Introduction

To advance electronics at the nanoscale, the exploration and exploitation of quantum degrees of freedom in materials become indispensable. Two-dimensional (2D) compounds form excellent systems, wherein these quantum degrees of freedom can be exploited toward electronic and spintronic applications. A particular group of interest is 2D Dirac materials, which have a Dirac cone with linear dispersion in their band structure1. These materials have been predicted to exhibit a large variety of exotic properties: massless fermions2, ultrahigh carrier mobility3, (fractional) quantum Hall effects (QHE)4,5, etc. Furthermore, there have been great advances in the preparation and growth of 2D materials in recent years6,7,8, which initiated the study of the largely unexplored group of 2D strongly correlated Dirac systems (SCDS). As the Dirac cone has shown to be instrumental for the realization of many nontrivial topological phases9,10, these SCDS uncover a rich playground where relativistic dispersion, electron correlations, and topological ordering meet.

In this work, we perform an ab initio study on the 2D single atomic layer of V2O3 with honeycomb–kagome (HK) lattice structure. First, we confirm the predicted11 ground-state properties of HK V2O3 and establish that it is an excellent SCDS candidate with a room-temperature QAHE. The second part of this work mainly focuses on the experimental feasibility by studying the structure under compressive and tensile biaxial strain as well as the graphene-supported V2O3 monolayer system.

Results

Structural stability and mechanical properties

The structurally optimized unit cell of HK V2O3 is shown in Fig. 1a with a planar HK lattice with a lattice constant 6.193 Å. A reduction of 0.15 Å of the V–O bond—from 1.94 to 1.79 Å—is found compared to the shortest V–O bond in bulk V2O312. A similar reduction is observed in other metal oxides13,14,15. This can be attributed to the reduction in coordination number of the V3+ cation in this 2D form, where V3+ has the electron configuration [Ar]3d2 with the two outer 4s electrons of vanadium ionically bonded to the oxygen. The structural stability is firstly examined by calculating the formation energy \({E}_{{\rm{form}}}={E}_{{\rm{{V}}_{2}{O}_{3}}}-2{E}_{{\rm{V}}}-3/2{\mu }_{{\rm{{O}}_{2}}}\) with \({E}_{{\rm{{V}}_{2}{O}_{3}}}\) and EV are the total energies of the HK V2O3 and V crystal structure (−8.934 eV/atom), while \({\mu }_{{\rm{{O}}_{2}}}=-4.9480\) eV/atom is the chemical potential oxygen gas. The obtained negative value of −1.81 eV/atom (compared to −2.29 eV/atom in bulk V2O3) together with earlier experimental work16 suggests the experimental feasibility of the system.

Fig. 1: V2O3 monolayer and its structural stability.
figure 1

a The (2 × 2) supercell of the HK V2O3 monolayer with the unit cell indicated by dashed lines and cyan and red representing the V and O atoms, respectively. b The calculated phonon band dispersion. c The energy fluctuations during AIMD and d the atomic structure obtained after 6000 ps.

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To study the dynamical stability of the monolayer system, the phonon spectrum was evaluated by DFPT. As shown in Fig. 1b, the spectrum shows no imaginary frequency phonon modes, confirming the dynamical stability. Furthermore, the AIMD simulation at a fixed temperature of 300 K also confirmed the thermal stability, since total energy fluctuations of only 0.3% are observed while planarity is preserved, as shown in Fig. 1c, d. In addition, the elastic properties and stability were verified (see Supplementary Table 1). Therefore, it can be concluded that the single atomic layer of V2O3 with HK lattice structure is stable on all three levels: dynamical, thermal, and mechanical.

