Prediction of two-dimensional Dirac materials with intrinsic magnetism, quantum anomalous Hall effect and high Curie temperature†

Xuming Wu ^ab, Chunhua Tian ^a, Lanhua Zhong ^a, Jun Quan ^a, Jie Yang ^a, Zhibin Shao *^d and Guoying Gao *^bc
aCollege of Physics Science and Technology, Lingnan Normal University, 524048 Zhanjiang, China
^bSchool of Physics, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail: guoying_gao@mail.hust.edu.cn
^cWuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China
^dPhysics Laboratory, Industrial Training Center, Shenzhen Polytechnic University, Shenzhen 518055, China. E-mail: zhibin_shao@szpt.edu.cn

First published on 30th July 2024

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1. Introduction

In recent years, two-dimensional (2D) materials have attracted considerable attention owing to their rich physical properties, such as spintronics,¹ valleytronics,^2,3 multiferroics,⁴ and photoelectrics.⁵ Many 2D materials enjoy the emergence of Dirac/Weyl fermions, and 2D Dirac materials such as graphene,⁶ which are characterized by linear energy dispersion at the Fermi level, can lead to half-integer/fractional/fractal quantum Hall effects (QHEs),^7–9 ultra-high carrier mobility,¹⁰ and many other novel physics. Interestingly, unlike conventional 2D nonmagnetic topological insulators, intrinsic 2D ferromagnetic materials exhibit broken time-reversal symmetry (TRS) via internal magnetization under spin–orbit coupling (SOC),¹¹ leading to the quantum anomalous Hall effect (QAHE) without an external magnetic field. In addition, chiral edge states are topologically protected and robust against scattering, providing a new way to fabricate topologically protected spin current. Thus, the QAHE would have great potential applications for designing low energy consumption and dissipation-less spintronic devices.

To date, many candidates have been predicted to exhibit the characteristics of the QAHE.^12–14 Excitingly, both the quantum spin Hall effect (QSHE) and QAHE have been confirmed in real materials,^11,15 but the QAHE was only realized in few quantum well systems at very low temperatures, such as the plateau of Hall conductance in V- or Cr-doped (Bi, Sb)₂Te₃ systems (<2 K);^11,16–19 magnetic disorder is very difficult to control by the magnetic doping approach, and the edge states are greatly affected by magnetic disorder. Thus, experimental conditions usually require an extremely low temperature in order to suppress magnetic disorders. In addition, the greatly accurate control of extrinsic impurities is required. Therefore, in order to overcome the shortcomings of magnetic disorder and doping concentration, intrinsic magnetic topological materials are desired. Very recently, the layered van der Waals compound MnBi₂Te₄ has been theoretically predicted and experimentally verified to host the intrinsic topologically antiferromagnetic state,^20,21 which has a large topologically nontrivial energy gap (∼0.2 eV). QAH plateaus can be discovered in few-layer MnBi₂Te₄ films at a record-high temperature of 4.5 K. In addition, the QAH insulator state was predicted in VSiGeN₄ monolayers due to the built-in electric field and strain.²² To date, most studies have mainly focused on the QAH states with low Chern number |C| = 1, and QAH insulators with |C| > 1 have been rarely reported. Remarkably, QAH insulators with high Chern numbers (|C| > 1) exhibit rich topological phases, which are expected to bring about new fundamental physics and potential applications. For example, the C-terminated 4H-SiC (0001) theoretically proposed has the QAHE with the Chern number C = 2, which does not need additional magnetic doping.²³ These findings inspired us to search new 2D intrinsic ferromagnetic QAH insulators with finite Chern number, large spin band gap, high Curie temperature (T_C), and high feasibility of topological phase transition.

