Recent advances in two-dimensional van der Waals magnets

Categories and synthesis of 2D van der Waals magnets

Generally, most magnet moments in 2D magnets originate from the orbital and spin angular momenta of the 3d or 4f electrons in the transition metals, and often interact with their periodic symmetric structures such as crystal field^[74]. Based on their crystalline period symmetries shown in Figure 2, the classification of the discovered 2D magnets can be mainly divided into hexagonal, triangular, tetragonal, and orthorhombic lattices^[75]. The first group is the layered transition metal hexagonal lattices, which include binary transition metal trihalides MX₃ (M = Cr, V or Ni; X is a trihalide)^[8,58,59], NPS₃^[76,77] (N = Mn, Fe or Ni) and Cr₂Z₂Te₆^[78] (Z = Si or Ge). Second, the structure of Fe₃GeTe₂^[79], MnBi_2nTe_3n+1^[73,80], and prismatic (2H) or octahedral (1T) binary transition metal dichalcogenides such as AY₂ (A = V, W or Mo; Y = S and Se)^[53,62,81]are classed as triangular lattice magnets. Third, square and rectangular lattices are also potential 2D magnets and include binary tetragonal iron monochalcogenides, trinary orthorhombic transition metal hetero-anionic halogenide chalcogenides, and trinary orthorhombic vanadium oxide bi-halogenides^[75]. Most of the currently known 2D FM materials can be classified using the above systems.

Figure 2. Classification of crystal structures of 2D ferromagnets and their popular synthesis routes, including chemical vapor transport^[83], flux melting^[84], atomic layer deposition^[86], chemical vapor deposition^[85] and molecular beam epitaxy^[87].

Owing to the fabrication of monolayer or few-layer magnetic crystals mainly referring to their crystal type and phase diagrams^[71], the selection of an effective synthesis method is a prerequisite for realizing high-crystallinity, high-quality, low- or non-defective, and ambient stable 2D magnets with controllable atomic thickness. The popular chemical vapor transport method is an effective and reliable technique that can crystallize macroscale high-crystallinity layered magnets, including MPS₃^[82], CrX₃^[59], VSe₂^[3], Cr₂Ge₂Te₆^[78], and Fe₃GeTe₂^[83], where transport agent and temperature zone control have significant impacts on the synthesis. Moreover, sublimation techniques have proved to be useful, direct, and low-cost self-transport methods that have been used to grow transition metal halides. Based on phase diagrams, a flux zone growth method, with the solution phase spontaneously nucleating and crystallizing, which requires high solubility, low competing phases, and the removal of exceeding flux, has been applied to achieve large and high-crystallinity 2D magnets^[84]. Specifically, the synthesis strategy relies on the characteristics of the material and its environmental demands. Generally, the above-mentioned products synthesized by the top-bottom strategy always require extra isolation, such as the use of Scotch tape, substrate-assisted exfoliation, and liquid exfoliation to obtain atomically thin films.

In contrast, bottom-top thin film synthesis can be a promising option, with chemical and physical vapor deposition^[85], atomic layer deposition^[86], molecular beam epitaxy (MBE)^[87], magnetron sputtering and thermal evaporation having been successful. These space-confined growth methods provide good thickness control to enable the fabricating restriction at a low crystallization speed via in-situ nucleation and growth dynamics. Among them, the chemical vapor transport and physical vapor deposition methods have been widely dedicated to the fabrication of various 2D magnets, including VSe₂^[64], VTe₂^[88], and CrI₃^[89]. The precursors directly react without additional reagents, and the thickness and size of the crystals show a strong relationship with time, temperature, carrier gas type, and flow rate^[90]. Another advanced approach producing high quality 2D magnets is the MBE technique, which is usually attached to characterization facilities, such as electron diffraction (ED), scanning tunneling microscopy (STM), angle-resolved photoemission spectroscopy (ARPES), etc., and thereby realize the in-situ characterization of the properties for the as-synthesized materials^[64,91].

Some novel 2D magnets are expected to be fabricated by adjusting the relevant synthesis parameters, such as reaction temperatures, substrate types, and dopants. In particular, when the total energy of the system restricts the crystal structural stability limit, the self-limited epitaxial growth can be used to directly obtain atomically thin films by challenging the intrinsic isotropic chemical environments and the unpaired dangling bonds at the vdW surfaces. This can effectively extend the synthesis scope of vdW layered materials for producing 2D ferromagnets^[92]. The theoretical and experimental study of the atomic distribution, time and spatial symmetry, and the atomic bonding features at crystal and vdW surfaces has a significant impact on the controllable fabrication of 2D magnets and has accelerated the design of novel and stable high-temperature 2D ferromagnets^[9]. Under the guidance of these methods and theories, 2D magnetic materials systems are currently rapidly expanding.

Theory of 2D magnetism

Among the effective interactions for neighboring spins, the indication of forming magnetic order lies in the magnetic moments aligning to a certain direction with the spontaneous breaking of the time-reversal symmetry (TRS)^[93]. This long-range arrangement is dominated by several general parameters and tends to be violated by the fluctuations at finite temperature (critical temperature), particularly the dimension factor in critical behavior^[94]. For low-dimensional magnets, the long-range ordered spin waves or magnons in this Goldstone mode can be destroyed by thermal fluctuations at low-energy excitations, particularly for 2D systems with more complex competitive relationships. The case of a gapless collective excitation, known as Goldstone modes^[71], is often associated with the broken TRS phase and is widely discussed in the contexts of Landau-Peierls instability^[71], Hohenberg superconductors and superfluids^[46] and the Mermin-Wagner theorem^[47]. For this description, the survival of long-range order at finite temperature basically depends on the level of phonon excitation and the spin dimensionality, thereby determining the critical magnetic characteristics of the system, such as the transition temperature and magnetic anisotropy, as shown in Figure 3A^[95].

Figure 3. (A) Magnon DOS variation with the system evolving from 2D isotropic to 3D and from isotropic to uniaxial magnetic anisotropy^[95]. (B) Schematic of spin orientation and critical temperature change with evolving spin dimensionality, where n = 1 correlates to the Ising Hamiltonian with strong uniaxial anisotropy and n = 2 and n = 3 correspond to an easy-plane anisotropy (XY Hamiltonian) and isotropic Heisenberg Hamiltonian, respectively^[94]. (C) Exchange interaction modes and preferred magnetic orientation showing diverse magnetic anisotropy^[95].

In view of spin-wave theory, the Mermin-Wagner-Hohenberg theorem claims that the gapless long-wavelength excitation can be easily excited at any non-zero temperature that would destroy the long-range ferro- and antiferromagnetism in the isotropic Heisenberg model^[47]. It is noteworthy that this proof is centered on a discussion regarding exchange interactions at finite temperature based on isotropic Heisenberg models that theoretically rule out the existence of intrinsic long-range crystalline and magnetic order in 2D and 1D systems. In other words, it does not preclude the magnetic order in situations where the spin dimensionality n is less than 3 or the possibility of stable magnetism by introducing external factors and intrinsic anisotropy^[96]. Consequently, the introduction of anisotropy (low spin dimension) may be applicable for protecting magnetism in low-dimensional systems. Therefore, according to relativistic effects, long-range magnetism in low-dimensional systems can be achieved by breaking the spin symmetry by introducing magnetic anisotropy. For example, for the spin dimensionality of 3 in Figure 3B, the isotropic Heisenberg model (n = 3) is absent of magnetic anisotropy, while the Ising model (n = 1) and XY (n = 2) circumstances enable strong easy-axis and easy-plane anisotropy at a non-zero temperature^[94]. Generally, for most magnetic insulators and conductors, simplifying a spin Hamiltonian is appropriate to discuss the magnetic properties to some extent. This description mainly contains four factors, namely, exchange (Heisenberg) interactions, anisotropic exchange, single-ion anisotropy, and the introduction of an external magnetic field. However, in specific situations, additional terms, such as Kitaev interactions^[97,98], biquadratic exchange^[99], Dzyaloshinsky-Moriya interactions^[100,101], dipolar interactions^[102], interlayer interactions^[103] and surface magnetic anisotropy^[96], should also be considered, which may also contribute to anisotropy.

