# Photovoltaic efficiency of transition metal dichalcogenides thin films by ab
initio excited-state methods

Enesio Marinho Jr<EMAIL_ADDRESS>Departamento de Física e Química,
Universidade Estadual Paulista (UNESP),
Av. Brasil, 56, Ilha Solteira, 15385-007 São Paulo, Brazil.  Instituto de
Física Teórica, Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, São Paulo, 01140-070 São Paulo, Brazil.
Cesar E. P. Villegas<EMAIL_ADDRESS>Departamento de Ciencias,
Universidad Privada del Norte, Lima 15434, Peru  Pedro Paulo de Mello
Venezuela<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
Fluminense (UFF),
Av. Gal. Milton Tavares de Souza, s/n, 24210-346 Niterói, Rio de Janeiro,
Brazil.  Alexandre Reily Rocha<EMAIL_ADDRESS>Instituto de Física
Teórica, Universidade Estadual Paulista (UNESP),
R. Dr. Bento Teobaldo Ferraz, 271, São Paulo, 01140-070 São Paulo, Brazil.

###### Abstract

Transition metal dichalcogenides (TMDCs) have garnered significant interest in
optoelectronics, owing to their scalability and thickness-dependent electrical
and optical properties. In particular, thin films of TMDCs could be used in
photovoltaic devices. In this work, we employ _ab initio_ many-body
perturbation theory within $G_{0}W_{0}$-BSE approach to accurately compute the
optoelectronic properties of thin films of 2H-TMDCs composed of Mo, W, S, and
Se. Subsequently, we evaluate their photovoltaic performance including exciton
recombination effects, and show this is a key ingredient. We obtain
efficiencies of up to 29% for a 200-nm thick film of WSe2, thus providing an
upper limit. We also include other phenomenological recombination mechanisms
that could be present in current samples. This slightly reduces efficiencies,
indicating that even with current synthesis technologies, there is still
potential for further enhancement of TMDCs’ performance in photovoltaic
applications.

###### keywords:

Suggested keywords

Keywords: Transition metal dichalcogenides $\cdot$ Photovoltaic efficiency
$\cdot$ Optoelectronics $\cdot$ Thin films $\cdot$ MBPT $\cdot$ $GW$-BSE

##  1 Introduction

Layered van der Waals (vdW) materials consist of stacked two-dimensional (2D)
sheets interacting via dispersion forces. Graphite is a well-known member of
this family that is broadly used in industry as components for electrodes,
lubricants, fibers, heat exchangers, and batteries 1. The experimental
isolation of its single-layer counterpart, graphene 2 has opened the way for a
new era of 2D materials with compelling optoelectronic properties 3, 4, 5.
However, the lack of an energy gap in graphene greatly limits its
applicability in semiconductor technologies. On the other hand, the
semiconducting properties of transition metal dichalcogenides (TMDCs), with
chemical formula MX2 (M = Mo, W, and others, and X = S, Se, Te) pushed the
frontier of vdW materials’ potential uses spanning a broad range of fields,
including nanoelectronics and nanophotonics, sensing and actuation in the
nanoscale, and photovoltaic solar cells 6, 7, 8, 9, 10, 11, 12.

The TMDCs monolayers present direct bandgaps ranging from ${\sim}1.4$ to $2.5$
eV 13, which is well-suited for solar energy absorption 14, particularly as
ultra-thin photon absorbers for applications in transparent and flexible solar
cells 15, 16. Theoretical predictions indicate that the TMDCs monolayers
should present absorption coefficients per unit thickness one to two orders of
magnitude greater than those of conventional semiconductors 17. This is
achieved due to the localized nature of electronic wave functions, leading to
van Hove singularities in the density of states. These strong peaks are
accountable for the heightened photoresponsivity observed in these 2D
materials 18. As a consequence, with a thickness of less than 1 nm, in
principle, they can absorb an amount of visible light comparable to 15 nm of
GaAs or 50 nm of Si 17, which makes them a promising choice for efficient
photovoltaic applications. However, in practice, the power conversion
efficiency (PCE) of single layer TMDC solar cells has typically not exceeded
2% 19, 20, 21, 22, 23, 24. This is mostly due to strong Fermi-level pinning at
the metal–semiconductor interface 25, and high probability of carrier
recombination caused by defective surfaces, which leads to a low open-circuit
voltage ($V_{\text{oc}}$) 15.

