⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 097501 Manipulating Skyrmion Motion on a Nanotrack with Varied Material Parameters and Tilted Spin Currents Jia Luo (罗佳)1, Jia-Hao Guo2, Yun-He Hou (侯云鹤)2, Jun-Lin Wang (王君林)3,4, Yong-Bing Xu (徐永兵)4, Yan Zhou (周艳)5*, Philip Wing Tat Pong6*, and Guo-Ping Zhao (赵国平)1,7* Affiliations 1College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China 2Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China 3School of Integrated Circuits, Guangdong University of Technology, Guangzhou 510006, China 4School of Physics, Engineering and Technology, University of York, York, YO10 5DD, United Kingdom 5School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China 6Department of Electrical and Computer Engineering, New Jersey Institute of Technology, NJ 07102, USA 7Center for Magnetism and Spintronics, Sichuan Normal University, Chengdu 610068, China Received 6 July 2023; accepted manuscript online 28 August 2023; published online 10 September 2023 *Corresponding authors. Email: zhouyan@cuhk.edu.cn; philip.pong@njit.edu; zhaogp@uestc.edu.cn Citation Text: Luo J, Guo J H, Hou Y H et al. 2023 Chin. Phys. Lett. 40 097501    Abstract Magnetic skyrmions are topological quasiparticles with nanoscale size and high mobility, which have potential applications in information storage and spintronic devices. The manipulation of skyrmion's dynamics in the track is an important topic due to the skyrmion Hall effect, which can deviate the skyrmions from the preferred direction. We propose a new model based on the ferromagnetic skyrmion, where the skyrmion velocity can be well controlled by adjusting the direction of the current. Using this design, we can avoid the annihilation of the skyrmion induced by the skyrmion Hall effect, which is confirmed by our micromagnetic simulation based on Mumax$^{3}$. In the meantime, we increase the average velocity of the skyrmion by varying the intrinsic material parameters in the track, where the simulations agree well with our analytical results based on the Thiele equation. Finally, we give a phase diagram of the output of the skyrmion in the T-type track, which provides some practical ways for design of logic gates by manipulating crystalline anisotropy through the electrical control.
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 DOI:10.1088/0256-307X/40/9/097501 © 2023 Chinese Physics Society Article Text Skyrmions were initially hypothesized in nuclear physics to characterize the resonance states of baryons,[1-7] which have been observed in a number of magnetic systems with broken inversion symmetries.[8,9] One kind of these systems, such as MnSi, Fe$_{1-x}$Co$_{x}$Si, and Cu$_{2}$OSeO$_{3}$ with the bulk Dzyaloshinskii–Moriya interaction (DMI), has a chiral crystal structure that supports skyrmions in the Bloch-type structure.[10-16] Another type of the system is ferromagnetic (FM) layer/heavy metal (HM) such as Fe/Ir,[17-19] Ni/Co,[20-22] and Ta/CoFeB/TaO$_{x}$,[23,24] which results in the interfacial DMI.[16,25] Due to the same topological form, a broad variety of magnetic configurations in the form of domain walls,[26-29] vortices,[30,31] bimeron,[32,33] and magnetic skyrmions have been intensely studied in magnetic materials in recent years. Furthermore, a magnetic skyrmion is a magnetic domain wall structure with topological protection in nanoscale.[34-36] As compared to magnetic domain walls, skyrmions have a small size on the order of 1 nm to 100 nm[37-39] and a low driven current density as low as $10^{6}$ A/m$^{2}$.[40,41] Skyrmions are also one of the essential topological solitons in solid-state physics, and they are particularly important in the future applications of spintronics. They are anticipated to be one of the possible information carriers in future high-density data storage devices and information processing devices, such as spin torque oscillators,[42-45] logic gates,[4,46,47] and racetrack memories.[48-50] According to these spintronic applications, the manipulation of skyrmions is crucial in order to address the growing need for information storage and processing technology.[34,51-53] The nucleation and stability of an isolated magnetic skyrmion, as well as the current-induced motion of magnetic skyrmions, have been thoroughly investigated so far.[54-56] In general, materials with strong spin-orbit interactions have large magnetic anisotropy. Since 2002, it has been demonstrated that perpendicular magnetic anisotropy (PMA)[57-60] exists at magnetic metal/oxide contacts. Subsequently, this PMA is observed in systems with moderate spin-orbit interactions, although its magnitude is equivalent to that measured at Co/Pt surfaces, a standard for substantial interfacial anisotropy.[61] In fact, this PMA was found to be rather common at magnetic metal/oxide interfaces, as it has been seen with a wide variety of amorphous or crystalline oxides.[62-65] A larger PMA can lead to a greater repulsive force because it generates an extra energy barrier that necessitates a higher current density to drive. In the meantime, the voltage was often used to modulate PMA constants in FM multilayer materials, which has been known as the voltage-controlled magnetic anisotropy (VCMA) effect.