Recent progress on 2D magnets: Fundamental mechanism, structural design and modification

Since the discovery of magnetic phenomena in ancient times, magnetism has attracted vast research interest due to its technological importance and theoretical complexity. From the technique point of view, permanent magnets, which are required for high coercivity and large energy product, are widely used in loudspeakers, earphones, electric meters, small motors, and wind power generation. The more esoteric applications of magnetism are in magnetic recording and storage devices of computers, as well as in audio and video systems. Only ten years after the discovery of interlayer exchange coupling and related giant magnetoresistance (GMR) effect in the magnetic multilayers, GMR devices were routinely used in the hard disk drives of computers.¹ They are also crucial for the emerging field of spintronics, which is regarded as the core of the next-generation information technology. By using electron spin rather than charge as the information carrier, spintronics possesses prominent advantages of speeding up data processing, high circuit integration density, and low energy consumption.^2,3

From the theoretical point of view, two fundamental concepts have been proposed to explain the fascinating magnetic phenomenon, namely, exchange interaction and spin-orbit coupling (SOC). The interplay between exchange interaction, spin-orbit coupling, and Zeeman effect is the essence of magnetism research. Together, they explain the origin of spin arrangement, orbital moment, and magnetocrystalline anisotropy, and the effect of external field on these quantities.⁴ Among them, the interatomic/interelectronic exchange interactions are at the heart of the phenomenon of long-range magnetic ordering. Parallel and antiparallel arrangements of spins constitute long-range ferromagnetic (FM) and antiferromagnetic (AFM) orderings. As the dimensionality decreases, the net magnetic moment per atom generally increases. This might be ascribed to lower coordination number, quantum confinement effect, less quenching of orbital magnetic moment, and so on. Meanwhile, in two-dimensional (2D) lattices, exchange interactions would be stronger along one or two spatial directions than others, showing large anisotropy. Depending on the type of magnetic ions and the electronic band structures, the traditional exchange mechanisms, including direct exchange interaction,^5–7 superexchange interaction,^8–14 double exchange interaction,¹⁵ itinerant electrons,^16–21 and Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanisms,²² are all found in various 2D systems, as well as three new interesting exchange interaction mechanisms, known as super-superexchange,^23,24 extended superexchange,^25,26 and multi-intermediate double exchange interactions.²⁷ These magnetic spin coupling mechanisms will be discussed in Sec. II.

The magnetism in 2D materials has been an emerging and rapidly growing research field. Much on-going progress on 2D magnets has been made mainly in the following three aspects. First, the experimentally unprecedented realizations of high quality 2D magnetic monolayer/multilayer materials using micromechanical cleavage, chemical vapor deposition (CVD), and molecular beam epitaxy (MBE) methods, including CrI_3,²⁸ VI_3,²⁹ CrCl_3,³⁰ CrBr_3,³¹ α-RuCl_3,^32,33 VSe_2,³⁴ VTe_2,³⁵ NbTe_2,³⁵ MnSe_x,³⁶ CrGeTe₃,³⁷ MPS₃ (M = Fe, Mn, Ni),^38–40 Fe_3,5GeTe_2,^41,42 and MnBi₂Te₄.⁴³ These exciting experiments not only bring about the missing magnetic member in the family of 2D materials but also solve a long-standing controversy about the existence of magnetism in low-dimensional systems. Based on Mermin-Wagner theorem,⁴⁴ 2D magnetic systems cannot have any long-range FM or AFM ordering at T > 0 for isotropic Heisenberg model. The main reason for the existence of magnetism is the magnetocrystalline anisotropy and the dipolar interaction. Although the strength of these interactions is of order of magnitude of ∼1 meV, they will have a crucial impact on the 2D materials by breaking the conditions of Mermin-Wagner theorem.

Second, optical and electronic means have also been widely developed to characterize the 2D magnets.⁴⁵ Spin-polarized scanning tunneling microscope (SP-STM), and x-ray magnetic circular dichroism (XMCD) are the standard experimental techniques. They are highlighted in the recently developed 2D van der Waals magnets. SP-STM can provide valuable and precise information about the magnetic ordering and differentiate magnetic structures as well as exchange interactions. Based on excitation of spin polarized carriers from core levels to Fermi levels via circularly polarized x rays, XMCD is an efficient photocurrent technique used to obtain the local electronic configuration of the valence states. Moreover, due to the theoretical development of the sum rules, the element-specific spin and orbital moments can be obtained separately.⁴⁶ XMCD can also yield the orbital moment anisotropy, which is related to the magnetocrystalline anisotropy energy (MAE) via perturbation model. X-ray magnetic linear dichroism (XMLD) has been proposed as a mean of measuring the anisotropy in the spin orbit interaction, which can be related to the MAE using a straight forward sum rule.⁴⁷ Inelastic light scattering provides access to the energy, symmetry, and statistics of lattice, electronic, and magnetic excitations in nanomaterials. In addition, Raman spectroscopy can also provide indirect evidence of magnetism, based on either spin-phonon coupling or magnons. Raman scattering from spin-phonon excitations yields incisive information on magnetic materials, which can be used to reveal magnetic ordering and phase transitions. For example, the studies of Kerr rotation versus the applied magnetic field (MOKE) produced the first indirect evidence of magnetic ordering in ultrathin films of Cr₂Ge₂Te₆³⁷ and CrI₃.²⁸ Specifically, single-layer FePS₃⁴⁸ and Cr₂Ge₂Te₆⁴⁹ at high temperatures showed a broad feature above the Curie temperature (T_C), which resulted from thermal magnetic fluctuations. In addition to the optical probes, the magnetic state of 2D itinerant magnets with conduction electrons can be detected by the anomalous Hall effect (AHE). So far, AHE has been applied to explore the magnetic ordering of layered Fe₃GeTe₂ down to the monolayer limit⁵⁰ and few-layer antiferromagnetic MnBi₂Te₄.⁴³ 

Third, due to the benefit of a continuous increase in computing power, it is possible to predict the magnetic ground state and corresponding magnetic parameters of a given 2D material from first-principles calculations. The most important quantities of 2D magnets include the magnetic moment, exchange integral, MAE, and T_C. The former three terms can be accurately determined by density functional theory (DFT) calculations of total energy and electronic structure with or without considering SOC.⁵¹ Commonly, combining DFT with Monte Carlo simulations implemented with Heisenberg model,⁵² one can trace the variation of magnetization with temperature and determine the magnetic transition temperature (T_C/T_N) of the 2D systems. For example, 2D CrI₃ has been predicted as a FM material by three theoretical groups during 2015 and 2016,^8,53,54 individually. The reported magnetic moment and Curie temperature were 3∼3.44 μ_B per formula unit and 61∼107 K, respectively, in good agreement with the successive experimental results reported in 2017.²⁸ To our knowledge, the magnetic behavior of α-RuCl₃, VSe₂, VTe₂, NbTe₂, MnSe_x, CrGe(Si)Te₃ single- or few-layer sheets have also been discussed by DFT calculations.^18,55–64

The powerful DFT tool and high-throughput computations would efficiently accelerate the screening of 2D magnetic candidate materials without having to synthesize them first.^65–68 Starting from the existing database of inorganic compounds, Mounet et al.⁶⁷ performed comprehensive DFT calculations to explore these 3D parent materials for possible exfoliation and finally identified 2662 prospective layered structures with weak interlayer binding energy. Among them, 36 ferromagnetic systems with relatively simple unit cells (less than six atoms) were identified with long-range magnetic order. After a further screening, monolayer materials of LaCl, YCl, ScCl, LaBr₂, and CrSBr have been identified as robust ferromagnets with T_C > 200 K.⁶⁵ Very recently, the ferromagnetic ordering of 2D CrSBr has been confirmed by experimental evidence, and the predicted T_C is very likely beyond 132 K.⁶⁹ In another study, a larger set of a 2D materials database (∼65 000) has been screened by high-throughput DFT calculations, yielding a total of 89 magnetic monolayer systems.⁶⁶ Among them, 56 showed FM ordering, while 33 prefer antiferromagnetic behavior. Furthermore, 24 FM monolayers were promising candidates possessing T_C higher than that of CrI₃.

To broaden the scope of intrinsic 2D magnets, atomic substitutions^65,70 and alloying^71,72 have been introduced in 2D materials and led to complex variations to the magnetic properties. In addition, many advanced techniques for exploring the potential energy landscapes can also be employed to theoretically design 2D magnets. For instance, the evolutionary algorithm and particle swarm optimization (PSO) approach have already demonstrated their validity in discovering potential 2D magnets,⁷³ including V₃X₈ (X = F, Cl, Br),⁷⁴ MnAs,⁷⁵ Co₂P,⁵ Fe₃P,⁷⁶ PtN_2,⁷⁷ CoB₆.⁷⁸ For 2D organic frameworks, bottom-up design is also a common strategy to achieve promising 2D magnets by properly selecting the building units and linkers.^79–82 

Driven by all these three aspects, the research of 2D magnets has boomed in the last few years. In recent years, a great deal of new 2D magnetic materials have been experimentally discovered and theoretically predicted. In this review, however, we restricted the term to the 2D intrinsic magnets. The 2D intrinsic nonmagnetic materials, which have already been concluded in the previous review,⁸³ will not be covered here. In this review, we classified these emerging 2D intrinsic magnets into binary transition metal halogenides; chalogenides; carbides; nitrides; oxides; borides; silicides; ternary transition metal compounds CrXTe₃, MPX₃, Fe-Ge-Te, MBi₂Te₄, and MXY (M = transition metal; X = O, S, Se, Te, N; Y = Cl, Br, I); f-state magnets; p-state magnets; and organic magnets such as metal organic framework (MOF) and covalent organic framework (COF). The most common prototypes of 2D inorganic magnetic materials are shown in Fig. 1. Their key magnetic properties, especially the magnetic ground state, magnetic moment, T_C, and MAE will be comprehensively described in Sec. III.

For 2D layered materials, their magnetic properties can be readily modulated by chemical compositions, functional groups, intercalation, and substrates. Moreover, the spin coupling to external perturbations like strain, electronic/magnetic fields, and carrier doping could be the other critical factors for tailoring the strength of exchange interactions or magnetic anisotropy. These modulation methods could be utilized on 2D intrinsic magnets to further enhance Curie temperature. The corresponding mechanisms for modulating the charge distributions, energy level, orbital occupation, symmetry, and hopping paths will be discussed in Sec. IV.

