Valleytronics in two-dimensional magnetic materials

Valleytronics uses valleys, a novel quantum degree of freedom, to encode information. It combines other degrees of freedom, such as charge and spin, to produce a more comprehensive, stable, and efficient information processing system. Valleytronics has become an intriguing field in condensed matter physics due to the emergence of new two-dimensional materials in recent years. However, in nonmagnetic valleytronic materials, the valley polarization is transient and the depolarization occurs once the external excitation is withdrawn. Introduction of magnetic field is an effective approach to realizing the spontaneous valley polarization by breaking the time-reversal symmetry. In hexagonal magnetic valleytronic materials, the inequivalent valleys at the K and –K(K′) Dirac cones have asymmetric energy gaps and Berry curvatures. The time-reversal symmetry in nonmagnetic materials can be broken by applying an external magnetic field, adding a magnetic substrate or doping magnetic atoms. Recent theoretical studies have demonstrated that valleytronic materials with intrinsic ferromagnetism, now termed as ferrovalley materials, exhibit spontaneous valley polarization without the need for external fields to maintain the polarization. The coupling of the valley and spin degrees of freedom enables stable and unequal distribution of electrons in the two valleys and thus facilitating nonvolatile information storage. Hence, ferrovalley materials are promising materials for valleytronic devices. In this review, we first briefly overview valleytronics and its related properties, the ways to realize valley polarization in nonmagnetic valleytronic materials. Then we focus on the recent developments in two-dimensional ferrovalley materials, which can be classified according to their molecular formula and crystal structure: MX2; M(XY)2, M(XY2) and M(XYZ)2; M2X3, M3X8 and MNX6; MNX2Y2, M2X2Y6 and MNX2Y6; and the Janus structure ferrovalley materials. In the inequivalent valleys, the Berry curvatures have opposite signs with unequal absolute values, leading to anomalous valley Hall effect. When the valley polarization is large, the ferrovalleys can be selectively excited even with unpolarized light. Intrinsic valley polarization in two-dimensional ferrovalley materials is of great importance. It opens a new avenue for information-related applications and hence is under rapid development.


Introduction
Finding new quantum degrees of freedom of electrons has been a fundamental goal in condensed matter physics research.The application of every new quantum degree of freedom usually led to significant development in science and technology.Traditional electronic devices are based on the manipulation of electron charge degree of freedom, but they are approaching the thermodynamic and quantum limits.To overcome these limits, spintronics, which processes information by manipulation of the spin degree of freedom, has emerged and is rapidly developing [1,2].In principle, any quantum degree of freedom, not only the charge and spin, can be used to encode information.The energies of the electron states form energy bands due to the interactions in the crystal, and the states in the energy bands near the minima of the conduction band and the maxima of the valence band are called valleys [3].Recently, it has been found that some two-dimensional (2D) materials, such as the H-phase monolayer MoS 2 , as shown in figures 1(a) and (b), have inequivalent valleys at the K and -K(K ′ ) points with opposite Berry curvatures, and opposite electrical, magnetic and optical properties [4][5][6][7][8][9].As a results, electrons in these crystals also have a valley degree of freedom, which is a new quantum degree of freedom.The field of research and the applications related to the utilization of valley degrees of freedom for storing and transmitting information is called valleytronics [10][11][12][13][14][15][16][17].Valleytronics utilizes the valley quantum degree of freedom in combination with other degrees of freedom, such as charge and spin of electrons to encode information, providing a richer, more stable and efficient solution for information processing [18,19].Valleytronics provides new opportunities for the development of next-generation electronic devices.
In H-phase monolayer MoS 2 , for instance, the spatial inversion symmetry is broken, and the Berry curvature of valleytronic materials is reversed at the inequivalent valleys, leading to a number of Berry-phase related exotic quantum phenomena [4,6], such as the valley Hall effect.Considering the effect of the Berry curvature Ω n (k), the conventional electron velocity is supplemented by a term related to Berry curvature: v = ∇ k E n (k)/ h + eE × Ω n (k), where E n (k) is the energy of the n th energy band at point k, and E is the applied electric field.Due to the term eE × Ω n (k), the carriers moves in the direction perpendicular to the electric field E, and the carriers at the inequivalent valleys, which have opposite Berry curvatures Ω n , move in opposite directions, as shown in figures 1(c) and (d).Therefore, the inequivalent valley carriers will be spatially separated in the absence of external magnetic field, thus realizing the manipulation of the valley carriers.
Valley-dependent optical selection rules are another important property related to the Berry phase.Hexagonal honeycomb valleytronic materials with asymmetric center inversion have inequivalent valleys at the K and -K(K ′ ) points.The optical transition induced by circularly polarized light at the valleys is related to the orbital angular momentum m of the valleys, whereas the orbital angular momentum in solids is related to the Berry curvature.It has been shown that the orbital magnetic moments is closely related the Berry phase at the valleys.The opposite Berry curvatures at the K and -K(K ′ ) valleys lead to opposite orbital magnetic moments [4], and hence the polarity of circularly polarized light absorbed or emitted at these two valleys is opposite.This circular dichroism can be exploited to selectively excite carriers at inequivalent valleys by light with opposite circular polarization.If the light is linearly instead of circularly polarized, the carriers in both K and -K(K ′ ) valleys will be excited and the valley polarization cannot be achieved.The coupling of the quantum degrees of freedom, valley and spin, can be realized if the system also has a large spin-orbit interaction [22].The spin splitting at inequivalent valleys is opposite, and optical transition occur only between the states with the same spin, so there is a spin selection rule on top of the valley selection rule for optical transition.As shown in figure 2, if the right circularly polarized light at a certain frequency excites only spin-up valley carriers at the -K(K ′ ) valley, then the left circularly polarized light at the same frequency excites only spin-down carriers at the K valley.This correlation between the valley polarization and a specific spin polarization facilitates the realization of encoding with spin and valleys.
Since the valley Hall effect and the valley selective optical excitation is based on nonzero Berry curvatures, certain symmetries of the system need to be broken to realize these effects.The time-reversal symmetry requires that and the spatial inversion symmetry requires It can be seen that if the system has both time reversal and spatial inversion symmetry, the Berry curvature is zero.To obtain nonzero Berry curvatures, either spatial inversion or time reversal symmetry or both of them has to be broken.Either of these two symmetries will result in the energy degeneration of the K and -K(K ′ ) valleys.

