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Magnetic anisotropy, exchange coupling and Dzyaloshinskii—Moriya interaction of two-dimensional magnets

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Abstract

The two-dimensional (2D) magnets provide novel opportunities for understanding magnetism and investigating spin related phenomena in several atomic thickness. Multiple features of 2D magnets, such as critical temperatures, magnetoelectric/magneto-optic responses, and spin configurations, depend on the basic magnetic terms that describe various spins interactions and cooperatively determine the spin Hamiltonian of studied systems. In this review, we present a comprehensive survey of three types of basic terms, including magnetic anisotropy that is intimately related with long-range magnetic order, exchange coupling that normally dominates the spin interactions, and Dzyaloshinskii—Moriya interaction (DMI) that favors the noncollinear spin configurations, from the theoretical aspect. We introduce not only the physical features and origin of these crucial terms in 2D magnets but also many correlated phenomena, which may lead to the advance of 2D spintronics.

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References

  1. N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17(22), 1133 (1966)

    Article  ADS  Google Scholar 

  2. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature 546(7657), 265 (2017)

    Article  ADS  Google Scholar 

  3. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature 546(7657), 270 (2017)

    Article  ADS  Google Scholar 

  4. M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M. H. Phan, and M. Batzill, Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substrates, Nat. Nanotechnol. 13(4), 289 (2018)

    Article  ADS  Google Scholar 

  5. D. J. O’Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K. Luo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta, D. W. McComb, and R. K. Kawakami, Room temperature intrinsic ferromagnetism in epitaxial manganese selenide films in the monolayer limit, Nano Lett. 18(5), 3125 (2018)

    Article  ADS  Google Scholar 

  6. Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2, Nature 563(7729), 94 (2018)

    Article  ADS  Google Scholar 

  7. J. Lee, S. Lee, J. H. Ryoo, S. Kang, T. Y. Kim, P. Kim, C. H. Park, J. G. Park, and H. Cheong, Ising-type magnetic ordering in atomically thin FePS3, Nano Lett. 16(12), 7433 (2016)

    Article  ADS  Google Scholar 

  8. G. Long, H. Henck, M. Gibertini, D. Dumcenco, Z. Wang, T. Taniguchi, K. Watanabe, E. Giannini, and A. F. Morpurgo, Persistence of magnetism in atomically thin MnPS3 crystals, Nano Lett. 20(4), 2452 (2020)

    Article  ADS  Google Scholar 

  9. A. Bedoya-Pinto, J. R. Ji, A. K. Pandeya, P. Gargiani, M. Valvidares, P. Sessi, J. M. Taylor, F. Radu, K. Chang, and S. S. P. Parkin, Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer, Science 374(6567), 616 (2021)

    Article  ADS  Google Scholar 

  10. T. Song, X. Cai, M. W. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, and X. Xu, Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures, Science 360(6394), 1214 (2018)

    Article  ADS  Google Scholar 

  11. X. Wang, J. Tang, X. Xia, C. He, J. Zhang, Y. Liu, C. Wan, C. Fang, C. Guo, W. Yang, Y. Guang, X. Zhang, H. Xu, J. Wei, M. Liao, X. Lu, J. Feng, X. Li, Y. Peng, H. Wei, R. Yang, D. Shi, X. Zhang, Z. Han, Z. Zhang, G. Zhang, G. Yu, and X. Han, Current-driven magnetization switching in a van der Waals ferromagnet Fe3GeTe2, Sci. Adv. 5(8), eaaw8904 (2019)

    Article  ADS  Google Scholar 

  12. H. Fu, C. Liu, and B. Yan, Exchange bias and quantum anomalous Hall effect in the MnBi2Te4/CrI3 heterostructure, Sci. Adv. 6(10), eaaz0948 (2020)

    Article  ADS  Google Scholar 

  13. Y. Li, J. Li, Y. Li, M. Ye, F. Zheng, Z. Zhang, J. Fu, W. Duan, and Y. Xu, High-temperature quantum anomalous Hall insulators in lithium-decorated iron-based superconductor materials, Phys. Rev. Lett. 125(8), 086401 (2020)

    Article  ADS  Google Scholar 

  14. D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng, N. Sivadas, B. Huang, E. Schmidgall, T. Taniguchi, K. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K. M. C. Fu, and X. Xu, Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics, Sci. Adv. 3(5), e1603113 (2017)

    Article  ADS  Google Scholar 

  15. K. Zollner, P. E. FariaJunior, and J. Fabian, Proximity exchange effects in MoSe2 and WSe2 heterostructures with CrI3: Twist angle, layer, and gate dependence, Phys. Rev. B 100(8), 085128 (2019)

    Article  ADS  Google Scholar 

  16. L. Ai, E. Zhang, J. Yang, X. Xie, Y. Yang, Z. Jia, Y. Zhang, S. Liu, Z. Li, P. Leng, X. Cao, X. Sun, T. Zhang, X. Kou, Z. Han, F. Xiu, and S. Dong, Van der Waals ferromagnetic Josephson junctions, Nat. Commun. 12(1), 6580 (2021)

    Article  ADS  Google Scholar 

  17. W. Zhao, Z. Fei, T. Song, H. K. Choi, T. Palomaki, B. Sun, P. Malinowski, M. A. McGuire, J. H. Chu, X. Xu, and D. H. Cobden, Magnetic proximity and nonreciprocal current switching in a monolayer WTe2 helical edge, Nat. Mater. 19(5), 503 (2020)