Magnetic ground state

To determine the magnetic ground state of the system, various magnetic configurations have been studied. By the use of noncollinear calculations with SOC, the total energy of the ion–electron system was determined for ferromagnetic (FM), nonmagnetic (NM), and four types of antiferromagnetic (AFM) states, depicted in Fig. 2a. By comparison of the total energies in Table 1, it can be concluded that the system is in a FM ground state, with the AFM stripy (AFM-ST) state as the second-lowest energy state. As a first approximation, we assume an isotropic exchange interaction (i.e., neglecting anisotropic exchange), and consider the following spin hamiltonian,

$$H=-\mathop{\sum}\limits_{\langle i,j\rangle }{J}_{1}{{\bf{S}}}_{i}\cdot {{\bf{S}}}_{j}-\mathop{\sum}\limits_{i}A{({{\bf{S}}}_{i}^{z})}^{2}$$
(1)

where i enumerates the magnetic ions. The first term is the nearest-neighbor exchange interaction with exchange parameter J1, while the second term corresponds to the single-ion anisotropy with the anisotropy parameter A. The interaction is ferromagnetic when J1 > 0 and positive A will favor an out-of-plane spin component Sz.

Fig. 2: Magnetic configurations and Curie temperature.
figure 2

a Top view of various possible AFM configurations. b The normalized magnetization as a function of the temperature from the MC simulations indicating a Tc = 440 K.

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Table 1 The DFT calculated energies (eV/f.u.) for different (out-of-plane) magnetic configurations with respect to the lowest FM energy state which is set to zero.
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The exchange parameter is evaluated by J1 = [EAFM − EFM]/2zS2 with the total spin moment S = 1, and with z = 3 neighboring spins. This gives J1 = 58 meV. On the other hand, the magnetocrystalline anisotropy energy (MAE) defined by EMAE = E[100] − E[001], with square brackets indicating the orientation of the spins, equals 2 meV/f.u. (i.e., A = 1 meV). This strong MAE can be understood from degenerate perturbation theory. The SOC-induced interaction involves the coupling between states with identical spin, with the most important interaction between the unoccupied and occupied d states. As can be seen in Fig. 3b, these states at the conduction band minimum (CBM) at Γ and valence band maximum (VBM) at K, respectively, consist of the \(({d}_{xy},{d}_{{x}^{2}-{y}^{2}})\)- and (dxz, dyz) orbitals. This SOC-induced interaction involves a change in the orbital angular momentum quantum number Lz from (dxz, dyz) with Lz =± 1 to \(({d}_{xy},{d}_{{x}^{2}-{y}^{2}})\) with Lz = ±2. This interaction between these out-of-plane spins leads to a maximization of the SOC energy stabilization, as described in ref. 17.

Fig. 3: Electronic band structure with derived orbital and topological character.
figure 3

a The electronic band structure within GGA+U without SOC and (b) with SOC. c The calculated Berry curvature projected onto the BZ. d The orbital- and spin-resolved DOS. e The derived orbital ordering and bonding of the spin-polarized V 3d orbitals.

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To obtain an estimate of the Curie temperature (Tc), the Curie–Bloch equation in the classical limit,

$$m(T)={\left(1-\frac{T}{{T}_{{\rm{c}}}}\right)}^{\beta }$$
(2)

with T the temperature and β a critical exponent, is fitted to the normalized magnetization obtained by the MC simulation (see Fig. 2b). The resulting Tc is equal to 440 K with β = 0.19, and can also be compared within the mean-field approximation for the honeycomb lattice (ignoring the ion anisotropy) by using the formula18,

$${T}_{{\rm{c}}}=\frac{3{J}_{1}}{2{k}_{{\rm{B}}}}{S}^{2}$$
(3)

with Boltzmann constant kB, resulting in Tc = 1010 K, which gives the usual overestimation of the critical temperature.

