Two-dimensional (2D) ferroic materials, including both 2D magnetic and polar layers, have become an emerging branch of condensed matter,1–41. M. An and S. Dong, APL Mater. 8, 110704 (2020). https://doi.org/10.1063/5.00318702. E. Torun, H. Sahin, S. K. Singh, and F. M. Peeters, Appl. Phys. Lett. 106, 192404 (2015). https://doi.org/10.1063/1.49210963. S. Zhou, L. You, H. L. Zhou, Y. Pu, Z. G. Gui, and J. L. Wang, Front. Phys. 16, 13301 (2021). https://doi.org/10.1007/s11467-020-0986-04. C. Gong and X. Zhang, Science 363, eaav4450 (2019). https://doi.org/10.1126/science.aav4450 since the experimental discoveries of ferromagnetism in CrI3, Cr2Ge2Te6, Fe3GeTe2,5–75. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden et al., Nature 546, 270 (2017). https://doi.org/10.1038/nature223916. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang et al., Nature 546, 265 (2017). https://doi.org/10.1038/nature220607. K. S. Burch, D. Mandrus, and J.-G. Park, Nature 563, 47 (2018). https://doi.org/10.1038/s41586-018-0631-z and ferroelectricity in SnTe, CuInP2S6, In2Se3.8–108. K. Chang, J. W. Liu, H. C. Lin, N. Wang, K. Zhao, A. M. Zhang, F. Jin, Y. Zhong, X. P. Hu, W. H. Duan, Q. M. Zhang, L. Fu, Q. K. Xue, X. Chen, and S. H. Ji, Science 353, 274 (2016). https://doi.org/10.1126/science.aad86099. F. Liu, L. You, K. L. Seyler, X. B. Li, P. Yu, J. H. Lin, X. W. Wang, J. D. Zhou, H. Wang, H. Y. He et al., Nat. Commun. 7, 12357 (2016). https://doi.org/10.1038/ncomms1235710. C. J. Cui, W. J. Hu, X. X. Yan, C. Addiego, W. P. Gao, Y. Wang, Z. Wang, L. Z. Li, Y. C. Cheng, P. Li et al., Nano Lett. 18, 1253 (2018). https://doi.org/10.1021/acs.nanolett.7b04852 Despite the fast growing number of 2D ferroics as revealed in experiments or predicted in calculations, there remain some tough issues to be solved. For example, due to the reduced coordination numbers in 2D lattices, the magnetic transition temperatures will be relatively low comparing with their three-dimensional (3D) counterparts. Thus, most experimentally verified ferromagnetic (FM) Curie temperatures (TC's) are below the room temperature.44. C. Gong and X. Zhang, Science 363, eaav4450 (2019). https://doi.org/10.1126/science.aav4450An alternative solution is to explore 2D antiferromagnets. In general, antiferromagnets are much abundant than ferromagnets, which provide more candidates to improve the transition temperatures. In addition, the importance of antiferromagnetic (AFM) spintronics has been gradually understood in recent years, which can be intrinsically more energy-saving and fast-operating in device applications.11,1211. T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016). https://doi.org/10.1038/nnano.2016.1812. T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wunderlich, and C. Felser, Nat. Phys. 14, 200 (2018). https://doi.org/10.1038/s41567-018-0063-6 However, the control and detection of antiferromagnetism remain challenging, since their net magnetization is fully compensated.Different from the plain ferromagnetism, AFM textures can be rather diverse. A very interesting one is the so-called A-type AFM (A-AFM) order, with antiferromagnetically coupled FM layers. This A-AFM order can be operated by electric voltages via the field effect, manifesting a carrier-density driving magnetoelectricity,1313. S. Dong and E. Dagotto, Phys. Rev. B 88, 140404(R) (2013). https://doi.org/10.1103/PhysRevB.88.140404 while other AFM orders are usually inactive to the field effect. For 2D magnets, Huang et al. and Wang et al. demonstrated the gate control of magnetism and tunneling magnetoresistance in the CrI3 bilayer with the A-AFM order.14,1514. B. Huang, G. Clark, D. R. Klein, M. David, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao et al., Nat. Nanotechnol. 13, 544 (2018). https://doi.org/10.1038/s41565-018-0121-315. Z. Wang, I. Gutiérrez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamoğlu, E. Giannini, and A. F. Morpurgo, Nat. Commun. 