Valley polarization in nonmagnetic valleytronics materials
In nonmagnetic valleytronic materials, time-reversal symmetry determines that the inequivalent valleys at K and -K(K ′ ) are energetically degenerate.Although it is possible to selectively excite carriers at different valleys with circularly polarized light, only a transient unbalanced distribution of carriers can be achieved at the inequivalent valleys through dynamic excitation.The lifetime of this unbalanced distribution is short, posing an obstacle to utilizing valley for stable information storage and processing.
The prerequisite for the application of valleytronics in nonvolatile devices is to break the energy degeneration of the inequivalent valleys, so that the valley gaps become different and the carrier distributions are not equal in the nondegenerate valleys, giving rise to robust polarization of valleys.In a magnetic field B, an energy shift of −µ • B is induced by the total magnetic moment µ, which includes the contribution from the spin magnetic moment, orbital magnetic moment and valley magnetic moment [24].The carriers at the inequivalent valleys have opposite orbital and valley magnetic moments.If there is spin splitting, the spin is also opposite in the inequivalent valleys.Therefore, the magnetic field will produce opposite energy shifts in the bands of the inequivalent valleys and hence make them nondegenerate [24].In recent years, in order to realize the spontaneous polarization of valleys, there are more and more studies on time-reversal symmetry breaking by magnetic field via, e.g.applying external magnetic field, doping magnetic atoms and forming heterostructures with magnetic materials.

Magnetic field
A direct way to generate magnetism is the addition of a magnetic field.It has been observed in experiments that after the addition of magnetic fields to MoSe 2 and WSe 2 [24][25][26], the peaks that were originally coincident in the circularly polarized photoluminescence spectra became non-coincident, which suggests that the energy degeneration of the valleys at K and -K(K ′ ) is broken.However, the problem with the applied magnetic field scheme is that it causes a very small difference in the energy of the inequivalent valleys, which is only 0.11-0.25 meV per Tesla.Recently, the valley splitting of WSe 2 in a magnetic field was calculated [27], being consistent with the experiments [25], as shown in figure 3(a).The schematic diagram for the effect of magnetic field B is shown in figure 3(b).However, the magnetic field required to produce a manageable valley splitting is too high to be realizable.
In the continuum approximation, it is found that, the Dirac-like Hamiltonian of monolayer graphene is valley dependent in the vicinity of the K and -K(K ′ ) point [30][31][32].The presence of magnetic field (in z direction) perpendicular to the graphene plane, which can be described by vector potential A using Landau gauge, will create Landau level spectrum.The eigenfunction can be written in terms of plane wave in y direction and quantum harmonic oscillator state in x direction.If we take into account spin and valley degeneracy, then the Landau level spectrum of monolayer graphene consists of fourfold-degenerate Landau levels (LL).The unconventional quantum Hall effect observed in graphene is deeply related with the presence of zero-energy LL [30].For K valley, the eigenfunction of zero energy only resides on the A sublattice, whereas for -K(K ′ ) valley, the zero-energy eigenfunction has nonzero amplitude only on the B sublattice.For the n = 0 level if an electron is in a particular valley then its spatial wavefunction resides on a single sublattice, A or B. If this zero-energy Landau level is 1/4-filled or 3/4-filled there will be a valley and spin polarized ground state.The spin polarization is from the Zeeman splitting and the system will form a valley polarized 'ferromagnet'-like state due to exchange correlations [33].In more complex systems, external reflectance contrast spectra at σ = 5.9 × 10 12 cm −2 under differing magnetic fields.The vertical bar shows the scale for contrast equal to 0.1 [35].magnetic fields play a key role in tuning valleytronic behavior (e.g.valley-dependent LLs and bound states, interplay of real and pseudomagnetic fields [34], etc).
In the quantum Hall regime, a large magnetic field is applied and series of LL are produced in the energy bands (figure 4(a)).The optical transition between them has to follow selection rules between the LLs.It is found that due to the spatial inversion asymmetry in transition metal dichalcogenides, such as MoS 2 and WSe 2 [35,36], the inter-LL selection rules is still valley-dependent, in contrast to the case of graphene.More interestingly, the selection rule is dependent on the magnetic field.When the magnetic field is flipped, the valley index of the valleys will be switched.Since the circular polarization is locked with the valley degree of freedom, the selection rule can be switched between the inequivalent valleys upon the change of the sign of the magnetic field.The valley splitting for LLs in the quantum Hall regime is depends linearly with the magnetic field and is comparable with the LL spacing (figure 4(b)).
Apart from the real magnetic fields, the strain induced pseudomagnetic field can also effectively tune the valleytronic properties.It was predicted that massless Dirac fermions in graphene monolayer can be confined by inhomogeneous magnetic fields (figure 5(a)), which requires a magnetic barrier about 15 nm thick and, at the same time, with the magnetic fields varied by tens of Tesla.Although this magnetic barrier is difficult to be realized by real magnetic fields, it can be achieved by nanoscale deformation of graphene, which will generate very large pseudomagnetic fields of tens of Tesla.(Figure 5(b)).Because the pseudomagnetic field, unlike the real magnetic field, observes the time-reversal symmetry, it has opposite signs in the two valleys of graphene.When a real external magnetic field is applied, the total effective magnetic fields in the two valleys become imbalanced and the effective magnetic barriers for quasiparticles in the two valleys are different (figure 5(c)).In this way, the valley-contrasting magnetic barriers are achieved and so is valley contrasting spatial confinement, lifting the valley degeneracy of the confined states [37].

