# Current-induced Helicity Switching of Frustrated Skyrmions on a Square Grid Obstacle Pattern

X Zhang, J Xia, O A Tretiakov, H T Diep, G Zhao, J Yang, Y Zhou, M

Ezawa, X Liu

► To cite this version:

X Zhang, J Xia, O A Tretiakov, H T Diep, G Zhao, et al. Current-induced Helicity Switching of Frustrated Skyrmions on a Square Grid Obstacle Pattern. Journal of Magnetics Society of Japan, 2023, 47 (1), pp.17-27. 10.3379/msjmag.2301R003. hal-04116507

HAL Id: hal-04116507

https://hal.science/hal-04116507

Submitted on 4 Jun 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Current-Induced Helicity Switching of Frustrated Skyrmions on a Square-Grid Obstacle Pattern

X. Zhang<fnref n="1" />, J. Xia<fnref n="1" />, O. A. Tretiakov<fnref n="2" />, H. T. Diep<fnref n="3" />, G. Zhao<fnref n="4" />, J. Yang<fnref n="5" />, Y. Zhou<fnref n="6" />, M. Ezawa<fnref n="7" />, and X. Liu<fnref n="1" />

Topological spin textures on artificial pinning landscape may show unique static and dynamic properties. Here, we computationally show that the helicity of frustrated skyrmions on an artificial square-grid obstacle pattern can be switched by a spin current pulse. The obstacle pattern is formed by defect lines with enhanced perpendicular magnetic anisotropy, which could protect the skyrmion from being annihilated at the sample edge. It is found that the skyrmion driven by a moderate current shows a circular motion guided by the boundary of the obstacle pattern, while it shows an almost straight motion toward the sample edge in the absence of the obstacle pattern. By applying a short pulse current to drive the skyrmion in a sample with the obstacle pattern, we find that the helicity of the skyrmion could be switched between Bloch-type configurations favored by the dipole-dipole interaction. Besides, we demonstrate the possibility of switching the helicity of an array of skyrmions on the square-grid obstacle pattern using the same method. Our results could be useful for the helicity control of topological spin textures, and may provide guidelines for building future helicity-based spintronic functions.

**Key words:** magnetic skyrmion, frustrated spin system, helicity, obstacle, pinning, spintronics, micromagnetics

## 1. Introduction

Skyrmions were predicted theoretically to exist in magnetic materials more than three decades ago <fnref n="1" />, <fnref n="2" />), which are now in the vanguard of next-generation spintronic technologies <fnref n="3" />⁻²²). They can be found in different types of magnetic materials <fnref n="3" />⁻²²), including chiral magnets and frustrated magnets. In chiral magnets, skyrmions are stabilized by the asymmetric exchange interaction ²³⁻²⁵), which is also called the Dzyaloshinskii-Moriya interaction ²⁶, ²⁷). In frustrated magnets, skyrmions are stabilized by the exchange frustration induced by the competing exchange interactions ²⁸⁻⁵⁹). The salient features of magnetic skyrmions include nanoscale size, low energy consumption, and most importantly, multiple degrees of freedom that can be controlled by external stimuli. Therefore, it is expected that skyrmions can be used as versatile building blocks in different information processing applications <fnref n="3" />⁻²²).

Most recently, it was suggested that frustrated skyrmions could be utilized as qubits in quantum computing ⁵⁵, ⁶⁰), where the manipulation of information is achieved through the precise control of the helicity degree of freedom. However, a frustrated skyrmion usually shows both center-of-mass and helicity dynamics when it is driven by an applied current ²⁹, ³⁰, ³⁶, ⁴², ⁴⁵, ⁵⁷). Namely, the helicity

dynamics is coupled to the center-of-mass dynamics. Such a feature may limit the applications of frustrated skyrmions in helicity-based quantum computing and relevant devices. Hence, it is vital to find an effective way to control and manipulate the helicity degree of freedom of frustrated skyrmions. We note that the helicity state of a nanoscale skyrmion can be experimentally observed by using the Lorentz transmission electron microscopy (TEM) ⁶¹).

Possible solutions to avoid the translational motion of a frustrated skyrmion include the pinning of the skyrmion by an artificial defect and the geometric confinement of the skyrmion. For example, a ferromagnetic skyrmion in a nanodisk with the diameter close to the skyrmion diameter may show limited translational motion due to the confinement effect provided by the skyrmion-edge repulsion ⁶²). However, skyrmions could be destroyed by touching the edge. Recently, it has been shown that a more effective strategy to confine, protect, and control skyrmions in magnetic thin films is to construct defect lines with different magnetic properties ⁶³⁻⁷⁴), which can be realized in engineered multilayer nanostructures ⁷⁵) or by using focused irradiation ⁷⁶). However, the helicity dynamics of a frustrated skyrmion confined by defect lines have not yet been explored. It is important to study such a problem, which is highly relevant to the realization of the skyrmion helicity-based applications.

