Skyrmion Number Research Articles

Dynamical switching of confined magnetic skyrmions under circular magnetic fields

Understanding the dynamical properties of magnetic skyrmions is of prime importance for realization of skyrmion based devices. Here we report on the dynamics of a confined magnetic skyrmion under application of circular magnetic fields (CMFs). We provide the complex map of the dynamical states and demonstrate that under some conditions CMFs can change the skyrmion number and lead to the switching of the core polarity. Combing the results of micromagnetic simulations with those of analytical calculations, based on the extended Thiele equation, we unravel the physical mechanism behind this dynamical switching. The core switching is associated with the dynamical motions of the skyrmion driven by a CMF. The formation of domain walls and their movement from the border towards the center leads to shrinking of the skyrmion and the creation of a skyrmion with opposite core polarity. The results provide a guideline for checking the skyrmion's stability as well as an efficient way for switching the core polarity using CMFs.

Research progress of room temperature magnetic skyrmion and its application

<sec>It has been found that many magnetic materials possess the properties arising from skyrmions at room temperature. In addition to the common interaction energy, chiral interaction is also needed to form the skyrmion in magnetic material. There are four chiral magnetic interactions, namely: 1) Dzyaloshinskii-Moriya (DM) interaction; 2) long-ranged magnetic dipolar interaction; 3) four-spin exchange interaction; 4) frustrated exchanged interaction. Through the competition between exchange interaction and chiral interaction, magnetic skyrmion can be realized in magnetic material subject to a certain magnetic field and temperature. The skyrmion generated by the DM interaction features small size (5–100 nm), which is easy to adjust. The skyrmion can be driven by magnetic field or ultralow current density. The magnetic materials with skyrmion can exhibit the properties related to the skyrmion Hall effect, the topological Hall effect and the emergent electrodynamics, which are closely related to the skyrmion number. The existence of skyrmion in the magnetic material can be indirectly measured by topological Hall effect. The movement of skyrmion can be driven by spin polarized current in the direction either parallel or perpendicular to the current direction. The movement of the skyrmion driven by spin polarized currents will continue when the current is present, and will disappear when the current disappears. </sec><sec>In previous studies, magnetic skyrmions were realized in a variety of materials. However magnetic skyrmions were found only in very limited types of single crystal materials at room temperature or near room temperature. In recent years, scientists have discovered a variety of magnetic skyrmion materials at room temperature, including film materials (such as multilayer materials, artificial skyrmion materials) and crystal materialssuch as <i>β</i>-Mn-type Co<sub>10–<i>x</i>/2</sub>Zn<sub>10–<i>x</i>/2</sub>Mn<i><sub>x</sub></i>, Fe<sub>3</sub>Sn<sub>2</sub>. Among all kinds of room temperature magnetic skyrmion materials, the most valuable one is the multilayer film material. The Skyrmion multilayer film has the advantages of small size, adjustable material type, simple preparation, good temperature stability, good device integration,etc. At the same time, skyrmion multilayer film is very easy to optimize by adjusting and constructing a special structure that has the wanted types of materials each with a certain thickness. Artificial skyrmion material obtains artificial skyrmion by constructing a micro-nano structure, therefore the artificial skyrmion with high-temperature stability can be realized by choosing high Curie temperature materials. There are a variety of materials which can realize the skyrmion above room temperature, such as Co<sub>9</sub>Zn<sub>9</sub>Mn<sub>2</sub> (300–390 K) and Fe<sub>3</sub>Sn<sub>2</sub> (100–400 K). These room temperature materials further widen the temperature application range of skyrmion. The room temperature materials can be prepared or characterized by a variety of techniquesincluding sputtering for fabrication and X-ray magnetic circular dichroism-photoemission electron microscopy (XMCD-PEEM) for characterization. </sec><sec>The discovery of the magnetic skyrmion materials at room temperature not only enriches the research content of materials science, but also makes the skyrmion widely applicable in novel electronic devices (such as racetrack memory, microwave detector, oscillators). Because the skyrmion has the advantages of small size, ultra-low driving current density, and topological stability, it is expected to produce racetrack memory based on the skyrmion with low energy consumption, non-volatile and high density. The MTJ microwave detector based on skyrmion can be achieved with no external magnetic field nor bias current but with low power input (< 1.0 μW); the sensitivity of the microwave detector can reach 2000 V·W<sup>–1</sup>. The frequency of the oscillator based on skyrmion can be tuned by magnetic field or current, and moreover, the oscillato is very easy to integrate with IC. In this paper, first, the basic characteristic of magnetic skyrmion is introduced; and then room temperature magnetic skyrmion is reviewed; finally the advances of the racetrack memory, microwave detectors and oscillators are introduced, highlighting the development trend of room temperature magnetic skyrmion. </sec>