9, 2516 (2018). https://doi.org/10.1038/s41467-018-04953-8 Similar phenomena have also been reported in other 2D magnetic bilayers.16–1816. H. F. Lv, Y. J. Niu, X. J. Wu, and J. L. Yang, Nano Lett. 21, 7050 (2021). https://doi.org/10.1021/acs.nanolett.1c0260417. S. J. Gong, C. Gong, Y. Y. Sun, W. Y. Tong, C. G. Duan, J. H. Chu, and X. Zhang, Proc. Natl. Acad. Sci. 115, 8511 (2018). https://doi.org/10.1073/pnas.171546511518. C. K. Tian, F. H. Pan, L. Wang, D. H. Ye, J. M. Sheng, J. C. Wang, L. J. Juan, J. L. Huang, H. X. Zhang, D. Y. Xu et al., Phys. Rev. B 104, 214410 (2021). https://doi.org/10.1103/PhysRevB.104.214410 However, the A-AFM states in all these bilayers rely on the weak van der Waals (vdW) interaction and special stacking configurations, both of which are very subtle and fragile. Natural robust A-AFM materials remain rare, which are urgently needed for nanoscale magnetoelectric devices based on 2D ferroic materials. Furthermore, the mutual exclusion between the tunability and robustness is also a challenging scientific question.MXenes, as an non-vdW family of 2D materials,19,2019. M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi, and M. W. Barsoum, Adv. Mater. 23, 4248 (2011). https://doi.org/10.1002/adma.20110230620. A. Miranda, J. Halim, M. W. Barsoum, and A. Lorke, Appl. Phys. Lett. 108, 033102 (2016). https://doi.org/10.1063/1.4939971 may provide an alternative solution to aforementioned questions. With a general chemical formula Mn+1XnTx (M is an early transition metal, X stands for carbon and/or nitrogen, and T is the surface terminations, such as O, OH, F, and Cl), MXenes can naturally accommodate multiple M layers in a unit sheet. Also, the neighboring M layers are connected via the M-X-M chemical bonds, implying much stronger interlayer coupling than aforementioned vdW ones. Thus, the structures of MXenes not only retain the low-dimensional characteristics but also provide a better platform to pursuit robust layered antiferromagnetism.Therefore, we will screen candidates from the n = 1 MXenes, which provides M-bilayer for possible A-AFM order. However, for many M2NT2, such as Ti2NO2, Cr2NO2, Mn2NF2, and Mn2N(OH)2, they are metallic and with ferromagnetic tendency.21,2221. H. Kumar, N. C. Frey, L. Dong, B. Anasori, Y. Gogotsi, and V. B. Shenoy, ACS Nano 11, 7648 (2017). https://doi.org/10.1021/acsnano.7b0257822. G. Wang, J. Phys. Chem. C 120, 18850 (2016). https://doi.org/10.1021/acs.jpcc.6b05224 Luckily, Cr2CCl2 owns desired properties, which will be systematically studied in the following. Even though its magnetic ground state was already reported to be Néel-type antiferromagnetism,2323. S. Li, J. J. He, L. Grajciar, and P. Nachtigall, J. Mater. Chem. C 9, 11132 (2021). https://doi.org/10.1039/D1TC02837E the buckled honeycomb Cr lattice can be considered as two stacking triangular layers with the intralayer ferromagnetic coupling and the interlayer AFM coupling, which can mimic the A-type antiferromagnetism. Thus, in the following, it is renamed as A′-AFM. In fact, similar idea was also used for the [111]-oriented Néel-type perovskite,2424. Y. K. Weng, L. F. Lin, E. Dagotto, and S. Dong, Phys. Rev. Lett. 117, 037601 (2016). https://doi.org/10.1103/PhysRevLett.117.037601 which is an effective approach to obtain robust layered antiferromagnetism. Its magnetic transition temperature is estimated to be very high (∼1300 K), but much lower than the previous estimation (6095 K).2323. S. Li, J. J. He, L. Grajciar, and P. Nachtigall, J. Mater. Chem. C 9, 11132 (2021). https://doi.org/10.1039/D1TC02837E Despite its robustness, this layered antiferromagnetism can be tuned by external electric field, or a proximate ferroelectric (FE) layer, which gives rise to the non-negligible magneto-optic Kerr effect (MOKE) signals even if its magnetization is fully compensated.First-principles calculations based on the density functional theory (DFT) are performed with the projector augmented-wave (PAW) pseudopotentials as implemented in the Vienna ab initio Simulation Package (VASP).2525. G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). https://doi.org/10.1103/PhysRevB.54.11169 For the exchange-correlation functional, the PBE parametrization of the generalized gradient approximation (GGA) is adopted,2626. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). https://doi.org/10.1103/PhysRevLett.77.3865 and the Hubbard U is applied using the Dudarev parametrization.2727. