1. Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China 2. School of Physics, Nanjing University, Nanjing 210093, China 3. Center of Materials Science and Optoeledctronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
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.
D. Mermin N., Wagner H.. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett., 1966, 17(22): 1133 https://doi.org/10.1103/PhysRevLett.17.1133
2
Gong C., Li L., Li Z., Ji H., Stern A., Xia Y., Cao T., Bao W., Wang C., Wang Y., Q. Qiu Z., J. Cava R., G. Louie S., Xia J., Zhang X.. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature, 2017, 546(7657): 265 https://doi.org/10.1038/nature22060
3
Huang B., Clark G., Navarro-Moratalla E., R. Klein D., Cheng R., L. Seyler K., Zhong D., Schmidgall E., A. McGuire M., H. Cobden D., Yao W., Xiao D., Jarillo-Herrero P., Xu X.. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature, 2017, 546(7657): 270 https://doi.org/10.1038/nature22391
4
Bonilla M., Kolekar S., Ma Y., C. Diaz H., Kalappattil V., Das R., Eggers T., R. Gutierrez H., H. Phan M., Batzill M.. Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substrates. Nat. Nanotechnol., 2018, 13(4): 289 https://doi.org/10.1038/s41565-018-0063-9
5
J. O’Hara D., Zhu T., H. Trout A., S. Ahmed A., K. Luo Y., H. Lee C., R. Brenner M., Rajan S., A. Gupta J., W. McComb D., K. Kawakami R.. Room temperature intrinsic ferromagnetism in epitaxial manganese selenide films in the monolayer limit. Nano Lett., 2018, 18(5): 3125 https://doi.org/10.1021/acs.nanolett.8b00683
6
Deng Y., Yu Y., Song Y., Zhang J., Z. Wang N., Sun Z., Yi Y., Z. Wu Y., Wu S., Zhu J., Wang J., H. Chen X., Zhang Y.. Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2. Nature, 2018, 563(7729): 94 https://doi.org/10.1038/s41586-018-0626-9
7
Lee J., Lee S., H. Ryoo J., Kang S., Y. Kim T., Kim P., H. Park C., G. Park J., Cheong H.. Ising-type magnetic ordering in atomically thin FePS3. Nano Lett., 2016, 16(12): 7433 https://doi.org/10.1021/acs.nanolett.6b03052
8
Long G., Henck H., Gibertini M., Dumcenco D., Wang Z., Taniguchi T., Watanabe K., Giannini E., F. Morpurgo A.. Persistence of magnetism in atomically thin MnPS3 crystals. Nano Lett., 2020, 20(4): 2452 https://doi.org/10.1021/acs.nanolett.9b05165
9
Bedoya-Pinto A., R. Ji J., K. Pandeya A., Gargiani P., Valvidares M., Sessi P., M. Taylor J., Radu F., Chang K., S. P. Parkin S.. Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer. Science, 2021, 374(6567): 616 https://doi.org/10.1126/science.abd5146
10
Song T., Cai X., W. Tu M., Zhang X., Huang B., P. Wilson N., L. Seyler K., Zhu L., Taniguchi T., Watanabe K., A. McGuire M., H. Cobden D., Xiao D., Yao W., Xu X.. Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures. Science, 2018, 360(6394): 1214 https://doi.org/10.1126/science.aar4851
11
Wang X., Tang J., Xia X., He C., Zhang J., Liu Y., Wan C., Fang C., Guo C., Yang W., Guang Y., Zhang X., Xu H., Wei J., Liao M., Lu X., Feng J., Li X., Peng Y., Wei H., Yang R., Shi D., Zhang X., Han Z., Zhang Z., Zhang G., Yu G., Han X.. Current-driven magnetization switching in a van der Waals ferromagnet Fe3GeTe2. Sci. Adv., 2019, 5(8): eaaw8904 https://doi.org/10.1126/sciadv.aaw8904
12
Fu H., Liu C., Yan B.. Exchange bias and quantum anomalous Hall effect in the MnBi2Te4/CrI3 heterostructure. Sci. Adv., 2020, 6(10): eaaz0948 https://doi.org/10.1126/sciadv.aaz0948
13
Li Y., Li J., Li Y., Ye M., Zheng F., Zhang Z., Fu J., Duan W., Xu Y.. High-temperature quantum anomalous Hall insulators in lithium-decorated iron-based superconductor materials. Phys. Rev. Lett., 2020, 125(8): 086401 https://doi.org/10.1103/PhysRevLett.125.086401
14
Zhong D., L. Seyler K., Linpeng X., Cheng R., Sivadas N., Huang B., Schmidgall E., Taniguchi T., Watanabe K., A. McGuire M., Yao W., Xiao D., M. C. Fu K., Xu X.. Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics. Sci. Adv., 2017, 3(5): e1603113 https://doi.org/10.1126/sciadv.1603113
15
Zollner K., E. Faria Junior P., Fabian J.. Proximity exchange effects in MoSe2 and WSe2 heterostructures with CrI3: Twist angle, layer, and gate dependence. Phys. Rev. B, 2019, 100(8): 085128 https://doi.org/10.1103/PhysRevB.100.085128
16
Ai L., Zhang E., Yang J., Xie X., Yang Y., Jia Z., Zhang Y., Liu S., Li Z., Leng P., Cao X., Sun X., Zhang T., Kou X., Han Z., Xiu F., Dong S.. Van der Waals ferromagnetic Josephson junctions. Nat. Commun., 2021, 12(1): 6580 https://doi.org/10.1038/s41467-021-26946-w
17
