Skyrmion-skyrmion Interaction Research Articles

Mechanical effect on skyrmion-skyrmion interaction

Understanding the dynamics of magnetic skyrmions is a prerequisite for their applications in advanced functional devices such as memory, logic, and computing devices. Especially, the skyrmion-skyrmion interaction is a key to comprehending diverse intrinsic skyrmion behaviors such as their ordering, melting, pinning, and moving. In literature, the skyrmions interactions are demonstrated to be affected by external fields like magnetic and temperature fields. However, as an important aspect, the mechanical effects on skyrmions interaction are still unclear due to the difficulty of studying complex magneto-elastic couplings. Here, we break the limitation by employing a phase-field simulation and demonstrate that strain-induced large skyrmion distortions drastically change the skyrmions interaction under applied mechanical loads. Then, an exponential equation is proposed to describe the skyrmions interaction under both positive and negative uniaxial strain loads with good accuracy. Therefore, we developed a phase-field simulation with high-order magneto-elastic coupling, provided an in-depth thermodynamic understanding of the mechanical effect of skyrmion dynamics, and proposed the according equation, which provided a new concept of mechanical-controlled magnetism and should be beneficial for further studies of skyrmion behaviors and advanced magnetic materials.

Precursor skyrmion states near the ordering temperatures of chiral magnets.

In noncentrosymmetric magnets, chiral Dzyaloshinskii-Moriya interactions (DMI) provide a distinctive mechanism for the stabilization of localized skyrmion states in two and three dimensions with a fixed sense of rotation. Near the ordering transition, the skyrmion strings develop attractive skyrmion-skyrmion interactions and ultimately become confined in extended clusters or textures [A. O. Leonov and U. K. Rößler, Nanomaterials, 2023, 13, 891], which is a consequence of the coupling between the magnitude and the angular part of the order parameter. Multi-skyrmionic states built from isolated skyrmions (IS) can form multiple modulated magnetic phases that may underlie the exotic magnetic phenomena of "partial order" or the field-driven "A-phase" observed in MnSi and other cubic helimagnets. Based on the standard phenomenological Dzyaloshinskii model, we obtain numerically exact solutions for skyrmion lattices (SkL), formulate their basic properties, and elucidate physical mechanisms of their formation and stability. Our detailed numerical studies show that the bound skyrmion states arise as hexagonal lattices of ±π-skyrmions (with the magnetization in the center along or opposite to the magnetic field) or square staggered lattices of π/2-skyrmions, which contain defect lines with zero modulus value and thus may form thermodynamically stable states only near the ordering temperature. In the simplest case of a two-dimensional (2D) skyrmionic texture, the structure is homogeneous in the third dimension (3D). The skyrmions preserve an ideal axisymmetric "double twist" core in condensed phases, while continuation into a space-filling texture is frustrated. The evolution of skyrmion lattices in an increasing magnetic field leads to a succession of phase transitions of first or second kind between diverse textures and finally ends due to the formation of isolated skyrmion-filaments with fixed radius and shape embedded in a homogeneously magnetized matrix. In the framework of the phenomenological model including only isotropic interactions (exchange, Zeeman, and DM energy contributions), the considered skyrmion lattices are only metastable states as the competing conical one-dimensional spiral forms the equilibrium state. But due to the weak couplings between skyrmions, secondary effects like anisotropies can stabilize skyrmionic textures as compared to simple helices. Also the topological nature of skyrmion condensates makes the magnetization processes in chiral magnets history-dependent and hysteretic.

Exponential Temperature Effects on Skyrmion-Skyrmion Interaction

Understanding the dynamic behaviors of magnetic skyrmions has great potential in the application of spintronics. This is especially the case for the interaction between two skyrmions. In the literature, the attractive and repulsive skyrmion-skyrmion interactions are demonstrated as strongly related to the distance of two skyrmions and external magnetic fields. However, as a natural property, the temperature effect on skyrmion-skyrmion interactions is unclear, which lacks understanding from thermodynamics, and accordingly its driving force equation is absent. Here, we employ a temperature-related phase-field simulation and demonstrate that the skyrmion-skyrmion interaction is dramatically changed by temperature (e.g., a temperature change of only a few kelvins results in an eightfold higher interaction), which can lead to critical temperature behavior of skyrmion phase. Detailed analysis shows the interaction potential decreases exponentially with an increase of temperature, and this pronounced temperature dependence originates from the temperature-related skyrmion structure. Then, an equation is proposed to describe the skyrmion interaction at finite temperatures. Therefore, this study provides a deeper understanding of the skyrmion kinetic behavior and explores the temperature control of skyrmion interactions coupled with other sources of interaction, which is a guidance for future study of skyrmions and their applications.

