Asymmetric magnetic switching owing to anisotropic Dzyaloshinskii–Moriya interaction

Abstract

The Dzyaloshinskii–Moriya interaction (DMI) plays a crucial role in the formation of chiral magnetic structures, such as chiral domain walls and magnetic skyrmions. Recent studies have revealed that anisotropic DMI can arise in specific systems or conditions, which is essential for the formation of three-dimensional spin textures. However, the impact of anisotropic DMI on magnetic moment switching has not been comprehensively studied. In this work, we systematically investigate the influence of anisotropic DMI on spin-orbit torque (SOT)-driven magnetization switching, employing a macrospin model to elucidate the underlying mechanisms. Our findings show that anisotropic DMI introduces a pronounced asymmetry in the magnetization reversal process. Simulations based on the Landau−Lifshitz−Gilbert equation further demonstrate that anisotropic DMI not only breaks the symmetry of the switching trajectory but also enhances switching efficiency by reducing the switching time. Furthermore, we demonstrate the realization of five distinct logic operations (AND, NAND, OR, NOR, NOT) within a single device, exploiting the asymmetric SOT-driven magnetization switching induced by anisotropic DMI. Overall, our results not only provide a comprehensive understanding of the role of anisotropic DMI in SOT-driven magnetic switching, but also open new avenues for the engineering of next-generation spintronic devices leveraging DMI.

Graphical abstract

Keywords

anisotropic Dzyaloshinskii–Moriya interaction / magnetization switching / logic operations / macrospin model

1 Introduction

Spintronic devices occupy a central position in the cutting-edge research on next-generation information technology, offering enormous potential for applications, including non-volatile memory, spin logic devices, and quantum computing [1−5]. The success of spintronic advancements, such as racetrack memories utilizing chiral domain walls (DWs) [6, 7] and systems leveraging magnetic skyrmions [8], relies heavily on the noncollinear spin textures orchestrated by the Dzyaloshinskii–Moriya interaction (DMI) [9−12]. In spintronics, current-induced spin−orbit torques (SOTs) have emerged as a powerful tool for manipulating skyrmions [13] and driving DW motion [14], enabling highly efficient spin-based information technologies. The discovery of DMI initially occurred in weak ferromagnetism of Fe₂O₃ [9] and later in non-centrosymmetric B20 bulk materials at low temperatures [15−19]. Subsequently, isotropic interfacial DMI was observed in heavy metal (HM)/ferromagnet (FM) bilayers with polycrystalline HMs [20−22]. However, recent studies have shown that DMI can exhibit anisotropic properties in specific systems or conditions [23−28]. Research based on point-group theory has demonstrated that anisotropic DMI (a-DMI) is prevalent in systems with point groups C₂, C_s, D₂, C_2v, C₄, S₄, D₄, D_2d, C₃, D₃, C₆ and D₆. Notably, point groups C₂ and C_s exhibit interlayer-DMI and out-of-plane DMI with inherent anisotropy [29]. This a-DMI is significant for the formation of novel topological spin structures, such as elliptical magnetic skyrmions [30, 31] and antiskyrmions [31, 32], which have attracted considerable attention not only for their fundamental physical exploration but also for their promising spintronic applications.

To date, three main sources of a-DMI have been experimentally identified: crystal symmetry, strain, and interlayer coupling, as illustrated in Fig.1(a). Anisotropic DMI is observed in several materials with high crystal symmetry, such as ultrathin epitaxial Au/Co/W(110) films with C_2v crystal symmetry [33], tetragonal Heusler compounds with D_2d symmetry [23], non-centrosymmetric tetragonal structures with S₄ symmetry [34], and two-dimensional magnets with P\overline{4}m2 symmetry [35]. Furthermore, research indicates that the strain can also induce a-DMI [24, 28, 36], typically through two methods: bending the film or controlling the electric field in hybrid ferroelectric-ferromagnetic systems. Recent studies have also shown that interlayer DMI in synthetic antiferromagnetic structures inherently exhibits anisotropic properties [37−39]. The common intrinsic mechanism behind all these forms of a-DMI is the breaking of symmetry along a specific symmetric direction within the materials. In addition to exploring the physical mechanisms of a-DMI, its potential applications have attracted significant attention. For example, a-DMI can stabilize three-dimensional topological structures [30−32]. However, the impact of a-DMI on magnetic moment switching is rarely reported, despite its critical importance for the design of spintronic devices [40].

