Xuming
Wu
^{ab},
Chunhua
Tian
^{a},
Lanhua
Zhong
^{a},
Jun
Quan
^{a},
Jie
Yang
^{a},
Zhibin
Shao
*^{d} and
Guoying
Gao
*^{bc}
^{a}College of Physics Science and Technology, Lingnan Normal University, 524048 Zhanjiang, China
^{b}School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail: guoying_gao@mail.hust.edu.cn
^{c}Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China
^{d}Physics Laboratory, Industrial Training Center, Shenzhen Polytechnic University, Shenzhen 518055, China. E-mail: zhibin_shao@szpt.edu.cn
First published on 30th July 2024
The intrinsic functionality of two-dimensional (2D) materials is crucial for both fundamental studies and practical applications in information processing and storage. In particular, 2D ferromagnets have emerged as a major research field, bringing in new concepts, physical effects, and device designs. More competitive ferromagnetic materials in 2D systems with the quantum anomalous Hall (QAH) state and room-temperature ferromagnetism are much desired. Herein, we predicted stable XC_{6} (X = V, Nb, and Cu) monolayers through first-principles calculations. Novel topological properties, including the gapless edge state, anomalous hall conductance, Chern number and Berry curvature, were systematically investigated. Without spin–orbit coupling, both VC_{6} and NbC_{6} monolayers are ferromagnetic Dirac half-metals, while CuC_{6} monolayers is a nonmagnetic Dirac semimetal. With spin–orbit coupling, both VC_{6} and NbC_{6} monolayers exhibit intrinsic QAH insulators with large out-of-plane magnetocrystalline anisotropy energy and a high Curie temperature of 425 K and 520 K, respectively, and the CuC_{6} monolayer is a quantum spin Hall (QSH) insulator. Our results provide a promising platform for realizing the QAH and QSH phases and the fantastic integration of Dirac physics, spintronics, and valleytronics.
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)_{2}Te_{3} 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 MnBi_{2}Te_{4} 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 MnBi_{2}Te_{4} films at a record-high temperature of 4.5 K. In addition, the QAH insulator state was predicted in VSiGeN_{4} 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 PC_{6} 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 XC_{6} (X ≠ P atom) monolayers. In the present work, we predict a novel family of 2D crystals in transition metal carbides XC_{6} (X = V, Nb, Cu) monolayers with hexagonal lattices. As expected, XC_{6} monolayers have a buckled graphene-like structure, which are highly similar to the PC_{6} monolayer. All of them show highly dynamic and thermodynamic stabilities. Further investigations of ferromagnetic and topological properties reveal that: (i) Without SOC, VC_{6} and NbC_{6} monolayers are ferromagnetic Dirac half-metals with high T_{C} (above 400 K), and CuC_{6} is a nonmagnetic Dirac semimetal. (ii) Remarkably, VC_{6} and NbC_{6} monolayers have nontrivial topological states and high Chern numbers (C = −3, 5) under the SOC effect, which are expected to possess the QAHE. CuC_{6} monolayer shows the QSH effect with the topological invariant Z_{2} = 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.
System | a (Å) | d _{TMC} (Å) | ΔE_{f} (eV per atom) | μ (μ_{B}) |
---|---|---|---|---|
VC_{6} | 6.839 | 1.856 | −1.543 | 1 |
NbC_{6} | 6.880 | 2.061 | −1.353 | 1 |
CuC_{6} | 6.862 | 1.984 | −1.506 | 0 |
In order to assess the stability of XC_{6} monolayers, we evaluate the formation energy, ΔE_{f} = [E(XC_{6}) − E(TM) – 6E(C)]/(1+6), where E(XC6), E(TM) and E(C) are the total energies of the XC_{6} 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 VC_{6}, NbC_{6} and CuC_{6}, respectively. These values are comparable or lower than those of some monolayer (Dirac) materials like FeC_{6} (−0.56 eV per atom),^{35} Be_{3}C_{2} (−0.99 eV per atom),^{36} InB_{6} (−1.20 eV per atom)^{37} and SbAsF_{2} (−1.25 eV per atom).^{38} Thus, the negative formation energies for XC_{6} monolayers indicate the possibility to realize them experimentally.
