#  _Ab initio_ calculation of the effective Coulomb interactions in $MX_{2}$
($M$=Ti, V, Cr, Mn, Fe, Co, Ni; $X$=S, Se, Te): intrinsic magnetic ordering
and Mott insulating phase

A. Karbalaee  Department of Physics, Faculty of Science, University of Zanjan,
Zanjan 45371-38791, Iran  S. Belbasi  Department of Physics, Faculty of
Science, University of Zanjan, Zanjan 45371-38791, Iran  H. Hadipour
<EMAIL_ADDRESS>Department of Physics, University of Guilan,
41335-1914, Rasht, Iran

(….)

###### Abstract

Correlated phenomena such as magnetism and Mott phase are a very controversial
issue in two-dimensional transition metal dichalcogenides (TMDCs). With the
aim of finding the value of correlation strength and understanding the origin
of ferromagnetic order in TMDCs, we first identify relevant low-energy degrees
of freedom on both octahedral T and trigonal prismatic H lattices in $MX_{2}$
($M$=Ti, V, Cr, Mn, Fe, Co, Ni; $X$=S, Se, Te) and then determine the strength
of the effective Coulomb interactions between localized $d$ electrons from the
first principles using the constrained random-phase approximation. The on-site
Coulomb interaction (Hubbard _U_) values lie in the range 1.4$-$3.7 eV
(1.1$-$3.6 eV) and depend on the ground-state electronic structure, _d_
-electron number, and correlated subspace. For most of the TMDCs we obtain
$1<U/W_{b}<2$ (the bandwidth $W_{b}$), which turn out to be larger than the
corresponding values in elementary transition metals. On the basis of the
calculated $U$ and exchange $J$ interaction, we have checked the condition to
be fulfilled for the formation of the ferromagnetic order by Stoner criterion.
The results indicate that experimentally observed Mn$X_{2}$ ($X$=S, Se) and
V$X_{2}$ ($X$=S, Se) have an intrinsic ferromagnetic behavior in pristine
form, although V-based materials are close vicinity to the critical point
separating ferromagnetic from paramagnetic phase.

###### pacs:

…..

##  I Introduction

Electron correlation has been an important issue in low-dimensional systems
after the experimental synthesis of graphene Geim ; Katsnelson  and other two-
dimensional (2D) materials Huang ; Gong ; yu . Generally 2D materials have
less bandwidth than the bulk due to lower continuity of the density of states,
as a consequence, strength of the correlation $U/W_{b}$, namely the ratio of
effective electron-electron interaction $U$ to bandwidth $W_{b}$, becomes
great Sasoglu . For 2D compounds containing transition metal (TM) atoms, in
addition to quantum confinement effects arising from reduced dimensionality,
correlation effects are expected to play a crucial role in determining the
electronic and magnetic properties due to the presence of narrow $t_{2g}$ or
$e_{g}$ states at the vicinity of Fermi level Yekta . One of the consequences
of a moderate correlation $U/W_{b}\sim 1$ or strong correlation $U/W_{b}\gg 1$
are inducing magnetic ordering Yasin ; Yekta , Mott insulatorYekta ; Isaacs ;
Wang-3 , and etc in low-dimensional systems. For instance, an intrinsic
ferromagnetism in the monolayers of TM halides such as Cr$X_{3}$ ($X$=Cl, Br,
I)McGuire ; Zhang ; Siena ; Seyler ; Shcherbakov ; Jiang , VI3 Son ; Kong ,
NiI2Kurumaji  has been observed experimentally and confirmed theoretically
Besbes ; Tian ; He ; Liu ; Torelli . Furthermore, due to the large correlation
strength $U/W_{b}>2$, most of the TM halides exhibit a Mott insulating
character especially in the systems with nearly half-filled 3$d$ bands Yekta .
Besides TM halides, other 2D materials such as in Cr2Ge2Te6 Gong , Fe3GeTe2
Deng ; Fei , V$X_{2}$ ($X$=Se,Te)Li ; Duvjir ; Bonilla ; Guo ; Ma ; Wang-2 ,
TiTe2Lasek , MnSe2 OHara , Cr$X_{2}$ ($X$=S, Se, Te)Houlong ; Freitas ; Sun-1
; Purbawati ; Habib ; Li-x ; Meng-1 , Co$X_{2}$($X$=S, Se, Te) have been found
to show magnetic orderings.

In the mentioned systems, synthesis of the 3$d$ TM dichalcogenides (TMDCs)
such as $MX_{2}$ ($M$=V, Cr, Mn; $X$=S, Se, Te) has led to a huge experimental
and theoretical interest due to the existence of incredibly rich correlated
phenomena ranging from room temperature intrinsic ferromagnetism yu ; Lasek ;
Bonilla ; Guo ; Ma ; OHara ; Wang-2  to charge density wave orderingChen-1 ;
Sugawara ; Feng-1 , and Mott phase Isaacs . Experimentally, even at room
temperature, ferromagnetic order were reported in Mn and V-based $MX_{2}$
compoundsBonilla ; OHara . The possibility of 100$\%$ spin polarization
indicates that $MX_{2}$ monolayers are promising material for spintronics
devices. Also, including the on-site Coulomb interaction leads to Mott
insulating behavior for V-based systems due to the presence of a half-filled
bandIsaacs . However, for most of the 3$d$ TMDCs such as Cr$X_{2}$, Mn$X_{2}$
systems and even V$X_{2}$, the origin of the intrinsic ferromagnetism remained
controversial and existence of strongly correlated phase is still in doubt
Fumega ; Wong ; Feng-1 ; Wong-1 ; Duvjir ; Chen-1 ; Vinai . In other words,
some other opposite studies indicate that the ferromagnetism seen in the TMDCs
is extrinsic 2D magnetism stemming from vacanciesChua-1 , or proximity effects
Zhang-2 ; Vinai  which are not able to be precisely eliminated in 2D crystal
growth. The formation of such extrinsic magnetic moments and long-range
magnetic order induced by atom vacancies in some other 2D nonmagnetic
materials such as graphene Yazyev ; Hadipour ; Sasioglu-1 ; Ugeda  and MoS2
Zhang-1 ; Cai  makes absence of intrinsic 2D magnetism for pristine TMDCs more
likely. From the theoretical side, magnetic ordering depends strongly on the
correlated subspace of the $d$-shell and its value of electron correlation
Wong . Anyway, controversy exists over the intrinsic magnetism in the
monolayer of 3$d$ TMDCs. Another interesting property of 3$d$ TMDCs such as
trigonal prismatic phase of Cr$X_{2}$ and V$X_{2}$ is valley
polarizationHoulong ; Tong . Most of the valleytronic $MX_{2}$ have a
appropriate spin-polarized bandgap of 1.0 eV and spontaneous valley
polarization are found to be about 40$-$90 meV Tong ; Chengan , large enough
for valleytronic devices. The presence of both large valley polarization and
spin splitting in a single material are attractive for the research on
valleytronic and spintronic applications.

To prevail over the problems of band gap underestimation in Density functional
theory (DFT) based on the local density approximation (LDA) or generalized
gradient approximation (GGA) and to improve band dispersion to find proper
valley polarization, GWA (in particular self-consistent GW + spin orbit
coupling) was performed to induce many-body correction Fuh ; Thygesen ; Filip
; Chen-1 ; Cazzaniga . However, due to presence of narrow $d$ bands in TMDCs
monolayer, the refined methods such as DFT+$U$ and DFT plus dynamical mean-
field theory (DFT + DMFT) will be needed. DFT +$U$ method has been employed to
study the electronic properties of some TMDCs Esters ; Houlong-1 ; Wang-1 , in
which the effective $U$ parameters are usually taken from the values found for
other materials including the same transition metal atom. Only a few works
have been performed the $abinitio$ linear response theory and constraint
random phase approximation (cRPA)Cococcioni ; Aryasetiawan ; Nomura  to
calculate Hubbard $U$ for VS2Isaacs ; SchoNhoff-1 . Indeed, the results of
this approach are not extensive and usually are not consistent with those
extracted from experiments. This motivates us to do fully _ab initio_
calculation to find effective Coulomb parameters for all 3$d$ TMDCs
monolayers. The $U$ parameters obtained from $abinitio$ calculations not only
provide a fundamental understanding of the correlated phenomena in TMDCs, but
these effective interaction can also be used in model Hamiltonians thus
increasing the predictive power of model calculations.