In this work, we investigate the influence of a-DMI on the SOT-driven magnetization reversal process, focusing on interlayer DMI in synthetic ferromagnet (SF)/synthetic antiferromagnet (SAF) systems through a macrospin model. Our results reveal that interlayer DMI can introduce asymmetric behaviors in the magnetization switching process. The influence of interlayer DMI on the dynamic precession of magnetic moments is elucidated through the Landau−Lifshitz−Gilbert (LLG) equation [41]. Based on the interlayer DMI-induced asymmetric SOT-driven magnetization switching, we demonstrate five types of logic gates (AND, NAND, OR, NOR, NOT) within a single device. This work not only deepens the understanding of the role of interlayer DMI in magnetization reversal, but also opens new avenues for the development of interlayer DMI-based spintronic devices.

2 Theoretical model

In conventional FM/HM bilayers, current-induced SOT can switch the perpendicular magnetic moments with the assistance of an in-plane magnetic field that breaks the symmetry [42, 43]. The processes of the magnetic moment switching from down to up and from up to down are symmetric due to the isotropic interfacial DMI. However, interlayer DMI can affect the magnetization configuration, and thus, the magnetization reversal process driven by SOT. To illustrate this process more intuitively, we analyze the impact of interlayer DMI on SOT-driven magnetization switching from the perspective of the effective DMI field H_DMI in SF/SAF, where the interlayer DMI is present between two FM layers separated by a non-magnetic metal spacer. As shown in Fig.1(b), when the interlayer DMI vector D is perpendicular to the current direction, the H_DMI exerted on the top magnetic moment by the bottom magnetic moment aligns with the current direction. During the switching of the magnetic moment from up to down, the H_DMI acting on the upper magnetic moment is directed along the positive x-axis, consistent with the direction of H_x, effectively increasing the assistant field and thus facilitating the switching. Conversely, when the magnetic moment switches from down to up, the situation is reversed, hindering the switching. As a result, the current-induced SOT-driven magnetization switching loops are asymmetric. Fig.1(b) specifically illustrates the scenario where the interlayer DMI vector D is perpendicular to the current direction, highlighting its maximum influence on SOT-driven magnetization switching. When the angle between the D vector and the current direction changes, the influence on the magnetic moment switching also varies due to the anisotropy of DMI.

To further explore the underlying physical mechanisms, we conducted simulations based on the macrospin model [43] (see Supplementary Material S1 for details). For simplicity, we adopt effective SOT field H_{\mathrm{S}\mathrm{O}\mathrm{T}} to generate the torque {\mathit{\tau }}_{\mathbf{S}\mathbf{O}\mathbf{T}}=-\mathit{M} \times {\mathit{H}}_{\mathbf{S}\mathbf{O}\mathbf{T}}=M{H}_{\mathrm{S}\mathrm{O}\mathrm{T}}\left(\mathit{m}\times \mathit{\sigma }\times \mathit{m}\right), where the amplitude of the SOT can be tuned by applied current. If a relatively small current is applied (i.e., small SOT), the orientations of M₁ and M₂ will remain confined to the x−z plane. Under this circumstance, all torques are aligned along the y-axis, simplifying the torque balance equations to