To better understand this complex, competitive behavior, the presence of magnetic interactions must be introduced, as described in Figure 3C^[95]. First, for the exchange interaction, the primary origin of this magnetic interaction lies in the effectiveness of the interaction, including the direct exchange amid neighboring spins and indirect exchange mediated by the intermediate states of the atoms. Herein, the direct exchange is heavily localized spatially, and indirect exchange is delicate to the energy difference and hopping energy of the intermediate state. The change in band energy by thickness modulation is responsible for the evolution of indirect exchange. Next, a source of anisotropic exchange derives from the two localized spins of bonding element and is mainly determined by spin-orbit coupling (SOC). With the SOC interaction, the crystal field breaks the degeneration orbital that could produce the unquenched orbital moments, resulting in magnetic anisotropy. Generally, the ligand anion generates large SOC that assists anisotropic exchange, and the SOC in the transition metal contributes to the magneto-crystalline anisotropy, as well as the orbital magnetic moment^[58]. Another case of easy-plane magnetocrystalline anisotropy mainly takes place in dipolar interaction Heisenberg magnets^[102] with a Kosterlitz-Thouless phase transition. Another case is the single-ion anisotropy from the interplay between the local crystal field and SOC, where its Hamiltonian depends on the high-spin state of the ion from the perturbation theory. With degeneracy, this anisotropy belongs to the first-order perturbation, whereas the second-order perturbation is another case of anisotropy without degeneracy^[71]. For example, for transition metals with a well-defined symmetric crystal field, the single-ion anisotropy is regarded to be stable, and this situation can be modulated by the Jahn-Teller effect in the octahedral structure. In short, magnetic interactions are an important origin of magnetic anisotropy and provide the main contribution to magnetism.

Magnetic states belong to an exhibition of the macroscopic features of ferromagnetic behavior from these interactions. First, as described above, the macroscopic magnetic characteristics are mainly attributed to the interactions between the local spin or the itinerant ferromagnetism of magnetic ions, including interlayer and intralayer accumulations/competitions. As shown in Figure 4, an FM state shows all spins in the lattice arranged in the same direction, while AFM states represent zero net moments where neighboring spins usually align in opposite directions on different sublattices. The magnetic state and order have a close relationship with the lattice parameters. Second, the magnetic anisotropy energy (MAE) represents the energy difference from the magnetic easy axis to the hard axis, showing the degree of stability of the long-range spin order in 2D materials. According to the magnetic moment arrangement orientations, they are classified as Heisenberg-, XY- and Ising-type magnets and are dominated by the interaction between the magnetic moments and the crystal field. Third, numerous internal or external conditions, including fields, fluctuations, and interactions, exhibit fierce competition at different temperatures or magnetic fields, leading to complex moments and multiple critical points that describe the critical magnetic phase transition behavior in 2D magnets. This evolutionary process is appealing for studying the interplay among 2D magnets.

Figure 4. Correlation of crystal structure and the corresponding spatial ordering distribution for each material family, including hexagonal (A) CrX₃, (B) Cr₂Ge₂Te₆ and (C) NPS₃ and triangular (D) Fe₃GeTe₂, (E) AY₂ and (F) MnBi_2nTe_3n+1.

Theoretical modeling is a powerful tool for predicting possible ferromagnets with the desired magnetic behaviors. For example, the group of Kan proposed several approval solutions by calculation predictions. The first step was to determine a certain magnetic semiconductor element A in the transition metal region, followed by a selection of another element B that should meet the same occupation state as A when embedded in the basis. Finally, the energy gap opened by the crystal field between A and B should be considerable^[9,104]. Furthermore, the spin-polarized materials only with s or p orbital hybridization should meet the coexistent demand of super-exchange and carrier-mediated interactions. This proposal also effectively promoted the research into the design of advanced ferromagnets.

The development of 2D magnets remains a work in progress, and the origin of the interactions and magnetic behavior at the 2D limit requires further study. In particular, a simplified model to describe the complex mechanism of interactions and the ability to predict novel 2D ferromagnets with good magnetic performance is required. These factors could help us to design and synthesize an ideal 2D vdW ferromagnet with robust magnetic behavior and high stability, thereby promoting the research into condensed physical properties.

Emergent physical properties and tunable ferromagnetism

In addition to ferromagnetism, other exceptional physical properties and their interactions with magnetism have also motivated many studies and potential applications. In Figure 5A, many fascinating magnetic properties can be explored in intrinsic 2D magnets^[105]. For example, considering the band energy in vdW materials, achieving the combination of ferromagnetic and semiconductor properties in ultrathin films has realized the long-term desire of spintronics. In contrast, atomically thick ferromagnetic half-metals and metals are also good choices for emerging robust spin orders, such as high critical temperatures^[4] and spin polarization ratios^[38]. Furthermore, many interesting topological quantum phenomena can be reached by controlling the spin parameters in 2D ferro- and antiferromagnets, such as axion and Chen insulator behavior with quantum anomalous Hall effects, as shown in Figure 5B^[106]. Magnons can represent a collective behavior of magnetic moments that can be basically regarded as a quantized spin wave and obey Bose-Einstein condensation^[107]. Therefore, they can associate spin states with magnon behavior in 2D vdW magnets. Dzyaloshinskii-Moriya interactions can stabilize topological spin textures known as skyrmions in 2D vdW magnets and have promising potential in spintronics technology, such as low-energy-consumption logic and memory devices^[108-110]. To describe the relaxation degree to the equilibrium constant value of magnetization in a 2D magnetic single crystal, which is mainly affected by the SOC band structure and electron scattering, the parameter of anisotropic Gilbert damping determines the performance of many spintronic devices^[111].

Figure 5. (A) Physical phenomena that can be studied with magnetic vdW materials^[105]. (B) Schematic of spin-orientation-dependent QAH and QSH effect associated with different AFM states^[106]. (C, D) Dynamics of time-resolved pump probing, phonon energy and delaying time spectra of NiPS₃^[112]. (E, F) Energy differences of AFM phase by unipolar doping and optical electron-hole doping. (G) Tunable 2D magnetism by optical doping of monolayer RuCl₃^[120].

