Giant and controllable nonlinear magneto-optical effects in two-dimensional magnets

The interplay of polarization and magnetism in materials with light can create rich nonlinear magneto-optical (NLMO) effects, and the recent discovery of two-dimensional (2D) van der Waals magnets provides remarkable control over NLMO effects due to their superb tunability. Here, based on first-principles calculations, we reported giant NLMO effects in CrI3-based 2D magnets, including a dramatic change of second-harmonics generation (SHG) polarization direction (90 degrees) and intensity (on/off switch) under magnetization reversal, and a 100% SHG circular dichroism effect. We further revealed that these effects could not only be used to design ultra-thin multifunctional optical devices, but also to detect subtle magnetic orderings. Remarkably, we analytically derived conditions to achieve giant NLMO effects and propose general strategies to realize them in 2D magnets. Our work not only uncovers a series of intriguing NLMO phenomena, but also paves the way for both fundamental research and device applications of ultra-thin NLMO materials.


INTRODUCTION
The interaction of light with the polarization and magnetization in matters could create profound nonlinear magneto-optical (NLMO) phenomena, such as the magnetization-induced second-harmonics generation (MSHG) and the photocurrent generation [1][2][3].Among them, NLMO effects related to second-harmonics generation (SHG) have distinctive advantages in both magnetization detection and light modulation.As an all-optical probe characterized by higher-rank tensors, SHG-related NLMO measurement is nondestructive with high spatial and temporal resolution [4][5][6][7][8], and shows great promise to characterize structural and magnetic signals in 2D magnets [9][10][11].Moreover, the rotation angle of SHGrelated NLMO effect is independent of sample thickness [12], which is in clear contrast to the thickness-dependent linear magneto-optical Faraday angle, and therefore has the potential to be used in miniature devices.
The most significant SHG signal originates from electric dipole transitions [8], in which the breaking of the inversion (P) symmetry is necessary.P symmetry can be broken by either crystal or magnetic structures, generating the corresponding crystal SHG and MSHG.These two types of SHG are described by an i-type tensor χ N which has even-parity under time-reversal (T ) operation and a c-type tensor χ M which is T -odd [13][14][15][16][17], respectively.Therefore, in non-centrosymmetric materials with T symmetry, only χ N survives, and in non-centrosymmetric magnetic materials with space-time inversion (PT ) symmetry, only χ M survives (Detailed derivations are in Supplementary Note 1A [18]).When both types of SHG coexist, that is in materials with simultaneous breaking of P, T and PT symmetries, a class of specific NLMO effects arises due to the interference of χ N and χ M under T -operation as where E(ω) is the electric field of the incident light and P(2ω) is the nonlinear polarization.The ± represents the influence of T -operation.We denoted this interference effect of χ N and χ M as the NLMO effect in the rest of the context.As illustrated in Fig. 1, The interference between the two types of SHG can change the intensity of second harmonic (SH) light through I(2ω) ∝ |P(2ω)| 2 ∝ |χ N ± χ M | 2 .Thus, in magnets with comparable crystal SHG and MSHG, notable NLMO effects can emerge.However, experimentally observed NLMO effects in bulk materials are usually small [6,12,[19][20][21][22][23][24], which seriously hinders the relevant research and applications of these effects.Additionally, a more profound theoretical understanding to explain and control the magnitude of NLMO effects is also absent.In contrast to bulk materials, recent experiments reveal many unusual SHG responses in two-dimensional (2D) materials, such as the large and tunable i-type SHG in MoS 2 [25,26], NbOI 2 /Cl 2 layers [27,28] and twisted h-BN [29,30], the giant pure c-type SHG in PT -symmetric ultrathin CrI 3 [1], MnPS 3 [31] and CrSBr [32], and the anomalous SHG of uncertain origin in MnBi 2 Te 4 thin films [33] and monolayer (ML) NiI 2 [34].These inspiring advances combined with the great tunability of atomically-thin materials made 2D materials an excellent platform for both fundamental research and design of miniature devices based on NLMO effects.
In this work, based on the computational method developed in Refs.[35,36] and symmetry analysis, we investigated the NLMO effects of representative 2D magnets possessing both χ N and χ M , including trilayer ABA stacking CrI 3 , ML Janus Cr 2 I 3 Br 3 and ML H-VSe 2 .We analytically derived and numerically calculated the NLMO angle for linearly-polarized incident light and the NLMO intensity asymmetry for circularly-polarized incident light at different frequencies.Remarkably, we uncovered giant NLMO effects due to the maximal interference between comparable χ N and χ M in CrI 3 -based 2D magnets, including a nearly 90 • polarization rotation or an on/off switching of certain light helicity of SH light upon magnetization reversal, and a maximal SHG circular dichroism (SHG-CD) effect within fixed magnetization.We also found that these NLMO effects are extremely sensitive to subtle changes in complex magnetic orders.Moreover, we revealed the influence of interlayer interaction, spin-orbit coupling (SOC), and synergistic effect of stacking and magnetic orders to NLMO effects, and proposed strategies to achieve these giant NLMO effects in more 2D materials.Our work not only uncovered a series of giant NLMO effects and corresponding candidate materials, but also provides strategies to manipulate these effects in other material systems, and further shed light on the application of NLMO effects in subtle magnetic orders detection and ultra-thin optical devices.

