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

NLMO effects exist in magnetic materials without $\mathcal{PT}$ symmetry and therefore, we consider two typical ferromagnetic (FM) semiconductors in the monolayer limit, CrI₃ and H-VSe₂ [37, 38, 39, 40, 41, 42, 43]. ML VSe₂ is non-centrosymmetric with magnetic symmetry $\bar{6}m^{\prime}2^{\prime}$ when the magnetization is along $\hat{z}$ axis. ML CrI₃ is centrosymmetric with magnetic symmetry $\bar{3}m^{\prime}$ [44] and AB stacking bilayer CrI₃ has $\mathcal{PT}$ symmetry as shown in Supplementary Figure 2 [18], therefore the ML does not have SHG and the bilayer only has the MSHG $\chi_{\rm M}$. Therefore, we use the strategy of multi-layer stacking and element replacement to break ${\mathcal{P}}$ and ${\mathcal{P}T}$ to enable NLMO effects in CrI₃ related materials. The simplest examples are ABA stacking trilayer CrI₃ with antiferromagnetic (AFM) interlayer coupling (denoted by ABA-AFM CrI₃) and ML Janus Cr₂I₃Br₃. Figure 2(a, c, e) shows the atomic and magnetic structures of ABA-AFM CrI₃, ML Cr₂I₃Br₃ and ML H-VSe₂ with their magnetic symmetries summarized in Tab. 1. Their band structures are shown in Supplementary Note 5 [18].

Figure 2(b, d, f) shows the influence of $\mathcal{T}$ operation to different SHG components $\chi^{abc}(2\omega;\omega,\omega)$ of the above mentioned materials. In the rest of the article, we use the shorthand notation $\chi^{abc}$ to represent $\chi^{abc}(2\omega;\omega,\omega)$, where $a,b,c$ are Cartesian directions. The two magnetic orders related by $\mathcal{T}$ symmetry are denoted as AFM/FM and tAFM/tFM, respectively. For SHG susceptibilities of trilayer CrI₃, 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 $\chi^{111}=-\chi^{122}=-\chi^{212}$ and $\chi^{222}=-\chi^{211}=-\chi^{112}$, where the subscripts 1 and 2 denote the Cartesian direction $\hat{x}$ and $\hat{y}$. Due to the presence of $m^{\prime}$ or $2^{\prime}$ symmetry in all three materials, one of the SHG component is $\mathcal{T}$-odd while the other is $\mathcal{T}$-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, $\chi_{\rm N}=(\chi+\mathcal{T}\chi)/2$ and $\chi_{\rm M}=(\chi-\mathcal{T}\chi)/2$, where $\mathcal{T}\chi$ is the SHG susceptibility of the time-reversal pair. The detailed expressions of $\chi_{\rm N}$ and $\chi_{\rm M}$ , as well as their symmetry requirements are summarized in Supplementary Note 1 [18]. The parity of each component under $\mathcal{T}$ is also summarized in Tab. 1.

The bottom panel of Fig. 2(b, d, f) clearly shows that the relative value of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ are distinctive in the three materials. Although $|\chi_{\rm M}|$ exceeds $|\chi_{\rm N}|$ in ABA-AFM CrI₃ at a wide frequency range and the opposite is observed in ML Cr₂I₃Br₃, there are still several intersections of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$. In contrast, $|\chi_{\rm N}|$ is much larger than $|\chi_{\rm M}|$ in ML H-VSe₂ without any intersections.

The coexistence of $\chi_{\rm N}$ and $\chi_{\rm M}$ 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 $\chi_{\rm N}$ and $\chi_{\rm M}$ to NLMO effects. Considering a normal incidence geometry where the material is in the $xy$-plane and the incident light propagates in the $-\hat{z}$ direction, the SHG polarization $\mathbf{P}(2\omega)$ of 2D materials has been given in Ref. [44]. As the intensity of the emitted SH light $I(2\omega)$ is proportional to $|\mathbf{P}(2\omega)|^{2}$ and is commonly measured, we calculated $|\mathbf{P}(2\omega)|^{2}$ to represent $I(2\omega)$.

For LPL characterized by $\mathbf{E}=E(\cos\theta,\sin\theta)$, where $\theta$ denotes the angle between light polarization and the $x$-axis of the sample, we investigated the polarization-resolved SHG as a function of $\theta$. The experiment measures the parallel ($||$) and perpendicular ($\bot$) components of the SHG signal with respect to the direction of the incoming light polarization while the sample rotates with the angle $\theta$. 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 $\pm\pi/6$ angle, as the black and red lines shown in Fig. 3(b). In addition, the angle $\theta_{\rm m}$ corresponding to the maximum (minimum) of $I_{||}$ also corresponds to the minimum (maximum) of $I_{\bot}$.

