Quantum Anomalous Hall Effect with Tunable Chern Numbers\\ in High-Temperature 1T-PrN$_2$ Monolayer

Xu-Cai Wu(吴绪才), Shu-Zong Li(李树宗), Jun-Shan Si(司君山),\\ Bo Huang(黄博), and Wei-Bing Zhang(张卫兵)$^{\hyperlink{s}{*}}$

doi:10.1088/0256-307X/41/5/057303

⇦Chinese Physics Letters, 2024, Vol. 41, No. 5, Article code 057303⇨ Quantum Anomalous Hall Effect with Tunable Chern Numbers in High-Temperature 1T-PrN$_2$ Monolayer Xu-Cai Wu (吴绪才), Shu-Zong Li (李树宗), Jun-Shan Si (司君山), Bo Huang (黄博), and Wei-Bing Zhang (张卫兵)* Affiliations Hunan Provincial Key Laboratory of Flexible Electronic Materials Genome Engineering, School of Physics and Electronic Sciences, Changsha University of Science and Technology, Changsha 410114, China Received 17 February 2024; accepted manuscript online 28 April 2024; published online 15 May 2024 ^*Corresponding author. Email: zhangwb@csust.edu.cn Citation Text: Wu X C, Li S Z, Si J S et al. 2024 Chin. Phys. Lett. 41 057303 Abstract Quantum anomalous Hall (QAH) insulators have highly potential applications in spintronic device. However, available candidates with tunable Chern numbers and high working temperature are quite rare. Here, we predict a 1T-PrN$_2$ monolayer as a stable QAH insulator with high magnetic transition temperature of above 600 K and tunable high Chern numbers of $C = \pm3$ from first-principles calculations. Without spin-orbit coupling (SOC), the 1T-PrN$_2$ monolayer is predicted to be a p-state Dirac half metal with high Fermi velocity. Rich topological phases depending on magnetization directions can be found when the SOC is considered. The QAH effect with periodical changes of Chern number ($\pm1$) can be produced when the magnetic moment breaks all twofold rotational symmetries in the $xy$ plane. The critical state can be identified as Weyl half semimetals. When the magnetization direction is parallel to the $z$-axis, the system exhibits high Chern number QAH effect with $C = \pm3$. Our work provides a new material for exploring novel QAH effect and developing high-performance topological devices.