In the final Sec. V, we will conclude with a discussion about future challenges and opportunities of 2D magnets. Four potential future directions regarding the practical applications have been proposed. In addition, this review article mainly focuses on the underlying mechanisms of exchange interaction, which not only determine the Curie temperature of 2D intrinsic magnets but also shed light on the strategies to manipulate the magnetic coupling. Therefore, this discussion is complementary to the other recent review articles on 2D magnets, which have discussed the history,⁸⁴ representative 2D magnets,^45,85,86 experimental synthesis,⁸⁷ probing techniques,⁸⁸ magnetic anisotropy,⁸⁹ device applications,^84,90–92 and their emergent phenomena.⁹³ 

To start discussing the magnetic properties of 2D materials, we should first know the magnetic ground state of a multi-electron atom, which is mainly determined by Hund's rules. Generally speaking, Hund's rules refer to a set of rules that describe spin-spin coupling, orbital-orbital coupling, and spin-orbit coupling, respectively. The first rule about the spin-spin coupling in a multi-electron atom, whose strength is at the order of c.a. 2 eV is especially important. Hence, the magnetic moment is mainly determined by the intra-atomic exchange interaction. The spin-orbit interaction describing the coupling between spin and orbital angular momentum further produces the energy level splitting with the order of several to hundreds mega-electron volts. We can estimate the moment per atom from this rule by maximizing spin (S), orbital momentum (L), and angular moment (J).

In a solid, exchange interaction and SOC are two most important concepts for magnetism. First of all, without interatomic exchange, there would be no spontaneous magnetization. Interatomic exchange interaction determines the long-range spin ordering, i.e., parallel or antiparallel spin alignment in a magnetic material. Meanwhile, spin-orbit interaction creates orbital magnetism and couples the spin to the lattice. Through the SOC interaction, spin and charge can talk to each other via exchanging energy and angular momentum, thereby establishing magnetic anisotropy. In addition, the other competition interactions have also been investigated to determine the size of the moment, including crystal fields, Hund's rule coupling, and onsite Coulomb repulsion.⁹⁴ These factors will be discussed in a following paper.

When a free ion is placed in a lattice and subjected to the interaction with its surrounding atoms, another key interaction arises, termed a ligand (crystal) field. The ligand effect on the central atom is entirely determined by symmetry and strength of field produced by these surrounding atoms. The symmetry and degenerate states of typical 2D magnets with 3d transition metals are summarized in Table I. It should be noted that the amplitude of splitting energy due to ligand field is comparable to the intra-atomic Coulomb and exchange interactions. Thus, the final ground state depends on the relative amplitude of them. For example, the d⁴-d⁷ configurations would show either low-spin or high-spin states in an octahedral ligand field.

In fact, ligand field theory provides a simple model to predict the magnetic behavior of 2D transition metal compounds, which is strongly influenced by the coordination environment and the number of d electrons. For example, the configurations of M₂NT₂ MXene are shown in Figs. 2(a) and 2(b).¹² For a transition metal ion located in an octahedral crystal field, its d orbitals split into the lower-energy t_2g states and higher-energy e_g orbitals. Similar to the transition metal dichalcogenides (TMDs), these nonbonding t_2g and e_g states of the MXenes are positioned between the bonding and antibonding states (σ and σ^*) of M–X and M–T bonds [Fig. 2(c)]. We assume a perfect bonding case as follows: (1) the nonmetal elements are in their nominal oxidation state, i.e., C^4–, N^3–, O^2–, F⁻, and OH⁻; (2) the M–X and M–T bonding states are filled; (3) the M–X and M–T antibonding states are empty. Therefore, only the electrons occupying the nonbonding d orbitals will contribute to the magnetism. For example, the nominal oxidation state of Cr ion in Cr₂CF₂ is +3. Based on Hund's rules, the remaining 3 electrons on Cr ion will occupy the t_2g band half filled, thereby giving a local magnetic moment of 3 μ_B. Similarly, the local magnetic moments of Mn₂CO₂, Mn₂CF₂, and Mn₂C(OH)₂ are 3 μ_B, 4 μ_B, and 4 μ_B, respectively. The electron spin arrangements on the transition metals in the nitride MXene are shown in the Fig. 2(d).¹² Similar analyses based on symmetry have also been used to identify the magnetic properties of binary transition metal halides, carbides, nitrides, oxides, borides, phosphides, silicides, arsenides, hydrides, and ternary transition metal compounds. The representative examples are shown in Table I. All these results are consistent with the results from DFT calculations.^{10,12,45,65,93,95–105}

Heisenberg model is a simple theoretical model to describe the physical effects of magnetic systems. Due to the strong local magnetic moment, the leading term is symmetric exchange interaction in Heisenberg model, which has the form

where i and j denote the lattice sites bearing a localized magnetic moment, and Ŝ_i or Ŝ_j is a quantum mechanical spin operator. J_ij is the exchange constant, which is the basis of most studies of magnetism. Evidently, J_ij will decrease rapidly with the distance between lattice i and j. The positive J_ij favors parallel spin alignment (i.e., FM), while the negative J_ij favors antiparallel spin alignment (i.e., AFM). Theoretically, the values of J_ij can be obtained from first-principles calculations. The simplest way is to calculate from the total energy difference between different spin orderings. In experiment, J_ij can be determined by fitting the inelastic neutron scattering data to the Heisenberg Hamiltonian.

Generally speaking, these microscopic parameters of J_ij define most of macroscopic magnetic properties, especially the Curie temperature and the magnetic response function to an external field. For example, one can estimate magnetic transition temperatures via mean field theory (MFT)¹⁰⁶ as follows:

where S is the atomic spin, J₀ is the sum of exchange interactions, and k_B is the Boltzmann constant. For 2D magnets, MFT usually overestimates the transition temperature by around 20% or even more, which is dependent on the coordination number.¹⁰⁶ Monte Carlo (MC) method gives a numerical solution to the Heisenberg model with reasonable accuracy. Therefore, it becomes the most commonly used method to predict the critical temperature in 2D magnets. However, MFT is still meaningful in establishing an upper limit of T_C at much less computational cost, which can be compared to MC results.

To explain the origin of spontaneous magnetization in metallic magnets with itinerant/localized electrons, such as single layer of Fe₃GeTe₂, T phase of TMDs, Cr₂C, and Fe₂C MXenes, another two models—Stoner model and RKKY model—have also been proposed, which will be discussed in Secs. II F and II G, respectively.

The exchange term is isotropic in the sense that the scalar product Ŝ_i·Ŝ_j in Eq. (1) does not change under any rotation applied to every spin in the system. Based on Mermin-Wagner theorem,⁴⁴ 2D materials can be neither FM nor AFM at nonzero temperature due to thermal fluctuations under isotropic Heisenberg model. However, considering the spin-orbit interaction, this symmetry is broken. Besides the symmetric exchange term H_ex discussed in Sec. II C, the spin orbital coupling gives rise to the antisymmetric anisotropic exchange terms. Generally, the strength of antisymmetric anisotropic exchange terms are far less than H_ex. However, they become crucial for the magnetic ground state of 2D systems.

So far, many simple anisotropic spin models have been proposed to interpret the emerging magnetism in 2D materials. A typical 2D magnet possesses easy-plane magnetic anisotropy, uniaxial anisotropy, and isotropic anisotropy, which can be described in 2D XY, Ising, and Heisenberg models (XXZ), respectively. Specifically, 2D XY-like behavior has been observed in Cr₂Ge₂Te_6,³⁷ MnPS_3,¹⁰⁷ CoGa₂X_4,⁹⁶ CaI_2,¹⁰⁸ and so on; 2D Ising FM behavior was also demonstrated in plenty of 2D magnets, such as Cr₂Si₂Te_6,¹⁰⁹ CrI_3,⁸ VI_3,²⁹ NiCl_3,¹¹⁰ NiCl_2,¹¹¹ H-VTe_2,⁵⁷ T-VTe_2,⁵⁷ T-MnTe_2,⁵⁷ H-FeTe_2,⁵⁷ and NiPS₃.¹¹² 

The antisymmetric anisotropic exchange terms contain various generic forms, for example, Dzyaloshinski-Moriya (DM) interaction term and single ion magnetocrystalline anisotropy term.^113,114 As stated above, both of them were determined by spin-orbit interaction. They have the following forms:

In 2D materials, the spin orbital interaction plays a similar role with non-metal atoms in superexchange interaction. As shown in Eq. (3), the DM interaction is characterized by the vector $D→ij$⁠, which is proportional to spin orbit coupling constant. It is also dependent on the position of the non-metal atom between the two magnetic atoms. Clearly, DM interaction favors an orthogonal alignment between spins.

In Eq. (4), A_i is the easy-axis single ion anisotropy factor. The magnetic anisotropy energy (MAE) is defined as the largest possible energy difference between two different magnetization directions. Within DFT framework, it can be easily obtained by performing total energy calculations including SOC.⁵¹ To clarify the origin of MAEs, the torque method¹¹⁵ was implemented in either all-electron full potential linearized augmented plane wave (FPLAPW) or Vienna Ab-initio Simulation Package (VASP) with plane wave basis sets.^116,117 To evaluate the contribution of SOC to the magnetic anisotropy, second-order perturbation theory has also been introduced.¹¹⁶ 

The DM interaction and single ion magnetocrystalline anisotropy term are typically a few percent of the isotropic term, producing a modest canting to symmetric exchange interactions. Nevertheless, the DM and single ion magnetocrystalline anisotropy terms are the good supplement to the 2D materials. A spontaneous rearrangement of atoms to favor the DM interaction can produce a large electric polarization in magnetoelectric materials. Meanwhile, the competition of spin-spin directions would be helpful to understand complicated magnetic behavior, such as magnetic skyrmions, and quantum spin liquid. The magnetic anisotropy is a prerequisite to realize FM or AFM states at the 2D limit.

A much-studied spin model (Kitaev model) is developed to describe the anisotropic spin exchange coupling for honeycomb spin lattice.¹¹⁸ It has the form

Neighboring spins couple depending on the direction of their bond γ with S_xS_x, S_yS_y, or S_zS_z. This exchange term and the symmetric Heisenberg term (Kitaev-Heisenberg model) together serve as a putative minima model for several materials, including α-RuCl₃¹¹⁹ and Li(Na)₂IrO₃.¹²⁰ 

Beyond these terms, one should also note that the real materials may have additional exchange couplings other than these couplings, including dipolar term, biquadratic interaction, and Zeeman coupling to the external magnetic field.¹²¹ The strength of these exchange interactions is also much lower than that of symmetric exchange interaction. Therefore, to simplify, only the dominated spin models are investigated and discussed in this review, which are roughly enough to describe the critical parameters of 2D magnets.

