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The first- and second-order magneto-optical effects and intrinsically anomalous transport in 2D van der Waals layered magnets CrXY (X = S, Se, Te; Y = Cl, Br, I)

Recently, the two-dimensional magnetic semiconductor CrSBr has attracted considerable attention due to its excellent air-stable property and high magnetic critical temperature. Here, we systematically investigate the electronic structure, magnetocrystalline anisotropy energy, first-order magneto-optical effects (Kerr and Faraday effects) and second-order magneto-optical effects (Schafer-Hubert and Voigt effects) as well as intrinsically anomalous transport properties (anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects) of two-dimensional van der Waals layered magnets CrXY (X = S, Se, Te; Y = Cl, Br, I) by using the first-principles calculations. Our results show that monolayer and bilayer CrXY (X = S, Se) are narrow band gap semiconductors, whereas monolayer and bilayer CrTeY are multi-band metals. The magnetic ground states of bilayer CrXY and the easy magnetization axis of monolayer and bilayer CrXY are confirmed by the magnetocrystalline anisotropy energy calculations. Utilizing magnetic group theory analysis, the first-order magneto-optical effects as well as anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects are identified to exist in ferromagnetic state with out-of-plane magnetization. The second-order magneto-optical effects are not restricted by the above symmetry requirements, and therefore can arise in ferromagnetic and antiferromagnetic states with in-plane magnetization. The calculated results are compared with the available theoretical and experimental data of other two-dimensional magnets and some conventional ferromagnets. The present work reveals that monolayer and bilayer CrXY with superior magneto-optical responses and anomalous transport properties provide an excellent material platform for the promising applications of magneto-optical devices, spintronics, and spin caloritronics.

Nernst effect (ANE) [31], and anomalous thermal Hall effect (ATHE) [32], remain largely unexplored. Since the family materials of CrXY (X = S, Se, Te; Y = Cl, Br, I) share the same crystal structure [15], atomically-thin films of CrXY should be easily obtained by the means of mechanical exfoliation, similarly to ML and BL CrSBr. A systematic study on the magneto-optical effects (including MOKE, MOFE, MOSHE, and MOVE) and intrinsically anomalous transport properties (including AHE, ANE, and ATHE) of ML and BL CrXY will contribute to further understanding of this class of 2D vdW magnets.
In this work, using the first-principles calculations, we systemically investigate the electronic, magnetic, optical, magneto-optical, and anomalous transport properties of ML and BL CrXY. We first reveal that CrSY and CrSeY are narrow band gap semiconductors, whereas CrTeY present multi-band metallic behaviors. Then, the magnetic ground state and the easy axis of magnetization for CrXY are confirmed by magnetocrystalline anisotropy energy (MAE) calculations. The results of MAE indicate that the easy axes of CrTeBr and CrTeI point along the out-of-plane direction (i.e., z-axis), while the other seven family members of CrXY prefer the in-plane magnetization. The BL CrSI, CrSeCl, and CrTeCl show strong interlayer FM coupling, whereas other six family members exhibit interlayer AFM coupling. Before calculating the magneto-optical and anomalous transport properties, we primarily analyze the shape of optical conductivity tensor by utilizing the magnetic group theory. The uncovered symmetry requirements indicate that the first-order magneto-optical effects (MOKE and MOFE) as well as the AHE, ANE, and ATHE can only arise in the FM state with out-of-plane magnetization. In order to explore the FM and AFM states with in-plane magnetization, we further study the second-order magneto-optical effects (MOSHE and MOVE). The calculated first-and second-order magneto-optical effects as well as intrinsically anomalous transport properties of CrXY are compared with other 2D magnets and some conventional ferromagnets. Our results show that 2D CrXY with superior magneto-optical responses and anomalous transport properties provide an excellent material platform for the promising applications of magneto-optical devices, spintronics, and spin caloritronics.

II. THEORY AND COMPUTATIONAL DETAILS
The magneto-optical effects, as a kind of fundamental magnetotransport phenomena in condensed matter physics, are usually considered to be a powerful probe of magnetism in low-dimensional systems [2,3]. The MOKE and MOFE can be described as the rotation of the polarization planes of reflected and transmitted lights after a linearly polarized light hits on the surface of magnetic materials, respectively. Here, we only consider the MOKE and MOFE in the so-called polar geometry (e.g., both the directions of incident light and mag-netization are along the z-axis), as shown in Fig. 1(a). The MOKE and MOFE are commonly regarded as firstorder magneto-optical effects as their magnitudes are linearly proportional to the strength of magnetization and their signs are odd in the direction of magnetization. On the other hand, if the magnetization lies on the 2D atomic plane (i.e., xy-plane) and the angle between the electric field (E) of incident light and the direction of magnetization (M) is 45 • , the polarization planes of reflected and transmitted lights still rotate, which are called MOSHE and MOVE [ Fig. 1(b)], respectively. The MOSHE and MOVE are considered to be second-order magneto-optical effects as their magnitudes are quadratic to the strength of magnetization and their signs are even in the direction of magnetization.
