The intrinsic quantum anomalous Hall effect in TaPdXTe (X = S, Se) monolayers†

Haofeng Wei^a, Yanzhao Wu^a, Junwei Tong^b, Li Deng^a, Xiang Yin^a, Zhijun Zhang^c and Xianmin Zhang*^a
aKey Laboratory for Anisotropy and Texture of Materials (Ministry of Education), School of Material Science and Engineering, Northeastern University, Shenyang, 110819, China. E-mail: zhangxm@atm.neu.edu.cn
^bDepartment of Physics, Freie Universität Berlin, Berlin, 14195, Germany
^cLiaoning Institute of Science and Technology, Benxi 117004, China

First published on 3rd September 2024

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1. Introduction

The quantum anomalous Hall (QAH) effect is a crucial concept in condensed matter physics,^1,2 which is induced by inherent magnetism and spin–orbit coupling (SOC) of materials,^3–6 and is characterized by quantized Hall conductance without an external magnetic field.^7,8 Due to their dissipationless electronic transport characteristics,⁹ QAH insulators show potential applications in spintronics, quantum computing, and low-power electronic devices.^10,11 Thus, obtaining QAH insulators is the key point for the development of condensed matter physics and materials science.^6,12 A QAH insulator was first reported in (Bi, Sb)₂Te₃ thin films with doping of chromium atoms.¹³ Then, the QAH effect was detected in twisted bilayer graphene¹⁴ and transition metal dichalcogenide heterobilayers.¹⁵ In recent years, the QAH effect has also been realized in some two-dimensional (2D) ferromagnetic (FM) materials by applying strain or modifying the electronic correlation strength.^16–20 However, due to their complex preparation processes and low controllability, the search for intrinsic QAH insulators has become a focus of topological physics.^21–25

Luckily, the spontaneous QAH effect has been theoretically predicted^21–23 and experimentally verified in MnBi₂Te₄ thin film at the temperature of 1.4 K,²⁴ which opens the door to the study of intrinsic QAH insulators. The QAH effect has been theoretically predicted in MFeSe (M = Tl, Ga)²⁶ and MnBr₃²⁷ monolayers with in-plane magnetic anisotropy. The QAH effect is reported in RuI₃,⁶ PdBr₃,²⁸ and PtBr₃²⁸ monolayers with small topological band gaps. Meanwhile, the high Chern number of 3 is demonstrated in the YN₂ monolayer with in-plane magnetic anisotropy²⁹ and Co₃Pb₃X₂ (X = S, Se) monolayer with a low Curie temperature (T_C = 51, 42 K).³⁰ Recently, the effect of layer stacking on the intrinsic properties of topological materials has also been studied.^31,32 However, the magnetic long-range ordering is restricted in 2D magnetic materials with in-plane magnetic anisotropy,³³ and the small topological band gap as well as the low T_C limit their practical application.³⁴ Therefore, it is urgent to seek high-performance QAH insulators for the development of advanced electronic devices.

In this study, by first-principles calculations, both TaPdSTe and TaPdSeTe monolayers show FM coupling with 100% spin-polarized electron states near the Fermi level. The easy axes of magnetization in TaPdSTe and TaPdSeTe monolayers are along the out-of-plane direction, which ensures the magnetic long-range ordering. Their T_C values are evaluated as 260 and 262 K by Monte Carlo (MC) simulations. Interestingly, both TaPdSTe and TaPdSeTe monolayers behave as intrinsic QAH insulators with topological gaps as high as 88 and 79 meV, respectively, which is induced by the energy band inversions of the d_xz and d_yz orbitals in Ta atoms. During −3 to 3% biaxial strains, TaPdSTe and TaPdSeTe monolayers always maintain the QAH effect, and their topological band gaps increase gradually from compressive to tensile strains. These excellent properties indicate that TaPdSTe and TaPdSeTe monolayers are two good candidates for high-performance QAH insulators.

