Manipulation of valley polarization and topology in monolayer MoSnC₂S₆ and MoPbGe₂Te₆

Abstract

The ferrovalley materials and their nontrivial band topological properties have recently attracted extensive interest in theoretical physics and their promising applications. Using first-principles calculations, we predict the valley polarization in monolayer MoSnC₂S₆ and MoPbGe₂Te₆. These materials possess a robust ferromagnetic ground state, with high Curie temperatures of 460 K and 319.5 K, respectively. The intrinsic valley polarization arises from the breaking of time-reversal symmetry and spatial inversion symmetry. Biaxial strain and electron correlation (U) can modulate the valley polarization and bandgap. The quantum anomalous Hall phase is driven by biaxial strain and U during the process of bandgap closing, opening, reclosing, and reopening. This can be demonstrated by the chiral edge states at the edges and the plateau in the anomalous Hall conductivity. During the closing and opening of the bandgap, the sign and magnitude of the Berry curvature also vary. Our work provides an ideal platform for valleytronics and the quantum anomalous Hall effect.

Graphical abstract

Keywords

first-principles calculations / ferrovalley / Curie temperature / quantum anomalous Hall effect

References

[1] D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics in graphene: Magnetic moment and topological transport, Phys. Rev. Lett. 99(23), 236809 (2007)

[2] Z. Song, X. Sun, J. Zheng, F. Pan, Y. Hou, M. H. Yung, J. Yang, and J. Lu, Spontaneous valley splitting and valley pseudospin field effect transistors of monolayer VAgP₂Se₆, Nanoscale 10(29), 13986 (2018)

[3] W. Yao, D. Xiao, and Q. Niu, Valley-dependent optoelectronics from inversion symmetry breaking, Phys. Rev. B 77(23), 235406 (2008)

[4] D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, Coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides, Phys. Rev. Lett. 108(19), 196802 (2012)

[5] Y. Liu, Y. Gao, S. Zhang, J. He, J. Yu, and Z. Liu, Valleytronics in transition metal dichalcogenides materials, Nano Res. 12(11), 2695 (2019)

[6] L. Cao, J. Zhong, J. Yu, C. Zeng, J. Ding, C. Cong, X. Yue, Z. Liu, and Y. Liu, Valley-polarized local excitons in WSe₂/WS₂ vertical heterostructures, Opt. Express 28(15), 22135 (2020)

[7] W. Zhou, Z. Yang, A. Li, M. Long, and F. Ouyang, Spin and valley splittings in Janus monolayer WSSe on a MnO(111) surface: Large effective Zeeman field and opening of a helical gap, Phys. Rev. B 101(4), 045113 (2020)

[8] G. Aivazian, Z. Gong, A. M. Jones, R. L. Chu, J. Yan, D. G. Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu, Magnetic control of valley pseudospin in monolayer WSe₂, Nat. Phys. 11(2), 148 (2015)

[9] D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson, A. Kormányos, V. Zólyomi, J. Park, and D. C. Ralph, Breaking of valley degeneracy by magnetic field in monolayer MoSe₂, Phys. Rev. Lett. 114(3), 037401 (2015)

[10] K. F. Mak, K. L. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS₂ by optical helicity, Nat. Nanotechnol. 7(8), 494 (2012)

[11] H. L. Zeng, J. F. Dai, W. Yao, D. Xiao, and X. D. Cui, Valley polarization in MoS₂ monolayers by optical pumping, Nat. Nanotechnol. 7(8), 490 (2012)

[12] Y. C. Cheng, Q. Y. Zhang, and U. Schwingenschlögl, Valley polarization in magnetically doped single-layer transition-metal dichalcogenides, Phys. Rev. B 89(15), 155429 (2014)

[13] R. Peng, Y. Ma, S. Zhang, B. Huang, and Y. Dai, Valley polarization in Janus single-layer MoSSe via magnetic doping, J. Phys. Chem. Lett. 9(13), 3612 (2018)

[14] L. Xu, M. Yang, L. Shen, J. Zhou, T. Zhu, and Y. P. Feng, Large valley splitting in monolayer WS₂ by proximity coupling to an insulating antiferromagnetic substrate, Phys. Rev. B 97(4), 041405 (2018)

[15] H. Zhang, W. Yang, Y. Ning, and X. Xu, Abundant valley-polarized states in two-dimensional ferromagnetic van der Waals heterostructures, Phys. Rev. B 101(20), 205404 (2020)

[16] C. Ke, Y. Wu, W. Yang, Z. Wu, C. Zhang, X. Li, and J. Kang, Large and controllable spin-valley splitting in two-dimensional WS₂/h-VN heterostructure, Phys. Rev. B 100(19), 195435 (2019)

[17] W. Y. Tong, S. J. Gong, X. Wan, and C. G. Duan, Concepts of ferrovalley material and anomalous valley Hall effect, Nat. Commun. 7(1), 13612 (2016)

