Anisotropic Electron-Hole Excitation and Large Linear Dichroism in Two-Dimensional Ferromagnet CrSBr with In-Plane Magnetization

The observation of magnetic ordering in atomically thin CrI$_3$ and Cr$_2$Ge$_2$Te$_6$ monolayers has aroused intense interest in condensed matter physics and material science. Studies of van de Waals two-dimensional (2D) magnetic materials are of both fundamental importance and application interest. In particular, exciton-enhanced magneto-optical properties revealed in CrI$_3$ and CrBr$_3$ monolayers have expanded the understanding of exciton physics in 2D materials. Unlike CrI$_3$ and CrBr$_3$ with out-of-plane magnetization, CrSBr has an in-plane magnetic moment, therefore, providing a good opportunity to study the magnetic linear dichroism and high-order magneto-optical effects. Here, based on the many-body perturbation method within density-functional theory, we have studied quasiparticle electronic structure, exciton, and optical properties in CrSBr monolayer. Strongly bounded exciton has been identified with the first bright exciton located at 1.35 eV, in good agreement with an experiment of photoluminescence (Nat. Mater. \textbf{20}, 1657 (2021)). Strong contrast in the optical absorption is found between the electric fields lying along the in-plane two orthogonal directions. In accordance with a typical and realistic experimental setup, we show that the rotation angle of linear polarized light, either reflected or transmitted, could be comparable with those revealed in black phosphorene. Such large linear dichroism arises mainly from anisotropic in-plane crystal structure. The magnetic contribution from the off-diagonal component of dielectric function to the linear dichroism in CrSBr is negligible. Our findings not only have revealed excitonic effect on the optical and magneto-optical properties in 2D ferromagnet CrSBr, but also have shown its potential applications in 2D optics and optoelectronics.


I. INTRODUCTION
Due to strong light-matter interaction, two-dimensional (2D) materials have demonstrated potential applications in semiconductor optoelectronics and photonics [1][2][3][4].The discovery of 2D van der Waals magnets Cr 2 Ge 2 Te 6 [5] and CrI 3 [6] monolayer, on the other hand, has provided an additional degree of freedom, where the out-of-plane magnetization could facilitate magneto-optical Kerr and Faraday effects in ultrathin limit [7,8].Based on many-body perturbation theory, Wu, Cao, Li, and Louie have revealed exciton effect on the magneto-optical effects in the monolayer of CrI 3 [9] and CrBr 3 [10].The Kerr rotation angle for the reflected light could reach as high as 0.9 • and the Fraday angle for the transmitted light could reach as high as 0.3 • .Recently, another kind of 2D magnet CrSBr with in-plane magnetization has been identified [11] and fabricated [12] successfully.The bilayer system shows great contrast in the optical responses between antiferromagnetic and ferromagnetic interlayer coupling [12].Considering the excitonic effect, such a magnetic ordering dependence of optical properties has been well explained by the many-body perturbation calculations [12].In monolayer limit, the system shows strong photoluminescence (PL) at 1.3 eV [12].In the meantime, the exciton-coupled coherent magnons will lead to efficient optical access to spin information [13].In contrast to Cr 2 Ge 2 Te 6 and CrI 3 (CrBr 3 ), when the magnetization in CrSBr monolayer is lying along the easy axis within the 2D plane, as demonstrated in Fig. 1, the former magneto-optical Kerr and Fraday effects, which measures rotation angle of linear polarized light, will be replaced by Schäfer-Hubert (SH) effect and Voigt effect, for reflected and transmitted lights, respectively [14].The original purpose of this paper is to provide a comprehensive framework of excitonic effect on the optical and high-order magneto-optical properties in 2D ferromagnetic CrSBr with in-plane magnetization.Generally, when the magnetization is pointing along y direction, it is believed that [14] and Here, θ SH and θ V are rotation angles, and η SH and η V are ellipticities.c is the velocity of light, n is the complex refraction index, and ε is the dielectric function.ω is the optical frequency, d is the thickness of magnetic material, and sub means the substrate.Clearly, such effects are closely related to the contrast between the dielectric properties of two in-plane diagonal components ε xx and ε yy as well as the magnitude of the off-diagonal component ε zx of the system of interest.Based on independent particle approximation (IPA), such effects in 2D magnets CrXY (X=S, Se, and Te; Y=Cl, Br, and I) were studied computationally [15].For the monolayer geometry with particular 2D dielectric screening, the inherent many-body correction to the quasiparticle band structure and optical properties could not be ignored [16][17][18][19][20][21].A huge excitonic effect with binding energy of as large as 1 eV (two orders larger than conventional semiconductors) will modify the optical spectrum from IPA strongly.Therefore, in this paper, based on density functional theory with many-body perturbation method, i.e., GW -BSE method (G for one-particle Green's function, W for screened Coulomb interaction, and BSE for Bethe-Salpeter equation) [22], we have studied the quasiparticle electronic structure, exciton, and optical properties in CrSBr monolayer.
The first bright excitonic state is found located at 1.35 eV, in good agreement with experiment [12].In the meantime, large linear dichroism (optical birefringence) from anisotropic in-plane crystal structure has been identified.The intrinsic magneto-optical effect from off-diagonal components of dielectric function, however, is found to be very small.CrSBr shows large difference between ε xx and ε yy which will dominate the linear dichroism, and consequently the magnitude of rotation angle θ and ellipticity η.It has also been revealed that excitonic effect has modified ε xx , ε yy , and ε xz dramatically, and consequently the linear dichroism and the high-order magneto-optical properties.When the magnetization is tuned by external perpendicular magnetic field towards out-of-plane direction, CrSBr monolayer could exhibit significant Kerr and Faraday effects, which are even larger than those in CrI 3 [9] and CrBr 3 [10].Furthermore, the short lifetime of the first bright exciton suggests its potential application in infrared light-emitting.Our findings not only have revealed excitonic effect on the optical and magneto-optical properties in 2D ferromagnet CrSBr, but also have shown its potential applications in 2D optics and optoelectronics.