In a pioneering theoretical study, Yong and Kane used the symmetry analysis and tight-binding model to reveal the possibility that Dirac points cannot be gapped by the SOC.²⁴ Later, Yao et al. proposed a realistic 2D HfGeTe monolayer, which hosts the so-called spin–orbit Dirac points (SDPs),²⁵ showing that SDPs are intrinsically robust against the SOC. A question is raised: can we obtain a pair of Dirac cones with the opposite spin channels, which are intrinsically robust against SOC in 2D magnetic materials? In this work, we propose a model to meet this requirement, as shown in Fig. 1. Without the SOC, two Dirac points are located at the Γ and K points in different spin channels, respectively. Importantly, two Dirac points cannot open band gap or only open a tiny gap within the SOC, and both of them show weak SOC strength; thus, we can get a pair of SDPs combining the feature of spin polarization, which can be called double spin–orbit Dirac points (DSDPs).

Previous calculations focused on the PC₆ monolayer have received much attention due to their anisotropic carrier mobility and excellent stability;²⁶ it was also predicted as an ideal anode candidate for potassium-ion batteries. A logical extension is to explore whether it is possible to form other stable XC₆ (X ≠ P atom) monolayers. In the present work, we predict a novel family of 2D crystals in transition metal carbides XC₆ (X = V, Nb, Cu) monolayers with hexagonal lattices. As expected, XC₆ monolayers have a buckled graphene-like structure, which are highly similar to the PC₆ monolayer. All of them show highly dynamic and thermodynamic stabilities. Further investigations of ferromagnetic and topological properties reveal that: (i) Without SOC, VC₆ and NbC₆ monolayers are ferromagnetic Dirac half-metals with high T_C (above 400 K), and CuC₆ is a nonmagnetic Dirac semimetal. (ii) Remarkably, VC₆ and NbC₆ monolayers have nontrivial topological states and high Chern numbers (C = −3, 5) under the SOC effect, which are expected to possess the QAHE. CuC₆ monolayer shows the QSH effect with the topological invariant Z₂ = 1 and a large band gap up to 45 meV. In general, our findings would expand the realm of topological states and open a new avenue to the fantastic integration of Dirac physics and spintronics.

2. Computational details

The electronic, magnetic, and topological properties of XC₆ were studied using the projector augmented wave (PAW) formalism in the Vienna ab initio simulation package (VASP).^27,28 We have used the generalized gradient approximation (GGA) in the Perdew–Burke and Ernzerhof (PBE) form for the exchange correlation potential. The cutoff of the plane-wave kinetic energy and the convergence of total energy were set to be 450 eV and 10⁻⁶ eV, respectively. A vacuum region larger than 20 Å was applied along the z axis to eliminate the interactions between neighboring layers. The Brillouin zone (BZ) integration was sampled using a 15 × 15 × 1 Monkhorst–Pack k-grid. Furthermore, the SOC effect and the screened exchange hybrid density functional by Heyd–Scuseria–Ernzerhof (HSE06) were adopted to check the electronic structure.²⁹ To determine the topological properties of XC₆ monolayers, an effective tight-binding (TB) Hamiltonian was constructed from the maximally localized Wannier function (MLWF), and the Berry curvature was calculated by the WANNIER90 package.^30,31 With the MLWFs, the topological properties were calculated using the software package of WannierTools.³² Phonon dispersion calculations were based on the supercell approach as implemented in the Phonopy code.³³ First-principles molecular dynamics (MD) simulations in the NVT ensemble lasted for 10 ps with a time step of 1.0 fs using the Nosé–Hoover method.³⁴

3. Results and discussion

The 2D buckled structure of the XC₆ monolayer is shown in Fig. 2; the structure is assumed to be similar to the hexagonal lattice of the PC₆ monolayer,²⁶ where the metal atoms X are bonded with three C atoms. In addition, we can see that the basic building blocks are two types of six-membered rings, in which the six edge-sharing XC₅ rings surround a middle C6 ring. The relaxed lattice constants are 6.839, 6.882, and 6.862 Å for VC₆, NbC₆ and CuC₆ monolayers (see Table 1) with layer thicknesses of 1.935, 2.693 and 2.145 Å, respectively.