Furthermore, the interactions between intrinsic magnetism and other physical quantities can produce numerous meaningful applications and novel phenomena. For instance, first, the behavior of magnetization and specific heat belongs to magnetocaloric effects, and the application of magnetic refrigeration can obtain ultralow temperatures. Second, in view of magneto-optic effects, one method is to measure the light reflection or transmission, including the Kerr, Faraday, and Cotton-Mouton effects, magnetic circular birefringence, and dichroism, while the other is to analyze the effect of light on the materials before and after the interaction with photons, as shown in Figure 5C and D^[112]. Polarization-resolved photoluminescence measurements represent a powerful technique for probing magnetism^[113], particularly for identifying the coupled valley pseudospin in valley physics by constructing heterostructures^[114]. Third, inspecting the interplay of spin-phonon coupling via inelastic light scattering from optical phonons is ideal for studying phonon performance^[115]. Fourth, another phenomenon related to magnetism and transport properties, including the measurement of lateral and longitudinal resistance, reveals the critical physical property of the spin-order effect on the electron scattering and potential charge density wave transitions. Furthermore, some indirect relationship between electric phenomena and magnetism could be used to investigate valley and spin splitting^[114]. Moreover, the effective interactions between magnetization and electric polarization bring about multiferroic behavior, which is promising in the practice of spintronics and microwave magneto-electrics. In addition, the research of interplay on the magnetic and superconducting fluctuations with van Hove singularities in the electronic spectrum enables us to further investigate the quantum-critical behaviors in 2D systems^[116]

Eventually, by understanding the above-mentioned physical properties in 2D vdW magnetic materials, the design and control of magnetism become optional and operative. First, the size and edge effect on the band structure suggests the existence of edge magnetism from a series of theoretical calculations^[117]. Second, the introduction of defects, including holes, impurities, dislocations, and domains, can effectively modulate the density of states of electrons in crystal lattices, thereby affecting the chemical environment and changing the magnetic order behavior. In particular, the proximity effect can drive various interesting physical properties by constructing asymmetric structures with two distinct surfaces in 2D magnets, including forming multiferroicity and ferroelasticity in 2D Janus magnets and superconductivity, frustrated magnetism and anisotropic spin texture through the adjacent regions in 2D heterostructures^[45,118]. Third, electric fields and external strain on the lattice usually influence the interactions among ions and tune the electronic band structures, hence changing the magnetic states, insulting states, critical points, and various other physical quantities^[119]. Finally, phase transitions can also be induced by free carriers and can produce electron-hole pairs by light, as shown in Figure 5E-G, and therefore form a proximate spin liquid, enabling a new tool to construct 2D magnetism^[120]. To summarize, regardless of the progress that has been made for emerging multiple physical behaviors, many of them are anticipated to be modified and further verified in the future.

Binary transition metal chromium trihalide family (CrX₃, X = Cl, Br, or I)

Theoretical and experimental studies indicate that exfoliated transition metal trihalides MX₃ (M = Cr, V or Ru; X = Cl, Br or I) have very interesting electronic and magnetic properties, including chromium trihalides CrX₃, the ferromagnet vanadium triiodide VI₃^[121] and ruthenium trihalides RuX₃^[122]. Due to their interesting topological and spintronic properties, these metal halides have received significant attention^[27]. For these chromium trihalides, they possess extinct magnetic diversity but share a common honeycomb iso-structure, with the edge-sharing octahedra formed by six halide anions at the corners and a trivalent chromium cation at the center. Calculations have shown that the localized d-orbital electrons in Cr³⁺ and the weak dielectric screening in the vdW structure can produce a large bandgap around the Fermi level, indicating that the CrX₃ family behaves like an insulator^[93].

Distinct differences in the cleavage energy and stiffness between the in-plane and out-of-plane structures enable a facile exfoliation of a few layered chromium trihalides, including CrCl₃, CrBr_3, and CrI₃. All three chromium trihalides are found to exhibit a strong intralayer FM coupling and multiple magnetic phases, as shown in Figure 6A, but these compounds hold three different magnetic behaviors^[123]. Firstly, CrBr₃ is a ferromagnet in both bulk and monolayer forms, with a layer-dependent T_c of 20-37 K, where the magnetic anisotropy easy axis is vertical to the vdW facet^[124]. Simultaneously, the CrI₃ ferromagnet displays a similar out-of-plane easy axis with the T_c ranging from 45 to 61 K from a single layer to a multilayer^[58]. It is also noteworthy that CrI₃ exhibits a ground state of AFM coupling when stacking over two layers, as well as an interesting layer-by-layer spin-flip transition at ~0.7 T^[94]. In contrast to the Ising model described for CrBr₃^[125] and CrI₃^[61], bilayer CrCl₃ can persist with an XY model form of easy-plane anisotropy interlayer AFM resonance, with a thickness-independent T_N of ~16.8 K^[84], which could facilitate research into topological vortices and the potential Berezinskii-Kosterlitz-Thouless (BKT) transition in this system. The substitution of halogen elements in the same main group has led to drastic changes in the magnetic behavior, indicating a series of mechanisms to be discovered and explained.

Figure 6. (A) Structures of vdW CrX₃^[123]. (B) Schematic diagram of super-exchange and direct exchange. (C) Spin displacement of two sublattice spins at the right-handed chiral wave in CrI₃, (D) spin-wave excitations at T = 52 K and (E) temperature dependencies of the stiffness of the acoustic mode of CrI₃^[126]. (F) 2D charge density difference for a certain plane. (G) Schematic diagram of exchange mechanism and d-orbit energy level splitting of V-V bonding. (H) On-site energies for d orbitals of nearest-neighbor V atoms with different strains in monolayer VICl^[17].

Considering the intralayer exchange in CrX₃ monolayers, the spin-spin interaction is mainly predicted by two interactions, as shown in Figure 6B-E^[126]. One is an AFM interaction formed by the nearest neighboring direct exchange interaction via the nearest Cr-Cr neighbor^[125]. The other is an FM form introduced by the next-nearest neighboring halide-mediated super-exchange pathway via the half-filled d-orbital t_2g of the Cr atom and empty p-orbital e_g of the halide ions. With the metallicity and atom radius increasing from Cl to I, the covalency of Cr-X can be strengthened and the distance of Cr-Cr is enlarged, resulting in a more powerful Cr-X-Cr super-exchange interaction and a weakened Cr-Cr direct exchange interaction^[127]. Furthermore, in the case of CrI₃ and CrBr₃, shown in Figure 6B, the heavier halide atoms lead to a stronger spin-orbit interaction to stabilize ferromagnetism within the intralayer. Therefore, a tendency of intralayer AFM-FM-AFM by the occupation from Cl to I can be explained by the synergy effect of the spin-orbit interaction and the competition between the nearest and second nearest neighbors. Interestingly, due to the lack of effective approaches that could obtain and then character monolayer magnetism in CrCl₃. The issue of the easy-plane magnetism in monolayer CrCl₃ was verified to be intralayer ferromagnetic until X-ray magnetic circular dichroism (XMCD) evidence of an MBE grown monolayer CrCl₃ in 2021^[128], thereby enriching the physics and universality scaling of phase transitions in low-dimensional systems.

Generally, the interlayer exchange interaction in few layers can be promoted to their bulk forms. However, the circumstances in CrI₃ and CrCl₃ are not as generally expected. Noting there are two stacking configurations of monoclinic (antiferromagnetic t_2g-t_2g dominated) and rhombohedral (ferromagnetic t_2g-e_g dominated) in CrX₃, possible speculation of the phase transition may occur not only with the temperature change, but also with the thickness accumulation. It is noteworthy that, by means of theoretical calculation^[103] and experimental results^[129], a thickness-dependent phase transition of monoclinic to rhombohedral at low temperature can exist in the CrI₃ system. Therefore, this emphasizes the importance of the stacking form for the regulation of ferromagnetism/antiferromagnetism. In addition, Figure 6F-Hshow a linear spin-wave for the magnon dispersion that has shown that a topological origin of the opened Dirac gap at the Γ-point stabilizes the magnetism^[17], due to the existence of both the single-ion uniaxial anisotropy and super-exchange interaction mechanism in the off-plane CrBr₃ and CrI₃. According to the Ising model, magnetic order is destroyed by spin-wave proliferation; therefore, the Curie temperature of CrBr₃ and CrI₃ is dominated by spin-wave proliferation^[130].