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998). https://doi.org/10.1103/PhysRevB.57.1505 As reported previously, a correction of Ueff=3 eV is imposed on Cr's 3d orbitals.2828. J. He, P. Lyu, L. Z. Sun et al., J. Mater. Chem. C 4, 6500 (2016). https://doi.org/10.1039/C6TC01287FFor the monolayer calculation, a vacuum space of 30 Å thickness is added along the c-axis direction to avoid layer interactions. The energy cutoff is fixed to 500 eV, and the Γ-centered 9×9×1 Monkhorst-Pack k-mesh is adopted for the monolayer and heterostructure, which can lead to a well convergence (see Fig. S1 in the supplementary material). The convergence criterion for the energy is 10−6 eV for self-consistent iteration, and the Hellman–Feynman force is set to 0.01 eV/Å during the structural optimization. The Gaussian smearing (ISMEAR = 0) in combination with a small SIGMA = 0.05 is utilized for both structural optimization and static calculations. Phonopy is adopted to calculate the phonon band structures.2929. A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015). https://doi.org/10.1016/j.scriptamat.2015.07.021 For the heterostructure calculation, the vdW correction DFT-D2 method is adopted.3030. S. Grimme, J. Comput. Chem. 27, 1787 (2006). https://doi.org/10.1002/jcc.20495The exciting code with the time-dependent DFT (TD-DFT) is adopted to calculate the MOKE signal.31,3231. A. Gulans, S. Kontur, C. Meisenbichler, D. Nabok, P. Pavone, S. Rigamonti, S. Sagmeister, U. Werner, and C. Draxl, J. Phys.: Condens. Matter 26, 363202 (2014). https://doi.org/10.1088/0953-8984/26/36/36320232. S. Sagmeister and C. Ambrosch-Draxl, Phys. Chem. Chem. Phys. 11, 4451 (2009). https://doi.org/10.1039/b903676h The spin–orbit coupling (SOC) is considered in the magnetocrystalline anisotropy, dielectric, and MOKE calculations.The Néel temperature TN for Cr2CCl2 is estimated based on the Heisenberg model using DFT-derived spin exchange parameters. A 48 × 48 2D honeycomb lattice with periodic boundary conditions is used in our Markov–chain Monte Carlo (MC) simulations.3333. D. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics ( Cambridge University Press, 2021). The initial 104 MC steps are discarded for thermal equilibrium, and the remaining 104 MC steps are reserved as statistical average in the simulation. TN is found as a maximum on the temperature dependent specific heat.The surfaces of MXenes are usually passivated by anions or chemical ligands. For Cr2CCl2 monolayer with Cl-terminated surfaces, there are four most possible sites for Cl adatoms,3434. M. Khazaei, M. Arai, T. Sasaki, C.-Y. Chung, N. S. Venkataramanan, M. Estili, Y. Sakka, and Y. Kawazoe, Adv. Funct. Mater. 23, 2185 (2013). https://doi.org/10.1002/adfm.201202502 as shown in Fig. S2 in the supplementary material. According to our calculations, model 2 is the most stable structure, which has the lowest energy among all considered configurations (see Table S1 in the supplementary material). Therefore, only the model 2 will be studied below in detail.As shown in Fig. 1, in the Cr2CCl2 monolayer, the carbon layer is sandwiched between two chromium layers and the outside surfaces are decorated by chlorines, which possesses a trigonal lattice with the space group (S.G.) P3¯m1. Its dynamical stability has been confirmed by phonon calculation, and no imaginary mode appears in the phonon spectrum over the entire Brillouin zone, as shown in Fig. S3 of the supplementary material.To determine its magnetic ground state, five most possible magnetic orders are compared, as shown in Figs. 1(b)–1(f). According to our calculation, the energy of A′-AFM is significantly lower than those of other four (as summarized in Table I), implying the ground state. The optimized in-plane lattice constant is also consistent with previous reported value (3.269 Å).2828. J. He, P. Lyu, L. Z. Sun et al., J. Mater. Chem. C 4, 6500 (2016). https://doi.org/10.1039/C6TC01287F

TABLE I. Five magnetic orders' energies (in units of meV/f.u.) and optimized lattice constants (in units of Å) of the Cr2CCl2 monolayer. The Y-AFM order is calculated in the 3×3×1 supercell. The FM and A′-AFM order use primitive cell, and the other two use 1×2×1 supercell. The energies are in relative to the A′-AFM one. The bandgap is in units of eV. M is the magnetic moment (in units of μB).

