Skyrmion Systems Research Articles

Solid–liquid transition in a skyrmion matter

We report Monte-Carlo studies of the orientational order and melting of a 2D skyrmion lattice containing more than one million spins. Two models have been investigated, a microscopic model of lattice spins with Dzyaloshinskii–Moryia interaction that possesses skyrmions, and the model in which skyrmions are treated as point particles with repulsive interaction derived from a spin model. They produce similar results. The skyrmion lattice exhibits a sharp one-step transition between solid and liquid phases on temperature and the magnetic field. This solid–liquid transition is characterized by the kink in the magnetization. The field-temperature phase diagram is computed. We show that the application of the field gradient to a 2D system of skyrmions produces a solid–liquid interface that must be possible to observe in experiments.

Observation of Topological Hall Effect in a Chemically Complex Alloy.

The topological Hall effect (THE) is the transport response of chiral spin textures and thus can serve as a powerful probe for detecting and understanding these unconventional magnetic orders. So far, the THE is only observed in either noncentrosymmetric systems where spin chirality is stabilized by Dzyaloshinskii-Moriya interactions, or triangular-lattice magnets with Ruderman-Kittel-Kasuya-Yosida-type interactions. Here, a pronounced THE is observed in a Fe-Co-Ni-Mn chemically complex alloy with a simple face-centered cubic (fcc) structure across a wide range of temperatures and magnetic fields. The alloy is shown to have a strong magnetic frustration owing to the random occupation of magnetic atoms on the close-packed fcc lattice and the direct Heisenberg exchange interaction among atoms, as evidenced by the appearance of a reentrant spin glass state in the low-temperature regime and the first principles calculations. Consequently, THE is attributed to the nonvanishing spin chirality created by strong spin frustration under the external magnetic field, which is distinct from the mechanism responsible for the skyrmion systems, as well as geometrically frustrated magnets.

Giant 2D Skyrmion Topological Hall Effect with Ultrawide Temperature Window and Low-Current Manipulation in 2D Room-Temperature Ferromagnetic Crystals

The discovery and manipulation of topological Hall effect (THE), an abnormal magnetoelectric response mostly related to the Dzyaloshinskii–Moriya interaction (DMI), are promising for next-generation spintronic devices based on topological spin textures such as magnetic skyrmions. However, most skyrmions and THE are stabilized in a narrow temperature window either below or over room temperature with high critical current manipulation. It is still elusive and challenging to achieve large THE with both wide temperature window till room temperature and low critical current manipulation. Here, using controllable, naturally oxidized sub-20 and sub-10 nm 2D van der Waals room-temperature ferromagnetic Fe3GaTe2−x crystals, we report robust 2D skyrmion THE with ultrawide temperature window ranging in three orders of magnitude from 2 to 300 K, in combination with giant THE of ∼ 5.4 μΩ⋅cm at 10 K and ∼ 0.15 μΩ⋅cm at 300 K, which is 1–3 orders of magnitude larger than that of all known room-temperature 2D skyrmion systems. Moreover, room-temperature current-controlled THE is also realized with a low critical current density of ∼ 6.2 × 105 A⋅cm−2. First-principles calculations unveil natural oxidation-induced highly enhanced 2D interfacial DMI reasonable for robust giant THE. This work paves the way to room-temperature electrically controlled 2D THE-based practical spintronic devices.

Perspective on unconventional computing using magnetic skyrmions

Learning and pattern recognition inevitably requires memory of previous events, a feature that conventional CMOS hardware needs to artificially simulate. Dynamical systems naturally provide the memory, complexity, and nonlinearity needed for a plethora of different unconventional computing approaches. In this perspective article, we focus on the unconventional computing concept of reservoir computing and provide an overview of key physical reservoir works reported. We focus on the promising platform of magnetic structures and, in particular, skyrmions, which potentially allow for low-power applications. Moreover, we discuss skyrmion-based implementations of Brownian computing, which has recently been combined with reservoir computing. This computing paradigm leverages the thermal fluctuations present in many skyrmion systems. Finally, we provide an outlook on the most important challenges in this field.