Stability of the 2pi-vertical-Bloch-line in a canted antiferromagnetic system

We investigate topological defects in canted antiferromagnetic materials with the orthoferrite as a model system, and demonstrate that a vertical Bloch line (VBL), a topological defect within magnetic domain walls, can couple to the other VBL to form a stable antibonding state. Actually, such a 2pi-VBL is a topologically non-trivial object with a skyrmion number equivalent to that of the skyrmion itself. We demonstrate that transitions to other topological objects are possible only when a strong enough magnetic field is applied; the 2pi-VBL can transform to two-independent pi-VBLs under the magnetic field applied along the orthorhombic a-axis, and it can be annihilated with the magnetic field applied along the c-axis. As long as the 2pi-VBL is in a stable condition, it can be driven by the magnetic field to move straightly with a high speed of about 2000 m/s. This unique property of the 2pi-VBL can provide us with a possibility to exploit it as another topological magnetic defect for the spintronic applications.

Skyrmions on 2D elastic surfaces with fixed boundary frame

We report simulation results of skyrmions on fluctuating 2D lattices, where the vertices ri(∈R3) are treated as a dynamical variable and, hence, there is no crystalline structure. On the fluctuating surfaces, an external magnetic field perpendicular to the surface, Dzyaloshinskii-Moriya and ferromagnetic interactions are assumed in addition to the Helfrich-Polyakov Hamiltonian for membranes. The surface (or frame) tension τ is calculated under both isotropic and uniaxial strain conditions, and this calculation clarifies a non-trivial dependence of τ on the skyrmion, stripe, and ferromagnetic phases. We find that the variation of τ with respect to the applied magnetic field in the skyrmion phase is accompanied by a variation of the total number of skyrmions. Moreover, we find that this total number variation is qualitatively consistent with a recent experimental result for the creation/annihilation of skyrmions of 3D crystalline material under uniaxial stress conditions. It is also found that the stripe phase is significantly influenced by uniaxial strains, while the skyrmion phase remains unchanged. These results allow us to conclude that the skyrmion phase is stable even on fluctuating surfaces.

Unconventional topological Hall effect in high-topological-number skyrmion crystals

Skyrmions with the topological number Q equal to an integer larger than 1 are called high-topological-number skyrmions or high-Q skyrmions. In this work, we theoretically study the topological Hall effect in square-lattice high-Q skyrmion crystals (SkXs) with Q = 2 and Q = 3. As a result of the emergent magnetic field, the Landau-level-like electronic band structure gives rise to quantized Hall conductivity when the Fermi energy is within the gaps between the adjacent single band and intertwined multiple bands. We found that different from the conventional (Q = 1) SkX, the Hall quantization number increases by 1/Q on average when the elevating Fermi energy crosses each band. We attribute the result to the fact that the Berry phase C is measured in the momentum space and the topological number of a single skyrmion Q is measured in the real space. The reciprocality does not affect the conventional SkX because Q = 1 = 1/Q.

Current-induced nucleation, manipulation, and reversible switching of antiskyrmioniums

Magnetic skyrmioniums are unique spin structures that can be viewed as two skyrmions with an opposite skyrmion number providing a zero topological charge. Here, we propose a unique method to nucleate magnetic antiskyrmioniums in ferromagnetic materials with D2d symmetry. Using micromagnetic simulations, we show the stabilization of isolated antiskyrmionium from ferromagnetic as well as antiskyrmionic ground states. By invoking material specific parameters, we have established the regions of the stable antiskyrmionium phase and investigated their dynamics under current in-plane as well as perpendicular to plane geometry. Our study shows that antiskyrmioniums can be moved with a large velocity at very small current density in comparison to antiskyrmions and their anisotropic magnetization distribution provides a distinct current driven dynamics in comparison to skyrmioniums. Most importantly, we demonstrate that an antiskyrmionium can be stabilized and switched reversibly from a single antiskyrmion. We have also shown that the antiskyrmionium can be driven efficiently in an anisotropy gradient, where the velocity varies inversely with the damping constant. The present results add a distinct direction to the realization of antiskyrmionium based racetrack devices using D2d symmetry based materials.