DOI:10.1088/0256-307X/41/5/057303 © 2024 Chinese Physics Society Article Text Quantum anomalous Hall (QAH) effect[1-4] has attracted extensive attention due to its peculiar dissipation-free chiral edge states. Experimentally, QAH was first observed in Cr-doped (Bi,Sb)$_2$Te$_3$. However, the working temperature is only 30 mK[5] since the reduced sample quality induced by doping.[6-8] Most recently, the intrinsic QAH effect was realized in MnBi$_2$Te$_4$ thin films, but it is still far away from practical applications due to the minimal band gap of 50 meV and the low observable temperature of 1.4 K.[9,10] On the other hand, it is well known that QAH can be characterized by the Chern number. Its magnitude and sign correspond to the number and direction of the dissipation-free conducting channels on the edge. Realizing materials with high Chern numbers allow for fabrication of high performance QAH devices. Various strategies such as changing the atomic layer number, doping concentrations, and the stacking order[11-15] have been proposed for adjusting the Chern numbers. It should be noticed that the Chern number for a specified structure is generally fixed. Recently, magnetization direction dependent Chern number without altering the structures has attracted much attention. Some QAH materials, such as NiAsO$_3$,[16] YN$_2$,[17-19] and rare-earth metal dinitrides,[20,21] have been reported. However, candidates with tunable high Chern numbers are still scarce at present. Therefore, it is crucial to explore QAH materials with both tunable high Chern number and high magnetization transition temperatures. In this Letter, we propose a 1T-phase PrN$_2$ monolayer as a stable QAH insulator with tunable Chern numbers and high magnetic transition temperature. When the magnetization lies in $xy$ plane and breaks all the three twofold rotational symmetries, the QAH states with low Chern numbers ($C = \pm1$) are generated. When the magnetization is parallel to $z$-axis, the band gap increases to 101.12 meV and enters the QAH phase with high Chern numbers ($C=\pm3$). These rich topological properties are predicted in the 1T-phase PrN$_2$ monolayer that is taken as one of the candidate materials for potential applications in spintronics. Computational Details. The present calculations have been performed using the Vienna ab initio simulation package code (VASP)[22,23] within the projector augmented-wave method.[24,25] General gradient approximations in the Perdew–Burke–Ernzerhof (PBE) implementation[26] are employed as the exchange-correlation functional. After convergence test shown in Figs. S1–S3 of the Supplementary Material, the plane-wave basis with a kinetic energy cutoff of 500 eV and $\varGamma$-centered Monkhorst–Pack grids of $21\times21\times1$[27] were used in our calculation. The structure relaxation including both the atomic positions and lattice vectors is performed by the conjugate gradient scheme until the maximum force on each atom is less than 0.001 eV/Å and the total energy was converged to $1\times10^{-8}$ eV with the Gaussian smearing method. The vacuum layer is set to 20 Å to avoid unnecessary interactions between the periodic layers. During calculations, we used the recommended pseudopotentials with the electronic configuration $2s^22p^3$ for N and $5s^25p^65d^16s^2$ for Pr. The phonon calculations have been performed using the finite displacement approach, as implemented in the Phonopy code,[28] in which a $3\times3\times1$ supercell and a displacement of 0.01 Å from the equilibrium atomic positions are employed. The ab initio molecular-dynamics (AIMD) simulations are performed with a $4\times4\times1$ supercell by using a Nosé heat bath scheme in a canonical ensemble at 300 K to examine the thermal stability of PrN$_2$ monolayer. The maximally localized Wannier functions (MLWFs) constructed by the WANNIER90[29] package are employed to obtain the Berry curvatures. The Chern numbers are calculated by the VASPBERRY code[30,31] and verified by WannierTools.[32] The edge states and the anomalous Hall conductivity (AHC) are calculated with WannierTools. Structure and Stability. As shown in Figs. 1(a)–1(c), the 1T PrN$_2$ monolayer possesses a hexagonal structure with the space group of $P\bar3m1$ (No. 164, $D^3_{3d}$). Each Pr atom forms an octahedral structure with six surrounding N atoms as shown in Fig. 1(a). The optimized lattice constant is 3.98 Å. Compared to previous studies, it is smaller than 4.06 Å of LaN$_2$[33] and larger than those of NdN$_2$ (3.776 Å),[21] GdN$_2$ (3.74 Å),[34] TbN$_2$-LuN$_2$ (3.78–3.64 Å).[20] This is due to the lanthanide contraction phenomenon, where the decrease in ionic radius of the lanthanide series elements from left to right is greater than the expected data. This phenomenon is attributed to the poor shielding effect of the $4f$ electrons on the nuclear charge, coupled with the anticipated periodic trend of increasing electronegativity and nuclear charge from left to right.[35,36]

Fig. 1. Geometry and stability of 1T-PrN$_2$. The octahedral structure unit (a), side view (b) and top view (c) of PrN$_2$ monolayer. Phonon spectrum of PrN$_2$ monolayer (d). Variations of the total free energy of PrN$_2$ monolayer (e) in the ($4\times4$) supercell with respect to simulation time of AIMD at 300 K. The inset shows the final state of the structure after the AIMD simulations.

In order to confirm the stability of PrN$_2$, we first calculate the cohesive energy expressed as $E_{\rm coh} = (E_{\rm PrN_2} - E_{\rm Pr} - 2E_{\rm N})/3$ where $E_{\rm PrN_2}$, $E_{\rm Pr}$, and $E_{\rm N}$ are the total energies of per unit cell of PrN$_2$ monolayer, Pr atom, and N atom, respectively. The negative cohesion energy of $-2.35$ eV indicates that the PrN$_2$ monolayer is energetically stable. Moreover, we also investigate the phonon dispersion of the PrN$_2$ monolayer. As shown in fig. 1(c), the negative frequency is absent in the whole Brillouin zone, indicating its dynamical stability. We also performed the MD simulation at 300 K. As shown in Fig. 1(d), the total free energy of PrN$_2$ remains almost invariant during the simulation and the structure remains intact at 300 K after 6 ps. This suggests that the PrN$_2$ monolayer is thermally stable at room temperature. Clearly, the dynamical, energetic and thermal stabilities suggest that the 1T PrN$_2$ monolayer can be realized experimentally even at room temperature. Magnetic Property. We now evaluate the magnetic properties of the PrN$_2$ monolayer. To reveal the magnetic ground state of monolayer PrN$_2$, four possible different magnetic configurations (Fig. 2), including ferromagnetic (FM), Néel antiferromagnetic (NAFM), stripy antiferromagnetic (SAFM), and zigzag antiferromagnetic (ZAFM), are considered in the calculations. The total energies of the NAFM, SAFM, and ZAFM states are 528.80, 325.82, and 385.49 meV per unit cell relative to that of the FM state, respectively, indicating that the magnetic ground state of the PrN$_2$ monolayer is ferromagnetism. The FM state exhibits 3 $\mu_{\scriptscriptstyle{\rm B}}$ magnetic moment per unit cell, mostly contributed by N atoms. As the Pauling electronegativity of the N atom (3.04) is much larger than that of the Pr atom (1.13), the Pr atom donates three outermost electrons ($5d^16s^2$) to the two N atoms. According to Hund's rule and the Pauli exclusion principle, total nine electrons in the two N atoms should exhibit an electronic configuration of ($\uparrow\downarrow\ \uparrow\downarrow\ \uparrow\downarrow\ \uparrow\ \uparrow\ \uparrow$), leaving three unpaired electrons with a magnetic moment of 3 $\mu_{\scriptscriptstyle{\rm B}}$.

