DOI: 10.1039/D4TC02809K
(Paper)
J. Mater. Chem. C, 2024, Advance Article

Haofeng Wei^{a},
Yanzhao Wu^{a},
Junwei Tong^{b},
Li Deng^{a},
Xiang Yin^{a},
Zhijun Zhang^{c} and
Xianmin Zhang*^{a}
^{a}Key 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
^{b}Department of Physics, Freie Universität Berlin, Berlin, 14195, Germany
^{c}Liaoning Institute of Science and Technology, Benxi 117004, China

Received
2nd July 2024
, Accepted 20th August 2024

First published on 3rd September 2024

The search for high-performance intrinsic quantum anomalous Hall (QAH) insulators is crucial for the development of topological electronics. Here, based on density–functional theory calculations, TaPdSTe and TaPdSeTe monolayers are demonstrated to be intrinsic QAH insulators with topological band gaps of 88 and 79 meV, respectively. Both TaPdSTe and TaPdSeTe monolayers show an out-of-plane magnetic anisotropy with the Curie temperatures of 260 and 262 K by Monte Carlo simulations, respectively. The calculated Chern number C for both materials is −1. The analysis of the electronic structures reveals that the ferromagnetic topological property is caused by the energy band inversions of d_{xz} and d_{yz} orbitals of Ta atoms. Additionally, the effects of biaxial strain on the magnetic and topological properties are discussed for the current TaPdSTe and TaPdSeTe monolayers. During −3 to 3% biaxial strains, TaPdSTe and TaPdSeTe monolayers maintain the QAH effect, but their topological band gaps increase gradually from compressive to tensile strains. This study presents two intrinsic topological insulators that can help develop low-power electronic devices.

Luckily, the spontaneous QAH effect has been theoretically predicted^{21–23} and experimentally verified in MnBi_{2}Te_{4} 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 MnBr_{3}^{27} monolayers with in-plane magnetic anisotropy. The QAH effect is reported in RuI_{3},^{6} PdBr_{3},^{28} and PtBr_{3}^{28} monolayers with small topological band gaps. Meanwhile, the high Chern number of 3 is demonstrated in the YN_{2} monolayer with in-plane magnetic anisotropy^{29} and Co_{3}Pb_{3}X_{2} (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.

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 effect^{46} 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) |

E_{FM} = E_{0} − (4J_{1} + 4J_{2} + 8J_{3})|S|^{2} − A|S|^{2} |

E_{AFM1} = E_{0} − (4J_{1} − 4J_{2} − 8J_{3})|S|^{2} − A|S|^{2} |

E_{AFM2} = E_{0} − (−4J_{1} − 4J_{2} + 8J_{3})|S|^{2} − A|S|^{2} |

E_{AFM3} = E_{0} − (−4J_{1} + 4J_{2} − 8J_{3})|S|^{2} − A|S|^{2}
| (2) |

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) |

(4) |

(5) |

Fig. 4 (a) Berry curvature, (b) AHC, and (c) chiral edge state for the TaPdSTe monolayer. (d) Berry curvature, (e) AHC, and (f) chiral edge state for the TaPdSeTe monolayer. |

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 −e^{2}/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†).

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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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