It should be noted that in some FM and AFM materials, the first-order magneto-optical effects are prohibited due to the symmetry requirements, but the second-order magneto-optical effects can exist. Here, we take ML FM Fe 3 GeTe 2 and BL AFM Fe 3 GeTe 2 with in-plane magnetization [33] as examples to discuss this point. The magnetic point group of ML FM Fe 3 GeTe 2 with in-plane magnetization (when the spin points to the x-axis) is m m2 , in which the mirror plane M x is perpendicular to the spin and is parallel to the z-axis. This mirror symmetry operation reverses the sign of the off-diagonal element of optical conductivity σ xy (≡ σ z ), thus indicating σ xy = 0. If the spin points to the y-axis, the magnetic point group of ML FM Fe 3 GeTe 2 is m m 2, which contains a combined symmetry T M z (T is the time-reversal symmetry, and M z is a mirror plane perpendicular to the z-axis and parallel to the spin). This combined symmetry operation reverses the sign of σ xy , also suggesting σ xy = 0. For BL AFM Fe 3 GeTe 2 , the magnetic point groups are 2 /m and 2/m when the spin points to the x-and y-axes, respectively. Both the two magnetic point groups have a combined symmetry T P (T and P are the time-reversal and spatial inversion symmetries, respectively), which forces σ xy = 0. Overall, in the cases of ML FM Fe 3 GeTe 2 and BL AFM Fe 3 GeTe 2 with in-plane magnetization, the vanishing off-diagonal element of optical conductivity (σ xy = 0) can not give rise to the firstorder magneto-optical effects (MOKE and MOFE), refer to Eqs. (1)-(3). In contrast, the second-order magnetooptical effects (MOSHE and MOVE) depend on both the off-diagonal and diagonal elements, see Eqs. (4)- (5). For the in-plane magnetization along either x-or y-axis, the two diagonal elements are not equivalent (σ xx = σ yy ) due to the anisotropy of the in-plane optical response, which definitively induces the second-order magneto-optical effects regardless of whether the off-diagonal element is zero or not.
The Kerr and Faraday rotation angles (θ K and θ F ), which are the deflection of the polarization plane with respect to the incident light, can be used to quantitatively characterize the magneto-optical performance. Additionally, the rotation angle θ K(F) and ellipticity ε K(F) are usually combined into the complex Kerr (Faraday) an- . ω, ω , and ω indicate the incident, reflected, and transmission lights, respectively. θK, θF, θSH, and θV represent the Kerr, Faraday, Schäfer-Hubert, and Voigt rotation angles, respectively. M and E label the magnetization and electric field of incident light, respectively. In (b), E and M form an angle of 45 • on the xy plane.
gle [34][35][36], and where c is the speed of light in vacuum, ω is frequency of incident light, d is the thickness of thin film, and n ± are the refractive indices for the right-and left-circularly polarized light, respectively. σ sub xx is the diagonal element of the optical conductivity tensor for a nonmagnetic substrate (e.g., SiO 2 /Si, as shown in Fig. 1). In our calculations, only the interface between CrXY and SiO 2 layers is considered, and the SiO 2 layer is assumed to be thick enough. In this case, the Si layer is unimportant as it does not influence the magneto-optical properties of CrXY. The σ sub xx of the SiO 2 layer can be written as i(1 − n 2 sub ) ω 4π , here the n sub is the energy-dependent refractive index of SiO 2 crystal (quartz) [37,38]. σ xy is the off-diagonal element of the optical conductivity tensor for a magnetic thin film and σ 0 = 1 2 (σ xx + σ yy ) due to in-plane anisotropy.
In the case of MOSHE and MOVE, the rotation angle θ SH(V) and ellipticity ε SH(V) can be also combined into the complex Schäfer-Hubert and Voigt angles. Considering the in-plane magnetization along the y-axis (Fig. 1), the complex Schäfer-Hubert angle can be expressed as and the complex Voigt angle is give by [39,40] Here, αβ = δ αβ + 4πi ω σ αβ with α, β ∈ x, y, z is the permittivity tensor (in cgs units), n and n ⊥ are the refractive indices of a magnetic thin film that are parallel and perpendicular to the direction of magnetization, respectively.