II. COMPUTATIONAL METHOD
Our first-principle calculations are performed using density-functional theory (DFT) as implemented in the Quantum Espresso package [23].We use the generalized gradient approximation with Perdew-Burke-Ernzerhof (PBE) [24] and norm-conserving pseudopotentials with a plane-wave cutoff of 80 Ry [25].The ground-state wave functions and eigenvalues are calculated within a k grid of 16 × 12 × 1.The structures are relaxed until the total forces are less than 0.01 eV/ Å and the convergence criterion for total energies is set to 10 −5 eV.
The quasiparticle band structure and excitonic properties are calculated with BerkeleyGW package [26][27][28][29] (see Appendix A).We have included spin-orbit coupling with spinor GW -BSE calculations [29].A slab model is used with vacuum layer of 15 Å along the out-of-plane direction and a truncated Coulomb interaction between CrSBr monolayer and its periodic image is adopted [30].Here, the calculations are based on one-shot G 0 W 0 with generalized plasmon pole model.The mean-field wave functions and eigenvalues within PBE are chosen as the starting point for G 0 W 0 , as a first guess for quasiparticle wave functions and eigenvalues.For the convergence of quasiparticle energies [31], we have tested the dependence on k -grid size, number of bands, as well as dielectric cutoff.We use a coarse k grid of 16 × 12 × 1, 1080 of number of bands, and the dielectric cutoff of 20 Ry (see Appendix B).The electron-hole excitations are then calculated by solving the BSE for each exciton state and frequency-dependent complex dielectric function ε(ω).For the BSE part, the fine k grid of 64 × 48 × 1 is used (see Appendix B).We use a Gaussian smearing with a broadening constant of 30 meV in optical absorbance spectrum.The number of bands for optical transitions is 6 valence and 8 conduction bands, which is sufficient to cover the span of the energies of visible light.Such kind of treatment of excited states is robust and has been applied successfully in a wide range of 2D materials [32], including graphene [33][34][35][36][37][38][39], graphyne [40], 2D transition metal dichalcogenides [41][42][43][44][45][46][47][48][49][50][51], black phosphorene [52][53][54][55][56], blue phosphorene [57,58], violet (Hittorf's) phosphorene [59,60], 2D monochalcogenides [61][62][63], 2D GaN [64], 2D boron nitrides [65][66][67], and 2D magnets [9,10,68].Experimentally, using second-harmonic generation technique, ferromagnetic order with magnetic moment pointing along y axis has been identified in CrSBr monolayer with a Curie temperature of T C = 146 ± 2 K [69].In the meanwhile, photoluminescence spectra show an obvious decrease at temperature between 130 K and 150 K, suggesting a phase transition therein [12].This Curie temperature in CrSBr monolayer is therefore much higher than Cr 2 Ge 2 Te 6 monolayer (@ 22 K under magnetic field of 0.075 T) [5] and CrI 3 monolayer (@ 45 K) [6].Theoretically, Cr 3+ ion has a magnetic moment of 3 µ B , pointing along the y axis of 2D plane.When the magnetization points along x or z axis, the total energy is 0.042 meV/Cr and 0.072 meV/Cr higher, respectively.And the value along z axis is in agreement with another theoretical result of 0.078 meV/Cr [70].It is noted that the calculated total energy for the state with magnetization along in-plane direction is 0.025 meV/Cr lower in Cr 2 Ge 2 Te 6 monolayer [7] and 0.263 meV/Cr higher in CrI 3 monolayer

