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Skyrmions are topological spin textures that exhibit particle-like dynamics under current-driven forces such as spin-orbit torques (SOT). In frustrated magnets, competing exchange interactions along with the Dzyaloshinskii-Moriya interaction (DMI) can aid in stabilizing textures of different topologies, like skyrmions and antiskyrmions [1], which exhibit different motion under SOT, such as trochoidal trajectories and skyrmion-antiskyrmion pair generation, in addition to the usual rectilinear motion. Such behavior results from a departure from the frozen-core limit, where additional dynamics in the (anti)skyrmion helicity couples to the core position [2]. Here, we discuss the role of thermally driven helicity dynamics in the Brownian motion of skyrmions under weak DMI and SOT. Depending on the magnitude of the DMI, which governs the strength of the restoring force experienced by the helicity, with respect to &lt;i&gt;k&lt;/i&gt;&lt;sub&gt;B&lt;/sub&gt;&lt;i&gt;T&lt;/i&gt;, the thermal drift of the skyrmion core under weak SOT can differ greatly from the dynamics at 0 K (Fig. 1). This behavior arises from the thermal drift and diffusion of the helicity under SOT, which is analogous to the well-known problem of Brownian motion in a tilted washboard potential, where the helicity can be assimilated to an oscillator phase in a periodic potential with a linear force term. Under weak SOT or strong DMI, we recover a &#39;&#39;phase-locked&#39;&#39; solution in which the helicity is pinned, resulting in rectlinear drift. For strong SOT or weak DMI, we find a &#39;&#39;free-running&#39;&#39; solution, where the helicity varies continuously in time and results in an amplified Brownian diffusion of the skyrmion core. In between these two regimes, thermally-activated hopping of the helicity between energy minima occurs. These results highlight the importance of helicity dynamics and may have bearing on dipole-stabilized bubbles for which drive-dependent skyrmion Hall angles [3] and low drift velocities have been reported [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/4499995d079e5c36223e0c31052e4788.jpg)


Fig. 1. Comparison between SOT-driven skyrmion motion at 0 K (black line) and 50 different realisations of the Brownian motion at 10 K (colored lines), along with the average thermal drift at 10 K (purple line).

2022 Joint MMM-INTERMAG

Anomalous thermal drift of skyrmions in frustrated magnets under spin-orbit torques

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

Skyrmions are topological spin textures that exhibit particle-like dynamics under current-driven forces such as spin-orbit torques (SOT). In frustrated magnets, competing exchange interactions along with the Dzyaloshinskii-Moriya interaction (DMI) can aid in stabilizing textures of different topologies, like skyrmions and antiskyrmions [1], which exhibit different motion under SOT, such as trochoidal trajectories and skyrmion-antiskyrmion pair generation, in addition to the usual rectilinear motion. Such behavior results from a departure from the frozen-core limit, where additional dynamics in the (anti)skyrmion helicity couples to the core position [2]. Here, we discuss the role of thermally driven helicity dynamics in the Brownian motion of skyrmions under weak DMI and SOT. Depending on the magnitude of the DMI, which governs the strength of the restoring force experienced by the helicity, with respect to kBT, the thermal drift of the skyrmion core under weak SOT can differ greatly from the dynamics at 0 K (Fig. 1). This behavior arises from the thermal drift and diffusion of the helicity under SOT, which is analogous to the well-known problem of Brownian motion in a tilted washboard potential, where the helicity can be assimilated to an oscillator phase in a periodic potential with a linear force term. Under weak SOT or strong DMI, we recover a ''phase-locked'' solution in which the helicity is pinned, resulting in rectlinear drift. For strong SOT or weak DMI, we find a ''free-running'' solution, where the helicity varies continuously in time and results in an amplified Brownian diffusion of the skyrmion core. In between these two regimes, thermally-activated hopping of the helicity between energy minima occurs. These results highlight the importance of helicity dynamics and may have bearing on dipole-stabilized bubbles for which drive-dependent skyrmion Hall angles [3] and low drift velocities have been reported [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/4499995d079e5c36223e0c31052e4788.jpg)


Fig. 1. Comparison between SOT-driven skyrmion motion at 0 K (black line) and 50 different realisations of the Brownian motion at 10 K (colored lines), along with the average thermal drift at 10 K (purple line).