Fig. 2. Possible magnetic configuration of the PrN$_2$ monolayer. FM, AFM-Néel, AFM-zigzag, and AFM-stripy are shown in (a)–(d), respectively. The red and blue arrows represent the spin direction of N atoms. The crystal cells used in the calculations are also shown.

Magnetocrystalline anisotropy energy (MAE) is one of the most fundamental properties of magnetic materials. We have calculated the total energies of the PrN$_2$ monolayer with the magnetic moment in different directions with an interval of 5$^\circ$. The direction-dependent energy landscapes are shown in Fig. 3. We find that the total energy of the PrN$_2$ monolayer with magnetization direction lying in the $xy$ plane is 132 meV lower than the case of the $z$-direction. This indicates that the PrN$_2$ monolayer belongs to the category of XY magnets.

Fig. 3. Magnetization-direction-dependent magnetocrystalline anisotropy energy (a) MAE as a function of the azimuthal angle $\theta$ between the magnetization direction and the positive direction of the $x$-axis in $xy$ plane. (b) MAE as a function of the azimuthal angle $\phi$ between the magnetization direction and the positive direction of the $z$-axis in $xz$ plane.

According to the Mermin–Wagner theorem, long-range magnetic order can not be generated in two-dimensional isotropic systems. However, the finite-size effects can stabilize the magnetic order, which was recently verified in the easy-plane magnets 1T-VSe$_2$[37] and CrCl$_3$[38] to the monolayer limit. In 2D XY magnets, the Berezinskii–Kosterlitz–Thouless (BKT) phase transition occurs near the critical temperature $T_{\rm BKT}$. Below the critical temperature, the system is in the ordered phase, while it enters the disordered phase above the critical temperature. To estimate the $T_{\rm BKT}$ of the PrN$_2$ monolayer, the XY model Hamiltonian is written as \begin{equation} H = -J_1\sum_{\langle i,j\rangle}{{{\boldsymbol S}_i}\cdot{{\boldsymbol S}_j}} -J_2\sum_{\langle i,j\rangle}{{{\boldsymbol S}_i}\cdot{{\boldsymbol S}_j}} -J_3\sum_{\langle i,j\rangle}{{{\boldsymbol S}_i}\cdot{{\boldsymbol S}_j}}, \tag {1} \end{equation} where ${\boldsymbol S}_i$ and ${\boldsymbol S}_j$ are the spin vectors in the $xy$ plane at sites $i$ and $j$; $J_1$, $J_2$, and $J_3$ are parameters of the magnetic exchange interaction between nearest-neighbor (NN), next-nearest-neighbor (NNN), next-next-nearest-neighbor (3NN), respectively. The magnetic energies of FM, NAFM, SAFM, and ZAFM can be explicitly expressed as \begin{align} &E_{\rm FM}= E_0+(-3J_1 -6J_2 -3J_3)\vert{\boldsymbol S}\vert^2, \tag {2}\\ &E_{\rm NAFM}= E_0+(3J_1 -6J_2 +3J_3)\vert{\boldsymbol S}\vert^2, \tag {3}\\ &E_{\rm SAFM}= E_0+(J_1 +2J_2 -3J_3)\vert{\boldsymbol S}\vert^2, \tag {4}\\ &E_{\rm ZAFM}= E_0+(-J_1 +2J_2 +3J_3)\vert{\boldsymbol S}\vert^2. \tag {5} \end{align} With the normalized magnetization of $\vert{\boldsymbol S}\vert=1$, the values of $J_1$, $J_2$, and $J_3$ are 58.64, 11.41, and 29.49 meV, respectively. They are all positive values, leading to an FM ground state for the PrN$_2$ monolayer. The BKT transition temperature can be estimated to be $0.89J_1/k_{\scriptscriptstyle{\rm B}}$,[16,27] where $k_{\scriptscriptstyle{\rm B}}$ is the Boltzmann constant. The estimated $T_{\rm BKT}$ of PrN$_2$ is about 605 K. Additionally, utilizing the Hamiltonian given by Eq. (1) we conducted magnetic Monte Carlo simulations on a $50\times50\times1$ supercell. The temperature-dependent magnetization and specific heat are shown in Fig. 4. Several canonical MC simulations[39,40] show that the maximum in the temperature dependence of the specific heat occurs at a temperature of about 15% higher than the Kosterlitz–Thouless transition temperature. The specific heat peak at 611 K, as shown in Fig. 4, also demonstrates the high BKT temperature of PrN$_2$.