    Article  ADS  Google Scholar 

  18. Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes, A. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q. C. Sun, E. J. Telford, H. H. Kim, M. Augustin, U. Vool, J. X. Yin, L. H. Li, A. Falin, C. R. Dean, F. Casanova, R. F. L. Evans, M. Chshiev, A. Mishchenko, C. Petrovic, R. He, L. Zhao, A. W. Tsen, B. D. Gerardot, M. Brotons-Gisbert, Z. Guguchia, X. Roy, S. Tongay, Z. Wang, M. Z. Hasan, J. Wrachtrup, A. Yacoby, A. Fert, S. Parkin, K. S. Novoselov, P. Dai, L. Balicas, and E. J. G. Santos, The magnetic genome of two-dimensional van der Waals materials, ACS Nano 16(5), 6960 (2022)

    Article  Google Scholar 

  19. D. Soriano, M. I. Katsnelson, and J. Fernández-Rossier, Magnetic two-dimensional chromium trihalides: A theoretical perspective, Nano Lett. 20(9), 6225 (2020)

    Article  ADS  Google Scholar 

  20. M. Nakano, Y. Wang, S. Yoshida, H. Matsuoka, Y. Majima, K. Ikeda, Y. Hirata, Y. Takeda, H. Wadati, Y. Kohama, Y. Ohigashi, M. Sakano, K. Ishizaka, and Y. Iwasa, Intrinsic 2D ferromagnetism in V5Se8 epitaxial thin films, Nano Lett. 19(12), 8806 (2019)

    Article  ADS  Google Scholar 

  21. C. Tang, L. Zhang, and A. Du, Tunable magnetic anisotropy in 2D magnets via molecular adsorption, J. Mater. Chem. C 8(42), 14948 (2020)

    Article  Google Scholar 

  22. C. Tang, K. Ostrikov, S. Sanvito, and A. Du, Prediction of room-temperature ferromagnetism and large perpendicular magnetic anisotropy in a planar hypercoordinate FeB3 monolayer, Nanoscale Horiz. 6(1), 43 (2021)

    Article  ADS  Google Scholar 

  23. M. Alsubaie, C. Tang, D. Wijethunge, D. Qi, and A. Du, First-principles study of the enhanced magnetic anisotropy and transition temperature in a CrSe2 monolayer via hydrogenation, ACS Appl. Electron. Mater. 4(7), 3240 (2022)

    Article  Google Scholar 

  24. V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group (I): Classical systems, Sov. Phys. JETP 32, 493 (1971)

    ADS  MathSciNet  Google Scholar 

  25. V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group (II): Quantum systems, Sov. Phys. JETP 34, 610 (1972)

    ADS  Google Scholar 

  26. J. M. Kosterlitz and D. J. Thouless, Long range order and metastability in two dimensional solids and super-fluids (application of dislocation theory), J. Phys. C 5(11), L124 (1972)

    Article  ADS  Google Scholar 

  27. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6(7), 1181 (1973)

    Article  ADS  Google Scholar 

  28. J. M. Kosterlitz, The critical properties of the two-dimensional xy model, J. Phys. C 7(6), 1046 (1974)

    Article  ADS  Google Scholar 

  29. H. L. Zhuang, P. R. C. Kent, and R. G. Hennig, Strong anisotropy and magnetostriction in the two-dimensional Stoner ferromagnet Fe3GeTe2, Phys. Rev. B 93(13), 134407 (2016)

    Article  ADS  Google Scholar 

  30. J. L. Lado and J. F. Rossier, On the origin of magnetic anisotropy in two dimensional CrI3, 2D Mater. 4 035002 (2017)

    Article  Google Scholar 

  31. L. Webster and J. A. Yan, Strain-tunable magnetic anisotropy in monolayer CrCl3, CrBr3, and CrI3, Phys. Rev. B 98(14), 144411 (2018)

    Article  ADS  Google Scholar 

  32. B. Yang, X. Zhang, H. Yang, X. Han, and Y. Yan, Nonmetallic atoms induced magnetic anisotropy in monolayer chromium trihalides, J. Phys. Chem. C 123(1), 691 (2019)

    Article  Google Scholar 

  33. B. Yang, X. Zhang, H. Yang, X. Han, and Y. Yan, Strain controlling transport properties of heterostructure composed of monolayer CrI3, Appl. Phys. Lett. 114(19), 192405 (2019)

    Article  ADS  Google Scholar 

  34. F. J. Dyson, General theory of spin-wave interactions, Phys. Rev. 102(5), 1217 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. F. Xue, Y. Hou, Z. Wang, and R. Wu, Two-dimensional ferromagnetic van der Waals CrCl3 monolayer with enhanced anisotropy and Curie temperature, Phys. Rev. B 100(22), 224429 (2019)

    Article  ADS  Google Scholar 

  36. Y. Li, Z. Jiang, J. Li, S. Xu, and W. Duan, Magnetic anisotropy of the two-dimensional ferromagnetic insulator MnBi2Te4, Phys. Rev. B 100(13), 134438 (2019)

    Article  ADS  Google Scholar 

  37. C. Xu, J. Feng, H. Xiang, and L. Bellaiche, Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers, npj Comput. Mater. 4, 57 (2018)

    Article  ADS  Google Scholar 

  38. Q. Cui, J. Liang, B. Yang, Z. Wang, P. Li, P. Cui, and H. Yang, Giant enhancement of perpendicular magnetic anisotropy and induced quantum anomalous Hall effect in graphene/NiI2 heterostructures via tuning the van der Waals interlayer distance, Phys. Rev. B 101(21), 214439 (2020)

    Article  ADS  Google Scholar 

  39. C. Gong and X. Zhang, Two-dimensional magnetic crystals and emergent heterostructure devices, Science 363(6428), eaav4450 (2019)

    Article  Google Scholar 

  40. S. Y. Park, D. S. Kim, Y. Liu, J. Hwang, Y. Kim, W. Kim, J. Y. Kim, C. Petrovic, C. Hwang, S. K. Mo, H. Kim, B. C. Min, H. C. Koo, J. Chang, C. Jang, J. W. Choi, and H. Ryu, Controlling the magnetic anisotropy of the van der Waals ferromagnet Fe3GeTe2 through hole doping, Nano Lett. 20(1), 95 (2020)