Suppression of skyrmion Hall effect via standing surface acoustic waves in hybrid ferroelectric/ferromagnetic heterostructures

Magnetic skyrmion is a promising information carrier for its low critical driven current density, topological stability, and small size, which has been proposed for various devices such as racetrack memory and logic gates. However, the skyrmion Hall effect originating from Magnus force leads to transverse motion, which hinders the development of skyrmionic device applications. Here, we propose artificial tracks built by standing surface acoustic waves (SSAWs) to suppress the skyrmion Hall effect through micromagnetic simulations. We systematically study the dynamics of an isolated skyrmion under SSAWs and driven currents in a prototype of the ferromagnetic skyrmion system. The skyrmion Hall angle changes from 80° to 0°, where the skyrmion motion is along the driven current. An analytical model considering magnetoelastic energy induced by SSAWs is developed, and a linear relation between the current density and the critical SSAW amplitude to eliminate the skyrmion Hall effect is achieved. Furthermore, a reconfigurable multichannel skyrmion racetrack is constructed through the change of SSAW wavelengths. Our work opens a feasible route for the suppression of skyrmion Hall effect via SSAWs.

Управление микромагнитной структурой многофазных тонких пленок CoPt путем варьирования толщин слоев

The magnetic properties and micromagnetic structure of [Co/Pt]10 multilayer magnetic films formed by independent variation of the Co and Pt layer thicknesses have been studied. The possibility of the film magnetization parameters manipulation was shown. It is found that the micromagnetic structure of the layers is significantly modified with a change in the thickness of the layers which correlates with the magnetization data. In particular, magnetic force microscopy revealed a system of magnetic skyrmions for a number of structures. The skyrmion density was found to be dependent on the growth conditions which in turn correlates with the shape of the magnetic hysteresis loop. Changing the thickness of the Co and Pt layers makes it possible to control the density of skyrmions in the range from 0.2 to 10.5 µm-2.

Manipulating the Micromagnetic Structure of Multiphase CoPt Thin Films by Varying Layer Thicknesses

The magnetic properties and micromagnetic structure of [Co/Pt]10 multilayer magnetic films formed by independent variation of the Co and Pt layer thicknesses have been studied. The possibility of the film magnetization parameters manipulation was shown. It is found that the micromagnetic structure of the layers is significantly modified with a change in the thickness of the layers which correlates with the magnetization data. In particular, magnetic force microscopy revealed a system of magnetic skyrmions for a number of structures. The skyrmion density was found to be dependent on the growth conditions which in turn correlates with the shape of the magnetic hysteresis loop. Changing the thickness of the Co and Pt layers makes it possible to control the density of skyrmions in the range from 0.2 to 10.5 mkm-2. Keywords: micromagnetic structure, skyrmions, CoPt films, electron beam evaporation.

Pattern recognition with neuromorphic computing using magnetic field-induced dynamics of skyrmions.

Nonlinear phenomena in physical systems can be used for brain-inspired computing with low energy consumption. Response from the dynamics of a topological spin structure called skyrmion is one of the candidates for such a neuromorphic computing. However, its ability has not been well explored experimentally. Here, we experimentally demonstrate neuromorphic computing using nonlinear response originating from magnetic field–induced dynamics of skyrmions. We designed a simple-structured skyrmion-based neuromorphic device and succeeded in handwritten digit recognition with the accuracy as large as 94.7% and waveform recognition. Notably, there exists a positive correlation between the recognition accuracy and the number of skyrmions in the devices. The large degrees of freedom of skyrmion systems, such as the position and the size, originate from the more complex nonlinear mapping, the larger output dimension, and, thus, high accuracy. Our results provide a guideline for developing energy-saving and high-performance skyrmion neuromorphic computing devices.

N-band photonic Hopf insulators based on 2D microring lattices.

