Magnetic skyrmions are key candidates for novel memory, logic, and neuromorphic computing. An essential property is their topological protection caused by the whirling spin texture as described by a robust integer winding number. However, the realization on an atomic lattice leaves a loophole for switching the winding number via concerted rotation of individual spins. Hence, understanding the unwinding microscopically is key to enhance skyrmion stability. Here, we use spin polarized scanning tunneling microscopy to probe skyrmion annihilation by individual hot electrons and obtain maps of the transition rate on the nanometer scale. By applying an in-plane magnetic field, we tune the collapse rate by up to four orders of magnitude. In comparison with first-principles based atomistic spin simulations, the experiments demonstrate a radial symmetric collapse at zero in-plane magnetic field and a transition to the recently predicted chimera collapse at finite in-plane field. Our work opens the route to design criteria for skyrmion switches and improved skyrmion stability.
Magnetic skyrmions are topologically-protected spin textures with attractive properties suitable for high-density and low-power spintronic device applications. Much effort has been dedicated to understanding the dynamical behaviours of the magnetic skyrmions. However, experimental observation of the ultrafast dynamics of this chiral magnetic texture in real space, which is the hallmark of its quasiparticle nature, has so far remained elusive. Here, we report nanosecond-dynamics of a 100 nm-size magnetic skyrmion during a current pulse application, using a time-resolved pump-probe soft X-ray im…
Quantum collapse of a small skyrmion in a thin magnetic film with Dzyalishinskii-Moriya (DMI) interaction has been studied. The energy of the skyrmion and the stability threshold determined by the DMI, the external magnetic field, and the underlying atomic lattice are investigated analytically and numerically. The Lagrangian describing the coupled dynamics of the skyrmion size and the chirality angle is derived. Equations of motion possess an instanton solution that corresponds to the skyrmion underbarrier contraction via quantum tunneling with subsequent collapse and decay of the topological …
Time dependence of the size and chirality of a ferromagnetic skyrmion in a Heisenberg model with the magnetic field on a square lattice has been studied analytically and numerically. The lattice and the magnetic field generate strong time dependence of the skyrmion chirality. Due to nonlinearity, the lattice alone also generates strong intrinsic damping that leads to the skyrmion collapse via the emission of spin waves. In the absence of the magnetic field the collapse is slow for a large skyrmion but it becomes exponentially fast in the presence of the Landau-Lifshitz damping when the field i…
Various transitions that a magnetic skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the skyrmion divides and forms two skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an im…
A path method is implemented in order to precisely and generally describe the collapse of isolated skyrmions in a Co/Pt(111) monolayer, on the basis of atomic scale simulations. Two collapse mechanisms with different energy barriers are found. The most obvious path, featuring a homogeneous shrinking gives the largest energy, whereas the lowest energy barrier is shown to comply with the outcome of Langevin dynamics under a destabilizing field of 0.25 T, with a lifetime of 20~ns at around 80~K. For this lowest energy barrier path, skyrmion destabilization occurs much before any topology change, …
We use the continuum micromagnetic framework to derive the formulas for compact skyrmion lifetime due to thermal noise in ultrathin ferromagnetic films with relatively weak interfacial Dzyaloshinskii-Moriya interaction. In the absence of a saddle point connecting the skyrmion solution to the ferromagnetic state, we interpret the skyrmion collapse event as ``capture by an absorber'' at microscale. This yields an explicit Arrhenius collapse rate with both the barrier height and the prefactor as functions of all the material parameters, as well as the dynamical paths to collapse.