[48,66-68] However, when driving the traditional ferromagnetic FM skyrmion with currents, due to the existence of the skyrmion Hall effect, the FM skyrmion will annihilate at the edge of the track causing signal loss. In order to overcome the skyrmion Hall effect, numerous solutions have been proposed. For example, researchers proposed the synthetic antiferromagnetic (SAFM) skyrmion[69] and the antiferromagnetic (AFM) skyrmion;[70-72] high magnetic anisotropy $K$ materials are added to the edge of the track.[73,74] In this Letter, a new model based on the FM skyrmion is proposed, in which we manipulate the direction of the skyrmion's velocity by adjusting the direction of the current. Using this method, we can avoid the annihilation of the skyrmion induced by the skyrmion Hall effect, which is confirmed by our micromagnetic simulation based on Mumax$^{3}$. In the meantime, we increase the average velocity of the skyrmion by varying the material parameters in some track areas, where the simulations agree well with our analytical results based on the Thiele equation. Finally, we give a phase diagram of the output of the skyrmion in the T-type track, which provides some practical ways for design of logic gates by manipulating crystalline anisotropy through the electrical control. Method and Model. Spin-orbit torque (SOT) is used to drive FM skyrmion. Both the analytical calculation of the Thiele equation and the numerical simulation of Mumax$^{3}$ are based on the Landau–Lifshitz–Gilbert (LLG) equation driven by the electric current, which incorporates Slonczewski-like spin torques. The definition is as follows:[75,76] \begin{align} \frac{d\boldsymbol{m}}{dt}=\,&-|\gamma|\boldsymbol{m}\times \boldsymbol{H}_{\rm eff}+\alpha \Big(\boldsymbol{m}\times \frac{d\boldsymbol{m}}{dt}\Big)\notag\\ &+|\gamma|\beta \epsilon \boldsymbol{m}\times (\boldsymbol{m}_{\rm p}\times \boldsymbol{m})-|\gamma|\beta \epsilon'\boldsymbol{m}\times \boldsymbol{m}_{\rm p}, \tag {1} \end{align} where $\boldsymbol{m}$ stands for the magnetization deduction $M/M_{\scriptscriptstyle{\rm S}}$, and $\boldsymbol{H} _{\rm eff}$ represents the effective field. The effective field $\boldsymbol{H} _{\rm eff}$ includes exchange interactions, DMI, PMA, and dipole-dipole interaction (DDI); $\boldsymbol{m} _{\rm p}$ is the polarization intensity vector; $\gamma$ and $\alpha$ denote the gyromagnetic ratio and Gilbert damping constant, respectively. In the third term, $\beta=|\frac{\hslash}{{2\mu}_{0}e}|\frac{j}{\tau M_{\scriptscriptstyle{\rm S}}}$, where $e$ is the electron charge in coulomb $C$, $\hslash$ is Planck's constant equal to $6.63 \times 10^{-34}$ J$\cdot$s, $j$ denotes the current density in A/m$^{2}$, $\tau$ is the FM layer thickness in meter, and $M_{\scriptscriptstyle{\rm S}}$ is the saturation magnetization in A/m. In the last term, $\epsilon' = \xi \epsilon$, with $\xi$ being the field-like SOT constant. In this study, we use the Cartesian coordinate system, where the $x$, $y$, and $z$ axes are along longitudinal, widthwise, and thickness directions of the track, respectively. The indication of the skyrmion movement will be explained as the analysis of a modified Thiele equation under the assumption of the stiffness of spin textures. In this work, we consider an isolated Neél skyrmion in the presence of interfacial DMI. The skyrmion dynamics is well described in terms of the Thiele equation[4,77] \begin{align} \boldsymbol{G\times v-}\alpha \boldsymbol{D}_{\rm diss}\cdot \boldsymbol{v-}\boldsymbol{F}_{j}=0. \tag {2} \end{align} The first term is the Magnus force with the gyrovector ${\boldsymbol G} = (4\pi Q\mu_{0}M_{\scriptscriptstyle{\rm S}}\tau/\gamma) {\boldsymbol e}_z$. It has been discovered that the Magnus force is proportional to the topological charge $Q$, which is an integer reflecting the number of times the spin direction wraps the unit sphere, i.e.,[78] \begin{align} Q =-\frac{1}{4\pi }\int \big[\boldsymbol{m}\cdot(\partial_{x}\boldsymbol{m}\times \partial_{y}\boldsymbol{m})\big] dxdy, \tag {3} \end{align} where the number of topologies, $Q$, is $\pm 1$ for an isolated FM skyrmion. In Eq. (2), the second term describes the dissipative force, the dissipation tensor $\boldsymbol{D} _{\rm diss}$ is given by \begin{align} \boldsymbol{D} _{\rm diss} = \frac{\mu_{0}M_{\scriptscriptstyle{\rm S}}\tau}{\gamma} \begin{pmatrix} d & 0\\ 0 & d\\ \end{pmatrix},\nonumber \end{align} with $d = \smallint (\partial_{x} \boldsymbol{m} \cdot\partial_{x} \boldsymbol{m}) dxdy$. The third term $\boldsymbol{F}_{j}$ is the current-induced force that can be written as \begin{align} \boldsymbol{F}_{j}=\frac{\mu_{0}B_{j}M_{\scriptscriptstyle{\rm S}}\tau}{\gamma} \begin{pmatrix}I_{x} \\I_{y}\end{pmatrix}, \nonumber \end{align} with $B_{j}=\gamma \hslash m_{\rm p}j/2\mu_{0}et_{z}M_{\scriptscriptstyle{\rm S}}$ and $I_{i}=\smallint [({\boldsymbol m}\times \hat{\boldsymbol{e}}_{{\boldsymbol{m}}_{\rm p}}) \cdot \partial_{i}{\boldsymbol m}] dxdy$, where $m_{\rm p}$ and $\boldsymbol{\hat{e}}_{{\boldsymbol{m}}_{\rm p}}$ represent the value and direction of the polarization vector $\boldsymbol{m}_{\rm p}$, respectively, i.e., $\boldsymbol{m} _{\rm p}=m_{\rm p} \boldsymbol{\hat{e}} _{\boldsymbol{m}_{\rm p}}$. For a typical SOT, the input current is in the HM layer along the film plane, which can induce the polarization current $\boldsymbol{j}$ in the HM layer through spin hall effect. In this case, $\boldsymbol{m}_{\rm p}$ is perpendicular to $\boldsymbol{j}$ with both $\boldsymbol{m} _{\rm p}$ and $\boldsymbol{j}$ in the plane of the film.