In this systematic study we first identify appropriate correlated subspaces by
constructing Wannier function and then by employing the cRPA approach within
the full-potential linearized augmented-plane-wave (FLAPW) method, we
determine the effective on-site Coulomb interaction of the $d$ electrons in
$MX_{2}$ ($M$=Ti, V, Cr, Mn, Fe, Co, Ni; $X$=S, Se, Te) for both H and T
phases. We find $1<U/W_{b}<2$ in most 3$d-MX_{2}$, making them moderately
correlated materials. In non-magnetic state, among the metallic TMDCs, the
correlation strength $U/W_{b}$ turn out to be 2.7 in T-Mn$X_{2}$ and 1.8 in
H-V$X_{2}$ with values much larger than the corresponding values in elementary
TMs. Based on $U$ and exchange interaction $J$ values, we discuss the tendency
of the electron spins to order ferromagnetically by Stoner criterion. The
results indicate that experimentally observed Mn$X_{2}$ ($X$=S, Se) and
V$X_{2}$ ($X$=S, Se) have an intrinsic ferromagnetic behavior in pristine
form. Moreover, V-based materials are close to the edge of paramagnetic to
ferromagnetic transition

Figure 1: Top view of the two-dimensional crystal of $MX_{2}$ for (a) H phase
(b) T phase. Side view of $MX_{2}$ for (c) H phase (d) T phase. (e) Trigonal
prismatic coordination of one TM and six chalcogen _X_ atoms in H-phase. (f)
Octahedral coordination in T phase where in each TM atoms are bound to six _X_
atoms. The blue and red spheres exhibit $M$ and $X$ atoms, respectively

##  II Computational details

There are two common types of crystal structures in monolayer of TMDCs, which
are trigonal prismatic (H) and octahedral (T) coordination Li-1 . The side and
top views of the crystal structures of T and H phase of $MX_{2}$ systems are
presented in Fig. 1(a-d). The difference in crystal field splitting generated
by surrounding chalcogen $X$ atoms for two lattice of H-$MX_{2}$ and
T-$MX_{2}$ leads to different correlated subspace and play a key role in
expressing the differences in the observed electronic and magnetic properties.
So, we first briefly describe the crystal structural of $MX_{2}$ in T and H
phases. The lattice of T-$MX_{2}$ consists of triangular nets of TM atoms so
that the $X$ atoms are arranged as octahedra with $M$ atoms in the center (see
Fig. 1(f)) . In this arrangement, $d$ electrons states splits into three
lower-energy orbitals $t_{2g}$ ($d_{xy}$, $d_{xz}$, $d_{yz}$) and two higher-
energy orbitals $e_{g}$ ($d_{z^{2}}$, $d_{x^{2}-y^{2}}$). Note that the
octahedron is tilted with respect to the standard Cartesian coordinate $x,y,z$
system, in such a way that $C_{4}$ axes (the axes goes through two opposite
vertices of the octahedron) is in the $z^{\prime}$ direction and two opposite
faces of the octahedron are parallel to the layers. As shown in Fig. 2(a) and
2(b), in fact, one of the eight triangles of octahedron is lying on the floor
and the $z$ axis is perpendicular to this triangle. In this situation, the
linear combination of the local $t^{\prime}_{2g}$ orbitals, namely
$d_{x^{\prime}y^{\prime}}$, $d_{x^{\prime}z^{\prime}}$,
$d_{y^{\prime}z^{\prime}}$ will be the low energy levels. Using the rotation
matrix reported in Fig. 2(c), the singlet $d_{z^{2}}$ orbital which is
oriented perpendicular to the layer, is given by
$(d_{x^{\prime}y^{\prime}}+d_{x^{\prime}z^{\prime}}+d_{y^{\prime}z^{\prime}})/\sqrt{3}$.
The other $t^{\prime}_{2g}$ orbitals,
$(d_{y^{\prime}z^{\prime}}-d_{x^{\prime}y^{\prime}})/\sqrt{2}$, and
$(d_{y^{\prime}z^{\prime}}+d_{x^{\prime}y^{\prime}}-2d_{x^{\prime}z^{\prime}})/\sqrt{6}$,
are corresponding to $e_{1g}$ doublet states. The two $d_{z^{\prime
2}}/d_{x^{\prime 2}-y^{\prime 2}}$ are denoted by high-energy degenerate
doublets $e_{2g}$ states. By these notation, the non-spin polarized density of
$d$ state for T-$VSe_{2}$ are presented in Fig. 2(d). Note that the bands are
not of pure $d_{z^{2}}$, $e_{1g}$, and $e_{2g}$ character but are mixtures.
Thus, this nominations like $d_{z^{2}}$ refer to their dominant orbital
character.

Figure 2: (a) Tilted octahedral crystal structure with respect to the standard
$x,y,z$ Cartesian coordinate in T phase where in two opposite faces of the
octahedron are parallel to the layers. (b) The same structure in the rotated
state $x^{\prime},y^{\prime},z^{\prime}$ so that it forms a conventional
octahedral phase. (c) The rotation matrix which convert $x,y,z$ to
$x^{\prime},y^{\prime},z^{\prime}$ coordinate. (d) DOS projected onto 3$d$
states of the $V$ atom in the non-spin-polarized calculation for T-$VSe_{2}$.
(e) The same as (d) for spin-polarized calculation. (f) The orbital-projected
band structures for 3$d$ electron of V atom of T-$VSe_{2}$ based on DFT-PBE.
(g) DFT-PBE (red) and Wannier interpolated band structures (blue) of
T-$VSe_{2}$ monolayer using $d_{z^{2}}+e_{g1}$ subspace.

For H-$MX_{2}$, because of the trigonal prismatic coordination Fig. 1(e), in
the crystal-field level, $d$-shell splits into a singlet $d_{z^{2}}$, a low-
energy doublet $e$ ($d_{xy}$ and $d_{x^{2}-y^{2}}$) and a high-energy doublet
$e^{\prime}$ ($d_{xz}$, $d_{yz}$). According to this notation, in Fig. 3(a) we
have presented the non-spin polarized density of $d$ state for H-phase of
H-$VSe_{2}$.

The basic unit cell is hexagonal in all TMDCs and consists of three atoms. We
consider $MX_{2}$ ($M$=Ti, V, Cr, Mn, Fe, Co, Ni; $X$=S, Se, Te) for both T
and H phases. $MX_{2}$ unit cells containing one $M$ and two $X$ atoms, are
simulated based on the slab model having a 25 $\AA$ vacuum separating them.
For the non-spin-polarized as well as for the spin-polarized DFT calculations,
the FLAPW method as implemented in the FLEUR code Marzari ; Freimuth ; Fleur
is used.

We employ GGA in the Perdew-Burke-Ernzerhof (PBE) parameterization Perdew  for
the exchange-correlation energy functional. Since, Coulomb matrix elements are
almost sensitive to the internal coordinates of the atoms and structural
distortions, for all calculations the lattice parameters and atomic positions
are chosen to be equal to the optimized parameters with considering the
relaxation of the atomic coordinates and possible distortions. The $z$
coordinate of two $X$ atoms are optimized with the residual force less than
0.01 eV/Å. In the scf calculation, $16\times 16\times 1$ $k$-point grids are
used for unit cells of all systems. A linear momentum cutoff of $G_{max}$= 4.5
bohr-1 is chosen for the plane waves. The maximally localized Wannier
functions (MLWFs) Marzari  are constructed with the Wannier90 library using
appropriate bands per $M$ atom. The DFT calculations are used as an input for
the SPEX code Mostofi ; Friedrich  to determine the strength of Coulomb
interaction between correlated electrons from the cRPA and RPA methods
Aryasetiawan ; Nomura . A dense $k$-point grid $12\times 12\times 1$ are used
in the cRPA and RPA calculations.

Figure 3: (a) DOS projected onto 3$d$ states of the $V$ atom in the non-spin-
polarized calculation for H-$VSe_{2}$. (b) The orbital-projected band
structures for 3$d$ electron of V atom of H-$VSe_{2}$ based on DFT-PBE. (c)
DFT-PBE (red) and Wannier interpolated band structures (blue) of H-$VSe_{2}$
monolayer using $d_{z^{2}}+e_{g}$ subspace. (d) The same as (a) for spin-
polarized calculation. DFT-PBE and Wannier interpolated band structures of
H-$VSe_{2}$ monolayer using $d_{z^{2}}+e_{g1}$ subspace for (e) spin up and
(f) spin down.

To find strength of screened Coulomb interaction we need to identify
correlated subspace which helps to construct Wannier functions properly. So,
at first we calculate both non-magnetic and magnetic orbital-resolved density
of states (DOS) for all systems which will discussed in next section by
details. Here, we have only presented the orbital projected band structure of
VSe2 in two phases ,namely T-VSe2 and H-VSe2, in Fig. 2(f) and Fig. 3(b)
respectively that resolve the contribution of different $d$ states. The reason
why we have chosen VSe2 case is that the origin of the ferromagnetism and Mott
phase in it’s monolayer limit is still under debate. Also, it was one of the
first material in TMDCs, in which ferromagnetism is detected experimentally in
monolayer.