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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Recently, ferromagnetic resonance (FMR) in insulating YIG film was shown to be driven by microwave current in the Pt film deposited on the top of YIG film and detected as a dc voltage in the Pt layer due to spin Hall effect and/or spin rectification[1,2]. Here, we demonstrate an alternative method to study FMR in ceramic YIG. A polycrystalline pellet was made from YIG nanopowder by sintering in the air at 900oC for 12 hr. Magnetization at 300K, shown in the main panel of Fig.1(a), affirms the soft ferromagnetic nature of YIG. The upper inset in (a) illustrates the sketch of our experimental setup. Since the resistance of YIG could not be measured by the impedance analyzer, a layer of silver paint was coated on the top surface of the sample and at two ends to allow radio-frequency current to flow. The sample bridges the signal line and the ground plane of the sample stage, which is connected to the impedance analyzer via a rigid coaxial cable. A microwave signal is injected into the Ag paint layer through the signal line. Resistive and reactance of the Ag-paint were measured while sweeping frequency at different values of the dc magnetic field,H. The lower inset shows magnetoresistance MR(b) and magnetoreactance MX(c) for five frequencies (f) as H decreased from 3 kOe to 0 Oe. The MR of the Ag-paint for 3 GHz exhibits a prominent peak at H = Hr around which MX crosses zero. Both the amplitude of the peak in MR and Hr decreases as f decreases. We argue that the peak in MR is caused by FMR. The inset shows the dependence of Hr on f and we are able to fit the data with Kittel’s relation for in-plane FMR. The observed electrical detection of FMR in YIG with colloidal silver particles having weak spin-orbit coupling than Pt deserves an urgent understanding. R. M. thanks the Ministry of Education, Singapore for support (Grant no. R144-000-442-114).

![](https://assets.underline.io/uploads/markdown_image/1/image/ae1fde754d75a41639f3df99d02d023b.jpg)

Fig. 1 (a) M-H isotherm of YIG sample at 300 K(main panel) and the schematic diagram of the experimental setup (upper inset). Field dependence of (b) magnetoresistance MR)and (c) magnetoreactance MX at selected frequencies of current excitation (bottom inset). Inset in (b) shows the frequency dependence of the resonance field (symbols) and Kittel's fit for FMR.

Electrically detected ferromagnetic resonance in polycrystalline YIG covered with a layer of Ag paint

The prospect of thermal imaging and control in magnetic particle imaging (MPI) is an important advancement for non-invasive medical diagnostics and therapeutics. Current MPI systems focus mainly on imaging from measurement of nanoparticle concentrations. Accurate derivation of temperature relies on detailed understanding of magnetization dynamics in the presence of AC drive fields and variable temperature [1]. Here, we report measurements of nanosecond magnetization dynamics using both time and frequency domain techniques. It has been reported that the imaging resolution of MPI is critically impacted by nanoparticle relaxation dynamics [2]; specifically, spin relaxation results in image blurring. The dynamic response (e.g. Néel and Brownian relaxation) of magnetic nanoparticles in magnetic fields is intricate, depending strongly on the particles’ inherent magnetic and structural properties, inter-particle interactions, and the local environment. We have developed an arbitrary-wave magnetic particle spectrometer with magnetic field amplitude up to 10 mTPk for characterizing magnetization dynamics over a broad temperature range (200 K to 350 K). We have characterized the dynamic AC susceptibility of nanoparticles ranging in diameter from 10 nm to 70 nm and in composition and observed peaks in the imaginary component (χ’’) at frequencies from as low as 50 Hz to 50 MHz. Figure 1 show the χ’’ peak frequency for 11.5 nm cobalt-doped ferrite nanoparticles at different temperatures. This peak reveals the effective relaxation timescale (τ ~ 1 μs). Additionally, we used pulsed excitation to characterize the magnetization dynamics with and without magnetic fields. Figure 2 shows a magnetization response of 7.5 nm cobalt-doped ferrites to a magnetic field pulse with 20 ns rise time. Monte Carlo simulations are used for analyzing both time and frequency domain measurements. This knowledge will eventually inform strategies for design and synthesis of magnetic nanoparticles with properties targeted for accurate and sensitive thermal imaging using MPI.