[77] Further, by solving Eq. (2), we can obtain the velocity of the skyrmion in the directions of $\boldsymbol{j}$ and $\boldsymbol{m} _{\rm p}$, expressed as[77] \begin{align} &v_{\boldsymbol{j}}=-\frac{\alpha dB_{j}I_{x}}{{(4\pi Q)}^{2}+\alpha^{2}d^{2}}, \tag {4}\\ &v_{\boldsymbol{m}_{\rm p}}=-\frac{4\pi QB_{j}I_{x}}{\left( 4\pi Q \right)^{2}+\alpha^{2}d^{2}}, \tag {5} \end{align} where $v_{\boldsymbol{j}}$ and $v_{\boldsymbol{m}_{\rm p}}$ represent the velocity components in the directions of $\boldsymbol{j}$ and $\boldsymbol{m} _{\rm p}$, respectively. The total velocity $\boldsymbol{v}=\sqrt {\boldsymbol{v}_{\boldsymbol{j}}^{2}+\boldsymbol{v}_{\boldsymbol{\hat{e}m}p}^{2}}$ of the FM skyrmion, in this case, can also be obtained via $v_{\boldsymbol{j}}$ and $v_{\boldsymbol{\hat{e}m}p}$. In the system studied here, the size of the skyrmion is about 10 nm, so $d$ and $I_{x}$ in Eqs. (4) and (5) can be approximated as[77] $d\approx 2\pi [(R/\sqrt{A/K})+(\sqrt{A/K}/R) ]$ and $I_{x} \approx \pi^{2}R$, where $R$ is the radius of the skyrmion. The $R$ is related to the material parameters, which can be approximated as[79] \begin{align} R\approx \frac{\sqrt{A/K}}{\sqrt {2-[\pi D/(2\sqrt {AK})]}}. \tag {6} \end{align} Now we can get the angle between the velocity $\boldsymbol{v}$ of the skyrmion and the current ${\boldsymbol j}$ by Eqs. (4) and (5), i.e., the Hall angle of the skyrmion $\theta_{\scriptscriptstyle{\rm SH}}$, which satisfies[80] \begin{align} \tan\theta_{\scriptscriptstyle{\rm SH}}=\frac{4\pi Q}{\alpha d}. \tag {7} \end{align} One can see that from Eq. (7), skyrmions will be deviated from the direction of the current by an angle $\theta_{\scriptscriptstyle{\rm SH}}$, which is called the skyrmion Hall effect in the literature, However, it has been rarely noticed that the direction of the skyrmion's velocity $\boldsymbol{v}$ can be adjusted by changing the current direction $\boldsymbol{j}$ so that the skyrmion can move in any direction we wanted. In particular, as we will show in the following, the annihilation of the skyrmion induced by the skyrmion Hall effect can be efficiently overcome if we design a suitable angle $\theta$ for the current $\boldsymbol{j}$. In the micromagnetic simulation part, we use the Mumax$^{3}$ with the DMI extension module,[81] with the LLG equation governing the time-dependent magnetization dynamics. As shown in Fig. 1, the skyrmion motion in the FM layer is driven by the spin current generated due to the spin Hall effect in the HM layer. For simplicity, we directly simulate the injection of the spin-polarized current with a certain current density and polarization rate. The polarization rate $m_{\rm p}$ of the spin current used in all simulations is 0.4. The structure is an ultrathin FM nanotrack on the HM with length $l = 500$ nm and width $w = 100$ nm with one modified PMA square area in the center of the nanotrack, with the length $a = 100$ nm. The PMA value $K_{uV}$ is controlled by applying voltage with a linearly varying relationship. The skyrmion will move on the FM free layer, and the interfacial DMI will be induced at the interface of the FM layer/HM layer. The starting location of the skyrmion is found at $x = 50$ nm, $y = 0$ nm, with the layer thickness set at 1 nm. As shown in Fig. 1(b), an angle $\theta$ is assumed to be between the direction of the driven current and the longitudinal direction of the track to manipulate the skyrmion motion efficiently. Since the direction of polarization $\boldsymbol{\hat{e}} _{\boldsymbol{m}_{\rm p}}$ induced by the HM layer is always perpendicular to the direction of the current $\boldsymbol{j}$, the $\boldsymbol{\hat{e}} _{\boldsymbol{m}_{\rm p}}$ direction can be expressed as $\boldsymbol{\hat{e}} _{\boldsymbol{m}_{\rm p}} = (\sin\theta, \cos \theta, 0)$. We set the PMA constant $K_{u0}$ in the center correction zone between 0.9$K_{u}$ and 1.05$K_{u}$, with $K_{u} = 0.8$ MJ/m$^{3}$. The default magnetic parameters are adopted as follows:[78] the saturation magnetization $M_{\scriptscriptstyle{\rm S}} = 580\times 10^{3}$ A/m, the gyromagnetic ratio $\gamma = -2.211 \times 10^{5}$ m/(A$\cdot$s), the Gilbert damping constant $\alpha = 0.1$, the current density $j = 4\times 10^{11}$ A/m$^{2}$, the PMA constant $K$, the exchange constant $A$, the DMI constant $D$. The discretization cell size is set as $1 \times 1\times 1$ nm$^{3}$.
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Fig. 1. (a) The simulated model of an FM skyrmion on an ultrathin FM/HM bilayer nanostructure. (b) The vertical view of the skyrmion motion on the nanotrack. The width $w = 100$ nm, length $l = 500$ nm, and the voltage-controlled-magnetic-anisotropy (VCMA) region length $a = 100$ nm, in which a voltage gate $K_{u0}$ is in the middle of the FM layer. The skyrmion is firstly located at $x = 50$ nm and $y = 0$ nm. The $\hat{e} _{m p}$ and $j$ indicate the polarization direction and the current direction, respectively, while $\theta$ represents the angle between $\hat{e} _{m p}$ and $j$. The PMA constant $K_{u} = 0.8$ MJ/m$^{3}$ and $K_{u0} = 0.72$–0.84 MJ/m$^{3}$, the exchange constant $A_{u} = A_{u0} = 15$ pJ/m, the DMI constant $D_{u} = D_{u0} = 3.0$ mJ/m$^{2}$.