As shown in Fig. 2(f) in T phase, in the non-magnetic calculation the mixture
of $d_{z^{2}}$ and $e_{1g}$ ($d_{xy}/d_{x^{2}-y^{2}}$) is significant which is
also observed in total DOS of $d$ states (see Fig. 2(d)) but $t^{\prime}_{2g}$
($d_{z^{2}}+e_{1g}$) states are well-isolated bands at the vicinity of Fermi
level. Although, we construct Wannier functions individually for three
$d_{z^{2}}$, $t^{\prime}{{}_{2}g}$, and the full $d$ shell orbitals as a
correlated subspace. In $t^{\prime}_{2g}$ and $d$ cases, the original and the
Wannier-interpolated bands agree very well. In Fig 2(g), we have presented a
comparison of the non-spin-polarized DFT-PBE band structures with the
corresponding Wannier-interpolated bands obtained with the $t^{\prime}_{2g}$
Wannier orbitals as a minimal subspace for T phase of VSe2. In other words, it
motivates a three-band model for VSe2 with $t^{\prime}_{2g}$ Wannier orbitals.
In a same way, correlated subspaces are defined for all considered T phase of
TMDCs which will be discussed in the next section.

In the H-VSe2, a single half-filled band which is predominantly $d_{z^{2}}$
orbital character is well separated from the other bands in the non-magnetic
calculation. Thus, to find minimal correlated subspace in the H phase we
construct Wannier functions for the $d_{z^{2}}$, $e_{g}$ orbitals. These
energy levels are mainly responsible for the electronic, magnetic, and
transport properties of H-VSe2. Note that $e^{\prime}_{g}$ states are far from
Fermi level and do not contribute to construct half-filled band at EF. The
comparison of the DFT-PBE band structure with Wannier interpolation (see Fig.
3(c))indicates that the best consistency can be obtained by $d_{z^{2}}+e_{g}$
Wannier orbitals as a minimal subspace.

Note that, since one of the aim of this paper is to determine effective
Coulomb parameters for low-energy model Hamiltonian of correlated TM
materials, the results before symmetry breaking take place, such as non-
magnetic $U$ should be calculated. Despite this, we have done spin-polarized
calculation for a few systems. As shown in Fig. 2(f) and 3(a), the crystal
field splitting is almost small in TMDCs materials. So, considering spin
polarization may cause to have problems with entangled bands. It complicates
the construction of Wannier function if we use $d_{z^{2}}+e_{g}$ subspace in
the magnetic calculation. As shown in the Fig. 3(e) and 3(f), the original and
the Wannier-interpolated bands do not agree very well. In this case we must go
beyond the $d_{z^{2}}+e_{g}$ minimal subspace, for instance, full $d$-shell
subspace, which wrongly eliminate a few specific states in the screening.

In the following, we briefly describe cRPA method. The fully screened Coulomb
interaction $\tilde{U}$ is related to the bare Coulomb interaction $V$ by

$\tilde{U}(\bm{r},\bm{r}^{\prime},\omega)=\int
d\bm{r}^{\prime\prime}\epsilon^{-1}(\bm{r},\bm{r}^{\prime\prime},\omega)V(\bm{r}^{\prime\prime},\bm{r}^{\prime}),$
(1)

where $\epsilon(\bm{r},\bm{r}^{\prime\prime},\omega)$ is the dielectric
function. The dielectric function is related to the electron polarizability
$P$ by

$\epsilon(\bm{r},\bm{r}^{\prime},\omega)=\delta(\bm{r}-\bm{r}^{\prime})-\int
d\bm{r}^{\prime\prime}V(\bm{r},\bm{r}^{\prime\prime})P(\bm{r}^{\prime\prime},\bm{r}^{\prime},\omega),$
(2)

where the RPA polarization function
$P(\bm{r}^{\prime\prime},\bm{r}^{\prime},\omega)$ is given by

$\begin{gathered}P(\bm{r},\bm{r}^{\prime},\omega)=\\\
2\sum_{m}^{\mathrm{occ}}\sum_{m^{\prime}}^{\mathrm{unocc}}\varphi_{m}(\bm{r})\varphi_{m^{\prime}}^{*}(\bm{r})\varphi_{m}^{*}(\bm{r}^{\prime})\varphi_{m^{\prime}}(\bm{r}^{\prime})\\\
\times\Bigg{[}\frac{1}{\omega-\Delta_{mm^{\prime}}+i\eta}-\frac{1}{\omega+\Delta_{mm^{\prime}}-i\eta}\Bigg{]}.\end{gathered}$
(3)

Here, $\varphi_{m}(\bm{r})$ are the single-particle DFT Kohn-Sham
eigenfunctions, and $\eta$ a positive infinitesimal.
$\Delta_{mm^{\prime}}=\epsilon_{m^{\prime}}-\epsilon_{m}$ with the Kohn-Sham
eigenvalues $\epsilon_{m}$. ß

In the cRPA approach, in order to exclude the screening due to the correlated
subspace, we separate the full polarization function of Eq. (3) into two parts

$P=P_{d}+P_{r},$ (4)

where $P_{d}$ includes only the transitions ($m\rightarrow m^{\prime}$)
between the states of the correlated subspace and $P_{r}$ is the remainder.
Then, the frequency-dependent effective Coulomb interaction is given
schematically by the matrix equation

$U(\omega)=[1-VP_{r}(\omega)]^{-1}V.$ (5)

It contains, in $P_{r}$, screening processes that would not be captured by the
correlated subspace and excludes the ones that take place within the subspace.

The matrix elements of the effective Coulomb interaction in the MLWF basis are
given by

$\begin{gathered}U_{\mathbf{R}n_{1},n_{3},n_{2},n_{4}}(\omega)=\\\ \int\int
d\bm{r}d\bm{r}^{\prime}w_{n_{1}\mathbf{R}}^{*}(\bm{r})w_{n_{3}\mathbf{R}}(\bm{r})U(\bm{r},\bm{r}^{\prime},\omega)w_{n_{4}\mathbf{R}}^{*}(\bm{r}^{\prime})w_{n_{2}\mathbf{R}}(\bm{r}^{\prime}),\end{gathered}$
(6)

where $w_{n\mathbf{R}}(\bm{r})$ is the MLWF at site $\mathbf{R}$ with orbital
index $n$, and the effective Coulomb potential
$U(\bm{r},\bm{r}^{\prime},\omega)$ is calculated within the cRPA as described
above. We define the average Coulomb matrix elements $U$, $U^{\prime}$, and
$J$ in the static limit ($\omega=0$) as follows Anisimov1; Anisimov2:

$U=\frac{1}{L}\sum_{m}U_{mm;mm}\,,$ (7)
$U^{\prime}=\frac{1}{L(L-1)}\sum_{m\neq n}U_{mn;mn}\,,$ (8)
$J=\frac{1}{L(L-1)}\sum_{m\neq n}U_{mn;nm}\,,$ (9)

where $L$ is the number of localized orbitals, i.e., two for $e_{g}$ and three
for $t_{2g}$ orbitals. This parametrization of partially screened Coulomb
interactions is the so-called Hubbard-Kanamori parametrization. Similar to the
definition of $U$ ($U^{\prime}$, $J$), we can also define the so-called fully
screened interaction parameters $\tilde{U}$ ($\tilde{U^{\prime}}$,
$\tilde{J}$) as well as unscreened (bare) $V$. The bare $V$ provides
information about the localization of Wannier functions and is a useful
parameter in the interpretation of the screened Coulomb interaction
parameters.

Figure 4: (a) The $d_{z^{2}}$ and $e_{1g}$-like MLWF for V atom of T-VSe2. (b)
The $d_{z^{2}}$ and $e_{g}$-like MLWF for V atom of H-VSe2. (c) The
$e_{g}$-like MLWF for Co atom of H-CoSe2

##  III Results and discussion

We start with the discussion of appropriate correlated subspace moving from
early Ti$X_{2}$ to late Ni$X_{2}$ for both T and H phases. To define
correlated subspace and also to identify the contribution of different $M$
atoms in the screening, in Fig. 5 we present non-magnetic orbital-resolved
DOSs for all considered M$X_{2}$ monolayers in T-phase. Similar to V-based
system, the comparison of the DFT-PBE band structure with Wannier
interpolation shows that for all compounds except Ni-based T-M$X_{2}$, the
bands with $t^{\prime}_{2g}$ ($d_{z^{2}}$+$e_{1g}$) character is the well-
defined correlated subspace. This results is to be expected because of the
existence of large $t^{\prime}_{2g}$ states near the EF for Ti$X_{2}$ to
Co$X_{2}$. In the case of T-Ni$X_{2}$ systems (see Fig. 5(h) for example), the
Fermi level are located in the energy gap between $t^{\prime}_{2g}$ and
$e_{2g}$ bands and the minimal correlated subspace depends on the type of
electron or hole doping. The original and the Wannier-interpolated bands do
not agree very well when we consider subspace $t^{\prime}_{2g}$. For these
systems, we have defined $e_{2g}$, and full $d$ states as correlated subspace.
It means $e_{2g}$ is well-defined subspace in zero-doping, but for optical
properties and other correlated phenomenon $d$ subspace would be necessary. As
shown in Fig. 5, the orbital-resolved DOS of T-$MX_{2}$ with $X$= S, Se, Te
look very similar, thus we have determined Hubbard $U$ parameters identically
moving from $X$=S to Te systems. The Fig. 5 exhibits a strong admixture of
chalcogen $p$ with $t^{\prime}_{2g}$ states which increase as one moves from
$M$=Ti to Ni-based systems and also from $X$=S to Te ones. To find correlated
subspace in these cases, we included a few more states in construction of the
Wannier functions. We find $U$ matrix elements of for example $p$-admixed
$t^{\prime}_{2g}$ states and pure $t^{\prime}_{2g}$ are nearly identical.
However, the delocalization effect arising from admixture of $p$ states will
be reflected in the value of coulomb parameters later.