From Eqs.(1)-(5), one would know that the key to calculate the first-and second-order magneto-optical effects is the optical conductivity, which can be obtained by the Kubo-Greenwood formula [41,42], where the superscripts 1 (2) indicates the real (imaginary) part of optical conductivity. ψ nk and ε nk are the Bloch function and interpolated energy at the band index n and momentum k, respectively. f nk , V , N k ,v x(y) , ω, and η are the Fermi-Dirac distribution function, volume of the unit cell, total number of k-points for sampling the Brillouin zone, the velocity operators, the photon energy, and energy smearing parameter, respectively. η was chosen to be 0.1 eV, corresponding to the carrier relaxation time of 6.5 fs, which is in the realistic range for multilayer transition metal chalcogenides [43,44]. In this work, a sufficiently dense k-mesh of 450×390×1 is used to calculate the optical conductivity. We also tested a denser k-mesh of 500×430×1 for the convergence of optical conductivity, see Figs. S1(a) and S1(b) in Supplemental Material [45]. The AHE, which is featured by the transverse voltage drop induced by a longitudinal charge current in the absence of external magnetic field, is another fundamental magnetotransport phenomenon in condensed mater physics. The contributions to AHE can be distinguished to extrinsic (skew-scattering and side-jump) and intrinsic mechanisms [30]. The extrinsic AHE is dependent on the scattering of electrons off impurities or due to disorder, while the intrinsic AHE originates from the Berry phase nature of the electrons in a perfect crystal. In this work, we only focus on the intrinsically anomalous transport, i.e., intrinsic AHE. The first-order magneto-optical effects and intrinsic AHE share the similar physical origin, and the dc limit of the real part of the off-diagonal element of optical conductivity, σ 1 xy (ω → 0), is nothing but the intrinsic anomalous Hall conductivity (AHC). Based on the linear response theory, the intrinsic AHC can be calculated [46], in which f nk is the Fermi-Dirac distribution function, V is volume of the unit cell, and Ω n xy (k) is the band-and momentum-resolved Berry curvatures, given by To calculate σ xy , a k-mesh of 600×520×1 was used. We also tested the convergence of σ xy on a denser k-mesh of 750×650×1, see supplemental Fig. S1(c) [45]. In addition to the AHE, the transverse charge current can also be induced by a longitudinal temperature gradient field, named ANE [31]. Additionally, a transverse thermal current arise under the longitudinal temperature gradient field, which is called ATHE or anomalous Righi-Leduc effect [32]. The magnetic materials that show large anomalous Nernst conductivity (ANC) and anomalous thermal Hall conductivity (ATHC) have great potentials to apply for spin caloritronics or thermoelectric devices. The AHE, ANE, and ATHE are closely related to each other, and can be expressed through the anomalous transport coefficients in the generalized Landauer-Büttiker formalism [47][48][49], where E, µ, and f are energy, chemical potential, and Fermi-Dirac distribution function, respectively. σ xy is the intrinsic AHC calculated by Eq. (7) at zero temperature. Then, the temperature-dependent ANC (α xy ) and ATHC (κ xy ) are written as To calculate α xy and κ xy , the integral in Eq. The first-principles density functional theory calculations were performed by utilizing the Vienna ab initio simulation package (vasp) [50,51]. The projector augmented wave (PAW) method [52] was used and the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof parameterization [53] was adopted to treat the exchange-correlation functional. Dispersion correction was performed at the van der Waals density functional (vdW-DF) level [54][55][56], where the optB86b functional was used for the exchange potential. For the ML and BL structures, the vacuum layer with a thickness of at least 15Å was used to avoid the interactions between adjacent layers. All structures were fully relaxed until the force on each atom was less than 0.001 eV·Å −1 . To describe the strong correlation effect of the d orbitals on Cr atom, the GGA+U method [57] was used by taking appropriate parameters of U and J in the range of 4.05 ∼ 4.40 eV and 0.80 ∼ 0.97 eV [15], respectively. The values of U and J used for each compound are listed in supplemental Tab. S1 [45]. To obtain accurate MAE, a large plane-wave cutoff energy of 600 eV and a Monkhorst-Pack k-mesh of 15×13×1 were used. The spin-orbit coupling (SOC) was included in the calculations of MAE, magneto-optical effects, and anomalous transport properties. The maximally localized Wannier functions were constructed by wannier90 package [41], and the d, p, and p orbitals of Cr, X (X = S, Se, Te), and Y (Y = Cl, Br, I) atoms were projected onto the Wannier functions, respectively. Totally, there are 44 and 88 Wannier functions for ML and BL CrXY, respectively, which converged adequately the results of magneto-optical effects and anomalous transport properties. Similarly, the computational technique employing the maximally localized Wannier functions can be in principle used to calculate the physical quantities that are closely related to orbital angular momenta [58][59][60][61]. The isotropy software [62] was used to identify magnetic space and point groups. The workflow of the various codes used in our work is depicted in supplemental Fig. S2 [45].

