# Nonconventional screening of Coulomb interaction in two-dimensional
semiconductors and metals: A comprehensive cRPA study of MX2 (M=Mo, W, Nb, Ta;
X=S, Se, Te)

H. R. Ramezani1 E. Şaşıoğlu2<EMAIL_ADDRESS>H. Hadipour1
<EMAIL_ADDRESS>H. Rahimpour Soleimani1 C. Friedrich3 S. Blügel3 I.
Mertig2 1Department of Physics, University of Guilan, 41335-1914, Rasht, Iran
2Institute of Physics, Martin Luther University Halle-Wittenberg, 06120 Halle
(Saale), Germany
3Peter Grünberg Institut, Forschungszentrum Jülich and JARA, 52425 Jülich,
Germany

###### Abstract

Two-dimensional (2D) semiconducting and metallic transition metal
dichalcogenides (TMDs) have attracted significant attention for their
promising applications in a variety of fields. Experimental observations of
large exciton binding energies and non-hydrogenic Rydberg series in 2D
semiconducting TMDs, along with deviations in plasmon dispersion in 2D
metallic TMDs, suggest the presence of a nonconventional screening of the
Coulomb interaction. The experimentally observed Mott insulating state in the
charge density wave (CDW) reconstructed lattice of TMDs containing $4d$ and
$5d$ elements further confirms the presence of strong Coulomb interactions in
these systems. In this study, we use first-principles electronic structure
calculations and constrained random-phase approximation to calculate the
Coulomb interaction parameters (partially screened $U$ and fully screened $W$)
between localized $d$ electrons in 2D TMDs. We specifically explore materials
represented by the formula MX2 (M=Nb, Ta, Mo, W, and X=S, Se, Te) and consider
three different phases ($1H$, $1T$, and $1T^{\prime}$). Our results show that
the short-range interactions are strongly screened in all three phases,
whereas the long-range interactions remain significant even in metallic
systems. This nonconventional screening provides a compelling explanation for
the deviations observed in the usual hydrogenic Rydberg series and
conventional plasmon dispersion in 2D semiconducting and metallic TMDs,
respectively. Our calculations yield on-site Coulomb interaction parameters
$U$ within the ranges of 0.8-2.5 eV, 0.8-1.9 eV, and 0.9-2.4 eV for the $1H$,
$1T$, and $1T^{\prime}$ structures, respectively. These values depend on the
specific chalcogen X, the number of $d$ electrons, and the correlated
subspace. Using the calculated $U$ parameters for the undistorted $1T$
structure, we extract the on-site effective $U^{\mathrm{eff}}_{00}$ and
nearest neighbor $U^{\mathrm{eff}}_{01}$ Coulomb interaction parameters for
reconstructed commensurate CDW NbX2 and TaX2 compounds. Furthermore, our
findings indicate a substantially high ratio of on-site effective Coulomb
interaction to bandwidth ($U^{\mathrm{eff}}_{00}/W_{b}\gg 1$) in CDW TMDs,
providing robust evidence for the experimentally observed strongly correlated
Mott phase. This work sheds light on the nonconventional screening of Coulomb
interactions in 2D TMDs, offering valuable insights into their electronic
properties and potential applications in emerging technologies. It advances
our fundamental understanding of these materials and holds promise for their
use in various applications.

###### pacs:

73.22.-f, 78.20.Bh, 71.15.-m, 71.10.Fd

##  I Introduction

Two-dimensional (2D) transition metal dichalcogenides (TMDs) have attracted
significant attention for their unique electronic [1, 2, 3], magnetic [4, 5],
optical [6, 7, 8], and valleytronic properties [9, 10, 11, 12]. Semiconducting
TMDs have tunable band gaps [13, 14] and high charge carrier mobility, making
them promising candidates for next-generation electronic devices that can
overcome the limitations of traditional silicon-based technology [15, 16].
TMDs also have strong light-matter interaction and valley-dependent electronic
properties, making them ideal for developing valleytronic devices such as
valley-selective photodetectors and light-emitting diodes [17, 18, 19]. In
addition to their potential applications in optoelectronics and valleytronics,
TMDs have also been shown to have promising thermoelectric properties [20, 21,
22, 23]. TMDs have a high thermoelectric figure of merit ($Z_{T}$), which is a
measure of their efficiency as thermoelectric materials. This makes them
promising candidates for applications in thermoelectric energy harvesting,
waste heat recovery, and refrigeration.

Metallic TMDs, such as ”cold metals” like 1$H$\- NbSe2 (see Fig. 1), have
opened up new possibilities for device innovation. One notable application is
their potential to enable tunnel diodes exhibiting negative differential
resistance with ultra-high peak-to-valley current ratios [24]. Cold metallic
TMDs have also been shown to achieve sub-60 mV/dec subthreshold swing in CMOS
transistors, which could lead to significant improvements in transistor
performance [25, 26, 27, 28, 29]. In addition to their applications in
nanoelectronics, metallic TMDs exhibit intriguing low-temperature properties
such as unconventional plasmon dispersion [30, 31], charge density wave (CDW)
phenomena [32, 33, 34], superconductivity [35, 36], and magnetism [37, 38,
39]. These diverse properties are governed by the crystal phases of MX2, which
can be trigonal prismatic $1H$, octahedral $1T$, or distorted octahedral
$1T^{\prime}$ (see Fig. 2) [40, 41]. TMDs are therefore a promising platform
for exploring a wide range of phenomena, with the potential to make
significant contributions to both fundamental science and technological
applications.

The screening of the Coulomb interaction in reduced dimensions is of
fundamental interest for practical applications, as it affects the transport
and optical properties of low-dimensional devices. In 2D materials containing
TM atoms, the dielectric screening of Coulomb interactions is significantly
reduced due to the confinement effect and presence of narrow $d$ electronic
states [42, 43]. This reduced screening has important consequences for the
properties of semiconducting TMDs, such as the MoS2 monolayer. For example, it
leads to the formation of tightly bound excitons with large binding energies
up to 1 eV [44, 45, 46, 47, 48, 49, 50]. Additionally, the exciton excitation
spectra in TMDs monolayer strongly deviates from the hydrogenic Rydberg series
[51, 52], which indicates a significantly reduced and non-local dielectric
screening of the Coulomb interaction. This is further supported by the fact
that monolayers of $1T$-NbSe2 and $1T$-TaS2 are Mott insulators, which is very
rare for $4d$ and $5d$ transition metal compounds [53]. For metallic 2D TMDs,
localized plasmons have been observed experimentally in the correlated 2D CDW
$1H$-TaSe2 and $1H$-NbSe2 materials [54, 30, 31, 55]. Additionally, electron
energy-loss spectra (EELS) for these materials have revealed a negative
dispersion of the plasmons in these materials [56], which is in contrast to
the results for a homogeneous 2D electron gas. This unconventional behavior of
the plasmon dispersion is attributed to the electronic correlation effects in
these materials.

Monolayer $1T^{\prime}$ TMDs, especially those with heavy elements like W and
Mo have strong spin-orbit coupling and are topological insulators with one-way
linear dispersion bands near the Fermi level EF [57, 58]. This crystal phase
exhibits an intriguing screening behavior, as electrons in linear bands do not
effectively screen long-range Coulomb interactions [59, 60, 61, 62]. This
property provides a unique perspective on the interplay between electronic
structure and screening phenomena in these materials. Additionally, the
topological insulating behavior of $1T^{\prime}$ TMDs adds another layer of
complexity to the interplay of electronic properties and their implications
for device applications and emergent physical phenomena.

Figure 1: (Colors online) Schematic representation of the density of states
for a semiconductor (a), a semimetal (b), a metal (c), and a cold metal (d).

The study of electronic screening effects and the calculation of Coulomb
matrix elements in transition metal (TM) materials, including TMDs, have been
the subject of several studies [63, 42, 43, 64]. Most of these studies have
focused on $3d$ TM compounds, which have narrow $t_{2g}$ and $e_{g}$ states
and strong correlation effects. A few works have investigated Coulomb
interactions in TMDs with $4d$ elements [65, 66, 67]. For example, in the
distorted structure of TaS2, the on-site Coulomb interaction $U_{00}$ was
calculated to be 0.65 eV [65], which is significantly lower than the value of
2.27 eV for undistorted $1T$-TaS2 [66]. Van Loon et al. used first-principles
calculations to determine an effective on-site Coulomb interaction
$U_{00}=1.8$ eV and a nearest-neighbor interaction $U_{01}=1$ eV for $1H$-NbS2
[67]. These effective interactions can be incorporated into model
Hamiltonians, which can improve the predictive power of model-based
calculations. This motivates our systematic and fully _ab-initio_ approach to
comprehensively compute effective Coulomb parameters for $4d$ and $5d$ TMD
monolayers. We anticipate that our work will provide new insights into the
complex interplay between electronic structure and interaction effects and
that it will enhance the predictive power and applicability of theoretical
models in this field.