For the H phase of $MX_{2}$, the situation for defining the proper subspace is
more complicated and differ from one system to another system. The behavior of
the electronic structure of 3$d$ orbitals across the TM atoms, from $X$=S to
Te, is similar to the case of T-phase. So, we only depicted the orbital
resolved DOS only for $M$Se2 materials in Fig. 6. For Ti- and V-based
$MX_{2}$, although single band with predominantly $d_{z^{2}}$ character is
well separated from the other states, but the best consistency are given by
$d_{z^{2}}+e_{g}$ states. However, because of the great importance of single
$d_{z^{2}}$ band which almost allows defining an effective single band low-
energy Hamiltonian, we have also reported the electron-electron interactions
for this single band correlated subspace. In $M$=Cr- to Fe-based $MX_{2}$
cases, we see the substantially contribution of both $d_{z^{2}}$ and $e_{g}$
states to the DOS around the Fermi energy and, as a consequence, bands with
predominantly $d_{z^{2}}+e_{g}$ character are obtained to construct Wannier
function properly. For the last two series, namely H-Co$X_{2}$ and
H-Ni$X_{2}$, we find $e_{g}$ and $e^{\prime}_{g}$ respectively as the minimal
correlated subspaces and these layers can be described by a two-orbital
effective low-energy model. Although a four-orbital $e_{g}+e^{\prime}_{g}$
low-energy model can be used for the Ni case with caution.

Figure 5: DOS projected onto 3$d$ states of the $M$ atom as well as on $p$
states of the $X$ atoms for T-$MX_{2}$ materials. The three distinct peaks of
$d$ bands correspond to $d_{z^{2}}$, $e_{1g}$ and $e_{2g}$ states.

Now, we come to the part we discuss the value of average on-site bare
(unscreened) Coulomb interaction $V$, the average on-site partially (fully)
screened interaction parameters $U$ ($\tilde{U}$), as well as exchange
interaction $J$($\tilde{J}$) for correlated subspace electrons of $MX_{2}$
systems. The results for T-$MX_{2}$ and H-$MX_{2}$ are presented in Table 1
and 2 respectively. In both H and T phases, bare interaction $V$ increases
with increasing the number of electrons in the $d$-shell. This is to be
expected because moving from the left Ti to the right Ni in periodic table,
the nuclear charge increases which leads to the contraction of $d$-wave
functions, and subsequently increase the localization of Wannier functions.
Note that, the values of V reach a maximum for Fe$X_{2}$ systems and drop off
sharply for Co$X_{2}$. For a better understanding, we plot the shape of
Wannier orbitals for early VSe2 and late CoSe2 systems in Fig. 4. As shown in
Fig. 4(c), the coupling of eg states to neighboring chalcogene $p$ states is
significant, which leads to a delocalization and, therefore, to smaller $V$
interaction. Furthermore, the results for the chalcogene series $MX_{2}$ with
$X$=S to Te shows a reduction in bare $V$, which is well understood by the
increase in the lattice constant, making the Wannier function more extended.

The calculated $U$ values for T-$MX_{2}$ (H-$MX_{2}$) lie between 1.4 and 3.7
eV (1.1 and 3.6 eV) and depend on the ground-state electronic structure,
$d$-electron number, chalcogene $X$, and correlated subspace. Since, all
screening channels with an excitation energy larger than a few eV around EF
are contributed in cRPA calculations of $MX_{2}$ systems, we see strong
screening with large difference $V-U$ values in these materials. As seen in
Table 1 and 2, the $U$ for $M$ sites tend to increase when $M$ is varied from
Ti to Fe. In each chalcogen $X$ and a particular subspace, Hubbard $U$
parameter is determined by two effects: i) Wannier localization effect due to
increasing $d$-electron number and ii) electronic structure effects. Similar
to the bare interaction, the first effect is important also in Hubbard $U$
which tend to enhance from Ti to Ni. For the second effect, the insight from
orbital-resolved DOS of T-phase (H-phase) depicted in Fig. 5 (Fig. 6) is that
below the $d$ states there is a broad peak of chalcogen $p$ orbitals, which
should contribute with $p\rightarrow t^{\prime}_{2g}$ ($p\rightarrow
d_{z^{2}}+e_{g}$) transitions substantially to the screening. Indeed, as we
move from Ti to Ni, the chalcogen $p$ states gradually move to the lower
energy and accompanied by TM 3$d$ bands. So, the contribution of $p\rightarrow
d$ transition into the polarization function does not change and even slightly
reduces when across the series Ti to Fe and give rise to the enhancement of of
$U$ parameters with increasing 3$d$ electron number in both phase of TMDCs.
Furthermore, there is another important screening processes via
$t^{\prime}_{2g}\rightarrow e_{2g}$ ($d_{z^{2}}+e_{g}\rightarrow
e^{\prime}_{g}$) in T-phase (H-phase) of $MX_{2}$ systems, which only reduce
the values of Coulomb parameters and does not change the trend across the TM
series.

SchoNhoff et al. SchoNhoff-1  determined Hubbard $U$ values for $V$S2 ($V$Se2)
by employing the cRPA method. The obtained $U$ values are 2.25 eV (2.40 eV),
which are smaller than the calculated Coulomb interactions with
$d_{z^{2}}+e_{g}$ subspace 3.12 eV (2.96 eV) and larger than the Coulomb
interactions with $d_{z^{2}}$ subspace 1.51 eV (1.43 eV) presented in Table 2.
Using the linear response approach, Isaacs et al. calculated $U$ values for H
phase of VS2 in non-magnetic state and obtained $U$ = 4.14 eV Isaacs . This
value is almost 1.0 eV larger than Hubbard interaction in Table 2. For the
magnetic state, they also determined $U$ interactions for both T and H phases
and obtained 3.84 eV for H phase and 3.99 eV for T phase. What subspace to use
in construction of the Wannier basis or how to define as the Hubbard $U$
parameter are the possible reasons for this disagreement.