In this section, we discuss the identified exchange mechanisms in the real 2D magnets based on the abovementioned three models, including direct exchange, superexchange, double exchange, super-superexchange, extended superexchange, multi-intermediate double exchange interaction, itinerant electrons, and RKKY (Fig. 3). Their regimes of applicability depend on the electronic band structures and the types of magnetic ions, i.e., metal or insulator, and localized or delocalized. However, it is really difficult to distinguish them completely. For example, 3d electrons are partially localized on the atomic sites and also partially delocalized in the crystal. In each exchange interaction, the magnetic ground state and Curie temperature will be further discussed, which are determined from the competition between the kinetic exchange energy and the Coulomb repulsion. It is worth noting that magnetic exchange interactions in the 2D magnets are rather complicated; thus, they are unable to be generalized under a one-theory umbrella. Although these types of exchange mechanism have been utilized to explain different materials appropriately, there are no clear borderlines between them. Moreover, several exchange interactions may possibly coexist in one real material. To understand the magnetic ordering and magnetic coupling strength, we always need to ask which one is dominant.

Direct exchange interaction is based on the overlap of electronic wavefunctions and is therefore very short ranged.¹²² It is always confined to electrons in the orbitals from the nearest neighboring atoms. Considering the distance of the magnetic atoms, the strength of direct exchange interaction is always very weak. As a consequence, the direct exchange interaction is neither the main source of magnetism nor can it appropriately describe the magnetic behavior in most of the reported 2D magnets. Even so, direct exchange interaction between the neighboring sites is still dominant in the magnetic materials with peculiar d-d, d-p, and p-p hybridizations. With sufficiently large overlap, the exchange integral J_ij tends to be antiferromagnetic, ferromagnetic, or ferrimagnetic, depending on symmetry relationship and occupation number of the orbitals.

As a specific example, Liu et al.⁵ proposed a novel class of 2D magnetic metal-shrouded materials, namely tetragonal transition metal phosphides (TM₂P), which showed peculiar coexistence of in-plane TM–P covalent bonds and interlayer TM–TM metallic bonds. For Co₂P, the spin-up d_z2 orbital contributes to the total on-site moment of ∼1 μ_B. The electrons can hop from the occupied d_z2 orbital to the empty d_z2 orbital via Co–Co metallic bonds; thus, the d-d exchange between the neighboring sites is ferromagnetic. However, the more active dxz/dyz orbitals dominate the AFM ground states in Fe₂P. There is overlap between dxz/dyz orbitals at two neighboring metal sites, and electrons would hop between these two active orbitals of Fe atoms. The resulting exchange interaction is antiferromagnetic and strong. Consequently, Fe₂P behaves as an antiferromagnetic material with T_N = 23 K, while Co₂P is a ferromagnetic material with T_C = 580 K. The corresponding AFM and FM direct interaction mechanisms are shown in Fig. 4(a).

To clarify the novel AFM ground state of ternary MPS₃ monolayer, the dominant electron hopping paths were analyzed. As a representative case, the first neighboring-to-neighboring interaction for MnPSe₃ monolayer is shown in Fig. 4(b).¹²³ Electrons hopping from two paths has been discussed—one is short-range direct interaction between the two neighboring Mn ions, and the other one is long-range Mn–Se–Mn superexchange with an angle of 84.1°. Owing to the strong interaction between neighboring Mn²⁺ cations and the large electron excitation energy from Se p orbital to Mn d orbital, direct AFM exchange interaction prevails over FM superexchange interaction.

Moreover, robust FM coupling with high Curie temperature was also observed in 1T-TaN₂¹²⁴ and 1T-YN₂ monolayers.⁶ In these systems, the N–N distances (∼1.74 Å) are short enough to generate the strong direct exchange interaction. In TaN₂, the magnetic moment arises mainly from the fully filled spin-up p_z orbitals and nearly unfilled spin-down p_z orbitals. Benefiting from the delocalized feature of p orbitals of N atoms, p-p direct exchange interaction [see Fig.4(c)] leads to strong long-range FM coupling. In 1T-YN₂, the J₁ parameter is 11.3 meV, confirming again that the direct interaction is FM coupling.

Robust ferrimagnetic ordering was proposed in 2D metal organic frameworks with conjugated electron acceptors diketopyrrolopyrrole (DPP) as organic linkers and transition metal Cr as nodes, namely, Cr-DPP [Fig. 4(d)].⁷ In 2D Cr-DPP, each Cr atom possesses a spin magnetic moment of around 4 μ_B, and each DPP unit has a spin magnetic moment of about 1 μ_B. Considering the symmetry matching rule, the majority of magnetic coupling between Cr and DPP can be ascribed to direct exchange interaction between d_xy↑ orbital of Cr and p↓ orbital (p_x or p_y) of the adjacent N atoms. Meanwhile, direct exchange between d_xz/d_yz↑ orbital of Cr and p_z↓ orbital of N contributes to the minority part due to the comparatively big energy gap between the two orbitals. The strong d-p direct exchange coupling of 2D Cr-DPP yields a Curie temperature of 316 K. Robust d-p direct exchange coupling was also found in 2D ferrimagnetic V-DPP⁷ and Cr-pentalene MOF,¹²⁵ and the corresponding T_C was 406 and 560 K, respectively.

Among the reported 2D magnets, there is a large number of transition metal compounds, such as binary/ternary transition metal halides, chalcogenides, borides, carbides, nitrides, oxides, hydrides, and silicides. Their detailed magnetic properties will be discussed in Sec. III. As we stated above, the direct overlap between d orbitals in these transition metal compounds is generally too small due to the large distance. Thus, d electrons can only move through hybridization with the ligand atoms between them, like 2p orbitals of B, C, N, O, H, and F. Such p-d hybridization provides a common type of exchange mechanism, known as superexchange interaction. That is to say, superexchange interaction arises from the non-neighboring magnetic ions mediated by the neighboring non-magnetic ions. Empirically, the magnetic ground state is determined by Goodenough-Kanamori-Anderson (GKA) rules,^126–128 which are based on the symmetry relationships and electron occupancy of the overlapping atomic orbitals. According to these rules: (1) A 180° superexchange interaction of two magnetic ions with partially filled d shells is AFM if virtual electron transfer occurs between the overlapping orbitals that are each half filled. (2) A 180° superexchange interaction of two magnetic ions with partially filled d shells is FM if virtual electron transfer occurs from a half-filled to an empty orbital or from a filled to a half-filled orbital. (3) A 90° superexchange interaction where the occupied d orbitals of metal atom overlap with different orthogonal p orbitals of the ligands results in weak ferromagnetism. In addition, the strength of superexchange coupling is also sensitive to two factors, i.e., (1) the degree of p-d hopping process and (2) the strength of SOC.

Based on the superexchange mechanism and GKA rules, the magnetic behavior of a variety of 2D magnets has been successfully explained, including the recently highlighted CrI_3,⁸ VI_3,⁹ VS_2,¹⁰ Cr₂Ge₂Te_3,¹¹ MXenes,^12,13 MnBi₂Te_4,¹⁴ GdI_2,¹²⁹ and so on. For instance, 2D CrI₃ is an insulator [Fig. 5(a)] with local spin of S = 3/2.⁸ Considering the local octahedral coordination field [Fig. 5(b)], the valence bands and the conduction bands consist of t_2g states and e_g states, respectively. The closed t_2g³ configuration indicates the formal Cr³⁺ charge state. The exchange splitting in 2D CrI₃ is about 3 eV, which is larger than the t_2g-e_g crystal field splitting. For the corner sharing I atoms, their in-plane p_x/p_y orbitals couple to the Cr d_x2-y2 orbital, and the out-of-plane I p_z to Cr d_xz/yz [Fig. 5(c)]. Owing to the closed t_2g³ subshell, the direct exchange interaction between two neighboring Cr³⁺ ions is AFM, which is associated with Pauli exclusion principle in the virtually excited t_2g²-t_2g⁴ state. However, the Cr–Cr distance is as large as 3.95 Å; thus the AFM exchange should be weak. As a consequence, the two nearly 90° Cr–I–Cr superexchange mechanisms dominate in 2D CrI₃. One originates from the stronger p-d hybridization via the orthogonal orbitals [Figs. 5(d) and 5(f)], and the other involves the relatively weak p-d hybridization via the same p_x orbital [Figs. 5(d) and 5(e)]. Both of them give an effective Cr–Cr FM coupling. According to the weak 90° d-p-d superexchange interaction, the Curie temperature of CrI₃ monolayer is 45 K.²⁸ In fact, the weak superexchange interaction also results in low Curie temperatures for the experimentally reported 2D magnets, such as 30 K for Cr₂Ge₂Te₆³⁷ and 34 K for CrBr₃.³¹ 

Besides direct exchange and superexchange, double exchange interaction always exists in the 2D magnets with high-spin states. It arises between the ions in different oxidation states. In double exchange pictures, the interaction occurs when one atom has an extra electron compared to the other one. The electron transfer from the neighboring sites should have the same direction of spin. Therefore, the magnetic coupling is ferromagnetic.¹³⁰ For example, Zhang et al.¹⁵ provided a double exchange model [Fig. 5(g)] in Cr₃X₄ monolayers (X = S, Se, Te), where seven hexagonal atomic layers are stacked in the sequence of X₁–Cr₁–X₂–Cr₂–X₂–Cr₁–X₁ along the z direction. The Cr₁ and Cr₂ atoms have different local coordination environments and show different valence states as Cr₁³⁺ and Cr₂²⁺, respectively. The X atom connects the nearest-neighboring Cr₁³⁺ and Cr₂²⁺ ions and gives up its spin-up or spin-down electron to Cr₁³⁺. Then its vacant orbital could be filled by an electron from Cr₂²⁺. Mediated by the X atom, double exchange process is revealed by the electron hopping from one Cr ion to the other neighboring Cr ion of different oxidation state. Such mechanism dominates in Cr₃Se₄ and Cr₃Te₄ monolayers, and strengthens the FM coupling. Consequently, high Curie temperatures of 370 and 460 K were reported for Cr₃Se₄ and Cr₃Te₄ monolayers, respectively.

The above three fundamental magnetic interactions have already explained the origin of most 2D magnetic insulators. However, the magnetic ground state of some complicated 2D materials are still too difficult to be determined from these theories. In such situations, a few new theories, i.e., extended superexchange interaction, super-superexchange interaction, and multi-intermediate double exchange interactions, have been proposed recently.