![](https://assets.underline.io/uploads/markdown_image/1/image/c09e0bfc2893d134e13e9ccf078bc358.jpg)

The imaginary component of AC susceptibility for 11.5 nm ferrite nanoparticles

![](https://assets.underline.io/uploads/markdown_image/1/image/cab02aa8ffbaedf3c9a626601584bbd6.jpg)

Pulsed magnetic field (black). Magnetization step-response of 7.5 nm cobalt-doped ferrite (red).

Magnetization dynamics of magnetic nanoparticles for thermal and magnetic particle imaging

New physics emerge from the interplay between spin waves and magnetic topological objects. Excitation modes permit to exploit the full potential of topological objects and increase the range of applications, in particular in the field of insulator-based spintronics. Recently, the number of proposals leveraging excitation modes of topological objects is steadily increasing, such as in microwave generation and detection, spin-wave guides, parametrons and in neuromorphic spintronic devices. In this talk I will present three ways to exploit excitations of magnetic topological objects. First, I will show that domain walls in thin films behave as flexible strings with an extra internal degree of freedom1. A direct application is the use of domain walls as spin-wave guides and hosts of magnetic domain wall skyrmions. Moreover, by considering closed domain wall strings, I will derive analytically deformations, instabilities2,3, and excitation modes of skyrmions4. Second, I will propose the use of topological objects as tunable nanoscale in-materio frequency multipliers for spintronic systems5. I will show that the excitation of topological objects by fractions of the eigenfrequencies presents advantages compared to the well-known parametric excitation as it requires less energy input and does not generate instabilities away from the excited topological objects. Third, I will discuss a mechanism to controllably manipulate the direction and speed of domain walls in kagome antiferromagnets via a single linearly polarized spin-wave source6. In this case, the coupling of spin wave modes inside the non-collinear domain wall allows for a highly tunable spin-wave induced domain wall motion with a frequency dependent domain wall velocity.

Exploiting excitations of magnetic topological objects

The establishment of chiral coupling in thin magnetic films with inhomogeneous anisotropy has led to the development of novel artificial systems of fundamental and technological interest. The chiral coupling itself is enabled by the Dzyaloshinskii-Moriya interaction (DMI) enforced by the patterned non-collinear magnetization [1-4]. Here, we create a domain wall track with out-of-plane (OOP) magnetization coupled to OOP stripes on each side via narrow parallel strips with in-plane magnetization as shown in Fig.1 (a). Since the chiral coupling prefers antiparallel OOP magnetization configuration over the parallel case, the regions of noncollinear magnetization in a single magnetic layer can be used to bias the domain wall velocity, as shown in series of Kerr images in Fig.1 (b). To tune the chiral torques, the design of the magnetic racetracks can be modified by varying the width of the tracks or the width of the transition region between noncollinear magnetizations, reaching effective chiral magnetic fields of up to 8 mT. Furthermore, we show how the magnitude of the chiral torques can be estimated by measuring asymmetric domain wall velocities. We also demonstrate spontaneous domain wall motion propelled by intrinsic torques even in the absence of any external driving force, which is of high importance for domain wall motion-based neuromorphic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/630ddf8c04b523dac1399ba71d81db43.jpg)

Figure 1: (a) Schematic of a DW racetrack with OOP magnetization delimited by two strips with IP anisotropy. The energy landscape of the DW racetrack is monotonous, except for local DW pinning sites. (b) Sequence of Kerr images of asymmetric DW motion in a racetrack with wOOP = 950 nm. The dashed lines correspond to the position of the IP magnetized strip while the full lines correspond to the boundary of the magnetic track. 80 current pulses are applied between each image; the pulse duration is 50 ns and the current density is 3.6×1011 A/m2.