    Article  ADS  Google Scholar 

  41. Y. P. Wang, X. Y. Chen, and M. Q. Long, Modifications of magnetic anisotropy of Fe3GeTe2 by the electric field effect, Appl. Phys. Lett. 116(9), 092404 (2020)

    Article  ADS  Google Scholar 

  42. X. G. Ye, P. F. Zhu, W. Z. Xu, N. Z. Shang, K. H. Liu, and Z. M. Liao, Orbit-transfer torque driven field-free switching of perpendicular magnetization chin, Phys. Lett. 39, 037303 (2022)

    Google Scholar 

  43. J. Seo, E. S. An, T. Park, S. Y. Hwang, G. Y. Kim, K. Song, W. Noh, J. Y. Kim, G. S. Choi, M. Choi, E. Oh, K. Watanabe, T. Taniguchi, J. H. Park, Y. J. Jo, H. W. Yeom, S. Y. Choi, J. H. Shim, and J. S. Kim, Tunable high-temperature itinerant antiferromagnetism in a van der Waals magnet, Nat. Commun. 12(1), 2844 (2021)

    Article  ADS  Google Scholar 

  44. X. Zhang, Q. Lu, W. Liu, W. Niu, J. Sun, J. Cook, M. Vaninger, P. F. Miceli, D. J. Singh, S. W. Lian, T. R. Chang, X. He, J. Du, L. He, R. Zhang, G. Bian, and Y. Xu, Room-temperature intrinsic ferromagnetism in epitaxial CrTe2 ultrathin films, Nat. Commun. 12(1), 2492 (2021)

    Article  ADS  Google Scholar 

  45. Y. Wang, M. E. Ziebel, L. Sun, J. T. Gish, T. J. Pearson, X. Z. Lu, A. E. Thorarinsdottir, M. C. Hersam, J. R. Long, D. E. Freedman, J. M. Rondinelli, D. Puggioni, and T. D. Harris, Strong magnetocrystalline anisotropy arising from metal—ligand covalency in a metal—organic candidate for 2D magnetic order, Chem. Mater. 33(22), 8712 (2021)

    Article  Google Scholar 

  46. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303(1), 2 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321(1), 2 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. C. Xu, J. Feng, M. Kawamura, Y. Yamaji, Y. Nahas, S. Prokhorenko, Y. Qi, H. Xiang, and L. Bellaiche, Possible Kitaev quantum spin liquid state in 2D materials with S = 3/2, Phys. Rev. Lett. 124(8), 087205 (2020)

    Article  ADS  Google Scholar 

  49. L. J. Sandilands, Y. Tian, K. W. Plumb, Y. J. Kim, and K. S. Burch, Scattering continuum and possible fractionalized excitations in α-RuCl3, Phys. Rev. Lett. 114(14), 147201 (2015)

    Article  ADS  Google Scholar 

  50. H.-S. Kim, V. Vijay Shankar, A. Catuneanu, and H. Y. Kee, Kitaev magnetism in honeycomb RuCl3 with intermediate spin—orbit coupling, Phys. Rev. B 91, 241110(R) (2015)

    Article  ADS  Google Scholar 

  51. A. Banerjee, C. A. Bridges, J. Q. Yan, A. A. Aczel, L. Li, M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler, Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet, Nat. Mater. 15(7), 733 (2016)

    Article  ADS  Google Scholar 

  52. S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales, Phys. Rev. B 93(21), 214431 (2016)

    Article  ADS  Google Scholar 

  53. H. S. Kim and H. Y. Kee, Crystal structure and magnetism in α-RuCl3: An ab initio study, Phys. Rev. B 93(15), 155143 (2016)

    Article  ADS  Google Scholar 

  54. M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections, Annu. Rev. Condens. Matter Phys. 9(1), 17 (2018)

    Article  ADS  Google Scholar 

  55. A. Banerjee, P. Lampen-Kelley, J. Knolle, C. Balz, A. A. Aczel, B. Winn, Y. Liu, D. Pajerowski, J. Yan, C. A. Bridges, A. T. Savici, B. C. Chakoumakos, M. D. Lumsden, D. A. Tennant, R. Moessner, D. G. Mandrus, and S. E. Nagler, Excitations in the field-induced quantum spin liquid state of α-RuCl3, npj Quant. Mater. 3, 8 (2018)

    Article  ADS  Google Scholar 

  56. Y. Kasahara, K. Sugii, T. Ohnishi, M. Shimozawa, M. Yamashita, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T. Shibauchi, and Y. Matsuda, Unusual thermal Hall effect in a Kitaev spin liquid candidate α-RuCl3, Phys. Rev. Lett. 120(21), 217205 (2018)

    Article  ADS  Google Scholar 

  57. H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1(4), 264 (2019)

    Article  Google Scholar 

  58. J. A. Sears, L. E. Chern, S. Kim, P. J. Bereciartua, S. Francoual, Y. B. Kim, and Y. J. Kim, Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl3, Nat. Phys. 16(8), 837 (2020)

    Article  Google Scholar 

  59. Y. Zhou, K. Kanoda, and T. K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89(2), 025003 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  60. H. J. Xiang, E. J. Kan, S. H. Wei, M. H. Whangbo, and X. G. Gong, Predicting the spin-lattice order of frustrated systems from first principles, Phys. Rev. B 84(22), 224429 (2011)

    Article  ADS  Google Scholar 

  61. B. Goodenough, Theory of the role of covalence in the perovskite-type manganites [La,M(II)] MnO3, Phys. Rev. 100(2), 564 (1955)