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.

Programmable Skyrmion Logic Gates Based on Skyrmion Tunneling

Magnetic skyrmions are promising candidates as elementary nanoscale bits in logic-in-memory devices, intrinsically merging high density memory and computing capabilities. Here we exploit the dynamics of skyrmions interacting with anisotropy energy barriers patterned by ion irradiation to design programmable logic gates. Using micromagnetic simulations with experimental parameters, we show that a fine tuning of the barrier height and width allows the selective tunneling of skyrmions between parallel nanotracks triggered by skyrmion-skyrmion interaction. This can be leveraged to design skyrmion De-multiplexer (DMux) logic gate which works solely using skyrmions as logic inputs. By cascading and connecting demultiplexer gates with a specific topology, we develop a fully programmable logic gate capable of producing any possible logic output as a sum of all minterms generated by a given set of inputs without requiring any complex additional electric/magnetic interconversion. The proposed design is fully conservative and cascadable and paves a new pathway for full skyrmionic-based logic-in-memory devices.

Skyrmion clusters and chains in bulk and thin-layered cubic helimagnets

Here, the author overviews the properties and the internal structure of isolated skyrmions within a cone phase of cubic helimagnets and chiral liquid crystals. The nascent conical state is shown to substantiate the attracting skyrmion-skyrmion interaction, which leads to the formation of skyrmion clusters and/or skyrmions chains. Whereas in bulk helimagnets skyrmion clusters exhibit the hexagonal arrangement of constituent skyrmions, in thin layers skyrmions align into polar skyrmion chains. Such behavior is thickness dependent and depends on the characteristic energy profiles.

Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets.

Existence of topological localized states (skyrmions and torons) and the mechanism of their condensation into modulated states are the ruling principles of condensed matter systems, such as chiral nematic liquid crystals (CLCs) and chiral magnets (ChM). In bulk helimagnets, skyrmions are rendered into thermodynamically stable hexagonal skyrmion lattice due to the combined effect of a magnetic field and, e.g., small anisotropic contributions. In thin glass cells of CLCs, skyrmions are formed in response to the geometrical frustration and field coupling effects. By numerical modeling, I undertake a systematic study of skyrmion or toron properties in thin layers of CLCs and ChMs with competing surface-induced and bulk anisotropies. The conical phase with a variable polar angle serves as a suitable background, which shapes skyrmion internal structure, guides the nucleation processes, and substantializes the skyrmion-skyrmion interaction. I show that the hexagonal lattice of torons can be stabilized in a vast region of the constructed phase diagram for both easy-axis bulk and surface anisotropies. A topologically trivial droplet is shown to form as a domain boundary between two cone states with different rotational fashion, which underpins its stability. The findings provide a recipe for controllably creating skyrmions and torons, possessing the features on demand for potential applications.

Field-driven metamorphoses of isolated skyrmions within the conical state of cubic helimagnets

Topologically stable field configurations appear in many fields of physics, from elementary particles to condensed matter and cosmology. Deep physical relations and common physical features of a large variety of very different solitonic states in these systems arise from the mathematical similarity of phenomenological equations. During the last decade, chiral liquid crystals and chiral magnets took on the role of model objects for the experimental investigation of topological solitons and the understanding of their nonsingular field configurations. This is directly related to the discovery of particle-like chiral skyrmions, which are also considered as promising ingredients for technological applications. Here we introduce a paradigm of facile skyrmionic networks with mutually orthogonal orientations of constituent isolated skyrmions. On the one hand, such networks are envisioned as a novel concept of spintronic devices based, e.g., on gapless skyrmion motion along each other, and are presumably responsible for precursor phenomena near the ordering temperatures of bulk cubic helimagnets. In particular, we demonstrate an interconversion between mutually orthogonal skyrmions: horizontal skyrmions may swirl into an intermediate spring-like state and subsequently squeeze into vertical skyrmions with both polarities. On the other hand, skyrmion tubes are considered as building blocks for particle-like states with more involved internal structure. A family of target-skyrmions, which includes a so far overlooked type with a multiple topological charge, is formed owing to the tendency to minimize the interaction energy between vertical and horizontal skyrmions. The conical phase serves as a suitable background for considered skyrmion evolution. It substantializes the attracting skyrmion-skyrmion interaction in the skyrmionic networks, shapes their internal structure, and guides the nucleation processes. Alternatively, intricate textural changes of isolated skyrmions result not only in the structural deformations of a host conical state, but may lead to the formation of an exotic skyrmion order with pairs of merons being the core of the game. Generically, the fundamental insights provided by this work emphasize a three-dimensional character of skyrmion metamorphoses and can also be extended to three-dimensional solitons, such as hopfions.