Magnetic substrate
The prerequisite for the scheme of forming heterostructures with magnetic materials is to find magnetic semiconductors, which was a challenge not long ago.In recent years, new 2D magnetic materials have been predicted and discovered, and attempts have been made to break the energy degeneration of the valleys by forming heterostructures with 2D magnetic materials and MX 2 type valleytronic materials.EuO and EuS are 2D ferromagnetic semiconductors, and it has been found that the valley splitting at inequivalent valley reaches up to 37.3 and 300 meV in the heterostructures of MoS 2 /EuS and MoTe 2 /EuO [38,39], respectively.Experimentally, the splitting of the inequivalent valleys in WSe 2 grows linearly with the magnetic field at a small rate of 0.2 meV Tesla −1 , which is increased by 10 times, i.e. to 2.5 meV Tesla −1 after the addition of a magnetic substrate EuS [39].CrI 3 is a newly discovered 2D ferromagnetic semiconductor, and experiments have found that the valley splitting at the top of the valence band in the WSe 2 /CrI 3 heterostructure reaches 5 meV per 1 Tesla applied [28].The crystal structure and band structure of WSe 2 /CrI 3 heterostructure are shown in figures 3(c)-(e) is a Schematic diagram of type-II band structure of CrI 3 /WSe 2 heterostructures.In addition, changing the magnetic field not only can tune the magnitude of the valley splitting, but also can invert the polarization of the K and -K(K ′ ) valleys, providing a means for manipulating the valley degrees of freedom to achieve information encoding.

Doping magnetic atoms
Another way to induce magnetism and valley polarization is by adsorption or doping of magnetic atoms [40].The calculations have shown that the adsorption of transition metals such as Sc, Fe, Mn, and Cu on MoS 2 results in a valley splitting of 16-40 meV.Valley polarization can also be realized in Janus monolayer MoSSe by magnetic doping.The crystal structures of pure and V-doped monolayer MoSSe are shown in figures 3(f) and (g).It is found that monolayer MoSSe has a direct band gap and strong coupling between spin and valley.Valley polarization in monolayer MoSSe can be achieved by Cr or V doping [29].V-doping can induce a valley splitting of 59 meV, as shown in figure 3(h).Recently, Sahoo et al investigated V-doped MoS 2 using two experimental approaches and found that a valley splitting of about 35 meV at room temperature [41].The first one is photoluminescent measurements based on a conventional chiral selective emission/excitation.The second one is reflection measurements based on spin Hall effect of light.This study thus opens up the possibility of developing atomically ultrathin optoelectronic spintronic devices.

Magnetic valleytronic materials: ferrovalley materials
The magnetic field required to create an experimentally meaningful valley splitting is too high and thus is not readily available.Adsorption and doping generally lead to severe scattering of carriers by impurities, which negatively affects their performance and renders their applications difficult.Although forming heterostructure with magnetic materials can also introduce magnetism, to find and prepare a suitable magnetic semiconductor is usually not simple.
In comparison with inducing valley polarization in nonmagnetic valleytronic materials by application of magnetic field, magnetic doping, and forming heterostructure with magnetic materials, magnetic valleytronic materials with intrinsic magnetism have more advantages, e.g.possessing large valley splitting and simultaneously free of impurity scattering.In 2016, Tong et al studied the magnetic monolayer of H-phase VSe 2 , extending the valleytronic concept to 2D ferromagnetic systems and proposing the concept of ferrovalley [42][43][44].When both time-reversion and spital-inversion symmetries are broken, the ferrovalley materials exhibits intrinsic valley polarization and the anomalous valley Hall effect.Based on the investigation of monolayer H-VSe 2 , a variety of possible ferrovalley materials have been proposed theoretically.In this review, we classify the ferrovalley materials according to their molecular formula and crystal structure, which are mainly categorized into: 1. MX 2 , 2. M(XY) 2 , M(XY 2 ) and M(XYZ) 2 , 3. M 2 X 3 , M 3 X 8 and MNX 6 , 4. MNX 2 Y 2 , M 2 X 2 Y 6 and MNX 2 Y 6 , 5. Janus structures constructed according to these ferrovalley materials.