    Article  ADS  Google Scholar 

  62. J. Kanamori, Superexchange interaction and symmetry properties of electron orbitals, J. Phys. Chem. Solids 10(2–3), 87 (1959)

    Article  ADS  Google Scholar 

  63. P. W. Anderson, New approach to the theory of superexchange interactions, Phys. Rev. 115(1), 2 (1959)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. C. Huang, J. Feng, F. Wu, D. Ahmed, B. Huang, H. Xiang, K. Deng, and E. Kan, Toward intrinsic room-temperature ferromagnetism in two-dimensional semiconductors, J. Am. Chem. Soc. 140(36), 11519 (2018)

    Article  Google Scholar 

  65. N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto, and D. Xiao, Magnetic ground state of semiconducting transition-metal trichalcogenide monolayers, Phys. Rev. B 91(23), 235425 (2015)

    Article  ADS  Google Scholar 

  66. Q. Pei and W. Mi, Electrical control of magnetic behavior and valley polarization of monolayer antiferromagnetic MnPSe3 on an insulating ferroelectric substrate from first principle, Phys. Rev. Appl. 11(1), 014011 (2019)

    Article  ADS  Google Scholar 

  67. T. Olsen, Magnetic anisotropy and exchange interactions of two-dimensional FePS3, NiPS3 and MnPS3 from first principles calculations, J. Phys. D 54(31), 314001 (2021)

    Article  ADS  Google Scholar 

  68. J. Li, J. Y. Ni, X. Y. Li, H. J. Koo, M. H. Whangbo, J. S. Feng, and H. J. Xiang, Intralayer ferromagnetism between S = 5/2 ions in MnBi2Te4: Role of empty Bi p states, Phys. Rev. B 101(20), 201408 (2020)

    Article  ADS  Google Scholar 

  69. C. Huang, J. Feng, J. Zhou, H. Xiang, K. Deng, and E. Kan, Ultra-high-temperature ferromagnetism in intrinsic tetrahedral semiconductors, J. Am. Chem. Soc. 141(31), 12413 (2019)

    Article  Google Scholar 

  70. 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 \(P\bar 4m2\) crystal symmetry, Nano Lett. 22(6), 2334 (2022)

    Article  ADS  Google Scholar 

  71. J. J. Zhang, L. Lin, Y. Zhang, M. Wu, B. I. Yakobson, and S. Dong, Type-II multiferroic Hf2VC2F2 MXene monolayer with high transition temperature, J. Am. Chem. Soc. 140(30), 9768 (2018)

    Article  Google Scholar 

  72. D. Amoroso, P. Barone, and S. Picozzi, Spontaneous skyrmionic lattice from anisotropic symmetric exchange in a Ni-halide monolayer, Nat. Commun. 11(1), 5784 (2020)

    Article  ADS  Google Scholar 

  73. J. Y. Ni, X. Y. Li, D. Amoroso, X. He, J. S. Feng, E. J. Kan, S. Picozzi, and H. J. Xiang, Giant biquadratic exchange in 2D magnets and its role in stabilizing ferromagnetism of NiCl2 monolayers, Phys. Rev. Lett. 127(24), 247204 (2021)

    Article  ADS  Google Scholar 

  74. H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin current and magnetoelectric effect in noncollinear magnets, Phys. Rev. Lett. 95(5), 057205 (2005)

    Article  ADS  Google Scholar 

  75. D. Khomskii, Classifying multiferroics: Mechanisms and effects, Physics (College Park Md.) 2, 20 (2009)

    ADS  Google Scholar 

  76. Y. Tokura, S. Seki, and N. Nagaosa, Multiferroics of spin origin, Rep. Prog. Phys. 77(7), 076501 (2014)

    Article  ADS  Google Scholar 

  77. Q. Song, C. A. Occhialini, E. Ergeçen, B. Ilyas, D. Amoroso, P. Barone, J. Kapeghian, K. Watanabe, T. Taniguchi, A. S. Botana, S. Picozzi, N. Gedik, and R. Comin, Evidence for a single-layer van der Waals multiferroic, Nature 602(7898), 601 (2022)

    Article  ADS  Google Scholar 

  78. D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano, E. Navarro-Moratalla, K. Watanabe, T. Taniguchi, S. Manni, P. Canfield, J. Fernández-Rossier, and P. Jarillo-Herrero, Probing magnetism in 2D van der Waals crystalline insulators via electron tunneling, Science 360(6394), 1218 (2018)

    Article  ADS  Google Scholar 

  79. Z. Wang, I. Gutiérrez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamoğlu, E. Giannini, and A. F. Morpurgo, Very large tunneling magnetoresistance in layered magnetic semiconductor CrI3, Nat. Commun. 9(1), 2516 (2018)

    Article  ADS  Google Scholar 

  80. H. H. Kim, B. Yang, T. Patel, F. Sfigakis, C. Li, S. Tian, H. Lei, and A. W. Tsen, One million percent tunnel magnetoresistance in a magnetic van der Waals heterostructure, Nano Lett. 18(8), 4885 (2018)

    Article  ADS  Google Scholar 

  81. S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Controlling magnetism in 2D CrI3 by electrostatic doping, Nat. Nanotechnol. 13(7), 549 (2018)

    Article  ADS  Google Scholar 

  82. S. Jiang, J. Shan, and K. F. Mak, Electric-field switching of two-dimensional van der Waals magnets, Nat. Mater. 17(5), 406 (2018)

    Article  ADS  Google Scholar 

  83. B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo-Herrero, and X. Xu, Electrical control of 2D magnetism in bilayer CrI3, Nat. Nanotechnol. 13(7), 544 (2018)