In order to overcome those performance issues, some TMDC-based solar cells
have been assembled by using few-layered or multi-layered stacks of 2H-TMDCs.
Lee et al. 26 have studied the photovoltaic response of graphene-sandwiched
MoS2/WSe2 heterostructures with different thicknesses. They reported a high
short-circuit current density of ${\sim}2.2$ A/cm2, measured under laser light
($\lambda=532$ nm). For comparison, the authors also reported the short-
circuit current of vertical heterojunctions composed of single atomic layers
of MoS2 ($n$-doped) and WSe2 (either $p$-doped or ambipolar) with a value of
${\sim}1$ mA/cm2. The external quantum efficiency (EQE) is also dependent on
the number of layers. The measured EQEs at 532 nm were 2.4%, 12%, and 34% for
monolayer, bilayer, and multilayer p–n junctions. Memaran et al. 27 have
studied the photovoltaic efficiency of the MoSe2 crystal composed of
${\sim}10$ atomic layers transferred onto a flat hBN crystal creating an
electrostatic p-n junction. They have reported photovoltaic efficiencies
surpassing 14% under AM-1.5 spectrum, with concomitant fill factors
approaching 0.7.

More recently, Cho et al. 28 proposed a transparent thin-film photovoltaic
cell composed by a heterojunction of WSe2/MoS2 and an indium tin oxide
electrode. Their device is highly transparent (${\sim}80\%$) and presents a
PCE of ${\sim}$10%. Also, high-specific-power flexible transition metal
dichalcogenide solar cells have been fabricated by employing transparent
graphene contacts and MoO_x capping for doping, passivation, and anti-
reflection 29. These lead to PCE of 5.1% and specific power of 4.4 Wg-1 for
flexible WSe2 solar cells.

These observations suggest that junctions composed of few-layered or
multilayered TMDCs can yield higher photovoltaic responses compared to
monolayers. Nonetheless, while monolayer TMDCs have been extensively studied
and well characterized for photovoltaic applications, their bulk counterparts
were less so, especially from a microscopic point of view. In this sense, a
highly accurate and reliable _ab initio_ prediction of the optoelectronic
properties and photovoltaic efficiencies of bulk 2H-TMDCs is crucial.

Herein, we employ _ab initio_ many-body perturbation theory within the
$GW$-BSE scheme to investigate the optoelectronic properties of bulk 2H-TMDCs
and then address their theoretical photovoltaic efficiencies. We analyze the
power-conversion efficiency of MX_2 TMDCs with M = Mo, W and X = S, Se,
modeling the inclusion of nonradiative recombination losses with exciton
eigenvalues computed by BSE, in an approach we denote as SLME-X which has
already been proposed by Ozório and co-workers 30, and compared those results
with another modeling in which we estimate the fraction of radiative
recombination processes using reported values obtained by photoluminescence
quantum yield experiments (SLME-PLQY). Applying the SLME-X approach, the
photovoltaic efficiencies of the TMDCs consistently fell within the range of
23% to 29%, reaching efficiencies analogous to those obtained using the
Shockley-Queisser formalism. On the other hand, when we employ the SLME-PLQY
framework, the power conversion efficiencies tend to be reduced, ranging from
19% to 24%.

##  2 Computational Methods

Ground-state atomic structures were obtained by first-principles calculations
based on density-functional theory (DFT) 31, 32 as implemented in the quantum
espresso package 33. The Perdew-Burke-Ernzerhof generalized-gradient
approximation 34 was employed to describe the exchange-correlation functional.
Fully relativistic optimized norm-conserving Vanderbilt pseudopotentials
(ONCVPSP) 35, considering semicore $s$ and $p$ states as valence electrons,
are used to describe the electron–ion interaction. A kinetic energy cutoff of
$84$ Ry is set to expand the Kohn-Sham orbitals. The charge density is
obtained in a $\Gamma$-centered Monkhorst-Pack _k_ -point sampling of
$8{\times}8{\times}4$. The structures are fully relaxed including van der
Waals corrections within the semi-empirical dispersion scheme (PBE-D2) as
proposed by Grimme 36, and also including spin-orbit coupling.

The many-body perturbation theory (MBPT) framework within the $GW$
approximation 37 has been applied to compute the quasiparticle energies of
bulk 2H-TMDCs by using the yambo code 38. The dielectric screening
$\epsilon_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q},\omega)$ is obtained
considering the Plasmon-Pole approximation in a $\mathbf{k}$-grid of $12\times
12\times 4$, with an energy cutoff of 60 Ry for the exchange potential and 30
Ry for the screened interaction $W_{0}$. Both the Green’s functions and the
screened Coulomb interaction are calculated including 600 bands. This number
of bands is sufficient to provide accurate results thanks to the
implementation of the Bruneval-Gonze terminators approach 39.

