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Frontiers of Physics

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Front. Phys.    2023, Vol. 18 Issue (1) : 13602    https://doi.org/10.1007/s11467-022-1217-7
TOPICAL REVIEW
Magnetic anisotropy, exchange coupling and Dzyaloshinskii–Moriya interaction of two-dimensional magnets
Qirui Cui1, Liming Wang1, Yingmei Zhu1, Jinghua Liang1, Hongxin Yang1,2,3()
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
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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.
Keywords magnetic anisotropy      exchange coupling      Dzyaloshinskii–Moriya interaction      two-dimensional magnets     
Corresponding Author(s): Hongxin Yang   
Issue Date: 30 November 2022
 Cite this article:   
Qirui Cui,Liming Wang,Yingmei Zhu, et al. Magnetic anisotropy, exchange coupling and Dzyaloshinskii–Moriya interaction of two-dimensional magnets[J]. Front. Phys. , 2023, 18(1): 13602.
 URL:  
https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1217-7
https://academic.hep.com.cn/fop/EN/Y2023/V18/I1/13602
Fig.1  The schematics of (a) magnetic anisotropy, (b) exchange interaction, and (c) Dzyaloshinskii−Moriya interaction in two-dimensional magnets. (a) When spins are aligned parallelly, there is always a preferred orientation of spin vectors, which is referred as the magnetic anisotropy. (b) Exchange coupling between two spins arises from electrons’ antisymmetric wave function and can be classified as direct (blue dashed line) and indirect (green and yellow dashed lines) types. The indirect exchange couplings can be established through connecting anion (green balls) or itinerant electrons (yellow balls). (c) For magnets lacking inversion symmetry, Dzyaloshinskii−Moriya interaction is allowed between neighboring spin sites, which favors the noncollinear spin configurations. The spin tilting is due to the spin−orbit scattering of the nonmagnetic electrons from the mediated site (the red ball). Left and middle panel is produced from Ref. [39].
Fig.2  (a) Schematics of magnetic anisotropy induced magnon excitation gap for an isotropic Heisenberg model. (b−e) Four most-widely applied models for describing the two-dimensional magnetism. (b) Heisenberg model: The spin vector can be along any directions in 3D space. This model is applied for describing ferromagnetism in 2D magnets with weak magnetic anisotropy, such as Cr2Ge2Te6 (left panel) and V5Se8 (right panel). (c) Ising model: The spin is either up or down along a given direction. This model is applied for describing magnetism in 2D magnets with strong magnetic anisotropy, such as CrI3 (left panel) and FePS3 (right panel). (d) 2D XY model: The spin is constrained in the XY plane due to the in-plane magnetic anisotropy. CrCl3 monolayer grown on Graphene/6H-SiC (0001) has been demonstrated as a 2D XY magnet. (e) Stoner model: Magnets exhibit the metallic feature and ferromagnetic order can be described by Stoner criteria. Stoner model is applied for describing magnetism in Fe3GeTe2 which even possesses the room-temperature ferromagnetism under ionic doping (left and middle panels). The left panel is the density of states of Fe3GeTe2 applied for calculating the Stoner criteria. (a) is produced from Ref. [39]. (b) is produced from Refs. [2, 20]. (c) is produced from Refs. [3, 7]. (d) is produced from Ref. [9]. (e) is produced from Refs. [6, 29].
Fig.3  Bilinear exchange coupling in 2D magnets. (a) Ferromagnetism in 2D magnets with octahedral crystal field, such as CrI3, normally arises from the indirect exchange coupling. Via tuning energy splitting between t2g and eg orbitals to decrease the electron hopping barrier, the FM exchange coupling could be obviously enhanced resulting in the enhancement of Curie temperature as shown in (d). (b) Some 2D magnets, such as NiI2 monolayer, possess strong spin frustration that the first-neighboring and second (third)-neighboring exchange coupling has the similar magnitude but opposite sign. Spin frustration could give rise the noncollinear spin configurations as shown in (e). (f) Moreover, the ferroelectricity probably originates from the specific noncollinear spin configurations, which thus makes system possess multiferroelectricity. (c) For vdW magnets, the bilinear exchange coupling also emerges between magnetic atoms in different layers, in other word, determining the interlayer magnetism. (g) Therefore, changing stacking configurations to modulate electron hopping path can results in the interlayer ferromagnetism-antiferromagnetism phases transition. The phenomena have been observed in CrI3 and CrBr3 bilayers by the spin polarized STM. (a) is produced from Ref. [64]. (b, d) is produced from Ref. [73]. (c) is produced from Ref. [86]. (e) is produced from Ref. [72]. (f) is produced from Ref. [77]. (g) is produced from Ref. [88].
Fig.4  High-order exchange coupling indicates that the superexchange process through the non-magnetic atom and the Coulomb repulsion at neighboring spin-sites involving more than one electron. (a, b) Biquadratic exchange coupling, which is also called as two-body fourth order interactions, belongs to two involving two electrons hopping between MA and MB sites. (c, d) Besides two-body fourth order interactions, three-body and four-body fourth order interactions also play crucial roles in determine magnetic features of materials such as Fe3GeTe2 monolayer. (e) Sizable biquadratic exchange coupling could favor the collinear spin configurations. For CrI3 monolayer, the model without considering the biquadratic underestimate the Curie temperature. (f) Monte Carlo simulations show that in NiCl2 monolayer, strong biquadratic exchange coupling transform the spin spiral ground state into the uniform ferromagnetic state. (g) Four-body fourth order effectively increase the energy barrier of skyrmion collapse, thus favoring the formation of non-collinear magnetism. (h) The high-order interactions, including two-body, three-body, and four-body interactions, play crucial roles in stabilizing topological magnetism in Fe3GeTe2 monolayer. (a, b, e) are produced from Ref. [96]. (c, d, h) are produced from Ref. [105]. (f) is produced from Ref. [73]. (g) is produced from Ref. [102].
Fig.5  DMI-induced topological quasiparticles. (a) The inversion symmetry breaking in 2D Janus magnets allows the isotropic DMI between neighboring spin sites as indicated by orange arrows. The Monte Carlo simulations show that topological magnetism, such as skyrmion lattice, is stabilized by the DMI. (b) P4¯m2 symmetry results in the anisotropic DMI as indicated by orange arrows. This anisotropic DMI could favor the formation of novel topological quasiparticles such as antiskyrmion. The inversion symmetry breaking and electric field-controllable magnetic properties in (c) multiferroic monolayer and (d) multiferroic materials based on transition-metal dichalcogenides open the opportunities for using electric field to manipulate the topological magnetism. The skyrmion particles is experimentally observed in vdW heterostructures based on the 2D magnets, (e) oxidized Fe3GeTe2/Fe3GeTe2 heterostructure and (f) Cr2Ge2Te6/Fe3GeTe2 heterostructure, by Lorenz transmission electron microscope and topological Hall effects, where noncollinear magnetism is stabilized by the interfacial DMI. (g) The next-neighboring DMI as indicated in (h) opens the non-trivial gap in magnon spectrum, which thus gives rise the topological magnon. (i) The non-trivial magnon gap is first observed in vdW magnets CrI3 by neutron-scattering microscopy. (a) is produced from Ref. [126]. (b) is produced from Ref. [142]. (c) is produced from Ref. [149]. (d) is produced from Ref. [150]. (e) is produced from Ref. [152]. (g) is produced from Ref. [153]. (g) is produced from Ref. [163]. (h, i) are produced from Ref. [97].
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