Bulk
[8] when compared with the magnetic state of out-of-plane magnetization.On the other hand, the antiferromagnetic coupling between the in-plane two Cr 3+ ions is 58.86 meV/Cr higher than the ferromagnetic ground state.This result agrees with previous ground-state calculations [70].For multilayer systems, the local magnetic moment still prefers y axis, with magnetic moment antiparallel with each other between neighboring layers [12].
As indicated by the band structure in Fig. 3, this monolayer of CrSBr is a direct band gap semiconductor with the top of valence band and the bottom of conduction band both located at Γ point.The band gap increases a little along x direction and increases sharply along orthogonal direction, i.e., Γ → Y. Clearly, this monolayer structure shows anisotropic band dispersion.On the other hand, the two spin channels are well separated in the ferromagnetic ground state.It should be pointed out that what we have simulated is a suspended monolayer, i.e., a single layer of CrSBr in vacuum.The strength of the Coulomb interaction in such material originates from weak dielectric screening in the 2D limit.For distances exceeding few nanometres, the screening is determined by the immediate surroundings of the material, which can be vacuum or air in the ideal case of suspended samples.Compared to MoS 2 [44] and blue phosphorene [57], such unique 2D dielectric screening in CrSBr should also be anisotropic.The static screened Coulomb interaction is constructed as [44] W GG (q; 0) = ε −1 GG (q; 0)v(q + G ). ( The effective static 2D dielectric function ε 2D (q) could be obtained [44] ε where the complicated details of the screening in the out-of-plane direction z have been integrated out.The dielectric screening in such a system obeys particular wavelength dependence.As demonstrated in Fig. 4(a), in the long-distance limit the electron-electron interacts like that in vacuum with ε = 1, while within the intermediate distance, the electronelectron interaction has effective dielectric screening by the 2D materials with ε ≥ 1.The label "others" in Fig. 4(a) refers to the other q points in q mesh which do not belong to (q x ,0) or (0,q y ).For the anisotropic nature, the dielectric screening changes along different directions.To accurately access the distance-dependence of 2D dielectric screening, we have obtained ε 2D (s) through Hankel transformation of ε 2D (q) and shown it in Fig. 4(b).
Clearly, if two charges are very close together, there is not enough space for the electronic cloud to polarize, so ε 2D (s → 0) = 1.On the other hand, if the two charges are very far away, the field lines connecting the charges travel mainly through the vacuum, so they are not much affected by the intrinsic dielectric environment of the quasi-2D semiconductor and Between these two ends, ε 2D (s) is influenced by the intrinsic material property of CrSBr monolayer and is larger than 1.At finite distance s max , ε 2D (s max ) will exhibit its maximum.Such kind of 2D dielectric screening has also been revealed in MoS 2 monolayer with s max = 1.5 Å and ε 2D (s max ) = 11 [44].For CrSBr monolayer, s max is little larger and ε 2D (s max ) is weaker.In the meantime, due to anisotropic crystal structure, the 2D dielectric screening in CrSBr monolayer also exhibits strong anisotropy.ε 2D (s max ) is 4.4 and 7.0 for ε 2D (s) along x and y axis, respectively.And corresponding s max is 3.0 Å and 4.0 Å, the same order with lattice constants.Interestingly, such kind of anisotropy persists to long distance.As shown in the inset of Fig. 4(b), the difference between ε 2D (s) along x and y axis does not vanish until 60 Å.So at the distance from 1 Å to ten times length of lattice constant, the anisotropic dielectric screening dominates Coulomb interaction in CrSBr monolayer.For 2D system, Coulomb potential could be described by the Keldysh model [17], where the potential between two charges has the form Here H 0 and Y 0 are, respectively, the Struve and Bessel functions of the second kind.ρ 0 is a screening length, which is ρ 0 = 2πα 2D , and α 2D is the 2D polarizability.Taking the 2D Fourier transform of Eq. 5 results in a dielectric function of the form We fit the Keldysh model to our ab initio effective dielectric function at small q, where the Coulomb potential approaches 1/s.The 2D polarizability, however, shows large difference between orthogonal two directions.α 2D is 4.2 Å for q along x axis and 10.4 Å for q along y axis.Based on above discussion, in sufficiently long range from 1 Å to 60 Å, the dielectric screening in CrSBr monolayer is anisotropic.Due to the unique dielectric environment (monolayer 2D material suspended in vacuum or other dielectric surroundings), electron-electron and electron-hole interactions in 2D materials are much stronger than conventional bulk materials, like GaAs, Si, and so on.Therefore, dielectric screening in 2D materials is reduced compared with conventional bulk materials.
With the reduced dielectric screening in 2D CrSBr monolayer, the quasiparticle correction to the electronic band structure is large.At Γ point, the quasiparticle band gap is 2.22 eV, where the value is 0.50 eV within PBE.In Fig. 5, we have shown the direct quasiparticle band gap in the Brillouin zone.The one-dimensional nature of the lowest transition energies is obvious.The optical absorbance in CrSBr monolayer is further calculated and shown in Fig. 6, where the difference between two in-plane orthogonal directions is obvious.Although the main optical absorption is located at 3.01 eV, the small peak located below could catch as high as 20 % of the incident light.The optical absorption edge is located at 1.35 eV when the electric field is along y direction.For the other direction, the absorption edge is   I. Obviously, these excitons process an increased real-space extension of the wave function.The effective electron-hole interaction is therefore reduced strongly, consistent with the smaller binding energy as shown in Table I.The difference in the oscillator strength between x direction and y direction is obvious for all the exciton states.With the above revealed anisotropic quasiparticle electronic band structure, electron-hole (f) (a)@1.351eV (b)@1.468eV (c)@1.586eV (e)@1.659eV (g)@1.673eV (h)@1.729eV excitation, and optical absorption, we continue to show the (magnetic) linear dichroism and MO SH and Vogit effects in CrSBr monolayer.The essence of a theoretical modeling of the MO effects lies in accurately accounting for the diagonal and off-diagonal frequencydependent macroscopic dielectric functions, which have been readily available from our G 0 W 0 -BSE calculations with electron-hole interaction included.As verified by the agreement between our theoretical calculations and experiments of CrSBr as well as the fact that exciton effect in ferromagnetic monolayer CrI 3 and CrBr 3 has strongly modified its MO responses [9,10], significantly different behaviors going beyond those from a treatment considering only transitions between non-interacting Kohn-Sham orbitals should be expected in CrSBr.To simulate the experimental setup, as demonstrated in Fig. 1, we consider CrSBr monolayer deposited on a dielectric substrate α-Al 2 O 3 , which has a wide band gap of 8.7 eV with refraction constant of 1.75 [10].Assuming a linearly polarized incident light, we calculate the SH and Voigt signals by analyzing the polarization plane of the reflection (transmission) light, which is in general elliptically polarized with a rotation angle and an ellipticity.The detailed calculations are based on the transfer matrix method and could be found in Appendix D. In Fig. 10, we have found that the rotation angle could be larger than 10 • for the reflected light.For transmitted light, the largest rotation angle is around 8 • .On the other hand, the revealed SH and Voigt effects dominate within the quasiparticle band gap.When the electron-hole interaction is not considered, both the amplitude and the position of the spectrum are modified significantly (see Fig. 15 in Appendix E).This verifies again the fact that it is the exciton effect that leads to the large linear dichroism in CrSBr monolayer.In addition, we can find that the off-diagonal component of dielectric function plays little role in the above revealed large linear dichroism in CrSBr, where as shown in Fig. 16 (Appendix E), θ SH and θ V do not change.We have also performed the calculations of magneto-optical effects when spin orientation is pointing along x and z axis.As shown in Fig. 17 and Fig. 18 (Appendix F), absorbance spectrum and exciton states are similar to those for the state with spin orientation along y axis.In the meantime, due to extremely small ε yz , the revealed rotation angle of long axis of the polarization ellipse for reflection and transmission light is still determined by the large ratio of ε xx /ε yy (see Fig. 19 in Appendix G).On the other hand, when spin orientation is tuned by external perpendicular magnetic field towards out-of-plane direction, i.e. z axis, the magneto-optical Kerr and Faraday effects will appear in CrSBr monolayer.In this case, a linearly polarized continuous-wave light is modified by the presence of out-of-plane magnetic field when propagating through CrSBr monolayer.The left and right-circularly polarized components will propagate with different refractive index and will pick up different optical path length and absorption.Therefore, the reflected or transmitted light becomes elliptically polarized (characterized by an ellipticity η), and the long axis of the polarization ellipse is rotated (characterized by a rotation angle θ).The detailed theoretical calculations of rotation angles and ellipticities are shown in Appendix H.Meanwhile, as shown in Fig. 21 (Appendix H), the maximal rotation angles of Kerr and Faraday effects are 1.1 • and 0.6 • , respectively, which are even larger than 0.9 • and 0.3 • in CrI 3 [9] and 0.3 • and 0.2 • in CrBr 3 [10].Similar to the cases with in-plane magnetization, the optical spectrum has been modified significantly for the exciton effects (see Fig. 22 in Appendix H).We list the maximal rotation angles in TABLE II for different spin orientations.Therefore, for both linearly and circularly polarized light, the large phase shift facilitates the applications of CrSBr monolayer as optical polarizers and waveplates.
TABLE II: Comparison of magneto-optical effects for reflection and transmission lights, polarization of lights propagating in CrSBr monolayer, off-diagonal elements, and maximums of corresponding rotation angles with electron-hole interaction when spin orientation is along x, y, and z axis in the energy range of 0 ∼ 3 eV.For black phosphorene, it is noted that the ratio between the long axis and short one is 1.4 for the anisotropic crystal structure and 1.5 eV energy difference between absorption edges for the electric field along two axes could be found (see Fig. 23 in Appendix I).For comparison, it is noted that the rotational angle in black phosphorene is of similar magnitude.