In this work, we report the current-induced dynamics of isolated frustrated skyrmions on a square-grid obstacle pattern, which is made of orthogonal defect lines with

<footnote n="1">¹Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan</footnote>

<footnote n="2">²School of Physics, The University of New South Wales, Sydney 2052, Australia</footnote>

<footnote n="3">³Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France</footnote>

<footnote n="4">⁴College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China</footnote>

<footnote n="5">⁵State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing, 100871, China</footnote>

<footnote n="6">⁶School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China</footnote>

<footnote n="7">⁷Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan</footnote>

<footnote>Corresponding author: Y. Zhou (e-mail: zhouyan@cuhk.edu.cn), M. Ezawa (e-mail: ezawa@ap.t.u-tokyo.ac.jp), X. Liu (e-mail: liu@cs.shinshu-u.ac.jp).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1 The total energy of a frustrated skyrmion in the systems with or without the DDI. (a) The total energy of a frustrated skyrmion as a function of the helicity number $\eta$ in the system with the DDI. The energies are given in units of $J_1 = 1$. (b) The total energy of a frustrated skyrmion as a function of the helicity number $\eta$ in the system without the DDI. (c) Schematic illustration showing the spin configurations of skyrmions with the helicity numbers of $\eta = 0, \pi/2, \pi, 3\pi/2$. The out-of-plane spin component $m_z$ is color coded: blue is into the plane, red is out of the plane, white is in-plane. The total energy is calculated by assuming a fixed helicity number $\eta$.</figcaption></figure>

enhanced perpendicular magnetic anisotropy (PMA). We show that the helicity states of a single isolated frustrated skyrmion as well as an array of isolated frustrated skyrmions can be switched between Bloch-type configurations by a short current pulse. Our results could be a basis for future storage and computing device applications based on the manipulation of the helicity degree of freedom of topological spin textures.

## 2. Simulation and system

We focus on the dynamics of isolated skyrmions in a two-dimensional (2D) frustrated magnetic spin system, where the skyrmions are stabilized by a delicate competition among ferromagnetic nearest-neighbor (NN), antiferromagnetic next-NN (NNN), and antiferromagnetic next-NNN (NNNN) exchange interactions. The system is a $J_1-J_2-J_3$ classical Heisenberg model on a simple square lattice$^{30, 33, 36, 49}$. The Hamiltonian $\mathcal{H}$ is given as

$$ \mathcal{H} = -J_1 \sum_{\langle i,j \rangle} \mathbf{m}_i \cdot \mathbf{m}_j - J_2 \sum_{\langle\langle i,j \rangle\rangle} \mathbf{m}_i \cdot \mathbf{m}_j \\ -J_3 \sum_{\langle\langle\langle i,j \rangle\rangle\rangle} \mathbf{m}_i \cdot \mathbf{m}_j - K_u \sum_i (m_i^z)^2 \\ -\frac{1}{2} M_S \sum_i \mathbf{B}_D \cdot \mathbf{m}_i, \qquad (1) $$

where $m_i$ represents the normalized spin at the site $i$, $|m_i| = 1$. $\langle i, j \rangle$, $\langle\langle i, j \rangle\rangle$, and $\langle\langle\langle i, j \rangle\rangle\rangle$ run over all the NN, NNN, and NNNN sites, respectively. $J_1, J_2$, and $J_3$ are the

<figure><img src="image_4.png" /><figcaption>Fig. 2 Typical current-induced motion of a frustrated skyrmion in the system with the DDI. (a) A typical trajectory of a frustrated skyrmion in the system with the DDI, which shows the skyrmion moves almost straight toward the left edge of the sample. A relaxed skyrmion with $\eta = \pi/2$ is placed at the sample center as the initial state at $t = 0$ ps. The sample size is 51 × 51 spins. The skyrmion is driven by a current density of $j = 70$ MA/cm$^2$. The insets show the zoomed-in view of the skyrmion at $t = 0$ ps and $t = 50$ ps. Note that the insets only focus on the skyrmion texture instead of the whole sample. The arrow indicates the motion direction of the skyrmion. (b) The helicity of the skyrmion as a function of time during $t = 0 - 50$ ps. The skyrmion helicity slightly varies with time due to the current-induced deformation of the skyrmion during its motion toward the left sample edge. The skyrmion is annihilated by touching the left sample edge at $t = 66$ ps, where the skyrmion position and helicity cannot be tracked due to the significant deformation and annihilation.</figcaption></figure>

constants for the NN, NNN, and NNNN Heisenberg exchange interactions, respectively. $K_u$ is the easy-axis magnetic anisotropy constant, and the z-axis direction is defined as the easy axis. $\mathbf{B}_D$ is the demagnetizing field resulting from the magnetic dipole-dipole interaction (DDI). $M_S$ denotes the saturation magnetization. We note that we do not consider the interface-induced Dzyaloshinskii-Moriya interaction in this work as it could be suppressed for an ultrathin monolayer$^{77}$.