RESULTS AND DISCUSSION
Structures and SHG of representative 2D magnets NLMO effects exist in magnetic materials without PT symmetry and therefore, we consider two typical ferromagnetic (FM) semiconductors in the monolayer limit, CrI 3 and H-VSe 2 [37][38][39][40][41][42][43].ML VSe 2 is noncentrosymmetric with magnetic symmetry 6m ′ 2 ′ when the magnetization is along ẑ axis.ML CrI 3 is centrosymmetric with magnetic symmetry 3m ′ [44] and AB stacking bilayer CrI 3 has PT symmetry as shown in Supplementary Figure 2 [18], therefore the ML does not have SHG and the bilayer only has the MSHG χ M .Therefore, we use the strategy of multi-layer stacking and element replacement to break P and PT to enable NLMO effects in CrI 3 related materials.The simplest examples are ABA stacking trilayer CrI 3 with antiferromagnetic (AFM) interlayer coupling (denoted by ABA-AFM CrI 3 ) and ML Janus Cr 2 I 3 Br 3 .Figure 2(a, c, e) shows the atomic and magnetic structures of ABA-AFM CrI 3 , ML Cr 2 I 3 Br 3 and ML H-VSe 2 with their magnetic symmetries summarized in Tab.I. Their band structures are shown in Supplementary Note 5 [18].
Figure 2(b, d, f) shows the influence of T operation to different SHG components χ abc (2ω; ω, ω) of the above mentioned materials.In the rest of the article, we use the shorthand notation χ abc to represent χ abc (2ω; ω, ω), where a, b, c are Cartesian directions.The two magnetic orders related by T symmetry are denoted as AFM/FM and tAFM/tFM, respectively.For SHG susceptibilities of trilayer CrI 3 , we scissor the band gap to 1.5 eV and our results show good agreement with the previous work [44] (Supplementary Figure 3 [18]).Due to the C 3z symmetry in all three representative materials, each material has only two independent in-plane SHG components, that is where the subscripts 1 and 2 denote the Cartesian direction x and ŷ.Due to the presence of m ′ or 2 ′ symmetry in all three materials, one of the SHG component is Todd while the other is T -even, as shown in the upper and middle panels of Fig. 2(b, d, f).Generally, the even and odd quantities can coexist in the same tensor component and in this case, χ N = (χ + T χ)/2 and χ M = (χ − T χ)/2, where T χ is the SHG susceptibility of the time-reversal pair.The detailed expressions of χ N and χ M , as well as their symmetry requirements are summarized in Supplementary Note 1 [18].The parity of each component under T is also summarized in Tab.I.
The bottom panel of Fig. 2(b, d, f

