# Two-dimensional honeycomb-kagome $\ce{V_{2}O_{3}}$:
a robust room-temperature magnetic Chern insulator interfaced with graphene

S. Mellaerts1, R. Meng1, M. Menghini1,3, V. Afanasiev1, J.W. Seo2 , M.
Houssa1,4 and J.-P. Locquet1 1Department of Physics and Astronomy, KU Leuven,
Celestijnenlaan 200D, 3001 Leuven, Belgium, 2Department of Materials
Engineering, KU Leuven, Kasteelpark Arenberg 44, 3001 Leuven, Belgium, 3IMDEA
Nanociencia, Calle Faraday 9, E28049, Madrid, Spain, 4Imec, Kapeldreef 75,
3001 Leuven, Belgium<EMAIL_ADDRESS>
###### Abstract

The possibility of dissipationless chiral edge states without the need of an
external magnetic field in the quantum anomalous Hall effect (QAHE) offers a
great potential in electronic/spintronic applications. The biggest hurdle for
the realization of a room-temperature magnetic Chern insulator is to find a
structurally stable material with a sufficiently large energy gap and Curie
temperature that can be easily implemented in electronic devices. This work
based on first-principle methods shows that a single atomic layer of
$\ce{V_{2}O_{3}}$ with honeycomb-kagome (HK) lattice is structurally stable
with a spin-polarized Dirac cone which gives rise to a room-temperature QAHE
by the existence of an atomic on-site spin-orbit coupling (SOC). Moreover, by
a strain and substrate study, it was found that the quantum anomalous Hall
system is robust against small deformations and can be supported by a graphene
substrate.

††preprint: APS/123-QED

##  I Introduction

To advance electronics at the nanoscale, the exploration and exploitation of
quantum degrees of freedom in materials becomes indispensable. Two-dimensional
(2D) compounds form excellent systems wherein these quantum degrees of freedom
can be exploited, towards electronic and spintronic applications. A particular
group of interest are 2D _Dirac materials_ , which have a Dirac cone with
linear dispersion in their band structure (DiracMaterial, ). These materials
have been predicted to exhibit a large variety of exotic properties: massless
fermions (MasslessFermion, ), ultrahigh carrier mobility (ultrahighmobility,
), (fractional) quantum Hall effects (QHE) (QHE, ; FQHE, ), etc. Furthermore,
there have been great advances in the preparation and growth of 2D materials
in recent years (Novoselov666, ; VdWMBE, ; EpitaxyTMD, ), 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 non-trivial topological phases (Haldane, ; KaneMele, ), these SCDS
uncover a new 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
$\ce{V_{2}O_{3}}$ with HK lattice structure. Firstly, we confirm the predicted
(HKV2O3, ) ground state properties of HK $\ce{V_{2}O_{3}}$ 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 $\ce{V_{2}O_{3}}$ monolayer system.

##  II Computational methods

All spin-polarized calculations were carried out within density-functional
theory (DFT), as implemented in the Vienna _ab initio_ simulation package
(VASP) (VASP, ). The generalized gradient approximation (GGA) in the form of
Perdew-Burke-Ernzerhof (PBE) (PBE, ), projected-augmented-wave (PAW) potential
(PAWmethod, ), and the plane-wave basis with 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 $12\times 12\times 1$
$\Gamma$-centered $k$-point mesh, with a vacuum space of $17\text{\,\AA}$
adopted along the normal of the atomic plane. To account for the localized
nature of the $3d$ electrons of the $\ce{V}$ cation a Hubbard correction $U$
is employed within the rotationally invariant approach proposed by Dudarev et
al. (Dudarev, ), where $U_{\text{eff}}=U-J$ is the only meaningful parameter.
The self-consistent linear response calculation introduced by Cococcioni _et
al._ (linearresponse, ) was adopted to determine $U$. In this way, a Hubbard
correction of $U=3.28$ eV is found (see Supplementary Material), which is
close to the $U_{Bulk}=3$ eV value found in bulk $\ce{V_{2}O_{3}}$ system
(HubbardV2O3, ), 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 $24\times 24\times 1$ $\Gamma$-centered $k$-point mesh. To test the
sensitivity of the results to the choice of the functional, the local density
approximation (LDA) (LDA, ) and the screened exchange hybrid density
functional by Heyd-Scuseria-Ernzerhof (HSE06) (HSE06, ) are employed.
Additionally, for the substrate calculations, the van der Waals (vdW)
interactions were taken into account by the use of Grimme’s DFT-D3 method
(Grimmecorrection, ).

Figure 1: $\ce{V_{2}O_{3}}$ monolayer and its structural stability. (a) The
$(2\times 2)$ supercell of the HK $\ce{V_{2}O_{3}}$ monolayer with the unit
cell indicated by dashed lines and cyan and red representing the
$\ce{\ce{{V}}}$ and $\ce{\ce{{O}}}$ atoms, respectively. (b) The calculated
phonon band dispersion. (c) The energy fluctuations during AIMD and (d) the
atomic structure obtained after $6000$ ps.

The phonon dispersion was calculated self-consistently on the basis of the
density functional perturbation theory (DFPT) and with the use of the PHONOPY
package (phonopy, ). The _ab initio_ molecular dynamic (AIMD) simulations were
carried out on a $4\times 4\times 1$ supercell at $300$ K in the canonical
ensemble using the Nose-Hoover thermostat approach (NV1, ; NV2, ) with $3000$
times steps of step size $2$ fs. The Curie temperature was estimated by Monte
Carlo (MC) simulations, as implemented in the VAMPIRE package (vampire, ), and
by the use of a fully quantum-mechanical method where the hamiltonian is
solved by Green’s functions (jorenmodel, ). For the former calculations a
rectangular $50\times 50\sqrt{{3}}$ supercell was used, where the spins were
thermalized for $10^{4}$ equilibrium steps, followed by $2\cdot 10^{4}$
averaging steps to calculate the thermal averaged magnetization for each
temperature. The atomic structures were visualized by the VESTA program
(VESTA, ).