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In order to assess the stability of XC₆ monolayers, we evaluate the formation energy, ΔE_f = [E(XC₆) − E(TM) – 6E(C)]/(1+6), where E(XC6), E(TM) and E(C) are the total energies of the XC₆ monolayer, V/Nb/Cu atom in the cubic crystal and C atom in the cubic crystal, respectively. The calculated formation energies are about −1.543, −1.353 and −1.506 eV per atom for VC₆, NbC₆ and CuC₆, respectively. These values are comparable or lower than those of some monolayer (Dirac) materials like FeC₆ (−0.56 eV per atom),³⁵ Be₃C₂ (−0.99 eV per atom),³⁶ InB₆ (−1.20 eV per atom)³⁷ and SbAsF₂ (−1.25 eV per atom).³⁸ Thus, the negative formation energies for XC₆ monolayers indicate the possibility to realize them experimentally.

In order to reveal the chemical bonding nature of XC₆ monolayers, their charge density difference projection onto an in-plane (ΔQ) and the electron localization function (ELF) were calculated. Taking the VC₆ monolayer as an example for analysis, the charge density difference can be defined as ΔQ = Q(VC₆) − Q(V) − Q(C₆). Fig. 3(a) shows abundant electrons around the C atoms and deficient electrons around the V atoms. The electron deficiency is mainly focused on the V atoms, and the electron accumulation is mainly localized in the C–C bonding regions. The projection line (see Fig. 3(b)) also confirms that the charge density difference along the z-direction of the V(C) atom increases (decreases), indicating that electrons are transferred from the V atom to the C atom. As a helpful method for the Bader charge analysis,^39,40 we find that the less electronegative V atoms lose 1.21 electrons per atom, and the more electronegative C atoms obtain 0.20 electrons per atom. Thus, VC₆ can be written as V^+1.21(C^−0.20)₆. The covalent bonding in the VC₆ monolayer can be further confirmed by the analysis of the ELF in Fig. 3(c) and (d). The average length of the C–C and V–C bonds in the VC₆ monolayer are 1.41 Å and 1.85 Å, respectively, which are comparable to those of the C–C bond in graphene (1.42 Å) and sp³ P–C bond length (1.81 Å) in the PC₆ monolayer;²⁶ these values indicate that C and C atoms have paired electrons with the local bosonic character (σ bond), and the strong covalent electron states are formed by the sp²-hybridized orbitals. The value 0.00 in V atoms refers to very low charge density, indicating that V atoms adopt a sp³ hybridization with C atoms, and the hybrid orbitals are filled with a lone electron pair. The above bonding configurations not only obey the chemical octet rule but can also enhance the structural stability. Note that the similar bonding configurations in NbC₆ and CuC₆ monolayers are presented in Fig. S1 (ESI†). In order to assess the dynamic stability of the XC₆ monolayer, we perform phonon spectrum calculations, and the phonon dispersions show that there are no imaginary frequencies (Fig. S2, ESI†), confirming the dynamic stability. We then calculate the thermal stability of the XC₆ monolayer by performing AIMD simulations. From Fig. S3 (ESI†), it can be seen that the average value of the total potential energy remains nearly constant during the entire simulations, and the structures nearly maintain the initial hexagonal honeycomb structure and have no obvious structure collapse in 300 K (Fig. S4, ESI†). These results demonstrate that XC₆ monolayers possess good thermal stability for room-temperature spintronic applications.

