# Accurate and efficient prediction of the band gaps and optical spectra of
chalcopyrite semiconductors from a non-empirical range-separated dielectric-
dependent hybrid: Comparison with many-body perturbation theory

Arghya Ghosh  Department of Physics, Indian Institute of Technology,
Hyderabad, India  Subrata Jana  Corresponding author<EMAIL_ADDRESS>Department of Molecular Chemistry and Materials Science Weizmann Institute of
Science, Rehovoth 76100, Israel  Dimple Rani  School of Physical Sciences,
National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Jatni, India  Manoar Hossain  Institut für Physik and IRIS
Adlershof, Humboldt-Universität zu Berlin, Zum Gro$\beta$en Windkanal 2, 12489
Berlin, Germany  Manish K Niranjan  Department of Physics, Indian Institute of
Technology, Hyderabad, India  Prasanjit Samal  School of Physical Sciences,
National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Jatni, India

###### Abstract

The accurate prediction of electronic and optical properties in chalcopyrite
semiconductors has been a persistent challenge for density functional theory
(DFT) based approaches. Addressing this issue, we demonstrate that very
accurate results can be obtained using a non-empirical screened dielectric-
dependent hybrid (DDH) functional. This novel approach showcases its
impressive capability to accurately determine band gaps, optical bowing
parameters, and optical absorption spectra for chalcopyrite systems. What sets
the screened DDH functional apart is its adeptness in capturing the many-body
physics associated with highly localized $d$ electrons. Notably, the accuracy
is comparable to the many-body perturbation based methods (such as
$G_{0}W_{0}$ or its various approximations for band gaps and Bethe-Salpeter
equation (BSE) on the top of the $G_{0}W_{0}$ or its various approximations
for optical spectra) with less computational cost, ensuring a more accessible
application across various research domains. The present results show the
predictive power of the screened DDH functional, pointing toward promising
applications where computational efficiency and predictive accuracy are
crucial considerations. Overall, the screened DDH functional offers a
compelling balance between cost-effectiveness and precision, making it a
valuable tool for future endeavors in exploring chalcopyrite semiconductors
and beyond.

##  I Introduction

Over the last two decades, ternary chalcopyrite semiconductors have attracted
a great deal of attention due to their applications in renewable and
sustainable technologies Green et al. (2019); Jackson et al. (2011); Walsh et
al. (2012); Feng et al. (2011); Rife et al. (1977); Alonso et al. (2001). In
particular, these materials are primarily used as absorbers in thin film solar
cells due to their great off-stoichiometric steadiness Feurer et al. (2019),
tunable electrical and thermal conductivity Plata et al. (2022), and
remarkable optoelectronic charecteristics Rife et al. (1977). The presence of
heavy metals (Cu, Ag, Be, etc.), coupled with their tunable electronic
structure, adds an intriguing dimension from a scientific perspective. The ab-
initio calculations for these ternary chalcopyrite semiconductors,
characterized by a complex electronic structure, are essential and have been
of prime importance. However, the ab-initio calculations have long predicted
that the band gap will be strongly influenced by the anion displacement ($u$)
from its primary mean position Vidal et al. (2010); Siebentritt et al. (2010).

Experimentally, acquiring the band gaps and optical spectra of Cu and Ag
chalcopyrite systems typically involves techniques like photoemission or
photoluminescence spectra Alonso et al. (2001); Shay et al. (1972); Tell et
al. (1971); Bellabarba et al. (1996). However, from a theoretical standpoint,
achieving an accurate treatment of excited states in these systems requires a
comprehensive approach. This often involves either a fully relativistic
treatment of many-body perturbation theory (MBPT) or a fully self-consistent
Green function-based method ($GW$) Hedin (1965); Aryasetiawan and Gunnarsson
(1998) accompanied by the solution of the Bethe-Salpeter equation (BSE)
Salpeter and Bethe (1951). While approaches like fully self-consistent $GW$
and BSE@$GW$ encompass crucial many-body effects, such as electron-
electron($e-e$) and electron-hole($e-h$) interactions Aryasetiawan and
Gunnarsson (1998); van Schilfgaarde et al. (2006); Kotani et al. (2007); Onida
et al. (2002), they are known to be computationally demanding and technically
challenging, posing significant hurdles to their widespread application.