##  III Results and discussions

###  III.1 Structural stability and mechanical properties

The structurally optimized unit cell of HK $\ce{V_{2}O_{3}}$ is shown in Fig.
1a with a planar honeycomb-kagome lattice with lattice constant
$6.193\>\text{\AA}$. A reduction of $0.15\>\text{\AA}$ of the
$\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond - from $1.94\>\text{\AA}$ to
$1.79\,\text{\AA}$ \- is found compared to the shortest V-O bond in bulk
$\ce{V_{2}O_{3}}$ (StructuralBulk, ). A similar reduction is observed in other
metal oxides (HKAl2O3, ; HKY2O3, ; 2DTMOscreening, ). This can be attributed
to the reduction in coordination number of the $\ce{\ce{{V}}}$3+ cation in
this 2D form, where $\ce{\ce{{V}}}$3+ has electron configuration
$[\ce{Ar}]3d^{2}$ with the two outer $4s$ electrons of vanadium ionically
bonded to the oxygen.

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.

Table 1: Comparison of the stiffness constants, Young’s modulus ($Y_{s}$), and Poisson’s ratio ($\nu$) of different 2D monolayer systems. |  $C_{11}$ (N/m) |  $C_{12}$ (N/m) |  $Y_{s}$ (N/m) | $\nu$
---|---|---|---|---
HK $\ce{V_{2}O_{3}}$ | $73.09$ | $31.84$ | $59.2$ | $0.44$
HK $\ce{Al_{2}O_{3}}$ (HKAl2O3, ) | $87.75$ | $59.35$ | $47.6$ | $0.68$
Graphene (HKAl2O3, ) | $353.68$ | $62.20$ | $342.7$ | $0.18$
h-$\ce{BN}$ (HKAl2O3, ) | $288.26$ | $63.54$ | $274.3$ | $0.22$

In addition to the stability verification, the elastic properties were
determined. By symmetry only two independent stiffness constants $C_{11}$ and
$C_{12}$ exist, where we neglect any out-of-plane bending contribution to the
elastic energy density, i.e. the vertical weight of the stifness constants
vanishes. The stiffness constants were determined by evaluating the total
energy of the system under uniform uni- and bi-axial deformations, over a
range of $-10$ % to $+10$ %. For all deformations, the atomic positions were
optimized and the corresponding energies were determined. In this way, the
stiffness constants, in-plane Young’s modulus
$Y_{s}=(C_{11}^{2}-C_{12}^{2})/C_{11}$ and Poisson’s ratio $\nu=C_{12}/C_{11}$
were calculated. The resulting values are summarized in Table 1, and compared
to other 2D materials.

The mechanical stability could also be confirmed by verifying the positivity
of the strain energy density function
$U(\epsilon)=\frac{1}{2}C_{ijkl}\epsilon_{ij}\epsilon_{kl}$ for all
$|\epsilon|\leq 10$ %. For the 2D hexagonal systems this translates into
satisfying the conditions $C_{11}>0$ and $C_{11}>|C_{12}|$. Hence, it can be
concluded that the single atomic layer of $\ce{V_{2}O_{3}}$ with HK lattice
structure is stable on all three levels: dynamical, thermal and mechanical.

###  III.2 Magnetic ground state

To determine the magnetic ground state of the system various magnetic
configurations have been studied. The total energy of the ion-electron system
was determined for ferromagnetic (FM), paramagnetic (PM), and four types of
antiferromagnetic (AFM) states, depicted in Fig. 2a. By comparison of the
total energies in Table 2, 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, the following spin hamiltonian was
considered,

$H=-\sum_{\langle
i,j\rangle}J_{1}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{i}A(\mathbf{S}_{i}^{z})^{2}$
(1)

where $i$ enumerates the magnetic ions. The first term is the nearest-neighbor
exchange interaction with exchange parameter $J_{1}$, while the second term
corresponds to the single-ion anisotropy with the anisotropy parameter $A$.
The interaction is ferromagnetic when $J_{1}>0$ and positive $A$ will favor an
out-of-plane spin component $\mathbf{S}^{z}$.

Table 2: The total energies (eV/f.u.) for different (out-of-plane) magnetic configurations with respect to the lowest FM energy state which is set to zero.  $\begin{aligned} \textrm{Magnetic}\\\ \textrm{ordering}\end{aligned}$ | FM | PM | AFM-N | AFM-ST | AFM-ZZ | AFM-N-ST
---|---|---|---|---|---|---
$\begin{aligned} \textrm{Energy}\\\ \textrm{(eV/f.u.)}\end{aligned}$ | $0$ | $1.54$ | $0.35$ | $0.18$ | $0.20$ | $0.42$

The exchange parameter is evaluated by $J_{1}=[E_{AFM}-E_{FM}]/2zS^{2}$ with
the total spin moment $S=1$, and with $z=3$ neighboring spins. This gives
$J_{1}=58$ meV. On the other hand, the magnetocrystalline anisotropy energy
(MAE) defined by $E_{MAE}=E_{[100]}-E_{[001]}$, 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 $\Gamma$ and valence band maximum (VBM) at K, respectively
consist of the $(d_{xy},d_{x^{2}-y^{2}})$\- and $(d_{xz},d_{yz})$-orbitals.
This SOC-induced interaction involves a change in the orbital angular momentum
quantum number $L_{z}$ from $(d_{xz},d_{yz})$ with $L_{z}=\pm 1$ to
$(d_{xy},d_{x^{2}-y^{2}})$ with $L_{z}=\pm 2$. This interaction between these
out-of-plane spins leads to a maximization of the SOC energy stabilization, as
described in (SOC_anisotropy, ).

Figure 2: Magnetic configurations and Curie temperature. (a) Top view of
various possible AFM configurations. (b) The normalized magnetization as a
function of the temperature from the MC simulations indicating a $T_{C}=447$
K.

To obtain an estimate of the Curie temperature ($T_{C}$), the Curie-Bloch
equation in the classical limit,

$m(T)=\left(1-\frac{T}{T_{C}}\right)^{\beta}$ (2)

with $T$ the temperature and $\beta$ a critical exponent, is fitted to the
normalized magnetization obtained by the MC simulation (see Fig. 2b). The
resulting $T_{C}$ is equal to $447$ K, and can also be compared within the
mean field approximation by using the formula

$T_{C}=\frac{{S(S+1)}}{3k_{B}}J_{1}$ (3)

with Boltzmann constant $k_{B}$ , which gives $449$ K. This is in relative
close agreement with the MC result, which can be attributed to the relatively
high MAE.