Zhao W., Fei Z., Song T., K. Choi H., Palomaki T., Sun B., Malinowski P., A. McGuire M., H. Chu J., Xu X., H. Cobden D.. Magnetic proximity and nonreciprocal current switching in a monolayer WTe2 helical edge. Nat. Mater., 2020, 19(5): 503 https://doi.org/10.1038/s41563-020-0620-0
18
H. Wang Q., Bedoya-Pinto A., Blei M., H. Dismukes A., Hamo A., Jenkins S., Koperski M., Liu Y., C. Sun Q., J. Telford E., H. Kim H., Augustin M., Vool U., X. Yin J., H. Li L., Falin A., R. Dean C., Casanova F., F. L. Evans R., Chshiev M., Mishchenko A., Petrovic C., He R., Zhao L., W. Tsen A., D. Gerardot B., Brotons-Gisbert M., Guguchia Z., Roy X., Tongay S., Wang Z., Z. Hasan M., Wrachtrup J., Yacoby A., Fert A., Parkin S., S. Novoselov K., Dai P., Balicas L., J. G. Santos E.. The magnetic genome of two-dimensional van der Waals materials. ACS Nano, 2022, 16(5): 6960 https://doi.org/10.1021/acsnano.1c09150
19
Soriano D., I. Katsnelson M., Fernández-Rossier J.. Magnetic two-dimensional chromium trihalides: A theoretical perspective. Nano Lett., 2020, 20(9): 6225 https://doi.org/10.1021/acs.nanolett.0c02381
20
Nakano M., Wang Y., Yoshida S., Matsuoka H., Majima Y., Ikeda K., Hirata Y., Takeda Y., Wadati H., Kohama Y., Ohigashi Y., Sakano M., Ishizaka K., Iwasa Y.. Intrinsic 2D ferromagnetism in V5Se8 epitaxial thin films. Nano Lett., 2019, 19(12): 8806 https://doi.org/10.1021/acs.nanolett.9b03614
21
Tang C., Zhang L., Du A.. Tunable magnetic anisotropy in 2D magnets via molecular adsorption. J. Mater. Chem. C, 2020, 8(42): 14948 https://doi.org/10.1039/D0TC04049E
22
Tang C., Ostrikov K., Sanvito S., Du A.. Prediction of room-temperature ferromagnetism and large perpendicular magnetic anisotropy in a planar hypercoordinate FeB3 monolayer. Nanoscale Horiz., 2021, 6(1): 43 https://doi.org/10.1039/D0NH00598C
23
Alsubaie M., Tang C., Wijethunge D., Qi D., Du A.. First-principles study of the enhanced magnetic anisotropy and transition temperature in a CrSe2 monolayer via hydrogenation. ACS Appl. Electron. Mater., 2022, 4(7): 3240 https://doi.org/10.1021/acsaelm.2c00476
24
L. Berezinskii V.. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group (I): Classical systems. Sov. Phys. JETP, 1971, 32: 493
25
L. Berezinskii V.. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group (II): Quantum systems. Sov. Phys. JETP, 1972, 34: 610
26
M. Kosterlitz J., J. Thouless D.. Long range order and metastability in two dimensional solids and superfluids (application of dislocation theory). J. Phys. C, 1972, 5(11): L124 https://doi.org/10.1088/0022-3719/5/11/002
27
M. Kosterlitz J., J. Thouless D.. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C, 1973, 6(7): 1181 https://doi.org/10.1088/0022-3719/6/7/010
L. Zhuang H., R. C. Kent P., G. Hennig R.. Strong anisotropy and magnetostriction in the two-dimensional Stoner ferromagnet Fe3GeTe2. Phys. Rev. B, 2016, 93(13): 134407 https://doi.org/10.1103/PhysRevB.93.134407
Webster L., A. Yan J., Strain-tunable magnetic anisotropy in monolayer CrCl3. CrBr3, and CrI3. Phys. Rev. B, 2018, 98(14): 144411 https://doi.org/10.1103/PhysRevB.98.144411
32
Yang B., Zhang X., Yang H., Han X., Yan Y.. Nonmetallic atoms induced magnetic anisotropy in monolayer chromium trihalides. J. Phys. Chem. C, 2019, 123(1): 691 https://doi.org/10.1021/acs.jpcc.8b09939
33
Yang B., Zhang X., Yang H., Han X., Yan Y.. Strain controlling transport properties of heterostructure composed of monolayer CrI3. Appl. Phys. Lett., 2019, 114(19): 192405 https://doi.org/10.1063/1.5091958
Xue F., Hou Y., Wang Z., Wu R.. Two-dimensional ferromagnetic van der Waals CrCl3 monolayer with enhanced anisotropy and Curie temperature. Phys. Rev. B, 2019, 100(22): 224429 https://doi.org/10.1103/PhysRevB.100.224429
36
Li Y., Jiang Z., Li J., Xu S., Duan W.. Magnetic anisotropy of the two-dimensional ferromagnetic insulator MnBi2Te4. Phys. Rev. B, 2019, 100(13): 134438 https://doi.org/10.1103/PhysRevB.100.134438
37
Xu C., Feng J., Xiang H., Bellaiche L.. Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers. npj Comput. Mater., 2018, 4: 57 https://doi.org/10.1038/s41524-018-0115-6
38
Cui Q., Liang J., Yang B., Wang Z., Li P., Cui P., Yang H.. 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, 2020, 101(21): 214439 https://doi.org/10.1103/PhysRevB.101.214439
39
Gong C., Zhang X.. Two-dimensional magnetic crystals and emergent heterostructure devices. Science, 2019, 363(6428): eaav4450 https://doi.org/10.1126/science.aav4450
40