    Article  ADS  Google Scholar 

  84. H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao, G. Nichols, F. Sfigakis, S. Zhong, C. Li, S. Tian, D. G. Cory, G. X. Miao, J. Shan, K. F. Mak, H. Lei, K. Sun, L. Zhao, and A. W. Tsen, Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides, Proc. Natl. Acad. Sci. USA 116(23), 11131 (2019)

    Article  ADS  Google Scholar 

  85. R. Xu and X. Zhou, Electric field-modulated magnetic phase transition in van der Waals CrI3 bilayers, J. Phys. Chem. Lett. 11(8), 3152 (2020)

    Article  Google Scholar 

  86. N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and D. Xiao, Stacking-dependent magnetism in bilayer CrI3, Nano Lett. 18(12), 7658 (2018)

    Article  ADS  Google Scholar 

  87. P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z. Y. Lu, and W. Ji, Stacking tunable interlayer magnetism in bilayer CrI3, Phys. Rev. B 99(14), 144401 (2019)

    Article  ADS  Google Scholar 

  88. W. Chen, Z. Sun, Z. Wang, L. Gu, X. Xu, S. Wu, and C. Gao, Direct observation of van der Waals stacking-dependent interlayer magnetism, Science 366(6468), 983 (2019)

    Article  ADS  Google Scholar 

  89. J. Xiao and B. Yan, An electron-counting rule to determine the interlayer magnetic coupling of the van der Waals materials, 2D Mater. 7, 045010 (2020)

    Article  Google Scholar 

  90. T. Li, S. Jiang, N. Sivadas, Z. Wang, Y. Xu, D. Weber, J. E. Goldberger, K. Watanabe, T. Taniguchi, C. J. Fennie, K. Fai Mak, and J. Shan, Pressure-controlled interlayer magnetism in atomically thin CrI3, Nat. Mater. 18(12), 1303 (2019)

    Article  ADS  Google Scholar 

  91. Y. Xu, A. Ray, Y. T. Shao, S. Jiang, K. Lee, D. Weber, J. E. Goldberger, K. Watanabe, T. Taniguchi, D. A. Muller, K. F. Mak, and J. Shan, Coexisting ferromagnetic —antiferromagnetic state in twisted bilayer CrI3, Nat. Nanotechnol. 17(2), 143 (2022)

    Article  ADS  Google Scholar 

  92. C. Wang, X. Zhou, L. Zhou, Y. Pan, Z. Y. Lu, X. G. Wan, X. Q. Wang, and W. Ji, Bethe—Slater-curve-like behavior and interlayer spin-exchange coupling mechanisms in two-dimensional magnetic bilayers, Phys. Rev. B 102, 020402(R) (2020)

    Article  ADS  Google Scholar 

  93. L. Wu, L. Zhou, X. Zhou, C. Wang, and W. Ji, In-plane epitaxy-strain-tuning intralayer and interlayer magnetic coupling in CrSe2 and CrTe2 monolayers and bilayers, Phys. Rev. B 106(8), L081401 (2022)

    Article  ADS  Google Scholar 

  94. J. C. Slonczewski, Fluctuation mechanism for biquadratic exchange coupling in magnetic multilayers, Phys. Rev. Lett. 67(22), 3172 (1991)

    Article  ADS  Google Scholar 

  95. N. S. Fedorova, C. Ederer, N. A. Spaldin, and A. Scaramucci, Biquadratic and ring exchange interactions in orthorhombic perovskite manganites, Phys. Rev. B 91(16), 165122 (2015)

    Article  ADS  Google Scholar 

  96. A. Kartsev, M. Augustin, R. F. L. Evans, K. S. Novoselov, and E. J. G. Santos, Biquadratic exchange interactions in two-dimensional magnets, npj Comput. Mater. 6, 150 (2020)

    Article  ADS  Google Scholar 

  97. L. Chen, J. H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Topological spin excitations in honeycomb ferromagnet CrI3, Phys. Rev. X 8(4), 041028 (2018)

    Google Scholar 

  98. L. Chen, J. H. Chung, M. B. Stone, A. I. Kolesnikov, B. Winn, V. O. Garlea, D. L. Abernathy, B. Gao, M. Augustin, E. J. G. Santos, and P. Dai, Magnetic field effect on topological spin excitations in CrI3, Phys. Rev. X 11(3), 031047 (2021)

    Google Scholar 

  99. D. A. Wahab, M. Augustin, S. M. Valero, W. Kuang, S. Jenkins, E. Coronado, I. V. Grigorieva, I. J. Vera-Marun, E. Navarro-Moratalla, R. F. L. Evans, K. S. Novoselov, and E. J. G. Santos, Quantum rescaling, domain metastability, and hybrid domain-walls in 2D CrI3 magnets, Adv. Mater. 33(5), 2004138 (2021)

    Article  Google Scholar 

  100. P. A. Lindgard, R. J. Birgeneau, J. Als-Nielsen, and H. J. Guggenheim, Spin-wave dispersion and sublattice magnetization in NiCl2, J. Phys. Chem. 8, 1059 (1975)

    Google Scholar 

  101. Z. Jiang, Y. Li, W. Duan, and S. Zhang, Half-excitonic insulator: A single-spin Bose—Einstein condensate, Phys. Rev. Lett. 122(23), 236402 (2019)

    Article  ADS  Google Scholar 

  102. S. Paul, S. Haldar, S. von Malottki, and S. Heinze, Role of higher-order exchange interactions for skyrmion stability, Nat. Commun. 11(1), 4756 (2020)

    Article  ADS  Google Scholar 

  103. B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu, Y. Yao, and W. Wang, Observation of magnetic skyrmion bubbles in a van der Waals ferromagnet Fe3GeTe2, Nano Lett. 20(2), 868 (2020)