Fig. 4. Magnetic moment and heat capacity with temperature obtaining in Monte Carlo simulation, in which $M = \sqrt{(m_x^2 + m_y^2)}$ and $C_v = (\langle E^2\rangle-\langle E\rangle^2)/T^2$.

Fig. 5. Electronic band structures of the PrN$_2$ monolayer without spin-orbit coupling (SOC), and partial density of states (PDOS) of the PrN$_2$ monolayer. (a) Spin-resolved band structure of the PrN$_2$ monolayer. The Fermi level is set to 0 eV. (b) Two-dimensional Brillouin zone of the PrN$_2$ monolayer with three twofold rotational axes, and 12 Dirac cones represented by orange funnel shapes. (c) PDOSs of the N and Y atoms for the PrN$_2$ monolayer. (d) PDOSs of the $s$ and $p$ orbitals of N atoms for the PrN$_2$ monolayer.

Now, we turn our attention to the electronic structure of the PrN$_2$ monolayer. The electronic energy band structure without SOC is shown in Fig. 5(a). The spin-up channel is insulating with a large direct gap (4.54 eV) at the $\varGamma$ point, while the spin-down channel has a linear Dirac point on the $M$–$K$ path at the Fermi energy level. Due to the symmetry, there are total 12 Dirac points in the first Brillouin zone, which are protected by the twofold rotational symmetries. As shown in Fig. 5(b), the Dirac cones on $K_1$–$K_2$ and $K_4$–$K_5$, on $K_2$–$K_3$ and $K_5$–$K_6$, on $K_3$–$K_4$ and $K_6$–$K_1$ are protected by axes $C_{2x}$, $C_2'$, $C_2''$, respectively. In order to further understand the formation of Dirac points, we calculated the projected density of states (PDOS) of the monolayer PrN$_2$. As shown in Figs. 5(c) and 5(d), we can see that the state near the Fermi level mainly gets contributions from the $p$ orbitals of N atoms, which indicates that the PrN$_2$ monolayer is a $p$ state Dirac half metal (DHM). Moreover, we also predicted a high Fermi velocity[17] of $4.42 \times 10^5$ m/s. This value is comparable to that of graphene ($8.2 \times 10^5$ m/s)[41] and silicene ($5.3 \times 10^5$ m/s).[42]

Fig. 6. Band gaps as a function of the angle $\theta$. Red points and lines are gaps on $K_1$–$K_2$/$K_4$–$K_5$, green points and lines are gaps on $K_2$–$K_3$/$K_5$–$K_6$, blue points and lines are gaps on $K_3$–$K_4$/$K_6$–$K_1$, points are the values calculated by DFT and solid lines are fitted by the function ${\rm gap} = 81.74\,{\rm meV} \times \sin\phi$, where $\phi$ represents the angle between the direction of the magnetic moment and the twofold rotational axes. Different color areas correspond to the different Chern numbers ($C=\pm1$).