For the 2D materials containing anions with different valence states, an extended superexchange theory was further proposed.²⁵ It has been validated in the representative CrOCl and FeOCl monolayers. Four possible superexchange paths (P1–P4) in CrOCl are displayed in Fig. 6(a). Similar to 2D CrI₃, the Cr atoms in 2D CrOCl are still located in a distorted octahedral crystal field. The d³ state (Cr³⁺) in CrI₃ can be written as t_2g³e_g,⁰ while the d³ state (Cr³⁺) and d⁴ state (Cr²⁺) in CrOCl is t_2g³e_g⁰ and t_2g³e_g¹, respectively. The extended superexchange theory indicates that (1) a 180° bond angle in the interaction path Cr–O–Cr (P4) favors strongly FM configuration through the dominated p_σ-p_σ bond; (2) a 180° bond angle in the P3 path corresponds to strong AFM, where the interaction is conducted through p_σ-p_σ bond; (3) For Cr–O–Cr path (P2) with 90° bond angle, the d³ state has a crucial unfilled e_g orbital; thus, the interaction is FM with weak coupling strength, owing to the competition of the two p_σ-p_π bonds; (4) For a 90° bond angle in the interaction path of Cr-Cl-Cr (P1), the d⁴ states interact through two p_σ-p_π bonds with moderate strength of AFM coupling. Based on first-principles calculations, they further clarified that monolayer CrOCl exhibited antiferromagnetic ordering.²⁵ On all accounts, this extended superexchange theory supports the strongly FM/AFM configurations, which are highly anticipated to design robust 2D magnetic materials with polyvalent anions. Based on this theory, the high Curie/Néel temperatures in 2D MXY (M = metal; X = S, Se, Te; Y = F, Cl, Br, I) compounds could be explained.^{26,65,70,131–133} According to above discussions, the extended superexchange theory may be regarded as a combination of superexchange and double exchange. Similar mechanism has also been proposed by Wang et al.²⁶ 

The super-superexchange interaction is different from the typical superexchange interactions by the mediated ligands.^23,24 As we stated above, single anion serves as an intermediate to bridge two magnetic cations in the superexchange interaction (M–X–M). In contrast, the super-superexchange interaction involves longer M–X⋯X–M hopping paths. According to the distance between magnetic ions, the strength of super-superexchange interaction is generally much weaker than those of direct interaction and superexchange interaction. Taking single layer CrAsS₄ as a representative, the exchange interaction parameter contributed by both short-range Cr–Cr direct exchange interaction and Cr–S–Cr superexchange interaction is 3.03 meV, while the value for long-range Cr–S–As–S–Cr super-superexchange interaction is –0.49 meV. Therefore, the super-superexchange interaction is usually neglected in most investigations. However, it could become stronger than superexchange interaction in some few-layer magnetic systems with non-covalent van der Waals (vdW) gaps. The importance of super-superexchange interaction is highlighted in the well-known bilayer CrI₃.^23,28,134 For bilayer CrI₃, the local magnetic moments form intralayer FM ordering below 45 K.²⁸ However, its interlayer magnetic coupling varies between FM and AFM, depending on local stacking geometry.²³ These significant results reveal that even weak overlap has a large impact on the interlayer magnetic coupling through exchange between two adjacent interlayer I atoms, which signifies a new exchange interaction mechanism, i.e., super-superexchange. It also means that the super-superexchange is basically the coupling between next-next nearest neighbor.

Sivadas et al.²³ carefully investigated the details of stacking-dependent interlayer exchange interaction for bilayer CrI₃ systems. The AB stacking (S₆ point group) from the low-temperature bulk structure and the AB′ stacking (C_2h point group) from the high-temperature bulk structure were considered. Figure 6(b) schematically shows different exchange interactions between Cr atoms in different layers. One can see that hopping from a t_2g to t_2g orbital is prohibited for FM alignment, while this is allowed for AFM alignment. Therefore, t_2g to t_2g hybridization leads to AFM ordering. On the other hand, hopping from t_2g-e_g leads to an exchange coupling that is predominantly FM because of the local Hund coupling. All these interlayer Cr−Cr exchange interactions are mediated by the hybridization between I p_z orbitals in different layers, which is the nature of super-superexchange interaction.

The stacking-dependent magnetism originates from a competition between different interlayer orbital hybridizations. Interlayer Cr−Cr nearest neighboring interaction J₁ and the second neighboring one J₂ in AB-stacking are shown in Fig. 6(c). One can see that J₁ is dominated by the virtual excitations between the half-filled t_2g orbitals of Cr and induces an AFM coupling, while J₂ is dominated by a virtual excitation from the half-filled t_2g orbitals of Cr to the empty e_g orbitals, resulting in a FM coupling [Fig. 6(c)]. Clearly, the second neighboring FM interlayer super-superexchange prevails as the AFM nearest neighboring interlayer exchange, making AB-stacking system ferromagnetic. A lateral shift of one layer to AB′-stacking of the bilayer system breaks the interlayer hybridization between I p states and generates a new type of hybridization, which in turn would reduce the strength of FM exchange interactions and result in an AFM ground state for AB′-stacking [Fig. 6(d)].

Compared to bilayer CrI₃ with the same Cr³⁺ ions, a mixture of Cr⁴⁺ and Cr³⁺ was found in bilayer CrS₂ due to the charge transfer from e_g to t_2g orbitals. This mixed valance state, together with delocalized S p orbitals and their resulting strongly interlayered S-S hopping, favor the double exchange interaction mechanism.¹³⁵ With further increase of the strength of interlayer coupling, a novel multi-intermediate double exchange interaction has been revealed by extensively investigating nine TMD bilayers MX₂ (M = V, Cr, Mn; X = S, Se, Te).²⁷ Taking 2D CrSe₂ as a prototype, one can see a distinct overlapped region (OR) at the interlayer area, as displayed in Fig. 6(e). In other words, OR could be effectively considered as an area accumulating an appreciable shared charge from the two adjacent interfacial Se sublayers. The OR can be regarded as a real atomic site and plays an important role in determining the interlayer magnetic coupling. In the interlayer FM configuration, the transferred spin-up charge of Se p_z to Cr leaves the spin-down component predominated at the OR [Fig. 6(e)]. Hence, the spin-up electrons of the bottom Cr atom could hop into the top Cr atom through 4p_z orbital of Se_ib atom [defined in Fig. 6(e)], and then through OR upon excitation, and further through Se_it 4p_z, as denoted by the wave-like red-dotted arrow [Fig. 6(e)]. Such electron hopping process largely reduces the kinetic energy of spin-up electron across the bilayer. The process in CrSe₂ is similar to double exchange interaction of CrS₂¹³⁵ but is mediated by multiple sites, which is termed as a new multi-intermediate double exchange interaction. Similarly, the interlayer AFM bilayer also has stacking-induced charge transfer and the interfacial Se p_z overlapping area. As displayed in Fig. 6(e), the spin-up electron of bottom Cr could still hop into the Se_ib 4p_z orbital and reach the OR. However, the next hopping step from OR to the Se_it 4p_z orbital is forbidden since the spin-up component is fully occupied. This appreciably lifts up the kinetic energy. Based on the modified interlayer Hubbard model, the competition between the interlayer hopping across the bilayer and the Pauli and Coulomb repulsions at OR will determine the MX₂ bilayers to have FM or AFM magnetic ground state.

In the above discussions, we mainly focus on 2D FM/AFM insulators. Their magnetism originates from local magnetic moments with exchange interactions that can be interpreted by the Heisenberg exchange mode. For 2D magnetic metals, a simplified model called Stoner model,¹⁶ can be formulated in terms of dispersion relations for the spin-up and spin-down electrons. In the Stoner model, there is a competition between kinetic energy and exchange energy due to Coulomb repulsion. In ferromagnetic metals, the exchange interaction will split the energy of states with different spins. This restructuring of the spins leads to a change in the energy of the system. The up spins occupying higher energy states would cost the kinetic energy. At the same time, the potential energy would decrease due to spin-spin exchange interaction. These changes in total energy provide the Stoner criterion for itinerant ferromagnetism. The Stoner criterion for itinerant ferromagnetic ordering is D(E_F)×I > 1, where D(E_F) is the total density of states at the Fermi level (E_F), and the Stoner parameter I can be estimated from dividing the exchange splitting of spin-up and spin-down bands by the corresponding magnetic moments. These two parameters reflect the competition between the exchange energy and kinetic energy. The former parameter D(E_F) is inversely proportional to the kinetic energy of electrons, whereas the latter one, I, describes the strength of electron exchange. Owing to the nature of itinerant electrons, Stoner model is applicable to illustrate the origin of the spontaneous magnetization in plenty of 2D metallic magnets, such as Fe₃GeTe_2,¹⁶ TMDs,^17,18,20,21 MXene,^19,97,136 as well as a variety of charge doped 2D materials.^137–142 Most of these reported 2D magnetic metals have excitingly high Curie temperatures.