    Article  ADS  Google Scholar 

  104. M. T. Birch, L. Powalla, S. Wintz, O. Hovorka, K. Litzius, J. C. Loudon, L. A. Turnbull, V. Nehruji, K. Son, C. Bubeck, T. G. Rauch, M. Weigand, E. Goering, M. Burghard, and G. Schütz, History-dependent domain and skyrmion formation in 2D van der Waals magnet Fe3GeTe2, Nat. Commun. 13(1), 3035 (2022)

    Article  ADS  Google Scholar 

  105. C. Xu, X. Li, P. Chen, Y. Zhang, H. Xiang, and L. Bellaiche, Assembling diverse skyrmionic phases in Fe3GeTe2 monolayers, Adv. Mater. 34(12), 2107779 (2022)

    Article  Google Scholar 

  106. X. Y. Li, F. Lou, X. G. Gong, and H. Xiang, Constructing realistic effective spin Hamiltonians with machine learning approaches, New J. Phys. 22(5), 053036 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  107. H. Yu, C. Xu, X. Li, F. Lou, L. Bellaiche, Z. Hu, X. Gong, and H. Xiang, Complex spin Hamiltonian represented by an artificial neural network, Phys. Rev. B 105(17), 174422 (2022)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  109. T. Moriya, New mechanism of anisotropic superexchange interaction, Phys. Rev. Lett. 4(5), 228 (1960)

    Article  ADS  Google Scholar 

  110. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120(1), 91 (1960)

    Article  ADS  Google Scholar 

  111. 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)

    Article  ADS  Google Scholar 

  112. P. M. Levy and A. Fert, Anisotropy induced by nonmagnetic impurities in Cu Mn spin-glass alloys, Phys. Rev. B 23(9), 4667 (1981)

    Article  ADS  Google Scholar 

  113. A. Kundu and S. Zhang, Dzyaloshinskii—Moriya interaction mediated by spin-polarized band with Rashba spin—orbit coupling, Phys. Rev. B 92(9), 094434 (2015)

    Article  ADS  Google Scholar 

  114. A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions: Advances in physics and potential applications, Nat. Rev. Mater. 2(7), 17031 (2017)

    Article  ADS  Google Scholar 

  115. 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)

    Article  ADS  Google Scholar 

  116. X. 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)

    Article  ADS  Google Scholar 

  117. X. 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)

    Article  ADS  Google Scholar 

  118. C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J. M. George, M. Weigand, J. Raabe, V. Cros, and A. Fert, Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature, Nat. Nanotechnol. 11(5), 444 (2016)

    Article  ADS  Google Scholar 

  119. A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce, A. K. C. Tan, M. Y. Im, A. P. Petrović, P. Ho, K. H. Khoo, M. Tran, C. K. Gan, F. Ernult, and C. Panagopoulos, Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers, Nat. Mater. 16(9), 898 (2017)

    Article  ADS  Google Scholar 

  120. O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza Chaves, A. Locatelli, T. O. Menteş, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigné, A. Stashkevich, S. M. Chérif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures, Nat. Nanotechnol. 11(5), 449 (2016)

    Article  ADS  Google Scholar 

  121. H. Yang, O. Boulle, V. Cros, A. Fert, and M. Chshiev, Controlling Dzyaloshinskii—Moriya interaction via chirality dependent atomic-layer stacking, insulator capping and electric field, Sci. Rep. 8(1), 12356 (2018)

    Article  ADS  Google Scholar 

  122. A. Kundu and S. Zhang, Dzyaloshinskii—Moriya interaction mediated by spin-polarized band with Rashba spin-orbit coupling, Phys. Rev. B 92(9), 094434 (2015)

    Article  ADS  Google Scholar 

  123. A. A. Ado, A. Qaiumzadeh, R. A. Duine, A. Brataas, and M. Titov, Asymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet, Phys. Rev. Lett. 121(8), 086802 (2018)

    Article  ADS  Google Scholar 

  124. H. Yang, G. Chen, A. A. C. Cotta, A. T. N’Diaye, S. A. Nikolaev, E. A. Soares, W. A. A. Macedo, K. Liu, A. K. Schmid, A. Fert, and M. Chshiev, Significant Dzyaloshinskii—Moriya interaction at graphene-ferromagnet interfaces due to the Rashba effect, Nat. Mater. 17(7), 605 (2018)

    Article  ADS  Google Scholar 

  125. A. Hallal, J. Liang, F. Ibrahim, H. Yang, A. Fert, and M. Chshiev, Rashba-type Dzyaloshinskii—Moriya interaction, perpendicular magnetic anisotropy, and skyrmion states at 2D materials/Co interfaces, Nano Lett. 21(17), 7138 (2021)

    Article  ADS  Google Scholar 

  126. J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev, A. Fert, and H. Yang, Very large Dzyaloshinskii-Moriya interaction in two-dimensional Janus manganese dichalcogenides and its application to realize skyrmion states, Phys. Rev. B 101(18), 184401 (2020)

    Article  ADS  Google Scholar 

  127. S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic scale magnetic skyrmion lattice in two dimensions, Nat. Phys. 7(9), 713 (2011)

    Article  Google Scholar 

  128. A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Blügel, and A. Manchon, Hund’s rule-driven Dzyaloshinskii—Moriya interaction at 3d—5d interfaces, Phys. Rev. Lett. 117(24), 247202 (2016)

    Article  ADS  Google Scholar 

  129. C. Xu, J. Feng, S. Prokhorenko, Y. Nahas, H. Xiang, and L. Bellaiche, Topological spin texture in Janus monolayers of the chromium trihalides Cr(I,X)3, Phys. Rev. B 101(6), 060404 (2020)

