Skyrmion Lattice Research Articles

Thermal Hysteresis and Ordering Behavior of Magnetic Skyrmion Lattices.

The physics of phase transitions in two-dimensional (2D) systems underpins research in diverse fields including statistical mechanics, nanomagnetism, and soft condensed matter. However, many aspects of 2D phase transitions are still not well understood, including the effects of interparticle potential, polydispersity, and particle shape. Magnetic skyrmions are chiral spin-structure quasi-particles that form two-dimensional lattices. Here, we show, by real-space imaging using in situ cryo-Lorentz transmission electron microscopy coupled with machine learning image analysis, the ordering behavior of Néel skyrmion lattices in van der Waals Fe3GeTe2. We demonstrate a distinct change in the skyrmion size distribution during field-cooling, which leads to a loss of lattice order and an evolution of the skyrmion liquid phase. Remarkably, the lattice order is restored during field heating and demonstrates a thermal hysteresis. This behavior is explained by the skyrmion energy landscape and demonstrates the potential to control the lattice order in 2D phase transitions.

Skyrmion lattice hosted in synthetic antiferromagnets and helix modes

Thin ferromagnetic films can possess unconventional magnetic properties, opening a new road for using them in spintronic technologies. In the present work exploiting three different methods, we comprehensively analyze phason excitations of a skyrmion lattice in synthetic antiferromagnets. To analyze phason excitations of the skyrmion lattice, we have constructed an analytical model based on three coupled helices and found a linear gapless mode. Micromagnetic simulations also support this result. Moreover, a similar result has been achieved within the rigid skyrmion lattice model based on the coupled Thiele's equations, when the coupling between skyrmions in different layers of the synthetic antiferromagnetic is comparable to or larger than the intralayer coupling. In addition, we also consider the orbital angular momentum and spin pumping current associated with phason excitations. Due to the gapless excitations in the case of skyrmion lattice, the pumping current is nonzero for the arbitrary frequency of pumping microwaves. In the case of individual skyrmions, no current is pumped when microwave frequency is inside the gap of the spectrum of individual skyrmions.

Confinement of Skyrmions in Nanoscale FeGe Device-like Structures.

Skyrmion-based devices have been proposed as a promising solution for low-energy data storage. These devices include racetrack or logic structures and require skyrmions to be confined in regions with dimensions comparable to the size of a single skyrmion. Here we examine skyrmions in FeGe device shapes using Lorentz transmission electron microscopy to reveal the consequences of skyrmion confinement in a device-like structure. Dumbbell-shaped elements were created by focused ion beam milling to provide regions where single skyrmions are confined adjacent to areas containing a skyrmion lattice. Simple block shapes of equivalent dimensions were also prepared to allow a direct comparison with skyrmion formation in a less complex, yet still confined, device geometry. The impact of applying a magnetic field and varying the temperature on the formation of skyrmions within the shapes was examined. This revealed that it is not just confinement within a small device structure that controls the position and number of skyrmions but that a complex device geometry changes the skyrmion behavior, including allowing skyrmions to form at lower applied magnetic fields than in simple shapes. The impact of edges in complex shapes is observed to be significant in changing the behavior of the magnetic textures formed. This could allow methods to be developed to control both the position and number of skyrmions within device structures.