NLMO effects under LPL are reflected by comparing the polarization-resolved SHG patterns before and after $\mathcal{T}$ operation which results from the sign reverse of $\chi_{\rm M}$. Under the symmetry of the three representative materials, this corresponds to a rotation of the polarization-resolved SHG pattern (Supplementary Figure 5b and 7b [18]), denoted as NLMO angle $\theta_{\mathcal{T}}$ (see details in Supplementary Note 2A [18]), with the expression

where $n$ is an integer originated from the rotation symmetry of the pattern. $\tan\delta=\frac{|\chi^{111}|}{|\chi^{222}|}\in(0,+\infty)$ is the magnitude ratio of $\chi^{111}$ and $\chi^{222}$, and $\Delta\varphi=-i\left(\ln{\frac{\chi^{111}}{|\chi^{111}|}}-\ln{\frac{\chi^{222}}{|\chi^{222}|}}\right)\in(-\pi,+\pi]$ is the phase difference between $\chi^{111}$ and $\chi^{222}$. The range of $\delta$ is (0, $\pi/2$). Eq. (3) does not hold when $|\chi^{111}|=|\chi^{222}|$ and $\Delta\varphi=\pm\pi/2$. In this situation, the polarization-resolved SHG is isotropic according to Eq. (2), therefore it does not show any change under $\mathcal{T}$ operation.

The maximum NLMO angle can be achieved is $\theta_{\mathcal{T}}=\pi/6+n\pi/3$ with the condition

and we denoted the corresponding frequency as the LPL characteristic frequency $\omega_{\rm LPL}$. At $\omega_{\rm LPL}$, the patterns of $I_{||}$ and $I_{\bot}$ are swapped exactly as shown in Fig. 3(b), which results in a remarkable polarization change of the SH light. If the magnitude of $I_{||}$ and $I_{\bot}$ differs greatly, the polarization of the SH light can rotate nearly $\pi/2$ under $\mathcal{T}$. The condition to achieve the $\pi/2$ polarization rotation at $\theta_{\rm m}=\pi/12+n\pi/6$ is

and the corresponding frequency is denoted as $\omega^{\prime}_{\rm LPL}$, which is a subset of $\omega_{\rm LPL}$. This condition is also graphically shown in the complex plane in Supplementary Figure 1 [18].

We numerically traced the $\omega$-dependence of NLMO angle $\theta_{\mathcal{T}}$ in the three representative materials, as shown in Fig. 3(a). As highlighted by the dashed horizontal line, $\theta_{\mathcal{T}}$ close to $\pi/6+n\pi/3$ corresponds to the largest NLMO angle. Remarkably, $\theta_{\mathcal{T}}$ is giant in ABA-AFM CrI₃ and ML Cr₂I₃Br₃ at several $\omega_{\rm LPL}$ frequencies. In contrast, $\theta_{\mathcal{T}}$ is always tiny in ML H-VSe₂. This distinction is consistent with their different ratio of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ susceptibilities observed in Fig. 2(b, d, f). Fig. 3(b) shows the dramatic change of SHG light from perpendicular to parallel polarization direction for incident light at $\theta_{m}=\pi/12$ (indicated by the yellow line) under $\mathcal{T}$ in this atomically thin Cr₂I₃Br₃ at $\omega^{\prime}_{\rm LPL}=1.71$ eV.

Next, we investigated the polarization and intensity of SH light under the illumination of the CPL. For the CPL characterized by $\mathbf{E}=E(1,\pm{i})$, where $\pm$ represents the left/right-handed ($\sigma^{\pm}$) CPL, the polarization of the SHG signal is

according to Tab. 1. 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 $I^{+}(\mathbf{M})$ is the SHG intensity generated by the magnetic order $\mathbf{M}$ when the helicity of the incident light is $\sigma^{+}$.