Next we discuss the band structure with SOC effect. Since it has easy plane with isotropic energy in the $xy$ plane, we need to study its energy band structure for different magnetic moment directions. The variation of band gap with angle $\theta$ is shown in Fig. 6, where $\theta$ represents the angle between the direction of the magnetic moment and the positive direction of the $x$-axis in $xy$ plane. We first focus on the red points calculated from DFT, which represent the band gap between $K_1$–$K_2$/$K_4$–$K_5$. The local band gaps at $\theta = 0^\circ$ and $\theta=180^\circ$ close, because of the direction of magnetic moment parallel to $C_{2x}$ axis preserving the $C_{2x}$ twofold rotational, protecting the degeneration on $K_1$–$K_2$ and $K_4$–$K_5$. As the angle $\phi$ between the magnetic moment and the $C_{2x}$ axis increases, the value of the band gap increases. It reaches the maximum (81.74 meV), when the magnetic moment is perpendicular to the $C_{2x}$ axis. It is easy to find that the values of local band gaps and angle $\phi$ satisfy the sinusoidal relationship. The local band gaps between $K_2$–$K_3$/$K_5$–$K_6$ and $K_3$–$K_4$/$K_6$–$K_1$ have similarly sinusoidal relationship with the angle between axes $C_2'$ and $C_2''$, as shown by the green lines and blue lines in Fig. 6. We can see that global band gap is largest to 40.87 meV when the magnetic moment direction is perpendicular to one of the twofold rotational axes. When the magnetic moment direction is parallel to one of the twofold rotational axes, the global band gap closes. Since it is well known that the PBE functional in generalized gradient approximation (GGA) suffers from the band-gap problem, we additionally employed the SCAN functional for our calculations. The band gaps calculated using the SCAN functional are approximately 1.3 times larger than those derived from GGA functionals. Moreover, as shown in Fig. S4(a) in the Supplementary Material, the bandgap obtained using the SCAN functional under in-plane magnetization also demonstrates a sinusoidal relationship with the angle between the magnetization direction and the twofold rotation axes. The closing and opening of the band gap are often accompanied by the topological phase transition. Thus, in order to study the topological property, the Chern numbers are computed according to \begin{align} C = \frac{1}{2\pi}\int_{\rm BZ}d^2k\varOmega(k), \tag {6} \end{align} where $\varOmega$ is the Berry curvature in the reciprocal space. The nonzero Chern numbers ($\pm1$) of the PrN$_2$ monolayer with in-plane magnetization suggest that the PrN$_2$ monolayer is in the QAH state, as shown in Fig. 6. The Chern numbers alternate the sign periodically every 60$^\circ$ as the magnetic moment direction changes. We further construct a Wannier-based tight-binding model by $p$ orbitals of N atoms to reveal the topological property of PrN$_2$ with different magnetic directions. Figures 7(a)–7(d) show the case of $\theta = 0^\circ$. The global band gap is closed on path $K_1$–$K_2$ ($K_4$–$K_5$) as shown in Fig. 7(b), since the magnetization direction preserves $C_{2x}$ symmetry. The local band gaps on path $K_2$–$K_3$ ($K_5$–$K_6$) and $K_3$–$K_4$ ($K_6$–$K_1$) are both 70.6 meV since the direction of magnetization is at a $30^\circ$ with respect to both $C_2'$ and $C_2''$. Because the crossing points at $K_1$–$K_2$/$K_3$–$K_4$ are double degeneracy and 100% spin polarization, the cross point can be recognized as Weyl-like points, which is also known as Weyl half semimetals (WHSMs).[20,43,44] We can clearly see a Fermi arc connecting the pair of gapless points in Fig. 7(c). Then, we focus on the case of $\theta=30^\circ$ and $\theta = 330^\circ$. As shown in Figs. 7(c)–7(h), they all have global band gaps as large as 40.87 meV ($40.87 \approx 81.74 \times \cos(30^\circ)$) since all three twofold rotational symmetries are broken. However, their edge states connecting the conduction and valence bands have opposite directions. In order to understand the mechanism of the topological phase transition, the Berry curvature is also investigated. As shown in Figs. 7(d), 7(h), and 7(l), the Berry curvature can be divided into three parts of $K_1$–$K_2$/$K_4$–$K_5$, $K_2$–$K_3$/$K_5$–$K_6$, and $K_3$–$K_4$/$K_6$–$K_1$. The red color in each part indicates that the contribution to Chern number is $+1$ and the blue color represents $-1$. The Chern numbers with $\theta= 30^\circ$ and $\theta= 330^\circ$ are $-1$ and $+1$ by adding them up. It is easy to see that the contributions of the three Berry curvatures to the Chern number are different only in the $K_1$–$K_2$/$K_4$–$K_5$ part, corresponding to the $C_{2x}$ axis. When the direction of magnetic moment is parallel to the $C_{2x}$ axis, it is a critical state, which corresponds to a zero contribution to the Chern number. When the magnetic moment directions are on either side of the $C_{2x}$ axis, their contributions to the Chern number in $K_1$–$K_2$/$K_4$–$K_5$ are opposite ($\pm1$), that is, the contributions to the Chern number in $K_1$–$K_2$/$K_4$–$K_5$ is $-1$ when $0 < ^\circ\theta < 180^\circ$ and $1$ when $180^\circ < \theta < 360^\circ$. For the same reason, the contributions to the Chern number in $K_2$–$K_3$/$K_5$–$K_6$ is $-1$ when $-120^\circ < \theta < 60^\circ$ and $1$ when $60^\circ < \theta < 240^\circ$, separated by the $C_2'$ axis. The contributions to the Chern number in $K_3$–$K_4/K_6$–$K_1$ is $-1$ when $120^\circ < \theta < 300^\circ$ and $1$ when $-60^\circ < \theta < 120^\circ$, separated by the $C_2''$ axis. Summing up these contributions produces a phase diagram of the Chern number, exhibiting a sign change every 60$^\circ$, consistent with Fig. 6.