The spin dynamics is governed by the Landau-Lifshitz-Gilbert equation augmented with the damping-like spin-orbit torque$^{78-81}$,

$$ \partial_t \mathbf{m} = -\gamma_0 \mathbf{m} \times \mathbf{h}_{\text{eff}} + \alpha (\mathbf{m} \times \partial_t \mathbf{m}) + \boldsymbol{\tau}_d, \qquad (2) $$

where $t$ is the time, $\gamma_0$ is the absolute gyromagnetic ratio, and $\alpha$ is the Gilbert damping parameter. $\mathbf{h}_{\text{eff}} = -\delta\mathcal{H}/(\delta\mathbf{m}\mu_0 M_S)$ is the effective field with $\mu_0$ and $\varepsilon$ being the vacuum permeability constant and average energy density, respectively. The saturation magnetization $M_S = 580$ kA/m. The damping-like spin-orbit torque is expressed as $\boldsymbol{\tau}_d = u (\mathbf{m} \times \mathbf{p} \times \mathbf{m})$, where the coefficient $u = (\gamma_0\hbar/\mu_0e)(j\theta_{\text{SH}}/2aM_S)$. $\hbar$ is the reduced Planck constant, $e$ is the electron charge, $a$ is the sample thickness, $j$ is the current density, and $\theta_{\text{SH}}$ is the spin Hall angle. The spin polarization direction $\mathbf{p} = +\hat{\mathbf{y}}$. For the sake of simplicity, we assume that $\theta_{\text{SH}} = 1$ so that the driving force is simply
<figure><img src="image_5.png" /></figure>

**Fig. 3** Current-induced motion of a frustrated skyrmion in a sample with the obstacle pattern. (a) The trajectory of a frustrated skyrmion in a sample with the square-shaped obstacle pattern, which is formed by edge spins with enhanced PMA (i.e., $K_o/K_u = 10$ with $K_o$ being the PMA of the obstacle defect line). A relaxed skyrmion with $\eta = \pi/2$ is placed at the sample center as the initial state at $t=0$ ps. The skyrmion moves in an almost circular path guided by the obstacle pattern, which is driven by a current density of $j = 40$ MA/cm$^2$. The sample size is 21 × 21 spins. The width of the obstacle line is 3 spins. (b) The helicity of the skyrmion as a function of time during $t=0-200$ ps. The skyrmion helicity changes between Bloch-type and Néel-type configurations during the motion of the skyrmion along the inner edge of the obstacle pattern. (c) Selected top-view snapshots showing the spin configurations of the skyrmion at $t=0, 43, 55$, and $75$ ps. The obstacle pattern with enhanced PMA is indicated by the orange area.

determined by $j$, i.e., $u \sim j$. In this work, we do not consider the effect of the field-like spin-orbit torque considering the helicity dynamics of a frustrated skyrmion is driven by the damping-like torque$^{30,36}$. The simulation is done by using the Object Oriented MicroMagnetic Framework (OOMMF)$^{82}$ upgraded with our extension modules for the $J_1$-$J_2$-$J_3$ classical Heisenberg model$^{30}$. In all simulations, the mesh size is $a^3$ with $a = 0.4$ nm being the lattice constant. Other default parameters are$^{30,36,45,51,52}$: $J_1 = 30$ meV, $J_2 = -0.8$ (in units of $J_1 = 1$), $J_3 = -0.6$ (in units of $J_1 = 1$), $K_u = 0.02$ (in units of $J_1/a^3 = 1$), $\alpha = 0.3$, and $\gamma_0 = 2.211 \times 10^5$ m/(As).

### 3. Results and discussion

The helicity is an important degree of freedom of frustrated skyrmions, which can be used to perform helicity-based storage and computing. As shown in Fig. 1, we first review the total energy of a frustrated skyrmion as function of its helicity in the systems with or without the DDI. In the simulation, we first place a single isolated skyrmion with the skyrmion number of $Q = -1$ at the center of a sample with 21 × 21 spins with open boundary

<figure><img src="image_6.png" /></figure>

**Fig. 4** Current-induced motion of a frustrated skyrmion in a sample without the obstacle pattern. Selected top-view snapshots showing the spin configurations of the frustrated skyrmion at $t=0, 10, 20, 30, 35, 38, 40, 42$, and $45$ ps in a sample without the obstacle pattern. A relaxed skyrmion with $\eta = \pi/2$ is placed at the sample center as the initial state at $t=0$ ps. The sample size is 21×21 spins. The skyrmion is driven by a current density of $j = 40$ MA/cm$^2$. The skyrmion moves toward the sample edge and annihilates when touching the sample edge.

conditions in the $x$ and $y$ directions. The skyrmion number is defined as $Q = 1/4\pi \int m \cdot (\hat{\sigma}_x m \times \hat{\sigma}_y m) dxdy$. As the skyrmion can be parametrized as $m(r) = m(\theta, \phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$, where $\phi = Q_v\varphi + \eta$ with $\varphi$ being the azimuthal angle ($0 \le \varphi < 2\pi$). Hence, the skyrmion considered in our work has a vorticity of $Q_v = \oint cd\phi/2\pi = +1$. $\eta \in [0, 2\pi)$ denotes the skyrmion helicity defined mod $2\pi$. It can be seen that the total energy of a frustrated skyrmion depends on its helicity number $\eta$, where the Bloch-type configurations are favored by the DDI. Namely, the total energy of a frustrated skyrmion is independent of $\eta$ only in the system without the DDI [see Fig. 1(b)]. Such a property is important for the dynamics of the frustrated skyrmion as the center-of-mass dynamics of frustrated skyrmions is coupled to the helicity dynamics$^{30}$. Namely, the skyrmion will move in a straight path with its helicity being fixed at the energetically favored Bloch-type configurations (i.e., $\eta = \pi/2, 3\pi/2$) in the system with the DDI$^{36}$.