Hopf insulators are topological insulators whose topological behavior arises from the nontrivial mapping from a 3D sphere to a 2D sphere, known as the Hopf map. The Hopf map, typically encountered in the study of spinor and Skyrmion systems, is classified topologically by an integer invariant called the Hopf index. Here we show that, owing to the periodic circulation of light inside each microring, a 2D lattice of microring resonators can emulate an N-band photonic Hopf insulator with nontrivial Hopf index. In particular, we show by numerical computation and direct analytical proof that the N-band Hopf index of the microring lattice is identical to its winding number. The result shows that the Hopf index is an alternative topological invariant for classifying 2D microring photonic lattices and establishes a correspondence between the Hopf insulator phase and the anomalous Floquet insulator phase of the lattice. More generally, our work shows that 2D microring lattices can provide a versatile nanophotonic platform for studying non-Abelian topological photonic systems.

Statics and dynamics of skyrmions interacting with disorder and nanostructures

Magnetic skyrmions are topologically stable nanoscale particle-like objects that were discovered in 2009. Since that time, intense research interest has led to the identification of numerous compounds that support skyrmions over a range of conditions spanning cryogenic to room temperatures. Skyrmions can be set into motion under various types of driving, and the combination of their size, stability, and dynamics makes them ideal candidates for numerous applications. Skyrmions represent a new class of system in which the energy scales of the skyrmion-skyrmion interactions, sample disorder, temperature, and drive can compete. A growing body of work indicates that the static and dynamic states of skyrmions can be influenced strongly by pinning or disorder in the sample; thus, an understanding of such effects is essential for the eventual use of skyrmions in applications. In this article we review the current state of knowledge regarding individual skyrmions and skyrmion assemblies interacting with quenched disorder or pinning. We outline the microscopic mechanisms for skyrmion pinning, including the repulsive and attractive interactions that can arise from impurities, grain boundaries, or nanostructures. This is followed by descriptions of depinning phenomena, sliding states over disorder, the effect of pinning on the skyrmion Hall angle, the competition between thermal and pinning effects, the control of skyrmion motion using ordered potential landscapes such as one- or two-dimensional periodic asymmetric substrates, the creation of skyrmion diodes, and skyrmion ratchet effects. We highlight the distinctions arising from internal modes and the strong gyroscopic or Magnus forces that cause the dynamical states of skyrmions to differ from those of other systems with pinning. We also discuss future directions and open questions related to the pinning and dynamics in skyrmion systems.

Dynamic phases and reentrant Hall effect for vortices and skyrmions on periodic pinning arrays

We consider the motion of superconducting vortices and skyrmions on a square substrate near the first commensurate matching field. Slightly above commensuration, a series of dynamic phases appears including interstitial flow, and there is a transition from fluid flow to soliton flow that generates negative differential conductivity. Slightly below commensuration, vacancy depinning occurs. The dynamic phase transitions produce features in the velocity–force curves, differential mobility, and velocity fluctuations. In this work we have gone to much longer simulation times than in previous work, allowing us to examine the differential conductivity curves over a larger range of vortex and pinning densities. When a Magnus force is also present, as in certain superconducting vortex or skyrmion systems, there is an expansion of the fluid state, and at lower drives there is a finite Hall angle in the fluid phase but a vanishing Hall angle in the soliton phase, giving rise to a reentrant Hall effect. We also find a regime where the Hall motion of the particles exhibits the same dynamic phases, including soliton motion at a finite angle that produces negative differential conductivity in the Hall response but not in the longitudinal response. Our results suggest that vortices driven over periodic pinning may be an ideal system for determining if a vortex Hall effect is occurring and would also be relevant for skyrmions at smaller Magnus forces.