2. Computational details

In this study, first-principles calculations based on density functional theory (DFT) are performed using the Vienna ab initio simulation package (VASP).^35,36 The d orbitals of Ta and Pd atoms are treated using the PBE+U_eff method,³⁷ with the Hubbard U_eff parameter set to 4 eV.³⁸ The phonon spectrum is calculated using the PHONOPY package.^39,40 Ab initio molecular dynamics (AIMD) simulations are performed under the NVT ensemble.⁴¹ The data for the calculations are processed using the VASPKIT package.⁴² A maximized localization function is created using the WANNIER90 package.⁴³ The topologically relevant properties are calculated using the WannierTools package.⁴⁴ More detailed calculation details are provided in the ESI.†

3. Results and discussion

3.1 Structure and stability

Fig. 1(a) shows the crystal structure of the TaPdXTe (X = S, Se) monolayer, where the X (X = S, Se) and Te atoms are located at the top and the bottom layer of the structure, respectively, and the Ta and Pd atoms are sandwiched in the middle layer. Both TaPdSTe and TaPdSeTe monolayers belong to the orthogonal crystal system, and show the Pmm2 space group. The optimized lattice constants of the TaPdSTe (TaPdSeTe) monolayer are a = 4.09 (4.14) Å and b = 4.19 (4.18) Å. Fig. 1(b) illustrates the high symmetry points of the TaPdSTe and TaPdSeTe monolayers in the first Brillouin zone (BZ). To investigate the stability of the TaPdSTe and TaPdSeTe monolayers, we also performed phonon spectra calculations and AIMD simulations. As shown in Fig. 1(c), all phonon modes are greater than zero, demonstrating the dynamic stability of the TaPdSTe monolayer. Meanwhile, the change of total energy is small during the AIMD simulations at 300 K, and the crystal structure keeps integrity at 6 ps, indicating the thermal stability of the TaPdSTe monolayer, as depicted in Fig. 1(d). Similarly, the dynamic and thermal stability of the TaPdSeTe monolayer is also demonstrated in Fig. 1(e) and (f), where all phonon modes are positive, and the total energy slightly fluctuates throughout the AIMD simulations and the structure remains an orthogonal structure at 6 ps.

3.2 Magnetic properties

After proving stability, the focus is now on investigating the magnetic properties of TaPdSTe and TaPdSeTe monolayers. To determine the magnetic ground state of the TaPdSTe and TaPdSeTe monolayers, one FM and three antiferromagnetic (AFM1/AFM2/AFM3) configurations are calculated using a 2 × 2 × 1 supercell, as shown in Fig. 2(a). The obtained energies of the AFM1, AFM2, and AFM3 configurations are 259 (272), 358 (374), and 319 (315) meV higher than that of the FM one, respectively, demonstrating the FM ground state of the TaPdSTe (TaPdSeTe) monolayer. Both magnetic moments of the TaPdSTe and TaPdSeTe monolayers are 1μ_B per f.u., which are mainly contributed by the Ta atoms.

In order to distinguish their type of magnets (XY or Ising), the magnetic anisotropy energies (MAE) of the TaPdSTe and TaPdSeTe monolayers were further studied, which are defined as: MAE = E_θ − E_[001]. Here, E_θ and E_[001] are the total energy of the magnetization orientation along the θ and [001] directions, respectively. Fig. S1 (ESI†) shows the spin vector S rotating at an angle θ from 0 to 180°. As shown in Fig. 2(b), the MAE of the TaPdSTe monolayer changes from 0 to 2.52 meV per f.u., from 0 to 0.36 meV per f.u., and from 2.52 to 0.36 meV per f.u. in the xz, yz, and xy planes, respectively. Similarly, the MAE of the TaPdSeTe monolayer varies between 0 to 1.79 meV per f.u., between 0 to 0.84 meV per f.u., and between 1.79 to 0.84 meV per f.u., respectively, as described in Fig. 2(c). These results clearly indicate that both TaPdSTe and TaPdSeTe monolayers show out-of-plane easy axes of magnetization with the MAE values of 0.36 and 0.84 meV per f.u., which indicates that both TaPdSTe and TaPdSeTe monolayers are Ising magnets.⁴⁵ This is crucial for an intrinsic QAH effect⁴⁶ and spontaneous valley polarization in magnetic materials.^47,48