Giant NLMO effects of representative 2D magnets
The coexistence of χ N and χ M can induce a variety of NLMO effects.In the following, we calculated the SHG responses under the linearly and the circularly polarized light (LPL and CPL), with an emphasis on the role of different ratios of χ N and χ M to NLMO effects.Consid-ering a normal incidence geometry where the material is in the xy-plane and the incident light propagates in the −ẑ direction, the SHG polarization P(2ω) of 2D materials has been given in Ref. [44].As the intensity of the emitted SH light I(2ω) is proportional to |P(2ω)| 2 and is commonly measured, we calculated |P(2ω)| 2 to represent I(2ω).
For LPL characterized by E = E(cos θ, sin θ), where θ denotes the angle between light polarization and the x-axis of the sample, we investigated the polarizationresolved SHG as a function of θ.The experiment measures the parallel (||) and perpendicular (⊥) components of the SHG signal with respect to the direction of the incoming light polarization while the sample rotates with the angle θ.In the three representative materials, the SHG intensity at the parallel and perpendicular polarization directions can be written as [44] ( Both the parallel and perpendicular components exhibit sixfold sunflower-like patterns and differ only by a ±π/6 angle, as the black and red lines shown in Fig. 3(b).
In addition, the angle θ m corresponding to the maximum (minimum) of I || also corresponds to the minimum (maximum) of I ⊥ .NLMO effects under LPL are reflected by comparing the polarization-resolved SHG patterns before and after T operation which results from the sign reverse of χ M .Under the symmetry of the three representative materials, this corresponds to a rotation of the polarizationresolved SHG pattern (Supplementary Figure 5b and 7b [18]), denoted as NLMO angle θ T (see details in Supple-mentary Note 2A [18]), with the expression where n is an integer originated from the rotation symmetry of the pattern.
and the corresponding frequency is denoted as ω ′ LPL , which is a subset of ω LPL .This condition is also graphically shown in the complex plane in Supplementary Figure 1 [18].
We numerically traced the ω-dependence of NLMO angle θ T in the three representative materials, as shown in Fig. 3(a).As highlighted by the dashed horizontal line, θ T close to π/6 + nπ/3 corresponds to the largest NLMO angle.Remarkably, θ T is giant in ABA-AFM CrI 3 and ML Cr 2 I 3 Br 3 at several ω LPL frequencies.In contrast, θ T is always tiny in ML H-VSe 2 .This distinction is consistent with their different ratio of |χ N | and |χ M | susceptibilities observed in Fig. 2(b, d, f).Fig. 3 according to Tab.I.The SH light is also CPL with helicity opposite to the incident light.Furthermore, the intensity of the SH light depends on the helicity of the incident light, which is called the SHG-CD effect [45][46][47].
The SHG-CD intensity asymmetry can be described by where I + (M) is the SHG intensity generated by the magnetic order M when the helicity of the incident light is σ + .T operation can also change the intensity of the SH light with certain helicity.We can define the NLMO intensity asymmetry at certain incident light helicity σ as where the ±M states are T -related, and I σ (+M) is the SHG intensity generated by the +M magnetic order.The range of η σ T is [-1,1], and η σ T = ±1 represents the situation in which an incident CPL with helicity σ can only generate SH light in the ±M magnetic state and are completely blocked in the opposite magnetic state.According to Eq. ( 6), the reverse of helicity and the reverse of magnetization are equivalent, and therefore the change of intensity under T operation can also be reproduced by changing the helicity of CPL at a fixed magnetization, i.e., η + T = η +M h .The expression of the NLMO intensity asymmetry in our case is (derivation is in Supplementary Note 2B [18].) As long as ∆φ ̸ = nπ, the NLMO intensity asymmetry is always nonzero, and the intensity of the SH light is always magnetization-dependent.In general, the maximum |η ± T | = 1 requires and we denoted the corresponding frequency as the CPL characteristic frequency ω CPL .This condition is also graphically shown in the complex plane in Supplementary Figure 1 [18].It is worth noting that the ω CPL is just where the θ T for LPL is ill-defined.
