Characteristics and Applications of Current-Driven Magnetic Skyrmion Strings

Zhaonian Jin(金兆年)$^{1}$, Minhang Song(宋敏航)$^{2}$, Henan Fang(方贺男)$^{2}$,\\ Lin Chen(陈琳)$^{2}$, Jiangwei Chen(陈将伟)$^{2}$, and Zhikuo Tao(陶志阔)$^{2\hyperlink{s}{*}}$

doi:10.1088/0256-307X/39/10/108502

⇦Chinese Physics Letters, 2022, Vol. 39, No. 10, Article code 108502⇨ Characteristics and Applications of Current-Driven Magnetic Skyrmion Strings Zhaonian Jin (金兆年)1, Minhang Song (宋敏航)2, Henan Fang (方贺男)2, Lin Chen (陈琳)2, Jiangwei Chen (陈将伟)2, and Zhikuo Tao (陶志阔)2* Affiliations 1Bell Honors School, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 2College of Electronic and Optical Engineering & College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210003, China Received 17 August 2022; accepted manuscript online 21 September 2022; published online 28 September 2022 ^*Corresponding author. Email: zktao@njupt.edu.cn Citation Text: Jin Z N, Song M H, Fang H N et al. 2022 Chin. Phys. Lett. 39 108502 Abstract We investigate the current-driven characteristics and applications of magnetic skyrmion strings by micromagnetic simulations. Under the spin-polarized driving current, the skyrmion string presents different moving trajectories in different layers due to the skyrmion Hall effect. Moreover, a series of skyrmion bobbers can be generated with a notch defect placed in the surface and the skyrmion bobbers will follow the skyrmion string. By varying the current density, the bobbers' characteristics such as number and velocity can be manipulated, which inspires us to propose a skyrmion string-based diode. In addition, an AND logic gate and an OR logic gate in the identical scheme based on the skyrmion string are proposed. AND logic and OR logic behaviors can be realized by varying the driving current densities. Our findings will contribute to further research of magnetic skyrmion strings for data storage, processing, and energy-efficient computing.