The electronic and magnetic properties of 2D Fe₃GeTe₂ have been investigated by Zhuang et al.¹⁶ The spin orbital projected band structures [Fig. 7(a)] show its metallic behavior. Several partially occupied d bands crossing the Fermi level contribute to the noninteger magnetic moment of Fe (1.484 μ_B). Both characters indicate the itinerant ferromagnetism in Fe₃GeTe₂. The two important parameters of Stoner model, i.e., D(E_F) and I, were determined from DOS as 1.56 states/eV per Fe atom and 0.71 eV, respectively. Therefore, Stoner's criterion of I×D(E_F) > 1 is satisfied, giving rise to the itinerant ferromagnetic ordering in monolayer Fe₃GeTe₂. The validity of Stoner model was also extended to the successively reported 2D Fe₅GeTe_2,¹⁴³ which has a Curie temperature of 270 K.⁴² 

Intrinsic vdW ferromagnets with T_C above 300 K have been experimentally reported in some binary transitional metal dichalcogenides (MX₂) with structural phases containing the octahedral units [Fig. 7(b)], including 1T-MnSe_x,¹⁴⁴ 1T-VSe_2,^34,145 1T-VTe_2,¹⁴⁶ and 1T-CrTe₂.¹⁴⁷ Both theoretical and experimental results suggested that exchange coupling due to the enhancement of itinerant type is responsible for their room-temperature ferromagnetism. The above four 2D room-temperature ferromagnets are compared to the corresponding elemental metals and some ferromagnetic elemental metals like Fe, Co, and Ni. As expected, only Fe, Co, and Ni among all the considered elemental metals meet the Stoner criterion to exhibit band ferromagnetism. In contrast, 2D VSe₂, CrTe₂, MnTe₂, and MnSe₂ sheets have much higher I×D(E_F) values than those of the corresponding elementary metals; thus the Stoner criterion is met for a band ferromagnetism in these 2D materials.¹⁴⁷ 

RKKY is a particular form of magnetic interaction that occurs in metals with localized magnetic moments. In the RKKY picture, the magnetic moments interact effectively through an indirect exchange process mediated by the conduction electrons. A localized magnetic moment induces spin polarization to the surrounding conduction electrons, and such polarization in turn couples to another neighboring localized moment. The coupling strength takes the form of distance-dependent exchange interaction given by

where k_F is the Fermi wave vector for two spin channels, and R_ij is the distance between the two magnetic atoms i and j. First, this interaction is of the long-range type. Second, FM or AFM ground states depend on the interatomic distance. Third, its strength oscillates with the distance. Similar to the Heisenberg model, the strength of exchange coupling is related to the magnetic transition temperature, but is not rooted in the symmetry. RKKY interaction is well understood in conventional 3D intermetallic compounds, which is the dominant coupling mechanism between rare earth ions. In 2D materials, however, only a few theoretical studies have directly discussed the existence of RKKY-type interaction.²² In this regard, Zhang et al.²² inferred that ferromagnetic interaction between Mn ions in MnSiTe₃ monolayer possibly originated from the RKKY coupling. Figure 7(c) displays the possible electron hopping paths between Mn ions of the nearest (N), next-nearest (NN), and next-next-nearest (NNN) interactions (J_N, J_NN, J_NNN). Unlike CrSiTe₃ and CrGeTe₃, the theoretical results showed that MnSiTe₃ monolayer is a half metal with the largest positive J_NNN, which is inconsistent with superexchange model. Benefiting from the half metallic nature, the mediation carriers between Mn ions are 100% spin-polarized free carriers. Figure 7(d) presents the RKKY interaction as a function of Mn–Mn distance. The RKKY contributions to the interaction for the N, NN, NNN neighbors in the model show good agreement with the results from first-principles calculations.²² Nevertheless, further theoretical and experimental efforts are still needed to exploit the RKKY mechanism in other 2D magnets.

Two-dimensional vdW magnets are defined as the layered materials that adopt ferromagnetic/antiferromagnetic ground states at finite temperature. The theoretical concept was first established 150 years ago.¹⁴⁸ Since 2004, many efforts have been devoted to searching for new 2D magnetic materials with spontaneous magnetization.⁸⁴ In 2017, for the first time, the intrinsic 2D ferromagnetism in atomically thin CrI₃ and Cr₂Ge₂Te₆ was demonstrated in an experiment.^28,37 Nowadays, finding 2D ferromagnets with high Curie temperatures remains an important issue for spintronics, and many promising 2D materials have been attained theoretically and experimentally over the past five years.

The first class of widely investigated 2D vdW magnets is binary transition metal halides. During the past 150 years, many bulk crystals of transition metal halides have been synthesized in the laboratory.¹⁴⁹ These bulk materials adopted simple AA or ABC layered stacking geometry with weak interlayer vdW interaction; thus, they can be mechanically exfoliated to fabricate 2D monolayer or few-layered sheets.^54,149 The compounds composed of transition metals with partially filled d shell are commonly observed, since they are easy to host local magnetic moments and produce ordered magnetism. Most of transition metal trihalides (MX₃), transition metal dihalides (MX₂), transition metal monohalides (MX), and other stoichiometric M-X monolayers reported to date are listed in Table II and categorized by their compositions as well as magnetic properties for discussion.

The successful exfoliation of bulk CrI₃ crystal into atomic monolayers opened a new era of 2D magnetism.²⁸ Using scanning magneto-optic Kerr microscopy, neutron scattering, and NMR spectroscopy, pristine monolayer CrI₃ has been proven to be an Ising ferromagnetic semiconductor with a bandgap of 1.2 eV,¹⁵⁰ a Curie temperature of 45 K,²⁸ and an out-plane MAE of 0.69 meV [Figs. 8(a)–(e)].⁵³ In fact, the study of magnetism of layered CrI₃ began even before that. In prior experiments, several theoretical groups already predicted robust long-range ferromagnetic ordering in the monolayer limit of CrI₃.^8,53,54 The reported magnetic moment, magnetic anisotropy energy, and Curie temperature of 2D CrI₃ were 3∼3.44 μ_B per formula unit, 0.65∼0.686 meV, and 61∼107 K, respectively.^8,53,54 The magnetic moment in monolayer CrI₃ with honeycomb lattice is contributed by the Cr³⁺ ions with electronic configuration of 3s⁰3d³ under edge sharing octahedral crystal field coordinated by six nonmagnetic I^¯ ions. In the octahedral crystal field, Cr³⁺ ions prefer S = 3/2 with three d electrons occupying the lower-energy t_2g triplet state via splitting. The long-range FM ordering is driven by the competition between direct antiferromagnetic exchange interaction of Cr–Cr sites and the superexchange interaction in the near 90° Cr–I–Cr bonds.⁸ The large magnetic anisotropy is attributed to the spin-orbit coupling of the intermediate I atom along the superexchange path.¹⁵¹ Using magneto-Raman spectroscopy, Xu et al.¹⁵² directly observed 2D magnons with acoustic magnon mode of 0.3 meV in monolayer CrI₃. Moreover, the magnetism in CrI₃ is layer dependent, that is, monolayer, trilayer, bulk CrI₃ systems are ferromagnetic, whereas the weak magnetic coupling between the individual FM monolayer leads to the AFM ground state in bilayer CrI₃ [Figs. 8(f)–8(h)].²⁸ An exfoliated thin film of CrI₃ sandwiched between graphene contacts acts as a spin-filter tunnel barrier, showing a record high tunneling magnetoresistance of 190 000%.^153,154

Motivated by the discovery of CrI₃, its sister transition metal compounds, i.e., chromium trihalides CrX₃ (X= F, Cl, Br) monolayers have been explored as potential 2D intrinsic magnetic semiconductors by many groups.^{30,53,54,104,155,156} Owing to the structural similarity with CrI₃, FM is also the ground state for monolayer CrX₃ (X= F, Cl, and Br), and the magnetic moment of each Cr³⁺ ion in these systems is about 3 μ_B. However, with the increase in the atomic radius of halogen, the theoretical bandgap reduces from 4.68 eV for CrF₃, to 3.44 eV for CrCl₃, to 2.54 eV for CrBr₃, and finally to 1.53 eV for CrI₃ at HSE06 level of theory.⁵³ Considering the microscopic origin of long-range FM ordering, a strongly increased hybridization between X-p and Cr-3d states will strengthen the Cr–X–Cr FM exchange interactions as X goes from F to I. However, direct AFM exchange interaction is expected to be weakened with increasing Cr–Cr distance along this series. Indeed, MC simulations based on classical Heisenberg model predicted Curie temperatures of CrF₃, CrCl₃, CrBr₃, and CrI₃ monolayers to be 41, 49, 73, and 95 K, respectively. Moving from Cl to I would increase the strength of spin-orbit coupling of halogen, which accounts for the enhancement of MAE from 0.031 meV for CrCl₃, to 0.186 meV for CrBr₃, and then to 0.686 meV for CrI₃.⁵³ Similar theoretical results were also reported in the other four papers.^{54,104,155,156} Experimentally, Gao et al.³¹ demonstrated ferromagnetism in 2D vdW CrBr₃ using direct d-d transition induced photoluminescence probing. They argued that spontaneous magnetization persists in monolayer CrBr₃ with Curie temperature of 34 K. The magnetic moment of each Cr³⁺ ion in monolayer CrBr₃ aligns in the out-of-plane direction, and the corresponding magnetic moment is about 3 μ_B. Although the Curie temperature and MAE are slightly smaller than those of monolayer CrI₃, monolayers of other chromium trihalides CrX₃ (X= F, Cl, Br) could be more stable in air.^28,157

Beyond chromium halides, the magnetic properties of vanadium trihalides (e.g., VI₃) monolayer were also intensively investigated as potential 2D magnetic materials. In the periodic table, vanadium locates as the left neighbor of chromium. Thus, V atom in VI₃ tends to adopt high-spin state of V³⁺ with 3d² electronic configuration, that is, two valence electrons occupy the triply degenerate t_2g state. Such partial occupancy could raise the possibility for weak Jahn-Teller distortion of the octahedra and the orbital ordering in vanadium trihalides, which would modify the superexchange interactions in its planar honeycomb lattice. Recently, two experimental groups confirmed that bulk VI₃ is a correlated Mott insulator with a bandgap of ∼1 eV.^29,158 In addition, the exfoliation energy from first-principles calculation revealed that VI₃ is more easily decoupled than CrI₃.¹⁵⁹ Hence, vanadium trihalides provide a new platform to investigate 2D magnets with S = 1. Using CVD method, Tian et al.²⁹ firstly confirmed layered VI₃ is a FM semiconductor with a Curie temperature of 50 K. It was further demonstrated to be a 2D Ising ferromagnet by DFT calculations, crystal field level diagrams, superexchange model analyses, and MC simulations.⁹ The monolayer limit of VI₃ has also been explored by first-principles calculations. The intrinsic ferromagnetism can persist and the Curie temperature is reduced to 17 K from the bulk value of 60 K.¹⁶⁰ Based on DFT calculations combined with self consistently determined Hubbard U approach, He et al.¹⁶¹ reported that monolayer VI₃ possesses not only intrinsic ferromagnetism but also exciting Dirac half-metallicity, and VCl₃ shows similar magnetic behavior. Their corresponding Curie temperatures were 80 and 98 K, respectively. For monolayer VBr₃, first-principles calculations predicted that it has intrinsic half-metallicity and high Curie temperature of 190 K. Moreover, topological nontrivial states, which were identified by calculations of Berry curvature and the corresponding edge states, surprisingly emerge at the Fermi level.¹⁶² 

Manganese atom is the right neighbor of chromium in periodic table. In the manganese trihalides MnX₃ (X = F, Cl, Br, I), low crystal field splitting is caused by the octahedral coordination of Mn ions. Thus, Mn ion has +3 oxide state with spin configuration of S = 2t_2g³e_g¹, exhibiting a magnetic moment of c.a. 4 μ_B. Sun and Kioussis predicted that 2D MnX₃ sheets are intrinsic Dirac half metals (DHMs).¹⁶³ The bandgaps of the minority spin channel from PBE+U calculations were 6.3, 4.33, 3.85, and 3.10 eV for MnF₃, MnCl₃, MnBr₃, and MnI₃, respectively. In-plane magnetization orientation was found in all MnX₃ systems and the magnetocrystalline anisotropy increases with increasing atomic size of halogen. Based on DFT derived exchange interaction parameters, the estimated Curie temperatures were greater than 450 K. Spin-polarized Dirac half metallic states in MnF₃, MnCl₃, and MnBr₃ were also confirmed by Tomar et al.¹⁶⁴ using DFT calculations with both PBE and HSE06 functionals. Based on Curie-Weiss mean field theory, MnF₃ was demonstrated as a room-temperature ferromagnet.