Types of ferrovalley materials 3.1.1. MX 2
The concept of valleys was first proposed in the study of semiconductors, such as silicon and aluminum arsenide.However, the vast majority of valleytronic materials discovered in research to date have graphene-like hexagonal structure (H-phase) with the molecular formula MX 2 (M is a transition metal element and X is a non-metallic element).Many H-phase monolayer MX 2 have been found to be ferrovalley materials, which break the spatial inversion symmetry with opposite Berry curvature at the K and -K(K ′ ) point.
Monolayer VSe 2 is a transition metal dichalcogenide with two main phases, T-and H-phase.The T-phase monolayer VSe 2 has a central inversion symmetry and is a 2D material with metallic properties.The H-phase monolayer VSe 2 has mirror symmetry and belongs to the D 3h point group with triple rotational symmetry.Theoretical studies have revealed that the monolayer H-VSe 2 is a ferromagnetic and valleytronic material [42,45].Since the ferromagnetism breaks the time reversal symmetry, the inequivalent valleys are no longer degenerate in H-VSe 2 , leading to spontaneous polarization of the valley, which is now referred to as the ferrovalley.H-VSe 2 is a new member of the ferroic family with ferrovalleys besides ferromagnetism and ferroelectricity.The energy band structure of H-VSe 2 is shown in figures 6(a)-(c).With the presence of SOC and the absence of magnetism, the energy band structure shows that the monolayer H-VSe 2 is a metallic material and has spin splitting of about 80 meV at the two valleys K and -K(K ′ ).If only the magnetism is considered without SOC, the spin-up and spin-down energy bands are completely separated near the Fermi energy level, and each cell has a magnetic moment of 1 µ B , with an upper and lower spin splitting of 930 meV.More interestingly, after considering both, there is not only a magnetic moment of 1 µ B , but also unequal spin splitting at K and -K(K ′ ) valleys.The band gap at -K(K ′ ) valley is larger than that at K valley, resulting in a valley splitting of 90 meV and thus leading to valley polarization, as a result of the intrinsic magnetism and spin-orbit coupling instead of extrinsic factors.When the direction of the magnetic moment is inverted, the larger gap is switched to the K valley and the smaller one goes to the -K(K ′ ) valley, indicating that valley polarization can be manipulated by changing the direction of the magnetic moment.For H-VSe 2 , as shown in figure 6(d), Berry curvatures of the inequivalent K and -K(K ′ ) valleys have opposite sign and different magnitudes.
In spite of the theoretical discovery of ferrovalley materials, the experimental realization of 2D magnetic materials is still difficult.In 2017, a breakthrough has been made in the experimental growth of 2D magnetic materials CrI 3 and Cr 2 Ge 2 Te 6 [46,47].But the experimental progress in preparation of ferrovalley materials is slower.2D VSe 2 was first prepared in T-phase, but not in the required H-phase [48].
Eventually, in 2022, You et al synthesized the monolayers of H-phase VSe 2 for the first time by chemical vapor deposition [49].First-principles calculations have predicted the phase stability and selectivity of VSe 2 monolayers of H-phase or T-phase [50], which have been confirmed by systematic synthesis studies.The honeycomb crystal structure was directly observed by scanning transmission electron microscopy, as evidenced by the analysis of selected area electron diffraction.The inversion symmetry breaking of the monolayer H-phase VSe 2 was further verified by giant second harmonic generation intensity mapping and angle-resolved tests.In addition, the H-phase VSe 2 exhibits typical p-type transport behavior.This progress marks a new stage in the study of 2D magnetic valleytronic materials.The experimental preparation process of H-phase monolayer VSe 2 and its transmission electron microscope image are shown in figure 7.
In addition, H-phase VS 2 and VTe 2 , which are structurally similar to VSe 2 , have been identified to be ferrovalley materials [51,52].Shen et al theoretically investigated the effects of spin direction and strain on the magnetic anisotropy, valley polarization, and magneto-optical Kerr effect in monolayers of ferrovalley H-VS 2 [52].It is found that H-VS 2 has a valley splitting of 123 meV when spin is aligned in the +z direction.
The valley splitting turns opposite if the spin polarization turns to the −z direction.Hence, the spin orientation can effectively modulate the valley polarization.VTe 2 monolayer has a Curie temperature of 377 K and a large valley splitting of 156.5 meV due to intrinsic magnetism, promising for room temperature applications.[51] Therefore, the V-based transition metal dichalcogenides, H-VS 2 , H-VSe 2 and H-VTe , are good candidates for valleytronic applications.
In addition to H-VSe 2 , H-VS 2 and H-VTe 2 , the main ferrovalley materials of H-MX 2 type that have studied theoretically to date include NbX 2 (X = S, Se), FeCl 2 , RuBr 2 , OsBr 2 , ScX 2 (X = Br, Cl, I), YX 2 (X = I, Br, Cl), LaX 2 (X = H, I, Br), GdX 2 (X = F, Cl, Br, I), LuX 2 (X = Cl, Br, I), and RI 2 (R = Sc, Y, La-Lu), etc., which have the same structure and observe the same symmetries.The detailed data of their magnetic moment, valley splitting, magnetic anisotropy and Curie temperature are listed in table 1.As discussed above, the valley splitting occurs only when the magnetization is not in-plane.There are H-MX 2 type ferrovalley materials, such as MBr 2 (M = Ru, Os) [53] and FeCl 2 [54], having native out-of-plane magnetic easy axes (along the z-axis), which can exhibit ferrovalley properties without the need to modulate the direction magnetic moment.The metallic elements and non-metallic elements that may form ferrovalley materials in the form of MX 2 are marked in yellow and brown, respectively, in table 2.