Nucleon-nucleon potential from skyrmion dipole interactions

We derive the nucleon-nucleon interaction from the Skyrme model using second order perturbation theory and the dipole approximation to skyrmion dynamics. Unlike previous derivations, our derivation accounts for the non-trivial kinetic and potential parts of the skyrmion-skyrmion interaction lagrangian and how they couple in the quantum calculation. We derive the eight low energy interaction potentials and compare them with the phenomenological Paris model, finding qualitative agreement in seven cases.

Fractal Analysis of Skyrmions Generated by Field-Assisted Fine-tuning of Magnetic Anisotropy

Magnetic skyrmions are topologically protected whirling spin textures nucleated from other ferromagnetic states that have great potential applications for nanoscale devices. To realize this potential, it is important to find useful methods to generate skyrmions in a controlled fashion and to identify tools to quantitatively describe their formation. Here we generate skyrmions in perpendicularly magnetized $\mathrm{Mg}\mathrm{O}/{\mathrm{Mn}}_{2}\mathrm{Co}\mathrm{Al}/\mathrm{Pd}$ ultrathin films and investigate their formation by fractal analysis. Using an in-plane field, we modify the effective magnetic anisotropy of the films to control the nucleation of skyrmions from two distinct domain states in the films: labyrinth domains typically found in perpendicular materials and a featureless monodomain ferromagnetic phase. By using fractal analysis, we obtain a magnetic phase diagram related to the fractal dimension of the domains that can describe the evolution of the magnetic states and can be used to categorize the formation of skyrmions. Moreover, we find that the skyrmion density is determined not only by the critical material parameter $\ensuremath{\kappa}$ but also by a repulsive skyrmion-skyrmion interaction at high in-plane magnetic fields. We thus establish a reliable and powerful method to generate skyrmions and a way to analyze the evolution of skyrmions.

Shear banding, intermittency, jamming, and dynamic phases for skyrmions in inhomogeneous pinning arrays

We examine driven skyrmion dynamics in systems with inhomogeneous pinning where a strip of strong pinning coexists with a region containing no pinning. For driving parallel to the strip, we find that the initial skyrmion motion is confined to the unpinned region and the skyrmion Hall angle is zero. At larger drives, a transition occurs to a phase in which motion also appears within the pinned region, creating a shear band in the skyrmion velocity, while the skyrmion Hall angle is still zero. As the drive increases further, the flow becomes disordered and the skyrmion Hall angle increases with drive until saturating at the highest drives when the system transitions into a moving crystal phase. The different dynamic phases are associated with velocity and density gradients across the pinning boundaries. We map out the dynamic phases as a function of pinning strength, skyrmion density, and Magnus force strength, and correlate the phase boundaries with features in the velocity-force curves and changes in the local and global ordering of the skyrmion structure. For large Magnus forces, the shear banding instability is replaced by large scale intermittent flow in the pinned region accompanied by simultaneous motion perpendicular to the direction of the drive, which appears as oscillations in the transport curves. We also examine the case of a drive applied perpendicular to the strip, where we find a jamming effect in which the skyrmion flow is blocked by skyrmion-skyrmion interactions until the drive is large enough to induce plastic flow.

Chiral edge currents for ac-driven skyrmions in confined pinning geometries

We show that ac driven skyrmion lattices in a weak pinning channel confined by regions of strong pinning exhibit edge transport carried by skipping orbits while skyrmions in the bulk of the channel undergo localized orbits with no net transport. The magnitude of the edge currents can be controlled by varying the amplitude and frequency of the ac drive or by changing the ratio of the Magnus force to the damping term. We identify a localized phase in which the orbits are small and edge transport is absent, an edge transport regime, and a fluctuating regime that appears when the ac drive is strong enough to dynamically disorder the skyrmion lattice. We also find that in some cases, multiple rows of skyrmions participate in the transport due to a drag effect from the skyrmion-skyrmion interactions. The edge currents are robust for finite disorder and should be a general feature of skyrmions interacting with confined geometries or inhomogeneous disorder under an ac drive. We show that similar effects can occur for skyrmion lattices at interfaces or along domain boundaries for multiple coexisting skyrmion species. The edge current effect provides a new method to control skyrmion motion, and we discuss the connection of these results with recent studies on the emergence of edge currents in chiral active matter systems and gyroscopic metamaterials.

