dynamics is required. Here, the authors show that in contrast to predictions the skyrmion Hall angle is independent of their diameter and motion is dominated by disorder and skyrmion-skyrmion interactions in the system.