Spin orientation Magneto-optical effects Polarization Off-diagonal elements Rotation angles (
However, CrSBr shows two broad peaks at the energies of both infrared and red lights, whereas black phosphorene exhibits a single narrow peak in this energy window (see Fig. 24 in Appendix I).Additionally, obvious linear dichroism has been revealed experimentally in black phosphorene and its few-layers [72][73][74][75].Therefore, polarization-sensitive broadband photodetector using a CrSBr vertical p-n junction could be constructed successfully for the application in 2D semiconductor optoelectronics.
To demonstrate other potential applications in 2D semiconductor optoelectronics, we further calculate the lifetime of the first bright exciton.Using "Fermi's golden rule", the radiative lifetime τ S (0) at 0 K of an exciton in state S is derived according to [76,77] τ S (0) = 2 c 8πe 2 E S (0) where c is the speed of light, A uc is the area of the unit cell, E S (0) is the energy of the exciton in state S, and is the square modulus of the BSE exciton transition dipole divided by the number of unit cells in this 2D system.The exciton radiative lifetime τ S at temperature T is obtained as where k B is Boltzmann constant, and M S = m * e + m * h is the exciton effective mass.The computed radiative lifetime of the first bright exciton in CrSBr monolayer at 4 and 300 K is 0.001 ns and 0.100 ns, respectively, which are even smaller than those in the conventional transition metal dichalcogenides, e.g., 0.004 ns and 0.270 ns in MoS 2 [77].In Table III, we have listed the exciton lifetime of some typical 2D semiconductors [56,57,59,64,[77][78][79][80][81][82][83][84][85][86], as well as conventional semiconductor GaInN/GaN quantum wells [87].Clearly, such a short lifetime in CrSBr monolayer together with its direct band gap shows its advantages in infrared light-emitting applications.Moreover, as shown in Fig. 11, the strong optical absorption of CrSBr monolayer covering the whole solar spectrum as well as its optimized band gap (1.35 eV) within the Shockley-Queisser limit also suggests efficient solar energy conversion.Therefore, from gapless graphene to narrow-band-gap black phosphorene, intermediate-band-gap transition-metal dichalcogenides, and wide-band-gap semiconductors of Hittorf's phosphorene, blue phosphorene, III-V monochalcogenides, and boron nitride, magnetic CrSBr could play an important role in infrared optoelectronics and 2D photovoltaics.WS 2 monolayer 0.002 [77] 0.17 [77] WSe 2 monolayer 0.004 [77] 0.29 [77] 0.002 [83], 0.004 [85] 4 [86] 2D GaN 0.6 [64] GaInN/GaN QW 10 2 [87] respectively.The solar spectrum [88] and averaged absorbance (dashed red line) of light polarized along x and y axis in CrSBr monolayer are also shown.