$\mathcal{T}$ 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 $\sigma$ as

where the $\pm\mathbf{M}$ states are $\mathcal{T}$-related, and $I^{\sigma}(+\mathbf{M})$ is the SHG intensity generated by the $+\mathbf{M}$ magnetic order. The range of $\eta_{\mathcal{T}}^{\sigma}$ is [-1,1], and $\eta_{\mathcal{T}}^{\sigma}=\pm 1$ represents the situation in which an incident CPL with helicity $\sigma$ can only generate SH light in the $\pm\mathbf{M}$ 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 $\mathcal{T}$ operation can also be reproduced by changing the helicity of CPL at a fixed magnetization, i.e., $\eta_{\mathcal{T}}^{+}=\eta_{\rm h}^{+\mathbf{M}}$.

The expression of the NLMO intensity asymmetry in our case is (derivation is in Supplementary Note 2B [18].)

As long as $\Delta\varphi\neq n\pi$, the NLMO intensity asymmetry is always nonzero, and the intensity of the SH light is always magnetization-dependent. In general, the maximum $|\eta_{\mathcal{T}}^{\pm}|=1$ requires

and we denoted the corresponding frequency as the CPL characteristic frequency $\omega_{\rm CPL}$. This condition is also graphically shown in the complex plane in Supplementary Figure 1 [18]. It is worth noting that the $\omega_{\rm CPL}$ is just where the $\theta_{\mathcal{T}}$ for LPL is ill-defined.

We numerically traced the $\omega$-dependence of $\eta_{\mathcal{T}}^{+}(2\omega)$ in the three representative materials in Fig. 3(c). Again, two CrI₃-based materials can achieve large $\eta_{\mathcal{T}}^{+}$ including $\pm 1$ at several frequencies while $|\eta_{\mathcal{T}}^{+}|$ is always below 0.4 in ML H-VSe₂. This contrast is another feature stemming from their distinct ratio of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$. We take ML Cr₂I₃Br₃ as an example to illustrate the intensity change of SH light under $\mathcal{T}$ operation for $\sigma^{\pm}$ CPL in Fig. 3(d). (The CPL SHG intensity of ABA-AFM CrI₃ and ML H-VSe₂ are shown in Supplementary Figure 4 and 8 [18].) The vertical dash line in Fig. 3(d) highlights one of the frequency $\omega_{\rm CPL}$ at which the intensity of SHG changes between a considerable value and zero under $\mathcal{T}$ operation. The upper ($\sigma^{+}$) and lower ($\sigma^{-}$) 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 $\omega_{\rm CPL}$.

The giant NLMO effects discovered in CrI₃-based materials enabled them to be used as magnetic-field-controlled atomically thin optical devices, as illustrated in Fig. 3(e). Utilizing incident light at frequencies $\omega^{\prime}_{\rm LPL}$, they can serve as optical polarization switchers for LPL as they can rotate the polarization of SH light by nearly $\pi/2$. At the incident frequency $\omega_{\rm CPL}$, they are ideal magneto-optical switches for CPL due to their large $\eta_{\mathcal{T}}$. 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 $\mathbf{P}(2\omega)$ generated in materials and the intensity of radiation is estimated by $I(2\omega)\propto|\mathbf{P}(2\omega)|^{2}$, 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, $\uparrow\uparrow\downarrow$ for ABA stacking trilayer CrI₃, which we denoted it as the MIX state and its $\mathcal{T}$-pair state as tMIX. ABA-MIX CrI₃ and ABA-AFM CrI₃ have the same net magnetic moment and tensor components as Tab. 1 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 CrI₃, each SHG tensor component contains both $\chi_{\rm N}$ and $\chi_{\rm M}$, and does not have a definite parity under $\mathcal{T}$, as shown in Fig. 4(a). As a result, for LPL, the polarization-resolved SHG shows both rotation and magnitude change under $\mathcal{T}$ operation as shown at the bottom two plots in Fig. 4(c). For CPL, the equivalence between helicity and magnetization reverse under $\eta_{\mathcal{T}}$ 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 $\omega_{\rm LPL}$ ($\omega_{\rm CPL}$) for ABA-AFM CrI₃, we can observe notable differences between ABA-AFM CrI₃ and ABA-MIX CrI₃, 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].

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 comparable $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$. However, in real materials, $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ may differ greatly which results in negligible NLMO effects. In the following, taking ABA-AFM CrI₃ as a model system, we demonstrated several strategies to enhance NLMO effects, in which $\chi^{111}$ is the $\mathcal{T}$-odd term $\chi_{\rm M}$ and $\chi^{222}$ is the $\mathcal{T}$-even term $\chi_{\rm N}$ as summarized in Tab. 1

The strength of interlayer interaction has great impact on properties of multilayer magnets [49, 50, 51], thus we firstly investigated the influence of interlayer distance $d$ in trilayer CrI₃, where $d_{0}$ is the interlayer distance of ABA-AFM CrI₃ at equilibrium. As shown in Fig. 5(a), both $|\chi_{\rm M}|$ and $|\chi_{\rm N}|$ change with $d$, but $|\chi_{\rm N}|$ is more sensitive to the variation of $d$. Therefore, a more comparable $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ can be realized by exerting hydrostatic pressure to multilayer CrI₃ along $z$-axis in experiments as shown in Fig. 5(d).