The mechanical properties of the XC₆ monolayer were also investigated by examining its elastic constants. The elastic constants were computed to be C₁₁ = 117.24/135.24/170.70 N m⁻¹, C₂₂ = 117.17/135.24/171.84 N m⁻¹, C₁₂ = 54.55/42.97/30.03 N m⁻¹ and C₄₄ = 11.98/46.15/65.61 N m⁻¹ for VC₆/NbC₆/CuC₆ monolayers, respectively. Evidently, the elastic constants satisfy the Born–Huang criteria C₁₁ > 0, C₂₂ > 0, C₄₄ > 0 and C₁₁C₂₂ > C₁₂², implying the mechanical stability. It is noted that the elastic coefficients of XC₆ monolayers are comparable to those of MoS₂ sheets (C₁₁ = 132.7 N m⁻¹, C₁₂ = 33.0 N m⁻¹).⁴¹ Based on these elastic constants, the corresponding in-plane Young's modulus E(θ) and Poisson ratio v along an arbitrary direction θ (θ is the angle relative to the positive x direction in the sheet) can be expressed as⁴²

	(1)

	(2)

where c = cosθ and s = sinθ. The polar diagrams of E(θ) and ν(θ) of the XC₆ monolayers are illustrated in Fig. 4. One can observe that both the in-plane Young's modulus and the Poisson ratio have very small anisotropy. Taking the CuC₆ monolayer as an example (Fig. 4(c)), the maximum Young's modulus and the minimum Poisson ration along the armchair/zigzag direction are 160 and 0.20 N m⁻¹, respectively, the E and ν range from 163 to 164 N m⁻¹ and 0.2001 to 0.2057 (Fig. 4), respectively, and the Young's modulus is smaller than that of graphene (340 N m⁻¹)⁴³ and hexagonal-BN monolayer (271 N m⁻¹),⁴⁴ suggesting that the CuC₆ monolayer has favorable mechanical flexibility. Among these systems, the Poisson's ratios of XC₆ monolayers are between 0.2 and 0.4, which are close to those of most previously reported 2D materials,⁴⁵ suggesting that XC₆ monolayers have potential applications in flexible electronics.

Using the optimized crystal structures, the calculated spin-polarized band structures of XC₆ monolayers without SOC are displayed in Fig. 5. Fig. 5(a) and (b) depict that both VC₆ and NbC₆ monolayers are Dirac half-metals, and the CuC₆ monolayer is a Dirac semimetal within GGA-PBE. For VC₆ and NbC₆ monolayers, the spin-up channels possess the gaps of 0.2 and 0.06 eV at the PBE level, respectively, while the spin-down channels show the linear band dispersion around the Fermi level, which creates an intrinsic full spin-polarized Dirac cone. Because the PBE functional generally underestimates the band gap, a more accurate hybrid functional HSE06 method is employed to calculate the nontrivial band structure. We find that the Dirac states for these systems are robust in the spin-down channel, and the band gaps of spin-up channel are 0.35 and 0.4 eV for VC₆ and NbC₆ monolayers, respectively, which are larger than those at the PBE level, as displayed in Fig. 5(d and e). The wide spin-up band gap reveals that VC₆ and NbC₆ monolayers have good stability of the full spin polarization around the Fermi level. Similar to graphene, the CuC₆ monolayer also exhibits a Dirac cone with the valence band (VB) and conduction band (CB) touching each other at the K point. The Fermi velocity v_f of these Dirac points can be evaluated using the expression: v_f= ∂E/ħ∂k. Our calculated v_f along the Γ–K direction for VC₆, NbC₆ and CuC₆ monolayers is 4.5 × 10⁵, 3.4 × 10⁵ and 3.6 × 10⁵ m s⁻¹, respectively. These values are closed to those of silicene (5.3 × 10⁵ m s⁻¹)⁴⁶ and graphene (≈10⁶ m s⁻¹).⁴⁷ The high Fermi velocity and the massless carrier character suggest that XC₆ monolayers are highly promising materials used in high-speed spintronic devices.