M(XY) 2 , M(XY 2 ) 2 and M(XYZ) 2
On the basis of MX 2 , new structures, M(XY) 2 , M(XY 2 ) 2 and M(XYZ) 2 can be formed by replacing the X atom with a non-metallic element group (XY, XY 2 , XYZ, etc.), and it has been revealed that the valleytronic properties still remain in a variety of such substitutions, among which, in particular, Fe(OH) 2 [77], MN 2 H 2  It is found by first-principles calculation that hydrogenation can help to stabilize the hexagonal phase of some materials, and for examples, the monolayer VN 2 H 2 , NbN 2 H 2 and TaN 2 H 2 in hexagonal phase are highly energetically, kinetically, and thermodynamically stable [78].The crystal and band structure of NbN 2 H 2 are shown in figure 8(a, b).With respect to the pristine MN 2 , the number of d-electrons is changed by hydrogenation and hence the hexagonal phase becomes favorable for MN 2 H 2 .VN 2 H 2 , NbN 2 H 2 and TaN 2 H 2 in hexagonal phase are ferromagnetic with Curie temperatures up to 364, 225, and 130 K, respectively, and further more they become bipolar magnetic semiconductors under magnetic interaction.Valley polarization exists in these monolayer H-MN 2 H 2 with valley splitting as high as 53, 110, and 350 meV in the V-, Nb-and Ta-based systems, respectively.Therefore, these monolayer H-MN 2 H 2 are promising magnetic valleytronic materials.
Recently, monolayers of MoSi 2 N 4 has been synthesized, which has stimulated extensive studies on MA 2 Z 4 materials [83,85] (M is transition metal; A, Z are non-metallic elements in group IV and group V, respectively).Several members of this family have been found to possess valleytronic properties [86].VSi 2 N 4 and VSi 2 P 4 have intrinsic ferromagnetism, resulting in spontaneous valley polarization [79,80,87].The crystal and band structure of VSi 2 N 4 are shown in figure 8(c, d).Monolayer VP 2 B 2 Te 2 tends to have in-plane magnetization, while monolayers VP 2 Al 2 S 2 and VP 2 GA 2 S 2 have out-of-plane easy magnetic axis.The out-of-plane magnetization together with SOC leads to the spontaneous valley polarization of monolayers VP 2 Al 2 S 2 and VP 2 GA 2 S 2 [84].The crystal and band structure of VP 2 GA 2 S 2 are shown in figure 8(e, f).

M 2 X 3 , M 3 X 8 and MNX 6
Monolayer Cr 2 Se 3, TM 2 (OM) 2 and Cr 2 CF 2 are found to be ferromagnetic semiconductors with valleytronic properties [88][89][90].Cr 2 Se 3 has robust out-of-plane magnetization.The crystal and band structure of Cr 2 Se 3 are shown in figure 9(a, b).Together with inversion symmetry breaking and strong spin-orbit coupling, monolayer Cr 2 Se 3 shows spontaneous valley polarization to exhibit the intriguing anomalous valley Hall effect.
Based on first-principles density-functional theory, Feng et al and Peng et al revealed that monolayer Nb 3 X 8 (X = Cl, Br, I) with a breathing kagome lattice are 2D multiferroic ferrovalley semiconductors [91,92].The crystal and band structure of Nb 3 I 8 are shown in figure 9(c, d).The spontaneous valley polarization of Nb 3 I 8 is as high as 107 meV, which is favorable for practical applications.This fascinating phenomenon can be attributed to the combined effect of intrinsic magnetic exchange interactions and strong spin-orbit coupling, superior to the valley polarization induced by external excitations.
Du et al reported the discovery of monolayer TiVI 6 , which is a promising 2D valleytronic material [93].Via first-principles calculations, it is found that monolayer TiVI 6 spontaneously exhibits spin polarization and valley polarization, resulting in 22 meV valley splitting in the valence band.The crystal and band structure of TiVI6 are shown in figure 9(e, f).The underlying physics is explored in terms of strong magnetic exchange interactions and large spin-orbit coupling.It is further revealed that this system can exhibit an anomalous valley Hall effect.By performing first-principles calculations, Zhao et al proposed that monolayer TcIrGe 2 S 6 is an excellent 2D valleytronic material [94].It is found that monolayer TcIrGe 2 S 6 is a ferromagnetic semiconductor with an out-of-plane magnetic easy axis.The large valley polarization, out-of-plane magnetization, and valley-contrast properties make monolayer TcIrGe 2 S 6 an ideal 2D valleytronic system.The crystal structure

Janus structure ferrovalley materials
2D Janus structures, especially those derived from transition metal dichalcogenides, have become a hot research topic in recent years.2D Janus structures with specular asymmetry in the z-direction show novel properties, such as Rashba effect and piezoelectric polarization [101][102][103][104][105][106].Monolayer transition metal dichalcogenides of the type H-MX 2 (M is a transition metal element; X is a chalcogen element) belong to D 3h space group, whereas monolayer of Janus H-MXY (M is a transition metal element; X and Y are chalcogen elements X ̸ = Y) has C 3v space group symmetry due to the loss of the mirror symmetry in the z-direction.Since the atomic radii and electronegativity of X and Y elements are different, the charge distribution between the M-X and M-Y layers is different [104].Despite the reduced symmetry, the phonon spectrum and molecular dynamics simulations show that the monolayer Janus MoSSe, WSSe, WSeTe, and WSTe are dynamically stable [102].
With the development of experimental techniques, the Janus structure is no longer only a theoretical model.Recently, Janus MoSSe [103,107], WSSe [108], and CrSSe [109] have been successfully synthesized experimentally by modified chemical vapour deposition method.In the synthesis of MoSSe, for instance, a monolayer of MoS 2 was prepared by chemical vapour deposition, then hydrogen plasma was passed to replace the S atoms in the top layer of the monolayer of MoS 2 with hydrogen atoms.Afterwards, the hydrogen plasma was stopped, and then the Se powder was sublimated at 350 • C-450 • C to replace the top hydrogen atoms with Se atoms.Finally, the Janus MoSSe was obtained.

