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Magnetic skyrmions are a prime example of topologically non-trivial spin textures observed in a variety of magnetic materials. They emerge when the exchange and anisotropy energies promoting parallel alignment of spins in a ferromagnet enter in competition with energies favoring non-collinear alignment of spins such as the Dzyaloshinskii-Moriya interaction (DMI), the long-range dipolar interaction or higher-order exchange interactions. The orthodox theory of skyrmions in ultrathin ferromagnetic layers with interfacial DMI relies on a model that accounts for the dipolar interaction through an effective anisotropy term, neglecting long-range effects. At the same time, in single ferromagnetic layers with interfacial DMI, large chiral skyrmions, also called skyrmionic bubbles have been observed, suggesting a non-trivial interplay between DMI and long-range dipolar effects[1]. The competition between these two energies also leads to the formation of skyrmions exhibiting spin rotations with intermediate angles between Neel and Bloch, a phenomenon also present in domain walls. In addition, there is a growing body of theoretical evidence that points to a need to take into account the long-range dipolar energy in the models describing magnetic skyrmions. The above considerations put into question the validity of the commonly used assumption that the long-range contribution of the dipolar interaction is negligible.&lt;br/&gt;Here we use rigorous mathematical analysis to develop a skyrmion theory that takes into account the full dipolar energy in the thin film regime and provides analytical formulas for compact skyrmion radius, rotation angle and energy[2,3]. We demonstrate that the DMI threshold at which a compact skyrmion loses its Néel character is a factor of 3 higher than that for a single domain wall. A reorientation of the skyrmion rotation angle from Néel to intermediate Néel-Bloch angles is predicted as the layer thickness is increased in the low DMI regime, which is confirmed by micromagnetic simulations. We will also present an extension of our model which includes applied magnetic field.

2022 Joint MMM-INTERMAG

Theory of Néel-Bloch transition for compact magnetic skyrmions

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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MMM 2020

Magnetic skyrmions are a prime example of topologically non-trivial spin textures observed in a variety of magnetic materials. They emerge when the exchange and anisotropy energies promoting parallel alignment of spins in a ferromagnet enter in competition with energies favoring non-collinear alignment of spins such as the Dzyaloshinskii-Moriya interaction (DMI), the long-range dipolar interaction or higher-order exchange interactions. The orthodox theory of skyrmions in ultrathin ferromagnetic layers with interfacial DMI relies on a model that accounts for the dipolar interaction through an effective anisotropy term, neglecting long-range effects. At the same time, in single ferromagnetic layers with interfacial DMI, large chiral skyrmions, also called skyrmionic bubbles have been observed, suggesting a non-trivial interplay between DMI and long-range dipolar effects[1]. The competition between these two energies also leads to the formation of skyrmions exhibiting spin rotations with intermediate angles between Neel and Bloch, a phenomenon also present in domain walls. In addition, there is a growing body of theoretical evidence that points to a need to take into account the long-range dipolar energy in the models describing magnetic skyrmions. The above considerations put into question the validity of the commonly used assumption that the long-range contribution of the dipolar interaction is negligible. Here we use rigorous mathematical analysis to develop a skyrmion theory that takes into account the full dipolar energy in the thin film regime and provides analytical formulas for compact skyrmion radius, rotation angle and energy[2,3]. We demonstrate that the DMI threshold at which a compact skyrmion loses its Néel character is a factor of 3 higher than that for a single domain wall. A reorientation of the skyrmion rotation angle from Néel to intermediate Néel-Bloch angles is predicted as the layer thickness is increased in the low DMI regime, which is confirmed by micromagnetic simulations. We will also present an extension of our model which includes applied magnetic field.