Y. Park S., S. Kim D., Liu Y., Hwang J., Kim Y., Kim W., Y. Kim J., Petrovic C., Hwang C., K. Mo S., Kim H., C. Min B., C. Koo H., Chang J., Jang C., W. Choi J., Ryu H.. Controlling the magnetic anisotropy of the van der Waals ferromagnet Fe3GeTe2 through hole doping. Nano Lett., 2020, 20(1): 95 https://doi.org/10.1021/acs.nanolett.9b03316
41
P. Wang Y., Y. Chen X., Q. Long M.. Modifications of magnetic anisotropy of Fe3GeTe2 by the electric field effect. Appl. Phys. Lett., 2020, 116(9): 092404 https://doi.org/10.1063/1.5144032
42
G. Ye X., F. Zhu P., Z. Xu W., Z. Shang N., H. Liu K., M. Liao Z.. Orbit-transfer torque driven field-free switching of perpendicular magnetization chin. Phys. Lett., 2022, 39: 037303 https://doi.org/10.1088/0256-307X/39/3/037303
43
Seo J., S. An E., Park T., Y. Hwang S., Y. Kim G., Song K., Noh W., Y. Kim J., S. Choi G., Choi M., Oh E., Watanabe K., Taniguchi T., H. Park J., J. Jo Y., W. Yeom H., Y. Choi S., H. Shim J., S. Kim J.. Tunable high-temperature itinerant antiferromagnetism in a van der Waals magnet. Nat. Commun., 2021, 12(1): 2844 https://doi.org/10.1038/s41467-021-23122-y
44
Zhang X., Lu Q., Liu W., Niu W., Sun J., Cook J., Vaninger M., F. Miceli P., J. Singh D., W. Lian S., R. Chang T., He X., Du J., He L., Zhang R., Bian G., Xu Y.. Room-temperature intrinsic ferromagnetism in epitaxial CrTe2 ultrathin films. Nat. Commun., 2021, 12(1): 2492 https://doi.org/10.1038/s41467-021-22777-x
45
Wang Y., E. Ziebel M., Sun L., T. Gish J., J. Pearson T., Z. Lu X., E. Thorarinsdottir A., C. Hersam M., R. Long J., E. Freedman D., M. Rondinelli J., Puggioni D., D. Harris T.. Strong magnetocrystalline anisotropy arising from metal–ligand covalency in a metal–organic candidate for 2D magnetic order. Chem. Mater., 2021, 33(22): 8712 https://doi.org/10.1021/acs.chemmater.1c02670
Xu C., Feng J., Kawamura M., Yamaji Y., Nahas Y., Prokhorenko S., Qi Y., Xiang H., Bellaiche L.. Possible Kitaev quantum spin liquid state in 2D materials with S = 3/2. Phys. Rev. Lett., 2020, 124(8): 087205 https://doi.org/10.1103/PhysRevLett.124.087205
49
J. Sandilands L., Tian Y., W. Plumb K., J. Kim Y., S. Burch K.. Scattering continuum and possible fractionalized excitations in α-RuCl3. Phys. Rev. Lett., 2015, 114(14): 147201 https://doi.org/10.1103/PhysRevLett.114.147201
50
Kim H.-S., Vijay Shankar V., Catuneanu A., Y. Kee H.. Kitaev magnetism in honeycomb RuCl3 with intermediate spin−orbit coupling. Phys. Rev. B, 2015, 91: 241110(R) https://doi.org/10.1103/PhysRevB.91.241110
51
Banerjee A., A. Bridges C., Q. Yan J., A. Aczel A., Li L., B. Stone M., E. Granroth G., D. Lumsden M., Yiu Y., Knolle J., Bhattacharjee S., L. Kovrizhin D., Moessner R., A. Tennant D., G. Mandrus D., E. Nagler S.. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater., 2016, 15(7): 733 https://doi.org/10.1038/nmat4604
52
M. Winter S., Li Y., O. Jeschke H., Valentí R.. Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales. Phys. Rev. B, 2016, 93(21): 214431 https://doi.org/10.1103/PhysRevB.93.214431
Hermanns M., Kimchi I., Knolle J.. Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections. Annu. Rev. Condens. Matter Phys., 2018, 9(1): 17 https://doi.org/10.1146/annurev-conmatphys-033117-053934
55
Banerjee A., Lampen-Kelley P., Knolle J., Balz C., A. Aczel A., Winn B., Liu Y., Pajerowski D., Yan J., A. Bridges C., T. Savici A., C. Chakoumakos B., D. Lumsden M., A. Tennant D., Moessner R., G. Mandrus D., E. Nagler S.. Excitations in the field-induced quantum spin liquid state of α-RuCl3. npj Quant. Mater., 2018, 3: 8 https://doi.org/10.1038/s41535-018-0079-2
56
Kasahara Y., Sugii K., Ohnishi T., Shimozawa M., Yamashita M., Kurita N., Tanaka H., Nasu J., Motome Y., Shibauchi T., Matsuda Y.. Unusual thermal Hall effect in a Kitaev spin liquid candidate α-RuCl3. Phys. Rev. Lett., 2018, 120(21): 217205 https://doi.org/10.1103/PhysRevLett.120.217205
57
Takagi H., Takayama T., Jackeli G., Khaliullin G., E. Nagler S.. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys., 2019, 1(4): 264 https://doi.org/10.1038/s42254-019-0038-2
58
A. Sears J., E. Chern L., Kim S., J. Bereciartua P., Francoual S., B. Kim Y., J. Kim Y.. Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl3. Nat. Phys., 2020, 16(8): 837 https://doi.org/10.1038/s41567-020-0874-0
J. Xiang H., J. Kan E., H. Wei S., H. Whangbo M., G. Gong X.. Predicting the spin-lattice order of frustrated systems from first principles. Phys. Rev. B, 2011, 84(22): 224429 https://doi.org/10.1103/PhysRevB.84.224429
61
Goodenough B.. Theory of the role of covalence in the perovskite-type manganites [La,M(II)] MnO3. Phys. Rev., 1955, 100(2): 564 https://doi.org/10.1103/PhysRev.100.564