Importantly, there are several limitations when higher-level methods are
applied to study the electronic structure of chalcopyrite systems. Firstly,
the strong $p-d$ hybridization near the top of the valence band closes the
Kohn-Sham (KS) bandgap, causes a divergence in the dipole transition matrix
element, which poses a significant challenge in constructing a reliable
response function in $GW$ or BSE@$GW$ calculations. Consequently, these higher
level-methods strongly depend on the chosen DFT wavefunctions and orbital
energies, and an inaccurate estimation of dielectric function may lead to a
wrong exciton effects Zhang et al. (2013); Aguilera et al. (2011). Secondly,
the band gaps of chalcopyrite systems depend crucially on the parameters
related to the crystal structure, in particular, on anion displacement ($u$)
Aguilera et al. (2011); Vidal et al. (2010). Thus, the incorrect estimation of
the structural parameters can affect the excitonic wave function.
Consequently, due to these factors, unphysical absorption peaks may appear in
the ab-initio optical spectrum.

Nevertheless, there is an emerging alternative method for tackling inherently
complex issues related to excited state electronic properties. This novel
method is based on the solution of Kohn-Sham (KS) equation or a generalized KS
scheme using the dielectric dependent hybrid functional (DDH) Brawand et al.
(2016); Chen et al. (2018); Brawand et al. (2017); Zheng et al. (2019); Jana
et al. (2023). In this paper, we employ dielectric dependent hybrid
functionals to investigate the electronic structure and optical properties of
chalcopyrite semiconductors. The DDH functional scheme is proposed as a viable
alternative to the highly demanding and complex $GW$ and BSE@$GW$ schemes,
offering a more accessible approach for exploring various systems Gerosa et
al. (2015a, b); Miceli et al. (2018); Zheng et al. (2019); Gerosa et al.
(2017); Chen et al. (2018); Hinuma et al. (2017); Brawand et al. (2017); Liu
et al. (2019); Ohad et al. (2022); Wing et al. (2019a); Ramasubramaniam et al.
(2019a); Wing et al. (2019b); Kronik and Kümmel (2018); Ramasubramaniam et al.
(2019b); Jana et al. (2023); Camarasa-Gómez et al. (2023). Notably, to the
best of our knowledge, no study has been reported on the performance of the
DDH functional in the case of ternary chalcopyrite semiconductors.

The findings presented in our work strongly suggest the efficacy of DDH
functional as a state-of-the-art scheme for accurately characterizing both
ground state and excited state electronic properties in highly localized $d$
electronic systems, particularly within the chalcopyrite framework where
describing the robust hybridization between $p$ and $d$ orbitals is crucial.
While modern meta-generalized gradient approximations (meta-GGAs) outperform
generalized gradient approximations (GGAs) in terms of structural and band gap
properties of semiconductors Jana et al. (2018a); Patra et al. (2019); Jana et
al. (2021); Patra et al. (2021a); Tran et al. (2021); Ghosh et al. (2022a);
Lebeda et al. (2023), they are plagued by well-known issues like many-electron
self-interaction and delocalization errors Perdew and Zunger (1981); Cohen et
al. (2012); Mori-Sánchez et al. (2008); Jana et al. (2018b), sometimes
inadequate in capturing the degree of electron correlation Ghosh et al.
(2021); Patra et al. (2021b); Ghosh et al. (2022b).

This paper evaluates the accuracy of the non-empirical screened DDH functional
in predicting band gaps and optical absorption spectra of chalcopyrite
semiconductors. Our results indicate that screened-DDH, addressing the
generalized Kohn-Sham (KS) scheme, agrees reasonably well with experimental
findings for both band gaps and optical spectra. Notably, the functional is
particularly effective for challenging Cu-based chalcopyrites. Our
calculations demonstrate the applicability of this method in describing the
$p-d$ hybridization where other high-level methods are either insufficient or
computationally demanding.

##  II Background of Methodologies

The Coulomb attenuated method (CAM) of the two-electron operator is used to
construct the screened range-separated hybrid (SRSH) as Kronik and Kümmel
(2018),

$\frac{1}{r_{ij}}=\frac{\alpha+\beta~{}erf(\mu
r_{ij})}{r_{ij}}+\frac{1-[\alpha+\beta~{}erf(\mu r_{ij})]}{r_{ij}}~{},$ (1)

where $r_{ij}=|{\bf{r}}_{i}-{\bf{r}}_{j}|$ is the relative position of two
electrons. Here, $0\leq\alpha\leq 1$ and $0\leq\beta\leq 1$ controls the
amounts of Fock (non-local) and semilocal ($sl$) GGA (PBE in this case) Perdew
et al. (1996) that are mixed to the full exchange-correlation (XC) functional
in its long-range ($lr$) or short-range ($sr$). $\mu$ is the screening
parameter determined later in this paper. Using Eq. 1, the resultant
generalized form of the SRSH XC expression becomes,