(1)\begin{array}{rl}{\tau }_{\mathrm{t}\mathrm{o}\mathrm{t}1}=\hat{\mathit{y}}\cdot {\mathit{\tau }}_{\mathrm{t}\mathrm{o}\mathrm{t}1}=& H_x{M}_{\mathrm{S}}{t}_1\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_1+{H_{k1}M}_{\mathrm{S}}{t}_1\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_1\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_1\\ & -H_{\mathrm{e}\mathrm{x}}{M}_{\mathrm{S}}{t}_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_2-{\theta }_1\right)\\ & +H_{\mathrm{D}\mathrm{M}\mathrm{I}}{M}_{\mathrm{S}}{t}_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_2-{\theta }_1\right)\\ & -{H_{\mathrm{S}\mathrm{O}\mathrm{T}}M}_{\mathrm{S}}{t}_1=0,\\ {\tau }_{\mathrm{t}\mathrm{o}\mathrm{t}2}=\hat{\mathit{y}}\cdot {\mathit{\tau }}_{\mathrm{t}\mathrm{o}\mathrm{t}2}=& H_x{M}_{\mathrm{S}}{t}_2\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_2+{H_{k2}M}_{\mathrm{S}}{t}_2\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_2\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_2\\ & +H_{\mathrm{e}\mathrm{x}}{M}_{\mathrm{S}}{t}_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_2-{\theta }_1\right)\\ & -H_{\mathrm{D}\mathrm{M}\mathrm{I}}{M}_{\mathrm{S}}{t}_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_2-{\theta }_1\right)\\ & -{H_{\mathrm{S}\mathrm{O}\mathrm{T}}M}_{\mathrm{S}}{t}_1=0.\end{array}

Here H_x is the magnetic field parallel to the sample surface; M_S is the saturation magnetization; H_k1(2) is the perpendicular magnetic anisotropy (PMA) effective field of the top (bottom) FM layer; t₁₍₂₎ is the thickness of top (bottom) FM layer; θ₁₍₂₎ is the angle between the magnetization of top (bottom) FM layer and z-axis; H_ex is the interlayer coupling field; H_DMI is the value of the interlayer DMI effective field in the sample. By inputting parameters into Eq. (1), one can predict the specific hysteresis loops resulting from SOT and interlayer DMI. Consequently, the impact of interlayer DMI on magnetization switching behavior can be determined. The material parameters used in the simulations are provided in Supplementary Material S1. For simplicity, we have normalized the effective fields of PMA, IEC, SOT and DMI with respect to H_{k1}.

3 Results and discussion

3.1 Asymmetric magnetic switching

Firstly, we simulated the hysteresis loops under SOT with interlayer DMI (see details in Supplemental Material S2). It is found that the critical switching field (H_sw) under a positive H_SOT is smaller than that under a negative H_SOT, indicating the chiral nature of the interlayer DMI. Next, we investigate the effect of interlayer DMI on SOT-driven magnetization switching, focusing on the case where the DMI vector D is perpendicular to both the current and the in-plane magnetic field. Fig.2(a) shows the SOT-driven magnetization switching loops under the in-plane magnetic field H_x for SF system with interlayer DMI. Clearly, these curves exhibit asymmetric switching, with a smaller positive critical SOT switching field \left(H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}\right) and a larger negative H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}, consistent with theoretical model depicted in Fig.1(b). In contrast, for the SAF sample, the positive H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} is larger than the negative one. For the SAF samples, the direction of H_DMI in the upper FM layer is opposite to that in the bottom layer, owing to the antiparallel magnetizations of the two FM layers, as shown in Fig.1(b). Thus, the distinct behaviors of SF and SAF samples can be attributed to the different interlayer couplings (FM/AFM coupling) between two FM layers. Besides, the SOT-driven magnetization switching curves are symmetric in the absence of interlayer DMI (see the Supplemental Material S3), in agreement with experimental observations [37, 44].

By varying the strength of H_DMI, a series of switching loops can be obtained. Fig.2(c) and (d) show that H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} progressively decreases as H_DMI increases, but the magnitudes of positive and negative H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} remain distinct at any given H_DMI. This reinforces the significant impact of interlayer DMI on the switching behavior, highlighting its chiral influence on the magnetization dynamics. To further quantify the asymmetry introduced by interlayer DMI, we define a difference ratio between the positive and negative H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} as

(2)\begin{array}{rl}{\mathrm{\Delta }H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}/H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}=& 2\left({|H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}+}|-|H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}-}|\right)/\left(\left|H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}+}\right|+{|H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}-}|\right)\\ =& 2\left(H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}+}+H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}-}\right)/\left(H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}+}-H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{C}-}\right).\end{array}

As shown in Fig.2(e, f), the magnitude of {\mathrm{\Delta }H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}/H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}} increases with H_DMI, indicating that stronger interlayer DMI leads to more asymmetric switching.