The properties of ferrovalley materials
When an in-plane electric field is applied, the interaction of the electric field with a non-zero Berry curvature will form an anomalous transverse current [10].In ferrovalley materials, such as Janus H-VSSe, the Berry curvature only has large values near the valleys.The Berry curvature of H-VSSe at the inequivalent valleys has opposite signs as well as unequal values, which can result in valley carriers moving in opposite directions and different velocities in the cross direction.When an electric field E is parallel to the monolayer H-VSSe with zero magnetic field B = 0. Since its Berry curvature is negative at the K valley and positive at the -K(K ′ ) valley, the carriers at the K and -K(K ′ ) valleys are turned to the left and right side, respectively, in the case of hole doping.In the electron doping case, the electrons occupy the bottom of conduction bands at the valleys, and the carrier movement in cross direction will be reversed with respect to the case of hole doping, because the Berry curvatures of the bottom of the conduction bands are opposite to those of the top of the valence bands in the same valley.However, due to the asymmetry of the Berry curvatures between the inequivalent valleys, the cross current to either side is not equal, leading to unbalanced charge accumulation at the edges and producing a Hall voltage.Although the inequivalent valleys in ferrovalley materials have the same spin, the unbalanced charge distribution will produce simultaneous charge polarization, spin polarization, and valley polarization.The anomalous Hall conductivity of MoS 2 is always zero because the Berry curvature of MoS 2 is an odd function in the Brillouin zone, the Hall current flowing to the two edges is the same and cancels each other, and therefore, there is no net Hall current in the cross direction.In H-VSSe, the odd-parity of the Berry curvature in Brillouin zone is broken, and thereby resulting in a non-zero anomalous Hall conductivity.The schematic of anomalous valley Hall effect are shown in figure 12(a).When the Fermi energy level is between the valence band edges V K and V −K of the two valleys, which corresponds to hole doping, the maximum anomalous Hall conductance is calculated to be 29.0 S cm −1 as shown in figure 12(b).The maximum anomalous Hall conductance is 7.9 S cm −1 when the Fermi energy level is between the conduction band edges of the two valleys, which corresponds to electron doping.Since charge Hall currents are more easily to be measured experimentally, the anomalous valley Hall effect offers a potential way to realize data storage using magnetic valleytronic materials.Figure 12(c) is a schematic of data storage based on the anomalous valley Hall effect.
In nonmagnetic transition-metal dichalcogenides, such as MoS 2 , there exists valley-selective circular dichroism that allows selective excitation of the valley carriers at K and -K(K ′ ) by oppositely circularly polarized light.In nonmagnetic MoS 2 , these two peaks overlap because of the energy degeneration of the valleys.When a magnetic field is applied to MoS 2 or MoSe 2 , a small splitting of the peaks can be observed.The splitting grows linearly with the magnetic field at the rate of 0.22 meV per Tesla [26,137].In order to study the optical property of valley Zeeman splitting in the ferromagnetic H-VSSe, the imaginary part of its dielectric function has been calculated.As shown in figures 13(a) and (b), the two valleys of H-VSSe have different optical absorption properties.The peaks of the red dotted and blue dashed curves correspond to the band gaps ∆ K and ∆ −K at the valley, respectively [116].The splitting between the peaks is 179 meV, which is equal to the valley splitting calculated by the GW method.Because of the large valley splitting, the difference of the excitation energies E K and E −K at the K and -K(K ′ ) valleys are large enough, allowing selective excitation of the valleys even with unpolarized light.In contrast, for nonmagnetic transition-metal dichalcogenides, the two valleys are degenerate in energy with the same bandgap.Therefore, to realize the selective optical excitation of the valley, it is necessary to use monochromatic light with the energy equaling to the valley gap and at the same time with circular polarization.Using unpolarized light with different frequencies to selectively excite the ferrovalleys in VSSe, which has large valley splitting, makes the application of valleytronics easier.
As shown in figures 13(c) and 11(d), the two valleys of H-VSe 2 have different optical absorption properties, K and -K(K ′ ) can only absorb and emit the left-handed and right-handed circularly polarized light, respectively.Similar to VSSe, the valley splitting of VSe 2 is large enough so that it is also possible to selectively excite the valleys in VSe 2 by unpolarized light with different energies.When the direction of the magnetic moment is reversed, the optical properties of the valleys are also reversed.