[18] P. Li, B. Liu, S. Chen, W. X. Zhang, and Z. X. Guo, Progress on two-dimensional ferrovalley materials, Chin. Phys. B 33(1), 017505 (2024)

[19] W. Xun, C. Wu, H. Sun, W. Zhang, Y. Z. Wu, and P. Li, Coexisting magnetism, ferroelectric, and ferrovalley multiferroic in stacking-dependent two-dimensional materials, Nano Lett. 24(11), 3541 (2024)

[20] P. Li, C. Wu, C. Peng, M. Yang, and W. Xun, Multifield tunable valley splitting in two-dimensional MXene Cr₂COOH, Phys. Rev. B 108(19), 195424 (2023)

[21] 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)

[22] J. Liu, W. J. Hou, C. Cheng, H. X. Fu, J. T. Sun, and S. Meng, Intrinsic valley polarization of magnetic VSe₂ monolayers, J. Phys. Condens. Matter 29(25), 255501 (2017)

[23] P. Zhao, Y. Dai, H. Wang, B. Huang, and Y. Ma, Intrinsic valley polarization and anomalous valley Hall effect in single-layer 2H-FeCl₂, ChemPhysMater 1(1), 56 (2022)

[24] 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)

[25] X. Li, T. Cao, Q. Niu, J. R. Shi, and J. Feng, Coupling the valley degree of freedom to antiferromagnetic order, Proc. Natl. Acad. Sci. U S A 110(10), 3738 (2013)

[26] W. Zhou, G. Zheng, A. Li, D. Zhang, and F. Ouyang, Orbital contribution to the regulation of the spin−valley coupling in antiferromagnetic monolayer MnPTe₃, Phys. Rev. B 107(3), 035139 (2023)

[27] R. Peng, Y. D. Ma, X. L. Xu, Z. L. He, B. B. Huang, and Y. Dai, Intrinsic anomalous valley Hall effect in single-layer Nb₃I₈, Phys. Rev. B 102(3), 035412 (2020)

[28] L. Feng, X. Chen, and J. Qi, Nonvolatile electric field control of spin-valley-layer polarized anomalous Hall effect in a two-dimensional multiferroic semiconductor bilayer, Phys. Rev. B 108(11), 115407 (2023)

[29] D. Zhang, A. Li, X. Chen, W. Zhou, and F. Ouyang, Tuning valley splitting and magnetic anisotropy of multiferroic CuMP₂X₆ (M = Cr, V; X = S, Se) monolayer, Phys. Rev. B 105(8), 085408 (2022)

[30] Y. Wu, J. Tong, L. Deng, F. Luo, F. Tian, G. Qin, X. Zhang, Realizing spontaneous valley polarization, and topological phase transitions in monolayer ScX₂ (X = Cl, Br, I), Acta Mater. 246, 118731 (2023)

[31] G. Zheng, S. Qu, W. Zhou, and F. Ouyang, Janus monolayer TaNF: A new ferrovalley material with large valley splitting and tunable magnetic properties, Front. Phys. (Beijing) 18(5), 53302 (2023)

[32] R. Yu, W. Zhang, H. J. Zhang, S. C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall effect in magnetic topological insulators, Science 329(5987), 61 (2010)

[33] Y. Shao, W. Lv, J. Guo, B. Qi, W. Lv, S. Li, G. Guo, and Z. Zeng, The current modulation of anomalous Hall effect in van der Waals Fe₃GeTe₂/WTe₂ heterostructures, Appl. Phys. Lett. 116(9), 092401 (2020)

[34] X. Zhu, Y. Chen, Z. Liu, Y. Han, and Z. Qiao, Valley-polarized quantum anomalous Hall effect in van der Waals heterostructures based on monolayer jacutingaite family materials, Front. Phys. (Beijing) 18(2), 23302 (2022)

[35] X. Deng, H. Yang, S. Qi, X. Xu, and Z. Qiao, Quantum anomalous Hall effect and giant Rashba spin−orbit splitting in graphene system co-doped with boron and 5d transition-metal atoms, Front. Phys. (Beijing) 13(5), 137308 (2018)

[36] H. Hu,W. Y. Tong,Y. H. Shen,X. Wan,C. G. Duan, Concepts of the half-valley-metal and quantum anomalous valley Hall effect, npj Comput. Mater. 6(1) (2020)

[37] X. Kou, S. T. Guo, Y. Fan, L. Pan, M. Lang, Y. Jiang, Q. Shao, T. Nie, K. Murata, J. Tang, Y. Wang, L. He, T. K. Lee, W. L. Lee, and K. L. Wang, Scale-invariant quantum anomalous Hall effect in magnetic topological insulators beyond the two-dimensional limit, Phys. Rev. Lett. 113(13), 137201 (2014)

[38] J. G. Checkelsky, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, Y. Kozuka, J. Falson, M. Kawasaki, and Y. Tokura, Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator, Nat. Phys. 10(10), 731 (2014)