Given that the interlayer DMI possesses in-plane anisotropy, we further studied the SOT-driven magnetization reversal process when applying current I and in-plane magnetic field H_IN at various angles in the plane. Once the sample is prepared, the direction of the interlayer DMI vector is fixed. Therefore, during the simulation process, we fixed the direction of the interlayer DMI effective field H_DMI, and defined the angle between its direction and H_IN as φ. As illustrated in Fig.3(a), when I and H_IN are applied along arbitrary in-plane directions, magnetic moments can be switched due to the SOT. However, the impact of interlayer DMI on magnetic moment reversal varies with φ. By examining the projection of the DMI along the direction of the magnetic moments, we analyzed the effect of the interlayer DMI on current-driven magnetic moment switching. As shown in Fig.3(b) and (c), in both SF and SAF structures, the H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} varies as a function of φ, reflecting the anisotropic nature of interlayer DMI. In particular, the φ-dependent H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}} can be fitted by the cosine function, with a π-phase shift between SF and SAF samples, corresponding to the differences in FM and AFM interlayer coupling. We also examined the variation of {\mathrm{\Delta }H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}/H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}} with the angle φ, which can be fitted by the cosine function, as shown in Fig.3(d) and (e). At 90° or 270°, {\mathrm{\Delta }H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}/H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}} is zero, indicating that the interlayer DMI has negligible effect on the magnetic moment reversal at these angles. However, at 0° or 180°, the absolute value of the {\mathrm{\Delta }H}_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{c}}/H_{\mathrm{S}\mathrm{O}\mathrm{T}}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}} is maximum, meaning that the effect of interlayer DMI on the magnetic moment reversal is the greatest. These phenomena arise from the different projections of the interlayer DMI, leading to varying symmetries in the SOT-driven magnetization reversal along different directions.

3.2 Dynamics of magnetic moment reversal

To gain deeper insight into the dynamics of magnetic moment reversal influenced by interlayer DMI, we simulated the precessional behavior of the magnetic moment driven by SOT and magnetic fields based on the LLG equation (see details in Supplemental Material S4). Fig.4(a, b) compare the temporal evolution of net magnetization in SF and SAF systems. In the absence of interlayer DMI, both configurations exhibit characteristic multi-precession switching trajectories before stabilization.

To further explore the asymmetry introduced by interlayer DMI, we varied the H_DMI from 0 to 0.06H_k1 and observed that interlayer DMI accelerates the precession rate, leading to different switching trajectories compared to the DMI-absent case, as illustrated in Fig.4(c) and (d). Through calculations, we also found that as interlayer DMI increases, the switching time decreases for both SF and SAF, as shown in Fig.4(e) and (f). This result highlights that interlayer DMI not only breaks the symmetry of the magnetic moment switching process, but also enhances switching efficiency. Under the same H_SOT = 0.3H_k1 and the same H_DMI, the SAF structure displays a notably longer switching time than that in the SF structure (see details in Supplemental Material S4), suggesting that the SF configuration has higher switching efficiency at comparable current levels relative to the SAF configuration. In the SF structure, the parallel alignment of magnetic moments in adjacent layers facilitates a synergistic effect between interlayer DMI and SOT, enhancing SOT efficiency and reducing the critical current density required for magnetization reversal, thus leading to shorter switching times. In contrast, the antiparallel alignment in the SAF structure results in opposite H_DMI on adjacent magnetic moments, reducing SOT efficiency. Thus, the difference of switching time is largely due to the different SOT efficiency between FM and AFM couplings. This difference in switching times between SF and SAF structures may have important implications for spintronic device design. The shorter switching time in SF structures is beneficial for applications requiring fast switching, such as high-speed memory devices or high-frequency oscillators. In contrast, SAF structures may be more suited for applications where stability and robustness against external disturbances are prioritized.

3.3 Realization of logic gates

Leveraging the interlayer DMI-induced asymmetric SOT-driven magnetization switching [Fig.5(a)], we successfully demonstrate five types of logic gates (NOT, AND, NAND, OR, NOR) within a single device. As depicted in Fig.5(b), the logic device consists of two current inputs (I_A and I_B) and one output. The output state is defined as “1” for the upward magnetization (+M_z) and “0” for the downward magnetization (−M_z). Although the output in the simulation is magnetization, it can be read by Hall voltage or tunnel magnetoresistance in the practical applications. Before logic operations, the device is initialized by a reset current I_{A/B}^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}}, which can be applied by either the A or B input.

