Quantum anomalous Hall effect in monolayers Ti₂X₂ (X = P, As, Sb, Bi) with tunable Chern numbers by adjusting magnetization orientation

Abstract

Despite extensive research, the achievement of tunable Chern numbers in quantum anomalous Hall (QAH) systems remains a challenge in the field of condensed matter physics. Here, we theoretically proposed that Ti₂X₂ (X = P, As, Sb, Bi) can realize tunable Chern numbers QAH effect by adjusting their magnetization orientations. In the case of Ti₂P₂ and Ti₂As₂, if the magnetization lies in the x−y plane, and all C₂ symmetries are broken, a low-Chern-number phase with C = 1 will manifest. Conversely, if the magnetization is aligned to the z-axis, the systems enter a high-Chern number phase with C = 3. As for Ti₂Sb₂ and Ti₂Bi₂, by manipulating the in-plane magnetization orientation, these systems can periodically enter topological phases (C = ±1) over a 60° interval. Adjusting the magnetization orientation from +z to −z will result in the systems’ Chern number alternating between ±1. The non-trivial gap in monolayer Ti₂X₂ (X = P, As, Sb, Bi) can reach values of 23.4, 54.4, 60.8, and 88.2 meV, respectively. All of these values are close to the room-temperature energy scale. Furthermore, our research has revealed that the application of biaxial strain can effectively modify the magnetocrystalline anisotropic energy, which is advantageous in the manipulation of magnetization orientation. This work provides a family of large-gap QAH insulators with tunable Chern numbers, demonstrating promising prospects for future electronic applications.

Graphical abstract

Keywords

tunable Chern number / ferromagnetic / quantum anomalous Hall effect / Ti₂X₂ (X = P, As, Sb, Bi) / first-principles

References

[1] L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98(10), 106803 (2007)

[2] P. Li, X. Yang, Q. S. Jiang, Y. Z. Wu, and W. Xun, Built-in electric field and strain tunable valley-related multiple topological phase transitions in VSiXN₄ (X = C, Si, Ge, Sn, Pb) monolayers, Phys. Rev. Mater. 7(6), 064002 (2023)

[3] H. Huan, Y. Xue, B. Zhao, G. Y. Gao, H. R. Bao, and Z. Q. Yang, Strain-induced half-valley metals and topological phase transitions in MBr₂ monolayers (M = Ru, Os), Phys. Rev. B 104(16), 165427 (2021)

[4] P. Li, Prediction of intrinsic two dimensional ferromagnetism realized quantum anomalous Hall effect, Phys. Chem. Chem. Phys. 21(12), 6712 (2019)

[5] B. I. Halperin, Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25(4), 2185 (1982)

[6] F. D. M. Haldane, Nobel Lecture: Topological quantum matter, Rev. Mod. Phys. 89(4), 040502 (2017)

[7] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[8] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[9] H. Chi and J. S. Moodera, Progress and prospects in the quantum anomalous Hall effect, APL Mater. 10(9), 090903 (2022)

[10] C. Z. Chang, C. X. Liu, and A. H. MacDonald, Colloquium: Quantum anomalous Hall effect, Rev. Mod. Phys. 95(1), 011002 (2023)

[11] J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe₂, Nature 622(7981), 63 (2023)

[12] 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. 14(1), 5558 (2023)

[13] G. M. Ferguson, R. Xiao, A. R. Richardella, D. Low, N. Samarth, and K. C. Nowack, Direct visualization of electronic transport in a quantum anomalous Hall insulator, Nat. Mater. 22, 1100 (2023)

[14] P. J. Guo, Z. X. Liu, and Z. Y. Lu, Quantum anomalous Hall effect in collinear antiferromagnetism, npj Comput. Mater. 9(1), 70 (2023)

[15] Y. L. Jiao, X. T. Zeng, C. Chen, Z. Gao, K. Y. Song, X. L. Sheng, and S. A. Yang, Monolayer and bilayer PtCl₃: Energetics, magnetism, and band topology, Phys. Rev. B 107(7), 075436 (2023)