Although numerous studies have attempted to explain the interplay in the vdW ferromagnet CrX₃, the genuine origin of the inter- and intra-layer Heisenberg exchange interaction needs to be given further attention in consideration of the detailed evolution of the extra exchange interactions, orbital environment, and Coulomb interaction. In addition, the magnetic anisotropy originating from various anisotropy Hamiltonians, such as anisotropic exchange, single-ion anisotropy, Kitaev exchange and the Dzyaloshinsky-Moriya interaction, should be further valued, especially the topologically protected item^[131].

Despite the tunable magnetism in the CrX₃ family, the application of vdW ferromagnets is still far from the requirement of maintaining a stable FM spin array that could survive over ambient temperature. A series of strategies for constructing novel geometries of atoms and modulating different electronic interactions have been applied to change the magnetic behavior in 2D materials. For example, as mentioned above, consideration of changing the bond angle and length may affect the electronic structure and magnetic behavior. A delicate design of mirror symmetry (out-of-plane) violation by constructing a Janus Y-C-X heterostructure (where C is generally a transition metal and X and Y are two different halide or chalcogenide elements) can effectively adjust the intrinsic and extrinsic behaviors, such as semiconducting^[115] and metallic/half-metallic behavior^[43,132], spin and valley polarization^[133,134], magnetic anisotropy^[135,136], robust ferromagnetism^[137], multiferroicity^[138], stabilized spin flips^[132] and the Dzyaloshinskii-Moriya interaction^[139,140]. Furthermore, more 2D magnetic crystals with a similar structure (such as VBr₃, VI_3, and NiBr₃) have been examined and have also attracted much interest because of their tunable magnetism and fascinating SOC behavior^[8,141]. By combining these two methods, another study revealed an enhanced T_c of 240 K for vanadium trihalides by creating a Janus V-X-Y configuration^[126]. As Figure 6H shows, with a 6% tensile strain, the T_c in this structure can be further promoted to 280 K, which represents an important step in enriching the family of ambient vdW ferromagnets.

Binary transition metal dichalcogenide family (ASe₂, A = V, Mn or W)

With more advanced approaches achieved in vdW ferromagnets, a variety of materials with a tendency for high T_C, as well as more accurate property detections, have emerged. In turn, the in-depth physics around the magnetoelectric interaction has been inspired. For example, layered vdW materials have shown a significant variation of characteristics when they have been thinned to a monolayer, including transitions between indirect and direct bandgap materials^[16], many-body quantum criticalities^[13] and field tunable properties of many-body states, as highlighted in Figure 7A^[42].

Figure 7. (A) Phase diagram and physical properties of TiSe₂ under electron doping^[42]. (B) Atomic-resolution STEM image and (C) intensity line profile of exfoliated VSe₂ flake and (D) layer-dependent Raman spectra (inset: mapping) of VSe₂ from monolayer to bulk^[143]. (E) Metal-insulator phase transition schematic of VSe₂^[149]. (F) Magnetoresistance of VSe₂ nanosheets with different thicknesses, showing WAL effect features by HLN fitting^[19].

As a versatile family of 2D materials, pristine 2D transition metal dichalcogenides possess trigonal 2H or 1T structures with potential features such as semiconducting^[142], metallic^[143] and charge density wave (CDW) behavior^[144]. Among them, VSe₂ is a six-fold symmetric hexagonal structure with the V atom located in the middle layer, separating two layers of Se atoms. XMCD results confirmed ferromagnetism in a 1T-VSe₂ monolayer flake at a T_c of 470 K, with a magnetic moment of 0.3 μ_B per V atom. From Figure 7B-D, for the vibration signals of VSe₂ in Raman spectra, when reducing the bulk thickness to a bilayer, the intensity ratio of the two peaks of E_2g (in-plane) and A_1g (out-of-plane) changes dramatically^[143]. When combined with the broadened XRD peaks of exfoliated VSe₂ nanosheets, the results suggest a highly crystalline c-axis orientation in the exfoliated VSe₂ nanosheets. With no evident shift or splitting for the 3d orbital of Se and the 2p orbital of V in the XPS peaks found in few-layer VSe₂ before and after the exposure to air, its air stability was further evidenced^[145].

Interestingly, experimental results exhibit paramagnetic characteristics in bulk VSe₂, whereas it can behave as a ferromagnet when thinned to a monolayer^[18], i.e., spontaneous magnetization exists in 2D VSe₂ rather than in 3D. An explanation of this phenomenon should be noted by the density of states (DOS) at the Fermi level (E_F) being greatly enhanced when tunning the critical thickness of VSe₂ to a bi- or monolayer^[146]. The Stoner criterion can be enhanced when the density difference of electrons between the spin-up and spin-down states is enlarged. Similar behavior was found in NbSe₂^[62], PtSe₂^[147] and MoSe₂^[148]. Furthermore, theoretical work has also predicted a metal-insulator transition, as shown in Figure 7E^[149], and a huge discrepancy in saturation magnetization with different substrates for the epitaxy of monolayer VSe₂. As Figure 7E shows, with the layer number increasing, the intraband coupling of the d value (blue line) ascends while the Coulomb energy of the U value (Red line) descends, and the competition between these values results in the change of bandgap from indirect to gapless. Furthermore, calculation results have shown that the SOC effect of monolayer VSe₂ induced a large MAE of ~1.1 meV.

For manipulating ferromagnetism, doping and stress tuning can lead to strong valley splitting by SOC, and structural rearrangement is able to induce a Peierls distortion to open up the energy gap at E_F, thereby frustrating the magnetic moment of the TMD monolayer^[15]. For example, results indicated that an Mn dopant reinforces the long-range ferromagnetism in some TMDs, including MoS₂, MoSe_2, and MoTe₂, which is much stronger than the substitution of Fe, Co, and Ni. The explanation for this lies in the balance of the AFM coupling from the direct exchange of two localized Mn 3d states and the FM coupling from the super-exchange of delocalized p states of the chalcogen with Mn^[150]. Another statement clarified a key factor of the short-range double-exchange mechanism when Mn dopants are inserted at Mo sites that controls the long-range magnetic order^[142]. Furthermore, Wang and co-workers demonstrated that a proper substitution at the Mo site at nonmagnetic MoS₂ or Mo vacancies favors the FM coupling, thereby transforming to an intrinsic diluted magnetic state^[151]. Furthermore, the tensile-based transport property of TMDs was investigated by DFT, and the results showed an evident difference in the bandgap, effective mass, elastic modulus, and deformation potential by applying tensile strain from different directions^[152].

However, doubts regarding whether the magnetism originates from a certain band structure spin splitting remain a controversial issue for the complex bandgap that indicates FM exchange-splitting^[75]. For example, on the one hand, due to the possibility of CDW, the phase transition temperature in this system can be suppressed^[153]. On the other hand, even though the reduction of the DOS at the Fermi surface would be aroused in the bulk form by changing the CDW, the phase transition temperature would also be lowered when the thickness is reduced^[15]. Additional evidence for this is the quasi-2D transport with magnetoresistance and the weak anti-localized (WAL) effect that was clearly observed in heptalayer VSe₂ below 10 K. The WAL results in Figure 7F fit well with the Hikami-Larkin-Nagaoka equation, indicating that the dephasing mechanism is electron-electron interaction dominated^[19]. In summary, judging the origin of magnetism in few-layer VSe₂ is a matter that requires attention and is awaiting a series of further experimental results to clarify it.