These MC simulations treat the spins as classical vector quantities.
Alternatively, the semi-classical Holstein-Primakoff approximation is often
employed. However, both approximations remain only justified for high spin
values and very low temperatures. Therefore, a fully quantum mechanical method
is employed, where the quantum Heisenberg hamiltonian is solved by Green’s
functions (jorenmodel, ). Since the Néel AFM phase is not the second lowest
energy state, it can be expected that second- and third-neighbor exchange
interactions ($J_{2},J_{3}$) are playing a dominant role in the magnetic
ordering. Therefore, the DFT total energies were mapped onto the Ising
hamiltonian to obtain $J_{1}$, $J_{2}$ and $J_{3}$ by the following four
equations:

$E_{FM/AFM-N}=E_{PM}-(\pm 3J_{1}+6J_{2}\pm 3J_{3})S^{2},$ (4)

and

$E_{AFM-ZZ/ST}=E_{PM}-(\pm J_{1}-2J_{2}\mp 3J_{3})S^{2}.$ (5)

The $J_{1}$, $J_{2}$ and $J_{3}$ values were found to be $41.9$ meV, $-3.8$
meV 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 Green’s functions, giving a Curie temperature equal to $488.46$
K.

###  III.3 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 _$\mathrm{K}$_ point and the existence of two flat-bands at
$\pm 1$ eV forming a degeneracy with the dispersive bands at the $\Gamma$
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 $\ce{V}$ $(d_{xz},d_{yz})$-orbitals. This means that the Dirac cone is
completely spin-polarized, making it a Dirac spin-gapless semiconductor (DSGS)
(WangSGS, ). By taking into account the intrinsic SOC, it is found that an
energy gap is opened at the $\mathrm{K}$ 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 Material), 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 process (atomicOnsiteSOC,
). This is in contrast to the second-order hopping SOC in the Kane-Mele model,
or first-order hopping SOC in the Xenes (Xenes, ). 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 SOC (OwnPaper, ;
MellaertsSimon2020TEi2, ).

Figure 3: Electronic band structure with derived orbital and topological
character. (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 $\ce{\ce{{V}}}$ $3d$ orbitals.

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 3, and the calculated band
structures can be found in the Supplementary Material. 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 3: The calculated direct and indirect energy gap $E_{g}$ values for the different exchange-correlation functionals with SOC included. | LDA+$U$ | GGA+$U$ | HSE06
---|---|---|---
Direct $E_{g}$ (eV) | $0.37$ | $0.54$ | $1.54$
Indirect $E_{g}$ (eV) | $0.19$ | $0.45$ | $1.52$

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 $\ce{\ce{{O}}}$ $2p$ orbitals and thus has the lowest energy state, the
($d_{xz},d_{yz})$-orbitals oriented out-of-plane will form a $\pi$-bond with
the oxygen $\ce{\ce{{O}}}$ $p_{z}$ as bridging ligand, and the
$(d_{xy},d_{x^{2}-y^{2}})$-orbitals will form an in-plane $\sigma$-bond with
bridging orbitals $\ce{\ce{{O}}}$ $(p_{x},p_{y})$. 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 $\ce{\ce{{O}}}$ $2p$
orbital weight, which confirms the localized nature of the two $\ce{V}$ $3d$
electrons. Based on Griffith’s crystal field theory (griffith1957ligand, ),
the spin state of the $\ce{\ce{{V}}}$3+ cation can be confirmed. By the
relative strengths of the exchange ($\Delta E_{\textrm{ex}}$) and crystal
field splitting ($\Delta E_{\textrm{CF}}$) with derived values of $3.34$ eV
and $0.8$ eV - it can be concluded that the $\ce{\ce{{V}}}$ cation has a high
($S=1$) spin state with magnetic moment $2\mu_{B}$.

To gain further insight in the nature of the chemical bond, the atomic charges
were determined by the Bader charge analysis code (Bader, ). It is found that
the atomic charges on $\ce{\ce{{V}}}$ and $\ce{\ce{{O}}}$ are resp.
$Q_{V}=1.59e$ and $Q_{O}=-1.07e$, confirming the ionic bonding character where
the electrons of the $\ce{\ce{{V}}}$ cation are attracted towards the
$\ce{\ce{{O}}}$ 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 $\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ 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 $\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$
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
TMOs (HKAl2O3, ; HKY2O3, ).

To study the topology of the band structure the Berry curvature
$\boldsymbol{\Omega}(\mathbf{k})$ of the system was calculated directly from
the DFT calculated wave functions by the VASPBERRY code, which is based on
Fukui’s method (VASPBERRY, ). The resulting Berry curvature, shown in Fig. 3c,
becomes non-zero at the Dirac points $\mathrm{K}$ and $\mathrm{K^{\prime}}$.
By integrating the Berry curvature $\boldsymbol{\Omega}_{n}(\mathbf{k})$ of
each $n$-th energy band over the whole BZ and summing over all occupied bands,
the Chern number

$C=\sum_{n}\frac{1}{2\pi}\int_{BZ}d^{2}\mathbf{k}\,\boldsymbol{\Omega}_{n}(\mathbf{k})$
(6)

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

Figure 4: Biaxial strain study. (a) The total energy of the system under
biaxial strain, taking into account (b) the possible structural deformations
($HK_{in}$, $HK_{out}$) with deformation parameters the bond angle $\theta$
and the buckling height $h$ respectively. (c) Top: the
$\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond length as a function of strain for all
three configurations. Bottom: the bond angle $\theta$ and and buckling height
$h$ in function of the strain for resp. $HK_{in}$ and $HK_{out}$. (d) The
energy gap and magnetic exchange interaction as a function of strain.

###  III.4 Biaxial strain

TMOs are known to exhibit a rich phase diagram as a function of strain, in
particular, for bulk $\ce{V_{2}O_{3}}$ the room-temperature metal-insulator
transition (MIT) between the paramagnetic insulating and metallic state can be
realized by the application of epitaxial strain (PiaThesis, ). Therefore, a
biaxial strain study on the 2D atomic layer of $\ce{V_{2}O_{3}}$ was
performed. The atomic positions were optimized for all biaxial strained
configurations from $-10$ % to $+10$ %. It was found that under compressive
strain, the lattice structure undergoes an in-plane buckling of the
$\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond ($HK_{in}$), as can be seen Fig. 4a-b.
This structural distortion is found by ionic relaxation with 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
$\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond ($HK_{out}$). Although these lattice
configurations are un/meta-stable, 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 substrates (NogueraSubstrate, ).