For the AND and NAND gates [Fig.5(c)], I_A and I_B are defined as input “1” (“0”) when they generate H_SOT of −0.15H_k1 (0). The gate state is reset by a current pulse I_reset, which generates a H_SOT of 0.3 H_k1. The device functions as an AND gate under the H_x = +0.2H_k1. The output is “1” only when both I_A and I_B inputs are “1” simultaneously. However, the device is a NAND gate under H_x = −0.2H_k1. The output returns “0” only when the I_A and I_B inputs are “1” at the same time.

For the OR and NOR gates [Fig.5(e)], I_A and I_B are defined as input “1” (“0”) when they generate SOT fields of 0.15H_k1 (0). The gate state is reset by I_reset, generating a H_SOT of −0.3 H_k1. The device is a NOR gate under H_x = +0.2H_k1. In this case, the output is “1” only when I_A and I_B inputs are “0” simultaneously. In contrast, the device functions as an OR gate under H_x = −0.2H_k1. If I_B is fixed as 0, a NOT logic gate can be realized under H_x = +0.2H_k1. When I_A-generated SOT field is 0.15H_k1 (0), I_A is defined as input “1” (“0”). During the logic operations, input 1 (“0”) favors −M_z (+M_z) state, resulting in output “0” (“1”). Therefore, five distinct controllable spin logic operations can be achieved by manipulating the two input (I_A, I_B) and the in-plane bias field.

For practical application, it is crucial to experimentally observe the significant effects of a-DMI on magnetization reversal. A synergistic strategy balancing PMA and a-DMI is essential: Utilizing FM materials with low PMA to reduce switching energy barriers, while enhancing a-DMI strength by interface engineering. This approach can facilitate asymmetric magnetic switching. Furthermore, a-DMI-enabled asymmetric switching paths facilitate logic operations. An ideal spin logic cell would be better based on a magnetic tunnel junction (MTJ) to improve the read margin [45]. Recent advances have demonstrated field-free SOT-driven MTJs with embedded magnetic layers in the SOT hard mask [46], enabling the operation of spin logic devices based on a-DMI-induced asymmetric switching without external magnetic fields. Future research directions include exploring two-dimensional magnets with intrinsically low PMA, dynamic modulation of a-DMI via electric or optical fields, and integrating these structures with CMOS architectures. This integrated approach could address both the fundamental mechanisms and practical challenges, advancing the use of a-DMI in spintronic devices.

4 Conclusion

In this work, we systematically investigated the influence of interlayer DMI on SOT-driven magnetization switching based on a macrospin model. The theoretical simulations reveal that interlayer DMI introduces an inherent asymmetry in SOT-driven magnetization reversal, and this asymmetry becomes increasingly pronounced with the enhancement of DMI. Additionally, we analyzed the behavior of SOT-induced magnetization switching along different in-plane directions, finding that the disparity between positive and negative critical SOT fields varies across different directions, owing to the distinct projections of interlayer DMI. These results highlight the chiral and anisotropic nature of interlayer DMI. Moreover, we simulated the precessional behavior of the magnetic moment using the LLG equation. The results demonstrate that interlayer DMI not only disrupts the symmetry of the magnetic moment switching process, but also improves the switching efficiency by reducing the switching time. Finally, we successfully demonstrated five types of logic gates within a single device based on the interlayer DMI-induced asymmetric SOT-driven magnetization switching. The flexibility and versatility of these logic operations highlight the potential of such devices for spintronic applications, offering a promising avenue for the development of more compact and energy-efficient logic circuits. In summary, our research offers a comprehensive understanding of the influence of interlayer DMI on SOT-driven magnetization switching, providing insights for optimizing spintronic systems with improved switching efficiency and multifunctional logic capabilities.