[16] J. Li and R. Q. Wu, Electrically tunable topological phase transition in van der Waals heterostructures, Nano Lett. 23(6), 2173 (2023)

[17] S. S. Li, X. Y. Li, W. X. Ji, P. Li, S. S. Yan, and C. W. Zhang, Quantum anomalous Hall effect with a high and tunable Chern number in monolayer NdN₂, Phys. Chem. Chem. Phys. 25(27), 18275 (2023)

[18] X. Y. Li, X. F. Xu, H. Zhou, H. X. Jia, E. Wang, H. X. Fu, J. T. Sun, and S. Meng, Tunable topological states in stacked Chern insulator bilayers, Nano Lett. 23(7), 2839 (2023)

[19] H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtz-mann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J. H. Chu, T. Cao, L. Fu, W. Yao, C. Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous Hall effect, Nature 622(7981), 74 (2023)

[20] P. Zhang, P. P. Balakrishnan, C. Eckberg, P. Deng, T. Nozaki, S. K. Chong, P. Quarterman, M. E. Holtz, B. B. Maranville, G. Qiu, L. Pan, E. Emmanouilidou, N. Ni, M. Sahashi, A. Grutter, and K. L. Wang, Exchange biased quantum anomalous Hall effect, Adv. Mater. 35(31), 2300391 (2023)

[21] S. Zhang, X. M. Zhang, Z. Q. He, L. Jin, C. Liu, Y. Liu, and G. D. Liu, Weyl nodal lines, Weyl points and the tunable quantum anomalous Hall effect in two-dimensional multiferroic metal oxynitride: Tl₂NO₂, Nanoscale 15(34), 14018 (2023)

[22] Y. F. Zhao, R. Zhang, J. Cai, D. Zhuo, L. J. Zhou, Z. J. Yan, M. H. W. Chan, X. Xu, and C. Z. Chang, Creation of chiral interface channels for quantized transport in magnetic topological insulator multilayer heterostructures, Nat. Commun. 14(1), 770 (2023)

[23] L. J. Zhou, R. B. Mei, Y. F. Zhao, R. X. Zhang, D. Y. Zhuo, Z. J. Yan, W. Yuan, M. Kayyalha, M. H. W. Chan, C. X. Liu, and C. Z. Chang, Confinement-induced chiral edge channel interaction in quantum anomalous Hall insulators, Phys. Rev. Lett. 130(8), 086201 (2023)

[24] Y. Gao, Y. Y. Zhang, J. T. Sun, L. Zhang, S. Zhang, and S. Du, Quantum anomalous Hall effect in two-dimensional Cu-dicyanobenzene coloring-triangle lattice, Nano Res. 13(6), 1571 (2020)

[25] 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 340(6129), 167 (2013)

[26] 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 367(6480), 895 (2020)

[27] J. Wang, B. Lian, H. Zhang, Y. Xu, and S. C. Zhang, Quantum anomalous Hall effect with higher plateaus, Phys. Rev. Lett. 111(13), 136801 (2013)

[28] C. Z. Chang, W. Zhao, D. Y. Kim, P. Wei, J. K. Jain, C. Liu, M. H. W. Chan, and J. S. Moodera, Zero-field dissipationless chiral edge transport and the nature of dissipation in the quantum anomalous Hall state, Phys. Rev. Lett. 115(5), 057206 (2015)

[29] Y. X. Wang and F. Li, High Chern number phase in topological insulator multilayer structures, Phys. Rev. B 104(3), 035202 (2021)

[30] M. Bosnar, A. Y. Vyazovskaya, E. K. Petrov, E. V. Chulkov, and M. M. Otrokov, High Chern number van der Waals magnetic topological multilayers MnBi₂Te₄/hBN, npj 2D Mater. Appl. 7(1), 33 (2023)