In this study, we used first-principles electronic structure calculations and
constrained random-phase approximation (cRPA) within the full-potential
linearized augmented-plane-wave (FLAPW) method to calculate the effective
Coulomb interaction parameters between localized $d$ electrons in 2D TMDs. We
specifically explored materials represented by the formula MX2 (M=Nb, Ta, Mo,
W, and X=S, Se, Te) and considered three different phases ($1H$, $1T$, and
$1T^{\prime}$). All compounds in the $1T$ and $1T^{\prime}$ phases are
metallic, while Mo and W-based compounds in the $1H$ phases are semiconductors
and Nb and Ta-based compounds in the same phase exhibit cold metallic behavior
(see Fig. 1 and Table 1). Our results show that the short-range interactions
are strongly screened in all three phases, whereas the long-range interaction
remains significant even in metallic systems. This nonconventional screening
provides a compelling explanation for the deviations observed in the usual
hydrogenic Rydberg series and conventional plasmon dispersion in 2D
semiconducting and metallic TMDs, respectively. Our calculations yield on-site
Coulomb interaction values within the ranges of 0.8-2.5 eV, 0.8-1.9 eV, and
0.9-2.4 eV for the $1H$, $1T$, and $1T^{\prime}$ structures, respectively. We
find that these values depend on the specific chalcogen X, the number of $d$
electrons, and the correlated subspace. Using the calculated $U$ parameters
for undistorted $1T$ structure, we extract the on-site effective
$U^{\mathrm{eff}}_{00}$ and nearest neighbor $U^{\mathrm{eff}}_{01}$ Coulomb
interaction parameters for reconstructed commensurate CDW NbX2 and TaX2
compounds. Furthermore, our findings indicate a substantially high ratio of
on-site Coulomb interaction to bandwidth ($U^{\mathrm{eff}}_{00}/W_{b}\gg 1$)
in CDW TMDs, providing robust evidence for the experimentally observed
strongly correlated Mott phase. The rest of the manuscript is organized as
follows: In Sec. II, we outline the computational method. Sec. III covers
results and discussions. Finally, Sec. IV presents conclusions and a summary.

##  II Computational Method

###  II.1 Crystal structure and ground state calculations

We consider 2D TMDs, which have the chemical formula MX2. Here, M represents
elements such as Mo, W, Nb, and Ta, and X represents chalcogen elements,
namely S, Se, and Te. Our study encompasses TMDs with distinct
crystallographic structures, including trigonal prismatic (1$H$), octahedral
(1$T$), and distorted octahedral (1$T^{\prime}$) structures. The crystal
structures are shown in Fig. 2 and the corresponding lattice parameters are
given in Table 1. In the 1$H$ and 1$T$ structures, the fundamental unit cell
has a hexagonal lattice configuration and it contains one M atom and two X
atoms, which are separated by a vacuum region of 20 Å. This is illustrated in
Fig. 2(a) and 2(b). For our investigation of the $1T^{\prime}$ structure, we
use an orthorhombic unit cell containing two M atoms and four X atoms, as
shown in Fig. 2(c).

Figure 2: Side and top views of the two-dimensional crystal structure of transition metal dichalcogenides MX2 in (a) 1$H$ structure, (b) 1$T$ structure, (c) 1$T^{\prime}$ structure, and (d) Star of David (SOD) reconstructed 1$T$ crystal structure. The purple and green spheres exhibit M and X atoms, respectively. Table 1: Lattice parameters and corresponding ground states of the studied MX2 (M=Mo, W, Nb, Ta and X=S, Se, Ta) compounds in 1$H$, 1$T$, and 1$T^{\prime}$ structures. Lattice parameters are taken from Ref. [68].  MX2 | Phase |  _a_ (Å) |  _b_ (Å) | Ground state
---|---|---|---|---
MoS2 | ${1H}$ | 3.18 | 3.18 | Semiconductor
| ${1T}$ | 3.19 | 3.19 | Metal
| ${1T^{\prime}}$ | 5.72 | 3.18 | Semimetal
MoSe2 | ${1H}$ | 3.32 | 3.32 | Semiconductor
| ${1T}$ | 3.28 | 3.28 | Metal
| ${1T^{\prime}}$ | 5.96 | 3.29 | Semimetal
MoTe2 | ${1H}$ | 3.55 | 3.55 | Semiconductor
| ${1T}$ | 3.49 | 3.49 | Metal
| ${1T^{\prime}}$ | 6.36 | 3.46 | Metal
WS2 | ${1H}$ | 3.19 | 3.19 | Semiconductor
| ${1T}$ | 3.21 | 3.21 | Metal
| ${1T^{\prime}}$ | 5.73 | 3.20 | Semimetal
WSe2 | ${1H}$ | 3.32 | 3.32 | Semiconductor
| ${1T}$ | 3.29 | 3.29 | Metal
| ${1T^{\prime}}$ | 5.95 | 3.30 | Semimetal
WTe2 | ${1H}$ | 3.55 | 3.55 | Semiconductor
| ${1T}$ | 3.51 | 3.51 | Metal
| ${1T^{\prime}}$ | 6.31 | 3.49 | Metal
NbS2 | ${1H}$ | 3.34 | 3.34 | Cold-metal
| ${1T}$ | 3.38 | 3.38 | Metal
NbSe2 | ${1H}$ | 3.47 | 3.47 | Cold-metal
| ${1T}$ | 3.48 | 3.48 | Metal
NbTe2 | ${1H}$ | 3.68 | 3.68 | Cold-metal
| ${1T}$ | 3.65 | 3.65 | Metal
TaS2 | ${1H}$ | 3.34 | 3.34 | Cold-metal
| ${1T}$ | 3.38 | 3.38 | Metal
TaSe2 | ${1H}$ | 3.47 | 3.47 | Cold-metal
| ${1T}$ | 3.50 | 3.50 | Metal
TaTe2 | ${1H}$ | 3.70 | 3.70 | Cold-metal
| ${1T}$ | 3.69 | 3.69 | Metal

The variation in crystal field splitting induced by the neighboring chalcogen
X atoms in the three lattice configurations of 1$H$-MX2, 1$T$-MX2, and
1$T^{\prime}$-MX2 yields distinct correlated subspaces, which will be
discussed in the following section. This phenomenon plays a key role in
expressing the differences in the observed electronic and optical properties
of the materials.

In the $1H$ structure, the chalcogens are aligned vertically along the
$z$-axis, with the transition metals sandwiched within the central plane along
the $x$-axis. This arrangement results in a Bernal (ABA) stacking
configuration, as depicted in the side view of Fig. 2(a). Conversely, the $1T$
structure exhibits rhombohedral (ABC) stacking, as evident from the side view
presented in Fig. 2(b)[69, 70]. In the $1H$ structure, the $d$ orbitals
experience a splitting into a singlet $d_{z^{2}}$, an intermediate-energy
doublet $e_{g}$ ($d_{x^{2}-y^{2}}$, $d_{xy}$), and a high-energy doublet
$e^{\prime}_{g}$ ($d_{yz}$, $d_{xz}$). On the other hand, in the case of the
1$T$ structure, the $d$ orbital splits into three lower-energy $t_{2g}$
($d_{xy}$, $d_{yz}$, $d_{xz}$) and two higher-energy $e_{g}$
($d_{x^{2}-y^{2}}$, $d_{z^{2}}$) states [71].

The 1$T^{\prime}$-MX2 structure of TMDs is a low-symmetry crystal phase that
can be considered as a periodically distorted structure of the 1$T$ structure.
In the 1$T^{\prime}$ structure, the two adjacent transition metal (TM) atoms
move towards each other in the $y$-direction, compared to the 1$T$ structure.
The 1$T^{\prime}$ structure is more stable than the 1$T$ structure, and the
energy barrier to separate the 1$T$ structure from the stable 1$T^{\prime}$
structure is nearly zero, leading to spontaneous structural distortions [72,
73]. This distortion causes an inversion of the band structure at the $\Gamma$
point between the $p_{x}$ orbital of the dichalcogenide and the $d$ orbital of
the TM, and a cone-like band structure is formed [74, 75].

The FLAPW method, as implemented in the FLEUR code [76], is used for the
ground-state calculations. We employ the generalized gradient approximation
(GGA) to the exchange-correlation potential, as parametrized by Perdew et al.
[77]. To ensure that the results are consistent, calculations are performed
using the same cutoff for the wave functions (kmax = 4 a.u.-1), and the same
16 $\times$ 16 $\times$ 1 (12 $\times$ 16 $\times$ 1) $\mathbf{k}$-point grid
for the 1$H$ and 1$T$ (1$T^{\prime}$) structures in the determination of the
ground states. These parameters have been verified to yield well-converged
Coulomb interaction parameters across all studied compounds. The maximally
localized Wannier functions (MLWFs) are constructed using the WANNIER90
library [78, 79, 80, 81]. The effective Coulomb potential is calculated within
the constrained random-phase approximation (cRPA) method [82, 83, 84, 85, 86],
as implemented in the SPEX code [87, 88]. We use a 14$\times$14$\times$1
(10$\times$14$\times$1) $\mathbf{k}$-point grid for 1$H$ and 1$T$
(1$T^{\prime}$) structures in the cRPA calculations.

###  II.2 The cRPA method and parametrization of the Coulomb matrix

The cRPA method is an efficient way to calculate the screened Coulomb
interaction between localized electrons. It allows us to determine individual
Coulomb matrix elements, such as on-site, off-site, intra-orbital, inter-
orbital, and exchange, as well as their frequency dependence. The fully
screened Coulomb interaction $W$ is related to the bare Coulomb interaction
$V$ by

$W(\mathbf{r},\mathbf{r^{\prime}};\omega)=\int\mathrm{d}\mathbf{r^{\prime\prime}}\epsilon^{-1}(\mathbf{r},\mathbf{r^{\prime\prime}},\omega)V(\mathbf{r^{\prime\prime}},\mathbf{r^{\prime}}),$
(1)

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

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

where the RPA polarization function is given by

$\displaystyle P\left(\mathbf{r},\mathbf{r^{\prime}};\omega\right)$
$\displaystyle=2\sum_{m}^{\mathrm{occ}}{\sum_{m^{\prime}}^{\mathrm{unocc}}}{\varphi_{m}(\mathbf{r})\varphi_{m^{\prime}}^{*}(\mathbf{r})\varphi_{m}^{*}(\mathbf{r^{\prime}})\varphi_{m^{\prime}}(\mathbf{r^{\prime}})}$
$\displaystyle\times\bigg{[}\frac{1}{\omega-\Delta_{mm^{\prime}}+i\eta}-\frac{1}{\omega-\Delta_{mm^{\prime}}-i\eta}\bigg{]},$
(3)

Here, $\varphi_{m}(\mathbf{r})$ 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 $\epsilon_{m}$. In the cRPA approach, to exclude the screening due
to the correlated subspace, we partition the full polarization function of Eq.
(II.2) 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)