Fig. 7. Band structures and topological properties for monolayer PrN$_2$ under SOC with $xy$ plane magnetization. The magnetization direction, band structures, edge states, and Berry curvatures with $\theta = 0^\circ$, $30^\circ$, $90^\circ$ are shown in (a)–(d), (e)–(h), (i)–(l), respectively.

Fig. 8. Band structures and topological properties for monolayer PrN$_2$ under SOC with out-of-plane magnetization. The band structure (a) of monolayer PrN$_2$. Berry curvature (b) of monolayer PrN$_2$. Edge state (c) of monolayer PrN$_2$. AHC $\sigma_{xy}$ of monolayer PrN$_2$.

As its small MAE, we can adjust the magnetization direction out of $xy$ plane by applying a small magnetic field. As shown in Fig. 8 for the case that the direction of the magnetic moment is pointing toward the $z$-axis. The PrN$_2$ monolayer opens up a band gap as large as 101.12 meV between the $M$–$K$ paths. Additionally, the value of band gap calculated by the SCAN functional is 132 meV as shown in Fig. S4(b) in the Supplementary Material, which is approximately 30% larger than that computed using the PBE functional. The contribution of the Berry curvature to the Chern number for all three parts is $-1$, so the PrN$_2$ monolayer enters a QAH state with high Chern number $C$ of $-3$. We can clearly see that there are three edge states connecting the conduction and valence bands as shown in Fig. 8(c). Moreover, the AHC $\sigma_{xy}$ of PrN$_2$ monolayer is shown in Fig. 8(d), which is exactly quantized to $e^2/h$. These indicate that monolayer PrN$_2$ can realize QAH effect with large band gap and high Chern number by manipulating magnetization direction. In conclusion, we have systematically investigated the structural stability, magnetic properties, electronic structure, and topological properties of monolayer PrN$_2$ by DFT calculations. It belongs to the category of XY magnets with high BKT temperature of 605 K. It is a $p$ state DHM with a high Fermi velocity of $4.42 \times 10^5$ m/s without SOC. When SOC is considered, rich topological states determined by the magnetization direction can be found. When the magnetization direction lying in the $xy$ plane, the WHSMs state and QAH state with low Chern number ($\pm1$) are formed. The size of the band gap can be precisely described by the sine function of the angle between the magnetization direction and the twofold rotational axes. When the direction of the magnetic moment turns to be out-of-plane, the large band gap (101.12 meV) and QAH state with higher Chern number ($\pm3$) can be obtained. Our study provides a new material with high magnetic transition temperature for tunable Chern number QAH state, which offers a promising platform for high-performance spintronics devices. Acknowledgments. This work was supported by National Natural Science Foundation of China (Grant No. 11874092), the Fok YingTong Education Foundation, China (Grant No. 161005), and the Science Fund for Distinguished Young Scholars of Hunan Province (Grant No. 2021JJ10039). References Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"Colloquium : Quantum anomalous Hall effectTopological insulators and superconductorsColloquium : Topological insulatorsExperimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological InsulatorImaging Dirac-mass disorder from magnetic dopant atoms in the ferromagnetic topological insulator Cr_x (Bi_0.1 Sb_0.9 )_2-x Te₃Visualization of superparamagnetic dynamics in magnetic topological insulatorsOrigin of magnetic inhomogeneity in Cr- and V-doped topological insulatorsQuantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi₂ Te₄Experimental Realization of an Intrinsic Magnetic Topological Insulator^*Quantum anomalous Hall effect with tunable Chern number in magnetic topological insulator filmQuantum Anomalous Hall Effect with Higher PlateausQuantum anomalous Hall effect with field-tunable Chern number near