    Article  ADS  Google Scholar 

  130. Y. Zhang, C. Xu, P. Chen, et al., Emergence of skyrmionium in a two-dimensional CrGe(Se,Te)3 Janus monolayer, Phys. Rev. B 102, 241107(R) (2020)

    Article  ADS  Google Scholar 

  131. S. Laref, V. Goli, I. Smaili, et al., Topologically stable bimerons and skyrmions in vanadium dichalcogenide Janus monolayers, arXiv: 2011.07813 (2011)

  132. J. Jiang, X. Liu, R. Li, and W. Mi, Topological spin textures in a two-dimensional MnBi2(Se,Te)4 Janus material, Appl. Phys. Lett. 119(7), 072401 (2021)

    Article  ADS  Google Scholar 

  133. Q. Cui, Y. Zhu, J. Jiang, J. Liang, D. Yu, P. Cui, and H. Yang, Ferroelectrically controlled topological magnetic phase in a Janus-magnet-based multiferroic heterostructure, Phys. Rev. Res. 3(4), 043011 (2021)

    Article  Google Scholar 

  134. W. Du, K. Dou, Z. He, Y. Dai, B. Huang, and Y. Ma, Spontaneous magnetic skyrmions in single-layer CrInX3 (X = Te, Se), Nano Lett. 22(8), 3440 (2022)

    Article  ADS  Google Scholar 

  135. P. Li, Q. Cui, Y. Ga, J. Liang, and H. Yang, Large Dzyaloshinskii—Moriya interaction and field-free topological chiral spin states in two-dimensional alkali-based chromium chalcogenides, Phys. Rev. B 106(2), 024419 (2022)

    Article  ADS  Google Scholar 

  136. F. Zhang, W. Mi, and X. Wang, Spin-dependent electronic structure and magnetic anisotropy of 2D ferromagnetic Janus Cr2I3X3 (X = Br, Cl) monolayers, Adv. Electron. Mater. 1900778 (2019)

  137. F. Zhang, H. Zhang, W. Mi, and X. Wang, Electronic structure, magnetic anisotropy and Dzyaloshinskii—Moriya interaction in Janus Cr2I3X3 (X = Br, Cl) bilayers, Phys. Chem. Chem. Phys. 22(16), 8647 (2020)

    Article  Google Scholar 

  138. R. Li, J. Jiang, X. Shi, W. Mi, and H. Bai, Two-dimensional Janus FeXY (X, Y = Cl, Br, and I, X ≠ Y) monolayers: Half-metallic ferromagnets with tunable magnetic properties under strain, ACS Appl. Mater. Interfaces 13(32), 38897 (2021)

    Article  Google Scholar 

  139. J. Jiang, R. Li, and W. Mi, Electrical control of topological spin textures in two-dimensional multiferroics, Nanoscale 13(48), 20609 (2021)

    Article  Google Scholar 

  140. Y. Xu, S. Qi, and W. Mi, Electronic structure and magnetic properties of two-dimensional h-BN/Janus 2H-VSeX (X = S, Te) van der Waals heterostructures, Appl. Surf. Sci. 537, 147898 (2021)

    Article  Google Scholar 

  141. S. Qi, J. Jiang, X. Wang, and W. Mi, Valley polarization, magnetic anisotropy and Dzyaloshinskii—Moriya interaction of two-dimensional graphene/Janus 2H-VSeX (X = S, Te) heterostructures, Carbon 174, 540 (2021)

    Article  Google Scholar 

  142. 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 \(P\bar 4m2\) crystal symmetry, Nano Lett. 22(6), 2334 (2022)

    Article  ADS  Google Scholar 

  143. Y. Ga, Q. Cui, Y. Zhu, D. Yu, L. Wang, J. Liang, and H. Yang, Anisotropic Dzyaloshinskii—Moriya interaction protected by D2d crystal symmetry in two-dimensional ternary compounds, npj Comput. Mater. 8, 128 (2022)

    Article  ADS  Google Scholar 

  144. F. Matsukura, Y. Tokura, and H. Ohno, Control of magnetism by electric fields, Nat. Nanotechnol. 10(3), 209 (2015)

    Article  ADS  Google Scholar 

  145. P. J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, and R. Wiesendanger, Electric-field-driven switching of individual magnetic skyrmions, Nat. Nanotechnol. 12(2), 123 (2017)

    Article  ADS  Google Scholar 

  146. C. Tang, L. Zhang, S. Sanvito, and A. Du, Electric-controlled half-metallicity in magnetic van der Waals heterobilayer, J. Mater. Chem. C 8(21), 7034 (2020)

    Article  Google Scholar 

  147. L. Zhang, C. Tang, S. Sanvito, Y. Gu, and A. Du, Hydrogen-intercalated 2D magnetic bilayer: Controlled magnetic phase transition and half-metallicity via ferroelectric switching, ACS Appl. Mater. Interfaces 14(1), 1800 (2022)

    Article  Google Scholar 

  148. C. Xu, P. Chen, H. Tan, Y. Yang, H. Xiang, and L. Bellaiche, Electric-field switching of magnetic topological charge in type-I multiferroics, Phys. Rev. Lett. 125(3), 037203 (2020)

    Article  ADS  Google Scholar 

  149. J. Liang, Q. Cui, and H. Yang, Electrically switchable Rashba-type Dzyaloshinskii—Moriya interaction and skyrmion in two-dimensional magnetoelectric multiferroics, Phys. Rev. B 102(22), 220409 (2020)

    Article  ADS  Google Scholar 

  150. Z. Shao, J. Liang, Q. Cui, M. Chshiev, A. Fert, T. Zhou, and H. Yang, Multiferroic materials based on transition-metal dichalcogenides: Potential platform for reversible control of Dzyaloshinskii—Moriya interaction and skyrmion via electric field, Phys. Rev. B 105(17), 174404 (2022)