The set $P_{r}$ comprises all transitions, excluding those occurring within
the correlated subspace The matrix elements of the effective Coulomb
interaction in the MLWF basis is given by

$\displaystyle{U}_{\mathbf{R}n_{1},n_{2},n_{3},n_{4}}(\omega)=$
$\displaystyle\int\int\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r^{\prime}}w_{n_{1}\mathbf{R}}^{*}(\mathbf{r})w_{n_{3}\mathbf{R}}(\mathbf{r})U(\mathbf{r},\mathbf{r^{\prime}},\omega)$
$\displaystyle\times
w_{n_{4}\mathbf{R}}^{*}(\mathbf{r^{\prime}})w_{n_{2}\mathbf{R}}(\mathbf{r^{\prime}}),$
(6)

where $w_{n\mathbf{R}}(\mathbf{r})$ is the MLWF at site $\mathbf{R}$ with
orbital index $n$, and the effective Coulomb potential
$U(\mathbf{r},\mathbf{r^{\prime}},\omega)$ is calculated within the cRPA
method as described above. We define the average on-site diagonal (direct
intra-orbital) $U$ and off-diagonal (direct and exchange inter-orbital)
$U^{\prime}$, $J$ matrix elements of the screened Coulomb potential in the
static limit $(\omega=0)$ as follows [89, 90]:

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

where $L$ is the number of localized orbitals, i.e., one for $d_{z^{2}}$,
three for $d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}$ and five for full $d$ orbitals.
We employ Eq. (7) to Eq. (9) for all subspaces discussed in this paper. One
can show that these Hubbard-Kanamori parameters define the full partially
screened Coulomb matrix [Eq. (II.2)] of subspaces formed by $t_{2g}$ and
$e_{g}$ orbitals (assuming spherical symmetry around the atoms). However, to
fully define Eq. (II.2) for the three-orbital
(${d_{z^{2}}+d_{xy}+d_{x^{2}\mbox{-}y^{2}}}$) and five-orbital ($d$)
subspaces, we need at least one additional matrix element. For reference, we
provide explicit values for several relevant matrix elements in the
supplementary material. Similar to the definition of $U$ ($U^{\prime}$, $J$),
we can also define the so-called fully screened interaction $W$ as well as
unscreened (bare) $V$. The bare Coulomb interaction parameter $V$ provides
information on the localization of Wannier functions. Several procedures have
been proposed in the literature to calculate the polarization function for
entangled bands [63, 91, 82, 83, 84, 85, 86]. In the present work, we use the
method described in Ref. [63, 91].

Calculating Coulomb interaction parameters for the $1H$, $1T$, and
$1T^{\prime}$ structures is relatively straightforward. However, the star-of-
David (SOD) reconstruction increases the number of atoms per 2D unit cell to
39, of which 13 are TM atoms. cRPA calculations for such systems would become
very demanding. For the sake of simplicity, we therefore utilize the results
from the undistorted $1T$ structure to make estimations of the on-site and
long-range effective Coulomb interactions [66, 92, 65] for the SOD
reconstructed systems. The estimated effective Coulomb interaction is given by
[65]

$U^{\mathrm{eff}}=\frac{1}{13^{2}}\sum_{\mathbf{R},\mathbf{R^{\prime}}}U_{\mathbf{R}-\mathbf{R^{\prime}}},$
(10)

when both the vectors $\mathbf{R}$ and $\mathbf{R^{\prime}}$ correspond to the
positions of TM atoms within a specific star in the 1$T$ structure,
$U^{\mathrm{eff}}$ represents the on-site effective interaction
$U^{\mathrm{eff}}_{00}$ for that star. On the other hand, to determine the
long-range interaction $U^{\mathrm{eff}}_{01}$ within the reconstructed
lattice, vector $\mathbf{R}$ should pertain to star A, while
$\mathbf{R^{\prime}}$ refers to the indices of TM atoms belonging to the
nearest neighboring star B [see a schematic representation of the star of
David in Fig. 2(d)].

##  III Results and discussion

###  III.1 Correlated subspace

In order to determine the strength of the screened Coulomb interaction, it is
important to identify the correlated subspace. This is the subspace of
electronic states that are most strongly interacting, and it is essential for
the accurate construction of Wannier functions and the corresponding effective
low-energy model Hamiltonian. As a first step, we performed electronic
structure calculations for all systems. The projected band structures (see
Figs. S1 and S2 in supplementary material) show that the $d$ orbitals of the
TM atom make a significant contribution to the bands near the Fermi level,
compared to the other orbitals from chalcogen atoms. Thus, all investigated
compounds can be described by an effective low-energy model based on only TM
atom $d$ electrons. In Fig. 3 and Fig. 4 we compare the DFT-PBE band
structures with the corresponding Wannier-interpolated bands for some selected
materials. The corresponding Wannier orbitals are depicted in Fig. 5. As seen
in Fig. 3, the 1$H$ phase of the MX2 (M=Mo, W; X=S, Se, Te) compounds can be
described well by a three-orbital ($d_{z^{2}}+e_{g}$) model, while the 1$T$
and 1$T^{\prime}$ phases of the same materials require a full $d$-orbital
effective model. On the other hand, 1$H$ MX2 (M=Nb, Ta; X=S, Se, Te) compounds
show cold metallic behavior and thus they can be described by a simple one-
orbital model, while the same materials in the 1$T$ structure require a three-
orbital model. For consistency, the Coulomb interaction parameters in the
following section will be presented for a one-orbital ($d_{z^{2}}$), for a
three-orbital ($d_{z^{2}}+d_{xy}+d_{x^{2}-y^{2}}$), and for five-orbital ($d$)
correlated subspaces.

Figure 3: (Color online) DFT-PBE (blue) and Wannier-interpolated band
structures (red) of (a) (b) (c) 1$H$-MoS2 and (d) (e) (f) 1$T$-MoS2, and (g)
(h) (i) 1$T^{\prime}$-WSe2. In each system, we considered three correlated
subspaces derived from one-orbital $d_{z^{2}}$, three-orbital
$d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}$, and full $d$ orbitals of the TM atom.
Figure 4: (Color online) DFT-PBE (blue) and Wannier-interpolated band
structures (red) of (a) (b) (c) 1$H$-NbS2 and (d) (e) (f) 1$T$-NbS2, (g) (h)
(i) 1$H$-TaS2, and (j) (k) (l) 1$H$-NbSe2. In each system, we considered three
correlated subspaces derived from one-orbital $d_{z^{2}}$, three-orbital
$d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}$, and full $d$ orbitals of the TM atom.
Figure 5: Plot of MLWFs for TM atoms in 1$H$-MoS2, 1$T$-MoS2,
1$T^{\prime}$-WSe2, 1$H$-NbS2, 1$T$-NbS2, 1$H$-TaS2, and 1$H$-NbSe2. First
column: The $d_{z^{2}}$-like MLWFs, considering only one-orbital subspace,
i-e., $d_{z^{2}}$. Second column: The $d_{xy}$-like MLWFs, considering a
three-orbital ($d_{z^{2}}+d_{xy}+d_{x^{2}\mbox{-}y^{2}}$) subspace. Third
column: The $d_{x^{2}\mbox{-}y^{2}}$like MLWFs, considering the full five
orbital $d$-space. Same isovalue is used in all cases.

###  III.2 Coulomb interaction parameters: MX2 (M=Mo, W; X=S, Se, Te)

Prior to discussing the effective Coulomb interaction parameters in in MX2
(M=Mo, W; X=S, Se, Te) compounds in 1$H$, 1$T$, and 1$T^{\prime}$ structures
it is worth noting that the screening of the Coulomb interaction in 2D
semiconductors has been extensively explored in recent years by numerous
researchers employing various methodologies. For instance, the quantum-
electrostatic heterostructure model [93] employs a monopole-dipole
approximation to estimate the dielectric function at zero frequency. Trolle et
al. [94] proposed a model 2D dielectric function to determine excitonic
binding energies. One difficulty in the calculation of dielectric functions of
2D systems with a 3D code is the unwanted but unavoidable screening
contribution of the periodic images of the layer in the neighboring
supercells. A possible solution is to truncate the Coulomb interaction in the
$z$ direction [95]. As an alternative, we employ an extrapolation formula that
yields the dielectric function for infinite inter-layer distances [96, 97].
The effect of the extrapolation can be seen in Fig. 6. In Fig. S5 of the
supplementary material, we show, for reference, the $q$-dependent static
dielectric function $\epsilon(\omega=0,q)$ calculated within RPA for 1H-MoS2
and a layer distance (supercell height) of 25Å, thus _including_ the inter-
layer screening. The calculated dielectric function exhibits a behavior
similar to the results obtained in other works [95, 98, 99, 100].

Figure 6: (Color online) Loss function for 1$H$-MoS2. The extrapolated curve
corresponds to a monolayer (infinite layer distance) and tends towards
$-\text{Im}(1/\varepsilon)$ and $\text{Im}(\varepsilon)$ for large and small
$q$, respectively.

The choice of correlated subspace can significantly impact the accuracy of the
calculated properties. For example, a minimal three-orbital low-energy model
might be sufficient for investigating transport properties in 1$H$ structure,
but the inclusion of the full $d$-orbital correlated subspace might be
necessary to accurately account for the complex interplay of electrons and
photons within the material when delving into optical properties, such as
absorption and emission spectra. This distinction highlights the nuanced
nature of these materials’ behaviors and the necessity of tailoring the
correlated subspace according to the specific properties under investigation.
As mentioned in the preceding section, we will consider a one-orbital
(${d_{z^{2}}}$), a three-orbital
(${d_{z^{2}}+d_{xy}+d_{x^{2}\mbox{-}y^{2}}}$), and five-orbital ($d$)
correlated subspace.