Huang C., Feng J., Wu F., Ahmed D., Huang B., Xiang H., Deng K., Kan E.. Toward intrinsic room-temperature ferromagnetism in two-dimensional semiconductors. J. Am. Chem. Soc., 2018, 140(36): 11519 https://doi.org/10.1021/jacs.8b07879
65
Sivadas N., W. Daniels M., H. Swendsen R., Okamoto S., Xiao D.. Magnetic ground state of semiconducting transition-metal trichalcogenide monolayers. Phys. Rev. B, 2015, 91(23): 235425 https://doi.org/10.1103/PhysRevB.91.235425
66
Pei Q., Mi W.. Electrical control of magnetic behavior and valley polarization of monolayer antiferromagnetic MnPSe3 on an insulating ferroelectric substrate from first principle. Phys. Rev. Appl., 2019, 11(1): 014011 https://doi.org/10.1103/PhysRevApplied.11.014011
67
Olsen T., Magnetic anisotropy, exchange interactions of two-dimensional FePS3. NiPS3 and MnPS3 from first principles calculations. J. Phys. D, 2021, 54(31): 314001 https://doi.org/10.1088/1361-6463/ac000e
68
Li J., Y. Ni J., Y. Li X., J. Koo H., H. Whangbo M., S. Feng J., J. Xiang H.. Intralayer ferromagnetism between S = 5/2 ions in MnBi2Te4: Role of empty Bi p states. Phys. Rev. B, 2020, 101(20): 201408 https://doi.org/10.1103/PhysRevB.101.201408
69
Huang C., Feng J., Zhou J., Xiang H., Deng K., Kan E.. Ultra-high-temperature ferromagnetism in intrinsic tetrahedral semiconductors. J. Am. Chem. Soc., 2019, 141(31): 12413 https://doi.org/10.1021/jacs.9b06452
70
Cui Q., Zhu Y., Ga Y., Liang J., Li P., Yu D., Cui P., Yang H.. Anisotropic Dzyaloshinskii–Moriya interaction and topological magnetism in two-dimensional magnets protected by P4¯m2 crystal symmetry. Nano Lett., 2022, 22(6): 2334 https://doi.org/10.1021/acs.nanolett.1c04803
71
J. Zhang J., Lin L., Zhang Y., Wu M., I. Yakobson B., Dong S.. Type-II multiferroic Hf2VC2F2 MXene monolayer with high transition temperature. J. Am. Chem. Soc., 2018, 140(30): 9768 https://doi.org/10.1021/jacs.8b06475
72
Amoroso D., Barone P., Picozzi S.. Spontaneous skyrmionic lattice from anisotropic symmetric exchange in a Ni-halide monolayer. Nat. Commun., 2020, 11(1): 5784 https://doi.org/10.1038/s41467-020-19535-w
73
Y. Ni J., Y. Li X., Amoroso D., He X., S. Feng J., J. Kan E., Picozzi S., J. Xiang H.. Giant biquadratic exchange in 2D magnets and its role in stabilizing ferromagnetism of NiCl2 monolayers. Phys. Rev. Lett., 2021, 127(24): 247204 https://doi.org/10.1103/PhysRevLett.127.247204
74
Katsura H., Nagaosa N., V. Balatsky A.. Spin current and magnetoelectric effect in noncollinear magnets. Phys. Rev. Lett., 2005, 95(5): 057205 https://doi.org/10.1103/PhysRevLett.95.057205
Song Q., A. Occhialini C., Ergeçen E., Ilyas B., Amoroso D., Barone P., Kapeghian J., Watanabe K., Taniguchi T., S. Botana A., Picozzi S., Gedik N., Comin R.. Evidence for a single-layer van der Waals multiferroic. Nature, 2022, 602(7898): 601 https://doi.org/10.1038/s41586-021-04337-x
78
R. Klein D., MacNeill D., L. Lado J., Soriano D., Navarro-Moratalla E., Watanabe K., Taniguchi T., Manni S., Canfield P., Fernández-Rossier J., Jarillo-Herrero P.. Probing magnetism in 2D van der Waals crystalline insulators via electron tunneling. Science, 2018, 360(6394): 1218 https://doi.org/10.1126/science.aar3617
79
Wang Z., Gutiérrez-Lezama I., Ubrig N., Kroner M., Gibertini M., Taniguchi T., Watanabe K., Imamoğlu A., Giannini E., F. Morpurgo A.. Very large tunneling magnetoresistance in layered magnetic semiconductor CrI3. Nat. Commun., 2018, 9(1): 2516 https://doi.org/10.1038/s41467-018-04953-8
80
H. Kim H., Yang B., Patel T., Sfigakis F., Li C., Tian S., Lei H., W. Tsen A.. One million percent tunnel magnetoresistance in a magnetic van der Waals heterostructure. Nano Lett., 2018, 18(8): 4885 https://doi.org/10.1021/acs.nanolett.8b01552
81
Jiang S., Li L., Wang Z., F. Mak K., Shan J.. Controlling magnetism in 2D CrI3 by electrostatic doping. Nat. Nanotechnol., 2018, 13(7): 549 https://doi.org/10.1038/s41565-018-0135-x
82
Jiang S., Shan J., F. Mak K.. Electric-field switching of two-dimensional van der Waals magnets. Nat. Mater., 2018, 17(5): 406 https://doi.org/10.1038/s41563-018-0040-6
83
Huang B., Clark G., R. Klein D., MacNeill D., Navarro-Moratalla E., L. Seyler K., Wilson N., A. McGuire M., H. Cobden D., Xiao D., Yao W., Jarillo-Herrero P., Xu X.. Electrical control of 2D magnetism in bilayer CrI3. Nat. Nanotechnol., 2018, 13(7): 544 https://doi.org/10.1038/s41565-018-0121-3
84
H. Kim H., Yang B., Li S., Jiang S., Jin C., Tao Z., Nichols G., Sfigakis F., Zhong S., Li C., Tian S., G. Cory D., X. Miao G., Shan J., F. Mak K., Lei H., Sun K., Zhao L., W. Tsen A.. Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides. Proc. Natl. Acad. Sci. USA, 2019, 116(23): 11131 https://doi.org/10.1073/pnas.1902100116