Therefore, in order to switch the helicity of a frustrated skyrmion in the system with the DDI, one will need to apply a driving current larger than a critical value to induce the helicity switching when the skyrmion can overcome the energy barrier between two different Bloch-type configurations$^{36}$. However, the critical value of the driving current density could be high, although the DDI-
induced energy barrier may be reduced by reducing the value of the saturation magnetization. Note that the skyrmion stability may also be dependent on the saturation magnetization. Meanwhile, the position of the skyrmion may not be controlled due to the center-of-mass motion coupled to the helicity dynamics<sup>36</sup>. For example, in Fig. 2 we show the current-induced dynamics of a frustrated skyrmion at a moderate driving current density in the system with the DDI, where the skyrmion with Bloch-type helicity moves in an almost straight path

toward the left edge of the sample. The trajectory and time-dependent helicity number of the skyrmion are given in Figs. 2(a) and 2(b), respectively. The skyrmion will be annihilated when it touches the edge of the sample. Hence, we propose an effective way to protect the frustrated skyrmion from being annihilated at the sample edge, and at the same time, to realize the current-induced switching of the skyrmion helicity in the system with the DDI.

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Helicity switching of a frustrated skyrmion in a sample with the obstacle pattern induced by a short current pulse. (a) The helicity of the skyrmion as a function of time during *t* = 0 – 300 ps. A relaxed skyrmion with *η* = π/2 is placed at the sample center as the initial state at *t* = 0 ps. The skyrmion is driven by a current density of *j* = 40 MA/cm² for 50 ps, followed by a relaxation of 250 ps. The sample size is 21 × 21 spins. The width of the obstacle line is 3 spins. The obstacle has an enhanced PMA (i.e., *K*₀/*K*_u = 10). The skyrmion helicity switches from *η* = π/2 to *η* = 3π/2 after the application of the current pulse. (b) The in-plane spin components *m*ₓ,ᵧ as functions of time during *t* = 0 – 300 ps. (c) The out-of-plane spin components *m*ₓ,ᵧ as a function of time during *t* = 0 – 300 ps. Note that *m*ₓ, ᵧ, and *m*ᵧ are the normalized in-plane and out-of-plane spin components for the whole sample. (d) Selected top-view snapshots showing the spin configurations of the skyrmion at *t* = 0, 30, 40, 50, and 300 ps. The obstacle pattern with enhanced PMA is indicated by the orange area.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Helicity switching of an array of nine frustrated skyrmions in a sample with the square-grid obstacle pattern induced by a short current pulse. Selected top-view snapshots showing the spin configurations of the frustrated skyrmions at $t = 0, 50,$ and $300$ ps. Nine relaxed skyrmions with $\eta = \pi/2$ are placed in the sample as the initial state at $t = 0$ ps. The system is driven by a current density of $j = 40$ MA/cm$^2$ for 50 ps, followed by a relaxation of 250 ps. The sample size is $65 \times 65$ spins. The width of the obstacle line is 5 spins. The obstacle has an enhanced PMA (i.e., $K_o/K_u = 10$). The helicity of all skyrmions switch from $\eta = \pi/2$ to $\eta = 3\pi/2$ after the application of the current pulse. The obstacle pattern with enhanced PMA is indicated by the orange area.</figcaption></figure>

As shown in Fig. 3, we first study the dynamics of a single isolated frustrated skyrmion driven by a moderate current of $j = 40$ MA/cm$^2$. The system is a sample of 21 × 21 spins, where an obstacle pattern is formed by defect lines at the four edges. We assume that the obstacle pattern has an enhanced PMA, i.e., $K_o/K_u = 10$, which could be realized by locally modifying the PMA in principle. Note that we assumed a large value of $K_o/K_u$ in order to make sure that the skyrmion cannot penetrate the obstacle. In real experiments, it is unnecessary to realize a large value of $K_o/K_u$. The width of the defect line is set to 3 spins. As the PMA of the obstacle area is larger than that in the interior of the sample, the skyrmion-obstacle interaction provides a repulsive force, which avoids the skyrmion penetrating the boundary of the obstacle. Such a feature also protects the skyrmion from be touching the edge of the sample, which may result in a better dynamic stability in the sample. It can be seen that the skyrmion moves in a circular path driven by the current and guided by the inner boundary of the obstacle pattern [see Fig. 3(a)]. We note that the skyrmion dynamics is affected by the obstacle pattern as soon as the current is applied, because the obstacle is initially very close to the skyrmion. Namely, the skyrmion is tightly confined by the obstacle pattern. When the obstacle is initially far from the skyrmion, the skyrmion should, in principle, first move to the left with a fixed helicity [see Fig. 2(a)] and then perform the circular motion with varying helicity when it is close to and guided by the obstacle.