Engineering of intrinsic chiral torques in magnetic thin films based on the Dzyaloshinskii-Moriya interaction

Magnetic multilayers of Ir/Co/Pt with interfacial Dzyaloshinskii-Moriya interaction (IDMI) [1] were deposited by magnetron sputtering with Co thickness d=1.8nm and repetitions N=6. Ferromagnetic resonance spectroscopy was applied to determine the values of effective magnetic uniaxial anisotropy, Hu = -0.45 T. Exploiting magneto-optical Kerr effect in longitudinal mode microscopy, magnetic force microscopy and vibrating sample magnetometery the magnetic field driven evolution of domain structures and hysteresis loops have been studied. Existence of weak stripe domains structure was deduced – large micrometer size domains with in-plane "core" magnetization modulated by small nano-domains with out-of-plane magnetization. Quantitative evaluation of IDMI was carried out using Brillouin light scattering (BLS) spectroscopy [2,3] from the difference between Stokes and anti-Stokes peak frequencies Δf. Due to the additive nature of IDMI, the asymmetric combination of Ir and Pt covers led to large values of surface DMI energy density Ds. Fig. 1 shows Stokes and anti-Stokes peaks for magnon scattering. It was found that Stokes and anti-Stokes frequencies as well as Δf, measured as a function of in-plane applied magnetic field, show hysteresis. These results are explained under the consideration of the influence of IDMI on the dynamics of in-plane magnetized “core” with weak stripe domains. In order to understand the spatial magnetization distribution and magnetization process, the micro-magnetic simulations were performed. Our findings leverage the impact of IDMI on domain structure. Large values of Ds can be useful for the stabilization of skyrmions [1] and studying the dynamical behavior of magnetic skyrmions in the in-plane magnetized sample [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/92e6a5942842efc48ac7a6e5bad30c22.jpg)

Fig. 1. BLS spectrum for d=1.8 nm where Stokes and anti-Stokes peaks are shown. Measurement was done for wavevector k=11.81 μm-1 and applied in-plane magnetic field µ0H=72.8 mT. Experimental data (black line) and its mirror curve (red line, for easy comparison) determine the frequency difference Δf ≈1.5 GHz.

Magnetization statics and dynamics in Ir/Co/Pt multilayers with Dzyaloshinskii-Moriya interaction

Microwave response of magnetic textures in ferromagnets offers rich physics and has been intensively investigated over the past decades for their potential application in magnonics. Eigenexcitations of films featuring bubble domains [1,2] and stripes [3] have been accounted for, broadly consisting of acoustic, optical and domain wall oscillation modes. Recently, topologically non-trivial chiral magnetic structures called skyrmions [4] have gathered a lot of interest owing to their robust, tiny and particle-like nature. They exhibit unique spectral signatures which have been observed mainly in bulk systems below room temperature [5,6]. The experimental observation and interpretation of their resonant dynamics in thin films remain elusive owing to high damping coefficients. Here we report room temperature investigation of resonant skyrmion dynamics in [Pt/CoFeB/AlOx]×20 multilayers. We optimize our sample by tuning the different layer thicknesses to host magnetic skyrmions at room temperature, along with a fairly low damping parameter α= 0.02. We use magnetic force microscopy to observe the static magnetic domain configuration consisting of randomly distributed skyrmions and dense skyrmion lattice as a function of an out-of-plane magnetic field. Broadband ferromagnetic resonance measurements are performed to study the dynamic response with an in-plane excitation field. Besides the Kittel mode (KM) above saturation, three distinct modes, a low frequency mode (LFM), an intermediate frequency mode (IFM) and a high frequency mode (HFM), appear with unique frequency-field dispersions in the non-saturated state (see Fig). Micromagnetic simulations reveal the characteristics of these modes, where the LFM features a mode localized in the skyrmion edges, while for the IFM skyrmion edge dynamics couples with the uniform background. In the HFM, the precession of skyrmion cores emits omnidirectional spin waves into uniform background, producing interference patterns in the lattice. Our results thus open up new avenues for skyrmion-based magnonic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/20e701a47ef97bc8c284fee76af80155.jpg)

Fig : Frequency-field dispersion map measured by FMR over a frequency range of 0.1 GHz to 20 GHz with an out-of-plane magnetic field swept from -500 mT to 500 mT.