IV. SUMMARY
To summarize, by considering explicitly many-body effects of the electron-electron and electron-hole interactions in 2D ferromagnet CrSBr, we have demonstrated unusual large optical anisotropy in this 2D material.The inherent in-plane magnetization with the consideration of spin-orbit coupling is found to contribute little to the revealed large linear dichroism.The giant SH and Voigt effects, which are comparable with those in black phosphorene, originate from the inherent anisotropic crystal structure and coherent onedimensional band dispersion for anisotropic electron-hole pairs excitation.Our calculation of the first exciton located at 1.35 eV agrees with a recent experiment of photoluminescence.
The in-gap exciton states of either bright or dark show the diversity of electron-hole excitation in this 2D magnet.Compared with black phosphorene, the relative delocalization of exciton in k -space indicates the nature of electron-hole excitation in CrSBr is in-between Wannier-Mott type and Frenkel type.Excitonic effect has modified ε xx , ε yy , and ε xz dramatically, and consequently the linear dichroism and the magneto-optical properties.With out-of-plane magnetization, Kerr and Faraday effects have been revealed in CrSBr monolayer, which are even larger than those in CrI 3 and CrBr 3 .Furthermore, the short exciton lifetime as well as the strong optical absorption covering the whole solar spectrum shows its advantages in application of light-emitting diode and solar cell.Our studies have provided a basic framework to account for the high-order magneto-optical effects in 2D magnets and also shown potential applications of CrSBr in 2D optics and optoelectronics.