DOI:10.1088/0256-307X/39/10/108502 © 2022 Chinese Physics Society Article Text Magnetic skyrmion has attracted much attention due to its topological characteristics since it has been first observed experimentally in chiral magnetic MnSi.[1] Skyrmions can be stabilized by the competition of several energies such as exchange interaction, Zeeman energy, anisotropy energy, and Dzyaloshinskii–Moriya interaction (DMI) energy. DMI energy, in particular, can lead to different topological magnetic skyrmion textures such as Bloch-type and Neel-type structures or different chiral characteristics as a result of inversion asymmetry and large spin-orbit coupling. Since skyrmion presents the nano-scale size and can be moved by a low-density spin-polarized current through the spin transfer torque (STT) effect, it is expected to be used in future spintronic data storage and data processing.[2-9] In particular, skyrmion diodes[10-13] and logic gates[6,9] are realized through computational methods. Especially, the reconfiguration logic gates or multi-logic gates in the same magnetic scheme are proposed, which present a vigorously potential application in designs of multiple functional devices.[9] Meanwhile, other topological spin textures related to skyrmion, such as two-dimensional bimerons,[14,15] antiskyrmions,[16] and ferrimagnetic skyrmions,[17,18] as well as three-dimensional skyrmion tubes[19-21] or strings,[22] skyrmion bobbers,[23-26] and Hopfions,[27] have also attracted much attention. Especially for the skyrmion string and other three-dimensional topological magnetic textures, the robust and flexible structures make them potential platforms for designs of new-type spintronic devices. In this work, we investigate the current-driven characteristics and applications of skyrmion strings. The moving profiles of the skyrmion string under spin-polarized current are presented. It is intriguing that skyrmion bobbers can be generated when the string passes a surface notch defect. The generation mechanism of the bobber is also discussed. Moreover, the number of bobbers, which are used to construct a one-way diode, can be manipulated through current densities. Furthermore, logic gates based on current-driven skyrmion strings are proposed. It is important that the functionality can be manipulated by changing the current density, and then tunable logic OR gate and AND gate based on the identical scheme are achieved. The numerical simulations were carried out in the object-oriented micromagnetic framework (OOMMF) by incorporating the DMI extension module.[28] To describe the magnetic system, we have considered exchange energy, Zeeman energy, DMI energy, perpendicular magnetic anisotropy energy, and demagnetization energy. The dynamics of time-dependent magnetization are governed by the Landau–Lifshitz–Gilbert (LLG) equation including the Zhang–Li spin transfer torque (STT) term and the Slonczewski STT term,[29,30] as follows: \begin{align} \frac{d\boldsymbol{m}}{dt}=\,&-\gamma \frac{1}{1+\alpha^{2}}\big\{\boldsymbol{m}\times \boldsymbol{B}_{\rm eff}+\alpha[\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{B}_{\rm eff})]\big\}\notag\\ &+\boldsymbol{\tau }_{\scriptscriptstyle{\rm ZL}}+\boldsymbol{\tau }_{\scriptscriptstyle{\rm Sl}}, \tag {1} \end{align} where $\boldsymbol{m}$ is the magnetic vector, $\gamma$ is the gyromagnetic ratio, $\alpha$ is the damping coefficient, $\boldsymbol{B}_{\rm eff}=\mu_{0}\boldsymbol{H}_{\rm eff}$, and $\boldsymbol{H}_{\rm eff}$ is the effective magnetic field. The last two terms denote the Zhang–Li STT term and the Slonczewski STT term. The Zhang–Li STT term can be expressed as \begin{align} \boldsymbol{\tau }_{\scriptscriptstyle{\rm ZL}}=\,&\frac{1}{1+\alpha^{2}}\big\{(1+\xi \alpha)\boldsymbol{m}\times[\boldsymbol{m}\times(\boldsymbol{u}\cdot \nabla)\boldsymbol{m}]\notag\\ &+(\xi -\alpha)\boldsymbol{m}(\boldsymbol{u}\cdot\nabla)\boldsymbol{m}\big\}, \tag {2} \end{align} where $\boldsymbol{u}=\frac{\mu_{\scriptscriptstyle{\rm B}}\mu_{0}}{2e\gamma {M_{\rm s}(1+\xi }^{2})}\boldsymbol{j}$, $\xi$ is the degree of non-adiabaticity, $\mu_{_{\scriptstyle \rm B}}$ is the Bohr magneton, $e$ is the electron charge, and $\boldsymbol{j}$ is the current density vector. The Slonczewski STT term can be expressed as \begin{align} \boldsymbol{\tau }_{\scriptscriptstyle{\rm Sl}}=\,&\frac{\beta }{1+\alpha^{2}}[-(\epsilon -\alpha \epsilon')\boldsymbol{m}\times (\boldsymbol{m}_{\rm p}\times \boldsymbol{m})\notag\\ &+(\epsilon'-\alpha \epsilon)\boldsymbol{m}\times \boldsymbol{m}_{\rm p}], \tag {3} \end{align} where $\boldsymbol{m}_{\rm p}$ is the electron polarization vector, $\epsilon$ and $\epsilon'$ are spin-torque terms, and $\beta$ is the parameter related to driving current and material properties. Here we only consider the Zhang–Li term because the spin-polarized driving current is in-plane biased. First, skyrmion strings can be generated by minimizing the energy of the system. As shown in Fig. 1(a), a skyrmion string with a length of 1000 nm and a radius of about 40 nm is constructed. It is known that the configuration and profile of the skyrmion string depend on the material parameters. In the simulations, we choose the typical parameters in the multilayer Co/Pt system as follows:[31-34] exchange stiffness constant $A=1.6 \times 10^{-11}$ J/m, magneto-crystalline anisotropy constant $K=0.51 \times 10^{6}$ J/m$^{3}$, saturation magnetization constant $M_{\rm s}=5.8\times 10^{5}$ A/m. Then we follow the procedure of energy-minimizing to establish an initial magnetic configuration for each combination of two key parameters, the DMI strength $D$ and external applied magnetic field $H$, systematically, which can be seen in Fig. S1 in the Supplementary Material. In the following simulation of skyrmion string, the external applied magnetic field is set to $5 \times 10^{5}$ A/m and DMI constant $D$ is set to $6.0 \times 10^{-3}$ J/m$^{2}$. In all simulations, a mesh size of $5 \times 5\times 5$ nm$^{3}$ is used. The simulations are performed at zero temperature unless stated otherwise. On the investigation of the Hall effect, the non-adiabaticity torque coefficient $\xi$ is changed from 0.1 to 0.5 while damping coefficient $\alpha$ is fixed at 0.3. On the investigation of the skyrmion bobber generation process, the coefficient $\xi$ is changed from 0.1 to 0.5, while the parameter $\alpha$ is fixed to the same value of $\xi$ from 0.1 to 0.5.