As the helicity dynamics is coupled to the center-of-mass dynamics of the skyrmion $^{30}$, the helicity of the skyrmion also varies with time during the circular

motion of the skyrmion, as shown in Fig. 3(b). In Fig. 3(c), we also show some selected top-view snapshots of the spin configurations of the sample, which show that the skyrmion helicity varies between Bloch-type and Néel-type configurations although the Bloch-type ones are favored by the DDI. As long as the driving current is applied, the skyrmion will move in the circular path with a varying helicity. However, it should be noted that the skyrmion driven by the same current of $j = 40$ MA/cm$^2$ will move toward the left edge and get annihilated from the edge in the sample without the obstacle pattern, as shown in Fig. 4. Namely, the circular motion and helicity switching of the skyrmion are a result of the skyrmion-obstacle interaction. We also note that the helicity switching is mainly controlled by the driving current, the dipole-dipole interaction, and the skyrmion-obstacle interaction. The frustrated skyrmion is stabilized by the exchange frustration instead of the asymmetric exchange interaction (i.e., the Dzyaloshinskii-Moriya interaction). The exchange frustration is realized by a delicate competition between different exchange interactions, therefore, the NNN and NNNN exchange interactions may affect the stability of the skyrmion. However, they may not affect the switching of the skyrmion helicity.

Based on the current-induced dynamic behaviors of a frustrated skyrmion in a sample with the obstacle pattern, we show in Fig. 5 that it is possible to switch the helicity of a frustrated skyrmion between the Bloch-type configurations favored by the DDI, i.e., $\eta = \pi/2 \rightarrow 3\pi/2$. To switch the skyrmion helicity, we apply a short current pulse to drive the dynamics of a single isolated frustrated
skyrmion in a sample with the obstacle pattern. The geometry and other system settings are the same as that used in the simulation in Fig. 3. The current density is $j = 40 \text{ MA/cm}^2$ and the pulse length is $\tau = 50 \text{ ps}$. The pulse length is determined based on the result given in Fig. 3(b). After the application of a single current pulse, the system is relaxed for 250 ps. Namely, the total simulation time is 300 ps. It can be seen in Fig. 5(a) that the helicity of the skyrmion varies from $\eta = \pi/2$ to $\eta \sim 3\pi/2$ during the application of the current pulse (i.e., $t = 0 - 50 \text{ ps}$). When the current is off, the helicity of the skyrmion approaches $\eta = 3\pi/2$ during the relaxation and finally remains as $\eta = 3\pi/2$ at $t = 300 \text{ ps}$. Namely, the helicity of the skyrmion is switched from the Bloch-type configuration $\eta = \pi/2$ to the other Bloch-type configuration $\eta = 3\pi/2$ after the application of a 50-ps-long current pulse.

In Figs. 5(b) and 5(c), we show the time-dependent in-plane and out-of-plane spin components of the system during the helicity switching and relaxation of the system. It shows that the system responds to the driving current instantly during $t = 0 - 50 \text{ ps}$ and then relatively slowly relaxes to the energetically favored state during $t = 50 - 300 \text{ ps}$. In Fig. 5(d), we show several selected top-view snapshots of the spin configurations of the sample. It can be seen that the skyrmion shows opposite Bloch-type helicities at the initial and final states. Namely, the skyrmion helicity is changed from the initial value $\eta = \pi/2$ to $\eta = 3\pi/2$ at the final state.

In Fig. 6, we further demonstrate that it is possible to switch the helicity of an array of frustrated skyrmions between the Bloch-type configurations on a square-grid obstacle pattern. The system is a sample of 65 × 65 spins, where an obstacle pattern is formed by orthogonal defect lines. The width of the defect line is set to 5 spins. Hence, the distance between two nearest-neighboring parallel defect lines is 15 spins. The geometry and other system settings are the same as that used in the simulation in Fig. 3. It should be noted that the width of the defect line will not affect the results as we assumed a strong anisotropy for the defect line in this work for the sake of simplicity, i.e., the skyrmion cannot penetrate the defect-line obstacle.

We put nine relaxed skyrmions with $\eta = \pi/2$ in the sample as the initial state at $t = 0 \text{ ps}$. To switch the skyrmion helicity, we apply a short current pulse to drive the system. The current pulse profile is the same as that used in Fig. 5, i.e., the current density is $j = 40 \text{ MA/cm}^2$ and the pulse length is $\tau = 50 \text{ ps}$. After the application of a single current pulse, the system is relaxed for 250 ps. As shown in Fig. 6, the helicity numbers of the nine skyrmions are switched from $\eta = \pi/2$ to $\eta = 3\pi/2$ in a uniform manner induced by the current pulse. We note that the nine skyrmions are far enough from each other, and the applied spin current is uniform in the whole sample. As a result, each skyrmion can be treated as an isolated skyrmion, and the dipole-dipole interaction will not lead to different dynamics for different skyrmions. Namely, the dynamics of each skyrmion is similar to the helicity switching

<figure><img src="image_9.png" /><figcaption>**Fig. 7** A phase diagram showing the helicity switching of a frustrated skyrmion induced by a short current pulse in a sample with the obstacle pattern. A relaxed skyrmion with $\eta = \pi/2$ is placed at the sample center as the initial state at $t = 0 \text{ ps}$. The skyrmion is driven by a current pulse, followed by a relaxation until $t = 500 \text{ ps}$. The sample size is 21 × 21 spins. The width of the obstacle line is 3 spins. The obstacle has an enhanced PMA (i.e., $K_o/K_u = 10$). The number in the phase diagram indicates the relaxed skyrmion helicity (in units of $\pi$) after the application of the current pulse.</figcaption></figure>

dynamics of a single skyrmion given in Fig. 5. Therefore, in principle, it is possible to switch the helicity of an array of many skyrmions in a large-scale sample with the square-grid obstacle pattern. Such an array of many skyrmions may be useful for building skyrmion-based multistate memory devices.