Results and Discussions. Under the model in Fig. 1, we first perform a qualitative analysis of the motion of the skyrmion and infer the approximate range of skyrmion Hall angles. In the next model (Fig. 2), the relationship between the velocity and position of the skyrmion will be discussed in detail. According to Eq. (6), an increase in the anisotropy constant $K$ can reduce the skyrmion's radius, in the case of our parameters. By simulation, we confirm that when the PMA constant $K_{u0}$ rises, the skyrmion's energy increases and its radius decreases. On the other hand, if the PMA constant goes down, the skyrmion's energy will decrease and its radius will increase. This phenomenon is consistent with the results in Ref. [82]. It can be seen from Eqs. (4) and (5) that the skyrmion velocity is roughly proportional to the skyrmion's radius for small $\alpha$. Thus, the skyrmion velocity will rise when the PMA constant drops. Therefore, a larger PMA generates an energy barrier so that the skyrmion velocity will decrease, whereas a smaller PMA generates an energy trap where the skyrmion moves faster. The simulation results show that the radius $R$ of the skyrmion is between 7 nm and 13 nm so that the dissipation constant $d$ is between 14 and 21. According to Eq. (7), the tan$\theta_{\scriptscriptstyle{\rm SH}}$ is between 5.99 and 8.8. Now we can manipulate the skyrmion motion in the FM track freely, i.e., by changing $\theta$, we can direct the skyrmion in any direction in the track we prefer. In particular, by designing a suitable angle $\theta$, the skyrmion will move along the longitudinal direction of the track and will not annihilate by the skyrmion Hall effect. A close examination shows that the skyrmion's direction is not sensitive to the actual value of tan$\theta$. Therefore, for convenience, we set tan$\theta$ of the current in the entire HM to 6.0, as shown in Fig. 2(b). In the meantime, we can set up multiple PMA zones in the track and change the local speed of the skyrmion in the track to achieve a faster average speed. Figure 2 shows the design of the skyrmion racetrack with two voltage gates and a top view of the track of the skyrmion transmission through two PMA constants' regions controlled by voltage gates. The PMA constants of the two modified areas are $K_{u1}$ and $K_{u2}$. Leaving the other parameters unchanged, we set $K_{u1}$ and $K_{u2}$ to 0.72 MJ/m$^{3}$ and 0.84 MJ/m$^{3}$, respectively.
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Fig. 2. (a) The simulated model of an FM skyrmion on an ultrathin FM/HM bilayer nanostructure. (b) The vertical view of the skyrmion motion on the nanotrack. The width $w = 100$ nm, length $l = 500$ nm, and the VCMA region length $a = 100$ nm, where two voltage gates $K_{u1}$ and $K_{u2}$ are on the FM layer. The skyrmion is firstly located at $x = 50$ nm and $y = 50$ nm. The PMA constant $K_{u} = 0.8$ MJ/m$^{3}$, $K_{u1} = 0.72$ MJ/m$^{3}$, and $K_{u2} = 0.84$ MJ/m$^{3}$. The exchange constant $A_{u} = A_{u1} = A_{u2} = 15$ pJ/m. The DMI constant $D_{u} = D_{u1} = D_{u2} = 3.0$ mJ/m$^{2}$.
The simulation results with spatially varied crystalline anisotropy are shown in Fig. 3. The dotted line in Fig. 3(a) denotes the simulated velocity of the skyrmion as a function of position, where the speed of the skyrmion in the $K_{u}$ area is maintained at about 100 m/s. When the skyrmion moves to the $K_{u1}$ region, it encounters an energy trap and the velocity of the skyrmion rises to 155 m/s. As the skyrmion leaves the center of the $K_{u1}$ region and moves towards the next $K_{u}$ region, this energy trap will prevent it from leaving, slowing down the soliton's speed to about 100 m/s. As a whole, the $K_{u1}$ region acts as an energy trap, which increases the average speed of the skyrmion. In contrary, when the skyrmion moves to the $K_{u2}$ region, this area acts as an energy barrier so that the average velocity of the skyrmion will decrease. The velocity of the skyrmion can also be obtained by solving the Thiele equation, as shown by the red line in Fig. 3(a), which is in good agreement with the numerical data. Slightly larger differences occur near the boundary of the regions, i.e., $x = 100$ nm, 200 nm, 300 nm, and 400 nm, because we assume the ultrafast response of the skyrmion with the variation of the PMA in the analytical calculation. However, the actual change of the skyrmion speed will be delayed as it needs time to reach the stable speed. Figure 3(b) shows the trajectory of the skyrmion, indicating that the skyrmion is roughly along the $x$ direction. As a result, our design of a tilted spin current as shown in Fig. 2 can avoid the annihilation of the skyrmion caused by the skyrmion Hall effect efficiently. It is noted, however, that there is a slight deviation from the $x$ direction between $x = 50$–100 nm and 200–500 nm, which is due to the fact that the tilted angle is designed assuming tan$\theta_{\scriptscriptstyle{\rm SH}} = 6.0$, whereas the actual tan$\theta_{\scriptscriptstyle{\rm SH}}$ is between 6.0 and 8.8.
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Fig. 3. (a) The skyrmion's velocity as a function of position. (b) The trajectory of the skyrmion. The parameters in (a) and (b) are $K_{u1} = 0.72\times 10^{6}$ J/m$^{3}$, $K_{u2} = 0.84\times 10^{6}$ J/m$^{3}$, $K_{u} = 0.80\times 10^{6}$ J/m$^{3}$; $A_{u}=A_{u1}=A_{u2} = 15$ pJ/m; $D_{u}=D_{u1}=D_{u2} = 3.0$ mJ/m$^{2}$. The dotted lines and the solid red lines are the results of numerical simulation and analytical calculation, respectively.
We also investigate the effects of exchange interactions and DMI on skyrmion motion in Fig. 4, by changing the exchange constant and DMI constant locally. Both numerical and analytical results illustrate that a large DMI and a small $A$ increase the velocity of the skyrmion, and vice versa, as shown in Figs. 4(a) and 4(c). According to Eq. (6), enhancement of DMI and reduction of $A$ here play a similar role on the decrease of the PMA in the above case, which can increase the radius of the skyrmion and hence enlarge the skyrmion speed. In particular, one can see from Fig. 4(c) that a slight increase of $D$ from 3.0 mJ/m$^{2}$ to 3.5 mJ/m$^{2}$, i.e., a 17% increase of $D$ can enhance the skyrmion velocity by about 120%. On the other hand, the decreasing $K$ and $A$ have much smaller effects on the skyrmion speed. A close examination of Eq. (6) shows that the skyrmion size $R$ is much more sensitive to the DMI value and hence explains the difference. Good agreement between numerical and analytical results justifies our explanation as well as our calculation. Similar to Fig. 3(b), Figs. 4(b) and 4(d) show that the skyrmions are roughly along the $x$ direction, illustrating the effect of the tilted spin current in overcoming the annihilation of the skyrmion.