Fig. 1. (a) Skyrmion string in a cuboid, (b) magnetization distributions along the $x$-axis at $y=0$ with different $z$-axis locations, (c) energies with different string lengths from 50 nm to 1000 nm.

As shown in Fig. 1(a), two-dimensional skyrmions are realized from the cross-sectional view at different $z$-axis locations with the magnetizations pointing towards $-z$ at the core and rotating towards its boundary by 180$^\circ$ which presents the Neel-type profile. Furthermore, the two-dimensional skyrmions at different $z$-axis locations present slightly different profiles as shown in Fig. 1(b), which presents the magnetization distributions along the $x$-axis. These slightly different profiles can also be seen in Fig. S2 in the Supplementary Material. We also calculate the energies with different string lengths as shown in Fig. 1(c), which presents the nearly linear relationship between energies and string lengths. Afterward, a spin-polarized current is applied to investigate the current-driven properties. The driving current with spin-polarization of 100% is along the $+x$ direction and distributes uniformly in the $y$–$z$ plane. The moving profiles under $j=3.84\times 10^{7}$ A/m$^{2}$ and $\xi =0.5$ are presented in Fig. 2.

Fig. 2. (a) Surface views of the moving skyrmion strings with $j=3.84\times 10^{7}$ A/m$^{2}$ in 4 ns. (b) Trajectories of guiding centers of skyrmions at $z=500$ nm, 600 nm, 700 nm, 800 nm, 900 nm, and 1000 nm. (c) Cross-sectional views of the moving skyrmion strings at $y=5$ nm in 4 ns. The colored bar denotes the magnetization of the $z$-axis component in Fig. 2(a) and the $y$-axis component in Fig. 2(c).