The mechanism of the band dispersion around the Fermi level of VC₆ and NbC₆ monolayers was worth clarifying, as revealed by the partial density of states (PDOS) in Fig. S5 (ESI†). For the CuC₆ monolayer, the valence band maximum (VBM) and conduction band minimum (CBM) around the Dirac cone are mainly contributed by the p_z orbital from C atoms. The highest occupied and lowest unoccupied d states of Cu atoms are far away from the Fermi level; therefore, there is almost no hybridization between the d orbitals of Cu atoms and the p orbital of C atoms around the Fermi level. As a result, the low-energy Dirac cone for the CuC₆ monolayer, originating from the C-p_z derived state, is similar to the corresponding p-derived states such as graphene and borophosphene,⁴⁸ suggesting that the nontrivial topological states will occur in the Dirac cone. Such a situation is in sharp contrast to the corresponding d-derived states such as PdCl₃,⁴⁹ ReBr₃, ReI₃,⁵⁰ and NiRuCl₆.⁵¹ However, for VC₆ and NbC₆ monolayers, both C-p and V(Nb)-d orbitals contribute to the band around the Fermi level. In terms of the effect of the crystal field,⁵² the 3d (4d) orbitals of V (Nb) atoms are split into a single a (d_z²) and two 2-fold degenerate e₁ (d_xy + d_x²−y²) and e₂ (d_xz + d_yz) orbitals. As displayed in Fig. S5 (ESI†), the projected electronic DOSs clearly show that the VBM and CBM around the Fermi level are primarily dominated by the e₁ and e₂ orbitals of V and Nb atoms and the p_z orbital of C atoms. Such a feature reveals the strong orbital hybridization between the d orbitals of V and Nb atoms and the p orbitals of C atom. In order to visualize the spatial distribution of spins in the VC₆ and NbC₆ lattices, we show in Fig. 6 the spin density distributions of VC₆ and NbC₆ monolayers in the ferromagnetic configuration. The difference between the electron densities of the two spin channels are calculated by Δρ = ρ↑ − ρ↓, which clearly shows that the magnetic moment is mainly from the V(Nb) atoms, and the spin density plotted analysis is also qualitatively consistent with the PDOS in Fig. S5 (ESI†).

It is interesting to explore the magnetic interaction between the V/Nb and C atoms. It is found that both the total magnetic moment of VC₆ and NbC₆ monolayers are 1μ_B; we can use the crystal field theory to understand it. The e₁ orbitals have the lowest energy, followed by the e₂ orbital and the a (d_z²) orbital. A V atom (3d³4s²) or Nb atom (4d⁴5s¹) has five valence electrons; three next-neighbor C atoms are in the sp² hybridization and form a carbon ring (C6). Additionally, according to the ELF of the VC₆ and NbC₆ monolayers (Fig. 3), the ELF values of the C–C bonds between the line of V–V (Nb–Nb) atoms is very high (∼1), indicating that the strong covalent electron states also occur in the C–C bonding, which can form the CC double bond in the C₂ unit; in turn, the remaining two C atoms have no paired electrons. Thus, each V(Nb) atom provides four electrons to couple with the remaining two C atoms, leaving only one electron in the spin-up channel. Consequently, the VC₆ and NbC₆ monolayers have an integer magnetic moment of 1μ_B per unit cell.

Next, we explore the magnetic ground state of VC₆ and NbC₆ monolayers; the 2 × 2 × 1 supercell with either FM (ferromagnetic) or AFM (antiferromagnetic) ordering of the out-site spin are considered, as illustrated in Fig. 7. Four possible spin configurations are initially set, including FM, Néel-AFM, zigzag-AFM and stripy-AFM states. The relative stability of the two magnetic coupling states can be evaluated from the energy difference (ΔE) between the FM and the lowest-AFM states, as listed in Table 2. After comparing the total energies, we find that the FM state has the lowest total energy for both the VC₆ and NbC₆ monolayers. We note that the bond angles of V–C–V and Nb–C–Nb are about 100.6° and 91.5° for VC₆ and NbC₆, respectively, which are close to 90°, indicating that the V–C–V (Nb–C–Nb) super-exchange interaction instead of the V–V (Nb–Nb) direct-exchange interaction favors the FM coupling according to the Goodenough–Kanamori–Anderson rule.⁵³ Similar phenomena were also found in our previous works on FM ordering in monolayer CrSeTe, MnSeTe and MnSTe.^54,55