We also point out that the current-induced switching process of the skyrmion helicity on the obstacle pattern depends on the applied current density and pulse length. As shown in Fig. 7, for higher current density, the minimum required pulse length to switch the helicity could be reduced. Namely, for lower current density, the minimum required pulse length to switch the helicity may be increased. However, when the current density is smaller than 30 MA/cm², the skyrmion helicity cannot be switched, indicating there is a critical current density beyond which the helicity switching can be achieved.

## 4. Conclusion

In conclusion, we have investigated the current-induced dynamics of a frustrated skyrmion in a sample with or without the obstacle pattern. The obstacle pattern with enhanced PMA could provide the skyrmion from being annihilated at the sample edge due to the skyrmion-obstacle repulsion, and more importantly, could lead to the circular motion of the frustrated
skyrmion with time-varying helicity driven by a moder-
ate current density. Namely, the repulsive interaction be-
tween the skyrmion and the defect-line obstacle plays an
important role in the helicity dynamics of the skyrmion.
In the system without the obstacle pattern, the frus-
trated skyrmion will move with a fixed Bloch-type helic-
ity due to the DDI and may be annihilated at the sample
edge. We have also studied the current-induced helicity
switching of a single frustrated skyrmion in a sample
with the obstacle pattern, where we find that the helicity
of the skyrmion can be switched from a Bloch-type con-
figuration to the other Bloch-type configuration driven by
a short current pulse. In addition, we show that the cur-
rent-induced switching of the skyrmion helicity can be re-
alized for an array of frustrated skyrmions on a square-
grid obstacle pattern. Our results suggest that the helic-
ity degree of freedom of topological spin textures may be
effectively controlled on nanostructures with well-de-
signed obstacle patterns. Our results also show the pos-
sibility that the helicity of skyrmions may be used in
spintronic functions.

**Acknowledgements** X.Z. was a JSPS International Research Fellow supported by JSPS KAKENHI (Grant No. JP20F20363). J.X. was a JSPS International Research Fellow supported by JSPS KAKENHI (Grant No. JP22F22061). O.A.T. acknowledges support from the Australian Research Council (Grant No. DP200101027), Russian Science Foundation (Grant No. 21-42-00035), NCMAS grant, and the Cooperative Research Project Program at the Research Institute of Electrical Communication, Tohoku University. G.Z. acknowledges support from the National Natural Science Foundation of China (Grants No. 51771127, No. 52171188, and No. 52111530143). J.Y. acknowledges support from the National Natural Science Foundation of China (Grant No. 51731001). Y.Z. acknowledges support from the Guangdong Special Support Project (Grant No. 2019BT02X030), Shenzhen Fundamental Research Fund (Grant No. JCYJ20210324120213037), Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403), Pearl River Recruitment Program of Talents (Grant No. 2017GC010293), and National Natural Science Foundation of China (Grants No. 11974298, No. 12004320, and No. 61961136006). M.E. acknowledges support from the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grants No. JP18H03676 and No. JP17K05490) and support from CREST, JST (Grants No. JPMJCR20T2 and No. JPMJCR16F1). X.L. acknowledges support by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grants No. JP20F20363, No. JP21H01364, No. JP21K18872, and No. JP22F22061).

References

1) A. N. Bogdanov and D. A. Yablonskii: *Sov. Phys. JETP*, **68**, 101 (1989).

2) U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer: *Nature*, **442**, 797 (2006).

3) N. Nagaosa and Y. Tokura: *Nat. Nanotechnol.*, **8**, 899 (2013).

4) M. Mochizuki and S. Seki: *J. Phys.: Condens. Matter*, **27**, 503001 (2015).

5) R. Wiesendanger: *Nat. Rev. Mater.*, **1**, 16044 (2016).

6) G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui: *J. Phys. D: Appl. Phys.*, **49**, 423001 (2016).

7) W. Kang, Y. Huang, X. Zhang, Y. Zhou, and W. Zhao: *Proc. IEEE*, **104**, 2040 (2016).

8) N. Kanazawa, S. Seki, and Y. Tokura: *Adv. Mater.*, **29**, 1603227 (2017).

9) W. Jiang, G. Chen, K. Liu, J. Zang, S. G. E. te Velthuis, and A. Hoffmann: *Phys. Rep.*, **704**, 1 (2017).

10) A. Fert, N. Reyren, and V. Cros: *Nat. Rev. Mater.*, **2**, 17031 (2017).

11) K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui: *J. Appl. Phys.*, **124**, 240901 (2018).

12) A. N. Bogdanov and C. Panagopoulos: *Nat. Rev. Phys.*, **2**, 492 (2020).

13) X. Zhang, Y. Zhou, K. Mee Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao, and S. Woo: *J. Phys.: Condens. Matter*, **32**, 143001 (2020).

14) Y. Fujishiro, N. Kanazawa, and Y. Tokura: *Appl. Phys. Lett.*, **116**, 090501 (2020).

15) C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert, M. Garst, T. Ma, S. Mankovsky, T. L. Monchesky, M. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. Pfleiderer, N. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von Bergmann, and J. Zang: *J. Phys. D: Appl. Phys.*, **53**, 363001 (2020).