Higher-order modulations in the skyrmion lattice phase of Cu2OSeO3

Using small angle neutron scattering, we have investigated higher-order peaks in the skyrmion-lattice phase of Cu$_2$OSeO$_3$, in which two different skyrmion lattices, SkX1 and SkX2, are known to form. For each skyrmion-lattice phase, we observed two sets of symmetrically inequivalent peaks at the higher-order-reflection positions with the indices $(110)$ and $(200)$. Under the condition where the SkX1 and SkX2 coexist, we confirmed the absence of the scattering at $\mathbf{Q}$ positions combining reflections from the two phases, indicating a significantly weak double-scattering component. Detailed analysis of the peak profile, as well as the temperature and magnetic-field dependence of the peak intensity, also supports the intrinsic higher-order modulation rather than the parasitic double scattering. The two higher-order modulations show contrasting magnetic-field dependence; the former $(110)$ increases as the field is increased, whereas the latter $(200)$ decreases. This indicates that, in Cu$_2$OSeO$_3$, skyrmions are weakly distorted, and the distortion is field-dependent in a way that the dominant higher-order modulation switches from $(110)$ to $(200)$ under field. Monte Carlo simulations under sweeping external magnetic field qualitatively reproduce the observed magnetic-field dependence, and suggests that the higher-order modulations correspond to the superlattices of weak swirlings appearing in the middle of the original triangular-latticed skyrmions.

Tuning the density of zero-field skyrmions and imaging the spin configuration in a two-dimensional Fe3GeTe2 magnet

With the advent of ferromagnetism, two-dimensional (2D) van der Waals (vdW) magnets have attracted particular attention in exploring topological spin textures, such as skyrmions used for next-generation spintronic devices. The discovery of magnetic skyrmions in Fe3GeTe2 (FGT) has sparked interest in investigating the spin configurations of skyrmions in FGT. Here, we used an in situ Lorentz microscope to directly demonstrate the generation and sustainability of Bloch-type skyrmions in a zero magnetic field over a wide temperature range in 2D vdW FGT. By tuning the value of the external magnetic field, the highest-density hexagonal skyrmion lattice emerges after reducing the magnetic field to zero. Moreover, by tilting the FGT nanosheet, we found that the field-free Bloch-type skyrmions in FGT can also represent an invisible contrast when the tilt angle is zero, but a reversed magnetic contrast emerges at a high tilt angle. On the basis of our experiments, we discuss the possible mechanisms for such variable magnetic contrast. These findings offer valuable insights into the spin configurations of skyrmions in 2D vdW FGT and shed light on the identification of spin configurations via Lorentz microscopy.

Road to zero-field antiferromagnetic skyrmions in a frustrated AFM/FM heterostructure

We demonstrate a mechanism of significant reduction, including complete elimination, of the external magnetic field required for the stabilization of a skyrmion lattice (SkX) phase in a frustrated triangular Heisenberg antiferromagnet (AFM) with the Dzyaloshinskii-Moriya interaction. It is achieved by coupling of such a AFM plane to a reference ferromagnetic (FM) layer, which generates an effective field cooperating with the external magnetic field. By Monte Carlo simulations we demonstrate that if the FM layer shows some axial single-ion anisotropy then the effective field can also be generated in zero external field due to a spontaneous FM long-range ordering. Then a sufficiently large interlayer coupling can fully substitute the external magnetic field and the SkX phase in the AFM layer can be stabilized even in zero external field.

Second-Harmonic Response in Magnetic Nodal-Line Semimetal Fe3GeTe2

We experimentally investigate second-harmonic transverse voltage response to ac electrical current for a magnetic nodal-line semimetal Fe3GeTe2 (FGT). For zero magnetic field, the observed second-harmonic voltage behaves as a square of the longitudinal current, as it should be expected for nonlinear Hall effect. The magnetic field behavior is found to be sophisticated: while the first-harmonic response shows the known anomalous Hall hysteresis in FGT, the second-harmonic Hall voltage is characterized by the pronounced high-field hysteresis and flat (B-independent) region with curves touching at low fields. The high-field hysteresis strongly depends on the magnetic field sweep rate, so it reflects some slow relaxation process. For the lowest rates, it is also accomplished by multiple crossing points. Similar shape of the second-harmonic hysteresis is known for skyrmion spin textures in nonlinear optics. Since skyrmions have been demonstrated for FGT by direct visualization techniques, we can connect the observed high-field relaxation with deformation of the skyrmion lattice. Thus, the second-harmonic Hall voltage response can be regarded as a tool to detect spin textures in transport experiments.