We numerically traced the ω-dependence of η + T (2ω) in the three representative materials in Fig. 3(c).Again, two CrI 3 -based materials can achieve large η + T including ±1 at several frequencies while |η + T | is always below 0.4 in ML H-VSe 2 .This contrast is another feature stemming from their distinct ratio of |χ N | and |χ M |.We take ML Cr 2 I 3 Br 3 as an example to illustrate the intensity change of SH light under T operation for σ ± CPL in Fig. 3(d).(The CPL SHG intensity of ABA-AFM CrI 3 and ML H-VSe 2 are shown in Supplementary Figure 4 and 8 [18].)The vertical dash line in Fig. 3(d) highlights one of the frequency ω CPL at which the intensity of SHG changes between a considerable value and zero under T operation.The upper (σ + ) and lower (σ − ) panels of Fig. 3(d) looks exactly the same except the line colors are swapped, which reflects the equivalency of helicity and magnetization reversal.Other than that, the difference of the red (blue) line between the upper and lower panels in Fig. 3(d) also indicates that this material exhibit a 100% SHG-CD effect at ω CPL .
The giant NLMO effects discovered in CrI 3 -based materials enabled them to be used as magnetic-fieldcontrolled atomically thin optical devices, as illustrated in Fig. 3(e).Utilizing incident light at frequencies ω ′ LPL , they can serve as optical polarization switchers for LPL as they can rotate the polarization of SH light by nearly π/2.At the incident frequency ω CPL , they are ideal magneto-optical switches for CPL due to their large η T .In addition, even without an external magnetic field, due to their large SHG-CD effects, they can serve as optical filters for CPL with a particular helicity.It is worth noting that the above discussed NLMO and SHG-CD effects are analyzed base on the nonlinear polarization P(2ω) generated in materials and the intensity of radiation is estimated by I(2ω) ∝ |P(2ω)| 2 , which is not specific for the reflected or the transmitted SH light.The exact value of reflected and transmitted SH light can be obtained by solving Maxwell equations with corresponding boundary conditions [48].Although the SH light in reflection or transmission may have different intensity, the above discussed NLMO and SHG-CD effects persist.
The materials we investigated so far have several particular symmetry features.In general, NLMO effects are highly sensitive to changes in magnetic orders and can be used as a powerful tool to distinguish subtle magnetic states.As an example, we considered a different magnetic order, ↑↑↓ for ABA stacking trilayer CrI 3 , which we denoted it as the MIX state and its T -pair state as tMIX.ABA-MIX CrI 3 and ABA-AFM CrI 3 have the same net magnetic moment and tensor components as Tab.I shows, and their linear optical responses are similar as shown in Supplementary Note 4 [18], which make them difficult to be distinguished experimentally.However, in ABA-MIX CrI 3 , each SHG tensor component contains both χ N and χ M , and does not have a definite parity under T , as shown in Fig. 4(a).As a result, for LPL, the polarization-resolved SHG shows both rotation and magnitude change under T operation as shown at the bottom two plots in Fig. 4(c).For CPL, the equivalence between helicity and magnetization reverse under η T is breaking, as reflected by the remarkable difference between the upper and lower panels in the Fig. 4(b).Therefore, under the illumination of LPL (CPL) at ω LPL (ω CPL ) for ABA-AFM CrI 3 , we can observe notable differences between ABA-AFM CrI 3 and ABA-MIX CrI 3 , as shown in Fig. 4(c).
In addition to the detection of full ordered magnetic ground states, the NLMO effects can also distinguish the influence from spin fluctuation at finite temperatures [23] and spin canting caused by an external field, as shown in Supplementary Figure 6 [18].