Table 1: Lattice parameter, orbital type of correlated subspace, bandwidth $W_{b}$, on-site intraorbital bare $V$, intraorbital partially (fully) screened $U(\tilde{U})$, interorbital partially (fully) screened $U^{\prime}(\tilde{U^{\prime}})$, partially (fully) screened exchange interaction $J(\tilde{J})$, correlation strength $U/W_{b}$, and the DOS at the Fermi level $D(E_{F})$ for T-$MX_{2}$ compounds. $MX_{2}$ | $a({\AA})$ | Orbitals |  $W_{b}$(eV) |  $V$(eV) |  $U(\tilde{U}$)(eV) |  $U^{\prime}(\tilde{U^{\prime}})$(eV) |  $J(\tilde{J})$(eV) | $U/W_{b}$ | $D(E_{F})$
---|---|---|---|---|---|---|---|---|---
TiS2 | 3.4176 |  $d_{z^{2}}$+$e_{1g}$ | 1.85 | 15.12 | 2.22(1.56) | 1.41(0.98) | 0.40(0.29) | 1.20 | 0.07
TiSe2 | 3.5441 |  $d_{z^{2}}$+$e_{1g}$ | 1.81 | 14.69 | 1.85(1.27) | 1.04(0.67) | 0.40(0.29) | 1.02 | 0.38
TiTe2 | 3.7409 |  $d_{z^{2}}$+$e_{1g}$ | 1.98 | 13.08 | 1.36(0.96) | 0.68(0.49) | 0.33(0.22) | 0.69 | 1.15
VS2 | 3.1917 |  $d_{z^{2}}$+$e_{1g}$ | 2.17 | 16.25 | 2.56(0.93) | 1.69(0.48) | 0.43(0.22) | 1.18 | 3.12
VSe2 | 3.3341 |  $d_{z^{2}}$+$e_{1g}$ | 1.92 | 16.09 | 2.14(0.83) | 1.29(0.40) | 0.42(0.21) | 1.11 | 3.26
VTe2 | 3.6020 |  $d_{z^{2}}$+$e_{1g}$ | 2.34 | 14.47 | 1.77(0.63) | 1.02(0.29) | 0.36(0.16) | 0.76 | 4.31
CrS2 | 3.0656 |  $d_{z^{2}}$+$e_{1g}$ | 2.07 | 16.68 | 2.78(1.12) | 1.93(0.16) | 0.42(0.21) | 1.35 | 0.30
CrSe2 | 3.2243 |  $d_{z^{2}}$+$e_{1g}$ | 2.16 | 16.14 | 2.49(0.60) | 1.75(0.12) | 0.38(0.17) | 1.16 | 5.80
CrTe2 | 3.6932 |  $d_{z^{2}}$+$e_{1g}$ | 2.12 | 15.59 | 2.27(0.48) | 1.52(0.30) | 0.37(0.14) | 1.07 | 4.23
MnS2 | 3.3507 |  $d_{z^{2}}$+$e_{1g}$ | 1.12 | 18.03 | 3.02(0.39) | 2.06(0.09) | 0.49(0.15) | 2.70 | 4.81
MnSe2 | 3.4912 |  $d_{z^{2}}$+$e_{1g}$ | 1.19 | 17.71 | 2.79(0.32) | 1.88(0.06) | 0.45(0.12) | 2.35 | 5.63
MnTe2 | 3.7448 |  $d_{z^{2}}$+$e_{1g}$ | 2.38 | 15.46 | 2.22(0.30) | 1.45(0.07) | 0.36(0.11) | 0.93 | 5.92
FeS2 | 3.2013 |  $d_{z^{2}}$+$e_{1g}$ | 1.51 | 18.34 | 3.13(0.60) | 2.19(0.37) | 0.46(0.11) | 2.07 | 4.13
FeSe2 | 3.3682 |  $d_{z^{2}}$+$e_{1g}$ | 1.69 | 17.98 | 2.81(0.56) | 1.86(0.32) | 0.46(0.12) | 1.66 | 4.59
FeTe2 | 3.6269 |  $d_{z^{2}}$+$e_{1g}$ | 2.55 | 12.80 | 1.63(0.44) | 1.07(0.26) | 0.29(0.08) | 0.64 | 3.61
CoS2 | 3.2281 |  $d_{z^{2}}$+$e_{1g}$ | 1.54 | 13.19 | 2.06(1.02) | 1.58(0.50) | 0.42(0.25) | 1.34 | 2.20
CoSe2 | 3.3704 |  $d_{z^{2}}$+$e_{1g}$ | 2.63 | 12.13 | 1.93(0.74) | 1.50(0.40) | 0.30(0.15) | 0.73 | 1.31
CoTe2 | 3.6227 |  $d_{z^{2}}$+$e_{1g}$ | 2.72 | 10.58 | 1.57(0.59) | 1.06(0.32) | 0.24(0.12) | 0.58 | 0.98
NiS2 | 3.3583 | $e_{2g}$ | 1.75 | 18.23 | 3.05(2.30) | 2.06(1.56) | 0.48(0.38) | 1.74 | 0.00
|  | $d$ | 3.58 | 21.46 | 3.67(2.62) | 2.41(1.33) | 0.63(0.54) | 1.02 | 0.00
NiSe2 | 3.5492 | $e_{2g}$ | 1.86 | 18.01 | 2.16(1.80) | 1.02(0.82) | 0.58(0.48) | 1.16 | 0.00
|  | $d$ | 3.49 | 20.28 | 3.09(2.39) | 1.97(1.06) | 0.61(0.50) | 0.88 | 0.00
NiTe2 | 3.7806 | $e_{2g}$ | 2.13 | 17.65 | 1.88(1.42) | 0.97(0.71) | 0.45(0.35) | 0.88 | 0.00
|  | $d$ | 4.32 | 18.45 | 2.65(1.87) | 1.62(0.92) | 0.53(0.41) | 0.61 | 0.00
Table 2: The same as Table 1 for H phase of $MX_{2}$. $MX_{2}$ | $a({\AA})$ | Orbitals |  $W_{b}$(eV) |  $V$(eV) |  $U(\tilde{U}$)(eV) |  $U^{\prime}(\tilde{U^{\prime}}$)(eV) |  $J(\tilde{J}$)(eV) | $U/W_{b}$ | $D(E_{F})$
---|---|---|---|---|---|---|---|---|---
TiS2 | 3.3376 |  $d_{z^{2}}$+$e_{g}$ | 2.23 | 14.47 | 2.88(2.62) | 2.02(1.74) | 0.52(0.41) | 1.29 | 0.00
|  | $d_{z^{2}}$ | 1.20 | 13.92 | 1.83 | —— | —— | 1.53 | 0.00
TiSe2 | 3.4739 |  $d_{z^{2}}$+$e_{g}$ | 2.39 | 14.18 | 2.55(2.26) | 1.70(1.32) | 0.51(0.39) | 1.07 | 0.00
|  | $d_{z^{2}}$ | 1.05 | 13.26 | 1.62 | —— | —— | 1.54 | 0.00
TiTe2 | 3.7277 |  $d_{z^{2}}$+$e_{g}$ | 2.78 | 13.43 | 2.03(1.59) | 1.25(0.88) | 0.47(0.36) | 0.73 | 0.00
|  | $d_{z^{2}}$ | 0.96 | 13.09 | 1.27 | —— | —— | 1.32 | 0.00
VS2 | 3.1650 |  $d_{z^{2}}$+$e_{g}$ | 2.97 | 15.05 | 3.12(1.16) | 2.23(0.59) | 0.55(0.34) | 1.05 | 1.52
|  | $d_{z^{2}}$ | 0.95 | 14.49 | 1.51 | —— | —— | 1.59 | 0.65
VSe2 | 3.3066 |  $d_{z^{2}}$+$e_{g}$ | 3.02 | 14.73 | 2.94(0.81) | 2.01(0.33) | 0.53(0.31) | 0.97 | 1.80
|  | $d_{z^{2}}$ | 0.76 | 14.24 | 1.43 | —— | —— | 1.88 | 0.62
VTe2 | 3.5458 |  $d_{z^{2}}$+$e_{g}$ | 3.40 | 14.05 | 2.45(0.65) | 1.40(0.25) | 0.49(0.29) | 0.72 | 1.72
|  | $d_{z^{2}}$ | 0.69 | 13.40 | 1.12 | —— | —— | 1.62 | 0.51
CrS2 | 3.0552 |  $d_{z^{2}}$+$e_{g}$ | 3.41 | 15.53 | 3.18(1.96) | 2.28(1.43) | 0.56(0.34) | 0.93 | 0.00
CrSe2 | 3.2108 |  $d_{z^{2}}$+$e_{g}$ | 3.62 | 15.31 | 2.93(1.67) | 2.01(1.15) | 0.53(0.33) | 0.81 | 0.00
CrTe2 | 3.4319 |  $d_{z^{2}}$+$e_{g}$ | 3.90 | 14.49 | 2.36(1.21) | 1.55(0.88) | 0.52(0.33) | 0.61 | 0.00
MnS2 | 3.0944 |  $d_{z^{2}}$+$e_{g}$ | 3.09 | 17.11 | 3.46(0.82) | 2.42(0.50) | 0.62(0.26) | 1.12 | 1.78
MnSe2 | 3.2335 |  $d_{z^{2}}$+$e_{g}$ | 3.32 | 16.92 | 3.10(0.68) | 2.08(0.42) | 0.61(0.24) | 0.93 | 1.65
MnTe2 | 3.5164 |  $d_{z^{2}}$+$e_{g}$ | 3.68 | 16.32 | 2.68(0.53) | 1.79(0.32) | 0.58(0.23) | 0.73 | 2.05
FeS2 | 3.1426 |  $d_{z^{2}}$+$e_{g}$ | 3.13 | 18.59 | 3.58(1.14) | 2.45(0.67) | 0.67(0.25) | 1.14 | 1.75
FeSe2 | 3.3040 |  $d_{z^{2}}$+$e_{g}$ | 3.18 | 18.46 | 3.31(1.06) | 2.19(0.62) | 0.67(0.24) | 1.04 | 1.94
FeTe2 | 3.5238 |  $d_{z^{2}}$+$e_{g}$ | 3.29 | 17.68 | 2.74(0.95) | 1.76(0.54) | 0.61(0.22) | 0.83 | 1.81
CoS2 | 3.2037 | $e_{g}$ | 2.45 | 13.40 | 1.88(0.67) | 1.40(0.44) | 0.44(0.29) | 0.77 | 2.07
CoSe2 | 3.3406 | $e_{g}$ | 2.51 | 12.93 | 1.72(0.61) | 1.26(0.38) | 0.43(0.28) | 0.68 | 1.96
CoTe2 | 3.5485 | $e_{g}$ | 2.73 | 11.50 | 1.59(0.53) | 1.11(0.36) | 0.43(0.28) | 0.58 | 2.18
NiS2 | 3.4782 |  $e_{g}$+$e^{\prime}_{g}$ | 2.96 | 16.99 | 3.07(0.99) | 2.36(0.75) | 0.45(0.32) | 1.04 | 0.74
|  | $e^{\prime}_{g}$ | 1.31 | 16.23 | 1.92(0.75) | 1.24(0.62) | 0.45(0.31) | 1.47 | 0.42
NiSe2 | 3.4263 |  $e_{g}$+$e^{\prime}_{g}$ | 3.15 | 15.17 | 2.61(0.82) | 1.94(0.63) | 0.39(0.31) | 0.83 | 0.67
|  | $e^{\prime}_{g}$ | 1.43 | 14.38 | 1.66(0.57) | 0.92(0.49) | 0.37(0.27) | 1.16 | 0.39
NiTe2 | 3.6122 |  $e_{g}$+$e^{\prime}_{g}$ | 3.23 | 13.22 | 1.91(0.62) | 1.40(0.47) | 0.36(0.27) | 0.59 | 0.69
|  | $e^{\prime}_{g}$ | 1.60 | 12.81 | 1.17(0.41) | 0.55(0.28) | 0.35(0.26) | 0.73 | 0.37

