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Topological Phase and Quantum Anomalous Hall Effect in Ferromagnetic Transition-Metal Dichalcogenides Monolayer $1T-\mathrm{V}{\mathrm{Se}}_{2}$

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## Abstract

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## 1. Introduction

## 2. Method

## 3. Result and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Side-view and (

**b**) top-view of the monolayer $1T-\mathrm{V}{\mathrm{Se}}_{2}$ lattice structure. (

**c**) Band structures given from PBE simulations. The red (blue) lines indicate the spin up (down) bands. (

**d**) Density of states (DOS) from PBE. The red (blue) region presents the spin up (down) contributions. (

**e**) Band structures and (

**f**) DOS obtained using HSE functional. The green circle indicates the Weyl point (WP).

**Figure 2.**(

**a**) HSE+SOC band structure of monolayer $1T-\mathrm{V}{\mathrm{Se}}_{2}$. The yellow region presents the continuous band gap and the green circle indicates the SOC band gap. (

**b**) Wilson loop of monolayer $1T-\mathrm{V}{\mathrm{Se}}_{2}$ with HSE functional and SOC. The red arrows highlight the sharp slope in the Wilson loop. The two crossings through the reference line (the orange dashed line) indicate that the Chern number is two ($C=2$).

**Figure 3.**(

**a**) 2D and 1D Brillouin zone of $1T-\mathrm{V}{\mathrm{Se}}_{2}$ monolayer and ribbon at the (010) edge, respectively. The blue dash lines indicate the relation between the high symmetry $k-$points in the 2D and 1D BZ. (

**b**) Band structure of $1T-\mathrm{V}{\mathrm{Se}}_{2}$ monolayer from the semi-infinite Green functions method. (

**c**) Band structure of $1T-\mathrm{V}{\mathrm{Se}}_{2}$ ribbon from the semi-infinite Green functions method. In comparison with (

**b**), four edge states (bright yellow curves) can be identified. Two of them are topological edge states as indicated by the red arrows (1) and (2). The other two edge states are topologically trivial as indicated by yellow arrows (3) and (4).

**Figure 4.**The intrinsic Hall conductivity from DFT simulations with temperature $T=200\phantom{\rule{0.222222em}{0ex}}K$. (

**a**) The Hall conductivity of spin up and down electrons from PBE simulations. The left (right) axis show the Hall conductivity in the unit $\frac{{e}^{2}}{2\pi \hslash}$ (${10}^{-4}\phantom{\rule{0.222222em}{0ex}}{\Omega}^{-1}$). (

**b**,

**c**) The Hall conductivity of spin up and down electrons from HSE simulations. The green arrow highlights the peak enhanced by the band crossing point.

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**MDPI and ACS Style**

Huang, A.; Chen, C.-H.; Chang, C.-H.; Jeng, H.-T.
Topological Phase and Quantum Anomalous Hall Effect in Ferromagnetic Transition-Metal Dichalcogenides Monolayer *Nanomaterials* **2021**, *11*, 1998.
https://doi.org/10.3390/nano11081998

**AMA Style**

Huang A, Chen C-H, Chang C-H, Jeng H-T.
Topological Phase and Quantum Anomalous Hall Effect in Ferromagnetic Transition-Metal Dichalcogenides Monolayer *Nanomaterials*. 2021; 11(8):1998.
https://doi.org/10.3390/nano11081998

**Chicago/Turabian Style**

Huang, Angus, Chin-Hsuan Chen, Ching-Hao Chang, and Horng-Tay Jeng.
2021. "Topological Phase and Quantum Anomalous Hall Effect in Ferromagnetic Transition-Metal Dichalcogenides Monolayer *Nanomaterials* 11, no. 8: 1998.
https://doi.org/10.3390/nano11081998