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/I…
Thermal collapse of an isolated skyrmion on a two-dimensional spin lattice has been investigated. The method is based upon solution of the system of stochastic Landau-Lifshitz-Gilbert equations for up $10^4$ spins. Recently developed pulse-noise algorithm has been used for the stochastic component of the equations. The collapse rate follows the Arrhenius law. Analytical formulas derived within a continuous spin-field model support numerically-obtained values of the energy barrier and the pre-exponential factor, and their dependence on the magnetic field. Our findings agree with experiments, as…
We investigate the topological change of a Belavin-Polyakov skyrmion under the action of a spin-polarized current. The dynamics is described by the Schrödinger equation for the electrons carrying the current coupled to the Landau-Lifshitz equation for the evolution of the magnetic texture in a square lattice. We show that the addition of an exchange dissipation term, tends to smooth the transition from the skyrmion state to the ferromagnetic state. We demonstrate that this topological change, in the continuum dissipationless limit, can be described as a self-similar finite-time singularity by …
Magnetic skyrmions are topological spin configurations in materials with chiral Dzyaloshinskii-Moriya interaction (DMI), that are potentially useful for storing or processing information. To date, DMI has been found in few bulk materials, but can also be induced in atomically thin magnetic films in contact with surfaces with large spin-orbit interactions. Recent experiments have reported that isolated magnetic skyrmions can be stabilized even near room temperature in few-atom thick magnetic layers sandwiched between materials that provide asymmetric spin-orbit coupling. Here we present the min…
The localized magnon modes of isolated $kπ$ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order $k$ a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields $2π$ and $3π$ skyrmions are destroyed via a burst instability connected to a breathing mode, while $1π$ skyrmions undergo an elliptic instability. At high fields …
Skyrmions, novel topologically stable spin vortices, hold promise for next-generation magnetic storage due to their nanoscale domains to enable high information storage density and their low threshold for current-driven motion to enable ultralow energy consumption. One-dimensional (1D) nanowires are ideal hosts for skyrmions since they not only serve as a natural platform for magnetic racetrack memory devices but also can potentially stabilize skyrmions. Here we use the topological Hall effect (THE) to study the phase stability and current-driven dynamics of the skyrmions in MnSi nanowires. Th…
Quantum states of a skyrmion in a 2D antiferromagnetic lattice are obtained by quantizing the scaling parameter of Belavin-Polyakov model. Skyrmion classical collapse due to violation of the translational invariance of the continuous spin-field model by the lattice is replaced in quantum mechanics by transitions between discrete energy levels of the skyrmion. Rates of transitions due to the emission of magnons are computed. Ways of detecting quantization of skyrmion states are discussed.
A new skyrmion topological transition driven by higher-order exchange interactions in Janus MnSeTe
CEMES, Université de Toulouse, CNRS（图卢兹大学CNRS CEMES）；Institute of Theoretical Physics and Astrophysics, University of Kiel（基尔大学理论物理与天体物理研究所）；Kiel Nano, Surface, and Interface Science (KiNSIS), University of Kiel（基尔大学基尔纳米、表面与界面科学研究所）
Magnetic phase transitions are a test bed for exploring the physics of non-equilibrium phenomena in condensed matter, which become even more complex when topological constraints are involved. In particular, the investigation of skyrmions and skyrmion lattices offers insight into fundamental processes of topological-charge creation and annihilation upon changing the magnetic state. Nonetheless, the exact physical mechanisms behind these phase transitions remain unresolved. Here, we have systematically compared ultra-thin films with isotropic and anisotropic Dzyaloshinskii-Moriya interactions (D…
Antiferromagnetic Skyrmion Scattering Revealed by Direct Time-Resolved Imaging of Collective Dynamics
Institute of Physics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学物理研究所）；Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University（东北大学电气通信研究所纳米电子与自旋电子学实验室）；Singulus Technologies AG（辛古卢斯科技有限公司）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie GmbH（柏林亥姆霍兹材料能源中心有限公司）
Magnetic skyrmions formed at temperatures around room temperature in square-based parallelepiped magnetic FePt nanoparticles with perpendicular magnetocrystalline anisotropy (MCA) were studied during the magnetization reversal using micromagnetic simulations. Finite Differences (FD) method were used for the solution of the Landau-Lifshitz-Gilbert equation. Magnetic configurations exhibiting Néel skyrmionic formations were detected. The magnetic skyrmions can be created in different systems generated by the variation of external field, side length and width of the squared-based parallelepiped m…
The breathing mode of a skyrmion, corresponding to coupled oscillations of its size and chirality angle is studied numerically for a conservative classical-spin system on a $500\times500$ lattice. The dependence of the oscillation frequency on the magnetic field is computed. It is linear at small fields, reaches maximum on increasing the field, then sharply tends to zero as the field approaches the threshold above which the skyrmion loses stability and collapses. Physically transparent analytical model is developed that explains the results qualitatively and provides the field dependence of th…
We compute annihilation rates of metastable magnetic skyrmions using a form of Langer's theory in the intermediate-to-high damping (IHD) regime. For a Néel skyrmion, a Bloch skyrmion, and an antiskyrmion, we look at two possible paths to annihilation: collapse and escape through a boundary. We also study the effects of a curved vs. a flat boundary, a second skyrmion and a non-magnetic defect. We find that the skyrmion's internal modes play a dominant role in the thermally activated transitions compared to the spin-wave excitations and that the relative contribution of internal modes depends on…
We report that in cylindrical nanorods skyrmion chains, i.e., three-dimensional spin textures, emerge dynamically, as revealed by micromagnetic simulations. The skyrmion-chain state occurs when the diameter of the rod is larger than the helical pitch length of the material, and the number of skyrmions on the chain is proportional to the length of the nanorod. This finding provides a deeper understanding of the interplay between geometry and skyrmionic symmetry, and shows how spatial confinement can stabilize spontaneous topological spin textures.