Figure 5: Graphene substrate study. (a) The top and side view of $(2\times 2)$
$\ce{V_{2}O_{3}}$ supercell on a $(5\times 5)$ graphene supercell. (b) The
calculated charge density difference $\Delta\rho$ 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 $\ce{V_{2}O_{3}}$ op graphene
within GGA+$U$+SOC. The color code is the same used in (c).

Identifying these different structural configurations under compressive strain
in the freestanding $\ce{V_{2}O_{3}}$ 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 $\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond develops as 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
$\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ 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 $HK_{out}$) will depend on external
conditions, only the most stable $HK_{in}$ 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 $HK_{in}$. This can be expected as the $\ce{\ce{{V}}}$
cations still form a honeycomb lattice, crucial for the existence of the Dirac
cone (existence, ). From these results, it can be concluded that the room-
temperature QAHE in 2D $\ce{V_{2}O_{3}}$ is robust against small structural
deformations.

###  III.5 Graphene substrate

Motivated by the successful deposition of $\ce{Y_{2}O_{3}}$ on graphene
(growthY2O3, ), the feasibility of using a graphene substrate for the growth
of the $\ce{V_{2}O_{3}}$ monolayer was studied. A $(2\times 2)$
$\ce{V_{2}O_{3}}$ supercell on a $(5\times 5)$ graphene supercell was
structurally optimized and that configuration is shown in Fig. 5a (see
Supplemental Material). Similar to $\ce{Al_{2}O_{3}}$ and $\ce{Y_{2}O_{3}}$
monolayers on graphene (HKAl2O3, ; HKY2O3, ), the $\ce{\ce{{V}}}$ cations are
located near the top sites of the $\ce{\ce{{C}}}$ atoms to maximize the
potential bonding. The planar HK lattice structure is conserved with a minimal
distance between graphene and $\ce{V_{2}O_{3}}$ of $3.43\>\text{\AA}$, while
the $\ce{\ce{{V}}}$-$\ce{\ce{{O}}}$ bond length remains approximately
preserved.

To obtain a first estimate of the interfacial interaction strength, the
adsorbtion energy $E_{\text{{ad}}}$ is determined,

$E_{\text{{ad}}}=E_{G}+E_{V_{2}O_{3}}-E_{G+V_{2}O_{3}}$ (7)

where $E_{G}$, $E_{V_{2}O_{3}}$ and $E_{G+V_{2}O_{3}}$ are the total energies
of the isolated graphene, isolated $\ce{V_{2}O_{3}}$, and the combined hybrid
structure, respectively. The adsorbtion energy equals $77$ meV per
$\ce{\ce{{C}}}$ atom indicating a weak interaction between both monolayers.

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
$\ce{\ce{{V}}}$ and $\ce{\ce{{O}}}$ remain unchanged, while the average excess
charge of the $\ce{\ce{{C}}}$ 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
$\Delta\rho=\rho_{G+V_{2}O_{3}}-\rho_{G}-\rho_{V_{2}O_{3}}$, shown in Fig. 5b.
The charge density difference shows a negligibly small inhomogeneous charge
distribution between both monolayers, which is mainly located between
$\ce{\ce{{V}}}$ and $\ce{\ce{{C}}}$ atoms. This indicates a small orbital
hybridization of the $\pi$-bonded orbitals of $\ce{V_{2}O_{3}}$ 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 for the
electronic properties at the interface (HKAl2O3, ; HKY2O3, ).

The electronic properties of the $\ce{V_{2}O_{3}}$ 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. The DOS contribution of
the $\ce{V_{2}O_{3}}$ remains approximately preserved, however from the
calculated band structure, it is clear that there is some hybridization of the
$\pi$-bonded orbitals. Nonetheless, the Dirac cone of $\ce{V_{2}O_{3}}$
remains preserved with a reduction in the SOC-induced direct energy gap to
$66$ meV.

##  IV Conclusions

In this work, we have confirmed that the single atomic layer of
$\ce{V_{2}O_{3}}$ 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
$\ce{\ce{{V}}}$ cations, such that the Dirac cone and the corresponding room-
temperature QAHE remain unaffected. Furthermore, this 2D $\ce{V_{2}O_{3}}$ 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 structure
(HKY2O3, ; HKAl2O3, ; growthY2O3, ; HKMn2O3, ; HKCr2O3, ; 2DTMOscreening, )
should encourage the experimental investigation of this group of 2D TMOs.

## Data Availability

The data that supports the findings of this study are provided here and the
Supplementary Material.

## Acknowledgements

Part of this work was financially supported by the KU Leuven Research Funds,
Project No. KAC24/18/056 and No. C14/17/080 as well as the Research Funds of
the INTERREG-E-TEST Project (EMR113) and INTERREG-VL-NL-ETPATHFINDER Project
(0559). Part of the computational resources and services used in this work
were provided by the VSC (Flemish Supercomputer Center) funded by the Research
Foundation Flanders (FWO) and the Flemish government.