technical paper

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

In magnetic trilayers composed of heavy-metal/ferromagnet/oxide (HM/FM/MOx), the structural inversion asymmetry gives rise to the so-called interfacial Dzyaloshinskii-Moriya interaction (iDMI) [1]. This antisymmetric exchange interaction is at the origin of non-collinear spin textures, such as spin spirals or magnetic skyrmions [2] with chiral Néel domain-walls (DWs). Their dynamics is directly related to their chirality, imposed by the iDMI sign [3]. Thus, controlling their current-induced motion (CIM) driven by spin-orbit torques can be achieved by controlling the iDMI. In this study, we propose techniques to control the skyrmion motion in a Ta/FeCoB/TaOx trilayer, through the control of the iDMI sign and amplitude. The skyrmion motion can be controlled, or even reversed, either by modulation of deposition parameters or dynamically by applying a gate voltage. We have found that modulation of the oxidation state at the FeCoB/TaOx interface (Fig. 1(a-b)) leads to an inversion of the iDMI sign, as measured by Brillouin light scattering. This iDMI sign crossover is accompanied by an inversion of the skyrmion CIM direction. Skyrmion motion reversal can also be triggered by a gate voltage through the control of iDMI sign (Fig. 1(c-d)). Moreover, our micromagnetic simulations show that a fine control of the iDMI amplitude and sign is feasible on individual nanometer-size magnetic skyrmion without annihilating it (Fig. 2(a-c)). The dipolar energy prevents the spin texture from annihilation when iDMI is close to zero, leading to a stable Bloch skyrmion (Fig 2b). The angle of the moments inside the DWs (Fig. 2d) evolves continuously from 0° to 180° when varying the iDMI value. This allows a fine tuning of the skyrmion CIM direction (Fig. 2e), allowing to suppress the skyrmion Hall angle, otherwise detrimental for device applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/0d5c24338e22f487f485d4062d620b55.jpg)

Figure 1: Control of iDMI sign : (a-b) Modulation of the FeCoB/TaOx interface oxidation and (c) Gate-controlled skyrmion motion (Magneto-Optical Kerr Effect images)

![](https://assets.underline.io/uploads/markdown_image/1/image/9276aafb08d714e75ddb64319fe097bf.jpg)

Figure 2: (a-c) Micromagnetic simulations : Adiabatic iDMI reversal on a nm-size magnetic skyrmion leading to (d) continuous transformation from CW (D<0) to CCW chirality (D>0). (e) Skyrmion's trajectory controlled by iDMI value

Get skyrmions back on track : Suppressing skyrmion Hall angle by material engineering or gate voltage

Magnetic Skyrmions are topologically protected swirling structures induced by chiral interactions in non-centrosymmetric magnetic compounds or thin films with broken inversion symmetry [1]. The spintronic devices based on these textures promise increased density and energy efficient data storage due to small nanometric size and topological protection [2][3]. In this work we present the effect of DMI, surface anisotropy, magnetic field and spin orbit torque (SOT) on the creation and annihilation of the skyrmions. Fig.1 (a) shows that the skyrmion density is increased as we increase the magnitude of DMI coefficient from 1mJ/m2 to 3mJ/m2. But with increasing the surface anisotropy the density is reduced and skyrmions are annihilated at some threshold anisotropy Kth which is proportional to the magnitude of the DMI coefficient. Furthermore, the size of skyrmion decreases with increasing anisotropy as shown in Fig. 1(b). In Fig. 1 (c-d) we study the effect of external magnetic field on the skyrmions. Fig. 1 (c) shows the MOKE images of the fabricated sample depicting the transition of magnetic texture from Labriynth domains to skyrmions around 8 Oe. The skyrmion density increases proportionally with field till 20 Oe and begins to roll off after 20 Oe. These results are supported by MuMax simulations as shown in Fig. 1(d). These results suggest that by proper optimization of material parameters and stack geometry we can tweak the skyrmion density for its application in memory and logic. In Fig. 2(a) we show MTJ with skyrmions in the free layer, using SOT control we propose this device for its application in neuromorphic computing. The SOT manipulation of skyrmion size, density and motion is shown in Fig. 2 (b). Increasing current Ic via the heavy metal increases the skyrmion density from 21µm-2 (at 2×1011 A/m2) to 40 21µm-2 (at 8×1011 A/m2). Further increase in Ic results in decrease of the skyrmion density. Finally, we implement a simple neuromorphic circuit using the proposed device.

![](https://assets.underline.io/uploads/markdown_image/1/image/0451b4c2cb6a4b63949ea900b0511b43.jpg)

Fig. 1: (a) Skyrmion density vs anisotropy and DMI coefficient. (b) Density and size of skyrmion vs anisotropy. (c) MOKE images (d) MuMax simulations.