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Fig. 4. (a) The skyrmion's velocity as a function of position. (b) The trajectory of the skyrmion. The parameters in (a) and (b) are $K_{u}=K_{u1}=K_{u2} = 0.80\times 10^{6}$ J/m$^{3}$; $A_{u} = 15$ pJ/m, $A_{u1}=A_{u2} = 13$ pJ/m. (c) The skyrmion's velocity as a function of position. (d) The trajectory of the skyrmion. The parameters in (c) and (d) are $K_{u}=K_{u1}=K_{u2} = 0.80\times 10^{6}$ J/m$^{3}$; $A_{u}=A_{u1}=A_{u2} = 15$ pJ/m; $D_{u} = 3.0$ mJ/m$^{2}$, $D_{u1}=D_{u2} = 3.5$ mJ/m$^{2}$. The dotted lines and the solid red lines are the results of numerical simulation and analytical calculation, respectively.
Using varied anisotropy and the tilted spin current, we can design a T-type track for magnetic skyrmion motion on nanomaterials. As illustrated in Fig. 5(a), the skyrmion is initially placed on the top left of the T-type track, where the PMA constants in the $K_{u4}$ area and the $K_{u3}$ area can vary. First of all, as shown in Fig. 5(b), we give the phase diagram of the motion of skyrmion with different tilted angles in the T-type track. From Fig. 5(b), we can see that there are three types of skyrmion motion, namely annihilation, entering the $K_{u4}$ region, and turning right into the lower $K_{u3}$ region. For convenience, we denote the two states of “entering the $K_{u4}$ region” and “entering the lower $K_{u3}$ region” as state “0” and state “1”, respectively. When the tilted angle of the current is between 70$^{\circ}$ and 90$^{\circ}$, the skyrmion will be in state 0. Increase of the tilted angle of the current to 100$^{\circ}$ will change the skyrmion state to 1. This remains for the angle below 110$^{\circ}$. Further, when the tilted angle of the current is larger than 110$^{\circ}$ or smaller than 70$^{\circ}$, the skyrmion will annihilate. Interestingly, we can see that the state of the skyrmion is unchanged for various anisotropy constants $K$ of the T-type track, indicating the stability of the present design of the logic gate. Furthermore, based on the results in Fig. 5(b), we improve the design of the ordinary VCMA control skyrmion by adding a constant to the tilted angles of the current $\theta = 90^{\circ}$. We consider the joint effect of skyrmion motion directions by adjusting PMA constants in $K_{u3}$ and $K_{u4}$ regions, with results shown in Fig. 5(c). The skyrmion motion on the FM layer shows two scenarios, including outputting data “0”, which indicates that the skyrmion goes into the $K_{u4}$ region and outputting data “1”, which denotes that the skyrmion will alter its motion direction and turn right into the lower $K_{u3}$ region. The phase boundary of two motion statuses shows the step shape, where the final position of the skyrmion is determined by the difference in parameters between the two regions. The smaller the difference between $K_{u3}$ and $K_{u4}$, the smaller the energy difference in the two regions, and hence the easier the passage of the skyrmion from the left to the right. The tilted spin current here avoids the annihilation of skyrmion signals caused by the skyrmion Hall effect, and facilitates the manipulation of the skyrmion.
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Fig. 5. (a) The track model of T-type. (b) Phase diagram of the motion of skyrmion with different tilted angles. (c) Phase diagrams of the motion of skyrmion with various (a) PMA constants $K_{u3}$ and $K_{u4}$. We fix PMA constant $K_{u5} = 1.2\times 10^{6}$ J/m$^{3}$, exchange energy constant $A = 15.0\times 10^{-12}$ J/m and DMI constants $D = 3.0$ mJ/m$^{2}$. In (b) and (c), the $\times$, red dots, and black squares denote annihilation, state 1, and state 0, respectively.
Compared with the design of existing spintronic devices,[4,42-50,73,74] our design is also to guide the skyrmion's motion by changing the energy potential. However, our design can manipulate the output of the device by adjusting the angle of the current, as shown in Fig. 5(b). In comparison to the method of VCMA manipulation skyrmion, our method removes the complex voltage control step and can realize the output of signal 0 or 1 by simply adjusting the current. In addition, under different parameters $K$, the output results of this method are basically unchanged and relatively stable. Therefore, our design is simpler and easily to be realized in the experiment. Further, according to the results of Fig. 5(c), our design can also have the functions of the original VCMA device. In summary, the present design can realize the control of the output signal from two ways, by adjusting the current in the plane and manipulating the VCMA out of the plane. Our results here open a new route for the efficient manipulation of skyrmions and the construction of skyrmion-based spintronics devices. In conclusion, we have proposed a new model based on the FM skyrmion, which allows for the manipulation of the skyrmion's velocity direction by adjusting the