The magnetization reversal in 330 nm triangular prismatic magnetic nanoelements with variable magnetocrystalline anisotropy similar to that of partially chemically ordered FePt is studied using micromagnetic simulations employing Finite Element discretizations. Several magnetic properties including the evaluation of the magnetic skyrmion number $S$ are computed in order to characterize magnetic configurations exhibiting vortex-like formations. Magnetic vortices and skyrmions are revealed in different systems generated by the variation of the magnitude and relative orientation of the magnetocry…
We address the problem of how magnetic skyrmions can propagate along a guided direction by parametric pumping. As evidenced by our micromagnetic simulations, skyrmions can hardly be driven by either a static electric field or a static magnetic field alone. Although the magnetic anisotropy can be modified by an electric field, parametric pumping with an oscillating electric field can only excite the breathing modes. On the other hand, a static magnetic field can break rotational symmetry through the Zeeman interaction, but it cannot serve as an energy source for propelling a skyrmion. Here we f…
We perform a direct comparison between Kramers' method in many dimensions -- i.e., Langer's theory -- adapted to magnetic spin systems, and a path sampling method in the form of forward flux sampling, as a means to compute collapse rates of metastable magnetic skyrmions. We show that a good agreement is obtained between the two methods. We report variations of the attempt frequency associated with skyrmion collapse by three to four orders of magnitude when varying the applied magnetic field by 5$\%$ of the exchange strength, which confirms the existence of a strong entropic contribution to the…
The atomic-scale spin structure of individual isolated skyrmions in an ultrathin film is investigated in real space by spin-polarized scanning tunneling microscopy. Their axial symmetry as well as their unique rotational sense is revealed by using both out-of-plane and in-plane sensitive tips. The size and shape of skyrmions change as a function of magnetic field. An analytical expression for the description of skyrmions is proposed and applied to connect the experimental data to the original theoretical model describing chiral skyrmions. Thereby, the relevant material parameters responsible f…
We show that thermal stability of magnetic skyrmions can be strongly affected by entropic effects. The lifetimes of isolated skyrmions in atomic Pd/Fe bilayers on Ir(111) and on Rh(111) are calculated in the framework of harmonic transition state theory based on an atomistic spin model parametrized from density functional theory. Depending on the system the attempt frequency for skyrmion collapse can change by up to nine orders of magnitude with the strength of the applied magnetic field. We demonstrate that this effect is due to a drastic change of entropy with skyrmion radius which opens a n…
The dynamics of a vortex in a thin-film ferromagnet resembles the motion of a charged massless particle in a uniform magnetic field. Similar dynamics is expected for other magnetic textures with a nonzero skyrmion number. However, recent numerical simulations revealed that skyrmion magnetic bubbles show significant deviations from this model. We show that a skyrmion bubble possesses inertia and derive its mass from the standard theory of a thin-film ferromagnet. Besides center-of-mass motion, other low energy modes are waves on the edge of the bubble traveling with different speeds in opposite…
We determine sizes and activation energies of kπ-skyrmions on a discrete lattice using the Landau- Lifshitz-Gilbert equation and the geodesic nudged elastic band method. The employed atomic material parameters are based on the skyrmionic material system Pd/Fe/Ir(111). We find that the critical magnetic fields for collapse of the 2π-skyrmion and 3π-skyrmion are very close to each other and considerably lower than the critical field of the 1π-skyrmion. The activation energy protecting the structures does not strictly decrease with increasing k as it can be larger for the 3π-skyrmion than for the…
We show that high concentration of skyrmions can be achieved in magnetic films with quenched disorder. A 2D system of Heisenberg spins with feromagnetic exchange, $J$, and random magnetic anisotropy of strength $D_R \ll J$ has been studied. Analytical theory for the dependence of the average skyrmion size on the magnetic field $H$, and for the stability of pinned skyrmions, is complemented by numerical studies of 2D lattices containing up to 40 million spins. At low fields the average size of the skyrmion, $λ$, is determined by the average size of Imry-Ma domains. On increasing the field the s…
We studied interactions between propagating spin waves (SWs) and a single skyrmion in a perpendicularly magnetized CoFeB nanostripe where the magnetic layer is interfaced with W and MgO. Micromagnetic numerical calculations revealed that robust interactions between the incident SWs and the skyrmion give rise to considerable forward skyrmion motions for specific SW frequencies (e.g., here: fsw = 12 - 19 GHz). Additionally, it was found that there exists a sufficiently low threshold field amplitude, e.g., 0.1 kOe for the fsw = 15 GHz SWs. This frequency-dependent interaction originated from the …