16) B. Göbel, I. Mertig, and O. A. Tretiakov: *Phys. Rep.*, **895**, 1 (2021).

17) S. Li, W. Kang, X. Zhang, T. Nie, Y. Zhou, K. L. Wang, and W. Zhao: *Mater. Horiz.*, **8**, 854 (2021).

18) S. Luo and L. You: *APL Mater.*, **9**, 050901 (2021).

19) Y. Tokura and N. Kanazawa: *Chem. Rev.*, **121**, 2857 (2021).

20) H. Vakili, J.-W. Xu, W. Zhou, M. N. Sakib, M. G. Morshed, T. Hartnett, Y. Quessab, K. Litzius, C. T. Ma, S. Ganguly, M. R. Stan, P. V. Balachandran, G. S. D. Beach, S. J. Poon, A. D. Kent, and A. W. Ghosh: *J. Appl. Phys.*, **130**, 070908 (2021).

21) C. H. Marrows and K. Zeissler: *Appl. Phys. Lett.*, **119**, 250502 (2021).

22) C. Reichhardt, C. J. O. Reichhardt, and M. V. Milošević: *Rev. Mod. Phys.*, **94**, 035005 (2022).

23) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni: *Science*, **323**, 915 (2009).

24) X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura: *Nature*, **465**, 901 (2010).

25) S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel: *Nat. Phys.*, **7**, 713 (2011).

26) I. Dzyaloshinskii: *J. Phys. Chem. Solids*, **4**, 241 (1958).

27) T. Moriya: *Phys. Rev.*, **120**, 91 (1960).

28) T. Okubo, S. Chung, and H. Kawamura: *Phys. Rev. Lett.*, **108**, 017206 (2012).

29) A. O. Leonov and M. Mostovoy: *Nat. Commun.*, **6**, 8275 (2015).

30) S.-Z. Lin and S. Hayami: *Phys. Rev. B*, **93**, 064430 (2016).
31) S. Hayami, S.-Z. Lin, and C. D. Batista: *Phys. Rev. B*, **93**, 184413 (2016).

32) L. Rozsa, A. Deak, E. Simon, R. Yanes, L. Udvardi, L. Szunyogh, and U. Nowak: *Phys. Rev. Lett.*, **117**, 157205 (2016).

33) C. D. Batista, S.-Z. Lin, S. Hayami, and Y. Kamiya: *Rep. Prog. Phys.*, **79**, 084504 (2016).

34) A. O. Leonov and M. Mostovoy: *Nat. Commun.*, **8**, 14394 (2017).

35) Y. A. Kharkov, O. P. Sushkov, and M. Mostovoy: *Phys. Rev. Lett.*, **119**, 207201 (2017).

36) X. Zhang, J. Xia, Y. Zhou, X. Liu, H. Zhang, and M. Ezawa: *Nat. Commun.*, **8**, 1717 (2017).

37) H. Y. Yuan, O. Gomonay, and M. Kläui: *Phys. Rev. B*, **96**, 134415 (2017).

38) Y. Hu, X. Chi, X. Li, Y. Liu, and A. Du: *Sci. Rep.*, **7**, 16079 (2017).

39) S. von Malottki, B. Dupe, P. F. Bessarab, A. Delin, and S. Heinze: *Sci. Rep.*, **7**, 12299 (2017).

40) A. Sonntag, J. Hermenau, S. Krause, and R. Wiesendanger: *Phys. Rev. Lett.*, **113**, 077202 (2014).

41) J. J. Liang, J. H. Yu, J. Chen, M. H. Qin, M. Zeng, X. B. Lu, X. S. Gao, and J. M. Liu: *New J. Phys.*, **20**, 053037 (2018).

42) U. Ritzmann, S. von Malottki, J.-V. Kim, S. Heinze, J. Sinova, and B. Dupé: *Nat. Electron.*, **1**, 451 (2018).

43) T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa, Y. Yamasaki, H. Sagayama, H. Nakao, Y. Taguchi, T.-h. Arima, and Y. Tokura: *Science*, **365**, 914 (2019).

44) L. Desplat, J. V. Kim, and R. L. Stamps: *Phys. Rev. B*, **99**, 174409 (2019).

45) J. Xia, X. Zhang, M. Ezawa, Z. Hou, W. Wang, X. Liu, and Y. Zhou: *Phys. Rev. Applied*, **11**, 044046 (2019).

46) R. Zarzuela, H. Ochoa, and Y. Tserkovnyak: *Phys. Rev. B*, **100**, 054426 (2019).

47) V. Lohani, C. Hickey, J. Masell, and A. Rosch: *Phys. Rev. X*, **9**, 041063 (2019).

48) B. Göbel, A. Mook, J. Henk, I. Mertig, and O. A. Tretiakov: *Phys. Rev. B*, **99**, 060407(R) (2019).

49) T. H. Diep: *Entropy*, **21**, 175 (2019).

50) I. F. Sharafullin and H. T. Diep: *Symmetry*, **12**, 26 (2020).

51) X. Zhang, J. Xia, L. Shen, M. Ezawa, O. A. Tretiakov, G. Zhao, X. Liu, and Y. Zhou: *Phys. Rev. B*, **101**, 144435 (2020).