 OrderEnergyS.G.abM(Cr)Gap A ′-AFM0P3¯m13.258±3.0991.67 FM495.5P3¯m13.2703.1350 Y-AFM547.9P3¯m15.620±2.9370.96 Z-AFM355.5C2/m3.2726.456±3.1211.16 S-AFM612.4C2/m3.2436.508±2.8120.55Based on the optimized structure of its ground state, the exchange couplings are derived by mapping the DFT energy to the Heisenberg model with normalized spins (|S|=1). The nearest-neighbor, next-nearest-neighbor, and next-next-nearest-neighbor exchanges (denoted as J1, J2, and J3, respectively) are estimated as 45.7, –37.7, and 45.2 meV, respectively. The chemical bonding Cr-C-Cr makes the J3 coupling even stronger than J2. Both J1 and J3 prefer AFM interlayer coupling, while J2 prefers intralayer ferromagnetism. Such a configuration of J's is not frustrated, which co-stabilize the layered A′-AFM order. These large J's are originated from the half-filled t2g orbitals of Cr3+ and the strong p-d hybridization of Cr-C bonds, both of which are advantage for a strong antiferromagnetic coupling. In addition, our J1/J2 is close to the values reported in the previous study,2828. J. He, P. Lyu, L. Z. Sun et al., J. Mater. Chem. C 4, 6500 (2016). https://doi.org/10.1039/C6TC01287F although they did not consider J3.Its magnetic anisotropy is also calculated by rotating the spin orientation. As shown in Fig. 2(a), the magnetic easy axis is along the [001]-axis, i.e., the out-of-plane direction. The magnetocrystalline anisotropy energy (MAE) is estimated to be 14 μeV/f.u., due to the weak SOC effect in the Cr2CCl2 monolayer.Based on the above DFT-derived coefficients, the MC method was employed to simulate the magnetic transition. A typical MC snapshot at 302 K, as shown in Fig. 2(b), suggests that an A′-AFM order has already been well established at room temperature. In fact, its Néel temperature TN is estimated to be ∼1300 K, as indicated by the peak of specific heat shown in Fig. 2(c). This high TN is reasonable considering the strong J's, but much lower than the previous estimation (6095 K),2323. S. Li, J. J. He, L. Grajciar, and P. Nachtigall, J. Mater. Chem. C 9, 11132 (2021). https://doi.org/10.1039/D1TC02837E which seems unreasonable.As mentioned before, such layered antiferromagnetism can be actively coupled with external electric fields along the c-axis. The magnetic point group of the Cr2CCl2 monolayer is centrosymmetric −3′m′, in which an electric field along the c-axis can generate an effective internal magnetic field.3535. H. J. Zhao, X. Liu, Y. Wang, Y. Yang, L. Bellaiche, and Y. Ma, Phys. Rev. Lett. 129, 187602 (2022). https://doi.org/10.1103/PhysRevLett.129.187602 This magnetoelectric effect generally works; even here, the Cr2CCl2 monolayer is a semiconductor with a moderate bandgap [see Figs. 3(a) and 3(b)].After applying an out-of-plane electric field (e.g., 0.3 V/Å), the electronic states, i.e., the density of states (DOS) and electronic bands, are slightly split between the spin-up and spin-down channels, as shown in Figs. 3(c) and 3(d). This electric field induced splitting is similar to the Zeeman splitting due to the magnetic field. However, this Zeeman-like splitting will not generate a net magnetization at zero temperature, since the existence of the bandgap.This Zeeman-like splitting can also be visualized in