Controllable NLMO effects of representative 2D magnets
The giant NLMO effects manifest many intriguing applications as we have learned from above results, a necessary condition to achieve those effects is to have com- Furthermore, we found stacking and magnetic orders can have synergistic effect on engineering the relative and absolute value of |χ N | and |χ M |.As shown in Fig. 5(c), in multilayer AB stacking AFM CrI 3 , χ N is forbidden in even-layer structures due to the presence of PT symme-try and remains almost the same in odd-layer structures, while χ M increases with the layer number.The above observations can be simply understood as χ N from each layer alternates the sign with similar magnitude, while χ M from each layer has the same sign, due to the PT symmetry between the A and the B layers [52].Inspired by this, as |χ M | is much smaller than |χ N | in ML H-VSe 2 , we calculated the SHG susceptibility of 7-layer (7L) AA ′ -AFM H-VSe 2 , which also has PT symmetry between A and A ′ layers.As expected, the peak value of |χ M | in 7-layer structure shown in Fig. 5(f) is almost 7 times of that in ML H-VSe 2 (Fig. 2f) while its |χ N | remains almost the same.|χ N | and |χ M | are comparable in 7L-AA ′ stacking H-VSe 2 , indicating it can host giant NLMO effects.Similarly, by tuning the stacking sequence and magnetic order, we can achieve P, T or neither symmetry between neighbouring layers and can realize an arbitrary control of the relative and absolute value of |χ N | and |χ M | (Supplementary Figure 9) [18].
To summarize, the simultaneous breaking of P, T and their joint symmetry allows NLMO effects with the electric-dipole origin.In several representative 2D magnets with similar symmetry, we investigated and demonstrated NLMO angle θ T under LPL, NLMO intensity asymmetry η T and SHG-CD η h under CPL.In particular, we discovered promising candidates with giant NLMO effects, including a near 90 • polarization rotation and an on/off switching of certain helicity of SH light upon magnetization reversal, as well as ±1 SHG-CD within a cer-tain magnetic configuration.These giant NLMO effects in candidate 2D magnets can be used not only to design atomically thin NLMO devices such as optical polarization switchers, switches and filters, but also to detect subtle magnetic orders in multilayer magnets such as ABA CrI 3 .We further derived that the comparable magnitude of |χ N | and |χ M | is indispensable to achieve giant NLMO effects.Lastly and most importantly, we found the interlayer distance, magnitude of SOC, and the synergistic effect of stacking and magnetic orders could be used to control the relative and absolute magnitude of |χ N | and |χ M |, which provides general design principles to achieve giant NLMO effects in 2D magnets.Our finding not only reveals several intriguing NLMO phenomena, but also pave the way to achieve subtle magnetization detection, giant and controllable NLMO effects in ultra-thin magneto-optical devices.