III. RESULTS AND DISCUSSION
A. Crystal, magnetic, and electronic structures Figure 2 shows the crystal structure of vdW layered materials CrXY (X = S, Se, Te; Y = Cl, Br, I). One can see that CrXY has a rectangular 2D primitive cell, in which a ML is made of two buckled planes of CrX sandwiched by two Y sheets, and a BL is formed by two MLs with the AA stacking order referring to experimental works [11,19,20,63]. The relaxed lattice constants of ML and BL structures are collected in supplemental Tab. S1 [45]. The calculated lattice constants of ML CrSBr, a = 3.54Å, b = 4.79Å, and thickness d = 8.00 A, are in good agreement with the experimental results, a = 3.50Å, b = 4.76Å [11], and d = 7.8 ± 0.3Å [19].
The magnetic anisotropy plays a vital role in stabilizing the long-range magnetic order of 2D systems. The MAE is defined as the difference of total energies between different magnetization directions (such as, along the x-, y-, and z-axis), which can be directly obtained by the first-principles calculations. To determine the magnetic ground states, we calculate the MAE of ML and BL CrXY. Since the FM ground state of ML CrXY was predicted by a recent theoretical work [15] and has been confirmed by two experimental works [19,20], here we directly adopt the FM configuration as the ground state of ML CrXY. Thus, the magnetic ground states for ML CrXY can also be determined by the Goodenough-Kanamori rules, i.e., superexchange mechanism [64][65][66]. The magnetic exchange interactions can be judged by the angle of the chemical bonds connecting the ligand and two nearest-neighboring magnetic atoms. Particularly, a system prefers to be ferromagnetic if the angle equals 90 • . Taking ML CrSBr as an example, the bond angle of Cr-Br-Cr is 88.96 • , and the bond angles of Cr-S-Cr are 94.24 • and 96.58 • , suggesting the ferromagnetic ground state. Our calculated MAE indicate that ML CrSI, CrSeY prefer to the in-plane magnetization along the x-axis, ML CrSCl, CrSBr, and CrTeCl are in favor of in-plane magnetization along the y-axis, while ML Cr-TeBr and CrTeI take for the out-of-plane z-axis magnetization. The evolution of the easy axis from in-plane to out-of-plane direction can be explained by the SOC strength of the systems. The atomic SOC strength increases gradually in the consequences of S → Se → Te and Cl → Br → I, and CrTeBr and CrTeI have the almost largest SOC strength among the other family compounds. It leads to the MAE results that CrTeBr and CrTeI have the out-of-plane z-axis magnetization, while the other family compounds have the in-plane x-or y-axis magnetization. This is similar to the case of 2D magnetic materials CrX 3 (X = Cl, Br, I) [67], in which the magnetization direction changes from in-plane (CrCl 3 ) to out-of-plane (CrI 3 ) with the increasing of SOC strength. that CrSI, CrSeCl, and CrTeCl exhibit the FM ground state, while the other six family members prefer to be AFM state. The easy magnetization axes of BL CrXY are identical to that of ML CrXY. The above results are summarized in supplemental Tab. S1 [45]. The easy magnetization axis of CrSBr is along the y-axis, which is in an accordance with the experimental reports [11,19,20].
After obtaining the magnetic ground states of CrXY, we shall further ascertain the corresponding magnetic groups as the group theory is a powerful tool to figure out the symmetry requirements for the magneto-optical effects and anomalous transport properties. The offdiagonal elements of optical conductivity tensor [σ αβ (ω)] fully determine the symmetry requirements of first-order magneto-optical effects [see Eqs. (1)- (3)]. At zero frequency limit, σ αβ (ω → 0) is nothing but the intrinsic AHC [i.e., Eq. (7)], and therefore the symmetry requirements of AHE, ANE, and ATHE [refer to Eqs. (9)-(11)] should be the same as that of σ αβ (ω). Here, we only need to analyze the symmetry results of σ αβ (ω) under relevant magnetic groups.