Ternary transition metal phosphorus trisulfide family (NPS₃, N = Mn, Fe or Ni)

The transition metal phosphorus trisulfide compounds NPS₃ (N = Fe, Mn, or Ni) are a class of vdW stacking AFM insulators (bandgap over 1.0 eV), which can be exfoliated down to the monolayer limit experimentally. These materials serve as good candidates for numerous applications in Li-ion batteries, optoelectronic materials processing, and visible-light water splitting, especially their intriguing magnetic properties. As shown in Figure 8A and B, NPS₃ materials possess a monoclinic C2/m space group that holds the (P₂S₆)₄ bipyramidal structure arranged in a triangular form^[154,155]. Inside, two-fold symmetry distributed transition metal atoms T are bonded with six S atoms on the upper and lower layers of the T plane with trigonal symmetry, where every six S atoms share a common connection with P atoms^[156].

Figure 8. Crystal and magnetic structure of NPS₃. NPS₃ lattice viewed along the (A) c-axis and (B) b-axis and (C) AFM structures of NPS₃^[155]. (D) Detailed and (E) whole resistivity survey of FePS₃ plotted against temperature at different pressures^[160].

Although these three compounds are isostructural, as shown in Figure 8C, varying the transition metal in NPS₃ results in different AFM orderings due to their different spin dimensionalities, thereby distinguishing the corresponding magnetic states. For MnPS₃, the magnetism contains a 3.5 µB interlayer FM ordering along the c-axis with a critical temperature of 78 K, which means the same spin could be reserved in two adjacent layers. In the ab plane, the spin of the two adjacent magnetic ions is reversed, showing Néel-type AFM^[76]. According to the crystal field model, the Mn⁺ ion is in the high-spin ⁶S ground state, and the spin-orbit coupling and triangular distortion of this ground state can be ignored, so the single-ion magnetic anisotropy is small and the magnetic dipolar anisotropy dominates. Therefore, the magnetic susceptibility for MnPS₃ is nearly isotropic due to the single-ion and dipolar anisotropy and can be described by the Heisenberg Hamiltonian^[56].

Interestingly, with the T_N at ~120 K, FePS₃ holds another AFM behavior that contains an interlayer AFM coupling and an intralayer zigzag-type AFM coupling, which means the pair of nearest two spins are aligned with the same orientation and opposite to their neighboring pairs. Due to the presence of high spin Fe ions, the triangular distortion and spin-orbit coupling effect are strong, resulting in FePS₃ showing a highly anisotropic magnetic susceptibility, which can be well described by the Ising Hamiltonian^[56].

For NiPS₃ (T_N of ~150 K), different from MnPS₃ and FePS₃, the spin lies in the ab plane. The interlayer coupling is FM, while the intralayer coupling is a zigzag type similar to FePS₃^[76]. Its magnetic anisotropy originates from the zero-field splitting of the Ni ion ³A_2g ground state caused by a triangular distortion. Since this is an indirect interaction, NiPS₃ exhibits weak anisotropy, which can be outlined with the anisotropic Heisenberg model^[56]. In summary, due to the diversity of the transition metal, the interaction in the crystal field is different. For NiPS₃ and FePS₃, the single-ion anisotropy is mainly discussed, while for MnPS₃, the magnetic dipolar anisotropy plays a key role.

Recent studies have shown that the magnetic properties of MnPS₃ and FePS₃ can be maintained in atomically thin monolayers. Using thickness-dependent Raman spectroscopy, Lee and co-workers reported that the Ising-type AFM order of FePS₃ persists as the thickness decreases to the 2D limit with almost unchanged transition temperatures^[154]. This indicates that the interlayer interaction hardly affects the magnetic ordering of FePS₃^[154]. Simultaneously, tunneling magnetoresistance measurements performed on MnPS₃ show that the magnetoresistance effect always exists with MnPS₃ cleaved from the bulk to the monolayer, and even at the monolayer limit, the process from the AFM ordered state to the spin-flop state with an increasing magnetic field can be observed. This suggests that in the MnPS₃ system, the antiferromagnetic order can exist in the monolayer and has a magnetic field dependence^[157]. This is further demonstrated by an explanation of the broken symmetry of space reversal and time reversal in the Néel ordered state of ultra-thin MnPS₃ down to at least 5.3 nm via optical second harmonic generation rotational anisotropy tests.^[155] A similar conclusion was found using Raman spectroscopy of MnPS₃ from bulk down to the bilayer, showing that as the temperature drops through the Néel temperature, the Raman spectra for the few-layer samples exhibit characteristic changes in the Mn ion vibration-related phonon modes, indicating the surprisingly stable magnetic ordering at a thin limit^[82]. For NiPS₃, good maintenance of the XY Hamiltonian down to the monolayer limit, its in-plane AFM ordering survival down to bilayer samples and with a slight change of the Néel temperature. However, it is almost completely suppressed when reaching the monolayer limit^[77].

In a 2D system, the interaction is very complex but also very interesting. For example, via giant magnetic field magneto-Raman spectroscopy over 30 T, the magnon-phonon coupling can be clearly and directly observed in the 2D AFM semiconductor FePS₃. Under the influence of a strong out-of-plane magnetic field, the magnon band observed in the Raman spectrum splits into two sub-bands corresponding to different spins. One of the sub-bands couples with nearby phonons to generate hybridized magnon-phonon quasiparticles. A single-ion magnetostriction-based magnetoelastic Hamiltonian was theoretically established to successfully simulate the experimental results, which clearly confirmed the magnon-phonon strong coupling observed in the 2D inverse model^[158]. Simultaneously, extensive spectroscopic measurements and theoretical tools, including spectroscopic ellipsometry, X-ray absorption, photoemission spectroscopy, and DFT calculations, showed that NiPS₃ had an abnormal shift in optical spectral weight under the Néel temperature, thereby reflecting the strong charge-spin coupling between the electronic and magnetic structures^[159].

We now consider the manipulation of the physical properties in the NPX₃ family^[160]. First, an insulator-metal transition was found in FePS₃ under high-pressure electrical transmission measurements [Figure 8D], with a significant resistivity reduction occurring when increasing the applied pressure above 3.0 GPa^[160]. Furthermore, when the applied pressure reaches 11.0 GPa in Figure 8E, the temperature-dependent resistivity of FPS₃ exhibits an obvious insulator-metal transition, which was further confirmed by high-pressure XRD data. Experiments have shown that the manipulation of pressure in 2D magnetic materials can significantly change the crystal structure and electronic properties. Another advanced technique is using light to control the magnetic anisotropy in NiPS₃. Afanasiev and collaborators used ultrashort pulses of light to optically pump NiPS₃ on the picosecond time scale^[112]. By changing the photon energy during orbital resonance, they could selectively activate the 2D magnon modes at terahertz frequencies, which are controlled by light-induced changes in the magnetic anisotropy axis. The most fundamental principle lies in the change of the magnetic anisotropy axis response to the photoexcitation of ground state electrons to transition to lower symmetry orbitals^[112], thereby confirming the control of magnetic anisotropy in 2D vdW AFM NiPS₃.