Table 2: Bare Coulomb interaction $V$, partially screened Hubbard-Kanamori parameters [$U$, $U^{\prime}$, $J$ (in eV)] and fully screened $W$ (in eV) for $d$ orbitals of TMs in MX2 (M=Mo, W; X=S, Se, Te) compounds. MX2 | Phase | Orbitals |  $V$(eV) |  $U$(eV) | ${U^{\prime}}(eV)$ | $J(eV)$ |  $W$(eV)
---|---|---|---|---|---|---|---
| ${1H}$ | ${d_{z^{2}}}$ | 8.84 | 1.95 | —– | —– | 1.95
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.47 | 2.33 | 2.04 | 0.15 | 1.54
|  | ${d}$ | 9.08 | 2.30 | 2.03 | 0.13 | 1.54
$\rm{MoS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 9.53 | 0.45 | —– | —– | 0.44
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 11.32 | 1.28 | 0.96 | 0.17 | 0.64
|  | ${d}$ | 10.39 | 1.27 | 1.01 | 0.13 | 0.60
| $\mathrm{1T^{\prime}}$ | ${d_{z^{2}}}$ | 8.37 | 1.66 | —– | —– | 1.54
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.93 | 2.00 | 1.57 | 0.22 | 1.41
|  | ${d}$ | 8.90 | 2.18 | 1.77 | 0.21 | 1.45
| $\mathrm{1H}$ | ${d_{z^{2}}}$ | 8.91 | 1.96 | —– | —– | 1.96
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.36 | 2.26 | 1.98 | 0.16 | 1.49
|  | ${d}$ | 9.16 | 2.26 | 1.99 | 0.13 | 1.51
$\rm{MoSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.94 | 0.49 | —– | —– | 0.49
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 10.55 | 1.41 | 1.06 | 0.18 | 0.65
|  | ${d}$ | 9.59 | 1.13 | 0.89 | 0.11 | 0.60
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 8.36 | 2.30 | —– | —– | 2.30
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.08 | 1.89 | 1.43 | 0.21 | 1.46
|  | ${d}$ | 9.57 | 2.28 | 1.77 | 0.23 | 1.57
| ${1H}$ | ${d_{z^{2}}}$ | 6.12 | 1.08 | —– | —– | 1.08
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 6.81 | 1.35 | 1.18 | 0.08 | 0.95
|  | ${d}$ | 7.28 | 1.46 | 1.25 | 0.09 | 1.04
$\rm{MoTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 6.69 | 0.49 | —– | —– | 0.49
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 7.03 | 0.52 | 0.35 | 0.07 | 0.60
|  | ${d}$ | 8.14 | 1.53 | 1.31 | 0.11 | 0.60
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 5.97 | 0.55 | —– | —– | 0.55
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 6.81 | 0.85 | 0.56 | 0.15 | 0.65
|  | ${d}$ | 7.96 | 1.14 | 0.78 | 0.18 | 0.81
| ${1H}$ | ${d_{z^{2}}}$ | 8.43 | 1.98 | —– | —– | 1.98
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.85 | 2.42 | 2.16 | 0.15 | 1.62
|  | ${d}$ | 8.47 | 2.39 | 2.14 | 0.12 | 1.62
$\rm{WS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.45 | 0.40 | —– | —– | 0.40
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.56 | 1.09 | 0.85 | 0.13 | 0.61
|  | ${d}$ | 8.85 | 1.31 | 1.09 | 0.12 | 0.58
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 7.94 | 1.40 | —– | —– | 1.40
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.06 | 1.90 | 1.51 | 0.21 | 1.33
|  | ${d}$ | 8.13 | 2.08 | 1.70 | 0.20 | 1.38
| ${1H}$ | ${d_{z^{2}}}$ | 8.31 | 2.04 | —– | —– | 2.04
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.87 | 2.42 | 2.16 | 0.17 | 1.61
|  | ${d}$ | 8.77 | 2.43 | 2.16 | 0.14 | 1.63
$\rm{WSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.67 | 0.43 | —– | —– | 0.43
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.09 | 1.10 | 0.84 | 0.15 | 0.62
|  | ${d}$ | 9.09 | 1.43 | 1.15 | 0.14 | 0.65
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 7.23 | 1.64 | —– | —– | 1.64
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.35 | 2.03 | 1.59 | 0.21 | 1.53
|  | ${d}$ | 9.17 | 2.41 | 1.88 | 0.23 | 1.70
| ${1H}$ | ${d_{z^{2}}}$ | 5.62 | 1.11 | —– | —– | 1.11
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 6.46 | 1.43 | 1.26 | 0.09 | 1.00
|  | ${d}$ | 6.86 | 1.54 | 1.33 | 0.09 | 1.08
$\rm{WTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 7.21 | 0.39 | —– | —– | 0.39
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 7.99 | 1.05 | 0.84 | 0.14 | 0.49
|  | ${d}$ | 7.47 | 1.08 | 0.89 | 0.11 | 0.47
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 5.38 | 0.57 | —– | —– | 0.57
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 6.07 | 0.85 | 0.59 | 0.12 | 0.68
|  | ${d}$ | 7.49 | 1.21 | 0.85 | 0.17 | 0.89

In Table 2, we present on-site Coulomb interaction parameters, including the
bare $V$, the partially screened Hubbard-Kanamori parameters $U$,
$U^{\prime}$, and $J$, as well as the fully screened $W$. Note that the
definitions of $V$ and $W$ are the same as $U$, i.e., the average diagonal
elements of the Coulomb matrix [see Eq.(7)]. To facilitate a more
comprehensive comparison, we visualize the unscreened $V$, partially screened
$U$, and fully screened $W$ parameters across all compounds and their
corresponding phases. This graphical representation is presented in Fig. 7,
enhancing our ability to discern the variations and trends in these
interactions more effectively.

Let us begin with a discussion of the bare (unscreened) Coulomb interaction.
The bare $V$ values range from 5.6 to 11.3 eV and depend on the principal
quantum number of the $d$ shell, chalcogen X, and symmetry of the structure.
Our calculated interaction $V$ for M=Mo and W with $4d$ and $5d$ orbitals,
respectively, are almost 4-5 eV smaller than the bare Coulomb interactions of
TMDs with $3d$ correlated subspaces [42]. Additionally, if we focus on a
specific subspace, the calculated $V$ parameters for WX2 are smaller than
MoX2. This is not unexpected, as the bare interaction $V$ generally decreases
when moving downward in the periodic table from $4d$ TM to $5d$ TM materials,
due to the lower degree of contraction of the $5d$ wave functions compared to
$4d$ and $3d$ ones. Furthermore, the results for the chalcogen series MX2 with
X=S to Te tend to show a reduction in bare $V$ (with exceptions). One might
attribute this to the increase in the lattice constant, making the Wannier
function more extended or more delocalized. This can be seen in Fig. 5, which
shows the shape of Wannier orbitals for Mo(W)X2 compounds. An analysis of the
shape of these Wannier functions and the projected band structures (see
supplementary material) indicates that the coupling of $d$ states to
neighboring chalcogen $p$ states is not negligible, which leads to
delocalization and, therefore, to smaller bare interaction $V$ parameters in
X=Se, Te compounds compared to X=S. From a symmetry point of view, the largest
value of bare interaction $V$ is observed in the $1T$ structure. This is
consistent with the stronger contraction of $d$ wave functions and weaker
admixture of chalcogen $p$ with $d$ states in the $1T$ structure.

Figure 7: (Color online) Comparison of the on-site Coulomb interaction
parameters for MX2 (M=Mo, W; X=S, Se, Te) in $1H$, $1T$, and $1T^{\prime}$
structures, considering three different correlated subspaces. Purple, green,
and orange indicate the values of $V$, $U$, and $W$, respectively.

In most of the TMDs considered, the Coulomb interaction is screened
efficiently due to the high density of M-$d$ and X-$p$ states near the Fermi
level $E_{F}$. As a consequence, the on-site $U$ and $W$ values are
considerably reduced. The calculated $U$ values for $1H$-MX2, $1T$-MX2, and
$1T^{\prime}$-MX2 lie in the range 1.3-2.4 eV, 1.1-1.4 eV, and 0.9-2.4 eV,
respectively. The calculated $U$ and $W$ values depend on the correlated
subspace, ground-state electronic structure, and chalcogen X atom. In contrast
to the bare interaction, the $U$ values are larger in the $1H$ structure. This
can be attributed to the band gap of the materials in this structure. For
example, considering a three-orbital correlated subspace, the $U$ values in
the $1T$ structure are smaller than in $1H$ and $1T^{\prime}$ due to a
metallic screening channel stemming from the chalcogen X-$p$ states in the
$1T$ and $1T^{\prime}$ structures (see Fig. 3 and the projected band structure
in the supplementary material).