The electron-hole (e-h) interaction is included in the calculation of the
optical absorption spectra of the 2H-TMDCs by means of the Bethe-Salpeter
equation (BSE), which can be written adopting the Tamm-Dancoff approximation
as an effective eigenvalue problem

$\sum_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}H^{\text{BSE}}_{\begin{subarray}{c}vc\mathbf{k}\\\
v^{\prime}c^{\prime}\mathbf{k}^{\prime}\end{subarray}}A^{S}_{v^{\prime}c^{\prime}\mathbf{k}^{\prime}}=\Omega_{S}A^{S}_{vc\mathbf{k}}\,,$
(1)

where $A^{S}_{vc\mathbf{k}}$ and $\Omega_{S}$ are the exciton eigenfunction
and the eigenvalues for the $S$-th exciton, respectively. In this form, the
BSE Hamiltonian is given by

$H^{\text{BSE}}_{\begin{subarray}{c}vc\mathbf{k}\\\
v^{\prime}c^{\prime}\mathbf{k}^{\prime}\end{subarray}}=\left(\epsilon^{\text{QP}}_{c\mathbf{k}}-\epsilon^{\text{QP}}_{v\mathbf{k}}\right)\variation_{c,c^{\prime}}\variation_{v,v^{\prime}}\variation_{\mathbf{k},\mathbf{k}^{\prime}}+\Xi_{vc\mathbf{k},v^{\prime}c^{\prime}\mathbf{k}^{\prime}}\,,$
(2)

where the first term comprises a diagonal part that contains the quasiparticle
energy differences, and $\Xi=\bar{v}-W$ is the e-h interaction kernel,
composed of a bare, repulsive short-range exchange term, and an attractive
static screened Coulomb potential 40.

To speed up the BSE calculations, we employ the double-grid method 41 in which
the kernel matrix elements are calculated on a coarse k-grid (the same used
for the $GW$ calculations), and then interpolated on a finer k-grid sampling
of 60 $\times$ 60 $\times$ 8, including eight valence bands and eight
conduction bands. These values are sufficient to obtain converged results for
the lowest peaks of the optical absorption. A full analysis of the convergence
of all the relevant parameters related to $GW$ and BSE calculations is
presented in the Supporting Information 42.

The optical absorption is calculated considering the imaginary part of the
dielectric function

$\epsilon_{2}(\omega)=\frac{4\pi^{2}}{VN_{k}}\sum_{S}\absolutevalue{\sum_{vc\mathbf{k}}A^{S}_{cv\mathbf{k}}\hat{\mathbf{e}}\cdot\matrixelement{v\mathbf{k}}{\mathbf{r}}{c\mathbf{k}}}^{2}\variation(\omega-\Omega_{S})\,,$
(3)

where $V$ is the volume, $N_{k}$ is the number of points in the Brillouin zone
sampling, $\hat{\mathbf{e}}$ is the light polarization vector, and
$\matrixelement{v\mathbf{k}}{\mathbf{r}}{c\mathbf{k}}$ are the single-particle
dipole matrix elements.

##  3 Results and Discussion

Figure 1: Top and side views of the crystal structure of bulk transition metal
dichalcogenides MX2 in trigonal prismatic (2H) phase, where M (blue) is the
transition metal Mo or W, and X (red) is the chalcogen atom S or Se. The first
Brillouin zone is also represented with high-symmetry $\mathbf{k}$-points and
paths. Table 1: Fully-relaxed lattice parameters for bulk 2H-TMDCs considering
PBE-D2 van der Waals corrections and fully including spin-orbit coupling. The
respective experimental values from Refs. 43, 44, 45 are shown for comparison.

|  MoS2 |  MoSe2 |  WS2 |  WSe2
---|---|---|---|---
PBE-D2-SOC |  |  |  |
$a$ (Å) | 3.19 | 3.32 | 3.19 | 3.34
$c$ (Å) | 12.42 | 13.03 | 12.16 | 12.81
Experiment |  |  |  |
$a$ (Å) | 3.16a | 3.29b | 3.15c | 3.28c
$c$ (Å) | 12.29a | 12.93b | 12.32c | 12.96c

a Ref. 43; b Ref. 45; c Ref. 44

The computed fully-relaxed crystal structure is shown in Fig. 1, and the
lattice parameters of the studied bulk systems are summarized in Table 1,
which are in good agreement with experimental results 43, 44, 45.