Appendix A: BASIC FRAMEWORK OF GW -BSE IN BERKELEYGW PACK-AGE
Generally, the quasiparticle self-energies are obtained by solving the Dyson equation [26].
where Σ is the self-energy operator within the GW approximation, and E QP nk and ψ QP nk are the quasiparticle energies and wavefunctions, respectively.The self-energy operator Σ is invoked for the quasiparticle behavior with quasiparticle energies E QP nk and wave functions ψ QP nk .Here, we use the many-body perturbation method with one-shot G 0 W 0 framework.The mean-field wave functions within DFT-PBE are used as quasiparticle wave functions and the quasiparticle energy is approached starting from DFT-PBE eigenvalue.
We first compute static polarizability based on random-phase approximation (RPA) [26]: where n, n are occupied and unoccupied band indices, k is wave vector, q is a vector in the first Brillouin zone, G is a reciprocal-lattice vector, and |n, k and E nk are the mean-field electronic eigenvectors and eigenvalues, respectively.
Then the dielectric matrix is constructed as [26] ε GG (q; 0) = δ GG − v(q + G)χ GG (q; 0), (A3) with the slab-truncated Coulomb interaction included [30], where z c is the truncation distance in the perpendicular direction.Such kind of treatment could guarantee the convergence of dielectric screening ("head" in ε GG ) to approach to unit in the long wavelength limit.
Within the generalized plasmon pole (GPP) model, the imaginary and real part inverse dielectric matrix with finite frequencies are given by [26] Imε and where Ω GG (q) and ωGG (q) are the effective bare plasma frequency and the GPP mode frequency defined as [26]: Here ρ is the electron charge density in reciprocal space and ω 2 p = 4πρ(0)e 2 /m is the classical plasma frequency.
Using the form of the above frequency-dependent dielectric function, the self-energy operator Σ is solved in two parts, Σ = Σ SEX + Σ COH , where Σ SEX is the screened exchange operator and Σ COH is the Coulomb-hole operator as [26] n, k|Σ and With the above obtained quasiparticle energies and static dielectric screening from RPA, the electron-hole excitations are then calculated by solving the BSE for each exciton state S [27]: where A S vck is the exciton wavefunction, Ω S is the excitation energy, and K eh is the electronhole interaction kernel.
The kernel contains two terms, a screened direct interaction and a bare exchange interaction, K eh = K d + K x , defined as [27]:

Appendix D: CALCULATIONS OF LINEAR DICHROISM (OPTICAL BIRE-FRIGENCE) AND SCHAFER-HUBERT AND VOIGT EFFECTS
As shown in Fig. 1, considering the in-plane magnetization along y axis (B = B êy ), the dielectric tensor is expressed as: The Fresnel equation for the propagation of electromagnetic wave is given by: For normal incidence, the complex refractory index n is: By solving the above Fresnel equations, we get the normal modes as the and ⊥ for linearly polarized plane waves which are parallel and perpendicular to êy (magnetization direction), with distinct refractive indices: and To mathematically describe how an electromagnetic wave interacts with such stratified and anisotropic media, we adopt a 4x4 formalism involving the in-plane components of both electric (E x , E y ) and magnetic fields (B x , B y ).We consider a monolayer material magnetized along the +y direction, and the second layer is vacuum.Within each layer (l = 0, 1, 2), we choose the four eigenmodes of light as follows: l) .The electric and magnetic fields of light in the first and second layers are given by: and (l) The electric and magnetic fields of light in the zeroth layer are given by, and The requirement of the continuity of the tangential field components at the interfaces connects the field amplitudes E (l) 0j between two neighboring layers.The dynamical matrix within each layer is given by a block-diagonal form: The propagation matrix is defined as a diagonal matrix: where δ ⊥ = ω c n ⊥ , δ = ω c n , and d is the thickness of layer 1, i.e. monolayer CrSBr.
In this two-interface setup, E (0) 0 and E (2) 0 are related by the transfer matrix M c in the basis of linearly polarized lights: M c has a simple block-diagonal form: where r mn and t mn are the Fresnel coefficients of the interface from the mth layer to the nth layer.And M has the same form.In the left and right layers, we adopt a basis transformation from the linearly polarized light (ê x ,ê y ) to the light (ê a ,ê b ) with êa = 1 l) , for l = 0, 2. This new basis of linearly polarized plane waves is denoted as {a →, a ←, b →, b ←}.In this basis of linearly polarized lights, the electric field amplitudes in the left and right layer are related by transfer matrix M : Because the angle of electric field of incident light and the direction of magnetization (+y) 0b← = 0.In addition, there are no reflecting lights from the zeroth medium (semiinfinite substrate) to CrSBr, which means E (0) 0a→ = E (0) 0b→ = 0.With these two conditions, by calculating Eq.D15, we can get: In this work, we rescale the calculated dielectric function in a slab model by the thickness of a monolayer material: and where l and d are thicknesses of the slab model along the out-of-plane direction and monolayer material, and εαα and εαβ are calculated dielectric functions in the BerkeleyGW package.
εαα and εαβ can be expressed as follows: and   In 2D magnets, the in-plane spin orientation could be tuned to out-of-plane direction by external perpendicular magnetic field.As shown in Fig. 20, here we consider the out-ofplane magnetization along z axis (B = B êz ).For an isotropic system the dielectric tensor is expressed as: The Fresnel equation for the propagation of electromagnetic wave is given by: For normal incidence, the complex refractory index n is: By solving the above Fresnel equations, we get the normal modes as the σ + and σ − for circularly polarized plane waves, with distinct refractive indices: For an anisotropic system, the distinct refractive indices are: To mathematically describe how an electromagnetic wave interacts with such stratified and anisotropic media, we adopt a 4x4 formalism involving the in-plane components of both electric (E x , E y ) and magnetic fields (B x , B y ).We consider a monolayer material magnetized along the +z direction, and the second layer is vacuum.Within each layer (l = 0, 1, 2), we choose the four eigenmodes of light as follows: − .The electric and magnetic fields of light in the first and second layers are given by: and The electric and magnetic fields of light in the zeroth layer are given by, and The requirement of the continuity of the tangential field components at the interfaces connects the field amplitudes E (l) 0j between two neighboring layers.The dynamical matrix within each layer is given by a block-diagonal form: The propagation matrix is defined as a diagonal matrix:  l) , for l = 0, 2. This new basis of linearly polarized plane waves is denoted as {x →, x ←, y →, y ←}.In this basis of linearly polarized lights, the electric field amplitudes in the left and right layer are related by transfer matrix M : Because the electric field of incident light is along the direction of magnetization (+z), E 0y← = 0.In addition, there are no reflecting lights from the zeroth medium (semi-infinite substrate) to CrSBr, which means E (0) 0x→ = E (0) 0y→ = 0.With these two conditions, by calculating Eq.G16, we can get:

FIG. 1 :
FIG. 1: SH and Vogit setup consisting of layers of vacuum, ferromagnetic monolayer CrSBr, substrate of semi-infinitely thick SiO 2 .Red arrow denotes the in-plane magnetization, which is along y axis.(b) Illustration of rotation angle and ellipticity for reflected (left panel) and transmitted (right panel) lights.