Real-space observation of skyrmion clusters with mutually orthogonal skyrmion tubes

We report the discovery and direct visualization of skyrmion clusters with mutually-orthogonal orientations of constituent isolated skyrmions in chiral liquid crystals and ferromagnets. We show that the nascent conical state underlies attracting inter-skyrmion potential, whereas an encompassing homogeneous state leads to the repulsive skyrmion-skyrmion interaction. The crossover between different regimes of skyrmion interaction could be identified upon changing layer thickness and/or the surface anchoring. We develop a phenomenological theory describing two types of skyrmions and the underlying mechanisms of their interaction. We show that isolated horizontal skyrmions with the same polarity may approach a vertical isolated skyrmion from both sides and thus constitute two energetically-different configurations which are also observed experimentally. In an extreme regime of mutual attraction, the skyrmions wind around each other forming compact superstructures with undulations. We also indicate that our numerical simulations on skyrmion clusters are valid in a parameter range corresponding to the A-phase region of cubic helimagnets.

Skyrmion clusters and conical droplets in bulk helimagnets with cubic anisotropy

Recent experimental findings in the bulk cubic helimagnet and Mott insulator ${\mathrm{Cu}}_{2}{\mathrm{OSeO}}_{3}$ highlight the paramount role of cubic anisotropy in stabilizing novel chiral and skyrmion phases. It was indeed found that if a magnetic field is applied along the easy $\ensuremath{\langle}001\ensuremath{\rangle}$ crystallographic direction, competing cubic and exchange anisotropies tilt the wave vector of the conical spiral away from the magnetic field [Qian et al., Sci. Adv. 4, eaat7323 (2018)]. Furthermore, in this configuration skyrmions have been observed in a broad range of temperatures and magnetic fields [Chacon et al., Nat. Phys. 14, 936 (2018)]. Starting from these experimental observations and on the basis of a phenomenological Dzyaloshinskii theory, we investigate additional implications of the cubic anisotropy for this specified field direction. By including cubic anisotropy we show that the phase transition between the conical and field-polarized or homogeneous states becomes first order. Furthermore, we show that this transition is accompanied by the formation of conical droplets---domains of the conical phase in the homogeneous state. We investigate the internal structure of these droplets, which at their boundaries encompass alternating regions of positive and negative energy densities with respect to the homogeneous state. We thus deduce that these droplets may be zipped and unzipped in phase during the first-order phase transition that occurs by either increasing or decreasing the magnetic field. On the other hand, we show that in the conical phase skyrmions may form clusters due to their attractive mutual interaction. However, in the homogeneous state, the skyrmion-skyrmion interaction becomes repulsive, and the skyrmion clusters expand and disperse isolated skyrmions. This mutual skyrmion repulsion prevents the stabilization of skyrmion clusters even if the energy of isolated skyrmions is lower than that of the homogeneous state. Yet this skyrmion dispersal may be prevented if skyrmions are surrounded by the circular spiral state and form skyrmion bags. Such a scenario could explain the existence of skyrmions in the field-polarized state reported experimentally.

The spin structures of interlayer coupled magnetic films with opposite chirality

Using Monte-Carlo simulations and micromagnetic simulations, we reveal how the spin structural correlation and the skyrmion dynamics are affected by the interlayer coupling in a chiral magnetic bilayer system, in which the two layers have opposite chirality. The interaction through interlayer coupling between chiral magnetic structures influences the static and dynamics properties profoundly. The competition between the Dzyaloshinskii-Moriya interaction and the interlayer interaction allows multiple magnetic structures to be energetically stable, which includes sole skyrmion states (skyrmion appears in only one of the layers) and skyrmion pair states (coupled skyrmions in top and bottom layers). When current driven spin transfer torques are applied to each state, the sole skyrmion state is mainly propelled by a spin transfer torque causing the skyrmion hall effect, but the skyrmion pair state is propelled by a torque from skyrmion-skyrmion interaction and not influenced by the skyrmion hall effect. Also upon application of an external magnetic field, we found the skyrmions in a skyrmion pair state extinguish in an exclusive way, as the annihilation of a skyrmion in one of the layers stabilizes the once paired skyrmion in the other layer, i.e. the skyrmion lattice sites have only one skyrmion in either layer.