In order to reveal the chemical bonding nature of XC_{6} monolayers, their charge density difference projection onto an in-plane (ΔQ) and the electron localization function (ELF) were calculated. Taking the VC_{6} monolayer as an example for analysis, the charge density difference can be defined as ΔQ = Q(VC_{6}) − Q(V) − Q(C_{6}). 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, VC_{6} can be written as V^{+1.21}(C^{−0.20})_{6}. The covalent bonding in the VC_{6} 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 VC_{6} 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 sp^{3} P–C bond length (1.81 Å) in the PC_{6} 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 sp^{2}-hybridized orbitals. The value 0.00 in V atoms refers to very low charge density, indicating that V atoms adopt a sp^{3} 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 NbC_{6} and CuC_{6} monolayers are presented in Fig. S1 (ESI†). In order to assess the dynamic stability of the XC_{6} 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 XC_{6} 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 XC_{6} monolayers possess good thermal stability for room-temperature spintronic applications.
The mechanical properties of the XC_{6} monolayer were also investigated by examining its elastic constants. The elastic constants were computed to be C_{11} = 117.24/135.24/170.70 N m^{−1}, C_{22} = 117.17/135.24/171.84 N m^{−1}, C_{12} = 54.55/42.97/30.03 N m^{−1} and C_{44} = 11.98/46.15/65.61 N m^{−1} for VC_{6}/NbC_{6}/CuC_{6} monolayers, respectively. Evidently, the elastic constants satisfy the Born–Huang criteria C_{11} > 0, C_{22} > 0, C_{44} > 0 and C_{11}C_{22} > C_{12}^{2}, implying the mechanical stability. It is noted that the elastic coefficients of XC_{6} monolayers are comparable to those of MoS_{2} sheets (C_{11} = 132.7 N m^{−1}, C_{12} = 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 as^{42}
(1) |
(2) |
Fig. 4 The polar diagrams of (a–c) Young's modulus E and (d–f) Poisson ratio v as a function of θ for the XC_{6} monolayer. |
Using the optimized crystal structures, the calculated spin-polarized band structures of XC_{6} monolayers without SOC are displayed in Fig. 5. Fig. 5(a) and (b) depict that both VC_{6} and NbC_{6} monolayers are Dirac half-metals, and the CuC_{6} monolayer is a Dirac semimetal within GGA-PBE. For VC_{6} and NbC_{6} 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 VC_{6} and NbC_{6} 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 VC_{6} and NbC_{6} monolayers have good stability of the full spin polarization around the Fermi level. Similar to graphene, the CuC_{6} 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 VC_{6}, NbC_{6} and CuC_{6} monolayers is 4.5 × 10^{5}, 3.4 × 10^{5} and 3.6 × 10^{5} m s^{−1}, respectively. These values are closed to those of silicene (5.3 × 10^{5} m s^{−1})^{46} and graphene (≈10^{6} m s^{−1}).^{47} The high Fermi velocity and the massless carrier character suggest that XC_{6} monolayers are highly promising materials used in high-speed spintronic devices.
The mechanism of the band dispersion around the Fermi level of VC_{6} and NbC_{6} monolayers was worth clarifying, as revealed by the partial density of states (PDOS) in Fig. S5 (ESI†). For the CuC_{6} 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 CuC_{6} 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 PdCl_{3},^{49} ReBr_{3}, ReI_{3},^{50} and NiRuCl_{6}.^{51} However, for VC_{6} and NbC_{6} 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 e_{1} (d_{xy} + d_{x2−y2}) and e_{2} (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 e_{1} and e_{2} 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 VC_{6} and NbC_{6} lattices, we show in Fig. 6 the spin density distributions of VC_{6} and NbC_{6} 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†).