Reservoir Computing with Spin Waves in a Skyrmion Crystal

The application of spintronics to physical reservoir computing has great potential, but development still suffers from inevitable technical complications in nanofabrication. This numerical study considers spin waves excited in a self-organized skyrmion lattice under a magnetic field in a chiral magnet---which would not require advanced manufacturing in practice---to make progress on the problem. Such a skyrmion lattice offers great levels of generalizability, memory capacity, and nonlinearity, fulfilling the fundamental requirements of reservoir computing. The results will promote engineering solutions to pave the way toward reliable, energy-conserving reservoir computing.

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Supplemental Videos and Datasets for: Three-Dimensional Neutron Far-Field Tomography of a Bulk Skyrmion Lattice

Investigations on antiferromagnetic skyrmion crystal generated by a staggered magnetic field

Antiferromagnetic skyrmions are promising candidates for future spintronic applications due to their particular robustness and ultrafast dynamics. In this work, we have used a staggered magnetic field to produce an antiferromagnetic (AFM) skyrmion crystal. The creation of such a skyrmion crystal is examined in two different situations: a regular clean system and a system containing impurity-like spin vacancies. We have found that defects decisively affect the AFM skyrmions creation, inducing skyrmions connections, the appearance of bimerons, and even the skyrmion lattice destruction at a critical defect density. In addition, we also explore the skyrmion dynamics in a hybrid racetrack which contains two distinct materials displayed in different areas (regions 1 and 2). A spin-polarized current can expel the skyrmions from region 1 (which sustains the skyrmion crystal) to region 2, initially empty of skyrmions. The organized collection of skyrmions of region 1 loses the traditional hexagonal symmetry when they penetrate the second region, composing a skyrmion gas. Nonetheless, this symmetry might be preserved if the spin-polarized current is high enough and reaches a determined value.

Rayleigh waves and cyclotron surface modes of gyroscopic metamaterials

We investigate the elastic normal modes of two-dimensional media with broken time-reversal and parity symmetries due to a Lorentz term. Our starting point is an elasticity theory that captures the low-energy physics of a diverse range of systems such as gyroscopic metamaterials, skyrmion lattices in thin-film chiral magnets and certain Wigner crystals. By focusing on a circular disk geometry we analyze finite-size effects and study the low-frequency shape oscillations of the disk. We demonstrate the emergence of the Rayleigh surface waves from the bottom of the excitation spectrum and investigate how the curvature of the disk-boundary modifies their propagation at long wavelengths. Moreover, we discover a near-cyclotron-frequency wave that is almost completely localized at the boundary of the disk, but is distinct from the Rayleigh wave. It can be distinguished from the latter by a characteristic excitation pattern in a small region near the center of the disk.

Phase diagram of superconductivity in the integer quantum Hall regime

An interplay between pairing and topological orders has been predicted to give rise to superconducting states supporting exotic emergent particles, such as Majorana particles obeying non-Abelian braid statistics. We consider a system of spin polarized electrons on a Hofstadter lattice with nearest-neighbor attractive interaction and solve the mean-field Bogoliubov-de Gennes equations in a self-consistent fashion, leading to gauge-invariant observables and a rich phase diagram as a function of the chemical potential, the magnetic field, and the interaction. As the strength of the attractive interaction is increased, the system first makes a transition from a quantum Hall phase to a skyrmion lattice phase that is fully gapped in the bulk but has topological chiral edge current, characterizing a topologically nontrivial state. This is followed by a vortex phase in which the vortices carrying Majorana modes form a lattice; the spectrum contains a low-energy Majorana band arising from the coupling between neighboring vortex-core Majorana modes but does not have chiral edge currents. For some parameters, a dimer vortex lattice occurs with no Majorana band. The experimental feasibility and the observable consequences of skyrmions as well as Majorana modes are indicated.