85
Xu R., Zhou X.. Electric field-modulated magnetic phase transition in van der Waals CrI3 bilayers. J. Phys. Chem. Lett., 2020, 11(8): 3152 https://doi.org/10.1021/acs.jpclett.0c00567
Jiang P., Wang C., Chen D., Zhong Z., Yuan Z., Y. Lu Z., Ji W.. Stacking tunable interlayer magnetism in bilayer CrI3. Phys. Rev. B, 2019, 99(14): 144401 https://doi.org/10.1103/PhysRevB.99.144401
88
Chen W., Sun Z., Wang Z., Gu L., Xu X., Wu S., Gao C.. Direct observation of van der Waals stacking-dependent interlayer magnetism. Science, 2019, 366(6468): 983 https://doi.org/10.1126/science.aav1937
89
Xiao J., Yan B.. An electron-counting rule to determine the interlayer magnetic coupling of the van der Waals materials. 2D Mater., 2020, 7: 045010 https://doi.org/10.1088/2053-1583/ab9ea4
90
Li T., Jiang S., Sivadas N., Wang Z., Xu Y., Weber D., E. Goldberger J., Watanabe K., Taniguchi T., J. Fennie C., Fai Mak K., Shan J.. Pressure-controlled interlayer magnetism in atomically thin CrI3. Nat. Mater., 2019, 18(12): 1303 https://doi.org/10.1038/s41563-019-0506-1
91
Xu Y., Ray A., T. Shao Y., Jiang S., Lee K., Weber D., E. Goldberger J., Watanabe K., Taniguchi T., A. Muller D., F. Mak K., Shan J.. Coexisting ferromagnetic–antiferromagnetic state in twisted bilayer CrI3. Nat. Nanotechnol., 2022, 17(2): 143 https://doi.org/10.1038/s41565-021-01014-y
92
Wang C., Zhou X., Zhou L., Pan Y., Y. Lu Z., G. Wan X., Q. Wang X., Ji W.. Bethe−Slater-curve-like behavior and interlayer spin-exchange coupling mechanisms in two-dimensional magnetic bilayers. Phys. Rev. B, 2020, 102: 020402(R) https://doi.org/10.1103/PhysRevB.102.020402
93
Wu L., Zhou L., Zhou X., Wang C., Ji W.. In-plane epitaxy-strain-tuning intralayer and interlayer magnetic coupling in CrSe2 and CrTe2 monolayers and bilayers. Phys. Rev. B, 2022, 106(8): L081401 https://doi.org/10.1103/PhysRevB.106.L081401
94
C. Slonczewski J.. Fluctuation mechanism for biquadratic exchange coupling in magnetic multilayers. Phys. Rev. Lett., 1991, 67(22): 3172 https://doi.org/10.1103/PhysRevLett.67.3172
95
S. Fedorova N., Ederer C., A. Spaldin N., Scaramucci A.. Biquadratic and ring exchange interactions in orthorhombic perovskite manganites. Phys. Rev. B, 2015, 91(16): 165122 https://doi.org/10.1103/PhysRevB.91.165122
96
Kartsev A., Augustin M., F. L. Evans R., S. Novoselov K., J. G. Santos E.. Biquadratic exchange interactions in two-dimensional magnets. npj Comput. Mater., 2020, 6: 150 https://doi.org/10.1038/s41524-020-00416-1
97
Chen L., H. Chung J., Gao B., Chen T., B. Stone M., I. Kolesnikov A., Huang Q., Dai P.. Topological spin excitations in honeycomb ferromagnet CrI3. Phys. Rev. X, 2018, 8(4): 041028 https://doi.org/10.1103/PhysRevX.8.041028
98
Chen L., H. Chung J., B. Stone M., I. Kolesnikov A., Winn B., O. Garlea V., L. Abernathy D., Gao B., Augustin M., J. G. Santos E., Dai P.. Magnetic field effect on topological spin excitations in CrI3. Phys. Rev. X, 2021, 11(3): 031047 https://doi.org/10.1103/PhysRevX.11.031047
99
A. Wahab D., Augustin M., M. Valero S., Kuang W., Jenkins S., Coronado E., V. Grigorieva I., J. Vera-Marun I., Navarro-Moratalla E., F. L. Evans R., S. Novoselov K., J. G. Santos E.. Quantum rescaling, domain metastability, and hybrid domain‐walls in 2D CrI3 magnets. Adv. Mater., 2021, 33(5): 2004138 https://doi.org/10.1002/adma.202004138
100
A. Lindgard P., J. Birgeneau R., Als-Nielsen J., J. Guggenheim H.. Spin-wave dispersion and sublattice magnetization in NiCl2. J. Phys. Chem., 1975, 8: 1059
101
Jiang Z., Li Y., Duan W., Zhang S.. Half-excitonic insulator: A single-spin Bose−Einstein condensate. Phys. Rev. Lett., 2019, 122(23): 236402 https://doi.org/10.1103/PhysRevLett.122.236402
102
Paul S., Haldar S., von Malottki S., Heinze S.. Role of higher-order exchange interactions for skyrmion stability. Nat. Commun., 2020, 11(1): 4756 https://doi.org/10.1038/s41467-020-18473-x
103
Ding B., Li Z., Xu G., Li H., Hou Z., Liu E., Xi X., Xu F., Yao Y., Wang W.. Observation of magnetic skyrmion bubbles in a van der Waals ferromagnet Fe3GeTe2. Nano Lett., 2020, 20(2): 868 https://doi.org/10.1021/acs.nanolett.9b03453
104
T. Birch M., Powalla L., Wintz S., Hovorka O., Litzius K., C. Loudon J., A. Turnbull L., Nehruji V., Son K., Bubeck C., G. Rauch T., Weigand M., Goering E., Burghard M., Schütz G.. History-dependent domain and skyrmion formation in 2D van der Waals magnet Fe3GeTe2. Nat. Commun., 2022, 13(1): 3035 https://doi.org/10.1038/s41467-022-30740-7