As Ising ferromagnets for the present TaPdSTe and TaPdSeTe monolayers,²⁸ the T_C is another important physical factor for practical application, which can be calculated by MC simulations based on the Heisenberg model. The spin Hamiltonian can be defined as:

	(1)

where A is the anisotropic energy parameter, S_i/j is the spin operator, S^Z_i is the spin vector along the z direction, and J is the exchange parameter. In this study, as indicated in Fig. S2 (ESI†), only the nearest neighbor J₁, near neighbor J₂, and next nearest neighbor J₃ are considered, which are obtained from following equations:

EFM = E0 − (4J1 + 4J2 + 8J3)|S|2 − A|S|2

EAFM1 = E0 − (4J1 − 4J2 − 8J3)|S|2 − A|S|2

EAFM2 = E0 − (−4J1 − 4J2 + 8J3)|S|2 − A|S|2

EAFM3 = E0 − (−4J1 + 4J2 − 8J3)|S|2 − A|S|2	(2)

where E_FM, E_AFM1, E_AFM2, and E_AFM3 represent the energies of TaPdSTe and TaPdSeTe monolayers under FM, AFM1, AFM2, and AFM3 states and E₀ is the total energy of the systems without magnetic coupling. Here, |S| is 1/2. Based on the above results, the J₁, J₂, and J₃ of the TaPdSTe (TaPdSeTe) monolayer are calculated to be 104.55 (104.48), 74.43 (82.72), and 27.45 (26.52) meV, respectively. Therefore, the T_C of the TaPdSTe and TaPdSeTe monolayers is estimated to be about 260 and 262 K, as shown in Fig. 2(d) and (e). The T_C values are larger than that in FeX₃ (X = Cl, Br, I) (116–175 K),⁴⁹ NiBr₃ (100 K),⁵⁰ and CoBr₂ (27 K)⁵¹ monolayers.

3.3 Energy band structure and QAH effect

The energy band structures of the TaPdSTe and TaPdSeTe monolayers were then calculated to investigate their electronic properties. As shown in Fig. 3(a), without the SOC effect, there is a Dirac cone around the Fermi level for the TaPdSTe monolayer, which is completely contributed by the spin-up channel, indicating a 100% spin-polarized half-metal characteristic. This is crucial for spintronics. Comparatively, the spin-down channels are insulated with a band gap of 66 meV. Notably, the electron states near the Fermi level mainly come from the Ta atoms, as indicated in Fig. S3(a) (ESI†). Moreover, as displayed in Fig. 3(b), the electron states near the Dirac cone mainly come from the d_xz and d_yz orbitals of the Ta atoms, and two orbitals simultaneously appear in the conduction and valence bands. This means that an energy band inversion occurs in the present TaPdSTe monolayer,⁵² indicating a QAH effect.⁴ Under the SOC effect, the Dirac cone is opened with a topological band gap as high as 88 meV, as described in Fig. 3(c). Similarly, as illustrated in Fig. 3(d)–(f), the TaPdSeTe monolayer also presents a QAH characteristic with a topological band gap up to 79 meV. The topological band gap of the TaPdSTe and TaPdSeTe monolayers is larger than that reported in RuI₃ (11 meV),⁶ MnBr₃ (35 meV),²⁷ and PtBr₃ (28 meV)²⁸ monolayers. Notably, there are two Dirac cones all over the BZ, as indicated in Fig. S4 (ESI†). The energy band structures obtained from the HSE06 method of the two monolayers are also studied, showing similar results, as indicated in Fig. S5 (ESI†). The out-of-plane magnetic anisotropy, high T_C, and large topological band gaps of the TaPdSTe and TaPdSeTe monolayers mean that the QAH effect can be observed at relatively higher temperatures than that of (Bi, Sb)₂Te₃ film (30 mK),¹³ MnBi₂Te₄ film (1.4 K),²⁴ and MoTe₂/WSe₂ heterobilayers (2.5 K).¹⁵