Cryogen free spin polarized scanning tunneling microscopy and magnetic exchange force microscopy with extremely low noise.

Spin polarized scanning tunneling microscopy (SP-STM) and magnetic exchange force microscopy (MExFM) are powerful tools to characterize spin structure at the atomic scale. For low temperature measurements, liquid helium cooling is commonly used, which has the advantage of generating low noise but has the disadvantage of having difficulties in carrying out measurements with long durations at low temperatures and measurements with a wide temperature range. The situation is just reversed for cryogen-free STM, where the mechanical vibration of the refrigerator becomes a major challenge. In this work, we have successfully built a cryogen-free system with both SP-STM and MExFM capabilities, which can be operated under a 9T magnetic field provided by a cryogen-free superconducting magnet and in a wide temperature range between 1.4 and 300K. With the help of our specially designed vibration isolation system, the noise is reduced to an extremely low level of 0.7pm. The Fe/Ir(111) magnetic skyrmion lattice is used to demonstrate the technical novelties of our cryogen-free system.

Commensuration effects on skyrmion Hall angle and drag for manipulation of skyrmions on two-dimensional periodic substrates

We examine the dynamics of an individually driven skyrmion moving through a background lattice of skyrmions coupled to a 2D periodic substrate as we vary the ratio of the number of skyrmions to the number of pinning sites across commensurate and incommensurate conditions. As the skyrmion density increases, the skyrmion Hall angle is nonmonotonic, dropping to low or zero values in commensurate states and rising to an enhanced value in incommensurate states. Under commensuration, the driven skyrmion is channeled by a symmetry direction of the pinning array and exhibits an increased velocity. At fillings for which the skyrmion Hall angle is zero, the velocity has a narrow band noise signature, while for incommensurate fillings, the skyrmion motion is disordered and the velocity noise is broad band. Under commensurate conditions, multi-step depinning transitions appear and the skyrmion Hall angle is zero at low drives but becomes finite at higher drives, while at incommensurate fillings there is only a single depinning transition. As the gyrotropic component of the skyrmion dynamics, called the Magnus force, increases, peaks in the velocity that appear in commensurate regimes cross over to dips, and new types of directional locking effects can arise in which the skyrmion travels along other symmetry directions of the background lattice. At large Magnus forces, and particularly at commensurate fillings, the driven skyrmion can experience a velocity boost in which the skyrmion moves faster than the applied drive due to the alignment of the Magnus-induced velocity with the driving direction. In some cases, an increase of the Magnus force can produce regimes of enhanced pinning when the skyrmion is forced to move along a nonsymmetry direction of the periodic pinning array. This is in contrast to systems with random pinning, where increasing the Magnus force generally reduces the pinning effect. We demonstrate these dynamics for both square and triangular substrates and map out the different regimes as a function of filling fraction, pinning force, and the strength of the Magnus force in a series of dynamic phase diagrams.

Phase degree of freedom and topology in multiple- Q spin textures

A periodic array of topological spin textures, such as skyrmions and hedgehogs, is called the multiple-$Q$ spin texture, as it is represented by a superposition of multiple spin density waves. Depending on the way of superposition, not only the magnetic but also the topological properties are modified, leading to a variety of quantum transport and optical phenomena caused by the emergent electromagnetic fields through the Berry phase. Among others, the phase degree of freedom of the superposed waves is potentially important for such modifications, but its effect has not been fully elucidated thus far. Here we perform systematic theoretical analyses of magnetic and topological properties of the multiple-$Q$ spin textures with the phase degree of freedom. By introducing a hyperspace with an additional dimension corresponding to the phase degree of freedom, we establish a generic framework to deal with the phase shift in the multiple-$Q$ spin textures. Applying the framework to the two-dimensional 3$Q$ spin textures, we clarify the complete topological phase diagram while changing the phase and magnetization, which depends on the types of the superposed waves. We also study the three-dimensional 4$Q$ spin textures and clarify even richer topological phase diagrams. In particular, we find novel topological phase transitions associated with the previously unidentified Dirac strings on which the hedgehogs and antihedgehogs cause pair creation and fusion. Moreover, we demonstrate that phase shifts are caused by an external magnetic field in both 3$Q$ and 4$Q$ cases by analyzing the numerical data in the previous studies. Our results illuminate the topological aspects of the skyrmion and hedgehog lattices with the phase degree of freedom, which would be extended to other multiple-$Q$ textures and useful for the exploration of topologically nontrivial magnetic phases and exotic quantum phenomena.