Subsequently, we compute the quasiparticle band structures and band gaps of
the bulk TMDCs using the $G_{0}W_{0}$ approximation including spin-orbit
coupling. The results are presented in Fig. 2 and Table 2. Our results, which
include both direct and indirect quasiparticle band gap values, align well
with the previously reported $G_{0}W_{0}$ findings 13. Moreover, our
calculations are in agreement with the band gaps and spin-splitting observed
for the valence band, denoted as $\Delta_{V}$, as measured through ARPES
according to Kim et al. 46. It can be seen that the band gaps of the TMDCs are
similar for those systems with the same chalcogen species.

Figure 2: Quasiparticle band structure of (a) MoS2, (b) MoSe2, (c) WS2 and (d)
WSe2 within $G_{0}W_{0}$ approximation including spin-orbit coupling. Table 2:
Quasiparticle fundamental (indirect) and direct band gap (in eV) at $K$ point
in the Brillouin zone, calculated within $G_{0}W_{0}$ approximation; spin-
splitting for the valence band $\Delta_{V}$ (in meV); ground-state and A
excitonic peak energy positions (X${}_{\text{gs}}$, XA), and Boltzmann factor
$\Delta_{\text{X}}$ for the fraction of the radiative e-h recombination in the
SLME-X approach for the bulk 2H-TMDCs (see text for further details). For
comparison, we present the band gaps and spin-splitting $\Delta_{V}$ obtained
by ARPES 46.

| $E_{\text{g}}^{i}$ | $E^{d}_{\text{g}}$ | $\Delta_{V}$ | X${}_{\text{gs}}$ | XA | $\Delta_{\text{X}}$
---|---|---|---|---|---|---
MoS2 | 1.34 | 1.99 | 0.20 | 1.846 | 1.877 | 0.031
1.14a | 1.82a | 0.16a |  |  |
MoSe2 | 1.16 | 1.69 | 0.24 | 1.583 | 1.607 | 0.024
1.25a | 1.57a | 0.20a |  |  |
WS2 | 1.30 | 2.02 | 0.47 | 1.915 | 1.956 | 0.041
1.25a | 1.82a | 0.44a |  |  |
WSe2 | 1.14 | 1.59 | 0.49 | 1.511 | 1.561 | 0.050
1.12a | 1.62a | 0.48a |  |  |

a Ref. 46

Turning to the optical properties, the computed imaginary part of the
dielectric function at both the independent particle (IP) level and including
e-h interaction are shown in Fig. 3. For comparison, we also show the
experimental optical absorption results (Refs. 47 and 48) for the respective
TMDCs (shadow areas).

We notice a good agreement between the BSE optical spectra and the
experimental ones, highlighting the excitons’ role in the optical absorption
of bulk TMDCs. Overall, our findings effectively capture the essential
characteristics of the experimental spectra, albeit with slightly diminished
accuracy for W-based TMDCs primarily attributable to the increased splitting
of electronic bands induced by higher spin-orbit coupling. The lack of
prominent peaks in the IP spectrum strongly suggests that the sharp redshifted
X peaks observed in BSE are associated with excitonic features. The absorption
spectra calculated within IP exhibit noticeably weaker and less defined
shoulder-like characteristics, indicating primarily band transitions. The two
lowest energy peaks correspond to the excitonic states labeled as A and B,
which are associated with interband transitions at the $K$ point in the
Brillouin zone, coming from the splitting of the valence band due to spin-
orbit coupling 49, 50.

Figure 3:  Optical spectra calculated at the BSE (blue solid line) and
independent particle (IP, red dot-dashed line) levels for the bulk (a) MoS2,
(b) MoSe2, (a) WS2, and (a) WSe2. The Kohn-Sham eigenvalues include the
quasiparticle corrections of the $G_{0}W_{0}$ calculations. The experimental
data (gray shadow areas) are taken from Refs. 47, 48.

Our results indicate that the ground-state exciton (X${}_{\text{gs}}$) for all
the studied systems corresponds to a dark state, that rises from transitions
between the valence band maximum and conduction band minimum at the $K$ point.
The corresponding excitonic peak energy position of X${}_{\text{gs}}$ and XA
for the TMDCs are summarized in Table 2.