Magnetic skyrmions are stable nanosized spin structures that can be displaced at low electrical current densities. Because of these properties, they have been proposed as building blocks of future electronic devices with unprecedentedly high information density and low energy consumption. The electrical detection of skyrmions via the Topological Hall Effect (THE), has so far been demonstrated only at cryogenic temperatures. Here, we report the observation of a skyrmion Topological Hall Effect near room temperature (276 K) in a mesoscopic lamella of FeGe. This region unambiguously coincides wit…
The competition between external drives and interparticle interactions are important for the static and dynamic properties of topological quasiparticles in confined geometries. Here, we study the dynamics of particlelike skyrmions under a current-induced compression, where skyrmions are stabilized in a confined space and form a skyrmion solid. We find that a lattice structural transition of nanoscale skyrmions could be induced by the compression, which depends on the initial lattice configuration and the compression level. The compression effect in the skyrmion solid with strong skyrmion-skyrm…
Polar skyrmions are predicted to emerge from the interplay of elastic, electrostatic and gradient energies, in contrast to the key role of the anti-symmetric Dzyalozhinskii-Moriya interaction in magnetic skyrmions. With the discovery of topologically-stable polar skyrmions, it is of both fundamental and practical interest to understand the microscopic nature and the possibility of temperature- and strain-driven phase transitions in ensembles of such polar skyrmions. Here, we explore the reversible transition from a skyrmion state (topological charge of -1) to a two-dimensional, tetratic lattic…
Transition-metal interfaces and multilayers are a very promising class of systems to realize nanometer-sized, stable magnetic skyrmions for future spintronic devices. For room temperature applications it is crucial to understand the interactions which control the stability of isolated skyrmions. Typically, skyrmion properties are explained by the interplay of pair-wise exchange interactions, the Dzyaloshinskii-Moriya interaction and the magnetocrystalline anisotropy energy. Here, we demonstrate that higher-order exchange interactions -- which have so far been neglected -- can play a key role f…
Axisymmetric solitonic states (chiral skyrmions) have been predicted theoretically more than two decades ago. However, until recently they have been observed in a form of skyrmionic condensates (hexagonal lattices and other mesophases). In this paper we report experimental and theoretical investigations of isolated chiral skyrmions discovered in PdFe/Ir(111) bilayers two years ago (Science 341 , 636 (2013)). The results of spin-polarized scanning tunneling microscopy analyzed within the continuum and discrete models provide a consistent description of isolated skyrmions in thin layers. The exi…
We demonstrate numerically on large spin lattices that one can write skyrmions in a thin magnetic film with a magnetic dipole of a few tens of nanometer in size. Nucleation of non-chiral skyrmions as well as chiral skyrmions formed by the Dzyaloshinskii-Moriya interaction has been investigated. Analytical model is developed that agrees with numerical results. It is shown that skyrmions can be written though a number of scenarios that depend on the experimental technique and parameters of the system. In one scenario, that branches into subscenarios of different topology, the magnetic dipole on …
Magnetic skyrmions are topological objects, which have recently been observed in thin films at room temperature. Sub 100-nm sizes and spin polarised current manipulation make them candidates for high density information storage and processing. Besides material thickness, skyrmion stability involves many contributions: ferromagnetic exchange, magnetic anisotropy, Dzyaloshinskii-Moryia exchange interaction (DMI) and demagnetising energy. Multidimensional phase diagrams have been reported in the litterature. In this work we propose a simple two-parameter model, exact for thin films in the negligi…
Magnetic skyrmions are topologically stable objects that are made with a bunch of spins tightly arranged in a smooth fashion. Their topological nature provides unusual and complex transport properties, such as the skyrmion Hall effect. Extensive Hall data have further revealed asymmetry between skyrmion and antiskyrmion Hall angles, which cannot be accounted by known mechanisms. Here, we explain this asymmetry by utilizing another universal transport coefficient called `Hall viscosity,' extensively studied in quantum Hall systems. Hall viscosity is modeled in steady-state skyrmions motion by g…
The skyrmion racetrack is a promising concept for future information technology. There, binary bits are carried by nanoscale spin swirls -- skyrmions -- driven along magnetic strips. Stability of the skyrmions is a critical issue for the realization of this technology. Here we demonstrate that the racetrack skyrmion lifetime can be calculated from first principles as a function of temperature, magnetic field and track width. Our method combines harmonic transition state theory extended to include Goldstone modes, with an atomistic spin Hamiltonian parametrized from density functional theory ca…
