Electric-field-tunable topological phases in valley-polarized quantum anomalous Hall systems with inequivalent exchange fields
Shiyao Pan, Zeyu Li, Yulei Han
Electric-field-tunable topological phases in valley-polarized quantum anomalous Hall systems with inequivalent exchange fields
Incorporating valley as a degree of freedom into quantum anomalous Hall systems offers a novel approach to manipulating valleytronics in electronic transport. Using the Kane−Mele monolayer as a concrete model, we comprehensively explore the various topological phases in the presence of inequivalent exchange fields and reveal the roles of the interfacial Rashba effect and external electric field in tuning topological valley-polarized states. We find that valley-polarized states can be realized by introducing Kane−Mele spin−orbit coupling and inequivalent exchange fields. Further introducing Rashba spin−orbit coupling and an electric field into the system can lead to diverse topological states, such as the valley-polarized quantum anomalous Hall effect with and valley-contrasting states with . Remarkably, different valley-polarized topological states can be continuously tuned by varying the strength and direction of the external electric field in a fixed system. Our work demonstrates the tunability of topological states in valley-polarized quantum anomalous Hall systems and provides an ideal platform for applications in electronic transport devices in topological valleytronics.
quantum anomalous Hall effect / topological phase transition / valleytronics / valley-polarized / topological states
[1] |
K.von KlitzingT.ChakrabortyP.Kim V.MadhavanX. DaiJ.McIverY.TokuraL.Savary D.SmirnovaA. M. ReyC.FelserJ.GoothX.Qi, 40 years of the quantum Hall effect, Nat. Rev. Phys. 2(8), 397 (2020)
|
[2] |
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. denNijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)
CrossRef
ADS
Google scholar
|
[3] |
F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61(18), 2015 (1988)
CrossRef
ADS
Google scholar
|
[4] |
C. Z. Chang, C. X. Liu, and A. H. MacDonald, Quantum anomalous Hall effect, Rev. Mod. Phys. 95(1), 011002 (2023)
CrossRef
ADS
arXiv
Google scholar
|
[5] |
K. He, Y. Wang, and Q. K. Xue, Topological materials: Quantum anomalous Hall system, Annu. Rev. Condens. Matter Phys. 9(1), 329 (2018)
CrossRef
ADS
Google scholar
|
[6] |
Y. Ren, Z. Qiao, and Q. Niu, Topological phases in two-dimensional materials: A review, Rep. Prog. Phys. 79(6), 066501 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[7] |
M. Onoda and N. Nagaosa, Quantized anomalous Hall effect in two-dimensional ferromagnets: Quantum Hall effect in metals, Phys. Rev. Lett. 90(20), 206601 (2003)
CrossRef
ADS
Google scholar
|
[8] |
C. Liu, X. Qi, X. Dai, Z. Fang, and S. C. Zhang, Quantum anomalous Hall effect in Hg1−yMnyTe quantum wells, Phys. Rev. Lett. 101(14), 146802 (2008)
CrossRef
ADS
arXiv
Google scholar
|
[9] |
C. Wu, Orbital analogue of the quantum anomalous Hall effect in p-band systems, Phys. Rev. Lett. 101(18), 186807 (2008)
CrossRef
ADS
arXiv
Google scholar
|
[10] |
R. Yu, W. Zhang, H. Zhang, S. C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall effect in magnetic topological insulators, Science 329(5987), 61 (2010)
CrossRef
ADS
arXiv
Google scholar
|
[11] |
Z.QiaoS. YangW.FengW.K. TseJ.Ding Y.YaoJ. WangQ.Niu, Quantum anomalous Hall effect in graphene from Rashba and exchange effects, Phys. Rev. B 82, 161414(R) (2010)
|
[12] |
Z. Wang, Z. Liu, and F. Liu, Quantum anomalous Hall effect in 2D organic topological insulators, Phys. Rev. Lett. 110(19), 196801 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[13] |
K. F. Garrity and D. Vanderbilt, Chern insulators from heavy atoms on magnetic substrates, Phys. Rev. Lett. 110(11), 116802 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[14] |