105
Xu C., Li X., Chen P., Zhang Y., Xiang H., Bellaiche L.. Assembling diverse skyrmionic phases in Fe3GeTe2 monolayers. Adv. Mater., 2022, 34(12): 2107779 https://doi.org/10.1002/adma.202107779
106
Y. Li X., Lou F., G. Gong X., Xiang H.. Constructing realistic effective spin Hamiltonians with machine learning approaches. New J. Phys., 2020, 22(5): 053036 https://doi.org/10.1088/1367-2630/ab85df
107
Yu H., Xu C., Li X., Lou F., Bellaiche L., Hu Z., Gong X., Xiang H.. Complex spin Hamiltonian represented by an artificial neural network. Phys. Rev. B, 2022, 105(17): 174422 https://doi.org/10.1103/PhysRevB.105.174422
Fert A., M. Levy P.. Role of anisotropic exchange interactions in determining the properties of spin-glasses. Phys. Rev. Lett., 1980, 44(23): 1538 https://doi.org/10.1103/PhysRevLett.44.1538
112
M. Levy P., Fert A.. Anisotropy induced by nonmagnetic impurities in Cu Mn spin-glass alloys. Phys. Rev. B, 1981, 23(9): 4667 https://doi.org/10.1103/PhysRevB.23.4667
113
Kundu A., Zhang S.. Dzyaloshinskii−Moriya interaction mediated by spin-polarized band with Rashba spin−orbit coupling. Phys. Rev. B, 2015, 92(9): 094434 https://doi.org/10.1103/PhysRevB.92.094434
114
Fert A., Reyren N., Cros V.. Magnetic skyrmions: Advances in physics and potential applications. Nat. Rev. Mater., 2017, 2(7): 17031 https://doi.org/10.1038/natrevmats.2017.31
115
Mühlbauer S., Binz B., Jonietz F., Pfleiderer C., Rosch A., Neubauer A., Georgii R., Böni P.. Skyrmion lattice in a chiral magnet. Science, 2009, 323(5916): 915 https://doi.org/10.1126/science.1166767
116
Yu X., Onose Y., Kanazawa N., H. Park J., H. Han J., Matsui Y., Nagaosa N., Tokura Y.. Real-space observation of a two-dimensional skyrmion crystal. Nature, 2010, 465(7300): 901 https://doi.org/10.1038/nature09124
117
Yu X., Kanazawa N., Onose Y., Kimoto K., Z. Zhang W., Ishiwata S., Matsui Y., Tokura Y.. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater., 2011, 10(2): 106 https://doi.org/10.1038/nmat2916
118
Moreau-Luchaire C., Moutafis C., Reyren N., Sampaio J., A. F. Vaz C., Van Horne N., Bouzehouane K., Garcia K., Deranlot C., Warnicke P., Wohlhüter P., M. George J., Weigand M., Raabe J., Cros V., Fert A.. Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nanotechnol., 2016, 11(5): 444 https://doi.org/10.1038/nnano.2015.313
119
Soumyanarayanan A., Raju M., L. Gonzalez Oyarce A., K. C. Tan A., Y. Im M., P. Petrović A., Ho P., H. Khoo K., Tran M., K. Gan C., Ernult F., Panagopoulos C.. Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers. Nat. Mater., 2017, 16(9): 898 https://doi.org/10.1038/nmat4934
120
Boulle O., Vogel J., Yang H., Pizzini S., de Souza Chaves D., Locatelli A., O. Menteş T., Sala A., D. Buda-Prejbeanu L., Klein O., Belmeguenai M., Roussigné Y., Stashkevich A., M. Chérif S., Aballe L., Foerster M., Chshiev M., Auffret S., M. Miron I., Gaudin G.. Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures. Nat. Nanotechnol., 2016, 11(5): 449 https://doi.org/10.1038/nnano.2015.315
121
Yang H., Boulle O., Cros V., Fert A., Chshiev M.. Controlling Dzyaloshinskii−Moriya interaction via chirality dependent atomic-layer stacking, insulator capping and electric field. Sci. Rep., 2018, 8(1): 12356 https://doi.org/10.1038/s41598-018-30063-y
122
Kundu A., Zhang S.. Dzyaloshinskii−Moriya interaction mediated by spin-polarized band with Rashba spin-orbit coupling. Phys. Rev. B, 2015, 92(9): 094434 https://doi.org/10.1103/PhysRevB.92.094434
123
A. Ado A., Qaiumzadeh A., A. Duine R., Brataas A., Titov M.. Asymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet. Phys. Rev. Lett., 2018, 121(8): 086802 https://doi.org/10.1103/PhysRevLett.121.086802
124
Yang H., Chen G., A. C. Cotta A., T. N’Diaye A., A. Nikolaev S., A. Soares E., A. A. Macedo W., Liu K., K. Schmid A., Fert A., Chshiev M.. Significant Dzyaloshinskii−Moriya interaction at graphene-ferromagnet interfaces due to the Rashba effect. Nat. Mater., 2018, 17(7): 605 https://doi.org/10.1038/s41563-018-0079-4
125
Hallal A., Liang J., Ibrahim F., Yang H., Fert A., Chshiev M.. Rashba-type Dzyaloshinskii−Moriya interaction, perpendicular magnetic anisotropy, and skyrmion states at 2D materials/Co interfaces. Nano Lett., 2021, 21(17): 7138 https://doi.org/10.1021/acs.nanolett.1c01713
126
Liang J., Wang W., Du H., Hallal A., Garcia K., Chshiev M., Fert A., Yang H.. Very large Dzyaloshinskii-Moriya interaction in two-dimensional Janus manganese dichalcogenides and its application to realize skyrmion states. Phys. Rev. B, 2020, 101(18): 184401 https://doi.org/10.1103/PhysRevB.101.184401