![](https://assets.underline.io/uploads/markdown_image/1/image/363eb0a3db7cbc306cb4f2f3e42c459d.jpg)

(a) Skyrmion-MTJ neuromorphic device. (b) Skyrmion density vs current density

Creation and Annihilation of Skyrmions for Neuromorphic Computing Applications

Skyrmion-based devices have been proposed as candidates for future low-power, ultra-dense nanocomputing and neuromorphic systems enabled by the characteristic properties (nanoscale size, non-volatility, robustness, low current density of propagation) of magnetic skyrmions, arising from their topology [1-2]. This is more pressing especially since the recent experimental demonstrations at room temperature (e.g., in technologically relevant magnetic multilayers (MMLs) [3-4]). At the same time, skyrmions have attracted extensive interest in CMOS-compatible nanodevices that can emulate biological synapses. Specifically, a multi-bit storage device using skyrmions as information carriers (bits) has been proposed in ideal conditions [5-6]. However, room temperature operation is required to integrate skyrmionic synapses in practical future devices. In our recent work [7], we propose a nanoscale skyrmionic synapse composed of MMLs that enables room temperature device operation (Fig. 1). We demonstrate that when embedding such MML skyrmionic synapses in a simple spiking neural network (SNN) with the spike-timing-dependent plasticity (STDP) learning rule, we can achieve only a ~75% classification accuracy in the MNIST handwritten data set. We propose that this performance can be significantly improved to ~98.61% by using a deep SNN with supervised learning rules. Our results illustrate that the proposed skyrmionic synapse can be a competitive candidate for future energy-efficient neuromorphic computing and edge computing.

![](https://assets.underline.io/uploads/markdown_image/1/image/ab5e21eb72c5f492aa06ba6e24bb2d6f.jpg)

Fig. 1. Schematic of biological neurons connected with a synapse, and the proposed nanoscale multilayer skyrmionic synapse device. The multilayer structure enables room-temperature operations.

Nanoscale room-temperature skyrmionic synapse for deep spiking neural networks

Skyrmionic devices show potential for low-power-low-heat nanocomputing with multiple information carriers, enabled by quasi-particles with a topological charge N, including skyrmions (N=±1) [1] and skyrmioniums (N=0) [2-3]. This is more pressing since the co-existence of stable and high-degree skyrmionic configurations with arbitrary value and sign of N has been experimentally demonstrated in liquid crystals and numerically predicted in ferromagnetic systems [4-5]. Hitherto, skyrmionic on-site memory, nanocomputing, and neuromorphic components have increasingly been investigated [6-7]. However, the “missing component” in skyrmionics so far has been an all-magnetic skyrmion-based interconnect, which may be relevant to future spintronic neuromorphic computing and quantum computing hardware. Our recent work investigates the dynamics of magnetic skyrmionic quasi-particles for a new concept for skyrmionic nanodevices, i.e., skyrmionic interconnect device [8]. We propose encoding sequences of information by using distinct skyrmionic quasi-particles as information carriers in a single device, such that we exploit topological filtering in order to enable automatic signal multiplexing (Fig. 1), circumventing the need for control signals. This results in saving signal routing by 50%, a major benefit compared to modern CMOS devices. We then propose to strategically etch notches in the device to introduce pipelining, which significantly boosts the signal throughput of the device. Our proposal widens the possibilities for spintronics-based nanocomputing.

![](https://assets.underline.io/uploads/markdown_image/1/image/28d051e7ad5fe9607f48995d96d5c5e1.jpg)

Fig. 1. Schematic drawing of the proposed skyrmionic interconnect device, and spin texture of a Néel skyrmion and a skyrmionium.