Machine learning techniques to construct detailed phase diagrams for skyrmion systems

Recently, there has been an increased interest in the application of machine learning (ML) techniques to a variety of problems in condensed matter physics. In this regard, of particular significance is the characterization of simple and complex phases of matter. Here, we use a ML approach to construct the full phase diagram of a well known spin model combining ferromagnetic exchange and Dzyaloshinskii-Moriya (DM) interactions where topological phases emerge. At low temperatures, the system is tuned from a spiral phase to a skyrmion crystal by a magnetic field. However, thermal fluctuations induce two types of intermediate phases, bimerons and skyrmion gas, which are not as easily determined as spirals or skyrmion crystals. We resort to large scale Monte Carlo simulations to obtain low temperature spin configurations, and train a convolutional neural network (CNN), taking only snapshots at specific values of the DM couplings, to classify between the different phases, focusing on the intermediate and intricate topological textures. We then apply the CNN to higher temperature configurations and to other DM values, to construct a detailed magnetic field-temperature phase diagram, achieving outstanding results. We discuss the importance of including the disordered paramagnetic phases in order to get the phase boundaries, and finally, we compare our approach with other ML algorithms.

Soliton motion in skyrmion chains: Stabilization and guidance by nanoengineered pinning

Using a particle-based model we examine the depinning motion of solitons in skyrmion chains in quasi-one-dimensional (1D) and two-dimensional (2D) systems containing embedded 1D interfaces. The solitons take the form of a particle or hole in a commensurate chain of skyrmions. Under an applied drive, just above a critical depinning threshold, the soliton moves with a skyrmion Hall angle of zero. For higher drives, the entire chain depins, and in a 2D system we observe that both the solitons and chain move at zero skyrmion Hall angle and then transition to a finite skyrmion Hall angle as the drive increases. In a 2D system with a 1D interface that is at an angle to the driving direction, there can be a reversal of the sign of the skyrmion Hall angle from positive to negative. Our results suggest that solitons in skyrmion systems could be used as information carriers in racetrack geometries that would avoid the drawbacks of finite skyrmion Hall angles. The soliton states become mobile at significantly lower drives than the depinning transition of the skyrmion chains themselves.

Skyrmions in a magnetic field and π0 domain wall formation in dense nuclear matter

We elucidate magnetic effects in the skyrmion system to probe into the dense nuclear matter. We find a deformed $\pi^0$ dipole structure of a baryon induced by a magnetic field. We then extend our scope to stacked Skyrme crystal layers to scrutinize phases of magnetized nuclear matter. We observe a quantized magnetic flux and identify a phase transition from a crystalline state to a $\pi^0$ domain wall corresponding to a topological transmutation from $\pi_3(S^3)$ to $\pi_1(S^1)$. We establish the phase diagram, which could be explored also in analogous systems with two-component Bose-Einstein condensates.

Effects of interlayer exchange on collapse mechanisms and stability of magnetic skyrmions

Theoretical calculations of thermally activated decay of skyrmions in systems comprising several magnetic monolayers are presented, with a special focus on bilayer systems. Mechanisms of skyrmion collapse are identified and corresponding energy barriers and thermal collapse rates are evaluated as functions of the interlayer exchange coupling and mutual stacking of the monolayers using transition state theory and an atomistic spin Hamiltonian. In order to contrast the results to monolayer systems, the magnetic interactions within each layer are chosen so as to mimic the well-established Pd/Fe/Ir(111) system. Even bilayer systems demonstrate a rich diversity of skyrmion collapse mechanisms that sometimes co-exist. For very weakly coupled layers, the skyrmions in each layer decay successively via radially-symmetric shrinking. Slightly larger coupling leads to an asymmetric chimera collapse stabilized by interlayer exchange. When the interlayer exchange coupling reaches a certain critical value, the skyrmions collapse simultaneously. Interestingly, the overall energy barrier for the skyrmion collapse does not always converge to a multiple of that for a monolayer system in the strongly coupled regime. For a certain stacking of the magnetic layers, the energy barrier as a function of the interlayer exchange coupling features a maximum and then decreases with the coupling strength in the strong coupling regime. Calculated mechanisms of skyrmion collapse are used to ultimately predict the skyrmion lifetime. Our results reveal a comprehensive picture of thermal stability of skyrmions in magnetic multilayers and provide a perspective for realizing skyrmions with controlled properties.