Fractional Skyrmion and Absence of Low-lying Andreev Bound States in a Micro Fractional-flux Quantum Vortex

We investigate quasi-particle excitation modes and the topological number of a fractional-flux quantum vortex in a layered (multi-component) superconductor. The Bogoliubov equation for a half-flux quantum vortex is solved to show that there is no low-lying Andreev bound state near zero energy in the core of a quantum vortex, which is surprisingly in contrast to the result for an inter-flux vortex. Related to this result, there are singular excitation modes that have opposite angular momenta, moving in the opposite direction around the core of the vortex. The topological index (skyrmion number) for a fractional-flux quantum vortex becomes fractional since the topological index is divided into two parts where one from the vortex (bulk) and the other from the kink (domain wall, boundary). The topological numbers for both the vortex and the kink (domain wall) are fractional, and their sum becomes an integer. This shows an interesting analogy between this result and the index theorem for manifolds with boundary. We argue that fractional-flux quantum vortices are not commutative each other and follow non-abelian statistics. This non-abelian statistics of vortices is different from that in p-wave superconductors.

Dynamics of antiskyrmions induced by the voltage-controlled magnetic anisotropy gradient

Magnetic skyrmions could serve as information carriers in next-generation information storage and processing devices. Effective manipulation of the skyrmion motion is a vital task for realizing skyrmion-based applications. Skyrmion motion driven by voltage-controlled magnetic anisotropy (VCMA) has been proposed, of which the investigation is mostly focused on the dynamics of skyrmions. Here, using micromagnetic simulations we show that antiskyrmions can be effectively driven by the VCMA gradient in the presence of the anisotropic Dzyaloshinskii-Moriya interaction. We find that the antiskyrmion shows a transverse motion, i.e., the skyrmion Hall effect, induced by the VCMA gradient. As the skyrmion number of an antiskyrmion is opposite to that of a skyrmion, the anisotropy gradient-induced motion of ferromagnetically coupled skyrmion-antiskyrmion pair and antiferromagnetically coupled skyrmion-skyrmion and antiskyrmion-antiskyrmion pairs do not show the skyrmion Hall effect. We find that the velocity-driving force relations of these three combinations are different. Our results may provide guidelines for building future skyrmion-based information processing applications.

Manipulating movement of skyrmion by strain gradient in a nanotrack

Abstract Magnetic skyrmions are swirling nanometric spin textures, which have attracted extensive attention by virtue of their potential applications as information carriers for future memory technology and spintronic devices. Here, we investigate the effect of strain gradient on the in-plane current-driven motion of magnetic skyrmion in a nanotrack. It is found that a longitudinal strain gradient will induce a transverse velocity with its value linearly proportional to the strain gradient. As well, we find that skyrmions can be accelerated or decelerated by the boundary of nanotrack. This behavior depends on the repulsive force induced by boundary and the topological number of skyrmion. Our results not only explain the effect of strain gradient, but also show the influence of the boundary on the skyrmion motion. This work may provide a guide for designing and developing the racetrack memory based on magnetic skyrmions.

Dynamics of a magnetic skyrmionium in an anisotropy gradient

A magnetic skyrmionium is a novel magnetization configuration with a zero skyrmion number, which can be viewed as a combination of two skyrmions with opposite skyrmion numbers. Here, we analytically and numerically study the dynamics of a skyrmionium under an anisotropy gradient. We find that the skyrmionium moves straight and can be efficiently driven by an anisotropy gradient with higher speed. Furthermore, skyrmionium deformation in a larger anisotropy gradient can be effectively suppressed by narrowing the width of the film. Our work shows an alternative driving method that may be promising for applications of skyrmionium-based racetrack memory.

Surface pinning and triggered unwinding of skyrmions in a cubic chiral magnet

In the cubic chiral magnet Fe$_{1-x}$Co$_{x}$Si a metastable state comprising of topologically nontrivial spin whirls, so-called skyrmions, may be preserved down to low temperatures by means of field cooling the sample. This metastable skyrmion state is energetically separated from the topologically trivial ground state by a considerable potential barrier, a phenomenon also referred to as topological protection. Using magnetic force microscopy on the surface of a bulk crystal, we show that certain positions are preferentially and reproducibly decorated with metastable skyrmions, indicating that surface pinning plays a crucial role. Increasing the magnetic field allows an increasing number of skyrmions to overcome the potential barrier, and hence to transform into the ground state. Most notably, we find that the unwinding of individual skyrmions may be triggered by the magnetic tip itself, however, only when its magnetization is aligned parallel to the external field. This implies that the stray field of the tip is key for locally overcoming the topological protection. Both the control of the position of topologically nontrivial states as well as their creation and annihilation on demand pose important challenges in the context of potential skyrmionic applications.