Direction of the magnetic moment
It is found that the orbital magnetic moment along z direction µ L is µ L (C ±K ) ≈ 0 and µ L (V ±K ) ≈ ±2µ B , respectively [116].µ B is the Bohr magneton.The spin of the valence-band edge almost remains parallel in the upward direction, producing an effective magnetic field B eff acting on µ L and inducing an energy shift of µ L B eff .Therefore the energy shift at the conduction valley edges is zero, whereas it is ±2µ B B eff at the valence edges of K and -K(K ′ ) valleys, respectively, leading to a valley splitting of 4µ B B eff .The angle θ between B eff and the orbital magnetic moment can be changed to tune the valley splitting.The spin quantization axis is rotated in the plane shown in figure 14(a), in which θ denotes the angle between B eff and the orbital angular moment (along z axis), yielding a valley splitting of 4µ B B eff cos θ.Thus the direction of spin, i.e. the effective magnetic field can be rotated to modulate the valley splitting.The θ dependent bands are investigated.As shown in figures 14(a) and (b), it is found that the band gaps ∆ K and ∆ −K at the K and -K(K ′ ) valleys increase and decrease, respectively, when θ varies from 0 • to 180 • .Therefore, the valley splitting can be continuously modulated.The schematic diagram of manipulation of valleytronic properties by tuning the direction of magnetic moment is shown in inset of figure 14(c).As shown in figure 14(c), it can be seen that when the direction of the effective magnetic field is perpendicular and parallel to the H-VSSe plane, the difference of the Berry curvature at the two valleys is maximum and zero, respectively.

Electric field
Applying an electric field can also effectively modulate the energy band structure.The response of the electronic properties of H-VSSe to the electric field is investigated by applying an electric field perpendicular to the plane of H-VSSe.In Janus H-VSSe, there is no mirror symmetry between the upper S atoms and the lower Se atoms, so the electric field in the positive and negative directions should have different effects.The GW energy bands under 0 and 0.7 eV Å −1 electric field are shown in figure 14(d) [116].It can be seen that the positive electric field E z makes the energy band shift upward, while the energy band structure shifts downward when the electric field is in the opposite direction.The band gaps ∆ K and ∆ −K first slowly vary linearly with a small electric field, while there is a sharp variation when the electric field reaches 0.4 eV Å −1 as shown in figure 14(e).However, the valley splitting remains almost constant throughout the process, as shown in the inset in figure 14(e).In H-VSSe, the band gap at the valley changes with the direction and magnitude of the electric field, which increases when the electric field is positive and decreases when the electric field is negative.In addition, the rate of change of the bandgap value is different for different directions of the electric field, which is quite different from the MX 2 -type transition-metal dichalcogenides.The calculations show that the valley bandgap of monolayer VSe 2 always increases with the value of the electric field |E z |, regardless of the direction of the electric field, because of the mirror symmetry of the two Se layers about the V layer.The Berry curvature at different electric fields has also been calculated, as shown in figure 14(f).It is found that the electric field has little effect on the Berry curvature at the valley.

Strain
In heterostructures or at different temperatures, the materials is generally subject to strain.As shown in figure 14(g), the effect of strain on the electronic structure of H-VSSe is investigated by calculating the GW energy bands in the plane at different strains [116].It is found that the tensile strain shifts the energy bands downward relative to the energy bands when unstrained [138], while the compressive strain shifts the energy bands upward.Strain modulates the band gap very effectively.When the strain is in the range of ±3%, the values of bandgap variation at the K and -K(K ′ ) valleys can reach 0.63 eV and 0.55 eV.The bandgap varies linearly with strain, decreasing in dilation and increasing in compression.The slopes of the two lines in figure 14(h) show that ∆ K has a larger slope than ∆ −K .Therefore, the energetic valley splitting increases with tensile strain and decreases with compressive strain.Over a strain range of ±3%, the band gap ∆ varies up to a value of 80 meV.There is also a considerable change in the Berry curvature at the valley, as shown in figure 14(i).The magnitudes of the Berry curvature at the K and -K(K ′ ) valleys and their difference increases (decreases) with tensile (compressive) strain.The anomalous Hall conductance is determined by the sum of the Berry curvatures in the entire Brillouin zone.Since the Berry curvature is not zero only near the valleys, it can be predicted that the anomalous Hall conductance will increase with increasing tensile strain, which increases the difference of the Berrry curvature of the inequivalent valleys.

Applications of 2D magnetic valleytronic materials
The rapid development of spintronics and valleyronics provides new possibilities for the next generation of nanoelectronic devices.The key to the application of valley electronics is to realize the valley polarization, so as to obtain two unequal valleys representing the states of '0' and '1' .To realize the polarization of valleys in nonmagnetic valleytronic materials, the current mainstream solution is to selectively polarize one of the valleys by circularly polarized light [139,140].However, because of the energy degeneration of the two valleys, as shown in figure 15(a), the unbalanced distribution of charge in the valley after the selective excitation just lasts for a short period of time and hence is difficult to be maintained, making the application of valleytronics difficult.Another way to achieve valley polarization is to break the time-reversal symmetry by applying an external magnetic field in order to lift the energy degeneration of two valleys.However, the valley polarization obtained by this method is very small.Usually 1 Tesla can only approximately produce a valley splitting of 0.25 meV [26,137], and very large magnetic fields are needed to achieve a detectable valley polarization.The application of a magnetic field to a heterojunction can greatly enhance the valley polarization produced per Tesla [141].Valleytronic materials with intrinsic magnetism have more desirable properties with large energy splitting of the inequivalent valleys and without impurity scattering [42,45,142].In the intrinsically magnetic valleytronic material, the two valley band gaps are not the same, as shown in figure 15(b), and the charge distribution in the valley at equilibrium is kept unbalanced, which is able to stably maintain the states of '0' and '1' , thus facilitating the information encoding.Moreover, the switching of different valley polarizations can be realized by changing the direction of the magnetic field, thus realizing the storage and transmission of information.