Since the off-diagonal elements of optical conductivity tensor can be regarded as a pseudovector (like spin), its vector-form notation, σ = [σ x , σ y , σ z ] = [σ yz , σ zx , σ xy ], is used here for convenience. In 2D systems, only σ z (≡ σ xy ) is potentially nonzero, while σ x (≡ σ yz ) and σ y (≡ σ zx ) are always zero. It can be easily understood from Eq. (6) as the zero velocity along the z-axis (i.e., v z = 0, meaning that the electrons can not move along the out-of-plane direction in 2D systems) definitely gives rise to vanishing σ yz and σ zx . For ML and BL FM CrXY with any magnetization directions (x-, y-, or z-axis), the magnetic space and point groups of are always identified to be P m m n and m m m [D 2h (C 2h ) in Schönflies notation], respectively. It is sufficient for us to analyze magnetic point group due to the translationally invariance of σ xy [68]. Hence, we only discuss the magnetic point group of m m m, in which there are one mirror plane M and two combined symmetries T M (T is the time-reversal symmetry). For the in-plane magnetization along the x-axis (y-axis), the mirror plane is M x (M y ), which is perpendicular to the spin magnetic moment, such a mirror operation reverses the sign of σ xy , indicating that σ xy should be zero. On the other hand, for out-of-plane magnetization, the mirror plane is M z and two combined symmetries are T M x and T M y . The spin magnetic moment is perpendicular to the M z , which does not change the sign of σ xy . Moreover, the mirror planes M x and M y are parallel to the spin, and both T M x and T M y preserve the sign of σ xy . A conclusion here is that σ xy is nonzero (zero) for ML and BL FM CrXY with out-of-plane (in-plane) magnetization.
In the case of the BL AFM CrXY, the magnetic space and point groups are P m m n and m m m [D 2h (D 2 ) in Schönflies notation] for out-of-plane magnetization, and are P m mn and mmm [D 2h (C 2v ) in Schönflies notation] for in-plane magnetization. There is a combined symmetry T P (P is the spatial inversion) in the two point groups of m m m and mmm , which forces σ xy to be zero. Overall, a nonzero σ xy can only exist in ML and BL FM CrXY with out-of-plane magnetization, which turn out to be our target systems for studying the MOKE, MOFE as well as AHE, ANE, and ATHE. It should be noted that the out-of-plane magnetization is not the magnetic ground states for some family members of CrXY, however, an applied external magnetic filed can easily tune the spin direction, as already realized in layered CrSBr [20]. The first-order magneto-optical effects (MOKE and MOFE) can not exist in FM and AFM CrXY with inplane magnetization, but it is not the case for the secondorder magneto-optical effects (MOSHE and MOVE). According to the Onsager relations, the diagonal elements of the permittivity tensor are even in magnetization (M), e.g., yy (−M) = yy (M). Furthermore, equations (4) and (5) show that the complex Schäfer-Hubert and Voigt angles depend on square of zx . Therefore, the MOSHE and MOVE must be even in M. In the case of FM and AFM CrXY with in-plane magnetization, although zx is always zero, the MOSHE and MOVE do exist because of the nonvanishing yy and xx (the relation of yy = xx also satisfies). Owing to this, we use the MOSHE and MOVE to characterize the in-plane magnetized ML FM and BL FM/AFM CrXY. The complex Schäfer-Hubert and Voigt angles are calculated by orienting the spins along the y-axis, and the results of x-axis in-plane magnetization can be simply obtained by putting a minus sign, refer to Eqs. (4) and (5).
The relativistic band structures of ML FM and BL FM/AFM CrXY with out-of-plane and in-plane magnetization are plotted in supplemental Figs. S4-S9 [45]. The results show that CrSY and CrSeY are narrow band gap semiconductors, while CrTeY are multi-band metals. All the band gaps of CrSY and CrSeY are summarized in supplemental Tab. S1 [45]. Here, we choose ML FM CrX Br (X = S, Se, Te) as representative examples to discuss the electronic properties of CrXY family materials, as shown in Fig. 3. A clear trend can be seen that the band gaps of CrX Br become smaller and eventually disappear with the increasing of atomic number (e.g., S → Se → Te). The orbital-decomposed density of states of ML FM CrX Br show that the d orbitals of Cr atom and p orbitals of X and Br atoms are dominant components near the Fermi level. For CrSBr, the calculated band gaps of ML FM, BL FM, and BL AFM states are 1.34 eV, 1.22 eV, and 1.30 eV, respectively, which are in good agreement with that of bulk material (∼ 1.25 eV) [11]. Since CrSY and CrSeY are narrow band gap semiconductors, the electrons and holes could bound in pairs to form quasiparticle excitons. Taking CrSBr as an example [20], the exciton binding energies for ML FM, BL FM, and BL AFM states are 0.5 eV, 0.37 eV, and 0.46 eV, respectively, which are apparently smaller than the exciton binding energies of ML FM CrI 3 (1.7 eV [69], 1.06 eV [70], and 0.84 [70]). It indicates that the relatively weak excitonic effects in CrSY and CrSeY may not significantly influence the optical and magneto-optical properties, in contrast to CrI 3 [69,70].