Localized and propagating spin-wave modes in thin film multilayers hosting skyrmions

The topological number of micromagnetic textures (Q) is a useful quantity for information encoding and transport [1,2], and determines the dynamics of magnetic objects such as skyrmions, merons and domain walls. Recent developments allow us to tune the strength of the interfacial Dzyaloshinskii-Moriya interaction, thereby inducing the formation of different states and generating a rich energy landscape for a given physical device, leading to new and potentially useful applications [3]. This energy landscape determines the thermal stability of a given state which is strongly dependent on the barrier that separates neighboring minima [4-6]. In this work we use the String Method for the Study of Rare Events to determine transition states and thermal barriers of a circular nanodisk at different magnitudes of the Dzialoshinskii-Moriya constant (D). Low values of D are associated with high energies of domain wall formation; in this regime mostly uniform magnetizations with isolated skyrmions correspond to the lowest energy states. The transition between these states occurs by the motion of entire skyrmions across the material's edge. At intermediate values of D, domain walls require less energy, skyrmions have large radii and leave the nanodisk via an intermediate minima in which a meron is expelled from the film (Figures 1 and 2). Very large D values cause complex textures to form but transtions are still mediated by meron motion across the device's edge. This work emphasizes the importance of considering meron-mediated-transitions to appropriately create the initial path used in methods that use chain-of-states to estimate transition barriers.

![](https://assets.underline.io/uploads/markdown_image/1/image/1eaa5816c8a78055e4675a6f428397e1.jpg)

String Method results showing energy and skyrmion number along the transition path. The images above correspond to points indicated in the images. The disk diameter is 80nm and its thickness is 0.4nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/43f1086b40436f45582a732b611bd679.jpg)

Skyrmion Number vs Energy for the same transition path of Fig.1. The skyrmion numbers of energy minima differ by roughly 1/2, highlighting the importance of meron motion across the edges as a mechanism for switching between energy minima. The values are not half-integer due to magnetization tilting at the sample's edge.

Thermally Activated Transitions in Confined Terromagnets with Perpendicular Magnetic Anisotropy as Function of the Interfacial Dzialoshinskii-Moriya Interaction Strength.

We theoretically study the dynamics of a domain wall in a ferromagnetic nanotube driven by electrons and magnons by investigating the electron-domain-wall and magnon-domain-wall interactions. Due to the geometry of the sample, domain walls are classified by the Skyrmion charge which counts the winding number of magnetic textures on the domain wall [1]. The domain wall with a non-zero Skyrmion charge generates an emergent magnetic field for interacting particles, which exerts the Lorentz force on moving electrons and magnons and thereby deflects their trajectories. This deflection is manifested as the generation of the finite orbital angular momenta of the electrons and magnons that traverse the domain wall [2,3]. For the case of electrons, we can interpret the exchange of orbital angular momenta between electrons and the domain wall as a current-induced torque. From this peculiar torque, the domain wall with the non-zero Skyrmion charge can be driven by an arbitrary small current without the Walker breakdown. For the case of magnons, we obtain the exact solution for the magnon on the Skyrmion-textured domain wall and also their scattering properties with the domain wall with the aid of supersymmetric quantum mechanics (SUSY QM). We also show that there is a critical wavenumber for the total reflection and it is discretized by the Skyrmion charge of the domain wall. Our results show that the orbital angular momenta of magnetic systems and quasi-particles can be intertwined in a curved geometry.

![](https://assets.underline.io/uploads/markdown_image/1/image/52936d7fd5815e871498a79c009adca1.jpg)

Fig. 1. A ferromagnetic nanotube with a domain wall possessing a finite Skyrmion charge. The arrows represent magnetizations of spin-wave excitations. The black curve lines the heads of the arrows. The axis of the nanotube is the z-axis.



