Lifetime of racetrack skyrmions

Writing and Detecting Topological Charges in Exfoliated Fe5-xGeTe2

Fe5-xGeTe2 is a promising two-dimensional (2D) van der Waals (vdW) magnet for practical applications, given its magnetic properties. These include Curie temperatures above room temperature, and topological spin textures─TST (both merons and skyrmions), responsible for a pronounced anomalous Hall effect (AHE) and its topological counterpart (THE), which can be harvested for spintronics. Here, we show that both the AHE and THE can be amplified considerably by just adjusting the thickness of exfoliated Fe5-xGeTe2, with THE becoming observable even in zero magnetic field due to a field-induced unbalance in topological charges. Using a complementary suite of techniques, including electronic transport, Lorentz transmission electron microscopy, and micromagnetic simulations, we reveal the emergence of substantial coercive fields upon exfoliation, which are absent in the bulk, implying thickness-dependent magnetic interactions that affect the TST. We detected a "magic" thickness t ≈ 30 nm where the formation of TST is maximized, inducing large magnitudes for the topological charge density (∼6.45 × 1020 cm-2), and the concomitant anomalous (ρxyA,max ≃22.6 μΩ cm) and topological (ρxyu,T 1≃5 μΩ cm) Hall resistivities at T ≈ 120 K. These values for ρxyA,max and ρxyu,T are higher than those found in magnetic topological insulators and, so far, the largest reported for 2D magnets. The hitherto unobserved THE under zero magnetic field could provide a platform for the writing and electrical detection of TST aiming at energy-efficient devices based on vdW ferromagnets.

Survival of skyrmions along granular racetracks at room temperature

Skyrmions can be envisioned as bits of information that can be transported along nanoracetracks. However, temperature, defects, and/or granularity can produce diffusion, pinning, and, in general, modification in their dynamics. These effects may cause undesired errors in information transport. We present simulations of a realistic system where both the (room) temperature and sample granularity are taken into account. Key feasibility magnitudes, such as the success probability of a skyrmion traveling a given distance along the racetrack, are calculated. The results are evaluated in terms of the eventual loss of skyrmions by pinning, destruction at the edges, or excessive delay due to granularity. The model proposed is based on the Fokker-Planck equation resulting from Thiele's rigid model for skyrmions. The results could serve to establish error detection criteria and, in general, to discern the dynamics of skyrmions in realistic situations.

Stabilization and racetrack application of asymmetric Néel skyrmions in hybrid nanostructures

Magnetic skyrmions, topological quasiparticles, are small stable magnetic textures that possess intriguing properties and potential for data storage applications. Hybrid nanostructures comprised of skyrmions and soft magnetic material can offer additional advantages for developing skyrmion-based spintronic and magnonic devices. We show that a Néel-type skyrmion confined within a nanodot placed on top of a ferromagnetic in-plane magnetized stripe produces a unique and compelling platform for exploring the mutual coupling between magnetization textures. The skyrmion induces an imprint upon the stripe, which, in turn, asymmetrically squeezes the skyrmion in the dot, increasing their size and the range of skyrmion stability at small values of Dzyaloshinskii-Moriya interaction, as well as introducing skyrmion bi-stability. Finally, by exploiting the properties of the skyrmion in a hybrid system, we demonstrate unlimited skyrmion transport along a racetrack, free of the skyrmion Hall effect.

Device geometry dependent deterministic skyrmion generation from a skyrmionium

A magnetic skyrmionium can be perceived as an association of two magnetic skyrmions with opposite topological charges. In this work, we have investigated the transformation of skyrmionium into multi-skyrmionic states via domain wall pairs in three different devices with variable geometric configurations. The same device geometries are considered for single ferromagnetic layer and synthetic antiferromagnetic system. It is observed that by tuning the current density, deterministic generation of skyrmions is possible via the spin transfer torque. The proposed device is efficiently adjustable to change the number of skyrmions also at room temperature. The results may lead to development of skyrmion-based devices for neuromorphic and unconventional computing.