Note that exclusive correlated subspaces must be incorporated into the $U$
calculation, thus, becomes meaningless to compare the Coulomb matrix elements
results of H-$MX_{2}$ ($M$=Ti,V,Co,Ni) and T-Ni$X_{2}$ monolayers with the
corresponding results of other $MX_{2}$ compounds. For instance, considering
the single band $d_{z^{2}}$ as a correlated subspace for H-phase of VSe2,
there is an extra sizable screening channels via $d_{z^{2}}\rightarrow e_{g}$
transition which can reduce Hubbard $U$ with respect to $U$ for the
$d_{z^{2}}+e_{g}$ subspace. On the other hand, the value of $U$=1.43 eV for
H-VSe2 is smaller than the $U$=1.62 eV for H-TiSe2, which does not follow the
increasing trend across the TM series. It can be described by the $M-3d$ and
$X-p$ projected DOS around the Fermi energy in Fig. 6. Indeed, as we go from
Ti to V in H-phase of $MX_{2}$, the Se-4$p$ states move towards $d_{z^{2}}$
resulting in more contribution into the polarization function. It compensates
the increase in $U$ caused by Wannier localization, leading to the reduction
of $U$ interaction with increasing 3$d$ electron number. The same behavior is
also observed in S- and Te-based H-$MX_{2}$ materials (see Table 2).

Figure 6: DOS projected onto 3$d$ states of the $M$ atom as well as on $p$
states of the $Se$ atoms for H-$MSe_{2}$ materials. The three distinct peaks
of $d$ bands correspond to $d_{z^{2}}$, $e_{g}$ and $e^{\prime}_{g}$ states.

On going from S to Te within each $MX_{2}$ systems, the lattice constant
increase, as a consequence, the longer bond lengths lead to smaller orbitals
overlap. It can bring the states closer together energetically as shown in
Fig. 5 for T phase, and thus smaller energy difference increase the
contribution of $p\rightarrow t^{\prime}_{2g}$ and $t^{\prime}_{2g}\rightarrow
e_{2g}$ transitions ($p\rightarrow d_{z^{2}}+e_{g}$ and
$d_{z^{2}}+e_{g}\rightarrow e^{\prime}_{g}$ transition in H-$MX_{2}$) into the
polarization functions. This means that Coulomb screening is enhanced in the
$M$S2-$M$Se2-$M$Te2 sequence and thus effective Coulomb interaction $U$ reduce
in the same sequence in both T and H phase. The situation for compounds with
other correlated subspaces is almost the same. For instance, considering the
$d_{z^{2}}$ correlated subspace the $U$ value is reduced in VTe2 with respect
to the VS2 case.

In Table 1, moving again from Ti to Ni, the same increasing trend is observed
for partially inter-orbital Coulomb interactions $U^{\prime}$. The obtained
$J$ parameters for $MX_{2}$ vary in the range 0.24-0.67 eV. Despite the very
different range of values in respect to $U$ and $U^{\prime}$, exchange $J$
gradually increase with the $d$-electron number, which are less affected by
the electronic structures. In cubic symmetry, Coulomb matrix elements, namely
$U$ and $U^{\prime}$, and $J$ fulfills the relation $U^{\prime}=U-2J$. Even
though TMDCs do not have cubic symmetry, this relation is nearly satisfied in
most of them.

So far, we focus on the partially screened Coulomb interactions $U$. To
analysis the screening within the correlated subspace, we also calculate fully
screened Coulomb interactions $\tilde{U}$ and are reported within parentheses
for T-$MX_{2}$ (H-$MX_{2}$) in Table 1 (Table 2). Except for the
semiconductors T-Ni$X_{2}$, H-Ti$X_{2}$, and H-Cr$X_{2}$, the efficient
metallic screening even causes more than 80$\%$ difference between $U$ and
$\tilde{U}$ parameters. Actually, the calculated $\tilde{U}$ parameters depend
strongly on the density of state at the vicinity of EF. For example, as seen
in Table 1, the $\tilde{U}$ value for VSe2 in T-phase (H-phase) is 0.83 eV
(0.81 eV) being about 60$\%$ smaller than $U$=2.14 eV (2.94) due to the large
DOS at EF and subsequently significant contribution of the
$t^{\prime}_{2g}\rightarrow t^{\prime}_{2g}$ ($d_{z^{2}}+e_{g}\rightarrow
d_{z^{2}}+e_{g}$) transitions to the polarization function. On the opposite
side, in H-Ti$X_{2}$, T-Ni$X_{2}$, H-Cr$X_{2}$, the screening within the
correlated subspace is very weak due to the presence of the band gap in the
electronic structures, and $(U-\tilde{U})/U$ reaches to 15-25$\%$ in these
materials. Therefore, the behavior of fully screened interactions $\tilde{U}$
is completely different from $U$, as they do not follow the ordering with
respect to increase of $d$-electron number.

In the following, we discuss the strength of the electronic correlations,
namely the ratio of the effective Coulomb interaction $U$ to the bandwidth
$W_{b}$ ($U/W_{b}$) in TMDCs. Note that the $U/W_{b}$ values are determined
for a non-spin-polarized state. Let us start by the results for T phase
presented in Table 1. For most of the H-$MX_{2}$ compounds, we find
$1<U/W_{b}<2$ which put these systems in the moderately correlated regime.
Therefore, Coulomb interaction play an important role in model Hamiltonian
study of the correlation effects in T-phase of TMDCs which induce correlated
phenomena like magnetic order. Starting from T-Ti$X_{2}$, the correlation
strength $U/W_{b}$ increases and reaches to the maximum in Mn-based TMDCs, and
then tends to decrease to Co-based systems. Moreover, electron correlation is
strong $U/W_{b}>2$ in T-Mn$X_{2}$ ($X$=S, Se) materials, as a results, beside
magnetic ordering, it becomes unstable to Mott phase.

The $U/W_{b}$ for the H-phase of TMDCs does not show any clear trend with
significant fluctuation around $U/W_{b}=1$ from one system to another. We find
that in H-$MX_{2}$ ($M$=Ti, V) with $d_{z^{2}}$ correlated subspace, the
correlation strength $U/W_{b}$ is maximum, bringing them closer to the Mott
phase.

It is interesting to compare our calculated $U/W_{b}$ values with reported
ones for elementary TMs and other layered materials containing TM atoms.
While, the obtained $1<U/W_{b}<2$ values for TMDC compounds turn out to be
larger than the corresponding values in elementary transition metals
$U/W_{b}<1$Sasoglu , they are significantly smaller than the ones calculated
for TM-halides $U/W_{b}>2$ Yekta . $U/W_{b}\sim 1$ were found in the case of
$M_{2}$C and $M_{2}$CO2 MX-enes, which reveals moderate electronic
correlations in these materialsYasin . Note that although the _ab initio_
Coulomb parameters of TMDCs are smaller than elementary 3$d$ TMs, MX-enes, the
narrow bands with $d_{z^{2}}$, $e_{1g}$, and $e_{g}$ character presented in
TMDCs result in a larger $U/W_{b}$ correlation strength.

Figure 7: Stoner criterion for both T and H phase of (a) $M$S2, (b) $M$Se2,
and (c) $M$Te2. Calculated magnetic moments (in units of $\mu_{B}$) of TM
atoms for (d) $M$S2, (e) $M$Se2, and (f) $M$Te2. We have also presented the
values in the case of $d_{z^{2}}$ correlated subspace with green star points
in H-phase.