Figure 2(a) shows the surface views of the moving skyrmion strings within 4 ns. We can see that the surface skyrmion travels with curved trajectories which are related to the skyrmion Hall effect. It has been proved that the translational motion can be described by the Thiele equation in which the skyrmion is treated as a rigid particle. The Thiele equation can be expressed as[35] \begin{eqnarray} \boldsymbol{G}\times \frac{d\boldsymbol{r}}{dt}+\boldsymbol{F}_{\scriptscriptstyle{\rm B}}+\boldsymbol{F}_{\alpha }+\boldsymbol{F}_{\scriptscriptstyle{\rm STT}}=0, \tag {4} \end{eqnarray} where $\boldsymbol{r}$ is the skyrmion position, and gyrocoupling vector $\boldsymbol{G}$ is (0, 0, $-4 \pi Q\mu_{0} dM_{\rm s}/\gamma$). We take $Q$ to represent the topological charge of skyrmion, \begin{eqnarray} Q=\frac{1}{4\pi }\iint {qdxdy},~~~~~q=\boldsymbol{m}\cdot \Big(\frac{\partial \boldsymbol{m}}{\partial x}\times \frac{\partial \boldsymbol{m}}{\partial y}\Big), \tag {5} \end{eqnarray} where $q$ represents the topological density. The four terms on the left-hand side of the Thiele equation represent Magnus force, edge repulsive force (boundary-induced force), damping force, and spin-polarized current driving force, respectively. The Magnus force is nonzero when skyrmion number $Q=\pm 1$ and leads to the skyrmion Hall effect which causes the transverse motion. The boundary-induced force can be described as $\boldsymbol{F}_{\scriptscriptstyle{\rm B}}=-\nabla U(\boldsymbol{r})$ with the potential energy $U(\boldsymbol{r})$ related to the boundary edge effect. Finally, the Magnus force can be balanced by the boundary-induced repulsive force. To illustrate the trajectories of different layers of the skyrmion string, the guiding centers ($R_{x}, R_{y}$) of skyrmions at different $z$-axis locations are calculated and presented in Fig. 2(b). The guiding centers ($R_{x}, R_{y}$) can be defined as[36-39] \begin{eqnarray} R_{x}=\frac{\iint {xqdxdy}}{\iint {qdxdy}},~~~~~ R_{y}=\frac{\iint {xqdxdy}}{\iint {qdxdy}}. \tag {6} \end{eqnarray} Because the skyrmions at various $z$-axis positions experience varied distributions of topological density, as visible in Fig. 1(b), it is evident that different dynamical properties can be achieved at various $z$-axis locations. Figure 2(c) shows the cross-sectional view at $y=5$ nm with driving times varying from 0 to 4 ns, which further illustrates the magnetic distribution and moving profiles. The moving profiles with $\xi$ of 0.2, 0.3, 0.4, and 0.6 are presented in Fig. S3 in the Supplementary Material. Furthermore, we investigate the current-driven properties of skyrmion string through the surface notch defect. Figure 3(a) shows the cross-sectional view. Figures 3(b) and 3(c) give the surface views in driving times of 4 ns when the current density $j$ is $3.84 \times 10^{8}$ A/m$^{2}$. We consider a cuboid notch defect located on the surface with $l=5$ nm and $h = 50$ nm as shown in Fig. 3(a). We have the non-adiabatic torque coefficient $\xi$ set to ignore the Hall effect at the same value as the damping parameter $\alpha$ to keep the skyrmion string traveling along the middle line.[40]

Fig. 3. (a) Cross-sectional views and [(b), (c)] surface views of the skyrmion strings and bobbers in driving times of 4 ns with $j=3.84\times 10^{8}$ A/m$^{2}$ and $\xi =\alpha =0.3$.

As shown in Fig. 3, the skyrmion strings can easily pass through the notch defect. The top of the string around the surface is deformed into a curved shape after passing the defect and then decoupled from the notch. More importantly, a series of skyrmion bobbers can be generated and move after the string. Figure 3(c) describes the moving features around the notch with the driving time ranging from 0.125 ns to 0.325 ns. We can see how the notch defect could pin the string or bobbers and then split them into pieces. Afterward, the pinned bobber is expanded with the flowing current and more bobbers will be generated subsequently. The skyrmion bobber is a new type of stable particle-like state, which was predicted theoretically by Rybakov et al.[24] and then observed experimentally by Zheng et al.[25] The profiles and configurations are shown in Fig. S4 in the Supplementary Material. It can be seen that the string can hold skyrmion configuration at different $z$ locations with almost the same radius. Meanwhile, the bobber can also hold skyrmion configuration but shrink to a Bloch point at the $z$ location of around 950 nm. For $j=3.84\times 10^{8}$ A/m$^{2}$, bobbers can be generated continuously for at least 10 ns (see Fig. S5 in the Supplementary Material), whereas a limited number of bobbers are generated when the driving current is decreased. Figure 4(a) presents the snapshots of moving strings and bobbers at 5 ns with different current densities $j$. The driving current is moving from left to right. The skyrmion strings are denoted by red arrows. It can be seen that fewer bobbers can be generated with decreasing $j$ while no bobbers are achieved when $j=1.92\times 10^{8}$ A/m$^{2}$. In the meantime, the bobber generation is related to the geometrical dimension of the defect. For example, the bobbers can be generated with any value of $l$ when $h$ is fixed at 50 nm and $j=3.84 \times 10^{8}$ A/m$^{2}$, while no bobbers are generated with $h$ larger than 100 nm when $l$ is fixed at 5 nm and $j=3.84 \times 10^{8}$ A/m$^{2}$ (see Fig. S6 in the Supplementary Material)

Fig. 4. Snapshots at 5 ns under different driving current densities $j$: (a) with a cuboid notch defect and (b) with an asymmetric triangle notch defect. (c) The output characteristics of skyrmion string-based diode.