References

[1] J. Jabeur, G. Di Pendina, F. Bernard-Granger, and G. Prenat, Spin orbit torque non-volatile flip-flop for high speed and low energy applications, IEEE Electron Device Lett. 35(3), 408 (2014)

[2] H. Yang, S. O. Valenzuela, M. Chshiev, S. Couet, B. Dieny, B. Dlubak, A. Fert, K. Garello, M. Jamet, D. E. Jeong, K. Lee, T. Lee, M. B. Martin, G. S. Kar, P. Sénéor, H. J. Shin, and S. Roche, Two-dimensional materials prospects for non-volatile spintronic memories, Nature 606(7915), 663 (2022)

[3] A. Chen, A review of emerging non-volatile memory (NVM) technologies and applications, Solid-State Electron. 125, 25 (2016)

[4] B. Behin-Aein, D. Datta, S. Salahuddin, and S. Datta, Proposal for an all-spin logic device with built-in memory, Nat. Nanotechnol. 5(4), 266 (2010)

[5] A. Steane, Quantum computing, Rep. Prog. Phys. 61(2), 117 (1998)

[6] S. S. Parkin, M. Hayashi, and L. Thomas, Magnetic domain-wall racetrack memory, Science 320(5873), 190 (2008)

[7] R. Tomasello, V. Puliafito, E. Martinez, A. Manchon, M. Ricci, M. Carpentieri, and G. Finocchio, Performance of synthetic antiferromagnetic racetrack memory: Domain wall versus skyrmion, J. Phys. D Appl. Phys. 50(32), 325302 (2017)

[8] M. Chauwin, X. Hu, F. Garcia-Sanchez, N. Betrabet, A. Paler, C. Moutafis, and J. S. Friedman, Skyrmion logic system for large-scale reversible computation, Phys. Rev. Appl. 12(6), 064053 (2019)

[9] I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4(4), 241 (1958)

[10] T. Moriya, New mechanism of anisotropic super-exchange interaction, Phys. Rev. Lett. 4(5), 228 (1960)

[11] Q. Cui, L. Wang, Y. Zhu, J. Liang, and H. Yang, Magnetic anisotropy, exchange coupling and Dzyaloshinskii−Moriya interaction of two-dimensional magnets, Front. Phys. (Beijing) 18(1), 13602 (2023)

[12] Y. H. Gao, X. P. Yao, F. Y. Li, and G. Chen, Spin-1 pyrochlore antiferromagnets: Theory, model, and materials’ survey, Front. Phys. (Beijing) 15(6), 63201 (2020)

[13] L. Caretta, M. Mann, F. Büttner, K. Ueda, B. Pfau, C. M. Günther, P. Hessing, A. Churikova, C. Klose, M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S. Eisebitt, and G. S. D. Beach, Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet, Nat. Nanotechnol. 13(12), 1154 (2018)

[14] K. Cai, Z. Zhu, J. M. Lee, R. Mishra, L. Ren, S. D. Pollard, P. He, G. Liang, K. L. Teo, and H. Yang, Ultrafast and energy-efficient spin−orbit torque switching in compensated ferrimagnets, Nat. Electron. 3(1), 37 (2020)

[15] A. Fert and P. M. Levy, Role of anisotropic exchange interactions in determining the properties of spin-glasses, Phys. Rev. Lett. 44(23), 1538 (1980)

[16] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiral magnet, Science 323(5916), 915 (2009)

[17] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Real-space observation of a two-dimensional skyrmion crystal, Nature 465(7300), 901 (2010)

[18] D. H. Kim, M. Haruta, H. W. Ko, G. Go, H. J. Park, T. Nishimura, D. Y. Kim, T. Okuno, Y. Hirata, Y. Futakawa, H. Yoshikawa, W. Ham, S. Kim, H. Kurata, A. Tsukamoto, Y. Shiota, T. Moriyama, S. B. Choe, K. J. Lee, and T. Ono, Bulk Dzyaloshinskii–Moriya interaction in amorphous ferrimagnetic alloys, Nat. Mater. 18(7), 685 (2019)

[19] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe, Nat. Mater. 10(2), 106 (2011)

[20] X. Ma, G. Yu, C. Tang, X. Li, C. He, J. Shi, K. L. Wang, and X. Li, Interfacial Dzyaloshinskii–Moriya interaction: Effect of 5d band filling and correlation with spin mixing conductance, Phys. Rev. Lett. 120(15), 157204 (2018)