To further prove the QAH characteristics of the TaPdSTe and TaPdSeTe monolayers, the corresponding Berry curvatures, anomalous Hall conductivity (AHC), and chiral edge states are calculated. The Berry curvature is calculated by the Kubo formula:^53,54

	(3)

where f_n(k) is the Fermi–Dirac distribution function with k being the electron wave vector, _x/y is the velocity operator, and E_nk/mk is the eigenvalue of the Bloch wave function ψ_nk/mk. As shown in Fig. 4(a), the Berry curvature of the TaPdSTe monolayer mainly exists on the S → X path. By integrating the Berry curvature over the first BZ, the AHC (σ_xy) can be calculated using the following equations:⁵⁵

	(4)

	(5)

As indicated in Fig. 4(b), the calculated AHC is −e²/ħ with quantized characteristics, reflecting the Chern number of −1. In addition, as shown in Fig. 4(c), one chiral edge state connecting the valence and conduction bands can be observed, which is consistent with C = −1. All this further demonstrates that the TaPdSTe monolayer is an intrinsic QAH insulator. Similarly, the TaPdSeTe monolayer shows an intrinsic QAH characteristic, as illustrated in Fig. 4(d)–(f).

3.4 Effect of biaxial strain on magnetic and topological properties

2D materials inevitably produce lattice mismatches with the substrate during preparation, which naturally causes strain inside these materials.⁵⁶ Therefore, we also study the magnetic and topological properties of the TaPdSTe and TaPdSeTe monolayer under −3 to 3% biaxial strains, which is defined as where a and a₀ are the lattice constants with and without strains, respectively. Negative and positive values represent compressive and tensile strains, respectively. As shown in Fig. 5(a) and (b), the ΔE (ΔE = E_AFM1 − E_FM) of the TaPdSTe and TaPdSeTe monolayers is always greater than zero within the strain range considered, indicating the FM ground state. The change of MAE with biaxial strain is then studied with E_[100] − E_[001] and E_[010] − E_[001], as illustrated in Fig. 5(c) and (d). It can be found that the E_[100] − E_[001] of the TaPdSTe monolayer increases monotonically from −3 to 3% strain, while the E_[010] − E_[001] of the TaPdSTe monolayer decreases with increasing tensile and compressive strain. Moreover, both E_[100] − E_[001] and E_[010] − E_[001] are greater than zero during the strain progress, indicating an out-of-plane easy axis of magnetization. Comparatively, for the TaPdSeTe monolayer during −3 to 3% strains, the E_[100] − E_[001] and E_[010] − E_[001] increase and decrease with the enlarging of the tensile and compressive strains, and always keeps a positive value, demonstrating an out-of-plane easy axis of magnetization. In addition, the topological band gaps of the TaPdSTe and TaPdSeTe monolayers increase gradually from −3 to 3% strain, as shown in Fig. 5(e) and (f). Therefore, the performance of the QAH effect in the TaPdSTe and TaPdSeTe monolayers can be further enhanced using biaxial tensile strains. Under biaxial strain, the orbit-resolved energy bands with the SOC effect of the TaPdSTe and TaPdSeTe monolayers are presented in Fig. S6 and S7 (ESI†).