## References

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  * [2] P. R. Wallace.  The band theory of graphite.  Phys. Rev., 71:622–634, May 1947.
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  * [5] Xu Du, Ivan Skachko, Fabian Duerr, Adina Luican, and Eva Y. Andrei.  Fractional quantum hall effect and insulating phase of dirac electrons in graphene.  Nature, 462(7270):192–195, 2009.
  * [6] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov.  Electric field effect in atomically thin carbon films.  Science, 306(5696):666–669, 2004.
  * [7] Ying-Hao Chu.  Van der waals oxide heteroepitaxy.  npj Quantum Materials, 2(1):67, 2017.
  * [8] Tanushree H. Choudhury, Xiaotian Zhang, Zakaria Y. Al Balushi, Mikhail Chubarov, and Joan M. Redwing.  Epitaxial growth of two-dimensional layered transition metal dichalcogenides.  Annual Review of Materials Research, 50(1):155–177, 2020.
  * [9] F. D. M. Haldane.  Model for a quantum hall effect without landau levels: Condensed-matter realization of the "parity anomaly".  Phys. Rev. Lett., 61:2015–2018, Oct 1988.
  * [10] C. L. Kane and E. J. Mele.  Quantum spin hall effect in graphene.  Phys. Rev. Lett., 95:226801, Nov 2005.
  * [11] H. P. Wang, Wei Luo, and H. J. Xiang.  Prediction of high-temperature quantum anomalous hall effect in two-dimensional transition-metal oxides.  Phys. Rev. B, 95:125430, Mar 2017.
  * [12] G. Kresse and J. Furthmüller.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.  Phys. Rev. B, 54:11169–11186, Oct 1996.
  * [13] John P. Perdew, Kieron Burke, and Matthias Ernzerhof.  Generalized gradient approximation made simple.  Phys. Rev. Lett., 77:3865–3868, Oct 1996.
  * [14] G. Kresse and D. Joubert.  From ultrasoft pseudopotentials to the projector augmented-wave method.  Phys. Rev. B, 59:1758–1775, Jan 1999.
  * [15] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton.  Electron-energy-loss spectra and the structural stability of nickel oxide: An lsda+u study.  Phys. Rev. B, 57:1505–1509, Jan 1998.
  * [16] Matteo Cococcioni and Stefano de Gironcoli.  Linear response approach to the calculation of the effective interaction parameters in the $\mathrm{LDA}+\mathrm{U}$ method.  Phys. Rev. B, 71:035105, Jan 2005.
  * [17] Daniel Grieger and Michele Fabrizio.  Low-temperature magnetic ordering and structural distortions in vanadium sesquioxide ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$.  Phys. Rev. B, 92:075121, Aug 2015.
  * [18] D. M. Ceperley and B. J. Alder.  Ground state of the electron gas by a stochastic method.  Phys. Rev. Lett., 45:566–569, Aug 1980.
  * [19] Aliaksandr V. Krukau, Oleg A. Vydrov, Artur F. Izmaylov, and Gustavo E. Scuseria.  Influence of the exchange screening parameter on the performance of screened hybrid functionals.  The Journal of Chemical Physics, 125(22):224106, 2006.
  * [20] Stefan Grimme, Jens Antony, Stephan Ehrlich, and Helge Krieg.  A consistent and accurate ab initio parametrization of density functional dispersion correction (dft-d) for the 94 elements h-pu.  The Journal of Chemical Physics, 132(15):154104, 2010.
  * [21] A Togo and I Tanaka.  First principles phonon calculations in materials science.  Scr. Mater., 108:1–5, Nov 2015.
  * [22] William G. Hoover.  Canonical dynamics: Equilibrium phase-space distributions.  Phys. Rev. A, 31:1695–1697, Mar 1985.
  * [23] Shuichi Nose.  A unified formulation of the constant temperature molecular dynamics methods.  The Journal of Chemical Physics, 81(1):511–519, 1984.
  * [24] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell.  Atomistic spin model simulations of magnetic nanomaterials.  Journal of Physics: Condensed Matter, 26(10):103202, Feb 2014.
  * [25] Joren Vanherck, Cihan Bacaksiz, Bart Soree, Milorad V. Milosevic, and Wim Magnus.  2d ferromagnetism at finite temperatures under quantum scrutiny.  Applied Physics Letters, 117(5):052401, 2020.
  * [26] Koichi Momma and Fujio Izumi.  VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data.  Journal of Applied Crystallography, 44(6):1272–1276, Dec 2011.
  * [27] Yuzheng Guo, Stewart J. Clark, and John Robertson.  Calculation of metallic and insulating phases of ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$ by hybrid density functionals.  The Journal of Chemical Physics, 140(5):054702, 2014.
  * [28] Ting Ting Song, Ming Yang, Jian Wei Chai, Martin Callsen, Jun Zhou, Tong Yang, Zheng Zhang, Ji Sheng Pan, Dong Zhi Chi, Yuan Ping Feng, and Shi Jie Wang.  The stability of aluminium oxide monolayer and its interface with two-dimensional materials.  Scientific Reports, 6(1):29221, 2016.
  * [29] Ting Ting Song, Ming Yang, Martin Callsen, Qing Yun Wu, Jun Zhou, Shao Feng Wang, Shi Jie Wang, and Yuan Ping Feng.  Graphene stabilized high-k dielectric ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ $(111)$ monolayers and their interfacial properties.  RSC Adv., 5:83588–83593, 2015.
  * [30] Heleen van Gog, Wun-Fan Li, Changming Fang, Rik S. Koster, Marjolein Dijkstra, and Marijn van Huis.  Thermal stability and electronic and magnetic properties of atomically thin 2d transition metal oxides.  npj 2D Materials and Applications, 3(1):18, 2019.
  * [31] Myung-Hwan Whangbo, Elijah E. Gordon, Hongjun Xiang, Hyun-Joo Koo, and Changhoon Lee.  Prediction of spin orientations in terms of homo-lumo interactions using spin–orbit coupling as perturbation.  Accounts of Chemical Research, 48(12):3080–3087, Dec 2015.
  * [32] X. L. Wang.  Proposal for a new class of materials: Spin gapless semiconductors.  Phys. Rev. Lett., 100:156404, Apr 2008.
  * [33] Cheng-Cheng Liu, Shan Guan, Zhigang Song, Shengyuan A. Yang, Jinbo Yang, and Yugui Yao.  Low-energy effective hamiltonian for giant-gap quantum spin hall insulators in honeycomb $\mathrm{X}$-hydride/halide $(\mathrm{X}=\mathrm{N}-\mathrm{Bi})$ monolayers.  Phys. Rev. B, 90:085431, Aug 2014.
  * [34] Alessandro Molle, Joshua Goldberger, Michel Houssa, Yong Xu, Shou-Cheng Zhang, and Deji Akinwande.  Buckled two-dimensional xene sheets.  Nature Materials, 16(2):163–169, 2017.
  * [35] S. Mellaerts, R. Meng, V. Afanasiev, J. W. Seo, M. Houssa, and J.-P. Locquet.  Quarter-filled kane-mele hubbard model: Dirac half-metals.  Phys. Rev. B under review, 2020.
  * [36] Simon Mellaerts.  Topological effects in 2d strongly correlated electron systems, 2020.
  * [37] JS Griffith and LE Orgel.  Ligand-field theory.  Quarterly Reviews, Chemical Society, 11(4):381–393, 1957.
  * [38] Min Yu and Dallas R. Trinkle.  Accurate and efficient algorithm for bader charge integration.  The Journal of Chemical Physics, 134(6):064111, 2011.
  * [39] Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki.  Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances.  Journal of the Physical Society of Japan, 74(6):1674–1677, 2005\.
  * [40] P Homm Jara.  Electronic, structural and optical phase transitions in strongly correlated oxide thin films grown by molecular beam epitaxy, 2019.
  * [41] J. Goniakowski and C. Noguera.  Properties of metal-supported oxide honeycomb monolayers: ${\mathrm{M}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{MM^{\prime}}}_{2}{\mathrm{O}}_{3}$ on $\mathrm{Me}(111)$ $(\mathrm{M},\mathrm{M}=\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Fe};\mathrm{Me}=\mathrm{Ag},\mathrm{Au},\mathrm{Pt})$.  The Journal of Physical Chemistry C, 124(15):8186–8197, Apr 2020\.
  * [42] Zhirong Liu, Jinying Wang, and Jianlong Li.  Dirac cones in two-dimensional systems: from hexagonal to square lattices.  Phys. Chem. Chem. Phys., 15:18855–18862, 2013.
  * [43] Rafik Addou, Arjun Dahal, and Matthias Batzill.  Growth of a two-dimensional dielectric monolayer on quasi-freestanding graphene.  Nature Nanotechnology, 8(1):41–45, 2013.
  * [44] Ping Li and Tian-Yi Cai.  Two-dimensional transition-metal oxides ${\mathrm{Mn}}_{2}{\mathrm{O}}_{3}$ realized the quantum anomalous hall effect.  The Journal of Physical Chemistry C, 124(23):12705–12712, Jun 2020\.
  * [45] Arqum Hashmi, Kenta Nakanishi, Muhammad Umar Farooq, and Tomoya Ono.  Ising ferromagnetism and robust half-metallicity in two-dimensional honeycomb-kagome ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ layer.  npj 2D Materials and Applications, 4(1):39, 2020.