Based on the realization of bobbers generated by the spin-polarized current driving skyrmion string through the surface defect, a skyrmion string-based diode is proposed. As shown in the inset of Fig. 4(c), an asymmetric triangle notch defect is located in the surface. Figure 4(b) shows snapshots at 5 ns with different driving current densities $j$ from $1.92 \times 10^{8}$ A/m$^{2}$ to $6.14 \times 10^{8}$ A/m$^{2}$. When the current density is less than $1.92 \times 10^{8}$ A/m$^{2}$, no bobbers will be generated, which is as same as the structure with a cuboid notch defect discussed above. When the current density is high than $1.92 \times 10^{8}$ A/m$^{2}$ and lower than $3.84 \times 10^{8}$ A/m$^{2}$, bobbers can be generated, but cannot be generated continuously. This presents different characteristics compared to the structure with a cuboid notch defect. Moreover, bobbers can be generated continuously when the current density is higher than $3.84 \times 10^{8}$ A/m$^{2}$ and lower than $5.38 \times 10^{8}$ A/m$^{2}$. Then, complex magnetic textures will be obtained when the current density increases up to $6.14 \times 10^{8}$ A/m$^{2}$. More importantly, no bobbers can be detected when the driving current moves from right to left with any current density (see Fig. S7 in the Supplementary Material). Additionally, the number of generated bobbers per nanosecond grows with increasing driving current (with driving current density $j$ from $3.84 \times 10^{8}$ A/m$^{2}$ to $5.38 \times 10^{8}$ A/m$^{2}$) when the bobbers are generated continuously. Therefore, if driving current density is chosen as the biased signal and the number of bobbers per nano-second is chosen as the output signal, a skyrmion string-based diode can be achieved, which is equivalent to the traditional one-way diode. The proposed diode will output continuous signals when the driving current with $j$ higher than $3.84 \times 10^{8}$ A/m$^{2}$ is forward biased. Furthermore, the diode will be collapsed when the driving current density increases up to $6.14 \times 10^{8}$ A/m$^{2}$ while the cutoff process is presented when the biased driving current is reversed. These different working states are denoted as I, II, III, and IV in Figs. 4(b) and 4(c). Thus, the one-way motion of skyrmion bobbers generated by skyrmion string is realized through a simple asymmetric structure that presents a skyrmion string-based diode.

Fig. 5. (a) Schematic diagram of the logic gate. (b)–(d) AND gate and (e)–(g) OR gate realized through current-driven skyrmion strings with $j=3.84 \times 10^{7}$ A/m$^{2}$ and $j=3.84 \times 10^{8}$ A/m$^{2}$, respectively.