The electronic structures, AHC, and chiral edge states of TaPdSTe and TaPdSeTe monolayers at −3% and 3% strains are further studied to demonstrate the robust QAH effect for different biaxial strains in the two monolayers. As shown in Fig. 6(a), under −3% compressive strain, the TaPdSTe monolayer maintains the energy band inversion characteristic with a band gap of 67 meV. The AHC is calculated as −e²/ħ, indicating the QAH effect with a Chern number of −1, as described in Fig. 6(b). The only one chiral edge state near the Fermi level further indicates the quantized characteristic of the TaPdSTe monolayer, as drawn in Fig. 6(c). As shown in Fig. 6(d)–(f), under 3% tensile strain, the TaPdSTe monolayer also presents the QAH effect with C = −1. Notably, the topological band gap increases to 109 meV. The topological band gaps of 67 and 109 meV are larger than the thermal fluctuation energy of 24 meV at room temperature.^10,52 Therefore, the QAH effect of the TaPdSTe monolayer is robust for biaxial strains. Similarly, the QAH effect of the TaPdSeTe monolayer is also robust for biaxial strains, as indicated in Fig. S8 (ESI†).

4. Conclusion

In summary, the stability, magnetic properties, electronic structure, and topological properties of TaPdXTe (X = S, Se) monolayers with and without strain are systematically studied using first-principles calculations. The TaPdXTe (X = S, Se) monolayer exhibits good dynamics and thermal stabilities and shows FM coupling with 100% spin-polarized. Moreover, the easy axes of magnetization in the TaPdSTe and TaPdSeTe monolayers are along the out-of-plane direction with MAE values of 0.36 and 0.84 meV, respectively. Their T_C values are evaluated as 260 and 262 K by MC simulations. Interestingly, both TaPdSTe and TaPdSeTe monolayers behave as intrinsic QAH insulators with topological gaps as high as 88 and 79 meV, respectively. The topological characteristic is induced by the energy band inversions of the d_xz and d_yz orbitals in the Ta atoms. From −3 to 3% biaxial strain, the TaPdSTe and TaPdSeTe monolayers always maintain the QAH effect, and their topological band gaps increase gradually. As a result, the out-of-plane magnetic anisotropy, large topological band gap, and intrinsic QAH characteristic indicate that TaPdSTe and TaPdSeTe monolayers are two excellent QAH candidates.

Data availability

All relevant data are within the main text and ESI.†

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (52271238), the Liaoning Revitalization Talents Program (XLYC2002075), and the Research Funds for the Central Universities (N2402002).