Orbital angular momentum of a domain wall and geometrically twisted magnons

We find numerically skyrmionic textures with skyrmion number Q=0 in ferromagnets with the Dzyaloshinskii-Moriya interaction perpendicular anisotropy and the magnetostatic field [1]. The configuration is asymmetric; the largest part has the form of a skyrmion and a smaller part has the form of an antiskyrmion. The texture is topologically trivial (Q=0) and it may be called a chiral droplet. Droplets are stable in an infinite film as well as in disk-shaped magnetic elements. They are stable also in a model without the magnetostatic field [2]. They are found in films for values of the parameters close to the transition from the ferromagnetic to the spiral phase. Chiral droplets are a valuable addition to a short list of skyrmionic textures [3]. The presence of topologically trivial textures (Q=0) along with usual skyrmions (Q=1) in the same magnetic system is rare. It is important because the dynamics of skyrmions is linked to their topology; it depends on the skyrmion number. Under spin-transfer torque (due to an in-plane current), droplets are accelerated in the direction of the current flow and they behave as solitary waves of Newtonian character with no Magnus force effect. This is in stark contrast to the Hall dynamics of the standard Q=1 skyrmion. In the case shown in the figure, the motion is initially complicated, with the droplet making a full π turn while its size is expanded. It is accelerated in the direction of the current and eventually, the velocity acquires a constant value, in very good agreement with the theoretical prediction. The dynamical behavior of a droplet can be compared to that of an axially symmetric skyrmionium [4] in order to show that the global topology rather than the symmetry of the configuration is crucial for skyrmion dynamics. We acknowledge support by the project ThunderSKY, funded by HFRI and GSRT, under Grant No. 871.

![](https://assets.underline.io/uploads/markdown_image/1/image/3fafac4601e2b98489be56ebef970d99.jpg)

The motion of a chiral droplet when an in-plane current is applied. The droplet is represented via the perpendicular component of the magnetization mz (color-coded). In the initial stage, the droplet is following a complicated path, it rotates and its shape adjusts to that of a propagating solitary wave. It eventually moves with no Magnus force effect.

Chiral droplets in ferromagnets, and their current-driven motion

Magnetic skyrmions, which are topologically protected particle-like spin textures, exhibit Brownian motion at ambient temperatures in a ferromagnetic thin film. The skyrmion system can be an ideal platform to design unconventional computers like stochastic/Brownian computers. Therefore, investigation of the Brownian motion of the magnetic skyrmion is now the subject of scientific and technical interest [2-5]. The motion of skyrmions driven by the current [6] and temperature gradient [7] have the chirality depending on the skyrmion number. In this study, we explore the effects of the chiral property of a skyrmion on its Brownian motion experimentally and theoretically. We fabricated the stacking structure as Ta(5)|Co16Fe64B20(1.26)|Ta(0.22)|MgO(1.5)|SiO2(5.0) and observed the position of the skyrmion x(t) = (x(t), y(t)) by using MOKE microscope. Fig. 1 shows the simultaneous velocity-position correlation function &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt;. After enough time has passed, &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt; correspond to the diffusion coefficient Dxx and the angular momentum of the skyrmionic diffusion, respectively, and the latter also has a sizable value. Fig. 2 represents the simulation trajectory in the periodic potential obtained by integrating the Thiele-Langevin eq. [8,9] including the thermal fluctuation and the mass term. The chirality of the Brownian motion is affected by the fluctuated potentials of the samples. This research and development work was supported by ULVAC, Inc., the Ministry of Internal Affairs and Communications, JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP20H05666, Japan and CREST (Non-classical Spin project, JPMJCR20C1) of the Japan Science and Technology Agency.

![](https://assets.underline.io/uploads/markdown_image/1/image/b2fe0462e46a0f673fe46f65f2392e00.jpg)

Fig. 1 velocity-position correlation function and <(v(t)×x(t))z>


![](https://assets.underline.io/uploads/markdown_image/1/image/6aa7d699583b7f1eef889ab6bb6a071e.jpg)

Fig. 2 The simulation result of the motion of the skyrmion in the periodic potential. The red arrows show the direction of the motion.

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