Two-dimensional intrinsic magnets are also predicted for many other trihalides of open-shell 3d transition metals (M = Ti, Fe, Co, Ni). For instance, TiCl₃ monolayer possesses weak interlayer interaction of 0.33 J/m², half metallicity with a bandgap of 0.6 eV in the majority spin channel, long-range FM ordering contributed by 3d valence states, and a high Curie temperature of 376 K.¹⁶⁵ FeX₃ monolayers were found to have antiferromagnetic ground state with Néel temperature of 70∼180 K.^162,164 Using DFT and DFT+U calculations, Sun et al.¹⁶² found Dirac spin-gapless half-metallic features in NiBr₃ monolayer, and the corresponding Curie temperature was 100 K. The NiCl₃ monolayer was also shown to have Dirac spin-gapless semiconducting characteristics and high-temperature ferromagnetism.¹¹⁰ The MC simulations based on Ising model demonstrated that the Curie temperature of NiCl₃ monolayer is as high as ∼400 K. The calculated Fermi velocity of Dirac fermions was about 4 × 10⁵ ms⁻¹. Among them, Ti, Fe, and Ni ions are still octahedrally coordinated to the halogen atoms, thus they can exist in either high-spin or low-spin state due to crystal field splitting. According to the calculated magnetic moment listed in Table II, low-spin state is observed in the cases of Ti³⁺ and Ni³⁺ with magnetic moment of about 1 μ_B. The magnetic moment of iron ion is found to be close to 4 μ_B, which corresponds to neither high-spin (5 μ_B) nor low-spin (1 μ_B) state of Fe³⁺. It could explain why AFM coupling is only found in FeX₃ series among the transition metal trihalides with CrI₃ type structure. For Co ion located in the octahedral coordination environments of bromides, non-magnetic behavior was observed in CoBr₃ with P-31m phase.¹⁶² However, DFT calculations¹⁶⁶ demonstrated that the P6/mmm phase of CoBr₃ monolayer hosts 2D intrinsic ferromagnetism with metallic behavior, Dirac cone, and quantum anomalous Hall effect simultaneously. Its Curie temperature was 264 K and Chern number was C = 2.

Another important group of transition metal trihalides contains the heavier 4d and 5d open-shell transition metal elements like Mo, Ru, Rh, Tc, Pd, Ir, Pt, and Os. Moving from 3d to 4d and 5d series increases the spin-orbit coupling effect, which is beneficial for designing 2D spintronic devices with large MAE, topological phenomena, and spin controlling. Antiferromagnetic coupling between Mo atoms was confirmed in the high-temperature phase of α-MoCl₃ by an combined experimental and theoretical study.¹⁶⁷ Unlike CrCl₃, α-MoCl₃ adopts the monoclinic AlCl₃ structure with space group of C2/m at room temperature. Originated from the magneto-structural phase transition, magnetic interactions in the high-temperature phase of α-MoCl₃ are stronger by at least one order of magnitude than those in the analogous CrCl₃ and CrBr₃. Similar coupled structural and magnetic transition is also expected in TcCl₃ and TiCl₃. Experimentally, the fractional Majorana fermion excitations of a Kitaev quantum spin liquid have been observed in α-RuCl_3,³² which leads to enormous amounts of research on the magnetic properties of layered α-RuCl₃. Motivated by the exfoliation of α-RuCl₃ monolayer from its 3D crystal,¹⁶⁸ the magnetic property has been further analyzed by DFT calculations and MC simulations.^55,156 It was demonstrated to be a stable 2D intrinsic ferromagnetic semiconductor. The obtained Curie temperature and MAE were 14.21 K and 0.95 meV, respectively. Using first-principles calculations, Kan et al.¹⁶⁹ predicted RuI₃ monolayer to be an intrinsic ferromagnetic quantum anomalous Hall (QAH) insulator with topologically nontrivial global bandgap of 11 meV. The Curie temperature and nearest-neighboring exchange coupling parameter were estimated to be 360 K and 82 meV, respectively. It was also found that FM RuCl₃ and RuBr₃ monolayers show similar electronic behavior as RuI₃ monolayer. However, their exchange energies are very small and sensitive to the choice of effective U value. For RuBr₃ and RuI₃ monolayers,¹⁷⁰ the bandgap, possible magnetic ground state, Curie temperature, and magnetic anisotropy energy have been reexamined by DFT calculations with PBE+U and inclusion of SOC. It was shown that they are FM semiconductors with indirect bandgaps of 0.7 and 0.32 eV, respectively. According to MC simulations, their magnetic transition temperatures from FM to PM were 13.0 and 2.1 K, respectively. The magnetic anisotropy energies obtained for RuBr₃ and RuI₃ were 5.26 and 12.88 meV, respectively. Robust intrinsic ferromagnetism has also been realized in 2D rhenium trihalides.¹⁷¹ The dynamic and thermodynamic stabilities were found in the heavier halides (Br and I), in contrast to the lighter halides (F and Cl). ReBr₃ and ReI₃ are half metals with large bandgap in the spin-up channel. Moreover, high Curie temperatures (390 and 165 K) and Chern number (C = –4) were obtained from DFT calculations with PBE functional. Both Weyl half semimetal and tunable QAH effects were simultaneously realized in monolayer PtCl_3,¹⁷² as signified by the in-plane magnetization, high Curie temperature, and mirror symmetry protected two 2D Weyl points. A room-temperature intrinsic QAH insulator was predicted in the ferromagnetic insulating OsCl₃ monolayer, which is characterized by an energy gap of 67 meV, a Chern number of C = 1, and a Curie temperature of 350 K.¹⁷³ 

Similar to transition metal trihalides, layered transition metal dihalides have also drawn significant attentions for exhibiting ferromagnetic, antiferromagnetic, and half-metallic characteristics. The structure of monolayer MX₂ is analogous to transition metal dichalcogenides, which contains a triangular lattice of transition metal cations. In these compounds, the metal ions are in the formal oxidation state of +2. Divalence and octahedral coordination renders V, Cr, Mn, Fe, Co, and Ni cations partially filled 3d³, 3d⁴, 3d⁵, 3d⁶, 3d⁷, and 3d⁸ electronic configurations, with S = 3/2, 2, 5/2, 2, 3/2, and 1, respectively. Considering their structural similarity, the sign of superexchange interaction is mainly determined by the orbital occupations, and thus a variety of magnetic ground states are anticipated.

The evolution of electronic and magnetic properties of these first-row transition metal dihalides MX₂ (M = V, Cr, Mn, Fe, Co, Ni; X = Cl, Br, I) has been systemically examined by first-principles calculations.^{100,111,142,174} Using PBE functional, T configuration (P-3m1) is energetically favorable for the monolayers of all considered cases, while correction by a Hubbard U term leads to inversion of the favorable monolayer configuration to H phase (P-6m2), as demonstrated for FeBr₂ and FeI₂.¹¹¹ Among them, FeCl₂, FeBr₂, and FeI₂ monolayers are ferromagnetic half metals, while CoCl₂, CoBr₂, NiCl₂, NiBr₂, and NiI₂ monolayers are ferromagnetic insulators. For VCl₂, VBr₂, VI₂, MnCl₂, MnBr₂, MnI₂, CrI₂, and CoI₂ monolayers, they were found to be antiferromagnetic semiconductors with bandgap in range of 0.2∼2.5 eV. Introducing U correction will largely enlarge their bandgap.¹¹¹ For example, the theoretical bandgaps of VBr₂ monolayer are 1.1 and 3.1 eV at PBE and PBE+U level, respectively. The easy axis of all eight abovementioned MX₂ monolayers in FM state is perpendicular to the basal plane with MAE in range of 0.02 to 0.89 meV. The highest T_C is observed in NiCl₂, which is 205 K predicted by Ising model.

Among the first-row transition metal dihalides, FeX₂ series have attracted more attentions, mainly due to the following two reasons: (1) largest out-of-plane MAE is found in FeCl₂, which is beneficial for the presence of 2D long-range magnetic ordering; (2) monolayer 1T-FeCl₂ films on Au(111) and graphite have been successfully synthesized using MBE technique.¹⁷⁵ Torun et al.¹⁷⁶ investigated the structural and magnetic properties of FeCl₂ monolayer using first-principles calculations. They found that 1T-FeCl₂ is more favorable than the 1H phase. Both PBE and HSE06 calculations demonstrated that 1T-FeCl₂ is an intrinsic half-metallic ferromagnet with Curie temperature of 17 K.¹⁷⁷ Hennig et al. have found that the Fe²⁺ ions in FeCl₂, FeBr₂, and FeI₂ are in a high-spin octahedral d⁶ configuration, resulting in a large magnetic moment of 4 μ_B.¹⁷⁸ A classical XY model with nearest neighboring coupling was used to estimate their critical temperatures, which range from 122 K for FeI₂ to 210 K for FeBr₂. Moreover, all three 2D FeX₂ materials as half metals were predicted to have appreciable electron densities of state at the Fermi level comparable to those of typical metals, suggesting good on/off ratios in spintronic devices.¹⁷⁸ 