Since the Néel AFM phase is not the second-lowest energy state, it can be expected that second- and third-neighbor exchange interactions (J2, J3) are playing a dominant role in the magnetic ordering. Therefore, the DFT total energies were mapped onto the Ising hamiltonian to obtain J1, J2, and J3 by the following four equations:

$${E}_{{\rm{FM/AFM-N}}}={E}_{{\rm{NM}}}-(\pm 3{J}_{1}+6{J}_{2}\pm 3{J}_{3}){S}^{2},$$
(4)

and

$${E}_{{\rm{AFM-ZZ/ST}}}={E}_{{\rm{NM}}}-(\pm {J}_{1}-2{J}_{2}\mp 3{J}_{3}){S}^{2}.$$
(5)

The J1, J2, and J3 values were found to be 41.9, −3.8, and 17.4 meV, respectively. Taking into account, these second- and third-neighbor exchange interactions in the spin hamiltonian (1), the hamiltonian is solved by the MC simulation, with the magnetization (shown in Fig. 2) fitted by Eq. (2), giving a Curie temperature equal to 320 K with β = 0.20.

Electronic structure

The electronic band structure and density of states (DOS) within GGA+U are depicted in Fig. 3. The electronic band structure shows two key features: a Dirac cone at the K point and the existence of two flat bands at ± 1 eV forming a degeneracy with the dispersive bands at the Γ point. From the DOS, it becomes clear that there exists a strong spin polarization with the two characteristic band features being derived from the spin-up V (dxz, dyz) orbitals. This means that the Dirac cone is completely spin-polarized, making it a Dirac spin-gapless semiconductor (DSGS)19. By taking into account the intrinsic SOC, it is found that an energy gap is opened at the K point, resulting in an indirect energy gap of 0.45 eV. It turns out that the energy gap is strongly dependent on the Hubbard correction U (see Supplementary Fig. 3), which has led to the idea of the cooperative effect between electron correlations and SOC. The relatively large energy gap opened at the Dirac point can be understood from the fact that the orbital degeneracy of the occupied states allows an atomic on-site SOC, involving no hopping process20. This is in contrast to the second-order hopping SOC in the Kane–Mele model, or the first-order hopping SOC in the Xenes21. Within this understanding, it is clear that an increase of U will force the electrons into a more localized state implying an increased effect of the atomic SOC22,23.

In addition to the GGA+U, the LDA and hybrid exchange-correlation functional HSE06 were used to study the electronic properties of the system. The resulting energy gap values are summarized in Table 2, and the calculated band structures can be found in Supplementary Fig. 4. It is noted that there is a strong variation of the energy gap depending on the chosen functional. This can be expected as these functionals are approximating the localized nature of the 3d orbitals very differently, which will be crucial for the atomic SOC effect.

Table 2 The calculated direct and indirect energy gap Eg values for the different exchange-correlation functionals with SOC included.
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This exchange-correlation functional dependence is informative and sheds light on the physical nature of the interplay of electron correlations and SOC in opening the gap at the Dirac cone. However, it should be kept in mind that the GGA+U (with U = 3 eV) is considered as the most accurate description as the electron correlations (U) have been estimated on the basis of a first-principle linear response method, while it is well-known that the HSE06 overestimates the bandgap of bulk V2O3 and other TMO24,25.

From the DOS, it can be inferred that the crystal field splits the d-orbitals into three orbital levels whose energy ordering can be derived from the orbital orientation. The \({d}_{{z}^{2}}\)-orbital level has negligible overlap with the O 2p orbitals and thus has the lowest energy state, the (dxz, dyz)-orbitals oriented out-of-plane will form a π-bond with the oxygen O pz as bridging ligand, and the \(({d}_{xy},{d}_{{x}^{2}-{y}^{2}})\)-orbitals will form an in-plane σ-bond with bridging orbitals O (px, py). This latter bond involves a strong orbital overlap and therefore the involved orbitals will be energetically less favorable, forming the highest energy state. The resulting orbital ordering is shown in Fig. 3e. It should be emphasized that in the vicinity of the Fermi level, there is approximately no O 2p orbital weight, which confirms the localized nature of the two V 3d electrons. Based on Griffith’s crystal-field theory26, the spin state of the V3+ cation can be confirmed. By the relative strengths of the exchange (ΔEex) and crystal-field splitting (ΔECF) with derived values of 3.34 eV and 0.8 eV—it can be concluded that the V cation has a high (S = 1) spin state with magnetic moment 2 μB.