Table 3: Long-range partially screened Coulomb interaction $U$ for MX2 (M=Mo, W; X=S, Se, Te) compounds. $U_{00}$ is the onsite interaction, $U_{01}$ is the nearest neighbor interaction, $U_{02}$ is the next nearest neighbor interaction, and so on, up to the sixth nearest neighbor interaction. MX2 | Phase | Orbitals |  $U_{00}$(eV) |  $U_{01}$(eV) |  $U_{02}$(eV) |  $U_{03}$(eV) |  $U_{04}$(eV) |  $U_{05}$(eV) |  $U_{06}$(eV)
---|---|---|---|---|---|---|---|---|---
| ${1H}$ | ${d_{z^{2}}}$ | 1.95 | 1.08 | 0.84 | 0.78 | 0.66 | 0.60 | 0.54
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.33 | 1.29 | 1.01 | 0.93 | 0.79 | 0.72 | 0.65
|  | ${d}$ | 2.30 | 1.29 | 1.01 | 0.93 | 0.79 | 0.72 | 0.65
$\rm{MoS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.45 | 0.09 | 0.03 | 0.02 | 0.01 | 0.01 | 0.00
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.28 | 0.26 | 0.09 | 0.07 | 0.03 | 0.02 | 0.01
|  | ${d}$ | 1.27 | 0.27 | 0.10 | 0.07 | 0.02 | 0.01 | 0.00
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 1.66 | 0.94 | 0.89 | 0.71 | 0.71 | 0.67 | 0.59
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.00 | 1.13 | 1.08 | 0.86 | 0.85 | 0.81 | 0.71
|  | ${d}$ | 2.18 | 1.24 | 1.16 | 0.93 | 0.91 | 0.87 | 0.76
| ${1H}$ | ${d_{z^{2}}}$ | 1.96 | 1.08 | 0.86 | 0.80 | 0.68 | 0.63 | 0.56
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.26 | 1.25 | 0.99 | 0.92 | 0.78 | 0.72 | 0.65
|  | ${d}$ | 2.26 | 1.25 | 0.99 | 0.92 | 0.78 | 0.72 | 0.65
$\rm{MoSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.49 | 0.11 | 0.05 | 0.04 | 0.02 | 0.01 | 0.00
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.41 | 0.33 | 0.15 | 0.11 | 0.05 | 0.03 | 0.01
|  | ${d}$ | 1.13 | 0.36 | 0.17 | 0.12 | 0.05 | 0.03 | 0.01
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 2.30 | 1.29 | 1.25 | 0.99 | 1.01 | 0.94 | 0.83
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.89 | 1.06 | 1.03 | 0.81 | 0.83 | 0.77 | 0.69
|  | ${d}$ | 2.28 | 1.24 | 1.18 | 0.94 | 0.93 | 0.88 | 0.77
| ${1H}$ | ${d_{z^{2}}}$ | 1.08 | 0.69 | 0.55 | 0.52 | 0.44 | 0.41 | 0.37
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.35 | 0.86 | 0.69 | 0.64 | 0.55 | 0.51 | 0.46
|  | ${d}$ | 1.46 | 0.84 | 0.69 | 0.64 | 0.55 | 0.51 | 0.46
$\rm{MoTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.49 | 0.21 | 0.17 | 0.17 | 0.15 | 0.14 | 0.13
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.52 | 0.22 | 0.19 | 0.18 | 0.16 | 0.15 | 0.14
|  | ${d}$ | 1.53 | 0.82 | 0.66 | 0.62 | 0.53 | 0.49 | 0.44
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 0.55 | 0.17 | 0.16 | 0.10 | 0.09 | 0.09 | 0.07
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.85 | 0.27 | 0.25 | 0.16 | 0.13 | 0.15 | 0.10
|  | ${d}$ | 1.14 | 0.32 | 0.28 | 0.17 | 0.14 | 0.15 | 0.10
| ${1H}$ | ${d_{z^{2}}}$ | 1.98 | 1.13 | 0.88 | 0.81 | 0.68 | 0.62 | 0.56
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.42 | 1.38 | 1.07 | 0.99 | 0.83 | 0.76 | 0.68
|  | ${d}$ | 2.39 | 1.38 | 1.07 | 0.99 | 0.83 | 0.76 | 0.68
$\rm{WS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.40 | 0.09 | 0.04 | 0.03 | 0.01 | 0.01 | 0.01
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.09 | 0.25 | 0.10 | 0.08 | 0.04 | 0.02 | 0.01
|  | ${d}$ | 1.31 | 0.40 | 0.18 | 0.14 | 0.05 | 0.03 | 0.00
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 1.40 | 0.77 | 0.73 | 0.58 | 0.56 | 0.55 | 0.47
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.90 | 1.04 | 0.99 | 0.79 | 0.76 | 0.74 | 0.64
|  | ${d}$ | 2.08 | 1.14 | 1.06 | 0.86 | 0.81 | 0.80 | 0.68
| ${1H}$ | ${d_{z^{2}}}$ | 2.04 | 1.14 | 0.90 | 0.84 | 0.71 | 0.65 | 0.58
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.42 | 1.36 | 1.07 | 0.99 | 0.84 | 0.77 | 0.69
|  | ${d}$ | 2.43 | 1.35 | 1.07 | 0.99 | 0.84 | 0.77 | 0.69
$\rm{WSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.43 | 0.09 | 0.04 | 0.03 | 0.02 | 0.01 | 0.01
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.10 | 0.24 | 0.10 | 0.08 | 0.04 | 0.03 | 0.01
|  | ${d}$ | 1.43 | 0.39 | 0.18 | 0.13 | 0.05 | 0.03 | 0.00
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 1.64 | 0.95 | 0.92 | 0.74 | 0.73 | 0.70 | 0.61
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.03 | 1.18 | 1.14 | 0.91 | 0.91 | 0.86 | 0.76
|  | ${d}$ | 2.41 | 1.31 | 1.24 | 0.99 | 0.97 | 0.93 | 0.81
| ${1H}$ | ${d_{z^{2}}}$ | 1.11 | 0.72 | 0.58 | 0.54 | 0.46 | 0.42 | 0.38
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.43 | 0.93 | 0.74 | 0.69 | 0.59 | 0.54 | 0.48
|  | ${d}$ | 1.54 | 0.91 | 0.74 | 0.69 | 0.59 | 0.54 | 0.48
$\rm{WTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.39 | 0.12 | 0.06 | 0.05 | 0.03 | 0.02 | 0.01
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.05 | 0.32 | 0.17 | 0.14 | 0.07 | 0.04 | 0.02
|  | ${d}$ | 1.08 | 0.36 | 0.19 | 0.15 | 0.08 | 0.05 | 0.02
| ${1T^{\prime}}$ | ${d_{z^{2}}}$ | 0.57 | 0.21 | 0.20 | 0.12 | 0.11 | 0.11 | 0.08
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.85 | 0.31 | 0.29 | 0.18 | 0.16 | 0.17 | 0.12
|  | ${d}$ | 1.21 | 0.37 | 0.33 | 0.20 | 0.17 | 0.17 | 0.11

In the $1T^{\prime}$ structure, all $d$ states are split due to its lower
symmetry compared to the $1T$ structure, which significantly modifies the
electronic structure [101]. In this context, the full $d$ orbital set is the
optimal correlated subspace for capturing the electronic characteristics of
this structure. The semimetallic behavior of the $1T^{\prime}$ structure
enhances electron-electron interactions compared to the $1T$ structure [102,
103, 101, 104]. While the $1T^{\prime}$ structure does not have a band gap,
the scarcity of metallic states near the Fermi level reduces the contribution
of X-$p$ $\longrightarrow$ M-$d$ transitions to screening, resulting in a $U$
value that is largely similar to that of the $1H$ structure.

Regardless of the correlated subspace or structural symmetry, electron
screening is enhanced in the MTe2 compared to MS2/MSe2, leading to a reduction
in both the Coulomb interaction parameters $U$ and $W$. For example, in the
three-orbital correlated subspace, the $U$ value decreases from 2.33 eV in
$1H$-MoS2 to 1.35 eV in $1H$-MoTe2. The transition from S to Te in each MX2
system contributes to the determination of the $U$ and $W$ parameters through
two mechanisms. First, similar to the effect seen in the bare interaction, the
Wannier localization effect causes a reduction in $U$ and $W$ from S to Te.
Second, the shift from S/Se to Te brings the X-$p$ states into closer energy
proximity with the M-$d$ states, as shown in Fig. S1 of the supplementary
material. This closeness in energy levels translates to a smaller energy
difference, which increases the contribution of X-$p$ $\longrightarrow$ M-$d$
transitions to the polarization function and as a consequence reduces the $U$
and $W$ parameters. On the other hand, a comparison of MoX2 and WX2 compounds
reveals that the screened interactions are nearly identical for most of them.
This is likely due to the similar atomic radii of Mo and W, which leads to a
similar degree of localization of the $d$ orbitals in both materials.

So far, we have only considered the on-site Coulomb interaction matrix
elements. The long-range off-site Coulomb interaction plays an important role
in determining the phase diagram of 2D materials. The long-range behavior of
the screened Coulomb interaction is shown in Fig. 8 for four MX2 compounds and
compared with the unscreened $1/r$ interaction. Table 3 reports the off-site
partially screened Coulomb interaction $U(r)$ for all considered Mo- and
W-based TMDs as a function of distance up to $r=5a$. As shown in Fig. 8(a),
the effective Coulomb interaction in $1H$-MoS2 reveals a significant long-
range part of $U$. This indicates that the short-range interaction is strongly
screened, while the long-range interaction is weakly screened. Furthermore,
due to reduced screening at large distances, in $1H$-MoS2 the calculated
$U(r)$ approaches the bare interaction $1/r$. The metallic states in the $1T$
structure [see Fig. 8(c)] give rise to a significant reduction in long-range
Coulomb interaction, and it is fully screened at a short distance of about
1.5a. In contrast, for the $1H$-MoS2 it takes a considerably larger value at a
short distance. We find that the $\epsilon(r)=V/U$ ratio for $1H$-MoS2 has a
strong $r$-dependence, i.e., $\epsilon(r_{1})$=3.4, $\epsilon(r_{2})$=2.1,
$\epsilon(r_{3})$=1.7, where the distance $r_{i}$ is given in units of lattice
constant $a$. This means that the $r$-dependent screening in $1H$-MX2 deviates
substantially from $1/\epsilon r$, i.e., $U(r)$ and $W(r)$ cannot be expressed
by a simple static dielectric constant $\epsilon$. The situation is quite
similar in other considered TMDs with $1H$ structure, where the dielectric
constant decreases with distance, in agreement with recent experiments. This
$r$-dependent nonconventional screening explains the large exciton binding
energies and deviations from the usual hydrogenic Rydberg series of energy
levels of the excitonic states in semiconducting monolayer TMDs [51, 52]. Note
that the long-range behavior of the $W(r)$ for bulk MoS2 can be fitted by a
static dielectric constant $\epsilon_{||}=9$ (see Fig. S3 in supplementary
material), revealing a conventional screening in three-dimensional
semiconductors. Indeed, the dielectric constant $\epsilon=9$ that we use is
very close to the experimental value of $\epsilon^{expt.}=10$ [105].