Skyrmionic Interconnect Device

Inversion symmetry breaking (ISB) is of great significance in governing many interesting physics. For example, the interfacial Dzyaloshinskii-Moriya interaction (DMI) exists at the heavy metal/ferromagnet (HM/FM) interface with the ISB along the sample normal direction, which holds promises for design and control of chiral spin textures in low-dimensional magnets with efficient current-driven dynamics [1,2]. In this talk, we discuss about a synthetic antiferromagnet (SAF) with built-in planar asymmetry which is attributed to both variable ferromagnetic-layer thickness and the spacer thickness. First we show how the sign and strength of the interlayer exchange coupling (IEC) and magnetic configurations depend on the ferromagnetic-layer thicknesses and the spacer thickness [3]. Interestingly, the IEC can be efficiently modified via changing the ferromagnetic-layer thickness which is essential to advancing magnetic memory and sensing [4] applications with efficient controls. Furthermore, we show the demonstration of the existence of interlayer DMI in our SAF with the built-in planar asymmetry, which supports a chiral magnetization across the SAF. We present the study of current-induced spin-orbit torque (SOT) switching of the chiral magnetization across the SAF and analyse the interlayer DMI’s effects on magnetization and its switching through a combination of experimental and numerical studies [5]. Based on our studies, we will give an outlook of potential applications of our findings in future SOT and domain wall-based devices.

Synthetic antiferromagnet with built-in planar asymmetry

Magnetic skyrmions are topological spin textures that can be stabilized by the Dzyaloshinskii-Moriya interaction [1,2]. There are particularly suitable for next generation spintronics devices, like the skyrmion-based racetrack memory [3]. Recent studies confirmed their current-driven skyrmion dynamics in ultrathin ferromagnets via spin-orbit torques [4], where the chirality and spin texture of the skyrmions are key. However, the topological Magnus effect leads to a transverse motion of ferromagnetic skyrmions due to their non-zero topological charge [5], which is disadvantageous for devices. Antiferromagnetically exchange-coupled skyrmions or compensated ferrimagnets could suppress this effect owing to an overall zero topological charge [6]. Here we explore a GdFeCo-based ferrimagnet system with perpendicular magnetic anisotropy. We demonstrate that in this system the chirality of worm domains as well as a magnetic skyrmionium can be observed using scanning electron microscopy with polarization analysis (SEMPA) (Figure) [7]. The high spatial resolution magnetic imaging technique reveals the domain wall (DW) spin structure as a function of temperature, with the observed structures found to be pure Néel-type spin textures, which is promising for devices. From the images we extract the DW width as a function of the temperature and we calculate the exchange stiffness for the system. Figure: Determination of the magnetization direction inside the DW of different ferrimagnetic chiral spin textures. a) The direction of the in-plane magnetization in the DW is displayed, as defined by the color wheel in the bottom left corner of the image. Scale bar in a): 1 µm. b) Distribution of the direction of the in-plane magnetization in the DW with respect to the local tangent at RT: angle Ψ (see inset where a clockwise angle is defined as negative following the usual convention). A Gaussian fit indicates a central value around −90°. c) Absolute in-plane magnetization intensity with the DW skeleton displayed in white and d) direction of the in-plane magnetization in the DW of a ferrimagnetic skyrmionium at 320 K. Scale bars in c,d): 500 nm. Extracted from [7].

![](https://assets.underline.io/uploads/markdown_image/1/image/92e18cd0f6b430b4b7a16c0928992f0f.jpg)