## Supplementary Material

### Stiffness constants

From continuum mechanics, a general equation for the elastic energy for a
deformation of a 2D material can be obtained:

$E=E_{0}+A_{0}\sum_{i}\sum_{j}\sigma_{ij}\epsilon_{ij}+\frac{A_{0}}{2}\sum_{i}\sum_{j}\epsilon_{ij}\cdot\left(\sum_{k}\sum_{l}C_{ijkl}\epsilon_{kl}\right)$
(8)

where $E_{0}$ is the equilibrium energy, $A_{0}$ the equilibrium unit cell
area, $\sigma_{ij}$ the rank $2$ stress tensor, $\epsilon_{ij}$ the rank $2$
strain tensor, and $C_{ijkl}$ the rank $4$ stiffness tensor. To simplify this
equation with high-rank tensors, the Voigt notation [voigt1928lehrbuch] will
be adopted. Furthermore, by the symmetries of the hexagonal lattice, the
stiffness matrix reduces to a $3\times 3$ matrix of the form
[SymmetrieElasticity]

$\left(\begin{array}[]{ccc}C_{11}&C_{12}&0\\\ C_{12}&C_{11}&0\\\
0&0&C_{33}\end{array}\right),$ (9)

with three independent stiffness constants $C_{11},$$C_{12}$ and $C_{33}$.
Since only in-plane deformations are considered, $C_{33}$ is set to zero. To
determine the stiffness constants, we consider a uni- and biaxial deformation
respectively given by the strain tensors

$\epsilon_{11}=\left(\begin{array}[]{cc}\delta\\\ 0\\\
0\end{array}\right),\>\epsilon_{12}=\left(\begin{array}[]{cc}\delta\\\
\delta\\\ 0\end{array}\right)$ (10)

with $\delta$ a dimensional quantity, representing the stength of the
deformation, defined by the relative change in the lattice constant along the
axis of deformation,

$\delta=\frac{|a-a_{0}|}{a}$ (11)

where $a_{0}$ and $a$ are the lattice constants along the deformation
direction of respectively the equilibrium and deformed structure. Using Eq. 8,
the energy change associated with both deformations is given by

$\displaystyle
E_{11}(\delta)=E_{0}+A_{0}\sigma_{1}\delta+\frac{A_{0}}{2}C_{11}\delta^{2}+O(\delta^{3}),$
(12) $\displaystyle
E_{12}(\delta)=E_{0}+A_{0}(\sigma_{1}+\sigma_{2})\delta+A_{0}(C_{11}+C_{12})\delta^{2}+O(\delta^{3}).$

The calculated total energies of the deformed configurations are shown in Fig.
6 and the corresponding functions in Eq. 12 are fitted to these energies.

Figure 6: The calculated total energies for the two types of deformations
shown on the right and the corresponding fit on the left.

### Estimate of the Hubbard correction

This linear response calculation is based on [16] and is divided into three
steps. First, a self-consistent electronic calculation is performed without
the presence of any perturbation $\alpha$. Secondly, a non-charge-self-
consistent calculation is performed with the presence of a non-zero
pertubartion $\alpha.$ In this calculation, the charge density is not allowed
to relax and thus a _bare_ (non-interacting) response is obtained. Lastly, a
charge-self-consistent claculation with non-zero $\alpha$ is employed where
the charge density is allowed to relax and to screen the perturbation. This
provides the _interacting_ response of the system. Both bare and interacting
response can be respectively quantified by the response density functions
[16]:

$\displaystyle\chi_{ij}=\frac{\partial^{2}E}{\partial\alpha_{I}\partial\alpha_{J}}=\frac{\partial
n_{I}}{\partial\alpha_{J}},$ (13)
$\displaystyle\chi_{IJ}^{0}=\frac{\partial^{2}E^{KS}}{\partial\alpha_{I}^{KS}\partial\alpha_{J}^{KS}}=\frac{\partial
n_{I}}{\partial\alpha_{J}^{KS}}$ (14)

where $I,$ $J$ are the $\ce{\ce{{V}}}$-sites in the supercell, and the $KS$
superscript stands for the non-interacting (Kohn-Sham) response.

Figure 7: (a) The DFT calculated interacting and non-interacting response
functions. (b) Indirect energy gap as a function of the Hubbard correction
$U$.