(a)  Figure 8: (Color online) Distance ($r$) dependence of the partially and
fully screened Coulomb interaction $U(r)$ and $W(r)$ for $t_{2g}$ electrons in
(a) $1H$-MoS2, (b) 1$T^{\prime}$-WTe2, and (c) $1T$-MoS2. Bare Coulomb
interaction $V(r)$ is depicted with a solid line. Panel (d) shows the behavior
of $U(r)$ and $W(r)$ for $1H$-MoS2 at much larger distances.

Most monolayer MoX2 and WX2 materials are stable only in the $1H$ structure.
However, recent work has shown that monolayer of WSe2 can also be grown in the
$1T^{\prime}$ structure, which has a topological gap of 129 meV as measured by
angle-resolved photoemission spectroscopy (ARPES) experiments [57, 58]. In the
absence of spin-orbit coupling, $1T^{\prime}$-WX2 is a semimetal, with two
bands meeting at the Fermi level (EF), similar to graphene. However, the
valence band maximum in $1T^{\prime}$-WX2 has a flattened non-parabolic shape
along the $\Gamma$-Y direction, while the conduction and valence bands in
graphene meet at the K point with linear dispersion. From the point of view of
screening, electrons in linear bands behave like electrons in an insulator,
and the long-range part of the effective Coulomb interaction is not screened
well [60, 59, 106]. In the $1T^{\prime}$ structure of MX2 compounds, the
screening also turns out to be non-local, even though the bands do not have
purely linear dispersion. For example, in the full $d$ subspace of MoS2, the
non-local sixth neighbor interaction $U_{06}$ for the $1T^{\prime}$ structure
is 0.76 eV, which is even larger than the corresponding value for the
semiconducting $1H$ structure ($U_{06}$ = 0.65 eV). These findings have
important implications for our understanding of the interaction effects in 2D
materials. It suggests that the relationship between long-range screening and
linear dispersion is not straightforward and that other factors such as the
density of correlated states around EF and the overlap of the $p$ orbitals
with these states also play a role.

###  III.3 Coulomb interaction parameters: MX2 (M=Nb, Ta; X=S, Se, Te)

The 2D Mo(W)X2 compounds discussed in the preceding section can have different
ground states, ranging from semiconducting to semimetallic to metallic,
depending on the crystal structure. In this section, we investigate the
screening of the Coulomb interaction in Nb- and Ta-based MX2 compounds (where
X=S, Se, Te). Nb and Ta atoms have one fewer electron than Mo and W atoms.
Their electronic structures show different metallic behaviors. In the $1H$
structure, these materials show cold metallic behavior, while in the $1T$
structure they are normal metals (see Fig. 1 and Fig. 4 ). As can be seen from
the Wannier interpolated band structures in Fig. 4 a minimal one-orbital low-
energy model may be sufficient for investigating transport properties in the
$1H$ structure of these materials. However, the $1T$ structure requires a more
comprehensive three-orbital low-energy model. For completeness, we provide
Coulomb interaction parameters for three correlated subspaces: a single
orbital ($d_{z^{2}}$), a three-orbitals ($d_{z^{2}}$ \+ $d_{xy}$ \+
$d_{x^{2}-y^{2}}$), and the full $d$ shell of the TM atom. The calculated
Coulomb interaction parameters, including the bare $V$, Hubbard-Kanamori
parameters ($U$, $U^{\prime}$, and $J$), and the fully screened $W$, are
systematically presented in Table 4 and Fig. 9 provides a visual
representation of the trends of $V$, $U$, and $W$ parameters across all
compounds and their respective phases, to facilitate the comparison.

Table 4: Bare Coulomb interaction $V$, partially screened Hubbard-Kanamori parameters [$U$, $U^{\prime}$, $J$ (in eV)] and fully screened $W$ (in eV) for $d$ orbitals of TMs in MX2 (M=Nb, Ta; X=S, Se, Te) compounds. MX2 | Phase | Orbitals |  $V$(eV) |  $U$(eV) | ${U^{\prime}}(eV)$ | $J(eV)$ |  $W$(eV)
---|---|---|---|---|---|---|---
| ${1H}$ | ${d_{z^{2}}}$ | 8.84 | 1.29 | —– | —– | 0.43
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.25 | 1.69 | 1.39 | 0.15 | 0.67
|  | ${d}$ | 8.92 | 2.30 | 2.01 | 0.13 | 0.67
$\rm{NbS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.36 | 1.23 | —– | —– | 0.41
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.99 | 1.08 | 0.83 | 0.14 | 0.61
|  | ${d}$ | 8.74 | 1.77 | 1.52 | 0.13 | 0.61
| ${1H}$ | ${d_{z^{2}}}$ | 9.21 | 1.11 | —– | —– | 0.37
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.58 | 1.59 | 1.29 | 0.17 | 0.62
|  | ${d}$ | 9.35 | 2.25 | 1.97 | 0.14 | 0.64
$\rm{NbSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 9.01 | 1.16 | —– | —– | 0.52
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.19 | 1.25 | 0.96 | 0.16 | 0.56
|  | ${d}$ | 9.19 | 1.29 | 1.03 | 0.14 | 0.58
| ${1H}$ | ${d_{z^{2}}}$ | 7.93 | 0.65 | —– | —– | 0.36
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.36 | 0.96 | 0.71 | 0.12 | 0.49
|  | ${d}$ | 8.05 | 0.94 | 0.70 | 0.10 | 0.49
$\rm{NbTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.18 | 0.60 | —– | —– | 0.35
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.86 | 0.86 | 0.59 | 0.13 | 0.53
|  | ${d}$ | 8.30 | 0.93 | 0.70 | 0.11 | 0.50
| ${1H}$ | ${d_{z^{2}}}$ | 8.12 | 1.13 | —– | —– | 0.41
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.66 | 1.76 | 1.47 | 0.16 | 0.69
|  | ${d}$ | 8.34 | 2.40 | 2.12 | 0.12 | 0.70
$\rm{TaS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.63 | 1.15 | —– | —– | 0.46
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.35 | 1.14 | 0.90 | 0.14 | 0.65
|  | ${d}$ | 7.87 | 1.89 | 1.66 | 0.12 | 0.61
| ${1H}$ | ${d_{z^{2}}}$ | 8.74 | 2.05 | —– | —– | 0.45
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 9.19 | 2.47 | 2.18 | 0.19 | 0.67
|  | ${d}$ | 8.96 | 2.45 | 2.17 | 0.14 | 0.69
$\rm{TaSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 8.58 | 1.14 | —– | —– | 0.38
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.90 | 1.21 | 0.95 | 0.16 | 0.62
|  | ${d}$ | 8.85 | 1.52 | 1.26 | 0.14 | 0.64
| ${1H}$ | ${d_{z^{2}}}$ | 7.67 | 0.90 | —– | —– | 0.33
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.15 | 1.17 | 0.92 | 0.15 | 0.52
|  | ${d}$ | 7.88 | 1.14 | 0.88 | 0.11 | 0.53
$\rm{TaTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 7.36 | 0.78 | —– | —– | 0.30
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 8.29 | 0.87 | 0.61 | 0.12 | 0.54
|  | ${d}$ | 7.62 | 0.94 | 0.72 | 0.11 | 0.50

We find that the calculated $V$ parameters are slightly smaller than the same
parameters in Mo(W)X2 compounds. This is because the Nb (Ta) atom has one
electron less than the Mo (W) atom (less nuclear charge), and thus the wave
functions are less contracted, leading to less localized Wannier functions and
as a consequence slightly smaller $V$ parameters. This is consistent with the
findings of previous studies [63, 42, 107]. We also compare the $V$ parameters
in Nb(Ta)X2 compounds with the corresponding ones in elementary transition
metals. We find that the $V$ values for the considered 4$d$/5$d$ TMDs
compounds, namely NbX2 and TaX2, are almost 2 eV smaller than the
corresponding ones in elementary Nb or Ta bulk systems [63]. This is likely
due to the coupling to neighboring chalcogen $p$ states in TMDs. The coupling
to $p$ states makes the TM $d$ orbitals spill into the $p$ states, giving rise
to delocalization and, therefore, to smaller $V$ parameters. This effect is
also reflected in the shape of the Wannier orbitals shown in Fig. 5 and in the
projected band structures (see supplementary material). An analysis of the
shape of the Wannier orbitals reveals that for any system in which the overlap
of the $d$ and $p$ orbitals is stronger, the Wannier functions spread to the
nearest neighboring atoms. Moreover as shown in Fig. S4, the coupling to
neighboring chalcogen $p$ states gets stronger in Te-based TMDs, which makes
the Wannier functions more extended.

Figure 9: (Color online) Comparison of the on-site Coulomb interaction
parameters for MX2 (M=Nb, Ta; X=S, Se, Te) in $1H$ and $1T$ structures,
considering three different correlated subspaces. Purple, green, and orange
indicate the values of $V$, $U$, and $W$, respectively.

Similarly to the bare Coulomb interaction parameters discussed above, the
Wannier localization effect is also important for the partially screened $U$
and fully screened $W$ parameters. The $U$ ($W$) parameter tends to decrease
from Mo/W to Nb/Ta TMDs systems because the Wannier functions are less
localized in Nb/Ta TMDs due to smaller value of nuclear charge. In addition,
for the $1H$ structure of MX2 (M=Nb, Ta), a flat half-filled band with mainly
$d_{z^{2}}$ character increases screening substantially. This results in
larger $V-W$ and $V-U$ differences compared to the corresponding differences
for the $1H$-MX2 (M=Mo, W) compounds. Note that although the interaction
parameter $U$ is smaller in metallic MX2 (M=Nb, Ta) compounds than in $4d/5d$
elementary TMs and other materials containing $4d/5d$ transition metals, such
as MXenes ($U=3.5-4$ eV) [64], the narrow bands with $d_{z^{2}}$, $t_{2g}$,
and $e_{g}$ character in TMDs result in a larger $U/W_{b}$ correlation
strength.