[21] A. W. Wells, P. M. Shepley, C. H. Marrows, and T. A. Moore, Effect of interfacial intermixing on the Dzyaloshinskii–Moriya interaction in Pt/Co/Pt, Phys. Rev. B 95(5), 054428 (2017)

[22] J. Yun, D. Li, B. Cui, X. Guo, K. Wu, X. Zhang, Y. Wang, J. Mao, Y. Zuo, and L. Xi, Current induced domain wall motion and tilting in Pt/Co/Ta structures with perpendicular magnetic anisotropy in the presence of the Dyzaloshinskii–Moriya interaction, J. Phys. D Appl. Phys. 51(15), 155001 (2018)

[23] Y. Ga, Q. Cui, Y. Zhu, D. Yu, L. Wang, J. Liang, and H. Yang, Anisotropic Dzyaloshinskii–Moriya interaction protected by D_2d crystal symmetry in two-dimensional ternary compounds, npj Comput. Mater. 8, 128 (2022)

[24] N. S. Gusev, A. V. Sadovnikov, S. A. Nikitov, M. V. Sapozhnikov, and O. G. Udalov, Manipulation of the Dzyaloshinskii–Moriya interaction in Co/Pt multilayers with strain, Phys. Rev. Lett. 124(15), 157202 (2020)

[25] R. Chen, F. Xu, Q. Cui, J. Liang, Y. Zhou, N. Wang, F. Pan, F. Xue, H. Yang, and C. Song, Anisotropic interlayer Dzyaloshinskii–Moriya interactions in synthetic ferromagnetic multilayers, Sci. Bull. (Beijing) 68(9), 878 (2023)

[26] S. Huang, C. Zhou, G. Chen, H. Shen, A. K. Schmid, K. Liu, and Y. Wu, Stabilization and current-induced motion of antiskyrmion in the presence of anisotropic Dzyaloshinskii-Moriya interaction, Phys. Rev. B 96(14), 144412 (2017)

[27] O. G. Udalov, I. S. Beloborodov, and M. V. Sapozhnikov, Magnetic skyrmions and bimerons in films with anisotropic interfacial Dzyaloshinskii–Moriya interaction, Phys. Rev. B 103(17), 174416 (2021)

[28] M. V. Sapozhnikov, R. V. Gorev, E. V. Skorokhodov, N. S. Gusev, A. V. Sadovnikov, and O. G. Udalov, Zigzag domains caused by strain-induced anisotropy of the Dzyaloshinskii–Moriya interaction, Phys. Rev. B 105(2), 024405 (2022)

[29] H. Niu, H. Y. Kwon, T. Ma, Z. Cheng, C. Ophus, B. Miao, L. Sun, Y. Wu, K. Liu, S. S. P. Parkin, C. Won, A. K. Schmid, H. Ding, and G. Chen, Reducing crystal symmetry to generate out-of-plane Dzyaloshinskii–Moriya interaction, Nat. Commun. 15(1), 10199 (2024)

[30] B. Cui,D. Yu,Z. Shao,Y. Liu,H. Wu, P. Nan,Z. Zhu,C. Wu,T. Guo,P. Chen, H. A. Zhou,L. Xi,W. Jiang,H. Wang,S. Liang, H. Du,K. L. Wang,W. Wang,K. Wu,X. Han, G. Zhang,H. Yang,G. Yu, Néel-type elliptical skyrmions in a laterally asymmetric magnetic multilayer, Adv. Mater. 33(12), 2006924 (2021)

[31] M. Hoffmann, B. Zimmermann, G. P. Müller, D. Schürhoff, N. S. Kiselev, C. Melcher, and S. Blügel, Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii–Moriya interactions, Nat. Commun. 8(1), 308 (2017)

[32] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. Rößler, C. Felser, and S. S. Parkin, Magnetic antiskyrmions above room temperature in tetragonal Heusler materials, Nature 548(7669), 561 (2017)