127
Heinze S., von Bergmann K., Menzel M., Brede J., Kubetzka A., Wiesendanger R., Bihlmayer G., Blügel S.. Spontaneous atomic scale magnetic skyrmion lattice in two dimensions. Nat. Phys., 2011, 7(9): 713 https://doi.org/10.1038/nphys2045
Xu C., Feng J., Prokhorenko S., Nahas Y., Xiang H., Bellaiche L.. Topological spin texture in Janus monolayers of the chromium trihalides Cr(I,X)3. Phys. Rev. B, 2020, 101(6): 060404 https://doi.org/10.1103/PhysRevB.101.060404
130
Zhang Y., Xu C., Chen P.. et al.. Emergence of skyrmionium in a two-dimensional CrGe(Se,Te)3 Janus monolayer. Phys. Rev. B, 2020, 102: 241107(R) https://doi.org/10.1103/PhysRevB.102.241107
131
Laref S.Goli V.Smaili I., et al.., Topologically stable bimerons and skyrmions in vanadium dichalcogenide Janus monolayers, arXiv: 2011.07813 (2011)
132
Jiang J., Liu X., Li R., Mi W.. Topological spin textures in a two-dimensional MnBi2(Se,Te)4 Janus material. Appl. Phys. Lett., 2021, 119(7): 072401 https://doi.org/10.1063/5.0057794
133
Cui Q., Zhu Y., Jiang J., Liang J., Yu D., Cui P., Yang H.. Ferroelectrically controlled topological magnetic phase in a Janus-magnet-based multiferroic heterostructure. Phys. Rev. Res., 2021, 3(4): 043011 https://doi.org/10.1103/PhysRevResearch.3.043011
134
Du W., Dou K., He Z., Dai Y., Huang B., Ma Y., Spontaneous magnetic skyrmions in single-layer CrInX3 (X = Te. Se). Nano Lett., 2022, 22(8): 3440 https://doi.org/10.1021/acs.nanolett.2c00836
135
Li P., Cui Q., Ga Y., Liang J., Yang H.. Large Dzyaloshinskii−Moriya interaction and field-free topological chiral spin states in two-dimensional alkali-based chromium chalcogenides. Phys. Rev. B, 2022, 106(2): 024419 https://doi.org/10.1103/PhysRevB.106.024419
136
Zhang F., Mi W., Wang X., Spin-dependent electronic structure, magnetic anisotropy of 2D ferromagnetic Janus Cr2I3X3 (X = Br. Cl) monolayers. Adv. Electron. Mater., 2019, 1900778 https://doi.org/10.1002/aelm.201900778
137
Zhang F., Zhang H., Mi W., Wang X., Electronic structure, magnetic anisotropy, Dzyaloshinskii–Moriya interaction in Janus Cr2I3X3 (X = Br. Cl) bilayers. Phys. Chem. Chem. Phys., 2020, 22(16): 8647 https://doi.org/10.1039/D0CP00174K
138
Li R., Jiang J., Shi X., Mi W., Bai H., Janus FeXY (X Two-dimensional, = Cl Y. Br, and I, X ≠ Y) monolayers: Half-metallic ferromagnets with tunable magnetic properties under strain. ACS Appl. Mater. Interfaces, 2021, 13(32): 38897 https://doi.org/10.1021/acsami.1c10304
139
Jiang J., Li R., Mi W.. Electrical control of topological spin textures in two-dimensional multiferroics. Nanoscale, 2021, 13(48): 20609 https://doi.org/10.1039/D1NR06266B
140
Xu Y., Qi S., Mi W., structure Electronic, properties of two-dimensional h-BN/Janus 2H-VSeX (X = S magnetic. Te) van der Waals heterostructures. Appl. Surf. Sci., 2021, 537: 147898 https://doi.org/10.1016/j.apsusc.2020.147898
141
Qi S., Jiang J., Wang X., Mi W.. Valley polarization, magnetic anisotropy and Dzyaloshinskii−Moriya interaction of two-dimensional graphene/Janus 2H-VSeX (X = S, Te) heterostructures. Carbon, 2021, 174: 540 https://doi.org/10.1016/j.carbon.2020.12.072
142
Cui Q., Zhu Y., Ga Y., Liang J., Li P., Yu D., Cui P., Yang H.. Anisotropic Dzyaloshinskii–Moriya interaction and topological magnetism in two-dimensional magnets protected by P4¯m2 crystal symmetry. Nano Lett., 2022, 22(6): 2334 https://doi.org/10.1021/acs.nanolett.1c04803
143
Ga Y., Cui Q., Zhu Y., Yu D., Wang L., Liang J., Yang H.. Anisotropic Dzyaloshinskii−Moriya interaction protected by D2d crystal symmetry in two-dimensional ternary compounds. npj Comput. Mater., 2022, 8: 128 https://doi.org/10.1038/s41524-022-00809-4
144
Matsukura F., Tokura Y., Ohno H.. Control of magnetism by electric fields. Nat. Nanotechnol., 2015, 10(3): 209 https://doi.org/10.1038/nnano.2015.22
145
J. Hsu P., Kubetzka A., Finco A., Romming N., von Bergmann K., Wiesendanger R.. Electric-field-driven switching of individual magnetic skyrmions. Nat. Nanotechnol., 2017, 12(2): 123 https://doi.org/10.1038/nnano.2016.234
146
Tang C., Zhang L., Sanvito S., Du A.. Electric-controlled half-metallicity in magnetic van der Waals heterobilayer. J. Mater. Chem. C, 2020, 8(21): 7034 https://doi.org/10.1039/D0TC01541E
147
Zhang L., Tang C., Sanvito S., Gu Y., Du A.. Hydrogen-intercalated 2D magnetic bilayer: Controlled magnetic phase transition and half-metallicity via ferroelectric switching. ACS Appl. Mater. Interfaces, 2022, 14(1): 1800 https://doi.org/10.1021/acsami.1c21848