Z_{2}

topological critical pointTopological phase transition from trigonal warping in van der Waals multilayersHigh Chern number quantum anomalous Hall effect tunable by stacking order in van der Waals topological insulatorsChern Number Tunable Quantum Anomalous Hall Effect in Monolayer Transitional Metal Oxides via Manipulating Magnetization OrientationYN2 monolayer: Novel p-state Dirac half metal for high-speed spintronicsQuantum anomalous Hall effect in a stable 1T-YN₂ monolayer with a large nontrivial bandgap and a high Chern numberFully spin-polarized Weyl fermions and in/out-of-plane quantum anomalous Hall effects in a two-dimensional d⁰ ferromagnetFerromagnetism with in-plane magnetization, Dirac spin-gapless semiconducting properties, and tunable topological states in two-dimensional rare-earth metal dinitridesQuantum anomalous Hall effect with a high and tunable Chern number in monolayer NdN₂Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setProjector augmented-wave methodFrom ultrasoft pseudopotentials to the projector augmented-wave methodGeneralized Gradient Approximation Made SimpleSpecial points for Brillouin-zone integrationsFirst-principles Phonon Calculations with Phonopy and Phono3pyWannier90 as a community code: new features and applicationsCircular Dichroism of Emergent Chiral Stacking Orders in Quasi-One-Dimensional Charge Density WavesWannierTools: An open-source software package for novel topological materialsMonolayer 1T-LaN2: Dirac spin-gapless semiconductor of p -state and Chern insulator with a high Chern number1T GdN2 monolayer — Spin-orbit induced magnetic Dirac semiconductor stable at room temperatureRelativistic effects in structural chemistryLanthanide and Actinide Contractions: Relativistic and Shell Structure EffectsStability and magnetism of strongly correlated single-layer

{VS}_{2}

Intrinsic 2D-XY ferromagnetism in a van der Waals monolayerMonte Carlo study of the planar spin modelMicrocanonical Monte Carlo simulations for the two-dimensional XY modelTwo-Dimensional Boron Hydride Sheets: High Stability, Massless Dirac Fermions, and Excellent Mechanical PropertiesStrain-induced self-doping in silicene and germanene from first-principlesTwo-dimensional Weyl half-semimetal and tunable quantum anomalous Hall effectTunable Weyl half-semimetals in two-dimensional iron-based materials

M FeSe (M = Tl, In, Ga)