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Using the energy difference between AFM and FM states, we can estimate the Curie temperature T_C employing the Monte Carlo simulations with the Heisenberg model. The spin Hamiltonian can be written as

	(3)

where J refers to the exchange-coupling parameters of the nearest- (J₁), next-nearest- (J₂), and next-next-nearest-neighbors (J₃), M is the spin magnetic moment per V(Nb) atom, and (i, j) denotes the summation over nearest neighbors. The calculated J₁, J₂ and J₃ for the VC₆/NbC₆ monolayer are 27.0/23.5, 5.9/7.6 and 5.7/2.1 meV, respectively. The positive J values mean that the FM state is the magnetic ground state for both monolayers.

We then used these J values to calculate the critical temperature by performing the Monte Carlo (MC) simulation⁵⁶ based on the Heisenberg model. A 60 × 60 × 1 supercell and 10 loops were adopted to carry out the MC simulation, and the average magnetic moment per formula unit was taken after the system reaches equilibrium at a given temperature. We plot the magnetic moment and specific heat vs. temperature in Fig. 7(e); the specific heat values are calculated as . We see that the magnetic moment of the system starts dropping at about 520/425 K for the VC₆/NbC₆ monolayer, which indicates that both systems undergo a transition from FM to the paramagnetic state, possessing the T_C values of 520 and 425 K, respectively. Correspondingly, from the simulated C_V (T) curve, a sharp peak in the plot of specific heat is found at about 520/425 K. Although MC simulations may overestimate the T_C value, such a high T_C implies the stable ferromagnetism of VC₆ and NbC₆ monolayers at room temperature. The stability of the ferromagnetism and the high T_C indicate that VC₆ and NbC₆ monolayers can provide an easily accessible platform for exploring the novel states of quantum matters and promising applications in spintronic devices.

We know that 2D FM materials with out-of-plane magnetic easy axis are very important to spintronic applications, e.g., the recently focused 2D CrI₃ and Cr₂Ge₂Te₆.^57,58 Thus, we now check the magnetic anisotropy energy (MAE) based on the PBE+SOC method for VC₆ and NbC₆ monolayers. Herein, three magnetization directions in-plane, namely, (100), (010) and (110) directions, and out-of-plane (001) direction are considered. The results show that the out-of-plane direction (001) is the easy axis for both VC₆ and NbC₆, and the calculated MAE for (100), (010) and (110) are 0.656, 0.657 and 0.657 meV per atom for the VC₆ monolayer, and 0.738, 0.794 and 0.876 meV per atom for the NbC₆ monolayer. These values are comparable with those of magnetic recording alloy FeCo (700–800 μeV per atom),⁵⁹ and monolayer Cr₂O₃ (512 μeV per atom),⁶⁰ FeAs (820 μeV per atom)⁶¹ and Fe₂Si (574 μeV per atom).⁶² The sizable MAE values render them suitable for magnetoelectronic applications.