[33] L. Camosi, S. Rohart, O. Fruchart, S. Pizzini, M. Belmeguenai, Y. Roussigné, A. Stashkevich, S. M. Cherif, L. Ranno, M. De Santis, and J. Vogel, Anisotropic Dzyaloshinskii–Moriya interaction in ultrathin epitaxial Au/Co/W (110), Phys. Rev. B 95(21), 214422 (2017)

[34] K. Karube, L. Peng, J. Masell, M. Hemmida, H. A. Krug von Nidda, I. Kézsmárki, X. Yu, Y. Tokura, Y. Taguchi, and Doping control of magnetic anisotropy for stable antiskyrmion formation in Schreibersite (Fe, Ni)₃P with S₄ symmetry, Adv. Mater. 34(11), 2108770 (2022)

[35] Q. Cui, Y. Zhu, Y. Ga, J. Liang, P. Li, D. Yu, P. Cui, and H. Yang, Anisotropic Dzyaloshinskii–Moriya interaction and topological magnetism in two-dimensional magnets protected by P4¯m2 crystal symmetry, Nano Lett. 22(6), 2334 (2022)

[36] O. G. Udalov and I. S. Beloborodov, Strain-dependent Dzyaloshinskii–Moriya interaction in a ferromagnet/heavy-metal bilayer, Phys. Rev. B 102(13), 134422 (2020)

[37] J. Yun, B. Cui, Q. Cui, X. He, Y. Chang, Y. Zhu, Z. Yan, X. Guo, H. Xie, J. Zhang, Q. Bai, Y. Zhai, H. Xu, Y. Zuo, D. Yang, C. Jia, G. Yu, H. Wu, H. Yang, D. Xue, and L. Xi, The Anisotropic interlayer Dzyaloshinskii–Moriya interaction in synthetic ferromagnetic/antiferromagnetic sandwiches, Adv. Funct. Mater. 33(33), 2301731 (2023)

[38] A. Fernández-Pacheco, E. Vedmedenko, F. Ummelen, R. Mansell, D. Petit, and R. P. Cowburn, Symmetry-breaking interlayer Dzyaloshinskii–Moriya interactions in synthetic antiferromagnets, Nat. Mater. 18(7), 679 (2019)

[39] E. Y. Vedmedenko, P. Riego, J. A. Arregi, and A. Berger, Interlayer Dzyaloshinskii–Moriya interactions, Phys. Rev. Lett. 122(25), 257202 (2019)

[40] W. He, C. Wan, C. Zheng, Y. Wang, X. Wang, T. Ma, Y. Wang, C. Guo, X. Luo, M. E. Stebliy, G. Yu, Y. Liu, A. V. Ognev, A. S. Samardak, and X. Han, Field-free spin−orbit torque switching enabled by the interlayer Dzyaloshinskii–Moriya interaction, Nano Lett. 22(17), 6857 (2022)

[41] S. Fukami,T. Anekawa,C. Zhang,H. Ohno, A spin−orbit torque switching scheme with collinear magnetic easy axis and current configuration, Nat. Nanotechnol. 11(7), 621 (2016)

[42] S. Emori, U. Bauer, S. M. Ahn, E. Martinez, and G. S. Beach, Current-driven dynamics of chiral ferromagnetic domain walls, Nat. Mater. 12(7), 611 (2013)

[43] L. Liu, O. J. Lee, T. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect, Phys. Rev. Lett. 109(9), 096602 (2012)

[44] K. Wang, L. Qian, S. C. Ying, and G. Xiao, Spin–orbit torque switching of chiral magnetization across a synthetic antiferromagnet, Commun. Phys. 4(1), 10 (2021)

[45] S. Jinnai, T. Butsurin, K. Watanabe, S. Fukami, and H. Ohno, Scaling magnetic tunnel junction down to single-digit nanometers — Challenges and prospects, Appl. Phys. Lett. 116(16), 160501 (2020)

[46] K. Garello,F. Yasin,H. Hody,S. Couet,L. Souriau,S. H. Sharifi,J. Swerts,R. Carpenter,S. Rao,W. Kim,J. Wu, K. K. V. Sethu,M. Pak,N. Jossart,D. Crotti,A. Furnémont,G. S. Kar, Manufacturable 300 mm platform solution for field-free switching SOT-MRAM, 2019 Symposium on VLSI Technology, pp T194–T195

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