Direct imaging of chiral domain walls in ferrimagnetic alloys

Skyrmions are excellent candidates for racetrack memory and synaptic devices1,2. However, their movement in ferromagnets (FM) is hindered by their transverse movement (skyrmions Hall effect, SkHE) and pinning at defects3. Ferrimagnetic (FiM) systems are proposed to overcome the transverse movement, while amorphous ferromagnets systems such as CoFeB have already shown significant reduction in pinning4,5 but yield skyrmions with larger diameters. Therefore, we propose a robust amorphous-ferrimagnetic system to bring forth the best of the two. So far, skyrmions have been generated in amorphous ferrimagnetic systems such as Pt/GdFeCo/MgO5 and Pt/CoGd/TaO6 where SkHE and pinning were considerably mitigated. However, these systems lack a certain robustness owing to the rapidly diminishing DMI at the FiM/Heavy metal (HM) interface with increase in rare-earth (RE) content. This hinders the concurrent attainment of small size, desired density, room temperature (RT) stability and high–mobility in skyrmions. To overcome this we followed a strategy previously used to generate dense, small skyrmions in FM7- employing two DMI generating interfaces of Ir/Fe-FiM and Co-FiM/Pt. The two DMIs at the interfaces being additive, strengthens the total DMI and provides a robust, tunable FiM system capable of addressing the above-mentioned challenges. Figure 1 shows the schematic, hysteresis loops and MFM images of a few of these multilayers. By tuning the rare-earth/3d transition magnetic material ratio and thicknesses in both the FiM layers, along with systematic change in the Tb/Gd ratio (tunes anisotropy) we were able to achieve a robust multilayer FiM system, with small (~50 nm), high-density skyrmions, and up to 88% reduction in the necessary energy to nucleate them. For instance, in 1c we show a considerably high-density of skyrmions, around ~72 nm in diameter, while the magnetization and nucleation fields are reduced by 70% and 88% respectively. Our multilayer system serves as a suitable template for designing desired skyrmion densities, sizes and reduced SkHE along with lower energy requirements for nucleation and possibly movement.

![](https://assets.underline.io/uploads/markdown_image/1/image/5bae4b2d492d8e480f3947ee0e9a336e.jpg)

Figure 1: Schematic, hysteresis loops, MFM (same scale in all images) of a) FM and b)-c) FiM

Robust and tunable rare-earth ferrimagnetic systems for dense, small and high-mobility skyrmions at room temperature

Skyrmions and antiskyrmions are topologically protected spin structures with opposite vorticities. Particularly in coexisting phases, these two types of magnetic quasiparticles may show fascinating physics and potential for spintronic devices. While skyrmions are observed in a wide range of materials until now antiskyrmions were exclusive to materials with D2d symmetry [1]. In our recent work, we have shown first and second-order antiskyrmions stabilized by magnetic dipole-dipole interaction in Fe/Gd-based multilayers [2]. Using Lorentz transmission electron microscopy imaging, we observed coexisting first and second-order antiskyrmions, Bloch skyrmions, and type-2 bubbles (Fig. 1) and determine the range of material properties and magnetic fields where the different spin objects form and dissipate. Phase diagrams of the spin objects were created in dependence on the magnetic out-of-plane field, temperature, saturation magnetization, and uniaxial magnetic anisotropy. We performed micromagnetic simulations to obtain more insight into the studied system and conclude that low values of saturation magnetization and uniaxial magnetic anisotropy lead to the existence of this zoo of different spin objects and that they are primarily stabilized by dipole-dipole interaction. Further, we investigated the nucleation process of antiskyrmions experimentally and theoretically revealing the necessity of a crossing point of three magnetic stripe domains to form an isolated antiskyrmion with an out-of-plane magnetic field. The previous unobserved second-order antiskyrmions and the disclosed coexistence of antiskyrmions and skyrmions potentially reachable for a whole range of different materials provide great potential for further studies on quasi-particle interactions, spin dynamics as well as for possible future applications in spintronics.

![](https://assets.underline.io/uploads/markdown_image/1/image/50c16dd6974021e442524cdd7ed4e0a3.jpg)

Fig. 1. Underfocused LTEM image of [Fe/Ir/Gd]2/[Fe/Gd]76/[Fe/Ir/Gd]2 showing five different coexisting spin objects (a). Bloch skyrmion with counter- (b) and with clockwise rotation (c), type-2 bubble (d), first-order antiskyrmion (e), and second-order antiskyrmion (f). Zoomed-in spin objects (b-f, I) and simulated LTEM contrast (b-f, II) caused by the theoretical spin textures (b-f, III).