current direction. This approach enables the skyrmion to move in any desired direction, where we have increased the average velocity of the skyrmion by modifying the material parameters in specific track areas. A large PMA, a large DMI, or a small $A$ can increase the velocity of the skyrmion by increasing its radius, within the parameters considered here. Moreover, we find that the skyrmion's size and speed are more sensitive to DMI than to PMA and exchange interactions. The simulation results agree well with the analytical results based on the Thiele equation. In addition, we design a T-type track using varied anisotropy and a tilted spin current and provided a phase diagram, where we can control the skyrmion's trajectory by adjusting the current direction or the PMA in different regions. Overall, our findings suggest that the control of skyrmion dynamics can be achieved by adjusting material parameters and designing specific track geometries. Further research is necessary to explore the full potential of these approaches and their practical applications. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 51771127, 52171188, 52111530143, 11974298, 12374123, and 12241403), the Central Government Funds of Guiding Local Scientific and Technological Development of Sichuan Province (Grant No. 2021ZYD0025), the Shenzhen Fundamental Research Fund (Grant No. JCYJ20210324120213037), Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403), the Key-Area Research & Development Program of Guangdong Province (Grant No. 2021B0101300003), and the Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2022A1515110863 and 2023A1515010837). References Magnetic skyrmions: advances in physics and potential applicationsMagnetic Skyrmion MaterialsSkyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applicationsMagnetic skyrmion logic gates: conversion, duplication and merging of skyrmionsSpontaneous skyrmion ground states in magnetic metalsThe baby Skyrme models and their multi-skyrmionsMESONS, SKYRMIONS AND BARYONSSkyrmions in magnetic multilayersSkyrmion Lattice in a Chiral MagnetDirect Observation of the Dzyaloshinskii-Moriya Interaction in a Pt/Co/Ni FilmStability and phase transition of skyrmion crystals generated by Dzyaloshinskii-Moriya interactionInvestigation of the Dzyaloshinskii-Moriya interaction and room temperature skyrmions in W/CoFeB/MgO thin films and microwiresChiral spin torque at magnetic domain wallsRole of the Dzyaloshinskii-Moriya interaction in multiferroic perovskitesTemperature dependence of the Dzyaloshinskii-Moriya interaction in ultrathin filmsFirst-principles calculations for Dzyaloshinskii–Moriya interactionTunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayersThe evolution of skyrmions in Ir/Fe/Co/Pt multilayers and their topological Hall signatureSkyrmions at the Edge: Confinement Effects in Fe / Ir ( 111 ) Lorentz TEM investigation of chiral spin textures and Néel Skyrmions in asymmetric [ Pt / ( Co / Ni ) M / Ir ] N multi-layer thin filmsEnergy-efficient generation of skyrmion phases in Co/Ni/Pt-based multilayers using Joule heatingCurrent-induced skyrmion generation and dynamics in symmetric bilayersX-ray photoelectron spectroscopy investigation of Ta/CoFeB/TaOx heterostructuresDriving skyrmions with low threshold current density in Pt/CoFeB thin filmStrain-Enabled Control of Chiral Magnetic Structures in MnSeTe MonolayerMagnetic Domain-Wall LogicTunable chiral spin texture in magnetic domain-wallsCurrent-driven magnetic domain-wall logicMagnetic Domain-Wall Racetrack MemoryDynamics of magnetic vorticesMagnetic vortices  The microscopic analogs of magnetic bubblesMagnetic bimerons as skyrmion analogues in in-plane magnetsCurrent-Induced Dynamics and Chaos of Antiferromagnetic BimeronsDirect Demonstration of Topological Stability of Magnetic Skyrmions via Topology ManipulationCurrent-driven dynamics of skyrmions stabilized in MnSi nanowires revealed by topological Hall effectEntropy-limited topological protection of skyrmionsA theory on skyrmion sizeSize and profile of skyrmions in skyrmion crystalsThe 20-nm Skyrmion Generated at Room Temperature by Spin-Orbit TorquesSkyrmion flow near room temperature in an ultralow current densityCurrent-Driven Skyrmion Dynamics and Drive-Dependent Skyrmion Hall Effect in an Ultrathin FilmA skyrmion-based spin-torque nano-oscillatorA ferromagnetic skyrmion-based nano-oscillator with modified profile of Dzyaloshinskii-Moriya interactionCurrent-induced magnetic skyrmions oscillatorCharacteristics and Applications of Current-Driven Magnetic Skyrmion StringsReconfigurable Skyrmion Logic GatesSkyrmions-based logic gates in one single nanotrack completely reconstructed via chirality barrierVoltage Controlled Magnetic Skyrmion Motion for Racetrack MemoryComplementary Skyrmion Racetrack Memory With Voltage ManipulationSkyrmion-skyrmion and skyrmion-edge repulsions in skyrmion-based racetrack memoryBipolar magnetic materials for electrical