Effect of the Magnus force on skyrmion relaxation dynamics

We perform systematic Langevin molecular dynamics simulations of interacting skyrmions in thin films. The interplay between Magnus force, repulsive skyrmion-skyrmion interaction and thermal noise yields different regimes during non-equilibrium relaxation. In the noise-dominated regime the Magnus force enhances the disordering effects of the thermal noise. In the Magnus-force-dominated regime, the Magnus force cooperates with the skyrmion-skyrmion interaction to yield a dynamic regime with slow decaying correlations. These two regimes are characterized by different values of the aging exponent. In general, the Magnus force accelerates the approach to the steady state.

Asymmetric isolated skyrmions in polar magnets with easy-plane anisotropy

We introduce a new class of isolated magnetic skyrmions emerging within tilted ferromagnetic phases of polar magnets with easy-plane anisotropy. The asymmetric magnetic structure of these skyrmions is associated with an intricate pattern of the energy density, which exhibits positive and negative asymptotics with respect to the surrounding state with a ferromagnetic moment tilted away from the polar axis. Correspondingly, the skyrmion-skyrmion interaction has an anisotropic character and can be either attractive or repulsive depending on the relative orientation of the skyrmion pair. We investigate the stability of these novel asymmetric skyrmions against the elliptical cone state and follow their transformation into axisymmetric skyrmions, when the tilted ferromagnetic moment of the host phase is reduced. Our theory gives clear directions for experimental studies of isolated asymmetric skyrmions and their clusters embedded in tilted ferromagnetic phases.

Spin excitation spectra of integral and fractional quantum Hall systems

Results are presented of detailed numerical calculations for the spin excitations of a two-dimensional electron gas confined to a quantum well (QW) of finite width w, at magnetic fields corresponding to both fractional and integral Landau level (LL) fillings $\ensuremath{\nu}\ensuremath{\approx}\frac{1}{3},$ 1, 3, and 5. The Hamiltonian matrix of up to $N=14$ interacting electrons is diagonalized exactly in Haldane spherical geometry, neglecting excitations to higher orbital LL's or to higher QW subbands. At sufficiently low Zeeman energy ${E}_{\mathrm{Z}},$ spin waves (SW's) and skyrmions are found at $\ensuremath{\nu}=\frac{1}{3}$ that are composite fermion analogues of the reversed-spin-electron--hole excitations known to occur at $\ensuremath{\nu}=1.$ Only for sufficiently large w are stable skyrmions found for $\ensuremath{\nu}=3$ and 5. Their stability depends upon the interaction pseudopotentials of their constituents. We propose a criterion on the pseudopotential needed for skyrmion stability, and construct phase diagrams (in the $w\ensuremath{-}{E}_{\mathrm{Z}}$ plane) for skyrmions of different size. The SW-SW and skyrmion-skyrmion interactions are also discussed, and the noninteracting SW condensates as well as the skyrmion fluids with Laughlin correlations (at intermediate skyrmion densities) are proposed.

Hopf Term and the Effective Lagrangian for the Skyrmions in a Two-Dimensional Electron Gas at SmallgFactor

We study interacting electrons in two dimensions moving in the lowest Landau level under the condition that the Zeeman energy is much smaller than the Coulomb energy and the filling factor is one. In this case, Skyrmion quasiparticles play an important role. Here, we present a simple and transparent derivation of the corresponding effective Lagrangian. In its kinetic part, we find a non-zero Hopf term the prefactor of which we determine rigorously. In the Hamiltonian part, we calculate, by means of a gradient expansion, the Skyrmion-Skyrmion interaction completely up to fourth order in spatial derivatives.

The structure of the skyrmion-skyrmion interaction

The structure of the skyrmion-skyrmion interaction is investigated. Using a zero-range boson-exchange interpretation, Skyrme models can be related to models including explicit boson stabilization. The structure of the potential can be understood in terms of conventional boson-exchange pictures and corrections to the conventional Skyrme model are made to estimate mesonic effects known to be important in the nucleon-nucleon interaction. The effects of soft-pion fluctuations and a sixth order term on the skyrmion-skyrmion interaction are assessed and interpreted in terms of ω and σ mesons. The modifications to the Skyrme model produce interactions in qualitative agreement with phenomenological models of the nucleon-nucleon interaction.