The Kerr and Faraday rotation angles (θ K and θ F ) at a given photon energy are obviously increasing with the thin-film thickness of CrXY, which can be easily seen from Tab. I and Figs. 4 and 5. In order to quantitatively describe the overall trend of Kerr and Faraday spectra, we here introduce a physical quantity called magneto-optical strengths (MOS) [75,80,81] MOS K and MOS F are plotted in Figs. 6(a) and 6(b), respectively, from which one can observe that the MOKE and MOFE of BL CrXY are nearly double that of ML CrXY. It is simply because that the MOKE and MOFE as the first-order magneto-optical effects are proportional to the magnetization, which is linearly dependent on the number of layers. This trend has also been observed in 2D van der Waals magnets CrI 3 [2] and Cr 2 Ge 2 Te 6 [73]. Moreover, a clear step-up trend of θ K and θ F can be seen in ML and BL CrXY by changing X (S → Se → Te) or Y (Cl → Br → I). It can be understood that the MOKE and MOFE are also proportional to the SOC strength, which increases with the atomic number. This trend can also be found in our previous work about hole-doped MX (M = Ga, In; X = S, Se, Te) monolayer, in which the MOKE and MOFE have a size consequence of InX > GaX due to the larger SOC strength of In atom.

C. Second-order magneto-optical effects
The first-order magneto-optical effects (MOKE and MOFE) are forbidden in ML and BL CrXY with in-plane magnetization, while the second-order magneto-optical effects (MOSHE and MOVE) can arise without the restriction by symmetry. Here, we illustrate the MOSHE and MOVE of ML FM and BL FM/AFM CrXY by taking the in-plane y-axis magnetization as an example, and the results of the x-axis magnetization can be obtained by putting a minus sign (refer to Eqs. (4) and (5) and note that zx equals zero due to the symmetry restriction). It has to be mentioned here that due to the in-plane crystal anisotropy of CrXY (a 2D rectangular primitive cell), the natural linear birefringence exists. This phenomenon is actually not related to magnetism, and can not be mixed into the second-order magneto-optical effects [82]. The diagonal elements of permittivity tensor can be expanded into a Taylor series in powers of magnetization M (up to second-order), αα is the part independent on magnetization,  αα is responsible for the natural linear birefringence originating from crystal anisotropy, and (2) αα is responsible for the MOSHE and MOVE originating from magnetism. We first calculate (0) αα in 2D nonmagnetic CrXY by forcing the magnetic moments to be zero during self-consistent field calculation, and then calculate αα (the sum of (0) αα and (2) αα ) in 2D magnetic CrXY. Finally, (2) αα can be simply obtained by subtracting (0) αα from αα . Thus, we take (2) αα instead of αα into Eqs. (4) and (5). The corresponding optical conductivities σ (2) αα are shown in supplemental Figs. S16 and S17 [45].
The calculated Schäfer-Hubert and Voigt rotation angles (θ SH and θ V ) of CrXY are plotted in Figs. 7 and 8, and the corresponding ellipticities (ε SH and ε V ) are plotted in supplemental Figs. S18 and S19 [45]. The spectra of θ SH and θ V for semiconducting CrSY and CrSeY are negligibly small in the low energy range far below the band gaps. However, in contrast to the Kerr and Faraday effects, θ SH and θ V take on finite values near the band edges [e.g., Figs. 7(a) and 8(a)]. The reason is that θ SH and θ V depend on the diagonal elements of optical conductivity, which are nonvanishing within the band gap (see Figs. S16 and S17 [45]). In the energy range larger than the band gaps, θ SH and θ V oscillate frequently, as shown in Figs. 7(a,b) and 8(a,b), respectively. For metallic CrTeY, the relatively larger θ SH and θ V can be found in Figs. 7(c) and 8(c), respectively. Taking CrSBr as an example, the maximal values of θ SH for ML FM, BL FM, and BL AFM states are -3.92 • , 6.01 • , and 9.87 • at the photon energies of 3.10 eV, 3.69 eV, and 2.24 eV, respectively; the maximal values of θ V for ML FM, BL FM, and BL AFM states are 0.40 • , -0.80 • , and -0.77 • at the photon energies of 3.13 eV, 2.66 eV, and 2.28 eV, respectively. The data of the largest θ SH and θ V as well as the related photon energies for all CrXY family members are summarized in Tab. I. The θ SH and θ V of CrXY are larger than that of some 2D AFMs, for example, 10 nm thick CoO thin-film (θ SH ∼ 0.007 deg) [76] and 10 nm thick CuMnAs thin-film (θ V ∼ 0.023 deg) [77].