Similar to two-dimensional skyrmions, skyrmion strings are non-trivial spin structures with topological protection. Therefore, spin logic gates can be realized based on these topological structures with nanoscale stability and low power.[6,9,41,42] As shown in Fig. 5, an AND logic gate and an OR logic gate are realized through current-driven skyrmion strings under different driving current densities $j$ in the identical scheme. Figure 5(a) is the schematic diagram of the logic gate. Figures 5(b)–5(d) are composed of three snapshots at a driving time of 0 ns, 1 ns, and 2 ns, respectively. Input with only one skyrmion string, up input as shown in Fig. 5(b), or down input as shown in Fig. 5(c) leads to no output of skyrmion string when $j=3.84 \times 10^{7}$ A/m$^{2}$. Both inputs with skyrmion strings can interact with each other, leading to a skyrmion string in output as shown in Fig. 5(d). Therefore, an AND logic gate can be realized if we define ‘0’ as the absence of a skyrmion string and ‘1’ as the presence of a skyrmion string. More importantly, an OR logic gate can be constructed in the identical scheme by increasing the driving current density. Figures 5(e)–5(g) are composed of three snapshots at a driving time of 0 ns, 0.5 ns and 1 ns, respectively, with a driving current density $j=3.84 \times 10^{8}$ A/m$^{2}$. As can be seen, any of the inputs with skyrmion strings can lead to a skyrmion string or a skyrmion bobber in output. Then, an OR logic gate is achieved as magnetic performance is defined as “1” and no magnetic signal is defined as “0”. Moreover, it is demonstrated that OR logic gate can be realized in this scheme by applying a current density larger than $7.68 \times 10^{7}$ A/m$^{2}$ (see Fig. S8 in the Supplementary Material). Experimentally, the readout of both skyrmion strings and bobbers can be realized by magnetoresistance effects such as GMR or TMR. Compared with other works on 2D skyrmion-based devices, the current density for driving skyrmion strings in our work is reduced significantly.[5-10] On the other hand, both AND logic and OR logic are realized in the identical scheme, which provides a potential application in designs of new multi-functional spintronics devices with engineering programmable logic gates. However, more logic functions such as NAND, NOR, or NOT should be explored in the same scheme based on the skyrmion string in the future work. In addition, thermal effects are very important for generation and manipulation of skyrmion strings and other skyrmion-related topological structures. Also, functionalities of skyrmion-based devices are affected by temperatures. By adding a fluctuating external magnetic field representing the irregular influence of temperature,[43] the temperature effect is investigated as shown in Fig. S9 in the Supplementary Material. We can see that skyrmion strings will split into several segments under finite temperatures. Furthermore, the string segments will annihilate with simulating time which is more obvious at higher temperatures. In summary, we have investigated dynamic characteristics of current-driven skyrmion strings through a micromagnetic simulation. A skyrmion string-based diode and a tunable skyrmion string logic gate are realized. It is found that when a skyrmion string moves through a surface notch defect, a series of skyrmion bobbers can be produced, and a skyrmion string-based diode is proposed. Moreover, a tunable skyrmion string-based logic device is constructed. By changing the driving current density, AND and OR logic gates can be realized without varying the structure. Our results imply that there exist broad applications for skyrmion strings in non-volatile memory and high-density logic devices. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No. 61574079). References Skyrmion Lattice in a Chiral MagnetSkyrmion-Electronics: An Overview and OutlookMagnetic skyrmions: advances in physics and potential applicationsBeyond skyrmions: Review and perspectives of alternative magnetic quasiparticlesReconfigurable Skyrmion Logic GatesSkyrmion-Based Programmable Logic Device with Complete Boolean Logic FunctionsCurrent-Induced Dynamics and Chaos of Antiferromagnetic BimeronsRealization of the skyrmionic logic gates and diodes in the same racetrack with enhanced and modified edgesA Skyrmion Diode Based on Skyrmion Hall EffectA ferromagnetic skyrmion-based diode with a voltage-controlled potential barrierMagnetic skyrmionium diode with a magnetic anisotropy voltage gatingA spin-wave driven skyrmion diode under transverse magnetic fieldsBound States of Skyrmions and Merons near the Lifshitz PointMagnetic bimerons as skyrmion analogues in in-plane magnetsMagnetic antiskyrmions above room temperature in tetragonal Heusler materialsCurrent-driven dynamics and inhibition of the skyrmion Hall effect of ferrimagnetic skyrmions in GdFeCo filmsSelf-focusing skyrmion racetracks in ferrimagnetsUnwinding of a Skyrmion Lattice by Magnetic MonopolesReal-space imaging of confined magnetic skyrmion tubesUnveiling the three-dimensional magnetic texture of skyrmion tubesPropagation dynamics of spin excitations along skyrmion stringsToggle-like current-induced Bloch point dynamics of 3D skyrmion strings in a room temperature nanowireNew Type of Stable Particlelike States in Chiral MagnetsExperimental observation of chiral magnetic bobbers in B20-type FeGeCurrent-driven transformations of a skyrmion tube and a bobber in stepped nanostructures of chiral magnetsBinding a hopfion in a chiral magnet nanodiskTable of ContentsDomain-wall dynamics driven by adiabatic spin-transfer torquesCurrent-driven excitation of magnetic multilayersSkyrmion motion driven by oscillating magnetic fieldOscillations of skyrmion clusters in Co/Pt multilayer nanodotsStrain-modulated magnetization precession in skyrmion-based spin transfer nano-oscillatorA ferromagnetic skyrmion-based nano-oscillator with modified perpendicular magnetic anisotropySteady-State Motion of Magnetic DomainsDynamics of magnetic vorticesSkyrmion ground state and gyration of skyrmions in magnetic nanodisks without the Dzyaloshinsky-Moriya interactionDriving magnetic skyrmions with microwave fieldsNanoscale Near-Field Steering of Magnetic VorticesNucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructuresSkyrmion Logic System for Large-Scale Reversible ComputationCascadable direct current driven skyrmion logic inverter gate