Electric field manipulation of spin chirality and skyrmion dynamic

The Dzyaloshinskii-Moriya interaction (DMI) is an antisymmetric exchange interaction that stabilizes spin chirality. One scientific and technological challenge is understanding and controlling the interaction between spin chirality and electric field. In this study, we investigate an unconventional electric field effect on interfacial DMI, skyrmion helicity, and skyrmion dynamics in a system with broken inversion symmetry. We design heterostructures with a 3d-5d atomic orbital interface to demonstrate the gate bias control of the DMI energy and thus transform the DMI between opposite chiralities. Furthermore, we use this voltage-controlled DMI (VCDMI) to manipulate the skyrmion spin texture. As a result, a type of intermediate skyrmion with a unique helicity is created, and its motion can be controlled and made to go straight. Our work shows the effective control of spin chirality, skyrmion helicity, and skyrmion dynamics by VCDMI. It promotes the emerging field of voltage-controlled chiral interactions and voltage-controlled skyrmionics.

Strain-Driven Zero-Field Near-10 nm Skyrmions in Two-Dimensional van der Waals Heterostructures

Magnetic skyrmions─localized chiral spin structures─show great promise for spintronic applications. The recent discovery of two-dimensional (2D) magnets opened new opportunities for topological spin structures in atomically thin van der Waals (vdW) materials. Despite recent progress in stabilizing metastable skyrmions in 2D magnets, their lifetime, essential for applications, has not been explored yet. Here, using first-principles calculations and atomistic spin simulations, we predict that compressive strain leads to stabilizing zero-field skyrmions with diameters close to 10 nm in a Fe₃GeTe₂/germanene vdW heterostructure. The origin of these unique skyrmions is attributed to the high tunability of Dzyaloshinskii-Moriya interaction and magnetocrystalline anisotropy energy by strain, which generally holds for Fe₃GeTe₂ heterostructures with buckled substrates. Furthermore, we calculate the energy barriers protecting skyrmions against annihilation and their lifetimes using transition-state theory. We show that nanoscale skyrmions in strained Fe₃GeTe₂/germanene can be stable for hours at temperatures up to 20 K.

A micromagnetic theory of skyrmion lifetime in ultrathin ferromagnetic films

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.

Electrical Generation and Deletion of Magnetic Skyrmion-Bubbles via Vertical Current Injection

The magnetic skyrmion is a topologically protected spin texture that has attracted much attention as a promising information carrier because of its distinct features of suitability for high-density storage, low power consumption, and stability. One of the skyrmion devices proposed so far is the skyrmion racetrack memory, which is the skyrmion version of the domain-wall racetrack memory. For application in devices, skyrmion racetrack memory requires electrical generation, deletion, and displacement of isolated skyrmions. Despite the progress in experimental demonstrations of skyrmion generation, deletion, and displacement, these three operations have yet to be realized in one device. Here, a route for generating and deleting isolated skyrmion-bubbles through vertical current injection with an explanation of its microscopic origin is presented. By combining the proposed skyrmion-bubble generation/deletion method with the spin-orbit-torque-driven skyrmion shift, a proof-of-concept experimental demonstration of the skyrmion racetrack memory operation in a three-terminal device structure is provided.

Visualizing the strongly reshaped skyrmion Hall effect in multilayer wire devices

Magnetic skyrmions are nanoscale spin textures touted as next-generation computing elements. When subjected to lateral currents, skyrmions move at considerable speeds. Their topological charge results in an additional transverse deflection known as the skyrmion Hall effect (SkHE). While promising, their dynamic phenomenology with current, skyrmion size, geometric effects and disorder remain to be established. Here we report on the ensemble dynamics of individual skyrmions forming dense arrays in Pt/Co/MgO wires by examining over 20,000 instances of motion across currents and fields. The skyrmion speed reaches 24 m/s in the plastic flow regime and is surprisingly robust to positional and size variations. Meanwhile, the SkHE saturates at ∼22^∘, is substantially reshaped by the wire edge, and crucially increases weakly with skyrmion size. Particle model simulations suggest that the SkHE size dependence — contrary to analytical predictions — arises from the interplay of intrinsic and pinning-driven effects. These results establish a robust framework to harness SkHE and achieve high-throughput skyrmion motion in wire devices.