[31] J. Ge, Y. Liu, J. Li, H. Li, T. Luo, Y. Wu, Y. Xu, and J. Wang, High-chernnumber and high-temperature quantum Hall effect without landau levels, Natl. Sci. Rev. 7(8), 1280 (2020)

[32] Y. F. Zhao, R. Zhang, R. Mei, L. J. Zhou, H. Yi, Y. Q. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth, M. H. W. Chan, C. X. Liu, and C. Z. Chang, Tuning the Chern number in quantum anomalous Hall insulators, Nature 588(7838), 419 (2020)

[33] 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. 125(8), 086401 (2020)

[34] 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 106(12), 125404 (2022)

[35] Z. Li, Y. Han, and Z. Qiao, Chern number tunable quantum anomalous Hall effect in monolayer transitional metal oxides via manipulating magnetization orientation, Phys. Rev. Lett. 129(3), 036801 (2022)

[36] L. Jin, L. R. Wang, X. M. Zhang, Y. Liu, X. F. Dai, H. L. Gao, and G. D. Liu, Fully spin-polarized Weyl fermions and in/out-of-plane quantum anomalous Hall effects in a two-dimensional d⁰ ferromagnet, Nanoscale 13(11), 5901 (2021)

[37] K. Wang, Y. Li, H. Mei, P. Li, and Z. X. Guo, Quantum anomalous Hall and valley quantum anomalous Hall effects in two-dimensional d⁰ orbital XY monolayers, Phys. Rev. Mater. 6(4), 044202 (2022)

[38] L. Li, X. Kong, X. Chen, J. Li, B. Sanyal, and F. M. Peeters, Monolayer 1T-LaN₂: Dirac spin-gapless semiconductor of p-state and Chern insulator with a high Chern number, Appl. Phys. Lett. 117(14), 143101 (2020)

[39] S. D. Guo, W. Q. Mu, and B. G. Liu, Valley-polarized quantum anomalous Hall insulator in monolayer RuBr₂, 2D Mater. 9(3), 035011 (2022)

[40] X. Feng, L. Cai, Z. Chen, Y. Dai, B. Huang, and C. Niu, Tunable second-order topological insulators in Chern insulators 2H-FeX₂ (X = Cland Br), Appl. Phys. Lett. 122(19), 193104 (2023)

[41] 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. 246, 118731 (2023)

[42] 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 95(20), 201402 (2017)

[43] J. He, X. Li, P. Lyu, and P. Nachtigall, Near-room-temperature Chern insulator and Dirac spin-gapless semiconductor: Nickel chloride monolayer, Nanoscale 9(6), 2246 (2017)

[44] S. D. Guo, W. Q. Mu, X. B. Xiao, and B. G. Liu, Intrinsic room-temperature piezoelectric quantum anomalous Hall insulator in Janus monolayer Fe₂IX (X = Cl and Br), Nanoscale 13(30), 12956 (2021)

[45] J. Y. You, C. Chen, Z. Zhang, X. L. Sheng, S. A. Yang, and G. Su, Two-dimensional Weyl half-semimetal and tunable quantum anomalous Hall effect, Phys. Rev. B 100(6), 064408 (2019)

[46] H. Zhang, W. Yang, P. Cui, X. Xu, and Z. Zhang, Prediction of monolayered ferromagnetic CrMnI₆ as an intrinsic high-temperature quantum anomalous Hall system, Phys. Rev. B 102(11), 115413 (2020)

[47] J. Y. You, Z. Zhang, B. Gu, and G. Su, Two-dimensional room-temperature ferromagnetic semiconductors with quantum anomalous Hall effect, Phys. Rev. Appl. 12(2), 024063 (2019)

[48] B. Zhang, F. Deng, X. Chen, X. Lv, and J. Wang, Quantum anomalous Hall effect in M₂X₃ honeycomb Kagome lattice, J. Phys. Condens. Matter 34(47), 475702 (2022)