Radial vortex core reversal driven by a spin-polarized current in a nanocontact structure

In this paper, we propose a reversal mechanism of magnetic radial vortex core in a Permalloy nanodisk driven by a spin-polarized current injected through a nanocontact. The micromagnetic simulation results indicate that the reversal process of radial vortex core is through the rotation and fusion of an edge soliton pair. Due to the effect of interfacial Dzyaloshinskii–Moriya interaction, the radial chirality and polarity of radial vortex are reversed simultaneously. The transformation is uncovered by detailed studies of magnetization configuration, skyrmion number, energy and effective field. This reversal mechanism of radial vortex differs entirely from the previous mechanisms reported in the literatures. This work not only provides deeper insights into the dynamic properties of magnetic radial vortices, but also has great potential for developing new kinds of spintronic devices using the radial vortices.

Current-Driven Dynamics of Frustrated Skyrmions in a Synthetic Antiferromagnetic Bilayer

We report the current-driven dynamics of frustrated skyrmions in an antiferromagnetically exchange coupled bilayer system, where the bilayer skyrmion consists of two monolayer skyrmions with opposite skyrmion numbers $Q$. We show that the in-plane current-driven bilayer skyrmion moves in a straight path, while the out-of-plane current-driven bilayer skyrmion moves in a circular path. It is found that the in-plane current-driven mobility of a bilayer skyrmion is much better than the monolayer one at a large ratio of $\beta/\alpha$, where $\alpha$ and $\beta$ denote the damping parameter and non-adiabatic spin transfer torque strength, respectively. Besides, the out-of-plane current-driven mobility of a bilayer skyrmion is much better than the monolayer one when $\alpha$ is small. We also reveal that one bilayer skyrmion (consisting of monolayer skyrmions with $Q=\pm 2$) can be separated to two bilayer skyrmions (consisting of monolayer skyrmions with $Q=\pm 1$) driven by an out-of-plane current. Our results may be useful for designing skyrmionic devices based on frustrated multilayer magnets.

Topological Hall effect in multilayers

Physics Skyrmions—topologically protected whirlpools of spin—are a candidate for information processing in magnetic materials. In early work, skyrmions were found in bulk materials at relatively low temperatures. More recently, room temperature skyrmions have been detected in thin multilayers, but understanding their behavior is tricky, thanks to a combination of complex electronic structure, disorder, and interfacial effects. Raju et al. used two complementary techniques—magnetic force microscopy and Hall transport—to tease out how skyrmions affect charge transport in these multilayers. The number of skyrmions seen through microscopy scaled with the topological contribution to the Hall effect, but a quantitative agreement between the results of the two probes requires taking into account interactions between skyrmions and the detailed properties of the system. Nat. Commun. 10, 696 (2019).

Observation of room-temperature polar skyrmions.

Complex topological configurations are fertile ground for exploring emergent phenomena and exotic phases in condensed-matter physics. For example, the recent discovery of polarization vortices and their associated complex-phase coexistence and response under applied electric fields in superlattices of (PbTiO3)n/(SrTiO3)n suggests the presence of a complex, multi-dimensional system capable of interesting physical responses, such as chirality, negative capacitance and large piezo-electric responses1-3. Here, by varying epitaxial constraints, we discover room-temperature polar-skyrmion bubbles in a lead titanate layer confined by strontium titanate layers, which are imaged by atomic-resolution scanning transmission electron microscopy. Phase-field modelling and second-principles calculations reveal that the polar-skyrmion bubbles have a skyrmion number of +1, and resonant soft-X-ray diffraction experiments show circular dichroism, confirming chirality. Such nanometre-scale polar-skyrmion bubbles are the electric analogues of magnetic skyrmions, and could contribute to the advancement of ferroelectrics towards functionalities incorporating emergent chirality and electrically controllable negative capacitance.

Dynamics of skyrmion in disordered chiral magnet of thin film form

Magnetic skyrmion is a topological spin texture characterized by the mapping from the two dimensional real space to the unit sphere. It is realized in chiral magnets under an external magnetic field in the plane perpendicular to it. In thin film samples, which are most relevant to the applications, the thickness of the system parallel to the magnetic field is finite, and a skyrmion turns into a skyrmion string, which is often assumed to be a straight rod. There are phenomena related to the internal degrees of freedom along the string, e.g., the monopole and anti-monopole creation/annihilation, corresponding to the change in the skyrmion number. However, the role of this finite thickness in the topological stability and dynamics has not been explored yet. Here we study theoretically the current-driven dynamics of a skyrmion string under disorder potential by systematically changing the thickness of the sample to reveal the dynamical phase diagram in the plane of current density and thickness. We found the three regions, i.e., (i) pinned skyrmion string, (ii) moving depinned skyrmion string, and (iii) annihilation of skyrmion string, for thin and thick limits while (iii) is missing in the intermediate case. This indicates that there is the optimal range of thickness for the topological stability of skyrmion string enhanced compared with a two-dimensional skyrmion. This result provides a way to design and control skyrmions in thin films and interfaces of finite thickness.