[1]	Mühlbauer S, Binz B, Jonietz F, Mühlbauerb S, Binz B, Jonietzc F, Pfleiderera C, Rosch A, Neubauer A, Georgii R, and Böni P 2009 Science 323 915
[2]	Kang W, Huang Y, Zhang X, Zhou Y, and Zhao W S 2016 Proc. IEEE 104 2040
[3]	Fert A, Reyren N, and Cros V 2017 Nat. Rev. Mater. 2 17031
[4]	Göbel B, Mertig I, and Tretiakov O A 2021 Phys. Rep. 895 1
[5]	Yu D X, Yang H X, Chshiev M, and Fert A 2022 Natl. Sci. Rev. 9 nwac021
[6]	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
[7]	Yan Z R, Liu Y Z, Guang Y, Yue K, Feng J F, Lake R K, Yu G Q, and Han X F 2021 Phys. Rev. Appl. 15 064004
[8]	Shen L C, Xia J, Zhang X C, Ezawa M, Tretiakov O A, Liu X X, Zhao G P, and Zhou Y 2020 Phys. Rev. Lett. 124 037202
[9]	Shu Y, Li Q R, Xia J, Lai P, Hou Z P, Zhao Y H, Zhang D G, Zhou Y, Liu X X, and Zhao G P 2022 Appl. Phys. Lett. 121 042402
[10]	Feng Y H, Zhang X, Zhao G P, and Xiang G 2022 IEEE Trans. Electron Devices 69 1293
[11]	Zhao L, Liang X, Xia J, Zhao G P, and Zhou Y 2020 Nanoscale 12 9507
[12]	Wang J L, Xia J, Zhang X C, Zheng X Y, Li G Q, Chen L, Zhou Y, Wu J, Yin H H, Chantrell R, and Xu Y B 2020 Appl. Phys. Lett. 117 202401
[13]	Song L L, Yang H H, Liu B, Meng H, Cao Y S, and Yan P 2021 J. Magn. Magn. Mater. 532 167975
[14]	Kharkov Y A, Sushkov O P, and Mostovoy M 2017 Phys. Rev. Lett. 119 207201
[15]	Göbel B, Mook A, Henk J, Mertig I, and Tretiakov O A 2019 Phys. Rev. B 99 060407(R)
[16]	Nayak A K, Kumar V, Ma T P, Werner P, Pippel E, Sahoo R, Damay F, Rößler U K, Felser C, and Stuart S P P 2017 Nature 548 561
[17]	Woo S, Song K, Zhang X C, Zhou Y, Ezawa M, Liu X X, Finizio S, Raabe J, Lee N J, Kim S, Park S Y, Kim Y, Kim J Y, Lee D, Lee O, Choi J W, Min B C, Koo H C, and Chang J 2018 Nat. Commun. 9 959
[18]	Kim S, Lee K, and Tserkovnyak Y 2017 Phys. Rev. B 95 140404
[19]	Milde P, Köhler D, Seidel J, Eng L M, Bauer A, Chacon A, Kindervater J, Mühlbauer S, Pfleiderer C, Buhrandt S, Schütte C, and Rosch A 2013 Science 340 1076
[20]	Birch M T, Cortés-Ortuño D, Turnbull L A, Wilson M N, Groß F, Träger N, Laurenson A, Bukin N, Moody S H, Weigand M, Schütz G, Popescu H, Fan R, Steadman P, Verezhak J A T, Balakrishnan G, Loudon J C, Twitchett-Harrison A C, Hovorka O, Fangohr H, Ogrin F Y, Gräfe J, and Hatton P D 2020 Nat. Commun. 11 1726