Dipolar-Stabilized First and Second-Order Antiskyrmions in Ferrimagnetic Multilayers

It has been predicted and measured that topologically non-trivial magnetisations can provide an extra contribution to the Hall resistivity (Rxy), called the topological Hall effect (THE)1. Skyrmions have a topology defined by the winding number, S, which takes an integer value dependent upon the number of total ‘twists’ in their magnetisation. The THE has been shown to be present in many B20 materials with bulk Dzyaloshinkii-Moriya Interaction (DMI) hosting skyrmions but the picture is less clear for multilayers with interfacial DMI. Observations by Maccariello et al. saw resistivity effects that could be accounted for by only the ordinary and anomalous Hall effects (OHE & AHE)2. In contrast, work by Zeissler et al. saw an extra contribution beyond the OHE and AHE that was larger than the THE predicted by theory3. In this work, Hall transport measurements combined with imaging of the magnetic texture were used to identify such contributions to understand discrepancies between the predicted1, and observed2,3, magnitudes of the topological contribution. Measurements of Rxy(B) were performed in [Pt|CoB|Ir]xN multilayers with different repeat numbers, N, over a range of temperatures from 290 K to 2 K. Figure 1 shows the temperature dependence of the anomalous Hall resistivity at saturation, ρA, with an increase in size at T &lt; 20 K, for a sample with N=4. The Rxy measurements for the same sample were analysed in combination with SQUIDVSM magnetometry data to produce Figure 2. The maximum excess Hall resistivity, ρexcess, is observed to increase with temperature and cannot be accounted for by the ordinary and anomalous Hall effects alone. This implies an extra Hall effect that may be topological in nature suggesting that the maximum density of topological magnetisation textures increases with temperature.

![](https://assets.underline.io/uploads/markdown_image/1/image/1a86da2275fb423bada15ee48b4609a9.jpg)

Fig. 1 Anomalous Hall resistivity at saturation, ρA, vs T showing an increase in size for T < 20 K, in samples of Ta|Pt|[CoB|Ir|Pt]x4 Pt multilayers.

![](https://assets.underline.io/uploads/markdown_image/1/image/d834fd8cf6941127ea0fd52e6ae23c9a.jpg)

Fig. 2 Excess Hall resistivity (possibly topological in nature) that cannot be accounted for by ordinary or anomalous Hall effects, ρexcess, vs T, showing an increase in the height of the maximum peak as temperature increases.

Observation of Increasing Topological Hall Resistivity with Temperature in Skyrmion Hosting Magnetic Multilayers with [Pt|CoB|Ir]xN Structure

The real-space Berry phase associated with conduction carriers has recently attracted much attention from the viewpoints of fundamental science and spin-electronics applications. The noncoplanar spin textures responsible for the Berry phase usually appear during the magnetization reversal, often occur in systems with Dzyaloshinskii-Moriya interactions (DMI) due to, broken inversion symmetry, and yield skyrmions in specific temperature and magnetic-field ranges [1, 2]. One key aim in skyrmion research is to realize small feature sizes, which lead to a pronounced topological Hall effect (THE), at high temperatures. Confined geometries such as nanoparticles or clusters are potential systems for this purpose because the competing exchange and magnetocrystalline anisotropy of nanoparticles are expected to cause substantially noncoplanar noncollinear spin structures even in the absence of DMI. Here, we use experiments and micromagnetic simulations to investigate spin textures in exchange-coupled Co nanoclusters produced by inert-gas condensation-type cluster deposition. The nanoclusters have an average size of 14 nm and exhibit a significant topological Hall effect (THE), as shown in the experimental Hall data of Fig. 1(a). The micromagnetic THE simulations, Fig. 1(b), predict the skyrmion number Q by integration over the skyrmion density and are in good agreement with the experimental data. THE is caused by the noncoplanar noncollinear spin structure of the Co nanoclusters, visualized in Fig. 1(c). A characteristic feature of both experimental THE and theoretical prediction (skyrmion number Q) is a drastic change over a relatively small field range, which we interpret as a topological phase transition analogous to the Lifshitz transition in metals. This work is supported by NSF-DMREF (No. 1729288), NSF-EQUATE (OIA-2044049), the NU Collaborative Initiative, and NCMN,

![](https://assets.underline.io/uploads/markdown_image/1/image/1dc29675931a0b1064caa936e9c420a1.jpg)

Fig. 1. Topological Hall effect (THE) in Co nanoclusters: (a) measured THE at 300 K and 10 K, (b) calculated Q as a function magnetic field and (c) simulated spin structure. The insets in (b) show the simulated spin textures at different fields. Red and blue colors represent spin-up and spin-down magnetization directions, respectively.

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