References

1. W. Wang, Y. Ou, C. Liu, Y. Wang, K. He, Q.-K. Xue and W. Wu, Direct evidence of ferromagnetism in a quantum anomalous Hall system, Nat. Phys., 2018, 14, 791–795 Search PubMed .
2. P. Deng, P. Zhang, C. Eckberg, S. K. Chong, G. Yin, E. Emmanouilidou, X. Che, N. Ni and K. L. Wang, Quantized resistance revealed at the criticality of the quantum anomalous Hall phase transitions, Nat. Commun., 2023, 14, 5558 Search PubMed .
3. J. Wang, B. Lian and S. C. Zhang, Quantum anomalous Hall effect in magnetic topological insulators, Phys. Scr., 2015, T164, 014003 CrossRef .
4. J. Wang, B. Lian, H. Zhang, Y. Xu and S.-C. Zhang, Quantum anomalous Hall effect with higher plateaus, Phys. Rev. Lett., 2013, 111, 136801 CrossRef PubMed .
5. X.-L. Sheng and B. K. Nikolić, Monolayer of the 5d transition metal trichloride OsCl₃: a playground for two-dimensional magnetism, room-temperature quantum anomalous Hall effect, and topological phase transitions, Phys. Rev. B, 2017, 95, 201402 CrossRef .
6. C. Huang, J. Zhou, H. Wu, K. Deng, P. Jena and E. Kan, Quantum anomalous Hall effect in ferromagnetic transition metal halides, Phys. Rev. B, 2017, 95, 045113 CrossRef .
7. J. Böttcher, C. Tutschku and E. M. Hankiewicz, Fate of quantum anomalous Hall effect in the presence of external magnetic fields and particle-hole asymmetry, Phys. Rev. B, 2020, 101, 195433 CrossRef .
8. J. Zou, Z. He and G. Xu, The study of magnetic topological semimetals by first principles calculations, npj Comput. Mater., 2019, 5, 96 CrossRef .
9. M. Nadeem, A. R. Hamilton, M. S. Fuhrer and X. Wang, Quantum anomalous Hall effect in magnetic doped topological insulators and ferromagnetic spin-gapless semiconductors—a perspective review, Small, 2020, 16, 1904322 CrossRef CAS PubMed .
10. Y. Li, J. Li, Y. Li, M. Ye, F. Zheng, Z. Zhang, J. Fu, W. Duan and Y. Xu, High-temperature quantum anomalous Hall insulators in Lithium-decorated iron-based superconductor materials, Phys. Rev. Lett., 2020, 125, 086401 CrossRef CAS .
11. C.-Z. Chang, C.-X. Liu and A. H. MacDonald, Colloquium: quantum anomalous Hall effect, Rev. Mod. Phys., 2023, 95, 011002 CrossRef CAS .
12. H. Zhang, T. Zhou, J. Zhang, B. Zhao, Y. Yao and Z. Yang, Quantum anomalous Hall effect in stanene on a nonmagnetic substrate, Phys. Rev. B, 2016, 94, 235409 CrossRef .
13. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma and Q.-K. Xue, Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science, 2013, 340, 167–170 CrossRef CAS .
14. M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science, 2020, 367, 900–903 CrossRef CAS .
15. T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. Devakul, K. Watanabe, T. Taniguchi, L. Fu, J. Shan and K. F. Mak, Quantum anomalous Hall effect from intertwined moiré bands, Nature, 2021, 600, 641–646 CrossRef CAS .
16. S.-D. Guo, W.-Q. Mu, J.-H. Wang, Y.-X. Yang, B. Wang and Y.-S. Ang, Strain effects on the topological and valley properties of the Janus monolayer VSiGeN₄, Phys. Rev. B, 2022, 106, 064416 CrossRef CAS .
17. S. D. Guo, Y. L. Tao, W. Q. Mu and B. G. Liu, Correlation-driven threefold topological phase transition in monolayer OsBr₂, Front. Phys., 2023, 18, 33304 CrossRef .
18. X. B. Peng, B. R. Xia, M. S. Si and D. Q. Gao, Electronic correlation-driven quantum anomalous valley Hall effect in intrinsic ferrovalley FeClBr, Appl. Phys. Lett., 2024, 124, 223101 CrossRef CAS .
19. Y. Wu, J. Tong, L. Deng, F. Luo, F. Tian, G. Qin and X. Zhang, Realizing spontaneous valley polarization and topological phase transitions in monolayer ScX₂ (X = Cl, Br, I), Acta Mater., 2023, 246, 118731 CrossRef CAS .