    Article  ADS  Google Scholar 

  151. Y. Wu, S. Zhang, J. Zhang, W. Wang, Y. L. Zhu, J. Hu, G. Yin, K. Wong, C. Fang, C. Wan, X. Han, Q. Shao, T. Taniguchi, K. Watanabe, J. Zang, Z. Mao, X. Zhang, and K. L. Wang, Néel-type skyrmion in WTe2/Fe3GeTe2 van der Waals heterostructure, Nat. Commun. 11(1), 3860 (2020)

    Article  ADS  Google Scholar 

  152. T. E. Park, L. Peng, J. Liang, A. Hallal, F. S. Yasin, X. Zhang, K. M. Song, S. J. Kim, K. Kim, M. Weigand, G. Schütz, S. Finizio, J. Raabe, K. Garcia, J. Xia, Y. Zhou, M. Ezawa, X. Liu, J. Chang, H. C. Koo, Y. D. Kim, M. Chshiev, A. Fert, H. Yang, X. Yu, and S. Woo, Néel-type skyrmions and their current-induced motion in van der Waals ferromagnet-based heterostructures, Phys. Rev. B 103(10), 104410 (2021)

    Article  ADS  Google Scholar 

  153. Y. Wu, B. Francisco, W. Wang, et al., A van der Waals interface hosting two groups of magnetic skyrmions, a van der Waals interface hosting two groups of magnetic skyrmions, Adv. Mater. 34(16), 2110583 (2022)

    Article  Google Scholar 

  154. W. Sun, W. Wang, H. Li, G. Zhang, D. Chen, J. Wang, and Z. Cheng, Controlling bimerons as skyrmion analogues by ferroelectric polarization in 2D van der Waals multiferroic heterostructures, Nat. Commun. 11(1), 5930 (2020)

    Article  ADS  Google Scholar 

  155. C. K. Li, X. P. Yao, and G. Chen, Writing and deleting skyrmions with electric fields in a multiferroic heterostructure, Phys. Rev. Res. 3(1), L012026 (2021)

    Article  Google Scholar 

  156. K. Dou, W. Du, Y. Dai, B. Huang, and Y. Ma, Two-dimensional magnetoelectric multiferroics in a MnSTe/In2Se3 heterobilayer with ferroelectrically controllable skyrmions, Phys. Rev. B 105(20), 205427 (2022)

    Article  ADS  Google Scholar 

  157. W. Sun, W. Wang, J. Zang, H. Li, G. Zhang, J. Wang, and Z. Cheng, Manipulation of magnetic skyrmion in a 2D van der Waals heterostructure via both electric and magnetic fields, Adv. Funct. Mater. 31(47), 2104452 (2021)

    Article  Google Scholar 

  158. W. Sun, W. Wang, H. Li, X. Li, Z. Yu, Y. Bai, G Zhang, and Z. Cheng, LaBr2 bilayer multiferroic moiré superlattice with robust magnetoelectric coupling and magnetic bimerons, npj Comput. Mater. 8, 159 (2022)

    Article  ADS  Google Scholar 

  159. J. Chen and S. Dong, Manipulation of magnetic domain walls by ferroelectric switching: Dynamic magnetoelectricity at the nanoscale, Phys. Rev. Lett. 126, 117603 (2021)

    Article  ADS  Google Scholar 

  160. Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Observation of the magnon Hall effect, Science 329(5989), 297 (2010)

    Article  ADS  Google Scholar 

  161. R. Matsumoto and S. Murakami, Theoretical prediction of a rotating magnon wave packet in ferromagnets, Phys. Rev. Lett. 106(19), 197202 (2011)

    Article  ADS  Google Scholar 

  162. R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh, R. I. Bewley, D. G. Nocera, and Y. S. Lee, Topological magnon bands in a kagome lattice ferromagnet, Phys. Rev. Lett. 115(14), 147201 (2015)

    Article  ADS  Google Scholar 

  163. F. Zhu, L. Zhang, X. Wang, F. J. dos Santos, J. Song, T. Mueller, K. Schmalzl, W. F. Schmidt, A. Ivanov, J. T. Park, J. Xu, J. Ma, S. Lounis, S. Blügel, Y. Mokrousov, Y. Su, and T. Brückel, Topological magnon insulators in two-dimensional van der Waals ferromagnets CrSiTe3 and CrGeTe3: Toward intrinsic gap-tunability, Sci. Adv. 7(37), eabi7532 (2021)

    Article  ADS  Google Scholar 

  164. X. Yu, X. Zhang, Q. Shi, S. Tian, H. Lei, K. Xu, and H. Hosono, Large magnetocaloric effect in van der Waals crystal CrBr3, Front. Phys. 14(4), 43501 (2019)

    Article  ADS  Google Scholar 

  165. Q. Pei, X. C. Wang, J. J. Zou, and W. B. Mi, Tunable electronic structure and magnetic coupling in strained two-dimensional semiconductor MnPSe3, Front. Phys. 13(4), 137105 (2018)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1405102), the National Natural Science Foundation of China (Grant Nos. 11874059 and 12174405), the Key Research Program of Frontier Sciences, CAS (Grant No. ZDBS-LY-7021), Ningbo Key Scientific and Technological Project (Grant No. 2021000215), “Pioneer” and “Leading Goose” R&D Program of Zhejiang Province under Grant 2022C01053, Zhejiang Provincial Natural Science Foundation (Grant No. LR19A040002), and Beijing National Laboratory for Condensed Matter Physics (Grant No. 2021000123).

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Cui, Q., Wang, L., Zhu, Y. et al. Magnetic anisotropy, exchange coupling and Dzyaloshinskii—Moriya interaction of two-dimensional magnets. Front. Phys. 18, 13602 (2023). https://doi.org/10.1007/s11467-022-1217-7

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