Anisotropy-driven response of the fractional antiferromagnetic skyrmion lattice in MnSc2S4 to applied magnetic fields

We theoretically and experimentally study the stability of the unconventional fractional antiferromagnetic skyrmion lattice (AF-SkL) in ${\mathrm{MnSc}}_{2}{\mathrm{S}}_{4}$ spinel under magnetic fields applied along the [1-10] crystal direction. By performing numerical Monte Carlo simulations for the minimal effective spin model that we proposed in S. Gao et al., Nature 586, 37 (2020), we show that the lattice is aligned within the equivalent and symmetric [1-11] or [1-1-1] planes, which are equally inclined to the applied magnetic-field $H$. We attribute this behavior to the magnetic anisotropy of the host material. Neutron single-crystal diffraction presents a very good agreement with the predictions of the effective model. It reveals that the topological spin texture gets destabilized at low temperatures and moderate magnetic fields and is replaced by a conical phase for $H$// [1-10]. The present study elucidates the central role of the magnetic anisotropy in the stabilization of AF-Sk states.

Search for stable skyrmion lattices at the ground state in a multiferroic nanofilm using artificial neural networks

Magnetoelectric nanofilms are of great interest as functional elements of ultra-dense memory cells. In the ground state they may contain various topological magnetic vortex structures of several nanometers in size. The qualitative and quantitative properties of such structures strongly depend on a set of physical parameters. To calculate the ground state configuration with given parameters, we use the steepest descent method. However, to study a large parametric space significant computational resources are required. To solve this problem, we propose the use artificial neural networks (ANN), which can help us uncover the relationship between combinations of parameters and the corresponding ground state configurations using a relatively small number of pre-computed configurations as the training data. The application of the ANN allows one to avoid excessive computational costs in the study of the parametric space and narrow down the parametric area in which the existence of stable non-trivial ground state configurations in the form of a stable skyrmion crystal is possible.

Annihilation dynamics of magnetic skyrmion lattice state under in-plane magnetic field and spontaneous recovery after field removal

Nanoscale magnetic skyrmions are potential candidates for information carriers, for which the nonvolatility under external stimuli is critical to the stability of recorded bits. Here, we present a micromagnetic study on the annihilation dynamics of a two-dimensional skyrmion lattice under a transverse magnetic field and the recoverability of the topological charge after removing the field. The temporal and spatial correlation functions and their Fourier transforms of the spin system reveal position-dependent dynamics under the transverse field. The slow modes $(<10\phantom{\rule{0.16em}{0ex}}\mathrm{GHz})$ hosted by all spins determine the spin configurations, and the fast ones $(>10\phantom{\rule{4pt}{0ex}}\mathrm{GHz})$ around the skyrmion core region control the dynamics of topological properties. A dynamic critical time ${t}_{\mathrm{cri}}$, when the field-induced ferromagnetic nuclei begin to be long-range correlated, is proved by statistical analyses of their spatial distribution over time. If turning off the transverse field before ${t}_{\mathrm{cri}}$, the collective motion of the spin system can restore the previously lost topological charge. Our results guide the realization of nonvolatile information storage in skyrmion lattice, thus widening the possibility of skyrmion-based devices for future spintronics-based nanocomputing.