Now we investigate the photovoltaic efficiency of the bulk TMDCs. Yu and
Zunger 51 developed the so-called spectroscopic limited maximum efficiency
(SLME) model to compute the maximum efficiency of the absorber in a single-
junction solar cell. In this formalism, the magnitude of the fundamental band
gap, the values of the direct and indirect electronic gaps, the standard solar
spectrum, the absorption coefficient of the material, and the thickness are
taken into account to estimate the PCE of the device

$\eta_{\text{SLME}}=\frac{P_{\text{m}}}{P_{\text{in}}}\,,$ (4)

where $P_{\text{in}}$ is the total incident power density with a value of
$1000.37$ W/m2, and $P_{\text{m}}$ is the maximum output power density which
can be obtained by maximizing the product of current density, $J$, and
voltage, $V$,

$P=JV=\left[J_{\text{sc}}-J_{0}(e^{eV/k_{\text{B}}T}-1)\right]V\,,$ (5)

where $k_{\text{B}}$ is the Boltzmann constant, and $T$ is the device
temperature taken as $T=298.15$ K throughout this work. Both the short circuit
current density $J_{sc}$ and the reverse saturation current density $J_{0}$
are calculated from the photon absorptivity $a(\omega)$ of the material as
follows

$a(\omega)=1-e^{-2\alpha(\omega)L}\,,$ (6)

where $\alpha(\omega)$ is the absorption coefficient computed from first
principles, and $L$ is the thickness of the photovoltaic absorber material
taken as 500 nm unless otherwise stated. In this regard, those current
densities can be described including the AM1.5G solar spectrum
$I_{\text{sun}}(\omega)$ and the black-body spectrum
$I_{\text{bb}}(\omega,T)$, respectively

$\displaystyle J_{\text{sc}}$ $\displaystyle=$ $\displaystyle
e\int_{E_{\text{g}}}^{\infty}a(\omega)I_{\text{sun}}(\omega)\,\differential\omega\,,$
(7) $\displaystyle J_{\text{0}}$ $\displaystyle=$
$\displaystyle\frac{J_{0}^{\text{r}}}{f_{\text{r}}}=\frac{e\pi}{f_{\text{r}}}\int_{E_{\text{g}}}^{\infty}a(\omega)I_{\text{bb}}(\omega,T)\,\differential\omega\,.$
(8)

$J_{0}$ corresponds to the total e-h recombination current at equilibrium in
the dark, regarding both radiative and non-radiative recombinations. Here,
$f_{\text{r}}$ is proposed as the fraction of the radiative recombination
current, which in the original SLME framework is modeled using a Boltzmann
factor

$f_{\text{r}}=e^{-\Delta_{r}/k_{\text{B}}T}\,,$ (9)

where $\Delta_{r}\equiv E_{\text{g}}^{\text{da}}-E_{\text{g}}$ is the
difference between the allowed direct band gap and the fundamental band gap,
respectively.

Note that the efficiency within the Shockley–Queisser (SQ) detailed balance
limit 52 ($\eta_{\text{SQ}}$) can be obtained from the equations above by
setting $f_{r}=1$ and considering the absorbance in Eq. (7) and (8) as a
Heaviside step function that vanishes for $E<E_{\text{g}}$ and is equal to $1$
for $E\geq E_{\text{g}}$, where $E_{\text{g}}$ is the optical band gap of the
photovoltaic absorber.

Figure 4: Photovoltaic properties of bulk 2H-TMDCs calculated using the
spectroscopic limited maximum efficiency (SLME) framework (a-c), modeling the
fraction of radiative recombination losses ($f_{r}$) by considering the
exciton eigenvalues from BSE (SLME-X) (d-f), and by extracting the fraction of
the radiative losses directly from photoluminescence quantum yield experiments
53 (SLME-PLQY) (g-i): $J$-$V$, $P$-$V$ curves, and power conversion efficiency
($\eta$) as a function of the thickness of the absorber. The results have been
computed for a device temperature of $T=298.15$ K. Table 3: Photovoltaic
properties of bulk TMDCs obtained within the Shockley-Queisser approach and
the spectroscopic limited maximum efficiency (SLME) standard framework, by
modeling the fraction of radiative recombination losses ($f_{r}$) by
considering the exciton eigenvalues from BSE (SLME-X), and by extracting the
fraction of the radiative losses directly from photoluminescence quantum yield
experiments 53 (SLME-PLQY): short-circuit current densities ($J_{\text{sc}}$),
open-circuit voltage ($V_{\text{oc}}$), fill factor (FF), and power-conversion
efficiency ($\eta$).