Ternary chromium silicon telluride (Cr₂Ge₂Te₆) family

Cr₂Ge₂Te₆ is a weak anisotropic ferromagnet semiconductor and possesses an all-direction collective spin moment orientation with a small energy difference below the T_c of 68 K^[10]. Along the c-axis of the 2D Cr₂Ge₂Te₆, the middle layer of transition metal Cr is sandwiched by two Ge-Te layers, where Te can be replaced by Se and Ge can be substituted by Si and Sn. Based on X-ray and neutron diffraction results, layered Cr₂Ge₂Te₆ exhibits a honeycomb structure along the c-axis in Figure 9A, with a vdW distance of 6.9 Å^[78]. Raman spectroscopy indicated that each Cr atom possesses a magnetic moment of 2.4 μ_B.

Figure 9. (A) Crystalline structure and (B) exchange coupling figuration of AB-stacked layer of Cr₂Ge₂Te₆^[78]. (C) Spin-polarized band structure and (D) calculated DOS of few-layered Cr₂Ge₂Te₆. (E) Magnetization reversal and spin realignment by carrier doping. (F) Simulated evolution of measured spin magnetization changes by tuning the hole and electron carrier density^[153]. (G) Magnetic susceptibility-temperature spectra of organics intercalated Cr₂Ge₂Te₆. (H) Comparison of super-exchange interaction in Cr₂Ge₂Te₆ and (I) double-exchange interaction in organics intercalated Cr₂Ge₂Te₆^[168].

Angle-dependent resistance confirmed that layered Cr₂Ge₂Te₆ possesses a uniaxial magnetic anisotropy Heisenberg magnet perpendicular to the isotropic ab-plane^[70,161]. Theoretical calculations predicted that the whole magnetic moment is localized at the Cr sites [Figure 9B], thereby leading to long-range ferromagnetism^[162]. Its magnetic anisotropic energy is computed to be ~0.1 meV/Cr between the magnetocrystalline and magnetic dipolar anisotropy energies. More critical exponents were fitted and summarized the 2D-Ising ferromagnetism in Cr₂Ge₂Te₆ and Cr₂Si₂Te₆^[163], and more results have shown that the ferromagnetic transition temperature of Cr₂Ge₂Te₆ has a relationship with the thickness. Computational results and Kerr rotation experiments have shown the monotonic increment in T_c from bilayer (~30 K) to multilayered bulk (~68 K)^[164]. In addition, for bilayer, trilayer, and six-layer samples, applying a slight external field arouses a notable change in T_c^[78]. Furthermore, with changing thickness, the electronic structure and magnetic properties of layered Cr₂Ge₂Te₆ vary readily, suggesting excellent potential for the construction of 2D vdW devices with facile manipulation.

Strain and defect engineering are also confirmed to have an impact on the electronic and magnetic characteristics of monolayer Cr₂Ge₂Te₆. For example, a semiconductor-to-metal transition in Cr₂Ge₂Te₆ can be realized by a compressive isostructural phase transition^[165], and an easy-axis direction can be switched by both the effect of Ge vacancies and thickness^[166]. Furthermore, for the carrier doping of Cr₂Ge₂Te₆, the electric-gate voltage tuning toward magnetization and the magnetic phase transition is found to be sensitive. In particular, the improvement in saturation magnetization can be interpreted by the enhancement of the magnetic moments on Cr atoms. Moreover, as shown in Figure 9C-G, by analyzing the DOS of the spin-polarized electrons, this circumstance can be explained as the Fermi level could be manipulated by moderate carrier doping by varying the amount of unpaired spin-up states, which consequently affects the magnetic moment^[153]. In addition, when an external magnetic field is applied vertically and parallel to the plane, the corresponding magnetoresistance effect on Cr₂Ge₂Te₆ changes from positive to negative, which exhibits an exchange interaction from carriers and local moments^[167].

Figure 9H and I show that for ferromagnetism in the Cr₂Ge₂Te₆ family, the spin moments on the Cr sitesmainly interact through the indirect 90° super-exchange coupling of the Cr dz²-Te p-Cr dz² path^[168]. The ligand p spin-orbit coupling contributes to weak FM coupling. Based on DFT calculations and the Hamiltonian and tight-binding models, the total Hamiltonian of the 2D Cr₂Ge₂Te₆ system contains two main factors, namely, the exchange interaction and single-ion anisotropy, with the former being regarded as a combination of the isotropy change and Kitaev interaction. Among them, it was found that the single-ion anisotropy and the Kitaev interaction could be attributed to the heavy ligand of Te that leads to SOC in Cr₂Ge₂Te₆^[161]. This may be the reason why the critical transition temperature in 2D Cr₂Ge₂Te₆ can be consciously tuned by an external magnetic field and thickness. Using DFT, the competitive relationship of MAE and dipolar anisotropy energy can explain the lack of spontaneous magnetization in monolayer Cr₂Ge₂Te₆^[169]. With a slight change in the Cr-Te bonding, a small out-of-plane anisotropy could be introduced in this 2D Heisenberg ferromagnet^[95]. Despite the existence of vdW spacing along the c-axis, its interlayer magnetic coupling cannot be ignored in view of the strong dimensional-dependent T_c of Cr₂Ge₂Te₆ for different thicknesses. Like Cr₂Ge₂Te₆, studies have shown that Cr₂Si₂Te₆^[57] and Cr₂Sn₂Te₆^[54] have similar performance in terms of magnetic anisotropy and phase transition temperature. Furthermore, topologically nontrivial magnetic spin states known as skyrmion bubbles with homochirality in layered Cr₂Ge₂Te₆ contribute to the dipolar interaction and uniaxial magnetic anisotropy^[170]. There are two kinds of spins isolated in each Bloch domain with a topological charge of ±1.

Manipulating and utilizing ferromagnetic performance has been of great importance for many model systems. For example, the driving of an external electric field enabled the tunable bipolar magnetism by rebalancing the spin-polarized band structure by shifting the Fermi level^[153]. A strain-induced structural distortion by detecting the change in spin-phonon coupling showed the modulation of spin-lattice coupling and magnetostrictions for Cr₂Si₂Te₆^[57], Cr₂Ge₂Te₆^[171], and Fe₃GeTe₂^[172]. Another approach is the employment of the magnetic proximity effect by constructing vdW moiré superlattices that tune the spin-dependent interlayer orbital coupling. This heteroatomic distribution by moiré periodicity via twisting or strain between the adjacent layers brings about a lateral modulation of a magnetic proximity field and develops a miniband splitting of spin^[173]. Furthermore, a theoretical investigation of the magneto-optical effect in layered Cr₂Ge₂Te₆ also described the calculations of optical conductivity and the magneto-optical Kerr and Faraday effects^[169]. Furthermore, according to the magnetic Cr dopant into Bi₂Se₃, a gap opening in the 2D Dirac surface states in a 3D topological insulator may give rise to quantum anomalous Hall states on 1D edges by TRS breaking^[174]. Moreover, endeavors have been attempted via growing topological insulators onto Cr₂Ge₂Te₆ by MBE growth^[175]. These fundamental behaviors suggest that 2D Cr₂Ge₂Te₆ presents multiple intriguing physical properties that can be widely used in various scientific fields.