J. Hu, Z. Zhu, and R. Wu, Chern half metals: A new class of topological materials to realize the quantum anomalous Hall effect, Nano Lett. 15(3), 2074 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[15] |
C. Fang, M. J. Gilbert, and B. A. Bernevig, Large-Chern-number quantum anomalous Hall effect in thin-film topological crystalline insulators, Phys. Rev. Lett. 112(4), 046801 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[16] |
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)
CrossRef
ADS
arXiv
Google scholar
|
[17] |
G. Zhang, Y. Li, and C. Wu, Honeycomb lattice with multiorbital structure: Topological and quantum anomalous Hall insulators with large gaps, Phys. Rev. B 90(7), 075114 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[18] |
H. Lu, A. Zhao, and S. Shen, Quantum transport in magnetic topological insulator thin films, Phys. Rev. Lett. 111(14), 146802 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[19] |
C. Liu, X. Qi, X. Dai, Z. Fang, and S. C. Zhang, Quantum anomalous Hall effect in Hg1−yMnyTe quantum wells, Phys. Rev. Lett. 101(14), 146802 (2008)
CrossRef
ADS
arXiv
Google scholar
|
[20] |
J. Ding, Z. Qiao, W. Feng, Y. Yao, and Q. Niu, Engineering quantum anomalous/valley Hall states in graphene via metal-atom adsorption: An ab-initio study, Phys. Rev. B 84(19), 195444 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[21] |
M. Ezawa, Valley-polarized metals and quantum anomalous Hall effect in silicene, Phys. Rev. Lett. 109(5), 055502 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[22] |
H. Zhang C. Lazo, S. Blügel, S. Heinze , Y. Mokrousov., Electrically tunable quantum anomalous Hall effect in graphene decorated by 5d transition-metal adatoms, Phys. Rev. Lett. 108(5), 056802 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[23] |
Z. Qiao, W. Ren, H. Chen, L. Bellaiche, Z. Zhang, A. H. MacDonald, and Q. Niu, Quantum anomalous Hall effect in graphene proximity coupled to an antiferromagnetic insulator, Phys. Rev. Lett. 112(11), 116404 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[24] |
S. C. Wu, G. Shan, and B. Yan, Prediction of near-room-temperature quantum anomalous Hall effect on honeycomb materials, Phys. Rev. Lett. 113(25), 256401 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[25] |
S. M. Huang, S. T. Lee, and C. Y. Mou, Ferromagnetism and quantum anomalous Hall effect in one-side-saturated buckled honeycomb lattices, Phys. Rev. B 89(19), 195444 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[26] |
H. Pan, Z. Li, C. C. Liu, G. Zhu, Z. Qiao, and Y. Yao, Valley-polarized quantum anomalous Hall effect in silicene, Phys. Rev. Lett. 112(10), 106802 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[27] |
Y. Zhao, R. Zhang, R. Mei, L. Zhou, H. Yi, Y. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth, M. H. W. Chan, C. Liu, and C. Z. Chang, Tuning the Chern number in quantum anomalous Hall insulators, Nature 588(7838), 419 (2020)
CrossRef
ADS
arXiv
Google scholar
|
[28] |
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)
CrossRef
ADS
arXiv
Google scholar
|
[29] |
J. E. Yang and H. Xie, Energy-resolved spin filtering effect and thermoelectric effect in topological-insulator junctions with anisotropic chiral edge states, Front. Phys. 17(6), 63504 (2022)
CrossRef
ADS
Google scholar
|
[30] |
Y. Han, Z. Yan, Z. Li, X. Xu, Z. Zhang, Q. Niu, and Z. H. Qiao, Large Rashba spin−orbit coupling and high-temperature quantum anomalous Hall effect in Re-intercalated graphene/CrI3 heterostructure, Phys. Rev. B 107, 205412 (2023)
CrossRef
ADS
arXiv
Google scholar
|
[31] |
L. Li, M. Wu, and X. Lu, Correlation, superconductivity and topology in graphene moiré superlattice, Front. Phys. 18(4), 43401 (2023)
CrossRef
ADS
Google scholar
|
[32] |
Y. Wang, F. Zhang, M. Zeng, H. Sun, Z. Hao, Y. Cai, H. Rong, C. Zhang, C. Liu, X. Ma, L. Wang, S. Guo, J. Lin, Q. Liu, C. Liu, and C. Chen, Intrinsic magnetic topological materials, Front. Phys. 18(2), 21304 (2023)
CrossRef
ADS
arXiv
Google scholar
|
[33] |