From Eqs. (4) and (5), one can know that θ SH , ε SH , θ V , and ε V are dominated by the two diagonal elements of permittivity tensor ( yy and xx ) as the offdiagonal element ( zx ) is restricted to be zero by symmetry. By plugging the optical conductivity ( yy(xx) = 1+ 4πi ω σ yy(xx) ) into Eq. (4), one shall find θ SH ∝ −σ 1 diff = −Re(σ yy − σ xx ) and ε SH ∝ −σ 2 diff = −Im(σ yy − σ xx ). Figure 9 plots the real (σ 1 diff ) and imaginary (σ 2 diff ) parts of the difference between the diagonal elements of optical conductivities (σ yy and σ xx ) for ML FM, BL FM, and BL AFM CrSBr. One can clearly observe that the spectral structure of θ SH is determined by σ 1 diff only differing by an opposite sign, comparing the middle panel of Fig. 7(a) with Fig. 9(a). The spectral structure of ε SH is thus determined by σ 2 diff , comparing the middle panel of supplemental Fig. S18(a) [45] with Fig. 9(b). On the other hand, there is not such a clear relation between θ V (ε V ) and σ 1 diff (σ 2 diff ) since equation (5) involves the square roots of yy and xx , though the spectral structures of θ V (ε V ) and σ 1 diff (σ 2 diff ) seem to be similar somewhere.   In this subsection, we shall discuss the AHE as well as its thermoelectric counterpart, ANE, and its thermal analogue, ATHE, for ML and BL FM CrXY with outof-plane magnetization. The ANE and ATHE are evaluated at 100 K below the Curie temperatures by the theoretical prediction (> 108 K) [15] and experimental measurements (> 140 K) [19,20]. The magnetic transition temperature can be in principle increased via doping effects. For example, magnetic atom-doped 2D transition metal dichalcogenides have been predicted to exhibit room-temperature ferromagnetism [83,84]. Additionally, the Curie temperature of trilayer Fe 3 GeTe 2 can be improved to room-temperature by ionic-liquid gating technique [7].
The calculated intrinsic AHC (σ xy ) as a function of energy are displayed in Fig. 10. For semiconducting CrSY and CrSeY, the AHC is zero within the band gaps, similarly to the magneto-optical Kerr and Faraday effects. For metallic CrTeY (Y = Cl, Br, I), σ xy takes the values of 0.14 e 2 /h, 0.75 e 2 /h, and 1.05 e 2 /h for ML FM states, and of 0.35 e 2 /h, 0.98 e 2 /h, and 1.56 e 2 /h for BL FM states at the Fermi energy (E F ), respectively. The magnetotransport properties like the AHE can be modulated by electron or hole doping, which shifts the E F upward or downward, respectively. According to the recent experimental works [7,85], the doping carrier concentration for 2D materials can reach up to 10 15 cm −2 via the gate voltage. Then, the variation range of the E F can be estimated in a reasonable range, see supplemental On the other hand for electron doping, σ xy can be reached to 1.60 e 2 /h, 2.12 e 2 /h, 3.61 e 2 /h for BL CrTeCl, CrTeBr, and CrTeI at 0.23 eV, 0.14 eV, and 0.11 eV, respectively. It is obvious that the metallic CrTeY exhibit more prominent AHE than the semiconducting CrSY and CrSeY. The largest σ xy for CrTeY are in the range of 544.46-827.95 S/cm (converted from the unit of e 2 /h by considering the effective thickness), which is larger than bulk Fe 3 GeTe 2 (140-540 S/cm) [86], thin film Fe 3 GeTe 2 (∼ 400 S/cm) [87], thin film Fe 4 GeTe 2 (∼ 180 S/cm) [88], and is even compared with bulk bcc Fe (751 S/cm [46], 1032 S/cm [89]).