Ternary iron germanium telluride Fe₃GeTe₂

Fe₃GeTe₂ is an itinerant ferromagnet that can reach a monolayer with strong c-axis out-of-plane anisotropy and has a T_c of ~220 K in the bulk form^[176]. There are two inequivalent Fe ions with different degrees of localization that contribute to different magnetic moments^[177]. Figure 10A and B show the hexagonal structure with three-atom-thick Fe₃Ge with Fe (I)-Fe (II) and Fe (I)-Ge bonding sandwiched by two weak vdW interaction layers of Te anions. In particular, the monolayer thick Fe₃GeTe₂ exhibits significant uniaxial magnetocrystalline anisotropy energy, with Fe atoms in each layer of 920 meV^[172]. With the existence of Fe vacancies, the T_c and magnetic anisotropy of the produced Fe₃GeTe₂ decrease^[178].

Figure 10. (A) Schematic of atomic structure of monolayer Fe₃GeTe₂. (B) AFM image, (C) optical image and (D) cross-sectional profile of the selected area in few-layer flakes of Fe₃GeTe₂. (E) Temperature-dependent sample resistance of Fe₃GeTe₂^[44]. (F) Photoemission electron microscopy topography image of magnetic-stripe domains of Fe₃GeTe₂ at different temperatures, sizes and shapes^[180]. (G) FM and AFM configurations and spin-density distribution in Fe₃GeTe₂ monolayer and their energy difference between the two couplings^[181]. (H) Large-scale and (I) atom resolved STM images and the (J) dI/dV curve of the surface of Fe₃GeTe₂^[182].

Magnetic analysis showed that the strong SOC counts for the c-axis magnetic anisotropy with ferromagnetic spin aligned in each layer^[44]. The anomalous Hall effect (AHE) test in Figure 10C-E shows an almost perfect square hysteresis loop under the application of an external field along the c-axis, particularly for thickness less than 200 nm, and its coercive field reduces with increasing temperature until totally disappearing over T_C, which suggests a hard magnetic behavior for bulk Fe₃GeTe₂^[37]. Furthermore, the ferromagnetic analysis showed a strong dependence on the thickness of Fe₃GeTe₂ and an indication of an effective interlayer coupling length that is less than five layers. The detection of a decreased T_c for monolayer Fe₃GeTe₂ by magnetic circular dichroism^[179] and AHE^[44] results [Figure 10C-F] revealed the survival of the magnetic order and may have been influenced by the difference in the electron density^[180]. As reported, Fe₃GeTe₂ is a metal with a carrier concentration of 1.2 × 10¹⁹ cm^-3 in MBE grown samples (with carrier mobility of ~50 cm²·V⁻¹·s⁻¹) and ~10²¹ cm^-3 (~8 × 10¹³ cm^-2 per layer) in the chemical vapor transport grown samples^[91]. An experiment evidenced a carrier doping method into trilayer Fe_3-xGeTe₂ via ionic gating contributed to a dramatic increase of T_c over ambient temperature^[179]. These recent advances accelerate the realization of ambient stable 2D magnets down to the monolayer limit.

The strong dimensionality dependence on the itinerant ferromagnetism in Fe₃GeTe₂ revealed a transition from the 3D to 2D Ising model when the thickness was less than five layers, accompanied by a rapid drop of T_c. The magnetic ground state of monolayer FGT is determined by direct exchange and super-exchange competition^[181]. On the one hand, the d orbitals on the nearest-neighbored Fe atoms overlap directly, making up the direct exchange interaction of AFM Fe-Fe coupling in FGT. On the other hand, the d orbitals of the nearest-neighbored Fe atoms overlap indirectly via the p orbitals of Ge or Te atoms, making up a super-exchange interaction of FM Fe-Ge/Te coupling. The Fe-Ge and Fe-Te bonds are stronger than that of Fe-Fe, and as a consequence, the FM super-exchange interaction prevails over the direct AFM exchange interaction, as shown in Figure 10G^[181].

Typically, the interlayer exchange coupling in Fe₃GeTe₂ is much weaker but may have a contribution to a 3D critical behavior in sufficient multiple layers. Considering the existence of intrinsic anisotropy, the long-range magnetic order in the monolayer limit is supposed to be maintained, leading to the diversity of Tc with dimension crossover from 3D to 2D. For example, the magnetic anisotropy originates from a strong SOC and magnetostatic dipole-dipole interaction that is strong enough to stabilize the 2D ferromagnetism. It is noteworthy that a heavy-fermion behavior from exchange splitting is confirmed by a massive spectral weight transfer in the 3d electrons of Fe₃GeTe₂, as shown in Figure 10H-J^[182], as well as a Kondo lattice behavior proved by the character of Fano resonance^[183]. These phenomena not only describe the relationship between the Kondo screening effect and Ising ferromagnetism but also reveal the origin of the heavy-fermion behavior in the 3d-electron vdW Fe₃GeTe₂^[183]. This coexistence of itinerant electrons and local magnetic moments possibly contributes to extensive research on strong correlation phases in 3d-electron systems^[95].

Owing to the unique properties of vdW Fe₃GeTe₂, it exhibits exceptional behavior that permits the development of many magnetism-related studies. For hole doping, the occupation degree and splitting of ferromagnetic Fe-Te bonding significantly influence the magnetic anisotropy in FGT^[184]. Furthermore, the strong magnetoelastic coupling in the magnetostrictive Fe₃GeTe₂ monolayer suggests an improvement of magnetic anisotropy energy via utilizing a 2% structural tensile strain^[172]. For example, a pulse-current induced magnetization reversal was recently found in a Fe₃GeTe₂ homojunction with reduced coercivity by spin-orbit torque and Joule heat^[185]. Moreover, when considering the magnetic proximity, the contact strength, and electric and magnetic behaviors, electronic and spin injection are determined by the bond and match type of their introduced substrates^[186]. Another interesting phenomenon in vdW Fe₃GeTe₂ lies in its shape of magnetic domain with different hysteresis loops. The thickness-dependent stripe domain phase in Figure 10F suggests the quasi-2D magnetic behaviors in Fe₃GeTe₂, as well as an indication of long-range dipolar interaction over the exchange interaction and magnetic anisotropy^[180,187]. In addition, at a certain thickness and temperature, a labyrinthine domain structure emerges^[188]. Interestingly, a topological spin texture of magnetic skyrmions was observed in Fe₃GeTe₂ vdW flakes and further revealed finite interfacial Dzvaloshinskii-Moriya interactions that confirm the spin chirality. Furthermore, a large anomalous Nernst effect, an evident spin-phonon coupling effect, and enhanced electronic specific heat enrich the complex phase diagram, enabling fertile grounds for massive emergent phenomena and applications of Fe₃GeTe₂.

The genuine magnetic origin of vdW Fe₃GeTe₂ is still debated. First, there is no consensus regarding long-range order over ferromagnetic or antiferromagnetic^[83]. Second, when describing the magnetic behavior in various conditions, the consistency among calculation and measurement results is not satisfactory. Finally, there is still a lack of clear evidence of describing the magnetic behavior after completely removing the unexpected ferromagnetic impurities.

Ternary manganese bismuth telluride (MnBi_2nTe_3n+1, n = 1, 2 or 3)

The layered manganese bismuth telluride (MnBi_2nTe_3n+1) family possesses combinations of intrinsic magnetism and nontrivial topological insulator (TI), hosting interesting physical effects, such as A-type antiferromagnetism [Figure 11A], intrinsic axion insulator states, a quantum anomalous Hall state and a quantized topological magnetoelectric effect^[189]. The magnetism coupling with topology accommodates a vast space to explore exotic topics related to quantum effects in condensed matter physics^[7].

Figure 11. (A) Crystal structure of bulk AFM MnBi₂Te₄ with high symmetry points in reciprocal space. Band structures profiles (B) without SOC and (c) with SOC. (D) SOC-dependent band gap energy at Γ, showing an evolution from a trivial insulator to a topological insulator by changing the SOC. (E) Wannier charge center evolving with k_y, referring to a nonzero topological invariant^[189].