| Shockley-Queisser |  | SLME
---|---|---|---
|  $J_{\text{sc}}$ $\left(\right.$mA/cm2$\left.\right)$ |  $V_{\text{oc}}$ (V) | FF (%) |  $\eta_{\text{SQ}}$ (%) |  |  $J_{\text{sc}}$ $\left(\right.$mA/cm2$\left.\right)$ |  $V_{\text{oc}}$ (V) | FF (%) |  $\eta_{\text{SLME}}$ (%)
MoS2 | 17.5 | 1.55 | 92.2 | 25.0 |  | 17.5 | 1.02 | 88.6 | 15.8
MoSe2 | 25.2 | 1.30 | 90.7 | 29.7 |  | 25.2 | 0.86 | 86.5 | 18.7
WS2 | 15.6 | 1.63 | 91.8 | 23.3 |  | 15.6 | 0.98 | 87.4 | 13.4
WSe2 | 26.9 | 1.26 | 90.1 | 30.5 |  | 26.9 | 0.84 | 86.4 | 19.5
| SLME-X |  | SLME-PLQY
|  $J_{\text{sc}}$ $\left(\right.$mA/cm2$\left.\right)$ |  $V_{\text{oc}}$ (V) | FF (%) |  $\eta_{\text{SLME-X}}$ (%) |  |  $J_{\text{sc}}$ $\left(\right.$mA/cm2$\left.\right)$ |  $V_{\text{oc}}$ (V) | FF (%) |  $\eta_{\text{SLME-PLQY}}$ (%)
MoS2 | 17.5 | 1.52 | 92.0 | 24.5 |  | 17.5 | 1.31 | 90.5 | 20.8
MoSe2 | 25.2 | 1.28 | 90.3 | 29.1 |  | 25.2 | 1.06 | 88.8 | 23.7
WS2 | 15.6 | 1.59 | 91.6 | 22.7 |  | 15.6 | 1.37 | 91.1 | 19.4
WSe2 | 26.9 | 1.21 | 89.8 | 29.2 |  | 26.9 | 1.01 | 88.1 | 23.8

It is important to notice that for indirect band gap materials with large
$\Delta_{r}$, as in the TMDCs studied here, the original SLME approach can
drastically underestimate the predicted PCE, given that the Boltzmann factor
rapidly decreases the fraction of radiative recombination current density
$f_{r}$. Consequently, the saturation current $J_{0}$ is overestimated,
yielding lower values for the open-circuit voltages. This effect was studied
in detail by Bercx _et al._ 54 for silicon, which possesses a
$\Delta_{r}=2.23$ eV. This leads to a PCE of zero regardless of the
temperature and thickness of the material.

First, we apply the standard SLME formalism to compute the PCEs considering
$\Delta_{r}$ as the difference between $X_{A}$ and $E_{\text{g}}^{i}$.
Employing this approach, for all the analyzed systems, the fraction of
radiative recombination current, $f_{r}$, is much smaller than 10-8, leading
to PCEs varying between 13% and 19% for the studied TMDCs, significantly
smaller than those obtained through Shockley-Queisser (Table 3). This result
suggests that, for these materials, employing the original SLME approach
implies that almost all e-h recombination would be non-radiative, even though
$J_{0}$ is modeled assuming a black-body spectrum, which relies on a radiative
principle. This issue motivated us to explore alternative models for
describing the recombination factor.

Delving into this possibility, and taking into account that the excitonic
effects in bulk TMDCs play a key role in their optical responses, we also
compute the PCE using a slightly modified SLME approach that has been proposed
by Ozório and co-workers 30. In their model, the fraction of the radiative
recombination current, $f_{\text{r}}$, is calculated by replacing the
$\Delta_{r}$ in Eq. (9) with $\Delta_{\text{X}}$, which is defined as

$\Delta_{\text{X}}=\text{X}_{A}-\text{X}_{\text{gs}}.$ (10)

We refer to this approach as exciton-based SLME or SLME-X. Within it, the
fraction of the radiative recombination current, $f_{r}$, calculated taking
into account the $\Delta_{\text{X}}$ values presented in Table 2. As a result,
the obtained $f_{r}$ values vary between 1.5$\times$10-2 and 3.9$\times$10-2.
It is important to point out that the presence of an indirect gap could lead
to lower energy dark excitons that lead to non-radiative decay. Thus, the most
effective method for obtaining the exciton ground state involves computing the
$q$-dependent BSE spectra 55. In that case, we expect that $f_{r}$ would have
even lower values, although still surpassing those obtained within the
standard SLME.

In Fig. 4a-b, we show the current and power density curves as a function of
the voltage, obtained with the SLME-X approach. The PCE as a function of
thickness is depicted in Fig. 4c. All the main solar cell parameters obtained
from these curves are summarized in Table 3.

We can verify that all the investigated photovoltaic properties are similar
for those TMDCs that are composed of the same chalcogen species. The
transition metal diselenides should have higher photovoltaic efficiencies than
disulfides due to their lower optical band gaps. In addition, our results
presented in Fig. 4(c) indicate that $\eta_{\text{SLME-X}}$ saturates for a
thickness around $10^{2}$ nm.