Strong interactions between localized electrons can lead to a transition from
a metal to an insulator and trigger a magnetic phase change [37, 38, 108,
109]. Magnetic ordering and Mott phases have been experimentally observed in
layered materials containing transition metal (TM) atoms, such as the TM
halides CrI3 [110, 111], VI3 [112, 43], the 3d transition metal
dichalcogenides (TMDs) $2H$-VX2 [113, 42], and the TM phosphorous
trichalcogenides NiPX3 (where X=S and Se) [114]. All of these materials have
flat bands in their energy spectra. In the case of $2H$-NbX2 and $2H$-TaX2,
there is an isolated, half-filled low-energy band with predominantly
$d_{z^{2}}$ character resulting in a large $U/W_{b}$ ratio. Although the $1H$
structure of NbX2 and TaX2 has stronger correlations due to a single, narrow
$d_{z^{2}}$ band, Mott-like insulating behavior has been observed in the
reconstructed $1T$ lattice.

Experimentally, compounds such as TaS2[115], NbSe2 [116], and TaSe2 [117]
exhibit metallic behavior at high temperatures in the 1$T$ crystal structure.
However, as the temperature is lowered, they undergo a transition to an
insulating phase characterized by a distinct atomic rearrangement pattern,
known as the ”star-of-David” motif [118, 119, 120, 121]. For example, the
reconstructed 1$T$-TaS2 monolayer has a Mott gap of 0.45 eV [108]. This
lattice reconstruction coincides with a commensurate CDW (CCDW) transition.

Table 5: The on-site effective Coulomb interaction $U^{\mathrm{eff}}_{00}$ and the off-site interaction $U^{\mathrm{eff}}_{01}$ are calculated for a specific atomic rearrangement pattern within the 1$T$ crystal structure, known as the star-of-David configuration. MX2 | Phase | Orbitals |  $U^{\mathrm{eff}}_{00}$(eV) |  $U^{\mathrm{eff}}_{01}$(eV)
---|---|---|---|---
NbS2 | $1T$ | $d$ | 0.71 | 0.40
NbSe2 | $1T$ | $d$ | 0.27 | 0.06
NbTe2 | $1T$ | $d$ | 0.18 | 0.04
TaS2 | $1T$ | $d$ | 0.83 | 0.47
TaSe2 | $1T$ | $d$ | 0.35 | 0.07
TaTe2 | $1T$ | $d$ | 0.18 | 0.04

The effective Coulomb interaction $U^{\mathrm{eff}}$ for the star-of-David
configuration in MX2 lattices has been reported in the literature by some
authors [66, 92, 65]. Following the same procedure [see Eq.(10)] and using the
results of the undistorted 1$T$ structure, we estimated the on-site and off-
site Coulomb interactions. The results are presented in Table 5. In the cases
of NbS2 and TaS2, we obtain $U^{\mathrm{eff}}_{00}$ values of 0.71 eV and 0.83
eV, respectively. This is 0.18 eV larger than the value of 0.65 eV reported
for TaS2 in a previous study using the same approach [65]. The on-site
interaction $U^{\mathrm{eff}}_{00}$ is large enough compared to the bandwidth
$W_{b}$=0.02 eV [66, 65] which induces a robust Mott insulator in CCDW
1$T$-TaS2. Moreover, $U^{\mathrm{eff}}_{00}$ for TaSe2 is also sizable,
revealing the importance of the correlation in other MX2 which have not been
quantitatively studied before.

The stabilization of the distorted CCDW 1$T$-MX2 (M=Nb, Ta) not only increases
the ratio of the on-site effective Coulomb interaction $U^{\mathrm{eff}}_{00}$
to the bandwidth $W_{b}$, but also leads to an increase in the ratio of the
long-range interaction $U^{\mathrm{eff}}_{01}$ to $U^{\mathrm{eff}}_{00}$ up
to 0.56. This could potentially lead to the emergence of exotic quantum
phenomena. The obtained on-site $U^{\mathrm{eff}}_{00}$ and off-site
$U^{\mathrm{eff}}_{01}$ Coulomb interaction parameters are important for use
in extended model Hamiltonians and methods beyond DFT, such as DFT+$U$ and DFT
plus dynamical mean-field theory (DFT+DMFT), which is used to obtain a
reliable electronic structure, Mott gap, and other properties.

We have discussed the on-site Coulomb interaction parameters in Nb(Ta)X2
(where X=S, Se, Te) compounds. As in Mo(W)X2 compounds, the screening of the
long-range Coulomb interaction is nonconventional in these compounds. Figure
10 shows the calculated partially and fully screened Coulomb interaction
parameters $U$ and $W$ for the $1H$ and $1T$ phases of NbS2 and TaS2 compounds
as a function of distance up to $r=5a$. Despite the metallic nature of these
compounds, the long-range Coulomb interactions do not screen well, and as a
consequence, long-range interactions remain sizable, as shown in Table 6. The
important consequence of this reduced screening and sizable long-range Coulomb
interaction in metallic systems is the appearance of CDW order and the
existence of intriguing plasmons in TMDs. The latter will be discussed in
detail below. Note that CDW appears to be stronger in the single-layer form of
1$T$-MX2 [122, 32, 33, 34]. The sizable $U_{0n}$ in the 1$T$ structure
indicates that the electron-electron interaction is one of the main reasons
for the occurrence of CDW order in TMDs. Although the value of off-site
$U_{0n}$ in the 1$T$ structure is not negligible, it is not as large as the
corresponding one in the 1$H$ structure. It would be interesting to take off-
site $U_{0n}$ into account within the extended Hubbard model and investigate
whether CDW order occurs for TaX2 and NbX2 or not.

(a)  Figure 10: (Color online) Distance ($r$) dependence of the partially screened Coulomb interaction $U(r)$ and fully screened Coulomb interaction $W(r)$ between $t_{2g}$ electrons in (a) 1$H$-NbS2, (b) 1$T$-NbS2 (c) 1$H$-TaS2, and (d) 1$T$-TaS2. The bare Coulomb interaction $V(r)$ is depicted with a solid line. Table 6: Long-range partially screened Coulomb interaction $U$ for MX2 (M=Nb, Ta; X=S, Se, Te) compounds. $U_{00}$ is the onsite interaction, $U_{01}$ is the nearest neighbor interaction, $U_{02}$ is the next nearest neighbor interaction, and so on, up to the sixth nearest neighbor interaction. MX2 | Phase | Orbitals |  $U_{00}$(eV) |  $U_{01}$(eV) | $U_{02}(eV)$ |  $U_{03}$(eV) | $U_{04}(eV)$ | $U_{05}(eV)$ | $U_{06}(eV)$
---|---|---|---|---|---|---|---|---|---
| ${1H}$ | ${d_{z^{2}}}$ | 1.29 | 0.42 | 0.22 | 0.17 | 0.09 | 0.06 | 0.03
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.69 | 0.55 | 0.28 | 0.22 | 0.11 | 0.07 | 0.03
|  | ${d}$ | 2.30 | 1.23 | 0.97 | 0.90 | 0.76 | 0.69 | 0.62
$\rm{NbS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 1.23 | 0.27 | 0.16 | 0.14 | 0.10 | 0.08 | 0.06
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.08 | 0.24 | 0.14 | 0.13 | 0.09 | 0.07 | 0.05
|  | ${d}$ | 1.77 | 0.79 | 0.61 | 0.57 | 0.48 | 0.45 | 0.40
| ${1H}$ | ${d_{z^{2}}}$ | 1.11 | 0.34 | 0.17 | 0.14 | 0.07 | 0.05 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.59 | 0.48 | 0.25 | 0.20 | 0.10 | 0.06 | 0.03
|  | ${d}$ | 2.25 | 1.19 | 0.95 | 0.89 | 0.76 | 0.70 | 0.63
$\rm{NbSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 1.16 | 0.28 | 0.18 | 0.15 | 0.11 | 0.09 | 0.06
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.12 | 0.27 | 0.17 | 0.15 | 0.10 | 0.08 | 0.06
|  | ${d}$ | 1.29 | 0.32 | 0.18 | 0.15 | 0.09 | 0.07 | 0.05
| ${1H}$ | ${d_{z^{2}}}$ | 0.65 | 0.16 | 0.09 | 0.08 | 0.05 | 0.03 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.96 | 0.24 | 0.14 | 0.11 | 0.07 | 0.05 | 0.03
|  | ${d}$ | 0.94 | 0.25 | 0.14 | 0.11 | 0.07 | 0.05 | 0.03
$\rm{NbTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.60 | 0.09 | 0.05 | 0.04 | 0.03 | 0.02 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.86 | 0.13 | 0.07 | 0.06 | 0.04 | 0.03 | 0.02
|  | ${d}$ | 0.93 | 0.20 | 0.11 | 0.09 | 0.05 | 0.04 | 0.03
| ${1H}$ | ${d_{z^{2}}}$ | 1.13 | 0.39 | 0.20 | 0.15 | 0.07 | 0.05 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.76 | 0.60 | 0.31 | 0.24 | 0.12 | 0.07 | 0.03
|  | ${d}$ | 2.40 | 1.32 | 1.03 | 0.96 | 0.80 | 0.73 | 0.65
$\rm{TaS_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 1.15 | 0.30 | 0.19 | 0.17 | 0.12 | 0.09 | 0.07
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.14 | 0.30 | 0.19 | 0.16 | 0.11 | 0.09 | 0.06
|  | ${d}$ | 1.89 | 0.95 | 0.73 | 0.68 | 0.58 | 0.53 | 0.48
| ${1H}$ | ${d_{z^{2}}}$ | 2.05 | 1.09 | 0.86 | 0.80 | 0.68 | 0.62 | 0.56
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 2.47 | 1.31 | 1.04 | 0.96 | 0.82 | 0.75 | 0.67
|  | ${d}$ | 2.45 | 1.31 | 1.03 | 0.96 | 0.82 | 0.75 | 0.67
$\rm{TaSe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 1.14 | 0.28 | 0.17 | 0.15 | 0.10 | 0.08 | 0.06
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.21 | 0.29 | 0.18 | 0.15 | 0.11 | 0.09 | 0.06
|  | ${d}$ | 1.52 | 0.44 | 0.24 | 0.20 | 0.12 | 0.09 | 0.06
| ${1H}$ | ${d_{z^{2}}}$ | 0.90 | 0.26 | 0.14 | 0.11 | 0.06 | 0.04 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 1.17 | 0.34 | 0.18 | 0.14 | 0.08 | 0.05 | 0.03
|  | ${d}$ | 1.14 | 0.33 | 0.17 | 0.14 | 0.07 | 0.05 | 0.03
$\rm{TaTe_{2}}$ | ${1T}$ | ${d_{z^{2}}}$ | 0.78 | 0.12 | 0.06 | 0.05 | 0.03 | 0.03 | 0.02
|  | ${d_{z^{2}},d_{xy},d_{x^{2}-y^{2}}}$ | 0.87 | 0.13 | 0.07 | 0.06 | 0.04 | 0.03 | 0.02
|  | ${d}$ | 0.94 | 0.21 | 0.11 | 0.09 | 0.05 | 0.04 | 0.03
(a)  Figure 11: Extrapolated electron energy loss spectra for selected wave
vectors along the $\Gamma$-M direction for (a) 1$H$-TaS2, (b) 1$T$-TaS2. (a)
Figure 12: Plasmon dispersion along the high symmetry line $\Gamma$-M in the
2D Brillouin zone for (a) 1$T$-TaS2, (b) 1$H$-NbS2, (c) 1$H$-NbSe2, and (d)
1$H$-TaS2.