Next, we investigated the influence of spin-orbit coupling (SOC) to NLMO effects by artificially tuning the magnitude of SOC strength $\lambda$. As the Fig. 5(b) shows, $\chi_{\rm M}$ is sensitive to the change of $\lambda$ while $\chi_{\rm N}$ changes slightly. As $|\chi_{\rm M}|$ is much larger than $|\chi_{\rm N}|$ in ABA-AFM CrI₃, we replaced I by Br to reduce SOC and found ABA-AFM CrBr₃ posses very comparable $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ in a broad frequency range, as shown in Fig. 5(e).

Furthermore, we found stacking and magnetic orders can have synergistic effect on engineering the relative and absolute value of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$. As shown in Fig. 5(c), in multilayer AB stacking AFM CrI₃, $\chi_{\rm N}$ is forbidden in even-layer structures due to the presence of $\mathcal{PT}$ symmetry and remains almost the same in odd-layer structures, while $\chi_{\rm M}$ increases with the layer number. The above observations can be simply understood as $\chi_{\rm N}$ from each layer alternates the sign with similar magnitude, while $\chi_{\rm M}$ from each layer has the same sign, due to the $\mathcal{PT}$ symmetry between the A and the B layers[52]. Inspired by this, as $|\chi_{\rm M}|$ is much smaller than $|\chi_{\rm N}|$ in ML H-VSe₂, we calculated the SHG susceptibility of 7-layer (7L) AA^′-AFM H-VSe₂, which also has $\mathcal{PT}$ symmetry between A and A^′ layers. As expected, the peak value of $|\chi_{\rm M}|$ in 7-layer structure shown in Fig. 5(f) is almost 7 times of that in ML H-VSe₂ (Fig. 2f) while its $|\chi_{\rm N}|$ remains almost the same. $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ are comparable in 7L-AA^′ stacking H-VSe₂, indicating it can host giant NLMO effects. Similarly, by tuning the stacking sequence and magnetic order, we can achieve $\mathcal{P}$, $\mathcal{T}$ or neither symmetry between neighbouring layers and can realize an arbitrary control of the relative and absolute value of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ (Supplementary Figure 9) [18].

To summarize, the simultaneous breaking of $\mathcal{P}$, $\mathcal{T}$ 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 $\theta_{\mathcal{T}}$ under LPL, NLMO intensity asymmetry $\eta_{\mathcal{T}}$ and SHG-CD $\eta_{\rm h}$ 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 $\pm 1$ SHG-CD within a certain 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 CrI₃. We further derived that the comparable magnitude of $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$ 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 $|\chi_{\rm N}|$ and $|\chi_{\rm M}|$, 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 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 3$d$ 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 CrI₃-based materials and H-VSe₂-based materials, respectively. The convergence criterion of force were set to 10 meV/Å  and 1 meV/Å for CrI₃-based materials and H-VSe₂-based materials, respectively. Total energy is converged within $10^{-6}$ eV. $k$-point samplings of $13\times 13\times 1$ were used for CrI₃-based materials and $15\times 15\times 1$ for H-VSe₂-based materials. Vacuum thickness about 20 Å was used in the calculations of 2D materials.

After getting the converged electronic structures, we generated the maximally localized Wannier functions using Wannier90 [58] to build the tight-binding Hamiltonian and calculate optical responses [35]. The $\mathcal{T}$ symmetry breaking is carefully treated in the calculations of SHG for magnets [36, 59]. We obtained 56 maximally localized orbitals for each layer of CrI₃ or Cr₂I₃Br₃ and 22 for each layer of H-VSe₂. The broadening factor of the Dirac delta function is taken to be 0.05 eV. We found $50\times 50\times 1$ and $150\times 150\times 1$ kmesh samplings are enough for the converged SHG susceptibilities for CrI₃-based materials and H-VSe₂-based materials. The band gap of trilayer CrI₃ has been scissored to 1.5 eV in the calculation of SHG susceptibility [44].