To gain further insight into the nature of the chemical bond, the atomic charges were determined by the Bader charge analysis code27. It is found that the atomic charges on V and O are resp. QV = 1.59e and QO = − 1.07e, confirming the ionic bonding character where the electrons of the V cation are attracted toward the O anion. Nevertheless, the atomic charges are slightly lower than in the bulk parent, which suggests that there is an increased electron delocalization and thus stronger bonding covalency compared to the bulk. This behavior can be linked to the reduced V–O bond length. On the other hand, the relative strong ionicity of the bond might explain the energetic stability of the planar configuration of these TMO monolayer systems, since the buckling of the V–O bond would result in an energetically unfavorable large dipole moment normal to the atomic plane. In this way, the reduced bond length and enhanced covalency can be explained as a means to suppress the possible large dipole moment. These trends and their explanations were already pointed out for other TMOs13,14.

To study the topology of the band structure, the Berry curvature Ω(k) of the system was calculated directly from the DFT calculated wave functions by the VASPBERRY code, which is based on Fukui’s method28. The resulting Berry curvature, shown in Fig. 3c, becomes nonzero at the Dirac points K and \({\rm{K}}^{\prime}\). By integrating the Berry curvature Ωn(k) of each nth energy band over the whole BZ and summing over all occupied bands, the Chern number

$$C=\mathop{\sum}\limits_{n}\frac{1}{2\pi }\int _{BZ}{d}^{2}{\bf{k}}\ {{\mathrm{\Omega }}}_{n}({\bf{k}})$$
(6)

is found to be C = 1. Therefore, it can be concluded that the system is a room-temperature Chern insulator.

Moreover, the Berry curvature and resulting Chern number C = 1 were found to be robust against the Hubbard value U. However, by evaluating the Berry curvature of the HSE06 band structure, a Chern number equal to zero was found (see Supplementary Fig. 6). Although the experimental validity of the HSE06 band structure can be critically questioned, it shows the intimate interplay of electron correlations and topology. The electron correlations causing a topological phase transition to a trivial state agree well with earlier theoretical studies22,29, and it opens great opportunities as the magnitude of electron correlations can be more easily controlled externally in Dirac materials30,31.

Biaxial strain

TMOs are known to exhibit a rich-phase diagram as a function of strain, in particular, for bulk V2O3 the room-temperature metal–insulator transition (MIT) between the paramagnetic insulating and metallic state can be realized by the application of epitaxial strain32. Therefore, a biaxial strain study on the 2D atomic layer of V2O3 was performed. The atomic positions were optimized for all biaxially strained configurations from − 10% to + 10%. It was found that under compressive strain, the lattice structure undergoes an in-plane buckling of the V–O bond (HKin), as can be seen in Fig. 4a, b. This structural distortion is found by ionic relaxation with a high force-convergence criterion of 0.005 eV/Å. On the other hand, by lowering the force-convergence criterion to 0.05 eV/Å, two metastable structures were found in the compressive region; one preserving the HK lattice structure (HK), while the other undergoes an out-of-plane buckling of the V–O bond (HKout). Although these lattice configurations are un/metastable, they could become stabilized under certain conditions and depending on underlying substrates. Moreover, a combination of in- and out-of-plane buckling can also be expected and was observed in an ab initio calculation of TMOs on metal substrates33.

Fig. 4: Biaxial strain study.
figure 4

a The total energy of the system under biaxial strain, taking into account (b) the possible structural deformations (HKin, HKout) with deformation parameters the bond angle θ and the buckling height h, respectively. c Top: the V–O bond length as a function of strain for all three configurations. Bottom: the bond angle θ and buckling height h in function of the strain for resp. HKin and HKout. d The energy gap and magnetic exchange interaction as a function of strain.