Conclusions and outlook
The main methods for valley polarization in nonmagnetic valleytronic materials include applying magnetic field, doping magnetic atoms, and forming heterostructure with magnetic materials.Ferrovalley materials with intrinsic magnetism have more desirable properties, having large valley splitting in inequivalent valleys and being free of the problem of impurity scattering.In this review, we have classified the ferrovalley materials according to their molecular formula and crystal structure, which are mainly categorized into: 1. MX 2 , 2. M(XY) 2 , M(XY 2 ) and M(XYZ) 2 , 3. M 2 X 3 , M 3 X 8 and MNX 6 , 4. MNX 2 Y 2 , M 2 X 2 Y 6 and MNX 2 Y 6 , 5. Janus structure ferrovalley materials.

C Luo et al
The ferrovalley materials possess unique novel properties.The Berry curvature has large values only near the valleys with opposite signs and unequal magnitudes in the two inequivalent valleys, leading to anomalous valley Hall effect.As a result of valley spontaneous polarization, the excitation energies E K and E −K at the K and -K(K ′ ) valleys are no longer equal, which allows selective excitation of the valleys even with unpolarized light.The properties of ferrovalley can be modulated by variation of the magnetizaiton, electric field and strain.
In addition to monolayer magnetic valleytronic materials, there are also a variety of magnetic valleytronic materials with multilayers [144][145][146][147] and those with magnetic substrate [148][149][150][151][152][153][154][155][156][157][158][159][160][161][162][163].The valley spontaneous polarization occurs only when the magnetization has an out-of-plane component.Although theoretical calculations have predicted some magnetic valleytronic materials with out-of-plane magnetic easy axis, the experimental synthesis has not been achieved.There have been a lot of active studies which have greatly promoted the development of valleytronics.It can be expected that more and more ferrovalley materials will be discovered and they will be closer and closer to realistic applications.

Figure 1 .
Figure 1.Crystal structure and properties of monolayer H-phase MoS2.(a) Crystal structure and high symmetry points in the Brillouin zone.(b) Band structure and Berry curvature, with two inequivalent valleys at high symmetry points K and -K(K ′ ).(c) Schematic diagram of the valley optical selection rule.(d) Schematic diagram of valley Hall effect [3, 6, 20, 21].

Figure 2 .
Figure 2. (a) Spin Hall and valley effects in electron and hole-doped systems.(b) Selection rules for valley and spin optical transitions.(c) Electron and hole spin and valley Hall effects when activated by a linearly polarized light field.(d) Electron and hole spin and valley Hall phenomena induced by two-color optical fields [23].

Figure 3 .
Figure 3. (a) Experimental results of valley splitting with magnetic field.(b) A schematic diagram of the bands near the valleys to show the energy shift in the magnetic field.(c) The top and side view of CrI3/WSe2 heterostructure.(d)The band structure of CrI3/WSe2 with atomic orbital projection.(e) Schematic diagram of type-II band structure of CrI3/WSe2 heterostructures.(f) The top and side view of Janus MoSSe, and the high symmetry points in the Brillouin zone.(g) The top view of V-doped monolayer MoSSe.(h) Band structures of V-doped monolayer Janus MoSSe [25, 27-29].

Figure 4 .
Figure 4.The schematic of the LL structure of monolayer WSe2.(a) The LL structure of monolayer WSe2 at 9 T for doping at 5.9 × 10 12 cm −2 .Bands v1, c1 and c2 are shown by solid lines with bands of the same electron spin in the same color.The LLs are represented by dashed lines.Black lines with an arrowhead show the inter-LL transitions.(b) The left-(l) and right-handed (r)reflectance contrast spectra at σ = 5.9 × 10 12 cm −2 under differing magnetic fields.The vertical bar shows the scale for contrast equal to 0.1[35].

Figure 6 .
Figure 6.The band structure of monolayer H-VSe2 (a) without magnetism but with SOC, (b) with magnetism but without SOC, (c) with SOC and +z magnetic moment.(d) The high symmetry points and Berry curvature of monolayer H-VSe2.[42, 45].

3. 1 . 4 .
MNX 2 Y 2 , M 2 X 2 Y 6 and MNX 2 Y 6 Sun et al found the valley splitting in monolayer MHfN 2 Cl 2 (M = V, Cr) is linearly modulated under different biaxial strain and Hubbard U values.The crystal structure and band structure of VHfN 2 Cl 2 are shown in figure 10(a).The maximum valley splitting is 175 and 62 meV for the monolayer VHfN 2 Cl 2 and CrHfN 2 Cl 2 , respectively.

Figure 12 .
Figure 12.(a) Schematic diagram of anomalous valley Hall effect.(b) Anomalous Hall conductivity of Janus H-VSSe.(c) Schematic diagram of data storage based on anomalous valley Hall effect [42, 116].

Figure 13 .
Figure 13.The optical properties of monolayer H-VSSe and H-VSe2.The k-resolved degree of optical polarization of (a) H-VSSe and (c) H-VSe2, respectively.(d) Optical absorption peak splitting and selection rules for optical transitions in two valleys of (c) H-VSSe and (d) H-VSe2, respectively [42, 45, 116].

Table 1 .
The magnetic moment, valley splitting, magnetic anisotropy and Curie temperature of different 2D ferrovalley materials MX2.

Table 2 .
Periodic table of elements.Those marked yellow and orange are metallic and non-metallic elements that may form ferrovalley materials MX2, respectively.