P. Deng, Y. Han, P. Zhang, S. K. Chong, Z. Qiao, and K. L. Wang, Tuning the number of chiral edge channels in a fixed quantum anomalous Hall system, Phys. Rev. B 109(20), L201402 (2024)
CrossRef
ADS
Google scholar
|
[34] |
S. Li, K. Wei, Q. Liu, Y. Tang, and T. Jiang, Twistronics and moiré excitonic physics in van der Waals heterostructures, Front. Phys. 19(4), 42501 (2024)
CrossRef
ADS
Google scholar
|
[35] |
D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics in graphene: Magnetic moment and topological transport, Phys. Rev. Lett. 99(23), 236809 (2007)
CrossRef
ADS
arXiv
Google scholar
|
[36] |
I. Martin, Y. M. Blanter, and A. F. Morpurgo, Topological confinement in bilayer graphene, Phys. Rev. Lett. 100(3), 036804 (2008)
CrossRef
ADS
arXiv
Google scholar
|
[37] |
R. V. Gorbachev, J. C. W. Song, G. L. Yu, A. V. Kretinin, F. Withers, Y. Cao, A. Mishchenko, I. V. Grigorieva, K. S. Novoselov, L. S. Levitov, and A. K. Geim, Detecting topological currents in graphene superlattices, Science 346(6208), 448 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[38] |
Z. Qiao, W. K. Tse, H. Jiang, Y. Yao, and Q. Niu, Two-dimensional topological insulator state and topological phase transition in bilayer graphene, Phys. Rev. Lett. 107(25), 256801 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[39] |
J. Zeng, R. Xue, T. Hou, Y. Han, and Z. Qiao, Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems, Front. Phys. 17(6), 63503 (2022)
CrossRef
ADS
Google scholar
|
[40] |
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. 18(5), 53302 (2023)
CrossRef
ADS
Google scholar
|
[41] |
S. D. Guo, Y. L. Tao, G. Wang, S. Chen, D. Huang, and Y. S. Ang, Proposal for valleytronic materials: Ferrovalley metal and valley gapless semiconductor, Front. Phys. 19(2), 23302 (2024)
CrossRef
ADS
Google scholar
|
[42] |
J. Tang, S. Wang, and H. Yu, Inheritance of the exciton geometric structure from Bloch electrons in two-dimensional layered semiconductors, Front. Phys. 19(4), 43210 (2024)
CrossRef
ADS
arXiv
Google scholar
|
[43] |
W. Xun, C. Wu, H. Sun, W. Zhang, Y. Wu, and P. Li, Coexisting magnetism, ferroelectric, and ferrovalley multiferroic in stacking-dependent two-dimensional materials, Nano Lett. 24(11), 3541 (2024)
CrossRef
ADS
Google scholar
|
[44] |
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)
CrossRef
ADS
Google scholar
|
[45] |
P. Li, B. Liu, S. Chen, W. Zhang, and Z. Guo, Progress on two-dimensional ferrovalley materials, Chin. Phys. B 33(1), 017505 (2024)
CrossRef
ADS
Google scholar
|
[46] |
H. Pan, X. Li, H. Jiang, Y. Yao, and S. A. Yang, Valley-polarized quantum anomalous Hall phase and disorder-induced valley-filtered chiral edge channels, Phys. Rev. B 91(4), 045404 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[47] |
Z. Yu, H. Pan, and Y. Yao, Electric field controlled spin- and valley-polarized edge states in silicene with extrinsic Rashba effect, Phys. Rev. B 92(15), 155419 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[48] |
C. C. Liu, J. J. Zhou, and Y. Yao, Valley-polarized quantum anomalous Hall phases and tunable topological phase transitions in half-hydrogenated Bi honeycomb monolayers, Phys. Rev. B 91(16), 165430 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[49] |
C.NiuG. BihlmayerH.ZhangD.WortmannS.BlügelY.Mokrousov, Functionalized bismuth films: Giant gap quantum spin Hall and valley-polarized quantum anomalous Hall states, Phys. Rev. B 91, 041303(R) (2015)
|
[50] |
J. Zhou, Q. Sun, and P. Jena, Valley-polarized quantum anomalous Hall effect in ferromagnetic honeycomb lattices, Phys. Rev. Lett. 119(4), 046403 (2017)
CrossRef
ADS
Google scholar
|
[51] |
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)
CrossRef
ADS
Google scholar
|
[52] |
H. Ai, D. Liu, J. Geng, S. Wang, K. H. Lo, and H. Pan, Theoretical evidence of the spin–valley coupling and valley polarization in two-dimensional MoSi2X4 (X = N, P, and As), Phys. Chem. Chem. Phys. 23(4), 3144 (2021)