In a $β$-Mn-type chiral magnet Co$_9$Zn$_9$Mn$_2$, we demonstrate that the magnetic field-driven collapse of a room temperature metastable topological skyrmion lattice passes through a regime described by a partial topological charge inversion. Using Lorentz transmission electron microscopy, the magnetization distribution was observed directly as the magnetic field was swept antiparallel to the original skyrmion core magnetization, i.e. negative magnetic fields. Due to the topological stability of skyrmions, a direct transition of the metastable skyrmion lattice to the equilibrium helical stat…
Emergent phenomena and functions arising from topological electron-spin textures in real space or momentum space are attracting growing interest for new concept of states of matter as well as for possible applications to spintronics. One such example is a magnetic skyrmion, a topologically stable nanoscale spin vortex structure characterized by a topological index. Real-space regular arrays of skyrmions are described by combination of multi-directional spin helixes. Nanoscale configurations and characteristics of the two-dimensional skyrmion hexagonal-lattice have been revealed extensively by …
Magnetic skyrmions are whirls in the magnetization whose topological winding number promises stability against thermal fluctuations and defects. They can only decay via singular spin configurations. We analyze the corresponding energy barriers of skyrmions in a magnetic monolayer for two distinct stabilization mechanisms, i.e., Dzyaloshinskii-Moriya interaction (DMI) and competing interactions. Based on our numerically calculated collapse paths on an atomic lattice, we derive analytic expressions for the saddle point textures and energy barriers of large skyrmions. The sign of the spin stiffne…
We theoretically study the dynamics of skyrmion crystals in electrically-insulating chiral magnets subjected to current-induced spin torques by adjacent metallic layers. We develop an elasticity theory that accounts for the gyrotropic force engendered by the non-trivial topology of the spin texture, tensions at the boundaries due to the exchange of linear and spin angular momentum with the metallic reservoirs, and dissipation in the bulk of the film. A steady translation of the skyrmion crystal is triggered by the current-induced tensions and subsequently sustained by dissipative forces, gener…
Magnetic chiral skyrmions are vortex like spin structures that appear as stable or meta-stable states in magnetic materials due to the interplay between the symmetric and antisymmetric exchange interactions, applied magnetic field and/or uniaxial anisotropy. Their small size and internal stability make them prospective objects for data storage but for this, the controlled switching between skyrmion states of opposite polarity and topological charge is essential. Here we present a study of magnetic skyrmion switching by an applied magnetic field pulse based on a discrete model of classical spin…
Topological order is important in many aspects of condensed matter physics, and has been extended to bosonic systems. In this Letter we report on the nontrivial topology of the magnon bands in two distinct quantum skyrmion crystals appearing in zero external magnetic field. This is revealed by nonzero Chern numbers for some of the bands. As a bosonic analog of the quantum anomalous Hall effect, we show that topological magnons can appear in skyrmion crystals without explicitly breaking time-reversal symmetry with an external magnetic field. By tuning the value of the easy-axis anisotropy at ze…
Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers
Université de Toulouse, CNRS, CEMES（图卢兹大学，法国国家科学研究中心，电子与分子材料中心）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）；Dipartimento di Matematica, Università di Pisa（比萨大学数学系）
The ground state of the two dimensional electron gas near $ν$=1 is investigated by inelastic light scattering measurements carried down to very low temperatures. Away from $ν$=1, the ferromagnetic spin wave collapses and a new low-energy spin wave emerges below the Zeeman gap. The emergent spin wave shows soft behavior as its energy increases with temperature and reaches the Zeeman energy for temperatures above 2 K. The observed softening indicates an instability of the two dimensional electron gas towards a magnetic order that breaks spin rotational symmetry. We discuss our findings in light …
Recent developments have led to an explosion of activity on skyrmions in three-dimensional (3D) chiral magnets. Experiments have directly probed these topological spin textures, revealed their nontrivial properties, and led to suggestions for novel applications. However, in 3D the skyrmion crystal phase is observed only in a narrow region of the temperature-field phase diagram. We show here, using a general analysis based on symmetry, that skyrmions are much more readily stabilized in two-dimensional (2D) systems with Rashba spin-orbit coupling. This enhanced stability arises from the competit…