Skyrmions - nanoscale, topological spin textures - are promising elements for next-generation computing due to their efficient coupling to currents in racetrack devices. Here, Tan et al. examine over 20,000 instances of current induced skyrmion motion to unveil a comprehensive picture of skyrmion dynamics across currents and fields.

Emergence of Nontrivial Spin Textures in Frustrated Van Der Waals Ferromagnets

In this work, first principles ground state calculations are combined with the dynamic evolution of a classical spin Hamiltonian to study the metamagnetic transitions associated with the field dependence of magnetic properties in frustrated van der Waals ferromagnets. Dynamically stabilized spin textures are obtained relative to the direction of spin quantization as stochastic solutions of the Landau-Lifshitz-Gilbert-Slonczewski equation under the flow of the spin current. By explicitly considering the spin signatures that arise from geometrical frustrations at interfaces, we may observe the emergence of a magnetic skyrmion spin texture and characterize the formation under competing internal fields. The analysis of coercivity and magnetic hysteresis reveals a dynamic switch from a soft to hard magnetic configuration when considering the spin Hall effect on the skyrmion. It is found that heavy metals in capped multilayer heterostructure stacks host field-tunable spiral skyrmions that could serve as unique channels for carrier transport. The results are discussed to show the possibility of using dynamically switchable magnetic bits to read and write data without the need for a spin transfer torque. These results offer insight to the spin transport signatures that dynamically arise from metamagnetic transitions in spintronic devices.

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.

Role of higher-order exchange interactions for skyrmion stability

Transition-metal interfaces and multilayers are a 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 for the stability of skyrmions. We use an atomistic spin model parametrized from first-principles and compare three different ultrathin film systems. We consider all fourth-order exchange interactions and show that, in particular, the four-site four spin interaction has a large effect on the energy barrier preventing skyrmion and antiskyrmion collapse into the ferromagnetic state. Our work opens perspectives to stabilize topological spin structures even in the absence of Dzyaloshinskii-Moriya interaction.

Entropy-reduced Retention Times in Magnetic Memory Elements: A Case of the Meyer-Neldel Compensation Rule

We compute mean waiting times between thermally activated magnetization reversals in a nanodisk with parameters similar to a free CoFeB layer used in magnetic random access memories. By combining Langer's theory and forward flux sampling simulations, we show that the Arrhenius prefactor can take values up to 10^{21} Hz, orders of magnitude beyond the value of 10^{9}  Hz typically assumed, and varies drastically as a function of material parameters. We show that the prefactor behaves like an exponential of the activation energy, which highlights a case of the Meyer-Neldel compensation rule. This suggests that modeling information retention times with a barrier-independent prefactor in such magnetic storage elements is not justified.

Theoretical investigation of antiferromagnetic skyrmions in a triangular monolayer

The chiral spin textures of a two-dimensional (2D) triangular system, where both antiferromagnetic (AF) Heisenberg exchange and chiral Dzyaloshinsky-Moriya interactions co-exist, are investigated numerically with an optimized quantum Monte Carlo method based on mean-field theory. We find that: helical, skyrmionic and vortical AF crystals can be formed when an external magnetic field is applied perpendicular to the 2D monolayer; the sizes of these skyrmions and vortices change abruptly at several critical points of the external magnetic field; each of these AF crystals can be decomposed into three periodical ferromagnetic sublattices. The quantum ingredient implemented into the theoretical framework helps to track the existence of AF skyrmion lattices down to low temperatures.

Dynamics of chiral state transitions and relaxations in an FeGe thin plate via in situ Lorentz microscopy

Studying the magnetic transition between different topological spin textures in noncentrosymmetric magnets under external stimuli is an important topic in chiral magnetism. Here, using in situ Lorentz transmission electron microscopy (LTEM) we directly visualize the thermal-driven magnetic transitions and dynamic characteristics in FeGe thin plates. A novel protocol-dependent phase diagram of FeGe thin plates was obtained via pulsed laser excitation. Moreover, by setting the appropriate specimen temperature, the relaxation of chiral magnetic states in FeGe specimens was recorded and analyzed with an Arrhenius-type relaxation mechanism. We present the field-dependent activation energy barriers for chiral state transitions and the magnetic transition pathways of these spin textures for FeGe thin plates. Our results unveil the effects of thermal excitation on the topological spin texture transitions and provide useful information about magnetic dynamics of chiral magnetic state relaxation.