In order to obtain an accurate estimate of $U$, the calculation of the
response density functions should be repeated for increasing supercell
dimensions, until there is convergence. However, a $2\times 2\times 1$
supercell containing $24$ atoms should suffice for this calculation, with
larger supercells giving minor improvements. Subsequently, the effective
interaction parameter $U$ can be determined by

$U=\left(\chi_{0}^{-1}-\chi^{-1}\right)_{IJ}$ (15)

where $IJ$ represents the evaluation of the response density for a
perturbation a site $J$ and its response measured at site $I$. These
calculations were performed for a perturbation range of $0.4$ eV with stepsize
$0.05$ eV. By linear fitting, illustrated in Fig. 7a, the response density
functions were determined from the slope and using Eq. 15 a Hubbard correction
of $U=3.28$ eV was found.

### Electronic structure in function of exchange-correlation functional

#### Effect of Hubbard correction

The electronic band structure was calculated for a Hubbard correction ranging
from $2.5$ eV to $4.5$ eV. Without the inclusion of the spin-orbit coupling
(SOC), it is observed that there is less overlap of the different orbital
levels for increasing $U$, nonetheless, the overall effect is relatively
small. By the inclusion of SOC, it is clear that the energy gap opened at the
Dirac point strongly depends on the Hubbard correction $U$, as shown in Fig.
7b.

Figure 8: The comparison of the calculated band structure within LDA+$U$,
GGA+$U$ and HSE06, with and without inclusion of SOC.

#### Exchange-correlation functional dependence

To test the sensitvity of the results of the electronic band structure to the
choice of the functional, the three most common exchange-correlation
functionals were employed. The generalized gradient approximation (GGA) in the
form of Perdew-Burke Ernzerhof (PBE) [13], the local density approximation
(LDA) [18] and the screened exchange hybrid density functional by Heyd-
Scuseria-Ernzerhof (HSE06) [19]. The calculated band structures for LDA+$U$
and HSE06 are shown in Fig. 8 and compared with the band structure obtained by
GGA+$U$.

### Topology of the band structure

In addition to the topological energy gap at the Fermi level with a Chern
number equal to one, the Chern number was also calculated for the gap opening
at the $\Gamma$ point around $-0.9$ eV below the Fermi level. The calculated
Berry cruvature is shown in Fig. 9 and was integrated by the VASPBERRY code,
which is based on Fukui’s method [39].

Figure 9: The calculated band structure and the corresponding Berry cruvature
projected onto the Brillouin zone at the Fermi level and at $-0.9$ eV.

### Graphene substrate

For the ionic relaxation of the graphene supported $\ce{V_{2}O_{3}}$ monolayer
system, several configurations of the $\ce{V_{2}O_{3}}$ monolayer with respect
to the graphene layer have been considered, as shown in Fig. 10. For all
configurations, the supercell was constructed with completely relaxed
structures of graphene and $\ce{V_{2}O_{3}}$ , with freestanding lattice
constants of $2.477\>\text{\AA}$ and $6.193\>\text{\AA}$, respectively. By
comparing the total energies of the different configurations, it was confirmed
that the other considered configurations are very close in energy ($<10^{-3}$
eV), and as the superstructure of graphene on $\ce{V_{2}O_{3}}$ is
incommensurate it is expected that for any configuration the $\ce{\ce{{V}}}$
cations will be located approximately on top of the $\ce{\ce{{C}}}$ atoms.
Furthermore, it was verified that the density of states remains approximately
preserved in these other configurations. At last, the initial distance between
both monolayers was varied to confirm absolute stability. The most stable
configuration was found to have unchanged structural parameters in both
monolayers, i.e. the vdW epitaxy does not induce any significant strain within
the system.

Figure 10: Two different superstructures with the $(2\times 2)$
$\ce{V_{2}O_{3}}$ monolayer with respect to the $(5\times 5)$ graphene layer.
The atom colors gray, cyan, and red corresponding to carbon, vanadium and
oxygen respectively.