In addition to first-row transition metals, the magnetic ground state of 4d/5d MX₂ monolayers in both 1T and 2H phases have been investigated by high-throughput first-principles calculations.¹⁰³ Among them, 23 out of 90 MX₂ monolayers exhibit robust magnetic ground states that are retained even after introducing the U terms. Besides the previously reported NiCl₂, VCl₂, MnCl₂, VBr₂, VI₂, MnI₂, and CoI₂, PtCl₂ is predicted to be a new noncollinear antiferromagnetic insulator. Meanwhile, AgCl₂, AgBr₂, and AuI₂ are found to be half metallic ferromagnets with spin splitting of 0.2∼0.5 eV. To find more 2D intrinsic magnets in MH₂ family, Shen et al.¹⁷⁹ also focused on 5d transition metal based MX₂ monolayers. They screened more than 6000 kinds of 2D electrenes (i.e., materials with excess electrons acting as anions) and found that LaBr₂ is one of most intriguing FM semiconductors with unusual long-range ferromagnetism induced by anions. From DFT calculations, its on-site moment, Curie temperature, and coercive field are 1 μ_B, 235 K, and 0.53 T, respectively.¹⁷⁹ For LaBr₂ monolayer, delocalized spin density in the intermediate region between La atoms was also unveiled in another theoretical paper by Jiang et al.⁶⁵ 

Among transition metal halides, MX compounds have the simplest stoichiometric ratio of 1:1. Without octahedral type crystal field, MX monolayer structures mainly consist of double hexagonal layers of metal atoms sandwiched by two hexagonal layers of halogen atoms. It is interesting to ask whether such MX monolayers still possess long-range magnetic ordering. By comprehensive first-principles calculations and MC simulations, Jiang et al.⁶⁵ have identified ScCl, YCl, and LaCl monolayers as ferromagnetic metals with appreciable Curie temperatures of 280, 240, and 260 K, respectively, while PBE+U calculations revised those values to 355, 460, and 260 K, respectively. Band structure analysis has shown that the spin density is relatively delocalized in the intermediate region between metal atoms, resulting in a small magnetic moment of 0.364, 0.333, and 0.317 μ_B per Sc, Y, and La atom, respectively. Wang et al.¹⁸⁰ also found that 2D ScCl monolayer is an intrinsic ferromagnet with large spin polarization. Their predicted Curie temperature was 185 K. Even so, both studies indicated that introduction of more transition metal in M-X systems would strongly quench its magnetic moment, while retaining Curie temperature at rather high value.

Niobium halides form another kind of potential 2D vdW magnets with an unusual composition—Nb₃X₈. In Nb₃X₈ monolayer,¹⁸¹ triangular Nb clusters are formed by Nb atoms; and consequently, every Nb atom is still arranged in a distorted octahedral environment. Therefore, both ferromagnetic ground state and semiconducting behavior were found in 2D Nb₃X₈ (X = Cl, Br, I) monolayers from GGA+U calculations. The Curie temperatures estimated by mean field approximation based on Heisenberg model were 31, 56, and 87 K for Nb₃Cl₈, Nb₃Br₈, and Nb₃I₈, respectively. Among them, the Nb₃I₈ monolayer has been successfully cleaved from its bulk phase.¹⁸² Without considering U term, Xiao et al.⁷⁴ investigated the magnetic properties of the family 2D V₃X₈ (X = F, Cl, Br, I) in the framework of DFT. They found that V₃Cl₈ monolayer is an intrinsic AFM semiconductor, while the other three systems are FM half metals. The estimated Curie temperatures from MC simulations were 77 and 103 K for 2D V₃F₈ and V₃I₈, respectively.

Two-dimensional binary transition metal chalcogenides, including transition metal dichalcogenides, transition metal monochalcogenides, and other stoichiometries in a general form of M_mX_n (M refers to transition metal, and X represents S, Se, and Te), have provided a gorgeous platform for exploring interesting electronic and magnetic properties, such as valley polarization and 2D magnetism.

Among 2D binary transition metal chalcogenides, the most widely studied ones are TMDs. Generally speaking, 2D TMDs form sandwich type structures in the X-M-X sequence, where transition metal atoms are sandwiched in between two layers of chalcogen atoms. There are four reported structural phases for TMDs, i.e., trigonal prismatic H-phase, octahedral T-phase, distorted octahedral 1T'-type, and T_d-type lattices.¹⁸³ In all these phases, each transition metal atom is surrounded by six chalcogen atoms. The five formerly degenerate d orbitals of 3d transition metal ion would split in energy as it is bonded to the chalcogen ligands. Under the crystal field with D_3h symmetry in H phase, the five degenerate 3d orbitals split into a single state a₁ (d_z2) and two twofold degenerate states e₁ (d_x2-y2/d_xy) and e₂ (d_xz/d_yz). While in T phase, the triangle sublattice of transition metal atoms gives rise to first-neighboring coordination number of 6, forming octahedral crystal field; thus, the d states split into t_2g and e_g manifolds. Due to the trigonal distortion, t_2g degeneracy is further lifted to form higher-lying a_1g level and twofold degenerate e_g states in T′ and T_d phases. Intuitively, the magnetic properties of 2D TMDs should be determined by splitting and filling behavior of d orbitals of the transition metal ions under various crystal fields. For the same transition metal ion, the electronegativity of a chalcogen atom also plays some role, such that the lighter chalcogen atom draws more electrons from the metal ions and affects their on-site magnetic moments.

It was computationally confirmed that intrinsic long-range magnetic ordering can be realized on the transition metal sites with respect to delocalized p states of S/Se/Te atoms in a variety of transition metal dichalcogenides. Early in 2002, extensive analyses of the stability of TMD monolayers based on DFT calculations predicted that, out of 88 combinations of TMDs compounds, 52 H or T structures can occur as the freestanding phase. Among them, H-phase TMDs with M = Cr, Mo, V, Mn, Co, and W, and X = S, Se, and Te were predicted to be ferromagnetic metals with net magnetic moment ranging from 0.2 to 3.0 μ_B per formula.⁵⁶ Similarly, Chen et al.⁵⁷ also systematically explored the magnetic properties of MTe₂ (M = Ti, V, Cr, Mn, Fe, Co, Ni) monolayers in both H and T phases. Their results indicated that H-VTe₂, T-MnTe₂ and H-FeTe₂ are ferromagnetic metals with magnetic moments of 0.78, 2.80, and 1.48 μ_B per formula unit (f.u.), respectively, while T-VTe₂ is an indirect bandgap semiconductor with a magnetic moment of 1.0 μ_B per formula. Using 2D Ising model and MFT, the estimated Curie temperatures were 301, 33, 88, and 229 K for H-VTe₂, T-VTe₂, T-MnTe₂, and H-FeTe₂ monolayers, respectively. Moreover, non-collinear DFT calculations revealed that H-VTe₂, T-VTe₂, and H-FeTe₂ monolayers have in-plane easy magnetization direction with MAE values of 0.51, 1.74, and 2.57 meV/f.u., respectively, while T-MnTe₂ has a perpendicular easy magnetization axis with MAE of 0.54 meV/f.u. DFT calculations within LDA+U approximation demonstrated that single-layer VS₂ of 1T phase is a strongly correlated material, where 2H structure of monolayer VS₂ is a ferromagnetic semiconductor.¹⁰ The magnetic moments are localized on the V atoms and couple ferromagnetically via superexchange interactions mediated by the S atoms. Calculations of magnetic anisotropy showed an easy plane for the magnetic moment in 2H VS₂. The magnetic properties of 2D NbS₂ and ReS₂ nanosheets have been investigated using the HSE06 hybrid functional.^184,185 Both of them are bipolar magnetic semiconductors with spin gaps of 0.27 and 1.63 eV. Furthermore, MC simulations predicted the Curie temperatures to be 141 and 157 K for 2D NbS₂ and ReS₂ systems, respectively.

Among the large family of potential 2D magnetic TMDs, VSe₂, VTe₂, MnSe_x, NbTe₂, NbSe₂, and CrTe₂ have attracted more and more attentions from both experimental and theoretical aspects, mainly because of the reported room-temperature T_C. In the following content, we will discuss the major developments that have paved the way to this point. First, all these six materials belong to the family of T phase [Fig. 9(a)] and exhibit metallic nature. The magnetic coupling mechanism stems from the competition between indirect superexchange and itinerant exchange interactions. Second, 2D metallic TMDs are well-known charge density wave (CDW) systems and the CDW phase transition steers the change of electronic structures in them. Therefore, the correlation effect and the possible competition between CDW ordering and magnetic ordering also needs to be clarified.

Monolayer VSe₂ material has been reported as one of the first room-temperature 2D ferromagnets.³⁴ From the electronic structure point of view, VSe₂ exhibits a 3d¹ configuration, which invokes both metallic and magnetic properties. Bonilla et al.³⁴ synthesized single- and few-layer VSe₂ sheets on HOPG and MoS₂ substrates using MBE and performed magnetic characterization by protecting the films with a Se capping layer. Their studies revealed a significant enhancement of magnetic moment in single-layer samples compared with multi-layer ones. Surprisingly, the ferromagnetic ordering is very robust and persists above room temperature. The room-temperature ferromagnetism was also observed on the exfoliated 2D VSe₂ flakes using superconducting quantum interference device (SQUID), XMCD [Figs. 9(b) and 9(c)], and magnetic force microscopy (MFM), where the monolayer flake displayed the strongest ferromagnetic properties.¹⁸⁶ First-principles calculations using both GGA and high-level LDA+DMFT (dynamical mean-field theory) approaches explained that it might be a ferromagnet with both itinerant and localized characters.^17,18,20 For example, LDA+DMFT calculations predicted ferromagnetic ordering in VSe₂ monolayer without CDW below 250 K. However, the origin of ferromagnetism in VSe₂ has spurred great controversies. First, the reported experimental magnetic moments (5∼15 μ_B)^34,187 are too large in comparison with the calculated values (0.365∼0.6 μ_B).^17,18,20 Second, no magnetic signal was detected in another XMCD experiment and the angle resolved photoemission spectrum showed no exchange splitting.^188,189 Third, several studies^{145,187,189–191} suggested CDW transition distortion can suppress the intrinsic ferromagnetic ground states in VSe₂. Wong et al.¹⁴⁵ observed traits of spin frustration in monolayer VSe₂ with long-range intrinsic ferromagnetism from complementary temperature- and field-dependent susceptibility measurements. They have also reported that the frustrated intrinsic magnetism in 2D VSe₂ can be lifted by the introduction of the Se-deficient defects.¹⁹² Yu et al. inferred that a defect-free sample is the key to verify the intrinsic ferromagnetism of VSe₂.¹⁸⁶ Nakano et al.¹⁹³ demonstrated the emergence of intrinsic ferromagnetism in V₅Se₈ (V_0.25VSe₂) epitaxial thin films grown by MBE, which can be classified as itinerant 2D Heisenberg ferromagnets with weak magnetic anisotropy.