Atomically thin MnBi₂Te₄ has a heptalayer structure, forming a Te-Bi-Te-Mn-Te-Bi-Te stacking along the out-of-plane axis. Similar to the architecture of 1T-ASe₂, the magnetic Mn cations and the nearest layered Te atoms form a triangular lattice at the center of the heptalayer. Similar materials, like MnBi_2nTe_3n+1, can be briefly considered as charge-balanced alternating stacking of monolayer MnBi₂Te₄ and few-layer Bi₂Te₃. Layered MnBi₂Te₄ presents intralayer FM and A-type interlayer AFM of Mn coupling along the c-axis below 25 K^[80]; for the MnBi₄Te₇ and MnBi₆Te₁₀, the Néel temperature T_N can be lowered down to 13 and 11 K, respectively. Viewing the opposite parities of studies shows that the multiple states of magnetism in MnBi_2nTe_3n+1 can be realized by applying different external fields and temperatures. When the outer magnetic field increases over 3.5 T at 2 K, the MnBi₂Te₄ experience a spin-flip transition from the layered AFM ground state to a canted AFM, with spin oriented out-of-plane and in-plane components aligning oppositely between the neighboring slabs. The same spin-flip can be seen at ~0.15 and 0.01 T for MnBi₄Te₇ and MnBi₆Te₁₀, respectively^[190], demonstrating a declined AFM interlayer exchange interaction as the interlayer distance increases. This further proves the favor of manipulating the FM coupling via changing the short-range adjacent layer AFM coupling and the strength of magnetic anisotropy.

In the bulk form of AFM MnBi₂Te₄, shown in Figure 11B and C, a band inversion of first close and then reopen behavior can be inferred in view of the opposite parities of valence band maximum and conduction band minimum at Γ point and the reversed signs with the strength changes in SOC^[189]. With the presence of spontaneous magnetic moments integrated by SOC that no requirement of the external magnetic field in Figure 11D and E, a topologically nontribal band structure could exhibit an exotic transport phenomenon of the quantum anomalous Hall effect. Together with the magnetic performance, the topological phases evolve with varying thickness of the stacking layer. DFT calculations indicated a quantum anomalous Hall (QAH) insulator behavior in odd-layered MnBi₂Te₄ and an axion insulator behavior in an even-layered sample^[12]. Furthermore, a QAH experiment on a hexalayered MnBi₂Te₄ sample had seen the axion insulator state with Hall resistivity was offset by the pair of top and bottom surfaces that cancel each other. Simultaneously, a nine-layer MnBi₂Te₄ device was also found to exhibit a Chern insulator phase^[73], i.e., the members of MnBi_2nTe_3n+1 reveal various topological states that can be complementarily and synthetically. In short, the magnetism coupling with topology would accommodate vast space to explore exotic topics related to quantum effects in condensed matter physics^[7].

Advances in heterostructure applications

With the advantages of easy access, good flexibility, controllable interlayer twisting and stacking, and few lattice mismatch, 2D vdW magnets provide a convenient platform for condensed matter physics and potential applications by interfacial engineering. In turn, to manipulate the magnetic properties, constructing heterostructures can effectively adjust the electronic structure in atomically thin films. Such artificial materials provide tremendous freedom to design, construct, manipulate and apply novel physical properties and devices^[191]. As shown in Figure 12, the elementary functions of 2D materials enable some advanced development of spintronics, including generating and detecting spin-charge conversion, spin transport, and manipulation, which requires a material itself to possess good nature with high carrier mobility, long spin lifetime, and long diffusion distances^[192].

Figure 12. Applications of 2D ferromagnets in vdW heterostructures and twisted lattices, including emergent physical properties of tuning the Schottky barrier height^[193], exchange bias^[196], valley manipulation^[118], anisotropic spin-triplet Andreev reflection^[198] and anisotropic strong correlation moiré flat bands^[202], as well as spintronic devices, such as spin valves^[204], spin-orbit torque^[209] and spin field-effect transistors^[36].

For interfacial engineering, exchange proximity represents one of the main effects in many vdW heterostructures devices, such as switching the Schottky barrier height in In₂Se₃/Fe₃GeTe₂^[193], giant tunneling magnetoresistance that accommodates spin-filter tunnel barriers^[33], adjusting the tunneling electroresistance in the 2D FM Schottky heterojunction^[194,195], generating exchange bias effect in CrCl₃/Fe₃GeTe₂^[196] and valley manipulation by CrI₃/WSe₂ heterostructures^[118]. In addition, combining ferromagnetic insulators and superconductors have opportunities to produce triplet and even topological superconductivity^[197], as well as the Andreev reflection at the interface of FM vdW/superconductor^[198,199]. Another consideration is to generate QAHE; due to the atomic flat and regardless of the electronic scattering, the vdW ferromagnet is ideal for introducing magnetism into quantum spin Hall effect topological insulator. Such efforts have been attempted by chemical vapor depositing Bi₂Te₃ on Cr₂Ge₂Te₆^[200] and epitaxy growth of Cr₂Ge₂Te₆ on (Bi,Sb)₂Te₃^[175]. Moreover, magnetic multilayers, moiré^[201,202] and twisted 2D magnet^[203] engineering can produce an additional degree of freedom in epitaxially growing metallic heterostructures and have an enormous potential for new physical phenomena.

Constructing magnetically coupled structures provides a promising platform for spintronics regardless of crystal mismatch. For instance, first, when a nonmagnetic layer has parallel/antiparallel spin polarization at its two sandwiching interfaces with a ferromagnet, it makes up a spin valve setup that corresponds to two switching configurations, such as variation of magnetoresistance and spin filtering effect^[204]. Currently, these spin filtering effect magnetic tunnel junction (MTJ) exhibit tunneling magnetoresistance in dual-gated quadra-layer CrI₃ spin-filter MTJ^[205] and tunneling spin-valve Fe₃GeTe₂/h-BN/Fe₃GeTe₂ heterostructure^[6], as well as a giant magnetoresistance effect in Fe₃GeTe₂/MoS₂/Fe₃GeTe₂ heterostructures^[206]. Interestingly, a prediction also claimed a similar FM insulator/superconductor/ FM insulator that can give insight into relations between superconductivity and ferromagnetism, which is waiting to be realized^[207]. Second, at the heavy-metal/ferromagnet interface, a spin-orbit torque can be produced by a spin Hall or Rashba effect that could generate an in-plane spin current and switch the magnetization direction^[208]. For example, the vertical magnetization of Fe₃GeTe₂ in Fe₃GeTe₂/Pt can be tuned by alternating the current direction in the Pt layer^[209]; a similar conclusion was drawn by Johansen’s theoretical discussion and proposed a qualitative relationship between current and the critical temperature that has the possibility to realize the BKT phase transition^[210]. Another heterostructure application of 2D ferromagnet potential is to attribute the spin nature to the spin logic gate devices, one of which solution is to form a spin field-effect transistor (s-FET). With ferromagnetic drain and source electrodes, the spin polarization of the current in the 2D electron gas channel can be modulated by the selected electrical gate voltage in a dual-gated graphene/CrI₃/graphene s-FET^[36]. Finally, in view of alternating the spin through the device requires only low energy dissipation within a very fast time, s-FET is considered a promising 2D spintronic candidate with low power consumption and high computing output.