Next, we focus on the topological electronic properties under the PBE+SOC for the XC₆ monolayers. The calculated SOC-induced gaps are 7 and 45 meV at the K point for VC₆ and CuC₆ monolayers, respectively (Fig. 8). Notably, the large SOC-induced gap for the CuC₆ monolayer suggests a realistic possibility for the utilization of topological effect at room temperature. Interestingly, the gaps at the Dirac points of K, K′ and Γ are 0.5, 0.5 and 4 meV for the NbC₆ monolayer, meaning that these Dirac points are intrinsically robust against SOC. Meanwhile, the Dirac points at Γ and K′/K coexist in two opposite spin channels around the Fermi level for the NbC₆ monolayer, i.e., the DSDPs appears. Then, we calculate the atomic- and orbital-resolved band structure of the NbC₆ monolayer at the PBE+SOC level to understand the DSDPs, as shown in Fig. S6 (ESI†). We know that the gap at the K/K′ points usually regulated by the d_x²−y² and d_xy(d_z²) orbitals for the valence (conduction) bands will be large. However, the valence (conduction) bands in the vicinity of the Fermi level for the NbC₆ monolayer at the K/K′ point are mainly composed by Nb-d_x²−y² and C-p_z (C-p_z and little Nb-d_z²) (Fig. S6, ESI†), i.e., there is almost no d_xy orbital at the K point around the Fermi level, and thus the gap is very small. So, the Dirac points at K and K′ are almost equivalent, which can be well preserved with near zero gap and do not appear at the opposite spin channels for K and K′ points (Fig. S7(a), ESI†). We know that the GGA-PBE usually underestimates the band gap, and the HSE06+SOC hybrid functional can be used to check the band gap of the NbC₆ monolayer (Fig. S7(b), ESI†). The band gaps of the Dirac cones at the K and K′ points are both 10 meV, which are still small. The Dirac cone at the Γ point opens a large gap of 360 meV, but the dispersion of the bands at the Γ point is still linear. The SOC induced large gaps were also found in Dirac monolayer WB₄ (266.9 meV) and Dirac monolayer Sn on Cu(111) (300 meV) with topological insulator characteristics.^63,64 Fig. S7(b) (ESI†) shows that an obvious band inversion occurs at the Γ point, indicating the nontrivial topological properties for the NbC₆ monolayer. We also consider the Coulomb interaction of Nb 4d electrons and use the PBE+U+SOC method to check the band structure of the NbC₆ monolayer. As shown in Fig. S8 (ESI†), the Dirac points at K and K′ are retained within all the U values considered, and the change in the gap at the K/K′ point with the U value is very small. However, the gap at the Γ point is greatly increased with the increasing U value, reaching 402 meV within the effective U value of 2.0 eV, which is consistent with that within the HSE+SOC discussed above. We note the similar phenomenon that the gap reaches 628 meV for Dirac monolayer Fe₂S₂ within PBE+SOC+U (U = 3 eV), which is significantly higher than that of 43 meV within PBE+SOC.⁶⁵ These large gaps in the 2D monolayers guarantee the QAH states.^63–65

The Berry curvatures are calculated in terms of the 80/92/88-band for the VC₆/NbC₆/CuC₆ Hamiltonian obtained from the tight-binding (TB) model, which can be used to understand the topological behavior for these monolayers with the out-of-plane magnetization. Based on orbital analysis near the Fermi level, the TB Hamiltonian with intrinsic SOC and exchange field can be written as

	(4)

where ε₀ is the on-site energy for both spin-up and spin-down channels, c⁺_ασi(c_ασi) represents the creation (annihilation) operator for an electron with spin α on site i. t_{ασi;β,σ′j} is the nearest-neighbor (NN) hopping integral parameter. The i, j in the symbol 〈i, j〉 run over all the NN hopping sites, and the second term H_soc is the intrinsic SOC Hamiltonian with the atomic SOC strength λ. The last term H_M is the exchange field with magnitude M, and s is the spin Pauli matrix. To ensure the computational accuracy of the TB model, we firstly show in Fig. 9 the TB band structure through the above Hamiltonian with SOC. Obviously, we can see that the calculated TB model bands are in good agreement with the DFT result. The band gaps of 6 meV and 55 meV can be opened at the K point for VC₆ and CuC₆ monolayer, respectively. For NbC₆, the band gaps of the Dirac cone are 4/4 meV at the K/Γ point. All these results of the TB model are in good agreement with those of DFT; therefore, both the DFT calculations of GGA, GGA+U, HSE06 and the TB model demonstrate that the intrinsic SOC in the XC₆ lattice can generate nontrivial topological properties to realize the QAH or QSH phase.