Fig. 6 Spin densities of VC_{6} and NbC_{6} monolayers. Yellow and purple red isosurfaces represent positive and negative spin densities (+ 0.001 e Å^{−3}), respectively. |
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 VC_{6} and NbC_{6} monolayers are 1μ_{B}; we can use the crystal field theory to understand it. The e_{1} orbitals have the lowest energy, followed by the e_{2} orbital and the a (d_{z2}) orbital. A V atom (3d^{3}4s^{2}) or Nb atom (4d^{4}5s^{1}) has five valence electrons; three next-neighbor C atoms are in the sp^{2} hybridization and form a carbon ring (C6). Additionally, according to the ELF of the VC_{6} and NbC_{6} 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 C_{2} 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 VC_{6} and NbC_{6} monolayers have an integer magnetic moment of 1μ_{B} per unit cell.
Next, we explore the magnetic ground state of VC_{6} and NbC_{6} 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 VC_{6} and NbC_{6} monolayers. We note that the bond angles of V–C–V and Nb–C–Nb are about 100.6° and 91.5° for VC_{6} and NbC_{6}, 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}
FM | Néel (eV) | Stripy (eV) | Zigzag (eV) | J _{1} (meV) | J _{2} (meV) | J _{3} (meV) | T _{C} (K) | |
---|---|---|---|---|---|---|---|---|
VC_{6} | 0 | 0.556 | 0.526 | 0.219 | 27.0 | 5.9 | 5.7 | 520 |
NbC_{6} | 0 | 0.514 | 0.496 | 0.260 | 23.5 | 7.6 | 2.1 | 425 |
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) |
We then used these J values to calculate the critical temperature by performing the Monte Carlo (MC) simulation^{56} 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 VC_{6}/NbC_{6} 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 VC_{6} and NbC_{6} monolayers at room temperature. The stability of the ferromagnetism and the high T_{C} indicate that VC_{6} and NbC_{6} 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 CrI_{3} and Cr_{2}Ge_{2}Te_{6}.^{57,58} Thus, we now check the magnetic anisotropy energy (MAE) based on the PBE+SOC method for VC_{6} and NbC_{6} 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 VC_{6} and NbC_{6}, and the calculated MAE for (100), (010) and (110) are 0.656, 0.657 and 0.657 meV per atom for the VC_{6} monolayer, and 0.738, 0.794 and 0.876 meV per atom for the NbC_{6} monolayer. These values are comparable with those of magnetic recording alloy FeCo (700–800 μeV per atom),^{59} and monolayer Cr_{2}O_{3} (512 μeV per atom),^{60} FeAs (820 μeV per atom)^{61} and Fe_{2}Si (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 XC_{6} monolayers. The calculated SOC-induced gaps are 7 and 45 meV at the K point for VC_{6} and CuC_{6} monolayers, respectively (Fig. 8). Notably, the large SOC-induced gap for the CuC_{6} 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 NbC_{6} 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 NbC_{6} monolayer, i.e., the DSDPs appears. Then, we calculate the atomic- and orbital-resolved band structure of the NbC_{6} 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 NbC_{6} 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 NbC_{6} 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 WB_{4} (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 NbC_{6} 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 NbC_{6} 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 Fe_{2}S_{2} 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}
Fig. 8 Spin-dependent band structures of VC_{6} (a), NbC_{6} (b) and CuC_{6} (c) monolayers within PBE+SOC. |
The Berry curvatures are calculated in terms of the 80/92/88-band for the VC_{6}/NbC_{6}/CuC_{6} 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) |
Fig. 9 Band structures of VC_{6} (a), NbC_{6} (b) and CuC_{6} (c) monolayers with SOC using the Wannier interpretation and Berry curvatures for the occupied bands along the high symmetry directions. |
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) |
(7) |
Fig. 10 The local DOS of the edge states of (a)–(c) and Berry curvature distributions (d)–(f) (in the arbitrary unit) of the semi-infinite boundary of the VC_{6}, NbC_{6} and CuC_{6} monolayer. |
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 XC_{6} 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 NbC_{6} monolayer. All these results suggest that XC_{6} monolayers may host topologically nontrivial states.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc00820k |
This journal is © The Royal Society of Chemistry 2024 |