[21]	Wolf D, Schneider S, Rößler U K, Kovács A, Schmidt M, Dunin-Borkowski R E, Büchner B, Rellinghaus B, and Lubk A 2022 Nat. Nanotechnol. 17 250
[22]	Seki S, Garst M, Waizner J, Takagi R, Khanh N D, Okamura Y, Kondou K, Kagawa F, Otani Y, and Tokura Y 2020 Nat. Commun. 11 256
[23]	Birch M T, Cortés-Ortuño D, Litzius K, Wintz S, Schulz F, Weigand M, Štefančič A, Mayoh D A, Balakrishnan G, Hatton P D, and Schütz G 2022 Nat. Commun. 13 3630
[24]	Rybakov F N, Borisov A B, Blügel S, and Kiselev N S 2015 Phys. Rev. Lett. 115 117201
[25]	Zheng F S, Rybakov F N, Borisov A B, Song D S, Wang S, Li Z, Du H F, Kiselev N S, Caron J, Kovács A, Tian M L, Zhang Y H, Blügel S, and Dunin-Borkowski R E 2018 Nat. Nanotechnol. 13 451
[26]	Zhu J, Wu Y D, Hu Q Y, Kong L Y, Tang J, Tian M L, and Du H F 2021 Sci. Chin. Phys. Mech. & Astron. 64 227511
[27]	Liu Y Z, Lake R K, and Zang J D 2018 Phys. Rev. B 98 174437
[28]	Beg M, Lang M, and Fangohr H 2022 IEEE Trans. Magn. 58 0101206
[29]	Li Z and Zhang S 2004 Phys. Rev. B 70 024417
[30]	Slonczewski J C 1996 J. Magn. Magn. Mater. 159 L1
[31]	Moon K W, Kim D H, Je S G, Chun B S, Kim W, Qiu Z Q, Choe S B, and Hwang C 2016 Sci. Rep. 6 20360
[32]	Tejo F, Velozo F, Elías R G, and Escrig J 2020 Sci. Rep. 10 16517
[33]	Yu G L, Xu X F, Qiu Y, Yang H, Zhu M M, and Zhou H M 2021 Appl. Phys. Lett. 118 142403
[34]	Guo J H, Xia J, Zhang X, Philip W T, and Zhou Y 2021 Phys. Lett. A 392 127157
[35]	Thiele A A 1973 Phys. Rev. Lett. 30 230
[36]	Papanicolaou N and Tomaras T N 1991 Nucl. Phys. B 360 425
[37]	Dai Y Y, Wang H, Tao P, Yang T, Ren W J, and Zhang Z D 2013 Phys. Rev. B 88 054403
[38]	Wang W W, Beg M, Zhang B, Kuch W, and Fangohr H 2015 Phys. Rev. B 92 020403(R)
[39]	Yu D X, Sui C W, Schulz D, Berakdar J, and Jia C L 2021 Phys. Rev. Appl. 16 034032
[40]	Sampaio J, Cros V, Rohart S, Thiaville A, and Fert A 2013 Nat. Nanotechnol. 8 839
[41]	Chauwin M, Hu X, Garcia-Sanchez F, Betrabet N, Paler A, Moutafis C, and Friedman J S 2019 Phys. Rev. Appl. 12 064053
[42]	Cheghabouri A M, Katmis F, and Onbasli M C 2022 Phys. Rev. B 105 054411
[43]	SPM Group 2002 Implementation of Temperature in Micromagnetic Simulations (Bristol: Institute of Applied Physics) http://www.nanoscience.de/group_r/stm-spstm/projects/temperature/theory.shtml