Due to strong 3d¹ electron coupling in the neighboring M⁴⁺–M⁴⁺ pairs (M = V, Nb, Ta) of 2D TMDs, metallic VTe₂, NbTe₂, NbSe₂, and TaTe₂ systems composed of group VB elements have been regarded as the potential intrinsic magnets. In some theoretical investigations, 2D VTe₂, NbTe₂, and TaTe₂ have been predicted to exhibit intrinsic magnetic ordering.^{58,59,194,195} In particular, VTe₂ monolayer was found to be a room-temperature ferromagnet with highest T_C value of 553∼618 K. Meanwhile, MC simulation of the hysteresis features of VTe₂ monolayer illustrated that it is possible to observe finite remanence and coercivity treatments nearly or well beyond room temperature.¹⁴⁶ Similar to the discussions on VSe₂, Wong et al.¹⁹⁶ also found that CDW order would rule out the ferromagnetic behavior in VTe₂ monolayer. Experimentally, the single crystalline ultrathin VTe₂, NbTe₂, and TaTe₂ sheets were synthesized using atmospheric pressure CVD approach.³⁵ The magnetic hysteresis (M‐H) measurements demonstrated that VTe₂ and NbTe₂ exhibit room‐temperature ferromagnetism [Figs. 9(d) and 9(e)]. The reported saturation magnetization, coercivity, and remnant magnetization values of VTe₂ were 0.3 emu g⁻¹, 1173.0 Oe, 0.16 emu g⁻¹ at 10 K, and 0.21 emu g⁻¹, 592 Oe, 0.10 emu g⁻¹ at 300 K, respectively. However, the competition between CDW and magnetic instability in VTe₂ is still under debate. On the one hand, formation of the CDW phase in 2D VSe₂ was observed in experiment, and three possible CDW transitions at 135, 240, and 186 K have been reported.^35,197,198 The CDW effect would suppress the magnetic instability and further lead to the absence of magnetic ordering. Through a combination of in situ microscopic and spectroscopic techniques, Wong et al.¹⁹⁶ observed a 4 × 4 CDW order and further excluded the intrinsic ferromagnetic ordering in VTe₂ by XMCD data. One the other hand, Sugawara and coworkers¹⁹⁹ recently found a large triangular Fermi surface at the K point that satisfies a nearly perfect nesting condition, whereas CDW is suppressed as highlighted by the observation of band crossing of the Fermi level at low temperature, in contrast to monolayer VSe₂. Combining DFT calculations with scanning tunneling microscopy and spectroscopy (STM/STS) measurements, ferrimagnetic ground state of 2D NbSe₂ was demonstrated with a magnetic moment of 1.09 μ_B.²⁰⁰ More importantly, their results also inferred that substrate is the key to verifying the intrinsic ferromagnetism of TMD materials.^196,200 It was shown that single-layer NbSe₂ does not display CDW instability unless a graphene layer is utilized as substrate.

Motivated by DFT calculations,^56,60 O'Hara et al.³⁶ also observed room-temperature ferromagnetism in manganese selenide (MnSe_x) films grown by MBE. Magnetic and structural characterizations provided strong evidence that, at the monolayer limit, ferromagnetism originates from the vdW MnSe₂ monolayer. Using DFT calculations combined with MC simulations, Kan et al.⁶⁰ have shown that 2D MnSe₂ sheets are ideal magnetic semiconductors with long-range magnetic ordering, where all Mn atoms are ferromagnetically coupled and the estimated T_C is 250 K [Fig. 9(f)]. Interestingly, first-principles calculations further revealed the great defect tolerance in MnSe₂. Despite the presence of high-density Se vacancies, the defective MnSe₂ monolayer can retain its stable ferromagnetic behavior.¹⁴⁴ Magnetic tunneling junctions based on monolayer MnSe₂ with room-temperature ferromagnetism were also observed with a large tunneling magnetoresistance of 725%.²⁰¹ Moreover, by optical and electronic measurements, Sun et al.¹⁴⁷ disclosed that the intrinsic ferromagnetically aligned spin polarization can hold up to 316 K in a metallic phase of 1T-CrTe₂. Detailed spin transport measurements suggested half-metallicity in its spin polarized band structure as well as in-plane room-temperature negative anisotropic magnetoresistance. Importantly, their study found that exchange coupling due to an enhancement of itinerant type was the source of room-temperature ferromagnetism in both bulk and few-layered Cr₂Te₃.²¹ 

Owing to the variable valence of transition metal elements, transition metal chalcogenides have diverse stoichiometric compositions. In addition to TMDs with M:X = 1:2, the 2D transition metal chalcogenide compounds with higher stoichiometries (M:X = 5:8, 2:3, 3:4, and 1:1) also exhibit interesting magnetic properties. In V₅S₈ nanosheets, an AFM to FM phase transition was observed when the thickness is down to 3.2 nm. Using DFT calculations, Zhang et al.²⁰² further investigated the thickness-dependent magnetic ordering in V₅S₈ thin films, and confirmed an antiferromagnetic to ferromagnetic phase transition when V₅S₈ is thinned down to 2.2 nm. The magnetic moments of the thin films in both antiferromagnetic and ferromagnetic states are mainly located on V atoms in the intermediate layer. Utilizing vdW epitaxy techniques, Cr₂S₃ sheets with one-unit cell thickness down to 1.78 nm have been successfully synthesized, which exhibited ferrimagnetic behavior with a Néel temperature of 120 K and the maximum saturation magnetic momentum of up to 65 emu.²⁰³ In addition, Lv et al.²⁰⁴ identified Co₂Se₃ as a 2D half-metal among a series of M₂Se₃ candidate materials, and the calculated T_C from the mean field theory was about 600 K. Using first-principles calculations, Ouyang and co-workers predicted a family of stable 2D honeycomb lattices of Cr₂X₃ (X = O, S, Se). Cr₂S₃ and Cr₂Se₃ are ferromagnetic half-metals with mirror symmetry protected nodal lines for the spin-down channels, while Cr₂O₃ layers are ferromagnetic semiconductors with large out-of-plane MAE.²⁰⁵ A hexagonal Ta₂S₃ sheet has also been predicted as 2D magnet from the spin-wave theory, which possesses sizeable out-of-plane MAE of 4.6 meV and high Curie temperature of 445 K.²⁰⁶ 

Based on first-principles calculations, a new composition of stable 2D transition metal chalcogenides, i.e., Cr₃X₄ (X = S, Se, Te) monolayers, has been predicted to possess fascinating magnetic properties.¹⁵ Among them, Cr₃S₄ monolayer is a ferrimagnetic semiconductor, while Cr₃Se₄ and Cr₃Te₄ monolayers are ferromagnetic half-metals with T_C of 370 and 460 K, respectively. Unlike the d–p–d superexchange interaction found in the other transition metal compounds, double exchange magnetic coupling mechanism is dominated in these 2D Cr₃X₄ sheets, finally leading to enhanced FM ordering and room temperature T_C. That is to say, a delocalized unpaired electron could hop between the two neighboring Cr ions with different oxidation states in 2D Cr₃X₄.

Using PSO technique combined with first-principles calculations, Zhang et al.²⁰⁷ predicted a new transition metal chalcogenide monolayer composed of cobalt and sulfur atoms—Co₂S₂. Their results revealed that a single-layer Co₂S₂ sheet is a ferromagnetic metal with a Curie temperature of 404 K. Two-dimensional ultrathin CrSe crystals were successfully synthesized on mica substrate via ambient pressure CVD method.²⁰⁸ Such CVD-grown 2D CrSe crystals exhibit evident ferromagnetic behavior at temperatures below 280 K. Kang et al.²⁰⁹ reported the synthesis of ultrathin FeTe film by CVD approach and discussed their structural and magnetic transition. Transport measurements revealed that tetragonal FeTe is an antiferromagnetic metal with T_N of about 71.8 K, while hexagonal FeTe is a ferromagnetic metal with T_C of around 220 K. Very recently, Yuan et al.²¹⁰ have grown FM MnSe monolayer on silicon substrate using MBE method. The thickness dependence of Curie temperature was found in the MnSe ultrathin films. The measured T_C was 54 K for monolayer, while it sharply increased to 225 K for three-layer MnSe, and 235 K for four-layer MnSe.

MXene, a category of 2D transition metal carbides, nitrides, and carbonitrides, possibly terminated by functional group (T) on the surface, with a general formula of M_n+1X_nT_x (M = transition metal; X = C and/or N; T = O, OH, F) are attractive additions to the family of 2D materials. Since the discovery of Ti₃C₂ in 2011, MXenes of Ti₂C, V₂C, Nb₂C, Mo₂C, Zr₃C₂, Nb₄C₃, Ta₄C₃, and Ti₄N₃, as well as of TiNbC, (Ti_0.5Nb_0.5)₂C, (V_0.5Cr_0.5)₃C₂, Ti₃CN, Mo₂TiC₂, Mo₂ScC₂, Cr₂TiC₂, Mo₂Ti₂C₃, (Nb_0.8Ti_0.2)₄C₃, and (Nb_0.8Zr_0.2)₄C₃ have already been fabricated in a laboratory. According to the large number of M-X compositions, more than a hundred MXenes have been theoretically predicted.²¹¹ 

The diverse compositions and controllable thickness of M_n+1X_nT_x systems provide an ideal playground to achieve 2D intrinsic magnetism (Table III).^92,211,212 As mentioned above, transition metal M atom usually has a partially filled d shell with unpaired electrons. First of all, the magnetic properties of MXene are influenced by the total number of d electrons. The M site can be also occupied by the ordered double transition metal species, i.e., M′M″. Generally, M′ atoms locate in the outer layers and M″ atoms occupy the middle layer, yielding [M′X]_nM″ arrangement. However, some in-plane ordered double transition metal MXenes have also been reported. The competition of electron hopping and electronic coupling between M′ and M″ ions will further modify the magnetic ground state of MXene. Each X (C or N) anion bounds with six transition metal M cations, forming XM₆ octahedral configuration. The transition metal atoms on the surface of M₂X are subjected to a C_3v ligand field contributed by the neighboring X atoms. Hence, the five 3d orbitals of transition metal atom would split into a single state a₁ (d_z2), two twofold degenerate states e₁ (d_x2-y2/d_xy) and e₂ (d_xz/d_yz). The splitting between a₁ and e₁/e₂ is determined by the strengths of M–X interactions. In addition, long-range spin interaction between the metal atoms always occurs through X atoms in MXene, which plays an important role in mediating the magnetic coupling.