Micromagnetic simulations study of skyrmions in magnetic FePt nanoelements

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 magnetocrystalline anisotropy direction, with respect to the normal to the triangular prism base. Micromagnetic configurations with skyrmion number greater than one have been detected for the case where magnetocrystalline anisotropy was normal to nanoelement's base. For particular magnetocrystalline anisotropy values three distinct skyrmions are formed and persist for a range of external fields. The simulation-based calculations of the skyrmion number S revealed that skyrmions can be created for magnetic nanoparticle systems lacking of chiral interactions such as Dzyaloshinsky-Moriya, but by only varying the magnetocrystalline anisotropy.

Realization of skyrmion subtracter and diverter in a voltage-gated synthetic antiferromagnetic racetrack

Magnetic skyrmions are promising candidates for next-generation information carriers. Many concepts and prototypes for skyrmion-based devices have been proposed, and most of the studies are based on the motion of isolated skyrmion. Nevertheless, so far, a comprehensive research of multi-skyrmion motion and related device design are still lacking. In this work, a skyrmion subtracter and a skyrmion diverter based on multi-skyrmion motion are numerically demonstrated in a voltage-gated synthetic antiferromagnetic racetrack. It shows that the number of skyrmions along an array n can be subtracted by an integer from 1 to n by proper control of the driving current density and the voltage-controlled magnetic anisotropy in a narrow racetrack, realizing the function of a skyrmion subtracter. Moreover, when the width of racetrack increases, a single array of skyrmions along the racetrack direction can redistribute along the voltage gate due to the blocking effect of the voltage-controlled magnetic anisotropy. As a result, the single array of skyrmions can split into several arrays, realizing the function of a skyrmion diverter. Our results thus provide guidelines for designing novel racetrack-type skyrmionic devices.

Magnetic skyrmions in FePt nanoparticles having Reuleaux 3D geometry: a micromagnetic simulation study.

The magnetization reversal in magnetic FePt nanoelements having Reuleaux 3D geometry is studied using micromagnetic simulations employing Finite Element discretizations. Magnetic skyrmions are revealed in different systems generated by the variation of the magnitude of the magnetocrystalline anisotropy which was kept normal to the nanoelement's base and parallel to the applied external field. The topological quantity of skyrmion number is computed in order to characterize micromagnetic configurations exhibiting skyrmionic formations. Micromagnetic configurations with a wide range of skyrmion numbers between -3 and 3 are indicative for the existence of one or multiple skyrmions that have been detected and stabilized in a range of external fields. Internal magnetic structures are shown consisting of Bloch type skyrmionic entities in the bulk altered to Néel skyrmions on the nanoelement's bottom and top base surfaces. The actual sizes of the formed skyrmions and the internal magnetization structures were computed. In particular, the sizes of the generated and persistent skyrmions were calculated as functions of the magnetocrystalline anisotropy value and of the applied external magnetic field. It is shown that the size of skyrmions is linearly dependent on the external field value. The slope of the linear curve can be controlled by the magnetocrystalline anisotropy value. The magnetic skyrmions can be created for FePt magnetic systems lacking of chiral interactions by designing the geometry-shape of the nanoparticle and by controlling the value of magnetocrystalline anisotropy.

Nonequilibrium phases and segregation for skyrmions on periodic pinning arrays

Using particle-based simulations, we examine the collective dynamics of skyrmions interacting with periodic pinning arrays, focusing on the impact of the Magnus force on the sliding phases. As a function of increasing pinning strength, we find a series of distinct dynamical phases, including an interstitial flow phase, a moving disordered state, a moving crystal, and a segregated cluster state. The transitions between these states produce signatures in the skyrmion lattice structure, the skyrmion Hall angle, the velocity fluctuation distributions, and the velocity-force curves. The moving clustered state is similar to the segregated state recently observed in continuum-based simulations with strong quenched disorder. The segregation arises from the drive dependence of the skyrmion Hall angle, and appears in the strong pinning limit when the skyrmions have nonuniform velocities, causing different portions of the sample to have different effective skyrmion Hall angles. We map the evolution of the dynamic phases as a function of the system density, the ratio of the Magnus force to the dissipative term, and the ratio of the number of skyrmions to the number of pinning sites.