real space. As illustrated in Figs. 3(e) and 3(f), for both spin-up and spin-down Cr's, the electronic cloud will be distorted by moving against the electric field direction. This distortion breaks the inversion symmetry between the lower layer Cr (spin-up) and upper layer Cr (spin-down), which results in the spin splitting.Although no net magnetization is induced by such electronic cloud distortions at zero temperature, this magnetoelectricity can be detected by some sensitive techniques, such as MOKE. In fact, the MOKE has been widely employed as a powerful tool for the characterization of low-dimensional magnetic materials.5–7,14,36,375. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden et al., Nature 546, 270 (2017). https://doi.org/10.1038/nature223916. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang et al., Nature 546, 265 (2017). https://doi.org/10.1038/nature220607. K. S. Burch, D. Mandrus, and J.-G. Park, Nature 563, 47 (2018). https://doi.org/10.1038/s41586-018-0631-z14. B. Huang, G. Clark, D. R. Klein, M. David, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao et al., Nat. Nanotechnol. 13, 544 (2018). https://doi.org/10.1038/s41565-018-0121-336. M. Diwekar, V. Kamaev, J. Shi, and Z. V. Vardeny, Appl. Phys. Lett. 84, 3112 (2004). https://doi.org/10.1063/1.171202737. K. Yang, W. T. Hu, H. Wu, M.-H. Whangbo, P. G. Radaelli, and A. Stroppa, ACS Appl. Electron. Mater. 2, 1373 (2020). https://doi.org/10.1021/acsaelm.0c00154Generally, the signal of MOKE is associated with the off diagonal components of the optical conductivity tensors σ.3737. K. Yang, W. T. Hu, H. Wu, M.-H. Whangbo, P. G. Radaelli, and A. Stroppa, ACS Appl. Electron. Mater. 2, 1373 (2020). https://doi.org/10.1021/acsaelm.0c00154 The optical conductivity has a relationship with the dielectric tensor ε: εij(ω)=δij+i4πωσij(ω).3838. D. Sangalli, A. Marini, and A. Debernardi, Phys. Rev. B 86, 125139 (2012). https://doi.org/10.1103/PhysRevB.86.125139 Therefore, the presence of MOKE directly depends on the imaginary components of the dielectric function. Under the electric field along the c-axis, the original magnetic point group −3′m′ is decreased to 3m′, whose dielectric tensor can be generally expressed as3939. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light ( Elsevier, 2013). ε=[εxxεxy0−εxyεxx000εzz].(1)Its frequency dependent dielectric function has been calculated. As shown in Fig. 3(g), for the A′-AFM phase, the imaginary components of off diagonal ε′′xy stay zero in the absence of an electric field. In contrast, it is nonzero for the FM phase, implying a detectable MOKE signal. The most interesting result is that the imaginary components of ε′′xy are also nonzero for the A′-AFM phase under an external electric field, although the net magnetization remains zero. As expected, a larger electric field leads to larger amplitude of imaginary components of ε′′xy. In short, the Zeeman-like band splitting in the Cr2CCl2 monolayer induced by an electric field can be detected via magneto-optical activity, even in the absence of net magnetization.

An alternative and feasible approach is to use a proximate FE layer to replace the external electric field, which can be even nonvolatile for the magnetoelectric switching.