Atomistic Perspective of Long Lifetimes of Small Skyrmions at Room Temperature

The current development to employ magnetic skyrmions in novel spintronic device designs has led to a demand for room-temperature-stable skyrmions of ever smaller size. We present extensive studies on skyrmion stability in atomistic magnetic systems in two- and three-dimensional geometries. We show that for materials described by the same micromagnetic parameters, the variation of the atomistic exchange between different neighbors, the stacking order, and the number of layers of the atomic lattice can significantly influence the rate of the thermally activated decay of a skyrmion. These factors alone are important considerations, but we show that their combination can open up novel avenues of materials design in the search for sub-10 nm skyrmions, as their lifetime can be extended by several orders of magnitude.

Efficient optimization method for finding minimum energy paths of magnetic transitions

Efficient algorithms for the calculation of minimum energy paths of magnetic transitions are implemented within the geodesic nudged elastic band (GNEB) approach. While an objective function is not available for GNEB and a traditional line search can, therefore, not be performed, the use of limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) and conjugate gradient algorithms in conjunction with orthogonal spin optimization (OSO) approach is shown to greatly outperform the previously used velocity projection and dissipative Landau-Lifschitz dynamics optimization methods. The implementation makes use of energy weighted springs for the distribution of the discretization points along the path and this is found to improve performance significantly. The various methods are applied to several test problems using a Heisenberg-type Hamiltonian, extended in some cases to include Dzyaloshinskii-Moriya and exchange interactions beyond nearest neighbours. Minimum energy paths are found for magnetization reversals in a nano-island, collapse of skyrmions in two-dimensional layers and annihilation of a chiral bobber near the surface of a three-dimensional magnet. The LBFGS-OSO method is found to outperform the dynamics based approaches by up to a factor of 8 in some cases.

Skyrmion flop transition and congregation of mutually orthogonal skyrmions in cubic helimagnets

Magnetic chiral skyrmions are particle-like excitations with a topological charge, which are currently considered as promising objects for the next generation of magnetic memory, logic, and neuromorphic devices. In three-dimensional systems, they can form rather complex topological structures. In bulk helimagnets, elongated skyrmion tubes can be ordered either perpendicularly or parallel to an external magnetic field and such configurations coexist in a specific range of fields. We have shown that with an increase in the magnetic field, the transition from perpendicular to parallel ordering in a 3D skyrmion dimer occurs through an intermediate state with mutually orthogonal skyrmion tubes. In the system with three and more skyrmion tubes, we uncovered a surprisingly large diversity of superstructures and systemized the principles of their formation. The nascent conical state is shown to induce the field-dependent switch between favored skyrmion clusters and underlies attracting inter-skyrmion potential. We argue that our numerical simulations on skyrmion clusters are valid in a parameter range corresponding to the A-phase region of cubic helimagnets. Moreover, skyrmionic superstructures constitute a novel concept of spintronic devices based on gapless skyrmion motion along with each other.

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.

Direct Demonstration of Topological Stability of Magnetic Skyrmions via Topology Manipulation

Topological protection precludes a continuous deformation between topologically inequivalent configurations in a continuum. Motivated by this concept, magnetic skyrmions, topologically nontrivial spin textures, are expected to exhibit topological stability, thereby offering a prospect as a nanometer-scale nonvolatile information carrier. In real materials, however, atomic spins are configured as not continuous but discrete distributions, which raises a fundamental question if the topological stability is indeed preserved for real magnetic skyrmions. Answering this question necessitates a direct comparison between topologically nontrivial and trivial spin textures, but the direct comparison in one sample under the same magnetic fields has been challenging. Here we report how to selectively achieve either a skyrmion state or a topologically trivial bubble state in a single specimen and thereby experimentally show how robust the skyrmion structure is in comparison with the bubbles. We demonstrate that topologically nontrivial magnetic skyrmions show longer lifetimes than trivial bubble structures, evidencing the topological stability in a real discrete system. Our work corroborates the physical importance of the topology in the magnetic materials, which has hitherto been suggested by mathematical arguments, providing an important step toward ever-dense and more-stable magnetic devices.

