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.2015.337In 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
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