Alternatively, non-radiative recombination losses can be experimentally
quantified by the photoluminescence quantum yield (PLQY) 56. We can also
estimate the maximum PCE by assuming that the PLQY is, in principle, equal to
the fraction of radiative recombination ($f_{r}$) 57. We denote this other
modified SLME approach as SLME-PLQY. Experimental studies have reported that
for single-layer TMDCs, the PLQY is close to unity 58, 59, 53. In contrast, it
has been demonstrated that the PLQY for their bulk counterpart is reduced by a
factor of 104 60.

We adopt the PLQY values reported by Mohamed and co-workers 53 as the
experimental-based $f_{r}$ for the TMDCs. These $f_{r}$ values are
6.8$\times$10-5, 7.9$\times$10-5, 4.9$\times$10-5, and 4.5$\times$10-5 for the
bulk MoS2, MoSe2, WS2 and WSe2, respectively.

Considering the SLME-PLQY approach, we also calculate the current density-
voltage, power density-voltage, and PCE-thickness curves of the studied TMDCs,
which are shown in Fig. 4d-f. For comparison with the SLME-X results, we
describe the solar cell parameters obtained within SLME-PLQY for the TMDCs in
Table 3. Given that the short-circuit current densities, $J_{sc}$, do not
depend on the fraction of the radiative recombination current, their values
are identical for both approaches. On the other hand, the open-circuit
voltages and fill factors are smaller in the SLME-PLQY approach in comparison
to the SLME-X one, leading to smaller values of PCE in the latter case.

All the obtained results of the photovoltaic efficiencies within the SQ and
SLME-based approaches are summarized in Fig. 5, where we present the maximum
PCE as a function of the optical band gap. For the analyzed TMDCs, the average
of the fraction of e-h radiative recombination is $f_{r}=3\times 10^{-8}$ for
the standard SLME, $f_{r}=6\times 10^{-5}$ for SLME-PLQY and $f_{r}=3\times
10^{-2}$ for SLME-X. In addition, we have estimated the error associated with
those average maximum PCEs, by introducing in the $f_{r}$ the error threshold
of the $GW$ calculations (see Fig. S5 in the Supporting Information). As a
result, regarding the SLME approach, for systems with optical gaps close to
1.9 eV (MoS2, WS2), the estimated error is ${\sim}$1%, whereas for those with
optical gaps around 1.6 eV (MoSe2, WSe2), the error can be approximately 2.4%.
Considering this uncertainty estimate, we expect comparable errors for the
remaining formalisms.

Based on these fractions of non-radiative e-h recombination, we compute the
efficiencies as a function of the optical energy gap. The shaded area in Fig.
5 represents the range of optical band gaps corresponding to the TMDCs
studied. We can verify that the SQ formalism, which considers only radiative
recombination processes ($f_{r}=1$), overestimates the photovoltaic efficiency
of the bulk TMDCs, with photovoltaic efficiencies ranging from ${\sim}23$% to
${\sim}29$% (Table S1 in the Supporting Information 42). On the other hand,
all the SLME-based approaches include a non-vanishing fraction of non-
radiative e-h recombination ($f_{r}<1$), decreasing the predicted photovoltaic
efficiencies of the absorbers.

Figure 5: Power conversion efficiency with respect to the optical energy gap
in Shockley-Queisser and SLME-based approaches. The plotted curves, derived
from the average computed fraction of e-h radiative recombination ($f_{r}$)
for the studied TMDCs, illustrate the impact of nonradiative recombination
losses in the PCE. Alongside the standard SLME, we incorporate considerations
from exciton eigenvalues obtained through BSE (SLME-X) and direct estimations
from photoluminescence quantum yield analyses experiments (SLME-PLQY). The
shaded area denotes the range of optical band gaps associated with the
analyzed TMDCs.

Our results suggest that even the highest quality TMDC-based thin film would
yield maximum efficiencies between ${\sim}23-29$%, if we take into account
only excitonic effects as a fundamental factor to describe the recombination
processes. Nevertheless, a more realistic upper limit of the photovoltaic
efficiency for the investigated TMDCs can be predicted by considering the
fraction of the radiative recombination directly from PLQY experiments, with
the TMDC thin-films reaching maximum PCEs between ${\sim}19-21$%. Based on it,
it is noticeable that PLQY-based efficiencies, reflecting the performance of
impurity-containing samples, are inferior to those derived from considering
only excitonic effects. This implies that there is still room for improving
the TMDC samples to attain greater PCEs.