As depicted in Fig. 10, the k-resolved Berry curvature obtained from the TB model is used to understand the topological feature. According to the Kudo equation, the Berry curvature can be expressed as the summation of all occupied contributions:⁶⁶

	(5)

	(6)

where the summation is over all the occupied states, the Greek letters α, β indicate Cartesian coordinates, E_n is the eigenvalue of the Bloch function |Ψ_nk〉, and f_n is the Fermi–Dirac distribution function, while v_α and v_β are the velocity operators. The results are plotted in Fig. 10(d)–(f); we can see that the Berry curvature distribution along high symmetry lines, which exhibits a large peak at the K point for VC₆, CuC₆ and NbC₆ monolayers, demonstrates nonzero Hall conductance. The anomalous Hall conductivity in the low energy range is displayed in Fig. S9 (ESI†). It shows the quantized value σ_xy, which confirms the characteristics of Chern insulators or topological insulators. As a consequence, its integration over the Brillouin zone must give rise to a nonzero Chern number (C) for the ferromagnetic VC₆ and NbC₆ monolayers, and a nontrival Z₂ number for nonmagnetic CuC₆ monolayer. The C can be written as

	(7)

where BZ stands for the integral over the Brillouin zone. By integrating the Berry curvature, the calculation on the Chern number gives C = −3 and 5 for VC₆ and NbC₆ monolayers, respectively. A non-zero Chern number suggests that the SOC-induced gap is topologically nontrivial and indicates that ferromagnetic VC₆ and NbC₆ monolayers are high Chern insulators and quantum anomalous Hall insulators. The characteristics of Dirac half-metals without SOC and quantum anomalous Hall insulators within the SOC for both VC₆ and NbC₆ monolayers are similar to those of monolayer 1T-YN₂ and Nb₂O₃.^67,68 Differently, for the nonmagnetic CuC₆ monolayer, the topological invariant Z₂ = 1 is obtained using the Wannier charge center (WCC) method,³² which signifies the existence of the QSH state.

Finally, topologically nontrivial edge states are further confirmed by the calculated edge states plotted in Fig. 10. As expected, the apparent nontrivial surface states in the XC₆ monolayer can be visualized, as shown by the highest color density in the ranges from −X to X path. The gapless chiral edge state near E_F connects the 2D valence and conduction bands, which demonstrates the characterized feature of QAHE and QSHE. The corresponding 2D Berry curvature distribution is shown in Fig. 10. We can see that the nonzero Berry curvature of the Dirac band is mainly around the K point for these systems. Moreover, the distribution of Berry curvatures is also high at the Γ point for the NbC₆ monolayer. All these results suggest that XC₆ monolayers may host topologically nontrivial states.

4. Conclusion

In conclusion, based on first-principles calculations, we predicted the new intrinsic ferromagnetic Dirac half-metals of VC₆, NbC₆ monolayer and nonmagnetic Dirac semimetal of the CuC₆ monolayer. The mechanical properties, phonon dispersion and AIMD simulations ensure the stability and the possibility of preparation of monolayer XC₆. In addition, the negative formation energies of XC₆ monolayers further guarantee their feasibility in experiment. The Fermi velocity (∼10⁵ m s⁻¹) is the same order of magnitude as that of graphene. More importantly, VC₆ and NbC₆ exhibit fascinating features of the QAH effect, including high Chern number and nontrivial edge states. Meanwhile, the CuC₆ monolayer shows the QSH state with a large nontrivial band gap up to 45 meV. Moreover, all the topological properties in the XC₆ monolayer can be further confirmed by a two-band k-p model. These findings are expected to provide fresh promising platforms for the fantastic integration of Dirac physics, spintronics, and valleytronics, offering new opportunities for the realization of applications in spintronic devices.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12174127 and 12004234), Post-doctoral Later-stage Foundation Project of Shenzhen Polytechnic University (No. 6023271020K), Shenzhen Polytechnic Research Fund (No. 6023310019K), the Young Innovative Talents Program in Guangdong Province Colleges and Universities (No. 2022KQNCX042), the Research Foundation for Advanced Talents of Lingnan Normal University (No. ZL22001), the Nature Science Foundation of Guangdong Province (No. 2023A1515011796, 2022A1515011137 and 2024A1515011908).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc00820k

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