To demonstrate this idea, a 2D vdW heterostructure Cr2CCl2/Sc2CO2 is constructed. Here, the Sc2CO2 monolayer is a ferroelectric MXene, with a spontaneous out-of-plane polarization of 1.60 μC/m2.4040. A. Chandrasekaran, A. Mishra, and A. K. Singh, Nano Lett. 17, 3290 (2017). https://doi.org/10.1021/acs.nanolett.7b01035 Also, the in-plane lattice geometry matches well between Sc2CO2 and Cr2CCl2 monolayers: similar trigonal framework and proximate lattice constants (a = 3.427 Å for Sc2CO2 and a = 3.258 Å for Cr2CCl2 according to our DFT structural optimization).Considering the FE polarization directions (P↑ vs P↓) and stacking modes (A vs B), here four possible structural configurations are considered, as shown in Fig. S4 of the supplementary material. According to our calculation, the configuration A always has a lower energy than the configuration B, for giving polarization (see Table S2 in the supplementary material). Hence, only the configuration A will be discussed in the following, as sketched in Figs. 4(a) and 4(b). The optimized distances between Cr2CCl2 and Sc2CO2 are d0=2.734 Å for P↑ and d0=2.868 Å for P↓, respectively. The P↑ state is lower in energy than that of P↓.Based on our DFT calculations, the A′-AFM phase remains the ground state of Cr2CCl2 upon the substrate polarization (see Table S3 in the supplementary material). In addition, the energy dependence of interlayer spacing is shown in Fig. 4(c). The saturation energies of heterostructure are only 0.213 and 0.175 J/m2 for P↑ and P↓, respectively, which are in the range of vdW materials.4141. M. An, Z. Yand, C. Jun, Z. H. Min, G. Y. Jun, and S. Dong, J. Phys. Chem. C 123, 30545 (2019). https://doi.org/10.1021/acs.jpcc.9b08706 Thus, the coupling between Cr2CCl2 and Sc2CO2 monolayers is the weak vdW interaction instead of the stronger chemical bonding. Furthermore, we have calculated the exchange J's based on the optimized A′-AFM structures of Cr2CCl2/P↑ and Cr2CCl2/P↓ heterostructures, as compared in Table S4 in the supplementary material. It demonstrated that the robust antiferromagnetism and high Néel temperature persist in the heterostructures.However, a significant tuning of bandgap (from 0.05 eV for P↑ to 1.33 eV for P↓) occurs when switching the polarization of the Sc2CO2 monolayer, as compared in Figs. 4(d) and 4(e). This is mostly due to the electrostatic field effect, which largely shifts the conducting band contributed by Cr's empty 3d orbitals. As a consequence, the optical properties will be largely different between the P↑ and P↓ conditions, which will be reflected in the MOKE behavior.The magnetic point group of the Cr2CCl2/Sc2CO2 heterostructure is 3m′, which does not change during the polarization switching. The optical conductivity can be expressed as Eq. (S6) in the supplementary material.3737. K. Yang, W. T. Hu, H. Wu, M.-H. Whangbo, P. G. Radaelli, and A. Stroppa, ACS Appl. Electron. Mater. 2, 1373 (2020). https://doi.org/10.1021/acsaelm.0c00154 The MOKE signal is expected in this heterostructure, characterized by the complex Kerr angle ϕK, as sketched in Fig. 4(f).The complex Kerr angle ϕK is consisted by the Kerr rotation angle θK and Kerr ellipticity ηK: ϕK=θK+iηK. As shown in Fig. 4(g), no MOKE signal appears in pristine Cr2CCl2, but ϕK emerges in the heterostructure. The ϕK's are asymmetric between the P↑ and P↓ conditions, since these two states are highly asymmetric in this heterostructure. Also, the magnitude of the predicted MOKE signal is within the detectable precision.42,4342. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). https://doi.org/10.1126/science.110551443. J. Lee, K. F. Mak, and J. Shan, Nat. Nanotechnol. 11, 421 (2016). https://doi.org/10.1038/nnano.2015.337

In summary, based on first-principles calculations, we have predicted the electric field induced Zeeman-like splitting of band structures and the magneto-optical Kerr effect in the layered collinear antiferromagnetic Cr2CCl2 monolayer. The effect also occurs in ferroelectric–magnetic heterostructures, such as Cr2CCl2/Sc2CO2. The high magnetic transition temperature of the Cr2CCl2 monolayer makes this magnetoelectric function available at room temperature. Our work opens a promising avenue for future studies of electrical tuning of low-dimensional antiferromagnetic spintronics.

See the supplementary material for more DFT results, including DFT energies, structures, model Hamiltonian, and MOKE equations.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12274069, 12274070, and 11834002) and the Big Data Computing Center of Southeast University.

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Xinyu Yang: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead). Ning Ding: Formal analysis (supporting); Methodology (supporting); Software (supporting). Jun Chen: Methodology (supporting); Software (supporting). Ziwen Wang: Software (supporting); Visualization (supporting). Ming An: Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Writing – original draft (equal). Shuai Dong: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

REFERENCES

1. M. An and S. Dong, APL Mater. 8, 110704 (2020). https://doi.org/10.1063/5.0031870, Google ScholarScitation, ISI2. E. Torun, H. Sahin, S. K. Singh, and F. M. Peeters, Appl. Phys. Lett.