20. K. Jia, X.-J. Dong, S.-S. Li, W.-X. Ji and C.-W. Zhang, Tunable abundant valley Hall effect and chiral spin–valley locking in Janus monolayer VCGeN₄, Nanoscale, 2024, 16, 8639–8649 RSC .
21. S.-J. Zhang, C.-W. Zhang, S.-F. Zhang, W.-X. Ji, P. Li, P.-J. Wang, S.-S. Li and S.-S. Yan, Intrinsic Dirac half-metal and quantum anomalous Hall phase in a hexagonal metal-oxide lattice, Phys. Rev. B, 2017, 96, 205433 CrossRef .
22. G. Xu, B. Lian and S.-C. Zhang, Intrinsic quantum anomalous Hall effect in the Kagome lattice Cs₂LiMn₃F₁₂, Phys. Rev. Lett., 2015, 115, 186802 CrossRef .
23. Q. Sun, Y. Ma and N. Kioussis, Two-dimensional Dirac spin-gapless semiconductors with tunable perpendicular magnetic anisotropy and a robust quantum anomalous Hall effect, Mater. Horiz., 2020, 7, 2071–2077 RSC .
24. Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen and Y. Zhang, Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi₂Te₄, Science, 2020, 367, 895–900 CrossRef CAS PubMed .
25. M. H. Wu, High-temperature intrinsic quantum anomalous Hall effect in rare Earth monohalide, 2D Mater., 2017, 4, 021014 CrossRef .
26. H. Huan, Y. Xue, B. Zhao, H. Bao, L. Liu and Z. Yang, Tunable Weyl half-semimetals in two-dimensional iron-based materials MFeSe (M = Tl, In, Ga), Phys. Rev. B, 2022, 106, 125404 CrossRef CAS .
27. F. Xie, Z. Yin, B. Zhou and Y. Ding, Tunable electronic band structure and magnetic anisotropy in two-dimensional Dirac half-metal MnBr₃ by external stimulus: strain, magnetization direction, and interlayer coupling, Phys. Chem. Chem. Phys., 2023, 25, 32515–32524 RSC .
28. J.-Y. You, Z. Zhang, B. Gu and G. Su, Two-dimensional room-temperature ferromagnetic semiconductors with quantum anomalous Hall effect, Phys. Rev. Appl., 2019, 12, 024063 CrossRef CAS .
29. X. Kong, L. Li, O. Leenaerts, W. Wang, X.-J. Liu and F. M. Peeters, Quantum anomalous Hall effect in a stable 1T-YN₂ monolayer with a large nontrivial bandgap and a high Chern number, Nanoscale, 2018, 10, 8153–8161 RSC .
30. Z. Zhang, J.-Y. You, X.-Y. Ma, B. Gu and G. Su, Kagome quantum anomalous Hall effect with high Chern number and large band gap, Phys. Rev. B, 2021, 103, 014410 CrossRef CAS .
31. X. Zou, R. Li, Z. Chen, Y. Dai, B. Huang and C. Niu, Engineering gapless edge states from antiferromagnetic Chern homobilayer, Nano Lett., 2024, 24, 450–457 CrossRef CAS PubMed .
32. Y. Tian, C.-B. Wang, B.-Y. Zhang, X. Kong and W.-J. Gong, Quantum layer spin Hall effect in sliding antiferromagnetic bilayers, J. Mater. Chem. C, 2024, 12, 10950–10959 RSC .
33. P. Jiang, L. Kang, X. Zheng, Z. Zeng and S. Sanvito, Computational prediction of a two-dimensional semiconductor SnO₂ with negative Poisson's ratio and tunable magnetism by doping, Phys. Rev. B, 2020, 102, 195408 CrossRef CAS .
34. J. He, X. Li, P. Lyu and P. Nachtigall, Near-room-temperature Chern insulator and Dirac spin-gapless semiconductor: nickel chloride monolayer, Nanoscale, 2017, 9, 2246–2252 RSC .
35. G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 1996, 6, 15–50 CrossRef CAS .
36. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186 CrossRef CAS PubMed .
37. F. Aryasetiawan, K. Karlsson, O. Jepsen and U. Schönberger, Calculations of Hubbard U from first-principles, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 125106 CrossRef .
38. S. He, R. Kang, P. Zhou, P. Lyu and L. Sun, Excellent intrinsic Chern insulators: monolayer PdTaX₂ (X = Se, Te), J. Mater. Chem. C, 2024, 12, 4062–4069 RSC .
39. S. Baroni, S. de Gironcoli, A. Dal Corso and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation Theory, Rev. Mod. Phys., 2001, 73, 515–562 CrossRef CAS .