First-principles calculations
First-principles calculations were performed by Vienna Ab initio Simulation Package [53] with SOC included.The exchange-correlation functional was parameterized in the Perdew-Burke-Ernzerhof form [54], and the projector augmented-wave potential [55] were used.For the 3d orbitals in magnetic ions Cr and V, the Hubbard U of 3 eV [38] and 1.16 eV [56] were used.For layered materials, we used DFT-D3 form van der Waals correction without damping [57].The cut-off energy of plane waves was set to 450 eV and 500 eV for CrI 3 -based materials and H-VSe 2 -based materials, respectively.The convergence criterion of force were set to 10 meV/ Å and 1 meV/ Å for CrI 3 -based materials and H-VSe 2 -based materials, respectively.Total energy is converged within 10 −6 eV.kpoint samplings of 13 × 13 × 1 were used for CrI 3 -based materials and 15 × 15 × 1 for H-VSe 2 -based materials.Vacuum thickness about 20 Å was used in the calculations of 2D materials.

FIG. 1 .
FIG. 1. Concept of NLMO effects.Magnets with non-centrosymmetric crystal structures exhibit both the crystal SHG and the MSHG due to the simultaneously breaking of P, T and PT symmetries.The crystal SHG is unaffected by the magnetization direction reversal while the MSHG changes the sign.Thus the two terms can constructively (left panel) or destructively (right panel) interfere with each other for the positive and negative magnetization directions, respectively, which leads to NLMO effects.The red and blue arrows represent magnetic orders related by T operation.

FIG. 2 .
FIG. 2. Atomic structures and SHG of the representative 2D magnets.(a, c, e) Top and side views of ABA CrI3 with AFM (↑↓↑) magnetic order, ML Cr2I3Br3 and ML H-VSe2 with FM (↑) magnetic order.The red/blue arrows represent the magnetic moment of each atom.(b, d, f) The real parts and the norm of SHG susceptibility tensor components.The red and blue lines represent results of AFM (↑↓↑)/FM (↑) orders and their T -related tAFM (↓↑↓) /tFM (↓) orders, respectively.

FIG. 3 .
FIG. 3. NLMO effects in representative 2D magnets.(a) The NLMO angle (c) NLMO intensity asymmetry as a function of frequency.The vertical dashed lines in the curves indicate the NLMO rotation is ill-defined at the corresponding frequencies and the polarization resolved SHG close to those frequencies in ABA-CrI3 and ML Cr2I3Br3 are shown in Supplementary Figure 5a and 7a [18].(b) Change of polarization-resolved SHG under T operation at a LPL characteristic frequency ω ′ LPL = 1.71 eV of ML Cr2I3Br3.The yellow line marks one of the LPL incident angle where the maximal polarization direction change of SHG happens under T operation.(d) CPL SHG intensity of FM and tFM states of Cr2I3Br3.The upper and lower panels are for incident helicity σ + and σ − .The difference between the red and blue curves in each panel reflects the NLMO intensity asymmetry.For a particular magnetic order, the difference between the upper and lower panels reflects the magnitude of SHG-CD.The dashed orange line indicates one of the CPL characteristic frequency ωCPL = 1.57eV of ML Cr2I3Br3.(e) Multi-degree controlling of NLMO in ML Cr2I3Br3.For linearly-polarized incident light at polarization angle θ = π/12 + nπ/6 and frequency ω ′ LPL , ML Cr2I3Br3 can act as an magneto-optical polarization switcher.For circularly-polarized incident light at frequency ωCPL, ML Cr2I3Br3 can act as a magneto-optical switch for circularly polarized SH light or an optical filter for circularly polarized SH light with a particular helicity.

FIG. 4 .
FIG. 4. SHG susceptibility, CPL SHG intensity and polarization-resolved SHG for ABA-MIX CrI3.(a) Real parts of SHG susceptibility tensor components of ABA-MIX CrI3.Red and blue lines represent results for MIX(↑↑↓) and tMIX(↓↓↑) magnetic orders, respectively.(b) SHG intensity of MIX and tMIX magnetic orders of ABA-MIX CrI3 under CPL illumination.The upper and lower panels are for incident helicity σ + and σ − .The difference between the red and blue curves in each panel reflects the ηT .For a particular magnetic order, the difference between the upper and lower panels reflects the magnitude of SHG-CD.(c) Polarization-resolved SHG of AFM, tAFM, MIX, and tMIX magnetic orders at one of the LPL frequency ω ′ LPL =1.22 eV of ABA-AFM CrI3.The red/blue arrow represents the magnetic moment in each CrI3 layer.

TABLE I
AFM CrI 3 at a wide frequency range and the opposite is observed in ML Cr 2 I 3 Br 3 , there are still several intersections of |χ N | and |χ M |.In contrast, |χ N | is much larger than |χ M | in ML H-VSe 2 without any intersections.
. Magnetic symmetries and symmetry properties of SHG tensor components of representative 2D magnets.χN and χM represent the T -even and T -odd SHG components.If a tensor component has both χN and χM parts, it means this tensor component does not have a definite parity under T .three materials.Although |χ M | exceeds |χ N | in ABA-