## References

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  * [2] P. R. Wallace.  The band theory of graphite.  Phys. Rev., 71:622–634, May 1947.
  * [3] K.I. Bolotin, K.J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H.L. Stormer.  Ultrahigh electron mobility in suspended graphene.  Solid State Communications, 146(9):351 – 355, 2008.
  * [4] Yuanbo Zhang, Yan-Wen Tan, Horst L. Stormer, and Philip Kim.  Experimental observation of the quantum hall effect and berry’s phase in graphene.  Nature, 438(7065):201–204, 2005.
  * [5] Xu Du, Ivan Skachko, Fabian Duerr, Adina Luican, and Eva Y. Andrei.  Fractional quantum hall effect and insulating phase of dirac electrons in graphene.  Nature, 462(7270):192–195, 2009.
  * [6] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov.  Electric field effect in atomically thin carbon films.  Science, 306(5696):666–669, 2004.
  * [7] Ying-Hao Chu.  Van der waals oxide heteroepitaxy.  npj Quantum Materials, 2(1):67, 2017.
  * [8] Tanushree H. Choudhury, Xiaotian Zhang, Zakaria Y. Al Balushi, Mikhail Chubarov, and Joan M. Redwing.  Epitaxial growth of two-dimensional layered transition metal dichalcogenides.  Annual Review of Materials Research, 50(1):155–177, 2020.
  * [9] F. D. M. Haldane.  Model for a quantum hall effect without landau levels: Condensed-matter realization of the "parity anomaly".  Phys. Rev. Lett., 61:2015–2018, Oct 1988.
  * [10] C. L. Kane and E. J. Mele.  Quantum spin hall effect in graphene.  Phys. Rev. Lett., 95:226801, Nov 2005.
  * [11] H. P. Wang, Wei Luo, and H. J. Xiang.  Prediction of high-temperature quantum anomalous hall effect in two-dimensional transition-metal oxides.  Phys. Rev. B, 95:125430, Mar 2017.
  * [12] G. Kresse and J. Furthmüller.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.  Phys. Rev. B, 54:11169–11186, Oct 1996.
  * [13] John P. Perdew, Kieron Burke, and Matthias Ernzerhof.  Generalized gradient approximation made simple.  Phys. Rev. Lett., 77:3865–3868, Oct 1996.
  * [14] G. Kresse and D. Joubert.  From ultrasoft pseudopotentials to the projector augmented-wave method.  Phys. Rev. B, 59:1758–1775, Jan 1999.
  * [15] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton.  Electron-energy-loss spectra and the structural stability of nickel oxide: An lsda+u study.  Phys. Rev. B, 57:1505–1509, Jan 1998.
  * [16] Matteo Cococcioni and Stefano de Gironcoli.  Linear response approach to the calculation of the effective interaction parameters in the $\mathrm{LDA}+\mathrm{U}$ method.  Phys. Rev. B, 71:035105, Jan 2005.
  * [17] Daniel Grieger and Michele Fabrizio.  Low-temperature magnetic ordering and structural distortions in vanadium sesquioxide ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$.  Phys. Rev. B, 92:075121, Aug 2015.
  * [18] D. M. Ceperley and B. J. Alder.  Ground state of the electron gas by a stochastic method.  Phys. Rev. Lett., 45:566–569, Aug 1980.
  * [19] Aliaksandr V. Krukau, Oleg A. Vydrov, Artur F. Izmaylov, and Gustavo E. Scuseria.  Influence of the exchange screening parameter on the performance of screened hybrid functionals.  The Journal of Chemical Physics, 125(22):224106, 2006.
  * [20] Stefan Grimme, Jens Antony, Stephan Ehrlich, and Helge Krieg.  A consistent and accurate ab initio parametrization of density functional dispersion correction (dft-d) for the 94 elements h-pu.  The Journal of Chemical Physics, 132(15):154104, 2010.
  * [21] A Togo and I Tanaka.  First principles phonon calculations in materials science.  Scr. Mater., 108:1–5, Nov 2015.
  * [22] William G. Hoover.  Canonical dynamics: Equilibrium phase-space distributions.  Phys. Rev. A, 31:1695–1697, Mar 1985.
  * [23] Shuichi Nose.  A unified formulation of the constant temperature molecular dynamics methods.  The Journal of Chemical Physics, 81(1):511–519, 1984.
  * [24] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell.  Atomistic spin model simulations of magnetic nanomaterials.  Journal of Physics: Condensed Matter, 26(10):103202, Feb 2014.
  * [25] Joren Vanherck, Cihan Bacaksiz, Bart Soree, Milorad V. Milosevic, and Wim Magnus.  2d ferromagnetism at finite temperatures under quantum scrutiny.  Applied Physics Letters, 117(5):052401, 2020.
  * [26] Koichi Momma and Fujio Izumi.  VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data.  Journal of Applied Crystallography, 44(6):1272–1276, Dec 2011.
  * [27] Yuzheng Guo, Stewart J. Clark, and John Robertson.  Calculation of metallic and insulating phases of ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$ by hybrid density functionals.  The Journal of Chemical Physics, 140(5):054702, 2014.
  * [28] Ting Ting Song, Ming Yang, Jian Wei Chai, Martin Callsen, Jun Zhou, Tong Yang, Zheng Zhang, Ji Sheng Pan, Dong Zhi Chi, Yuan Ping Feng, and Shi Jie Wang.  The stability of aluminium oxide monolayer and its interface with two-dimensional materials.  Scientific Reports, 6(1):29221, 2016.
  * [29] Ting Ting Song, Ming Yang, Martin Callsen, Qing Yun Wu, Jun Zhou, Shao Feng Wang, Shi Jie Wang, and Yuan Ping Feng.  Graphene stabilized high-k dielectric ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ $(111)$ monolayers and their interfacial properties.  RSC Adv., 5:83588–83593, 2015.
  * [30] Heleen van Gog, Wun-Fan Li, Changming Fang, Rik S. Koster, Marjolein Dijkstra, and Marijn van Huis.  Thermal stability and electronic and magnetic properties of atomically thin 2d transition metal oxides.  npj 2D Materials and Applications, 3(1):18, 2019.
  * [31] Myung-Hwan Whangbo, Elijah E. Gordon, Hongjun Xiang, Hyun-Joo Koo, and Changhoon Lee.  Prediction of spin orientations in terms of homo-lumo interactions using spin–orbit coupling as perturbation.  Accounts of Chemical Research, 48(12):3080–3087, Dec 2015.
  * [32] X. L. Wang.  Proposal for a new class of materials: Spin gapless semiconductors.  Phys. Rev. Lett., 100:156404, Apr 2008.
  * [33] Cheng-Cheng Liu, Shan Guan, Zhigang Song, Shengyuan A. Yang, Jinbo Yang, and Yugui Yao.  Low-energy effective hamiltonian for giant-gap quantum spin hall insulators in honeycomb $\mathrm{X}$-hydride/halide $(\mathrm{X}=\mathrm{N}-\mathrm{Bi})$ monolayers.  Phys. Rev. B, 90:085431, Aug 2014.
  * [34] Alessandro Molle, Joshua Goldberger, Michel Houssa, Yong Xu, Shou-Cheng Zhang, and Deji Akinwande.  Buckled two-dimensional xene sheets.  Nature Materials, 16(2):163–169, 2017.
  * [35] S. Mellaerts, R. Meng, V. Afanasiev, J. W. Seo, M. Houssa, and J.-P. Locquet.  Quarter-filled kane-mele hubbard model: Dirac half-metals.  Phys. Rev. B under review, 2020.
  * [36] Simon Mellaerts.  Topological effects in 2d strongly correlated electron systems, 2020.
  * [37] JS Griffith and LE Orgel.  Ligand-field theory.  Quarterly Reviews, Chemical Society, 11(4):381–393, 1957.
  * [38] Min Yu and Dallas R. Trinkle.  Accurate and efficient algorithm for bader charge integration.  The Journal of Chemical Physics, 134(6):064111, 2011.
  * [39] Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki.  Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances.  Journal of the Physical Society of Japan, 74(6):1674–1677, 2005\.
  * [40] P Homm Jara.  Electronic, structural and optical phase transitions in strongly correlated oxide thin films grown by molecular beam epitaxy, 2019.
  * [41] J. Goniakowski and C. Noguera.  Properties of metal-supported oxide honeycomb monolayers: ${\mathrm{M}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{MM^{\prime}}}_{2}{\mathrm{O}}_{3}$ on $\mathrm{Me}(111)$ $(\mathrm{M},\mathrm{M}=\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Fe};\mathrm{Me}=\mathrm{Ag},\mathrm{Au},\mathrm{Pt})$.  The Journal of Physical Chemistry C, 124(15):8186–8197, Apr 2020\.
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