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Identifying these different structural configurations under compressive strain in the freestanding V2O3 monolayer allows a further study on the structural stability and provides a better understanding of the involved chemical bonds governing the stability of this planar single atomic layer. From the structural in- and out-of-plane distortions, it is noted that the deformation of the V–O bond develops to prevent a further reduction of this bond length (see Fig. 4c). This behavior can be linked to the ionic character of the bond. On the other hand, from the energetic point of view, the system prefers to undergo an in-plane deformation rather than the out-of-plane deformation. This can be attributed to the energy cost of the formation of a dipole moment under a buckling of the V–O bond.

In addition to this structural study, the electronic and magnetic properties as a function of the biaxial strain were studied. Since the stabilization of the metastable states (HK and HKout) will depend on external conditions, only the most stable HKin configuration is considered for the compressive region. The energy gap and the magnetic exchange interaction are shown in Fig. 4. It is clear that within the strain range of −5 to +5%, the electronic and magnetic properties are sufficiently preserved to maintain the room-temperature magnetic Chern insulating phase. It was also confirmed that the Chern invariant C remains equal to 1, even in the presence of the in-plane distortion of HKin. This can be expected as the V cations still form a honeycomb lattice, crucial for the existence of the Dirac cone34. From these results, it can be concluded that the room-temperature QAHE in 2D V2O3 is robust against small structural deformations.

Graphene substrate

Motivated by the successful deposition of Y2O3 on graphene35, the feasibility of using a graphene substrate for the growth of the V2O3 monolayer was studied. A (2 × 2) V2O3 supercell on a (5 × 5) graphene supercell was structurally optimized and that configuration is shown in Fig. 5a (see Supplementary Note 2). Similar to Al2O3 and Y2O3 monolayers on graphene13,14, the V cations are located near the top sites of the C atoms to maximize the potential bonding. The planar HK lattice structure is conserved with a minimal distance between graphene and V2O3 of 3.43 Å, while the V–O bond length remains approximately preserved.

Fig. 5: Graphene substrate study.
figure 5

a The top and side view of (2 × 2) V2O3 supercell on a (5 × 5) graphene supercell. b The calculated charge density difference Δρ where blue and red correspond to excess and depleted charge density, respectively. c The DOS within GGA+U and (d) band structure of 2D HK V2O3 op graphene within GGA+U+SOC. The color code is the same used in (c).

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To obtain a first estimate of the interfacial interaction strength, the adsorption energy Ead is determined,

$${E}_{\text{ad}}={E}_{{\rm{G}}}+{E}_{{\rm{{V}}_{2}{O}_{3}}}-{E}_{{\rm{G+{V}}_{2}{O}_{3}}}$$
(7)

where EG, \({E}_{{\rm{{V}}_{2}{O}_{3}}}\), and \({E}_{{\rm{G+{V}}_{2}{O}_{3}}}\) are the total energies of the isolated graphene, isolated V2O3, and the combined hybrid structure, respectively. The adsorption energy equals 77 meV per C atom indicating a weak interaction between both monolayers, albeit stronger than for HK Y2O3 on graphene (42 meV per C atom)14.

To preserve the electronic properties, it is important to prevent charge transfer between both monolayers, therefore the atomic charges of the systems were determined for the hybrid system. The average atomic charges of both V and O remain unchanged, while the average excess charge of the C atom is only 0.004e. This indicates negligible electron transfer between both monolayers, which can also be confirmed by the calculation of the charge density difference \({{\Delta }}\rho ={\rho }_{{\rm{G+{V}}_{2}{O}_{3}}}-{\rho }_{{\rm{G}}}-{\rho }_{{\rm{{V}}_{2}{O}_{3}}}\), shown in Fig. 5b. The charge density difference shows a negligibly small inhomogeneous charge distribution between both monolayers, which is mainly located between V and C atoms. This indicates a small orbital hybridization of the π-bonded orbitals of V2O3 and graphene. This is similar to earlier studies of HK TMOs on graphene where it was shown that vdW interactions and orbital hybridization play an important role in the electronic properties at the interface13,14.