FIG. 2 :
FIG. 2: Illustration of crystal structure of 2D magnet CrSBr monolayer.Local magnetic moment at Cr atoms pointing along y axis.

FIG. 3 :
FIG.3: PBE band structure with colors denoting the direction of spin polarization along the y axis.

FIG. 4 :
FIG. 4: (a) Frequency-dependence of 2D dielectric screening in CrSBr monolayer under ab initio calculation.(b) Distance-dependence of 2D dielectric screening in CrSBr monolayer under ab initio calculation (solid line) and Keldysh model (dashed line).Inset of (b) shows the difference between ε 2D (s) along y axis and x axis.

1 .FIG. 6 :FIG. 7 :
FIG. 6: Anisotropic optical absorbance with electric polarization along x axis (a) and y axis (b) and corresponding exciton spectrum (c) in CrSBr monolayer.Here, green lines are for the bright excitons and the gray lines are for dark excitons.The dotted lines are for the optical absorption based on IPA.Inset shows the experiment data of PL spectrum of CrSBr monolayer under magnetic field along y axis [12].
FIG. 8: Real-space plots (upper panel) of modulus squared of the exciton wave function and corresponding reciprocal-space plots (down panel) for the first eight excitons.Here, the hole (black circle) is fixed at Cr.

FIG. 9 :
FIG. 9:The distribution of free electron-hole pair with electron energy at E c and hole energy at E v for selected exciton states weighted by module squared exciton envelope function for each interband transitions between states |vk and |ck , with quasiparticle energies of E QP vk and E QP ck , respectively.

FIG. 12 :FIG. 13 :
FIG. 12: Convergence of the quasiparticle (G 0 W 0 ) band gap at three high symmetry k points with respect to the scale of coarse k grid (a), number of bands (b), and dielectric cutoff energy (ε cutoff ) (c).In (a), we use a ε cutoff of 20 Ry and 1080 bands; in (b), we use a k grid of 16 × 12 × 1 and 1080 bands; and in (c), we use a ε cutoff of 20 Ry and a k grid of 16 × 12 × 1.With the above consideration, we use a 16 × 12 × 1 k grid, a ε cutoff of 20 Ry, and 1080 (20 times larger) bands for our G 0 W 0 calculations.

FIG. 14 :
FIG. 14: The up panel shows the orbital-decomposed band structure (DFT-PBE level) of CrSBr from maximum localized Wannier orbitals using Wannier90 [71].The partial charge at Γ point is shown in the down panel.

FIG. 16 :
FIG. 16: Comparison of rotation angles for both (a) reflected and (b) transmitted lights in CrSBr monolayer between the situations with and without the inclusion of off-diagonal component ε xz .

FIG. 17 :
FIG. 17: Anisotropic optical absorbance in CrSBr monolayer with spin orientation along x axis for the electric polarization along x (a) and y (b) directions, respectively.(c) Exciton spectrum.

FIG. 18 :
FIG. 18: Anisotropic optical absorbance in CrSBr monolayer with spin orientation along z for the electric polarization along x (a) and y (b) directions, respectively.(c) Exciton spectrum.

FIG. 22 :FIG. 23 :
FIG. 22: Magneto-optical Kerr and Faraday effects: Rotation angles of (a) reflected and (b) light in CrSBr monolayer for the state with spin orientation along z axis without the inclusion of electron-hole interactions.Insets are the corresponding ellipticities.

TABLE I :
Excitonic properties of first eight excitons with excitation energy (E xct ), exciton binding energy (E b ), and oscillator strength for electric field along x and y directions.

TABLE III :
Collection of exciton radiative lifetime τ (ns) for various 2D materials.τ LT S and τ RT S are the computed radiative lifetimes at low temperature (≈ 4 K) and room temperature, respectively.