Spin structure relation to phase contrast imaging of isolated magnetic Bloch and Néel skyrmions

Magnetic skyrmions are promising candidates for future storage devices with a large data density. A great variety of materials have been found that host skyrmions up to the room-temperature regime. Lorentz microscopy, usually performed in a transmission electron microscope (TEM), is one of the most important tools for characterizing skyrmion samples in real space. Using numerical calculations, this work relates the phase contrast in a TEM to the actual magnetization profile of an isolated Néel or Bloch skyrmion, the two most common skyrmion types. Within the framework of the used skyrmion model, the results are independent of skyrmion size and wall width and scale with sample thickness for purely magnetic specimens. Simple rules are provided to extract the actual skyrmion configuration of pure Bloch or Néel skyrmions without the need of simulations. Furthermore, first differential phase contrast (DPC) measurements on Néel skyrmions that meet experimental expectations are presented and showcase the described principles. The work is relevant for material sciences where it enables the engineering of skyrmion profiles via convenient characterization.

Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets

Room-temperature skyrmions in ferromagnetic films and multilayers show promise for encoding information bits in new computing technologies. Despite recent progress, ferromagnetic order generates dipolar fields that prevent ultrasmall skyrmion sizes, and allows a transverse deflection of moving skyrmions that hinders their efficient manipulation. Antiferromagnetic skyrmions shall lift these limitations. Here we demonstrate that room-temperature antiferromagnetic skyrmions can be stabilized in synthetic antiferromagnets (SAFs), in which perpendicular magnetic anisotropy, antiferromagnetic coupling and chiral order can be adjusted concurrently. Utilizing interlayer electronic coupling to an adjacent bias layer, we demonstrate that spin-spiral states obtained in a SAF with vanishing perpendicular magnetic anisotropy can be turned into isolated antiferromagnetic skyrmions. We also provide model-based estimates of skyrmion size and stability, showing that room-temperature antiferromagnetic skyrmions below 10 nm in radius can be anticipated in further optimized SAFs. Antiferromagnetic skyrmions in SAFs may thus solve major issues associated with ferromagnetic skyrmions for low-power spintronic devices.

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

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

Chiral spin ordering of electron gas in solids with broken time reversal symmetry

In this work we manifest that an electrostatic disorder in conducting systems with broken time reversal symmetry universally leads to a chiral ordering of the electron gas giving rise to skyrmion-like textures in spatial distribution of the electron spin density. We describe a microscopic mechanism underlying the formation of the equilibrium chiral spin textures in two-dimensional systems with spin-orbit interaction and exchange spin splitting. We have obtained analytical expressions for spin-density response functions and have analyzed both local and non-local spin response to electrostatic perturbations for systems with parabolic-like and Dirac electron spectra. With the proposed theory we come up with a concept of controlling spin chirality by electrical means.

Spin Torque Efficiency and Analytic Error Rate Estimates of Skyrmion Racetrack Memory

In this paper, the thermal stability of skyrmion bubbles and the critical currents to move them over pinning sites were investigated. For the used pinning geometries and the used parameters, the unexpected behavior is reported that the energy barrier to overcome the pinning site is larger than the energy barrier of the annihilation of a skyrmion. The annihilation takes place at boundaries by current driven motion, as well as due to the excitation over energy barriers, in the absence of currents, without forming Bloch points. It is reported that the pinning sites, which are required to allow thermally stable bits, significantly increase the critical current densities to move the bits in skyrmion-like structures to about jcrit = 0.62 TA/m². The simulation shows that the applied spin transfer model predicts experimentally obtained critical currents to move stable skyrmions at room temperature well, which is in contrast to simulations based on spin orbit torque that predict significantly too low critical currents. By calculating the thermal stability, as well as the critical current, we can derive the spin torque efficiency η = ΔE/Ic = 0.19 kBT300/μA, which is in a similar range to the simulated spin torque efficiency of MRAM structures. Finally, it is shown that the stochastic depinning process of any racetrack-like device requires an extremely narrow depinning time distribution smaller than ~6% of the current pulse length to reach bit error rates smaller than 10-9.

Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method

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 important avenue for the creation of Skyrmions in future magnetic devices.