40. A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater., 2015, 108, 1–5 CrossRef CAS .
41. G. J. Martyna, M. L. Klein and M. Tuckerman, Nosé–Hoover chains: the canonical ensemble via continuous dynamics, J. Chem. Phys., 1992, 97, 2635–2643 CrossRef .
42. V. Wang, N. Xu, J.-C. Liu, G. Tang and W.-T. Geng, VASPKIT: a user-friendly interface facilitating high-throughput computing and analysis using VASP code, Comput. Phys. Commun., 2021, 267, 108033 CrossRef CAS .
43. A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt and N. Marzari, wannier90: a tool for obtaining maximally-localised wannier functions, Comput. Phys. Commun., 2008, 178, 685–699 CrossRef CAS .
44. Q. Wu, S. Zhang, H.-F. Song, M. Troyer and A. A. Soluyanov, WannierTools: an open-source software package for novel topological materials, Comput. Phys. Commun., 2018, 224, 405–416 CrossRef CAS .
45. R. Basnet, T. Patel, J. Wang, D. Upreti, S. K. Chhetri, G. Acharya, M. R. U. Nabi, J. Sakon and J. Hu, Understanding and tuning magnetism in layered Ising-type antiferromagnet FePSe₃ for potential 2D magnet, Adv. Electron. Mater., 2024, 10, 2300738 CrossRef CAS .
46. Y. Gao, K. Liu and Z.-Y. Lu, Intrinsic ferromagnetic and antiferromagnetic axion insulators in van der Waals materials MnX₂B₂T₆ family, Phys. Rev. Res., 2022, 4, 023030 CrossRef CAS .
47. T. Zhang, X. Xu, B. Huang, Y. Dai and Y. Ma, 2D spontaneous valley polarization from inversion symmetric single-layer lattices, npj Comput. Mater., 2022, 8, 64 CrossRef .
48. K. Sheng, B. Zhang and Z.-Y. Wang, Valley polarization in a two-dimensional high-temperature semiconducting TiInTe₃ honeycomb ferromagnet, Acta Mater., 2024, 262, 119461 CrossRef CAS .
49. P. Li, Prediction of intrinsic two dimensional ferromagnetism realized quantum anomalous Hall effect, Phys. Chem. Chem. Phys., 2019, 21, 6712–6717 RSC .
50. J. Sun, X. Zhong, W. Cui, J. Shi, J. Hao, M. Xu and Y. Li, The intrinsic magnetism, quantum anomalous Hall effect and Curie temperature in 2D transition metal trihalides, Phys. Chem. Chem. Phys., 2020, 22, 2429–2436 RSC .
51. P. Chen, J.-Y. Zou and B.-G. Liu, Intrinsic ferromagnetism and quantum anomalous Hall effect in a CoBr₂ monolayer, Phys. Chem. Chem. Phys., 2017, 19, 13432–13437 RSC .
52. Y. Z. Wu, L. Deng, J. W. Tong, X. Yin, Z. J. Zhang, F. B. Tian and X. M. Zhang, High-performance quantum anomalous Hall effect in monolayer Ti₂Sb₂KRb and Ti₂Bi₂NaK, Appl. Phys. Lett., 2024, 124, 223104 CrossRef CAS .
53. K. Sheng, B. Zhang, H.-K. Yuan and Z.-Y. Wang, Strain-engineered topological phase transitions in ferrovalley 2H-RuCl₂ monolayer, Phys. Rev. B, 2022, 105, 195312 CrossRef CAS .
54. Y. Wu, J. Tong, L. Deng, F. Luo, F. Tian, G. Qin and X. Zhang, Coexisting ferroelectric and ferrovalley polarizations in bilayer stacked magnetic semiconductors, Nano Lett., 2023, 23, 6226–6232 CrossRef CAS PubMed .
55. D. Xiao, M.-C. Chang and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys., 2010, 82, 1959–2007 CrossRef CAS .
56. X. Yu, Z. Peng, L. Xu, W. Shi, Z. Li, X. Meng, X. He, Z. Wang, S. Duan, L. Tong, X. Huang, X. Miao, W. Hu and L. Ye, Manipulating 2D materials through strain engineering, Small, 2024, 2402561,  DOI:10.1002/smll.202402561.  , Early View.

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Footnote

† Electronic supplementary information (ESI) available: The more detailed computational details and supporting data can be found. See DOI: https://doi.org/10.1039/d4tc02809k

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