148
Xu C., Chen P., Tan H., Yang Y., Xiang H., Bellaiche L.. Electric-field switching of magnetic topological charge in type-I multiferroics. Phys. Rev. Lett., 2020, 125(3): 037203 https://doi.org/10.1103/PhysRevLett.125.037203
149
Liang J., Cui Q., Yang H.. Electrically switchable Rashba-type Dzyaloshinskii−Moriya interaction and skyrmion in two-dimensional magnetoelectric multiferroics. Phys. Rev. B, 2020, 102(22): 220409 https://doi.org/10.1103/PhysRevB.102.220409
150
Shao Z., Liang J., Cui Q., Chshiev M., Fert A., Zhou T., Yang H.. Multiferroic materials based on transition-metal dichalcogenides: Potential platform for reversible control of Dzyaloshinskii−Moriya interaction and skyrmion via electric field. Phys. Rev. B, 2022, 105(17): 174404 https://doi.org/10.1103/PhysRevB.105.174404
151
Wu Y., Zhang S., Zhang J., Wang W., L. Zhu Y., Hu J., Yin G., Wong K., Fang C., Wan C., Han X., Shao Q., Taniguchi T., Watanabe K., Zang J., Mao Z., Zhang X., L. Wang K.. Néel-type skyrmion in WTe2/Fe3GeTe2 van der Waals heterostructure. Nat. Commun., 2020, 11(1): 3860 https://doi.org/10.1038/s41467-020-17566-x
152
E. Park T., Peng L., Liang J., Hallal A., S. Yasin F., Zhang X., M. Song K., J. Kim S., Kim K., Weigand M., Schütz G., Finizio S., Raabe J., Garcia K., Xia J., Zhou Y., Ezawa M., Liu X., Chang J., C. Koo H., D. Kim Y., Chshiev M., Fert A., Yang H., Yu X., Woo S.. Néel-type skyrmions and their current-induced motion in van der Waals ferromagnet-based heterostructures. Phys. Rev. B, 2021, 103(10): 104410 https://doi.org/10.1103/PhysRevB.103.104410
153
Wu Y.Francisco B.Wang W., 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)
154
Sun W., Wang W., Li H., Zhang G., Chen D., Wang J., Cheng Z.. Controlling bimerons as skyrmion analogues by ferroelectric polarization in 2D van der Waals multiferroic heterostructures. Nat. Commun., 2020, 11(1): 5930 https://doi.org/10.1038/s41467-020-19779-6
155
K. Li C., P. Yao X., Chen G.. Writing and deleting skyrmions with electric fields in a multiferroic heterostructure. Phys. Rev. Res., 2021, 3(1): L012026 https://doi.org/10.1103/PhysRevResearch.3.L012026
156
Dou K., Du W., Dai Y., Huang B., Ma Y.. Two-dimensional magnetoelectric multiferroics in a MnSTe/In2Se3 heterobilayer with ferroelectrically controllable skyrmions. Phys. Rev. B, 2022, 105(20): 205427 https://doi.org/10.1103/PhysRevB.105.205427
157
Sun W., Wang W., Zang J., Li H., Zhang G., Wang J., Cheng Z.. Manipulation of magnetic skyrmion in a 2D van der Waals heterostructure via both electric and magnetic fields. Adv. Funct. Mater., 2021, 31(47): 2104452 https://doi.org/10.1002/adfm.202104452
158
Sun W., Wang W., Li H., Li X., Yu Z., Bai Y., Zhang G, Cheng Z.. LaBr2 bilayer multiferroic moiré superlattice with robust magnetoelectric coupling and magnetic bimerons. npj Comput. Mater., 2022, 8: 159 https://doi.org/10.1038/s41524-022-00833-4
159
Chen J., Dong S.. Manipulation of magnetic domain walls by ferroelectric switching: Dynamic magnetoelectricity at the nanoscale. Phys. Rev. Lett., 2021, 126: 117603 https://doi.org/10.1103/PhysRevLett.126.117603
160
Onose Y., Ideue T., Katsura H., Shiomi Y., Nagaosa N., Tokura Y.. Observation of the magnon Hall effect. Science, 2010, 329(5989): 297 https://doi.org/10.1126/science.1188260
161
Matsumoto R., Murakami S.. Theoretical prediction of a rotating magnon wave packet in ferromagnets. Phys. Rev. Lett., 2011, 106(19): 197202 https://doi.org/10.1103/PhysRevLett.106.197202
162
Chisnell R., S. Helton J., E. Freedman D., K. Singh D., I. Bewley R., G. Nocera D., S. Lee Y.. Topological magnon bands in a kagome lattice ferromagnet. Phys. Rev. Lett., 2015, 115(14): 147201 https://doi.org/10.1103/PhysRevLett.115.147201
163
Zhu F., Zhang L., Wang X., J. dos Santos F., Song J., Mueller T., Schmalzl K., F. Schmidt W., Ivanov A., T. Park J., Xu J., Ma J., Lounis S., Blügel S., Mokrousov Y., Su Y., Brückel T.. Topological magnon insulators in two-dimensional van der Waals ferromagnets CrSiTe3 and CrGeTe3: Toward intrinsic gap-tunability. Sci. Adv., 2021, 7(37): eabi7532 https://doi.org/10.1126/sciadv.abi7532
164
Yu X., Zhang X., Shi Q., Tian S., Lei H., Xu K., Hosono H.. Large magnetocaloric effect in van der Waals crystal CrBr3. Front. Phys., 2019, 14(4): 43501 https://doi.org/10.1007/s11467-019-0883-6
165
Pei Q., C. Wang X., J. Zou J., B. Mi W.. Tunable electronic structure and magnetic coupling in strained two-dimensional semiconductor MnPSe3. Front. Phys., 2018, 13(4): 137105 https://doi.org/10.1007/s11467-018-0796-9