The electronic properties of the V2O3 monolayer on the graphene substrate are shown in Fig. 5c, d. From the band structure and DOS, it is noted that the conduction band minimum of graphene is about 0.2 eV below the Fermi level, indicating that graphene is slightly n-doped. A similar n-doping of graphene was found for HK Y2O314. The DOS contribution of the V2O3 remains approximately preserved, however from the calculated band structure, it is clear that there is some hybridization of the π-bonded orbitals. Nonetheless, the Dirac cone of V2O3 remains preserved with a reduction in the SOC-induced direct energy gap to 66 meV.

Overall the HK V2O3 monolayer on graphene shows very similar behavior to HK Y2O314 in terms of vdW bonding—as can be seen from the Bader charge analysis, charge density difference, and the effect on the electronic structure. Therefore, based on the successful deposition of Y2O3 on graphene35 together with the results in the section “structural stability and mechanical properties” and early experimental work16, it is expected that graphene might be an optimal substrate for the growth of the V2O3 monolayer.

Discussion

In this work, we have confirmed that the single atomic layer of V2O3 with HK lattice structure is a structurally stable room-temperature magnetic Chern insulator. This system features the coexistence of topological order and strong electron correlations, and might therefore be an excellent SCDS candidate. It was shown that the system can undergo small structural deformations which preserve the honeycomb lattice of the V cations, such that the Dirac cone and the corresponding room-temperature QAHE remain unaffected. Furthermore, this 2D V2O3 can be further stabilized by the support of a graphene substrate, where there is only a small interfacial interaction mainly attributed to orbital hybridization, but preserving the Dirac cone of both materials. These observations together with earlier studies on TMO with HK lattice structure13,14,15,35,36,37,38,39 should encourage the experimental investigation of this group of 2D TMOs.

Methods

DFT calculations

All spin-polarized calculations were carried out within density-functional theory (DFT), as implemented in the Vienna ab initio simulation package (VASP)40. The generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE)41, projected-augmented-wave (PAW) potential42, and the plane-wave basis with an energy cutoff of 550 eV were used. For the structural relaxation, a force-convergence criterion of 0.005 eV/Å was used with the Brillouin zone (BZ) sampled by a 12 × 12 × 1Γ-centered k-point mesh, with a vacuum space of 17 Å adopted along the normal of the atomic plane. To account for the localized nature of the 3d electrons of the V cation, a Hubbard correction U is employed within the rotationally invariant approach proposed by Dudarev et al.43, where Ueff = U − J is the only meaningful parameter. The self-consistent linear response calculation introduced by Cococcioni et al.44 was adopted to determine U. In this way, a Hubbard correction of U = 3.28 eV is found (see Supplementary Methods), which is close to the UBulk = 3 eV value found in bulk V2O3 system45, which is then applied throughout the paper.

The magnetic and electronic self-consistent calculations were performed with a total energy convergence criterion of 10−6 eV with the BZ sampled by a denser 24 × 24 × 1Γ-centered k-point mesh. The spin–orbit coupling (SOC) was included by a second variational procedure on a fully self-consistent basis. To test the sensitivity of the results to the choice of the functional, the local density approximation (LDA)46 and the screened exchange hybrid density functional by Heyd–Scuseria–Ernzerhof (HSE06)47 are employed. In addition, for the substrate calculations, the van der Waals (vdW) interactions were taken into account by the use of Grimme’s DFT-D3 method48. The phonon dispersion was calculated self-consistently on the basis of the density-functional perturbation theory (DFPT) and with the use of the PHONOPY package49.

Monte-Carlo simulations

The ab initio molecular dynamic (AIMD) simulations were carried out on a 4 × 4 × 1 supercell at 300 K in the canonical ensemble using the Nose–Hoover thermostat approach50,51 with 3000 times steps of step size 2 fs. The Curie temperature was estimated by Monte-Carlo (MC) simulations, as implemented in the VAMPIRE package52. For these calculations, a rectangular \(100\times 100\sqrt{3}\) supercell was used, where the spins were thermalized for 104 equilibrium steps, followed by 2 × 104 averaging steps to calculate the thermal-averaged magnetization for each temperature. The atomic structures were visualized by the VESTA program53.