[39] C. Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S. C. Zhang, C. Liu, M. H. W. Chan, and J. S. Moodera, High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator, Nat. Mater. 14(5), 473 (2015)

[40] S. Zhang, C. Zhang, S. Zhang, W. Ji, P. Li, P. Wang, S. Li, and S. Yan, Intrinsic Dirac half-metal and quantum anomalous Hall phase in a hexagonal metal-oxide lattice, Phys. Rev. B 96(20), 205433 (2017)

[41] H. Sun, S. S. Li, W. Ji, and C. W. Zhang, Valley-dependent topological phase transition and quantum anomalous valley Hall effect in single-layer RuClBr, Phys. Rev. B 105(19), 195112 (2022)

[42] B. Wu, Y. Song, W. Ji, P. Wang, S. Zhang, and C. Zhang, Quantum anomalous Hall effect in an antiferromagnetic monolayer of MoO, Phys. Rev. B 107(21), 214419 (2023)

[43] 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)

[44] K. Dolui, S. Ray, and T. Das, Intrinsic large gap quantum anomalous Hall insulators in LaX (X = Br, Cl, I), Phys. Rev. B 92(20), 205133 (2015)

[45] C. W. Hu, . Realization of an intrinsic ferromagnetic topological state in MnBi₈Te₁₃, Sci. Adv. 6(30), eaba4275 (2020)

[46] 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 367(6480), 900 (2020)

[47] S. D. Guo, Y. L. Tao, W. Q. Mu, and B. G. Liu, Correlation-driven threefold topological phase transition in monolayer OsBr₂, Frontiers of Physics 18(3), 33304 (2023)

[48] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136(3B), B864 (1964)

[49] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169 (1996)

[50] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59(3), 1758 (1999)

[51] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50(24), 17953 (1994)

[52] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77(18), 3865 (1996)

[53] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13(12), 5188 (1976)

[54] A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015)

[55] G. J. Martyna, M. L. Klein, and M. Tuckerman, Nosé–Hoover chains: The canonical ensemble via continuous dynamics, J. Chem. Phys. 97(4), 2635 (1992)

[56] V. Wang,N. Xu,J. C. Liu,G. Tang,W. T. Geng, VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code, Comput. Phys. Commun. 267, 108033 (2021)

[57] A. A. Mostofi,J. R. Yates,Y. S. Lee, I. Souza,D. Vanderbilt,N. Marzari, wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178(9), 685 (2008)

[58] Q. S. Wu, S. N. Zhang, H. F. Song, M. Troyer, and A. A. Soluyanov, Wannier tools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018)

[59] L. Liu, X. Ren, J. H. Xie, B. Cheng, W. K. Liu, T. Y. An, H. W. Qin, and J. F. Hu, Magnetic switches via electric field in BN nanoribbons, Appl. Surf. Sci. 480, 300 (2019)

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

[61] P. W. Anderson, New approach to the theory of superexchange interactions, Phys. Rev. 115(1), 2 (1959)

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

[63] S. Yang, Y. Chen, and C. Jiang, Strain engineering of two‐dimensional materials: Methods, properties, and applications, InfoMat 3(4), 397 (2021)

[64] S. Bertolazzi, J. Brivio, and A. Kis, Stretching and breaking of ultrathin MoS₂, ACS Nano 5(12), 9703 (2011)

[65] H. J. Conley, B. Wang, J. I. Ziegler, R. F. Jr Haglund, S. T. Pantelides, and K. I. Bolotin, Bandgap engineering of strained monolayer and bilayer MoS₂, Nano Lett. 13(8), 3626 (2013)

[66] D. Wang, R. Wu, and A. J. Freeman, First-principles theory of surface magnetocrystalline anisotropy and the diatomic-pair model, Phys. Rev. B 47(22), 14932 (1993)

[67] M. Cococcioni and S. de Gironcoli, Linear response approach to the calculation of the effective interaction parameters in the LDA+U method, Phys. Rev. B 71(3), 035105 (2005)

[68] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)

[69] D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82(3), 1959 (2010)

Declarations

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.15302/frontphys.2025.044210.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 52073308, 12164046, and 12304097), the Key Project of the Natural Science Program of Xinjiang Uygur Autonomous Region (Grant No. 2023D01D03), the Tianchi Distinguished Professor Research Fund of Xinjiang Uygur Autonomous Region, the Tianchi-Talent Project for Young Doctors of Xinjiang Uygur Autonomous Region (No. 51052300570), China Postdoctoral Science Foundation (Grant Nos. 2022TQ0379 and 2023M733972), Hunan Provincial Natural Science Foundation of China (Grant Nos. 2023JJ40703 and 2021JJ30864), the Fundamental Research Funds for the Central Universities of Central South University (No. 2023ZZTS0385), and the State Key Laboratory of Powder Metallurgy at Central South University. This work was carried out in part using computing resources at the High-Performance Computing Center of Central South University.

RIGHTS & PERMISSIONS

2025 Higher Education Press