A theory of skyrmion crystal formation.

A generic theory about skyrmion crystal (SkX) formation in chiral magnetic thin films and its fascinating thermodynamic behaviours is presented. A chiral magnetic film can have many metastable states with an arbitrary skyrmion density up to a maximal value when the parameter κ, which measures the relative Dzyaloshinskii-Moriya interaction (DMI) strength, is large enough. The lowest energy state of an infinite film is a long zig-zag ramified stripe skyrmion occupying the whole film in the absence of a magnetic field. Under an intermediate field perpendicular to the film, the lowest energy state has a finite skyrmion density. This is why a chiral magnetic film is often in a stripy state at a low field and a SkX only around an optimal field when κ is above a critical value. The lowest energy state is still a stripy helical state no matter with or without a field when κ is below the critical value. The multi-metastable states explain the thermodynamic path dependences of the various metastable states of a film. The decrease of the κ value with the temperature explains why SkXs become metastable at low temperatures in many skyrmion systems. These findings open a new avenue for SkX manipulation and skyrmion-based applications.

Dynamics of flowing 2D skyrmions

We investigate, numerically, the effects of externally imposed material flows on the structure and temporal evolution of liquid crystal (LC) skyrmions. The dynamics of a 2D system of skyrmions is modeled using the Ericksen–Leslie theory, which is based on two coupled equations, one for material flow and the other for the director field. As the time scales of the velocity and director fields differ by several orders of magnitude for realistic values of the system parameters, we have simplified the calculations by assuming that the velocity relaxes instantaneously when compared to the relaxation of the director field. Thus, we have used a finite-differences method known as artificial compressibility with adaptive time step to solve the velocity field and a fourth-order Runge-Kutta method for the director field. We characterized the skyrmion shape or configuration as a function of the time and the average velocity of the flow field. We found that for velocities above a certain threshold, the skyrmions stretch in the direction perpendicular to the flow, by contrast to the regime of weak flows where the skyrmions stretch along the streamlines of the flow field. These two regimes are separated by an abrupt (first-order) dynamical transition, which is robust with respect to e.g., the LC elastic anisotropy. Additionally, we have found how the presence of a second skyrmion affects the evolution of the shape of the skyrmions, by comparing the evolution of pairs of skyrmions to the evolution of a single-skyrmion.

Theory of Magnetic-Texture-Induced Anomalous Hall Effect on the Surface of Topological Insulators

The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate the skyrmion-induced Hall conductivity and confirm that it depends on the shape of skyrmions, such as Bloch-type or N\'eel-type skyrmions. The results of this study can be applied not only to conventional skyrmion systems but also to more general magnetic structures.

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.

Megahertz dynamics in skyrmion systems probed with muon-spin relaxation

We present longitudinal-field muon-spin relaxation (LF $\mu$SR) measurements on two systems that stabilize a skyrmion lattice (SkL): Cu$_2$OSeO$_3$, and Co$_x$Zn$_y$Mn$_{20-x-y}$ for $(x,y)~=~(10,10)$, $(8,9)$ and $(8,8)$. We find that the SkL phase of Cu$_2$OSeO$_3$ exhibits emergent dynamic behavior at megahertz frequencies, likely due to collective excitations, allowing the SkL to be identified from the $\mu$SR response. From measurements following different cooling protocols and calculations of the muon stopping site, we suggest that the metastable SkL is not the majority phase throughout the bulk of this material at the fields and temperatures where it is often observed. The dynamics of bulk Co$_8$Zn$_9$Mn$_3$ are well described by $\simeq~2$ GHz excitations that reduce in frequency near the critical temperature, while in Co$_8$Zn$_8$Mn$_4$ we observe similar behavior over a wide range of temperatures, implying that dynamics of this kind persist beyond the SkL phase.