52) J. Xia, X. Zhang, M. Ezawa, O. A. Tretiakov, Z. Hou, W. Wang, G. Zhao, X. Liu, H. T. Diep, and Y. Zhou: *Appl. Phys. Lett.*, **117**, 012403 (2020).

53) X. Zhang, J. Xia, M. Ezawa, O. A. Tretiakov, H. T. Diep, G. Zhao, X. Liu, and Y. Zhou: *Appl. Phys. Lett.*, **118**, 052411 (2021).

54) R. Zarzuela, D. Hill, J. Sinova, and Y. Tserkovnyak: *Phys. Rev. B*, **103**, 174424 (2021).

55) C. Psaroudaki and C. Panagopoulos: *Phys. Rev. Lett.*, **127**, 067201 (2021).

56) X. Zhang, J. Xia, O. A. Tretiakov, H. T. Diep, G. Zhao, J. Yang, Y. Zhou, M. Ezawa, and X. Liu: *Phys. Rev. B*, **104**, L220406 (2021).

57) J. Xia, X. Zhang, O. A. Tretiakov, H. T. Diep, J. Yang, G. Zhao, M. Ezawa, Y. Zhou, and X. Liu: *Phys. Rev. B*, **105**, 214402 (2022).

58) M. Miyata, J.-i. Ohe, and G. Tatara: *Phys. Rev. Applied*, **18**, 014075 (2022).

59) I. F. Sharafullin, M. K. Kharrasov, and H. T. Diep: *Phys. Rev. B*, **99**, 214420 (2019).

60) J. Xia, X. Zhang, X. Liu, Y. Zhou, and M. Ezawa: *arXiv preprint arXiv:2204.04589*, (2022).

61) X. Yu: *J. Magn. Magn. Mater.*, **539**, 168332 (2021).

62) S. Rohart and A. Thiaville: *Phys. Rev. B*, **88**, 184422 (2013).

63) C. Reichhardt, D. Ray, and C. J. O. Reichhardt: *New J. Phys.*, **17**, 073034 (2015).

64) C. Reichhardt and C. J. O. Reichhardt: *Phys. Rev. B*, **94**, 094413 (2016).

65) J. Müller: *New J. Phys.*, **19**, 025002 (2017).

66) I. Lima Fernandes, J. Bouaziz, S. Blügel, and S. Lounis: *Nat. Commun.*, **9**, 4395 (2018).

67) L. González-Gómez, J. Castell-Queralt, N. Del-Valle, A. Sanchez, and C. Navau: *Phys. Rev. B*, **100**, 054440 (2019).

68) J. Castell-Queralt, L. González-Gómez, N. Del-Valle, A. Sanchez, and C. Navau: *Nanoscale*, **11**, 12589 (2019).

69) S. Bhatti and S. N. Piramanayagam: *Phys. Status Solidi RRL*, **13**, 1900090 (2019).

70) X. Zhang, J. Xia, K. Shirai, H. Fujiwara, O. A. Tretiakov, M. Ezawa, Y. Zhou, and X. Liu: *Commun. Phys.*, **4**, 255 (2021).

71) J. C. B. Souza, N. P. Vizarim, C. J. O. Reichhardt, C. Reichhardt, and P. A. Venegas: *Phys. Rev. B*, **104**, 054434 (2021).

72) J. Souza, N. Vizarim, C. Reichhardt, C. Reichhardt, and P. Venegas: *arXiv preprint arXiv:2204.11906*, (2022).

73) X. Zhang, J. Xia, and X. Liu: *Phys. Rev. B*, **106**, 094418 (2022).

74) N. Del-Valle, J. Castell-Queralt, L. González-Gómez, and C. Navau: *APL Mater.*, **10**, 010702 (2022).

75) K. Ohara, X. Zhang, Y. Chen, Z. Wei, Y. Ma, J. Xia, Y. Zhou, and X. Liu: *Nano Lett.*, **21**, 4320 (2021).

76) R. Juge, K. Bairagi, K. G. Rana, J. Vogel, M. Sall, D. Mailly, V. T. Pham, Q. Zhang, N. Sisodia, M. Foerster, L. Aballe, M. Belmeguenai, Y. Roussigné, S. Auffret, L. D. Buda-Prejbeanu, G. Gaudin, D. Ravelosona, and O. Boulle: *Nano Lett.*, **21**, 2989 (2021).

77) Y.-K. Park, J.-S. Kim, Y.-S. Nam, S. Jeon, J.-H. Park, K.-W. Kim, H.-W. Lee, B.-C. Min, and S.-B. Choe: *NPG Asia Mater.*, **12**, 38 (2020).

78) W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann: *Science*, **349**, 283 (2015).

79) J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert: *Nat. Nanotechnol.*, **8**, 839 (2013).

80) R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio: *Sci. Rep.*, **4**, 6784 (2014).

81) X. Zhang, J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, and M. Ezawa: *Phys. Rev. B*, **94**, 094420 (2016).

82) M. J. Donahue and D. G. Porter, *OMMMF User's Guide, Version 1.0, Interagency Report NO. NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (1999) http://math.nist.gov/oommf*.

Received Oct. 05, 2022; Revised Oct. 31, 2022; Accepted Nov. 10, 2022