manipulation of spin-polarization orientationThermal generation, manipulation and thermoelectric detection of skyrmionsSolution Synthesis of Layered van der Waals (vdW) Ferromagnetic CrGeTe3 Nanosheets from a Non-vdW Cr2 Te3 TemplateAntiferromagnetic Skyrmion: Stability, Creation and ManipulationNucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructuresThermal skyrmion diffusion used in a reshuffler deviceConverting a two-dimensional ferromagnetic insulator into a high-temperature quantum anomalous Hall system by means of an appropriate surface modificationA ferromagnetic skyrmion-based nano-oscillator with modified perpendicular magnetic anisotropyMaterials with perpendicular magnetic anisotropy for magnetic random access memoryAbundant valley-polarized states in two-dimensional ferromagnetic van der Waals heterostructuresPerpendicular magnetic anisotropy in Pd/Co and Pt/Co thin-film layered structuresPerpendicular magnetic anisotropy in Pd/Co thin film layered structuresPrediction and confirmation of perpendicular magnetic anisotropy in Co/Ni multilayersPerpendicular Magnetic Anisotropy of Co-Au Multilayers Induced by Interface SharpeningCo-Cr recording films with perpendicular magnetic anisotropyLow-power non-volatile spintronic memory: STT-RAM and beyondVoltage-Driven High-Speed Skyrmion Motion in a Skyrmion-Shift DeviceVoltage-induced perpendicular magnetic anisotropy change in magnetic tunnel junctionsMagnetic bilayer-skyrmions without skyrmion Hall effectCurrent-Induced Dynamics of the Antiferromagnetic Skyrmion and SkyrmioniumDynamics of an antiferromagnetic skyrmion in a racetrack with a defectHigh-Temperature Quantum Anomalous Hall Effect in n − p Codoped Topological InsulatorsAn Improved Racetrack Structure for Transporting a SkyrmionAll-magnetic control of skyrmions in nanowires by a spin waveNumerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equationDirect Solution of the Landau-Lifshitz-Gilbert Equation for MicromagneticsSpin torque nano-oscillators based on antiferromagnetic skyrmionsTopological charge analysis of ultrafast single skyrmion creationSkyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interactionParticle-like skyrmions interacting with a funnel obstacleThe design and verification of MuMax3Magnetic skyrmion shape manipulation by perpendicular magnetic anisotropy excitation within geometrically confined nanostructures
[1] Fert A, Reyren N, and Cros V 2017 Nat. Rev. Mater. 2 17031
[2] Tokura Y and Kanazawa N 2021 Chem. Rev. 121 2857
[3] Zhang X C, Zhou Y, Song M K, Park T E, Xia J, Ezawa M, Liu X, Zhao W, Zhao G P, and Woo S 2020 J. Phys.: Condens. Matter 32 143001
[4] Zhang X C, Ezawa M, and Zhou Y 2015 Sci. Rep. 5 9400
[5] Rößler K U, Bogdanov A, and Pfleiderer C 2006 Nature 442 797
[6] Weidig T 1999 Nonlinearity 12 1489
[7] Ball R D 1990 Int. J. Mod. Phys. A 5 4391
[8] Jiang W J, Chen G, Liu K, Zang J D, te Velthuis S G E, and Hoffmann A 2017 Phys. Rep. 704 1
[9] Mühlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubauer A, Georgii R, and Böni P 2009 Science 323 915
[10] Di K, Zhang L V, Lim S H, Ng C S, Kuok H M, Yu J, Yoon J, Qiu X, and Yang H 2015 Phys. Rev. Lett. 114 47201
[11] El H S, Bailly-Reyre A, and Diep H 2018 J. Magn. Magn. Mater. 455 32
[12] Jaiswal S, Litzius K, Lemesh I, Büttner F, Finizio S, Raabe J, Weig M, Lee K, Langer J, and Ocker B 2017 Appl. Phys. Lett. 111 22409
[13] Ryu K S, Thomas L, Yang S H, and Parkin 2013 Nat. Nanotechnol. 8 527
[14] Sergienko A I and Dagotto E 2006 Phys. Rev. B 73 94434
[15] Zhou Y F, Mansell R, Valencia S, Kronast F, and van Dijken S 2020 Phys. Rev. B 101 54433
[16] Yang H X, Liang J H, and Cui Q R 2023 Nat. Rev. Phys. 5 43
[17] Soumyanarayanan A, Raju M, Oyarce G A, Tan K A, Im M Y, Petrović A, Ho P, Khoo K, Tran M, and Gan C 2017 Nat. Mater. 16 898
[18] Raju M, Yagil A, Soumyanarayanan A, Tan K A, Almoalem A, Ma F, Auslaender O M, and Panagopoulos C 2019 Nat. Commun. 10 696
[19] Hagemeister J, Iaia D, Vedmedenko Y E, von Bergmann K, Kubetzka A, and Wiesendanger R 2016 Phys. Rev. Lett. 117 207202
[20] Li M, Lau D, de Graef M, and Sokalski V 2019 Phys. Rev. Mater. 3 64409
[21] Brock A J, Montoya A S, Im M Y, and Fullerton E E 2020 Phys. Rev. Mater. 4 104409
[22] Hrabec A, Sampaio J, Belmeguenai M, Gross I, Weil R, Chérif M S, Stashkevich A, Jacques V, Thiaville A, and Rohart S 2017 Nat. Commun. 8 15765
[23] Syamlal S K, Kalal S, Perumal P H, Kumar D, Gupta M, and Sinha J 2021 Mater. Sci. Eng. B 272 115367
[24] Ojha B, Mallick S, Sharma M, Thiaville A, Rohart S, and Bedanta S 2021 arXiv:2106.2407 [cond-mat.mtrl-sci]
[25] Wang Z W, Liang J H, and Yang H X 2023 Chin. Phys. Lett. 40 017501
[26] Allwood A D, Xiong G, Faulkner C, Atkinson D, Petit D, and Cowburn R 2005 Science 309 1688
[27] Franken H J, Herps M, Swagten J H, and Koopmans B 2014 Sci. Rep. 4 5248
[28] Luo Z C, Hrabec A, Dao P T, Sala G, Finizio S, Feng J, Mayr S, Raabe J, Gambardella P, and Heyderman J L 2020 Nature 579 214