As seen above, most of the metallic TMDC compounds having almost large
correlation strength are expected to display correlation phenomena, such as
ferromagnetic ordering. So, in the following, we discuss the appearance of
ferromagnetism in these materials. Note that ignoring semiconductors
T-Ni$X_{2}$, H-Ti$X_{2}$, and H-Cr$X_{2}$, the correlated subspace of all
compounds are partially filled in the non-spin-polarized calculation, thus the
Stoner model is well suited to explain the origin of the ferromagnetism of
metallic TMDCs. Based on this model, the instability of the paramagnetic state
towards ferromagnetic ordering is given by the Stoner criterion
$I.D(E_{F})>1$, where $I$ is the Stoner parameter and $D(E_{F})$ is the DOS at
the Fermi energy in the nonmagnetic state. Solving the multiorbital Hubbard
model, Stollhoff et al. found that relationship between the Stoner parameter
$I$, Hubbard $U$, and exchange $J$ is given by $I=3(U+6J)/25$ Stollhoff . On
the basis of the calculated effective Coulomb parameters $U$ and $J$ presented
in Table 1 and Table 2, the Stoner criterion $I.D(E_{F})>1$ for all compounds
are presented in Fig. 7. Among the metallic $MX_{2}$, only H-Ni$X_{2}$($X$=S,
Se, Te), T-Ti$X_{2}$ ($X$=S, Se, Te), CrS2, and Co$X_{2}$($X$=Se, Te) do not
satisfy the Stoner criterion. Almost all theoretically predicted ferromagnetic
TMDCs fulfill the Stoner criterion, which is reasonably consistent with the
our results of spin-polarized total energy calculations and the sizable
magnetic moments presented in right panels of Fig. 7. Note that simple Stoner
model does not predict correctly the ground state magnetic phase of
H-Co$X_{2}$ ($X$=S, Se, Te) and T-CoS2. Despite the value of $I.D(E_{F})$ for
these materials are larger than 1, they have a non-magnetic ground state.

H-VSe2 and T-MnS2 were among the first TMDCs materials in which room
temperature ferromagnetism were detected experimentally in the monolayer
limit. Despite this, a few works indicated that the ferromagnetism observe in
the TMDCs is not intrinsic and stem from defects or proximity effectsChua-1 ;
Zhang-2 ; Vinai  Also, controversy exists over the Mott insulating behavior in
the monolayer of 3$d$ TMDCs. From the theoretical side, the electronic and
magnetic ground state depend strongly on taking the correct Hubbard $U$
parameter of the $d$ electrons into account in model Hamiltonian or first-
principle calculation. As shown in Fig. 7, in the case of T-MnS2, the
criterion $I.D(E_{F})>1$ is easily satisfied that explains why relatively
strong ferromagnetic order is observed in the experiment. Also, it’s
correlation strength $U/W_{b}$=2.7 reinforces this idea that the preferred
magnetic order is ferromagnetic. The situation in the case of H phase of VSe2
is not straightforward. We find $I.D(E_{F})\sim 1$ which puts system close to
the edge of paramagnetic to ferromagnetic transition (see left panels of Fig.
7) . It may explain why there is no agreement in the magnetic phase of
V$X_{2}$ materials.

##  IV Conclusion

We have systematically determined the correlated subspaces and effective on-
site and interorbital Coulomb interactions between localized electrons in 3$d$
$MX_{2}$ ($M$=Ti, V, Cr, Mn, Fe, Co, Ni and $X$=S, Se and Te) by employing
first _ab initio_ calculations in conjunction with a parameter-free cRPA
scheme. These Coulomb interactions not only provide a fundamental
understanding of the correlated phenomena such as magnetic ordering, charge
density wave, or Mott phase in TMDCs, but these effective cRPA parameters can
also be used in model Hamiltonians thus increasing the predictive power of
model calculations. Based on Hubbard $U$, exchange interactions $J$, and
electronic structure, we find that for most of the TMDC compounds the
correlation strength $U/W_{b}$ are large enough to be able to put them in
moderate and even strong correlated regime. Among TMDC materials, the maximum
value $U/W_{b}$=2.7 eV is obtained in MnS2 which is almost three time larger
than the corresponding value in elementary Mn. We thus expect electron
correlations play an important role in model Hamiltonian studies of the
3$d$-TMDC. Since, there is no agreement on the magnetic ordering of these
materials, in particular experimentally observed systems like VSe2 and MnS2,
we have checked the condition to be fulfilled for the formation of
ferromagnetic order by Stoner criterion $I.D(E_{F})>1$. The results indicate
that both Mn$X_{2}$ ($X$=S, Se) and V$X_{2}$ ($X$=S, Se) have an intrinsic
ferromagnetic behavior in pristine form, with difference that V-based
materials are close vicinity to the critical point separating ferromagnetic
from paramagnetic phase