The plasmon dispersion of monolayer TMDs has been studied extensively in
recent years [54, 30, 31, 55]. While the dispersion of plasmons in three-
dimensional bulk systems commences with finite energy at a wavevector of
$q=0$, conventional plasmons in 2D metallic systems follow a
$\omega_{p}\propto\sqrt{q}$ relationship. To ascertain plasmon excitations, we
identify the peaks in the loss function
$L(\mathbf{q},\omega)=-\text{Im}(1/\epsilon_{m}(\mathbf{q},\omega))$, wherein
$\epsilon_{m}(\mathbf{q},\omega)=1/\epsilon_{00}^{-1}(\mathbf{q},\omega)$
represents the macroscopic dielectric function. Fig. 12 displays the
calculated plasmon dispersion for the monolayers of four compounds: 1$T$-TaS2,
1$H$-NbS2, 1$H$-NbSe2, and 1$H$-TaS2 and compared with the $\sqrt{q}$ plasmon
dispersion. Surprisingly, materials of the form MX2 (M=Nb, Ta) deviate from
the anticipated $\sqrt{q}$ plasmon dispersion observed in conventional 2D
metallic systems. Particularly striking is the nearly linear dispersion of
plasmons in the monolayer metallic 1$H$-NbS2 at small $\mathbf{q}$, which
transitions to a nearly dispersionless behavior within an intermediate range
of wavevectors ($\mathbf{q}=0.1$ Å-1 to 0.3 Å-1). Moreover, we observe a
negative slope in the dispersion relation for 1$H$-MX2 at larger $\mathbf{q}$.
This deviation from the $\sqrt{q}$ behavior is most pronounced in the 1$H$
structure due to its comparatively greater long-range interaction compared to
the 1$T$ structure. Notably, these findings align well with electron energy
loss spectroscopy (EELS) measurements conducted by other researchers, who
noted a negative slope in the plasmon dispersion of 1$H$-TaS2, 1$H$-TaSe2, and
1$H$-NbSe2 [56]. The negative plasmon dispersion in TMDs has been attributed
to the strong electron-electron interactions in these materials. These
interactions can lead to the formation of collective charge fluctuations,
which can couple to the plasmons and modify their dispersion. The results of
our study provide new insights into the plasmon properties of TMDs. These
insights could be used to design TMD-based plasmonic devices with novel
functionalities.

###  III.4 Frequency dependency of screened Coulomb interaction

(a)  Figure 13: (Color online) Frequency dependence of the on-site and off-
site Coulomb interaction parameters $U(\omega)$ for (a) 1$H$-MoS2, (b)
1$H$-NbS2, and (c) 1$T$-NbS2. The real and imaginary parts of $U(\omega)$ for
different correlated subspaces are presented individually.

In this section, we investigate the frequency-dependent behavior of the
partially $U(\omega)$. We focus on the semiconducting 1$H$-MoS2, metallic
1$H$-NbS2, and metallic 1$T$-NbS2 compounds. We analyze these materials with
respect to two distinct correlated subspaces. The real and imaginary
components of the computed on-site interaction $U_{00}(\omega)$, as well as
the interactions of the first and second nearest neighbors, namely
$U_{01}(\omega)$ and $U_{02}(\omega)$, are shown in Fig. 13 for these
materials. Given the structural and compositional similarities among them, the
frequency-dependent behaviors of the screened Coulomb interactions exhibit
comparable trends across all three systems.

In the case of 1$H$-MoS2, the $U(\omega)$ profile displays smooth behavior
with minor fluctuations up to a frequency of 15 eV. Beyond this point, it
experiences linear growth, ultimately peaking at the plasmon frequency of
approximately 25 eV (which is also evident as a peak in the imaginary part of
$U(\omega)$). Subsequently, as the frequency increases further, it
asymptotically approaches the static value of 9.5 eV characteristic of
1$H$-MoS2.

For 1$H$-NbS2 and 1$T$-NbS2, both metallic systems, the Coulomb interactions
exhibit more pronounced variations at lower frequencies. These strong
variations can be attributed to the narrow bands around the Fermi level in
these materials. Notably, the frequency dependency of $U_{00}(\omega)$ shows
similar trends to the $U_{01}(\omega)$ and $U_{02}(\omega)$ cases. The decline
in $U(\omega)$ at lower frequencies, around 5 eV, in metallic systems arises
from effective screening influenced by $d$ states around the Fermi level,
which forms a bandwidth of approximately 5 eV. This behavior extends to the
off-site Coulomb interactions (nearest-neighbor $U_{01}(\omega)$ and next
nearest-neighbor $U_{02}(\omega)$), as shown by the dashed lines. One can
imagine that the variations in the effective Coulomb interaction at low
frequencies will average out, so that the static limit $U(\omega=0)$ may still
be appropriate for model Hamiltonian studies.

##  IV Conclusions

In this work, we have employed the random phase approximation within the FLAPW
method to study the on-site and $r$-dependent screening of the Coulomb
interactions $U(r)$ (partially screened) and $W(r)$ (fully screened) in 2D
TMDs MX2 (M=Mo, W, Nb, Ta; X = S, Se, Te). Our results show that the
$r$-dependent screening in semiconducting compounds like MoS2 deviates
substantially from the conventional behavior, i.e., $U(r)$ and $W(r)$ cannot
be expressed by a simple static dielectric constant $\epsilon$. We found that
the short-range interactions are strongly screened, while the long-range
interactions are weakly screened, i.e., they decay much slower than the bare
$1/r$ interaction. This nonconventional screening of the Coulomb interaction
in 2D TMDs can be attributed to the reduced dimensionality of these materials.
This $r$-dependent nonconventional screening explains the large exciton
binding energies and deviations from the usual hydrogenic Rydberg series of
energy levels of the excitonic states in semiconducting monolayer TMDs.

Our results also show that metallic TMDs like NbS2 in the $1H$ structure
exhibit a correlation strength $U/W_{b}\sim 2$, which is significantly larger
than the corresponding values in elementary TMs. This is due to the strong
$r$-dependent screening, which leads to a larger effective Coulomb
interaction. The large $U/W_{b}$ ratio suggests that these materials are prone
to Mott insulating behavior, which has been experimentally observed in the
reconstructed $1T$ lattice. Using the calculated $U$ parameters for
undistorted $1T$ structure, we extract the on-site effective
$U^{\mathrm{eff}}_{00}$ and nearest neighbor $U^{\mathrm{eff}}_{01}$ Coulomb
interaction parameters for reconstructed CCDW NbX2 and TaX2 compounds.
Strictly speaking, for the reconstructed star of David $1T$-MX2 (M=Nb, Ta)
structure, the large $U^{\mathrm{eff}}_{00}\sim 0.8$ eV estimated for $4d/5d$
electrons compared with the relatively small bandwidth $W_{b}\sim$ 0.02 eV
satisfies the condition of $U^{\mathrm{eff}}_{00}/W_{b}\gg 1$. Furthermore, we
found that the long-range Coulomb interactions remain sizable in metallic
TMDs, despite the metallic nature of these materials. This is due to the
reduced screening at long distances. The long-range Coulomb interactions can
lead to the existence of intriguing plasmons in the monolayer of TMDs and the
appearance of CDW order.

This study presents a comprehensive understanding of the Coulomb interactions
in both semiconducting and metallic 2D TMDs. Furthermore, the derived Coulomb
interaction parameters can be effectively employed in model Hamiltonians and
DFT+$U$ (DFT+$U$+$V$) methods, consequently boosting the predictive power of
these techniques.

###### Acknowledgements.

E.Ş and I.M. acknowledge support from Sonderforschungsbereich TRR 227 of
Deutsche Forschungsgemeinschaft (DFG) and funding provided by the European
Union (EFRE), Grant No: ZS/2016/06/79307.

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