[1]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015
[2]	Chang C Z, Liu C X, and MacDonald A H 2023 Rev. Mod. Phys. 95 011002
[3]	Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[4]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[5]	Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, and Xue Q K 2013 Science 340 167
[6]	Lee I, Kim C K, Lee J, Billinge S J L, Zhong R, Schneeloch J A, Liu T, Valla T, Tranquada J M, Gu G, and Davis J C S 2015 Proc. Natl. Acad. Sci. USA 112 1316
[7]	Lachman E O, Young A F, Richardella A, Cuppens J, Naren H R, Anahory Y, Meltzer A Y, Kandala A, Kempinger S, Myasoedov Y, Huber M E, Samarth N, and Zeldov E 2015 Sci. Adv. 1 e1500740
[8]	Qi S, Liu Z, Chang M, Gao R, Han Y, and Qiao Z 2020 Phys. Rev. B 101 241407
[9]	Deng Y, Yu Y, Shi M Z, Guo Z, Xu Z, Wang J, Chen X H, and Zhang Y 2020 Science 367 895
[10]	Gong Y, Guo J, Li J, Zhu K, Liao M, Liu X, Zhang Q, Gu L, Tang L, Feng X, Zhang D, Li W, Song C, Wang L, Yu P, Chen X, Wang Y, Yao H, Duan W, Xu Y, Zhang S C, Ma X, Xue Q K, and He K 2019 Chin. Phys. Lett. 36 076801
[11]	Jiang H, Qiao Z, Liu H, and Niu Q 2012 Phys. Rev. B 85 045445
[12]	Wang J, Lian B, Zhang H, Xu Y, and Zhang S C 2013 Phys. Rev. Lett. 111 136801
[13]	Duong L Q, Lin H, Tsai W F, and Feng Y P 2015 Phys. Rev. B 92 115205
[14]	Zeng J, Ren Y, Zhang K, and Qiao Z 2017 Phys. Rev. B 95 045424
[15]	Zhu W, Song C, Bai H, Liao L, and Pan F 2022 Phys. Rev. B 105 155122
[16]	Li Z, Han Y, and Qiao Z 2022 Phys. Rev. Lett. 129 036801
[17]	Liu Z, Liu J, and Zhao J 2017 Nano Res. 10 1972
[18]	Kong X, Li L, Leenaerts O, Wang W, Liu X J, and Peeters F M 2018 Nanoscale 10 8153
[19]	Jin L, Wang L, Zhang X, Liu Y, Dai X, Gao H, and Liu G 2021 Nanoscale 13 5901
[20]	Yu Y, Chen X, Liu X, Li J, Sanyal B, Kong X, Peeters F M, and Li L 2022 Phys. Rev. B 105 024407
[21]	Li S, Li X, Ji W, Li P, Yan S, and Zhang C 2023 Phys. Chem. Chem. Phys. 25 18275
[22]	Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15
[23]	Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[24]	Blöchl P E 1994 Phys. Rev. B 50 17953
[25]	Kresse G and Joubert D 1999 Phys. Rev. B 59 1758
[26]	Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[27]	Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188
[28]	Togo A 2023 J. Phys. Soc. Jpn. 92 012001
[29]	Pizzi G, Vitale V, Arita R, Blügel S, Freimuth F, Géranton G, Gibertini M, Gresch D, Johnson C, Koretsune T, Ibañez-Azpiroz J, Lee H, Lihm J M, Marchand D, Marrazzo A, Mokrousov Y, Mustafa J I, Nohara Y, Nomura Y, Paulatto L, Poncé S, Ponweiser T, Qiao J, Thöle F, Tsirkin S S, Wierzbowska M, Marzari N, Vanderbilt D, Souza I, Mostofi A A, and Yates J R 2020 J. Phys.: Condens. Matter 32 165902
[30]	Kim H J 2018 VASPBERRY: Berry curvature calculation code for VASP output, Zenodo
[31]	Kim S W, Kim H J, Cheon S, and Kim T H 2022 Phys. Rev. Lett. 128 046401
[32]	Wu Q, Zhang S, Song H F, Troyer M, and Soluyanov A A 2018 Comput. Phys. Commun. 224 405
[33]	Li L, Kong X, Chen X, Li J, Sanyal B, and Peeters F M 2020 Appl. Phys. Lett. 117 143101
[34]	Lin Z Z and Chen X 2020 Appl. Surf. Sci. 529 147129
[35]	Pyykkö P 1988 Chem. Rev. 88 563
[36]	Seth M, Dolg M, Fulde P, and Schwerdtfeger P 1995 J. Am. Chem. Soc. 117 6597
[37]	Zhuang H L and Hennig R G 2016 Phys. Rev. B 93 054429
[38]	Bedoya-Pinto A, Ji J R, Pandeya A K, Gargiani P, Valvidares M, Sessi P, Taylor J M, Radu F, Chang K, and Parkin S S P 2021 Science 374 616
[39]	Tobochnik J and Chester G V 1979 Phys. Rev. B 20 3761
[40]	Ota S, Ota S B, and Fahnle M 1992 J. Phys.: Condens. Matter 4 5411
[41]	Jiao Y, Ma F, Bell J, Bilic A, and Du A 2016 Angew. Chem. 128 10448
[42]	Wang Y and Ding Y 2013 Solid State Commun. 155 6
[43]	You J Y, Chen C, Zhang Z, Sheng X L, Yang S A, and Su G 2019 Phys. Rev. B 100 064408
[44]	Huan H, Xue Y, Zhao B, Bao H, Liu L, and Yang Z 2022 Phys. Rev. B 106 125404