### Acknowledgements

……

## References

  * (1) A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007).
  * (2) M. I. Katsnelson, Mater. Today 10, 20 (2007).
  * (3) B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, Di Xiao, P. Jarillo-Herrero, and X. Xu, Nature (London) 546, 270 (2017).
  * (4) Ch. Gong, L. Li, Zh. Li, H. Ji, A. Stern, Y. Xia, T. Cao,W. Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature (London) 546, 265 (2017).
  * (5) W. Yu , J. Li, TS. Herng, et al. , Adv Mater. 31, 40, 1903779 (2019).
  * (6) E. Sasoglu, C. Friedrich, and S. Blugel, Phys. Rev. B 83, 121101(R) (2011); Phys. Rev. Lett. 109, 146401 (2012).
  * (7) H. Hadipour and Y. Yekta, Phys. Rev. B 100, 195118 (2019).
  * (8) Y. Yekta Kia, H. Hadipour, E. Sasoglu, C. Friedrich, S. A. Jafari, S. Blugel, and I. Mertig, Phys. Rev. Materials 5, 030001 (2021).
  * (9) Y. D. Wang, W. L. Yao, Z. M. Xin, T. T. Han, Z. G. Wang, L. Chen, C. Cai, Yuan Li, and Y. Zhang Nature Communications, 11, 4215 (2020).
  * (10) Eric B. Isaacs and Chris A. Marianetti ,Phys. Rev. B 94, 035120 (2016).
  * (11) M. A. McGuire, G. Clark, S. KC,W. M. Chance, G. E. Jellison, V. R. Cooper, X. Xu, and B. C. Sales, Phys. Rev. Materials 1, 014001 (2017).
  * (12) Zh. Zhang, J. Shang, Ch. Jiang, A. Rasmita, W. Gao, and T. Yu, Nano Lett. 19, 3138 (2019).
  * (13) M. C. De Siena, S. E. Creutz, A. Regan, P. Malinowski, Q. Jiang, K. T. Kluherz, Gu. Zhu, Zh. Lin, J. J. De Yoreo, X. Xu, J. H. Chu, and D. R. Gamelin, Nano Lett. 20, 2100 (2020).
  * (14) K. L. Seyler, D. Zhong, D. R. Klein, Sh. Gao, X. Zhang, B. Huang, E. Navarro-Moratalla, L. Yang, D. H. Cobden, M. A. McGuire, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nat. Phys. 14, 277 (2018).
  * (15) D. Shcherbakov, P. Stepanov, D.Weber, Y.Wang, J. Hu, Y. Zhu, K. Watanabe, T. Taniguchi, Zh. Mao, W. Windl, J. Goldberger, M. Bockrath, and Ch. N. Lau, Nano Lett. 18, 4214 (2018).
  * (16) Sh. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Nat. Nanotechnol. 13, 549 (2018).
  * (17) S. Son, M. J. Coak, N. Lee, J. Kim, T. Y. Kim, H. Hamidov, H. Cho, Ch. Liu, D. M. Jarvis, Ph. A. C. Brown, J. H. Kim, Ch. H. Park, D. I. Khomskii, S. S. Saxena, and J. G. Park, Phys. Rev. B 99, 041402(R) (2019).
  * (18) T. Kong, K. Stolze, E. I. Timmons, J. Tao, D. Ni, S. Guo, Z. Yang, R. Prozorov, and R. J. Cava, Adv. Mater. (Weinheim) 31, 1808074 (2019).
  * (19) T. Kurumaji, S. Seki, S. Ishiwata, H. Murakawa, Y. Kaneko, and Y. Tokura, Phys. Rev. B 87, 014429 (2013).
  * (20) O. Besbes, S. Nikolaev, N. Meskini, and I. Solovyev, Phys. Rev. B 99, 104432 (2019).
  * (21) Sh. Tian, J. F. Zhang, Ch. Li, T. Ying, Sh. Li, X. Zhang, K. Liu, and H. Lei, J. Am. Chem. Soc. 141, 5326 (2019).
  * (22) J. He, Sh. Ma, P. Lyua, and P. Nachtigall, J. Mater. Chem. C 4, 2518 (2016).
  * (23) J. Liu, Q. Sun, Y. Kawazoed, and P. Jenac, Phys. Chem. Chem. Phys. 18, 8777 (2016).
  * (24) D. Torelli, K. S. Thygesen, and T. Olsen, 2D Mater. 6, 045018 (2019).
  * (25) Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Nature 563, 94 (2018).
  * (26) Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J.-H. Chu, and X. Xu, Nat. Mater. 17, 778 (2018).
  * (27) J. Li, B. Zhao, P. Chen, R. Wu, B. Li, Q. Xia, G. Guo, J. Luo, K. Zang, Z. Zhang, H. Ma, G. Sun, X. Duan, and X. Duan, Adv. Mater. 30, 1801043 (2018).
  * (28) G. Duvjir, B. K. Choi, I. Jang, et al., Nano Lett. 18, 5432 (2018).
  * (29) M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M. H. Phan, and M. Batzill, Nat. Nanotechnol. 13, 289 (2018).
  * (30) Yuqiao Guo,Haitao Deng, Xu Sun, et al., Adv. Mater. 29, 29, 1700715 (2017).
  * (31) Y. Ma, Y. Dai, M. Guo, C. Niu, Y. Zhu, and B.Huang, ACS Nano 6, 1695 (2012).
  * (32) Xiong Wang, Dian Li, Zejun Li, Changzheng Wu, Chi-Ming Che, Gang Chen, and Xiaodong Cui, ACS Nano 15, 10, 16236 (2021).
  * (33) K. Lasek , PM. Coelho , K. Zberecki , et al. , ACS Nano. 14, 8473 (2020).
  * (34) D. J. OHara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K. Luo, Ch. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta, D. W. McComb, and R. K. Kawakami, Nano Lett. 18, 3125 (2018).
  * (35) Houlong L. Zhuang, Michelle D. Johannes, Michael N. Blonsky, and Richard G. Hennig ,Appl. Phys. Lett. 104, 022116 (2014).
  * (36) D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. Strobel, F. Gay, J. Marcus, and M. Nsunez-Regueiro, J. Phys. Condens. Matter 27, 176002 (2015).
  * (37) X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang, L. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V. Bouchiat, M. Nunez Regueiro, N. Rougemaille, J. Coraux, A. Purbawati, A. Hadj-Azzem, Z. Wang, B. Dong, X. Wu, T. Yang, G. Yu, B.Wang, Z. Han, X. Han, and Z. Zhang, Nano Research 13, 3358 (2020).
  * (38) A. Purbawati, J. Coraux, J. Vogel, A. Hadj-Azzem, N. Wu, N. Bendiab, D. Jegouso, J. Renard, L. Marty, V. Bouchiat, A. Sulpice, L. Aballe, M. Foerster, F. Genuzio, A. Locatelli, T. O. Mentes, Z. V. Han, X. Sun, M. Nunez-Regueiro, and N. Rougemaille, ACS Appl. Mater. Interfaces 12, 30702 (2020).
  * (39) Mohammad Rezwan Habib, Shengping Wang, Weijia Wang, Han Xiao, Sk Md Obaidulla, Anabil Gayen, Yahya Khan, Hongzheng Chen , and Mingsheng Xu, Nanoscale 11, 20123 (2019).
  * (40) B. Li , Z. Wan , C. Wang, et al.,Nature Materials 20, 818 (2021).
  * (41) L. Meng , Z. Zhou, M. Xu, et al., Nat Commun 12, 809 (2021).
  * (42) P Chen, Y. -H. Chan, X. -Y. Fang, Y Zhang, M Y Chou, S. -K. Mo, Z Hussain, A. -V. Fedorov, and T. -C. Chiang Nature Communications 6, 8943 (2015).
  * (43) Katsuaki Sugawara, Yuki Nakata, Ryota Shimizu, Patrick Han, Taro Hitosugi, Takafumi Sato, and Takashi Takahashi, ACS Nano 10, 1, 1341 (2016).
  * (44) J. Feng , D. Biswas, A. Rajan , et al. , Nano Lett. 18, 4493 (2018).
  * (45) P. Chen , WW. Pai, YH. Chan, et al., Phys. Rev. Lett. 121, 196402 (2018).
  * (46) A. O. Fumega, M. Gobbi, P. Dreher, W. Wan,C. Gonzalez- Orellana, M. Pena-Diaz, C. Rogero, J. Herrero-Martin, P. Gargiani, M. Ilyn, et al., J. Phys. Chem. C 123, 27802 (2019).
  * (47) P. K. J. Wong, W. Zhang, F. Bussolotti, X. Yin, T. S. Herng, L. Zhang, Y. L. Huang, G. Vinai, S. Krishnamurthi, D. W. J. A. M. Bukhvalov, et al. Adv. Mater. 31, 1901185 (2019).
  * (48) PKJ. Wong, W. Zhang, F. Bussolotti, et al., Adv. Mater. 31, 23 1901185 (2019).
  * (49) G. Vinai, C. Bigi, A. Rajan, et al, Phys. Rev. B 101, 035404 (2020).
  * (50) R. Chua , J. Yang , X. He, et al., Adv. Mater. 32, 202000693 (2020).
  * (51) W. Zhang, L. Zhang, PKJ. Wong , et al, ACS Nano, 13, 8997 (2019).
  * (52) OV. Yazyev and L.Helm , Phys. Rev. B 75, 125408 (2007).
  * (53) H. Hadipour, Phys. Rev. B 99, 075102 (2019).
  * (54) E. Sasioglu, H. Hadipour, C. Friedrich, S. Blugel, and I.Mertig, Phys. Rev. B 95, 060408(R) (2017).
  * (55) M. M. Ugeda, I. Brihuega, F. Guinea, and J. M. Gomez-Rodriguez, Phys. Rev. Lett. 104, 096804 (2010).
  * (56) Z. Zhang , X. Zou, VH. Crespi, and BI. Yakobson, ACS Nano 7, 10475 (2013).
  * (57) L. Cai, J. He, Q. Liu, et al., J Am Chem Soc. 137, 2622 (2015).
  * (58) Wen-Yi Tong, Shi-Jing Gong, Xiangang Wan,and Chun-Gang Duan, Nature Communications 7, 13612 (2016).
  * (59) L. Chengan, M. Yandong, Zh. Ting, X. Xilong, H. Baibiao, D. Ying, New J. Phys. 22 033002 (2020).
  * (60) Huei-Ru Fuh, Ching-Ray Chang, Yin-Kuo Wang, Richard F. L. Evans, Roy W. Chantrell, and Horng-Tay Jeng, Sci. Rep. 6, 32625 (2016).
  * (61) Kristian Sommer Thygesen, 2D Mater 4 ,022004 (2017).
  * (62) Filip A. Rasmussen and Kristian S. Thygesen, J. Phys. Chem. C 119, 13169 (2015)
  * (63) M. Cazzaniga, H. Cercellier, M. Holzmann, C. Monney, P. Aebi, G. Onida, and V. Olevano, Phys. Rev. B 85, 195111 (2012).
  * (64) Marco Esters, Richard G. Hennig, and David C. Johnson Phys. Rev. B 96, 235147 (2017) .
  * (65) Houlong L. Zhuang and Richard G. Hennig Phys. Rev. B 93, 054429 (2016).
  * (66) Cong Wang, Xieyu Zhou, Yuhao Pan, Jingsi Qiao, Xianghua Kong, Chao-Cheng Kaun, and Wei Ji, Phys. Rev. B 97, 245409 (2018).
  * (67) M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105,(2005).
  * (68) F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein, Phys. Rev. B 70, 195104 (2004); F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Schönberger, ibid. 74, 125106 (2006); T. Miyake, F. Aryasetiawan, and M. Imada, ibid. 80, 155134 (2009).
  * (69) Y. Nomura, M. Kaltak, K. Nakamura, C. Taranto, S. Sakai, A. Toschi, R. Arita, K. Held, G. Kresse, and M. Imada, Phys. Rev. B 86, 085117 (2012); B.-C. Shih, Y. Zhang, W. Zhang, and P. Zhang, ibid. 85, 045132 (2012).
  * (70) G. SchoNhoff, M. RoSner, E. Kamil, J. Berges, and T. Wehling, APS March Meeting Abstracts, 2018, S35-007 (2018).
  * (71) Angus Huang , Chin-Hsuan Chen , Ching-Hao Chang , and Horng-Tay Jeng, Nanomaterials 11, 1998 (2021).
  * (72) Li. Fengyu, Tu. Kaixiong, and Zhongfang Chen, J. Phys. Chem. C 118, 21264 (2014).
  * (73) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
  * (74) F. Freimuth, Y. Mokrousov, D. Wortmann, S. Heinze, and S. Blügel, Phys. Rev. B 78, 035120 (2008).
  * (75) http://www.flapw.de/MaX-5.0/
  * (76) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * (77) A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008).
  * (78) C. Friedrich, S. Blügel, and A. Schindlmayr, Phys. Rev. B. 81, 125102 (2010).
  * (79) G. Stollhoff, A. M. Oles, and V. Heine, Phys. Rev. B 41, 7028 (1990).