The obtained efficiencies can be compared with previous theoretical studies.
For instance, Ozdemir and Baroni 61 investigated the photovoltaic efficiency
of bulk TMDCs based on ab initio many-body perturbation theory calculations,
including the e-h interaction. In their work, the absorptivity is calculated
as $A=1-T-R$, where $T$ is the transmitivity and $R$ is the reflectivity, and
those properties are computed from Fresnel’s equations. For TMDCs with a
thickness of $10^{3}$ nm, the same ones analyzed in our work, they reported
values of PCE in a range of $\sim$ 21% to $\sim$ 26%, being the selenides the
most efficient systems, consistent with our findings, especially those that
account for nonradiative recombination through PLQY experiments.

Furthermore, Chouldhary et al. 62 have reported theoretical results of SLME
based on DFT without taking e-h interactions into account. The
$\eta_{\text{SLME}}$ computed for a thickness of 500 nm were $18.6\%$,
$27.6\%$, $15.2\%$, and $21.5\%$ for bulk MoS2, MoSe2, WS2, and WSe2,
respectively. Although in that work the authors have used band gaps calculated
at the DFT level with meta-GGA functionals, their SLME efficiencies lie
between our predicted values modeled by the SLME-X and SLME-PLQY approaches.

Finally, despite the large range of efficiencies, we verify that the selenides
present higher efficiencies, with MoSe2 and WSe2 photovoltaic efficiencies
being almost the same. Our results highlight the importance of the fraction of
radiative e-h recombination factor ($f_{r}$) to determine the maximum PCEs of
bulk TMDCs, which indeed can be extrapolated to other indirect band gap
absorbers. In particular, we present a range of values for this relevant solar
cell parameter and their corresponding efficiencies, based on several
approaches that can represent reliable upper limits for the photovoltaic
efficiencies of single-junction solar devices produced with TMDC thin films.

##  4 Conclusions

We carried out first-principles calculations based on DFT combined with many-
body perturbation theory within the $G_{0}W_{0}$-BSE framework to study the
optical response and the photovoltaic properties of bulk 2H-MX2 (M$\,=\,$Mo,
W, and X$\,=\,$S, Se), using for the latter the spectroscopic limited maximum
efficiency (SLME) formalism. Our results indicated that an accurate
description of the optical spectrum within the $GW$-BSE framework requires
mapping out the Brillouin zone in very fine _k_ -grid samples. The agreement
obtained when comparing the BSE optical spectra and the experimental ones
highlights the excitons’ roles in the optical absorption of bulk TMDCs. The
calculated power conversion efficiencies (PCEs) of the analyzed TMDCs were
overestimated with the Shockley-Queisser formalism, as expected since in this
approach one includes only radiative e-h recombination losses. On the other
hand, the standard SLME approach tends to underestimate the PCE of these van
der Waals materials, due to a quite small estimate of the fraction of
radiative recombination. In other words, for the standard SLME, almost all the
recombination processes should correspond to nonradiative losses. These
findings have motivated us to explore alternative methods for modeling PCEs
that account for radiative and nonradiative recombination processes in more
realistic ways. Thus, we investigate the modeling of the fraction of e-h
recombination processes by considering the energy range from exciton
eigenvalues (SLME-X), and also by employing experimental results obtained for
photoluminescence quantum yield (SLME-PLQY). As a result, we predict that when
we adopt the exciton energy ranges to compute the PCE within the SLME-X
approach, the maximum PCEs vary between ${\sim}23-29$%, whereas taking the
experimental fraction of radiative e-h recombination from PLQY experiments
within the SLME-PLQY approach, the bulk TMDC maximum PCEs reached values
between ${\sim}19-24$%. Regardless of the approach used to determine the PCE,
our results indicate that the selenide-based TMDCs exhibit superior
photovoltaic efficiency in comparison to the sulfide-based ones. In
conclusion, we believe that our comprehensive analysis of the effects of the
fraction of e-h radiative recombination in the TMDC efficiencies can be used
as accurate predictions of upper limits for the photovoltaic efficiencies of
solar devices based on few-layer or multi-layer TMDCs, paving the way for the
development of efficient nanoscopically thin solar cells.

## Acknowledgements

E.M.Jr and A.R.R. acknowledge the financial support from the Brazilian agency
FAPESP, Grant No. 20/13172-8 and 2017/02317-2. This research was supported by
resources supplied by CENAPAD-SP and the Center for Scientific Computing
(NCC/GridUNESP) of the UNESP.

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