[49] H. P. Wang, W. Luo, and H. J. Xiang, Prediction of high-temperature quantum anomalous Hall effect in two-dimensional transition-metal oxides, Phys. Rev. B 95(12), 125430 (2017)

[50] P. Li and Z. X. Guo, The Dirac half-semimetal and quantum anomalous Hall effect in two-dimensional Janus Mn₂X₃Y₃ (X, Y = F, Cl, Br, I), Phys. Chem. Chem. Phys. 23(35), 19673 (2021)

[51] P. Li, Y. Ma, Y. Zhang, and Z. X. Guo, Room temperature quantum anomalous Hall insulator in a honeycomb-Kagome lattice, Ta₂O₃, with huge magnetic anisotropy energy, ACS Appl. Electron. Mater. 3(4), 1826 (2021)

[52] M. Aapro, M. N. Huda, J. Karthikeyan, S. Kezilebieke, S. C. Ganguli, H. G. Herrero, X. Huang, P. Liljeroth, and H. P. Komsa, Synthesis and properties of monolayer MnSe with unusual atomic structure and antiferromagnetic ordering, ACS Nano 15(8), 13794 (2021)

[53] J. B. Goodenough, Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO₃, Phys. Rev. 100(2), 564 (1955)

[54] J. Kanamori, Superexchange interaction and symmetry properties of electron orbitals, J. Phys. Chem. Solids 10(2−3), 87 (1959)

[55] J. Kanamori, Crystal distortion in magnetic compounds, J. Appl. Phys. 31(5), S14 (1960)

[56] W. E. Pickett and J. S. Moodera, Half metallic magnets, Phys. Today 54(5), 39 (2001)

[57] I. Žutić, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76(2), 323 (2004)

[58] Z. Liu, M. Amani, S. Najmaei, Q. Xu, X. Zou, W. Zhou, T. Yu, C. Qiu, A. G. Birdwell, F. J. Crowne, R. Vajtai, B. I. Yakobson, Z. Xia, M. Dubey, P. M. Ajayan, and J. Lou, Strain and structure heterogeneity in MoS₂ atomic layers grown by chemical vapour deposition, Nat. Commun. 5(1), 5246 (2014)

[59] G. H. Ahn, M. Amani, H. Rasool, D. H. Lien, J. P. Mastandrea, J. W. III Ager, M. Dubey, D. C. Chrzan, A. M. Minor, and A. Javey, Strain-engineered growth of two-dimensional materials, Nat. Commun. 8(1), 608 (2017)

[60] C. Zhang, M. Y. Li, J. Tersoff, Y. Han, Y. Su, L. J. Li, D. A. Muller, and C. K. Shih, Strain distributions and their influence on electronic structures of WSe₂‒MoS₂ laterally strained heterojunctions, Nat. Nanotechnol. 13(2), 152 (2018)

[61] Z. Dai, L. Liu, and Z. Zhang, 2D materials: Strain engineering of 2D materials: Issues and opportunities at the interface, Adv. Mater 31(45), 1970322 (2019)

[62] Z. Li, Y. Lv, L. Ren, J. Li, L. Kong, Y. Zeng, Q. Tao, R. Wu, H. Ma, B. Zhao, D. Wang, W. Dang, K. Chen, L. Liao, X. Duan, X. Duan, and Y. Liu, Efficient strain modulation of 2D materials via polymer encapsulation, Nat. Commun. 11(2), 1151 (2020)

Declarations

The authors declare no competing interests and no conflicts.

Electronic supplementary materials

The computational details, theoretical models and details, structural stabilities, relative magnetic properties, relative properties of different magnetization directions, which are available in the online version of this article at https://doi.org/10.15302/frontphys.2025.024203 and are accessible for authorized users.

Acknowledgements

The authors acknowledge the support by the National Natural Science Foundation of China (Grant Nos. 30930852, 62371397, and 62374134) and the start-up funds from Northwestern Polytechnical University.

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