CrossRef
ADS
Google scholar
|
[53] |
M. U. Rehman, Z. Qiao, and J. Wang, Valley-symmetry-broken magnetic topological responses in (Pt/Pd)2HgSe3/CrGeTe3 and Pd2HgSe3/CrI3 through interfacial coupling, Phys. Rev. B 105(16), 165417 (2022)
CrossRef
ADS
Google scholar
|
[54] |
F. Zhan, Z. Ning, L. Y. Gan, B. Zheng, J. Fan, and R. Wang, Floquet valley-polarized quantum anomalous Hall state in nonmagnetic heterobilayers, Phys. Rev. B 105(8), L081115 (2022)
CrossRef
ADS
arXiv
Google scholar
|
[55] |
Y. M. Xie, C. P. Zhang, J. X. Hu, K. F. Mak, and K. T. Law, Valley-polarized quantum anomalous Hall state in moiré MoTe2/WSe2 heterobilayers, Phys. Rev. Lett. 128(2), 026402 (2022)
CrossRef
ADS
arXiv
Google scholar
|
[56] |
F. Zhan, J. Zeng, Z. Chen, X. Jin, J. Fan, T. Chen, and R. Wang, Floquet engineering of nonequilibrium valley-polarized quantum anomalous Hall effect with tunable Chern number, Nano Lett. 23(6), 2166 (2023)
CrossRef
ADS
arXiv
Google scholar
|
[57] |
S. D. Guo, Y. L. Tao, W. Q. Mu, and B. G. Liu, Correlation-driven threefold topological phase transition in monolayer OsBr2, Front. Phys. 18(3), 33304 (2023)
CrossRef
ADS
Google scholar
|
[58] |
P. Li, X. Yang, Q. Jiang, Y. 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)
CrossRef
ADS
Google scholar
|
[59] |
A. Marrazzo, M. Gibertini, D. Campi, N. Mounet, and N. Marzari, Prediction of a large-gap and switchable Kane-Mele quantum spin Hall insulator, Phys. Rev. Lett. 120(11), 117701 (2018)
CrossRef
ADS
arXiv
Google scholar
|
[60] |
A. Marrazzo, M. Gibertini, D. Campi, N. Mounet, and N. Marzari, Relative abundance of Z2 topological order in exfoliable two-dimensional insulators, Nano Lett. 19(12), 8431 (2019)
CrossRef
ADS
arXiv
Google scholar
|
[61] |
I. Cucchi, A. Marrazzo, E. Cappelli, S. Ricco’, F. Y. Bruno, S. Lisi, M. Hoesch, T. K. Kim, C. Cacho, C. Besnard, E. Giannini, N. Marzari, M. Gibertini, F. Baumberger, and A. Tamai, Bulk and surface electronic structure of the dual-topology semimetal Pt2HgSe3, Phys. Rev. Lett. 124(10), 106402 (2020)
CrossRef
ADS
arXiv
Google scholar
|
[62] |
Z. Liu, Y. Han, Y. Ren, Q. Niu, and Z. Qiao, Van der Waals heterostructure Pt2HgSe3/CrI3 for topological valleytronics, Phys. Rev. B 104(12), L121403 (2021)
CrossRef
ADS
Google scholar
|
[63] |
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. 18(2), 23302 (2023)
CrossRef
ADS
Google scholar
|
[64] |
C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95(22), 226801 (2005)
CrossRef
ADS
Google scholar
|
[65] |
D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82(3), 1959 (2010)
CrossRef
ADS
arXiv
Google scholar
|
[66] |
D.CeresoliT. ThonhauserD.VanderbiltR.Resta, Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals, Phys. Rev. B 74(2), 024408 (2006)
|
[67] |
See Supplemental Material at https://doi.org/10.15302/frontphys.2025.014207 for more information about the origin of difference in phase diagrams between points B and C, the two contributions of orbital magnetization and the analysis of slope dMz/dµ.
|
[68] |
C. C. Liu, H. Jiang, and Y. Yao, Low-energy effective Hamiltonian involving spin-orbit coupling in silicene and two-dimensional germanium and tin, Phys. Rev. B 84(19), 195430 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[69] |
P. Wang, L. Lu, and K. Bertoldi, Topological phononic crystals with one-way elastic edge waves, Phys. Rev. Lett. 115(10), 104302 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[70] |
S. Mousavi, A. Khanikaev, and Z. Wang, Topologically protected elastic waves in phononic metamaterials, Nat. Commun. 6(1), 8682 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[71] |
A. B. Khanikaev and G. Shvets, Two-dimensional topological photonics, Nat. Photonics 11(12), 763 (2017)
CrossRef
ADS
Google scholar
|
[72] |
S. Mittal, V. V. Orre, D. Leykam, Y. D. Chong, and M. Hafezi, Photonic anomalous quantum Hall effect, Phys. Rev. Lett. 123(4), 043201 (2019)
CrossRef
ADS
arXiv
Google scholar
|
/
〈 | 〉 |