Strikingly, a peak of σ xy appears in BL CrSY and CrSeY near the bottom of conduction bands, and grows larger with the increasing of atomic number (Cl → Br → I), as displayed in Figs. 10(a) and 10(b). We interpret the underlying physical reasons by taking BL CrSeY as examples. At the energy position of the peaked σ xy , there exists a band crossing in the band structure that is dominated by Cr d x 2 −y 2 orbitals [see the left inset in Fig. 10(b)]. The band crossing is a Weyl-like point, which is protected by the glide symmetry M z τ ( 1 2 , 1 2 ,0) . It preserves even after including SOC because the glide symmetry still exists. Pinning the E F at the band crossing point, the Berry curvature exhibits hot spots around the Γ point [see the right inset in Fig. 10(b)], which definitely gives rise to a large σ xy . With the increasing of atomic number Y , the region of the hot spots of Berry curvature enlarges gradually, which enhances the peak of σ xy .
The ANE, the thermoelectric counterpart of AHE, is a celebrated effect from the realm of the spin caloritronics [90,91]. At the low temperature, the ANC (α xy ) is connected to the energy derivative of σ xy via the Mott relation [92], which can be derived from Eq. The ATHE, the thermal analogue of AHE, has been widely used to study the charge neutral quasiparticles in quantum materials [95][96][97][98][99][100][101]. The calculated ATHC (κ xy ) of ML and BL CrXY are plotted in Fig. 12. Similarly to σ xy and α xy , κ xy is vanishing within the band gaps of CrSY and CrSeY. The calculated κ xy at the true

IV. SUMMARY
In summary, we systematically investigated the band structures, magnetocrystalline anisotropy energy, firstand second-order magneto-optical effects, and intrinsically anomalous transport properties in monolayer and bilayer CrXY (X = S, Se, Te; Y = Cl, Br, I) by employing the first-principles calculations. From the band structures, monolayer and bilayer CrSY and CrSeY are identified to be narrow band-gap semiconductors, whereas monolayer and bilayer CrTeY are multi-band metals. The results of magnetocrystalline anisotropy energy show that CrTeBr and CrTeI prefer to out-of-plane magnetization, whereas the other seven family members of CrXY are in favor of in-plane magnetization. The magnetic ground states for bilayer CrXY are confirmed, that is, bilayer CrSI, CrSeCl, and CrTeCl present strong interlayer ferromagnetic coupling, whereas the other ones show interlayer antiferromagnetic coupling. The magnetic group theory identifies the nonzero elements of optical conductivity tensor, which in turn determines the symmetry of magneto-optical Kerr and Faraday effects as well as the anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects. The above physical effects can only exist in monolayer and bilayer ferromagnetic CrXY with outof-plane magnetization. The magneto-optical strength for the Kerr and Faraday effects of bilayer CrXY are nearly two times that of monolayer CrXY due to the doubled magnetization, and the magneto-optical strength of both monolayer and bilayer CrXY increase distinctly with the increasing of atomic number in X or Y because of the enhanced spin-orbital coupling. The largest Kerr and Faraday rotation angles in the moderate energy range (1 ∼ 5 eV) are found in bilayer CrSeI, which turns out to be 1.49 • and 0.20 • , respectively. The anomalous transport properties of CrXY are considerably large under proper hole or electron doping. For instance, the anomalous Hall, anomalous Nernst, and anomalous thermal Hall conductivities reach up to their maximums of 3.61 e 2 /h, -3.71 AK −1 m −1 , and 0.17 WK −1 m −1 in bilayer CrTeI, monolayer CrSBr, and bilayer CrTeI, respectively. In the case of in-plane magnetization, the second-order magneto-optical effects do exist in both ferromagnetic and antiferromagnetic CrXY, in contrast to the first-order ones. The calculated Schäfer-Hubert and Voigt effects are considerable large in CrXY. For example, the largest Schäfer-Hubert and Voigt rotation angles are found to be 13.39 • and -1.67 • in bilayer ferromagnetic CrTeCl, and to be 11.31 • and -1.50 • in bilayer antiferromagnetic CrTeCl, respectively.
Our results show that 2D van der Waals layered magnets CrXY may be excellent material candidates for promising applications to magneto-optical devices, spintronics, and spin caloritronics. For example, the semiconducting CrXY can be used to make van der Waals magnetic heterostructures, similarly to semiconducting ferromagnetic CrI 3 , which realizes the quantum anomalous Hall effect [102] and valley manipulation [103,104]. The metallic CrXY can also be used to make heterostructures, similarly to metallic Fe 3 GeTe 2 , which can increase the critical temperatures of 2D magnetic materials [105], fabricate giant magnetoresistance (GMR) devices [106], design metal electrodes [107], and tailor the anomalous transport properties [108].