[29] Parkin S S, Hayashi M, and Thomas L 2008 Science 320 190
[30] Papanicolaou N and Tomaras T 1991 Nucl. Phys. B 360 425
[31] Ivanov B, Stephanovich V, and Zhmudskii A 1990 J. Magn. Magn. Mater. 88 116
[32] Göbel B, Mook A, Henk J, Mertig I, and Tretiakov A O 2019 Phys. Rev. B 99 60407
[33] Shen L C, Xia J, Zhang X C, Ezawa M, Tretiakov A O, Liu X, Zhao G P, and Zhou Y 2020 Phys. Rev. Lett. 124 37202
[34] Je G S, Han H S, Kim K S, Montoya A S, Chao W, Hong I S, Fullerton E E, Lee K S, Lee K J, and Im M Y 2020 ACS Nano 14 3251
[35] Liang D, DeGrave P J, Stolt J M, Tokura Y, and Jin S 2015 Nat. Commun. 6 8217
[36] Wild J, Meier N T, Pöllath S, Kronseder M, Bauer A, Chacon A, Halder M, Schowalter M, Rosenauer A, and Zweck J 2017 Sci. Adv. 3 e1701704
[37] Wang X, Yuan H, and Wang X 2018 Commun. Phys. 1 31
[38] Wu H, Hu X, Jing K, and Wang X 2021 Commun. Phys. 4 210
[39] Liu J H, Wang Z D, Xu T, Zhou H A, Zhao L, Je S G, Im M Y, Fang L, and Jiang W J 2022 Chin. Phys. Lett. 39 017501
[40] Yu X, Kanazawa N, Zhang W, Nagai T, Hara T, Kimoto K, Matsui Y, Onose Y, and Tokura Y 2012 Nat. Commun. 3 988
[41] Juge R, Je S G, de Souza D, Buda-Prejbeanu D L, Peña-Garcia J, Nath J, Miron M I, Rana G K, Aballe L, and Foerster M 2019 Phys. Rev. Appl. 12 44007
[42] Garcia-Sanchez F, Sampaio J, Reyren N, Cros V, and Kim J 2016 New J. Phys. 18 75011
[43] Guo J H, Xia J, Zhang X C, Pong P W T, Wu Y M, Chen H, Zhao W S, and Zhou Y 2020 J. Magn. Magn. Mater. 496 165912
[44] Zhang S F, Wang J B, Zheng Q, Zhu Q Y, Liu X Y, Chen S J, Jin C D, Liu Q F, Jia C L, and Xue D S 2015 New J. Phys. 17 23061
[45] Jin Z N, Song M H, Fang H N, Chen L, Chen J W, and Tao Z K 2022 Chin. Phys. Lett. 39 108502
[46] Luo S J, Song M, Li X, Zhang Y, Hong J M, Yang X F, Zou X C, Xu N, and You L 2018 Nano Lett. 18 1180
[47] Yu D X, Yang H X, Chshiev M, and Fert A 2022 Natl. Sci. Rev. 9 nwac021
[48] Kang W, Huang Y, Zheng C, Lv W, Lei N, Zhang Y, Zhang X C, Zhou Y, and Zhao W 2016 Sci. Rep. 6 23164
[49] Kang W, Zheng C, Huang Y, Zhang X C, Zhou Y, Lv W, and Zhao W 2016 IEEE Electron Device Lett. 37 924
[50] Zhang X C, Zhao G P, Fangohr H, Liu P J, Xia W, Xia J, and Morvan F 2015 Sci. Rep. 5 7643
[51] Li X X and Yang J L 2013 Phys. Chem. Chem. Phys. 15 15793
[52] Wang Z D, Guo M H, Zhou H A, Zhao L, Xu T, Tomasello R, Bai H, Dong Y Q, Je S G, and Chao W L 2020 Nat. Electron. 3 672
[53] Yang H, Wang F, Zhang H S, Guo L H, Hu L Y, Wang L F, Xue D J, and Xu X H 2020 J. Am. Chem. Soc. 142 4438
[54] Zhang X C, Zhou Y, and Ezawa M 2016 Sci. Rep. 6 24795
[55] Sampaio J, Cros V, Rohart S, Thiaville A, and Fert A 2013 Nat. Nanotechnol. 8 839
[56] Zázvorka J, Jakobs F, Heinze D, Keil N, Kromin S, Jaiswal S, Litzius K, Jakob G, Virnau P, and Pinna D 2019 Nat. Nanotechnol. 14 658
[57] Zhang H S, Qin W, Chen M X, Cui P, Zhang Z Y, and Xu X H 2019 Phys. Rev. B 99 165410
[58] Guo J H, Xia J, Zhang X C, Pong Philip W T, and Zhou Y 2021 Phys. Lett. A 392 127157
[59] Sbiaa R, Meng H, and Piramanayagam S 2011 Phys. Status Solidi RRL 5 413
[60] Zhang H S, Yang W J, Ning Y H and Xu X H 2020 Phys. Rev. B 101 205404
[61] Carcia P 1988 J. Appl. Phys. 63 5066
[62] Carcia P, Meinhaldt A, and Suna A 1985 Appl. Phys. Lett. 47 178
[63] Daalderop G H O, Kelly P J, and den Broeder F J A 1992 Phys. Rev. Lett. 68 682
[64] den Broeder F J A, Kuiper D, van de Mosselaer A P, and Hoving W 1988 Phys. Rev. Lett. 60 2769
[65] Iwasaki I S and Ouchi K 1978 IEEE Trans. Magn. 14 849
[66] Wang K L, Alzate J G, and Amiri P K 2013 J. Phys. D 46 074003
[67] Liu Y Z, Lei N, Wang C X, Zhang X C, Kang W, Zhu D Q, Zhou Y, Liu X X, Zhang Y G, and Zhao W S 2019 Phys. Rev. Appl. 11 14004
[68] Nozaki T, Shiota Y, Shiraishi M, Shinjo T, and Suzuki Y 2010 Appl. Phys. Lett. 96 022506
[69] Zhang X C, Yan Z, and Ezawa M 2016 Nat. Commun. 7 10293
[70] Shen L C, Li X G, Zhao Y L, Xia J, Zhao G P, and Yan Z 2019 Phys. Rev. Appl. 12 064033
[71] Liang X, Zhao G P, Shen L C, Xia J, Zhang X C, and Zhou Y 2019 Phys. Rev. B 100 144439
[72] Qi S F, Qiao Z H, Deng X Z, Cubuk E D, Chen H, Zhu W G, Kaxiras E, Zhang S B, Xu X H, and Zhang Z Y 2016 Phys. Rev. Lett. 117 056804
[73] Lai P, Zhao G P, Tang H, Ran N, Wu S Q, Xia J, Zhang X C, and Zhou Y 2017 Sci. Rep. 7 45330
[74] Zhang X C, Ezawa M, Xiao D, Zhao G P, Liu Y, and Zhou Y 2015 Nanotechnology 26 225701
[75] Bartels S, Ko J, and Prohl A 2008 Math. Comput. 77 773
[76] Nakatani Y, Uesaka Y, and Hayashi N 1989 Jpn. J. Appl. Phys. 28 2485
[77] Shen L C, Xia J, Zhao G P, Zhang X C, Ezawa M, Tretiakov O A, Liu X X, and Zhou Y 2019 Appl. Phys. Lett. 114 042402
[78] Yin G, Li Y, Kong L, Lake K R, Chien C L, and Zang J 2016 Phys. Rev. B 93 174403
[79] Rohart S and Thiaville A 2013 Phys. Rev. B 88 184422
[80] Zhang X C, Xia J, and Liu X X 2022 Phys. Rev. B 106 094418
[81] Vansteenkiste A, Leliaert J, Dvornik M, Helsen M, Garcia-Sanchez F, and van Waeyenberge B 2014 AIP Adv. 4 107133
[82] Mehmood N, Wang J B, Zhang C L, Zeng Z Z, Wang J N, and Liu Q F 2022 J. Magn. Magn. Mater. 545 168775