$\begin{split}E_{xc}^{SRSH}(\alpha,\beta;\mu)=(1-\alpha)E_{x}^{sr-
sl,\mu}+\alpha E_{x}^{Fock-sr,\mu}\\\ +[1-(\alpha+\beta)]E_{x}^{sl-
lr,\mu}+(\alpha+\beta)E_{x}^{Fock-lr,\mu}+E_{c}^{sl}\end{split}$ (2)

and the corresponding potential,

$\displaystyle V_{xc}^{SRSH}(\alpha,\beta;\mu)$ $\displaystyle=$
$\displaystyle[\alpha+\beta Erf(\mu r)]V_{x}^{Fock}+\beta V_{x}^{sl-sr,\mu}$
(3) $\displaystyle+$
$\displaystyle[1-(\alpha+\beta)]V_{x}^{sl}+V_{c}^{sl}~{}.$

Here, $E_{c}$ and $V_{c}$ are the PBE correction energy and potential
functionals, respectively. Eq. 3 can be seen as the generalized form of the
“CAM” type hybrid used extensively for finite and extended systems Gerosa et
al. (2015a, b); Miceli et al. (2018); Zheng et al. (2019); Gerosa et al.
(2017); Chen et al. (2018); Hinuma et al. (2017); Brawand et al. (2017); Liu
et al. (2019); Ohad et al. (2022); Wing et al. (2019a); Ramasubramaniam et al.
(2019a); Wing et al. (2019b); Kronik and Kümmel (2018); Ramasubramaniam et al.
(2019b); Jana et al. (2023); Camarasa-Gómez et al. (2023). However, the naming
of the functionals becomes different based on how one determines the
parameters. Typically, one can consider $\alpha+\beta=\gamma$, where $\gamma$
is an another parameter. In terms of $\alpha$ and $\gamma$ Eq. 3 becomes,

$\displaystyle V_{xc}^{SRSH}(\alpha,\gamma;\mu)$ $\displaystyle=$
$\displaystyle[\alpha-(\alpha-\gamma)Erf(\mu r)]V_{x}^{Fock}$ (4)
$\displaystyle-$ $\displaystyle(\alpha-\gamma)V_{x}^{sl-
sr,\mu}+(1-\gamma)V_{x}^{sl}+V_{c}^{sl}~{}.$

In particular, the following choices are important for bulk solids: (i) for
$\alpha=0.25$, $\gamma=0$, and $\mu=0.11$ Bohr-1 in Eq. 4, resulting in the
recovery of the HSE06 like functionals Heyd et al. (2003); Krukau et al.
(2006); Heyd and Scuseria (2004); Jana et al. (2020a, b, 2018c); Jana and
Samal (2018, 2019),

$\displaystyle V_{xc}^{HSE06}(0.25,0;0.11)$ $\displaystyle=$ $\displaystyle
0.25[1-Erf(0.11r)]V_{x}^{Fock}$ $\displaystyle-$ $\displaystyle 0.25V_{x}^{sl-
sr,0.11}+V_{x}^{sl}+V_{c}^{sl}~{},$

and (ii) for $\alpha=1$ and $\gamma=\epsilon_{\infty}^{-1}$, where
$\epsilon_{\infty}$ is the high-frequency macroscopic static dielectric
constant or ion-clamped static (optical) dielectric constant or electronic
dielectric constant, the resultant functional becomes,

$\displaystyle V_{xc}^{DDH}(1,\epsilon_{\infty}^{-1};\mu)$ $\displaystyle=$
$\displaystyle[1-(1-\epsilon_{\infty}^{-1})Erf(\mu r)]V_{x}^{Fock}$
$\displaystyle-$ $\displaystyle(1-\epsilon_{\infty}^{-1})V_{x}^{sl-
sr,\mu}+(1-\epsilon_{\infty}^{-1})V_{x}^{sl}+V_{c}^{sl}~{},$

which is named dielectric-dependent range-separated hybrid functional based on
the CAM (DD-RSH-CAM) Chen et al. (2018) or simply DDH (used throughout this
paper).

In particular, the model dielectric function for bulk solid is defined
according to Eq. 4 with $\gamma=\epsilon_{\infty}^{-1}$ as,

$\varepsilon^{-1}(|{\bf{G}}|)=\alpha-(\alpha-\epsilon^{-1}_{\infty})e^{-|{\bf{G}}|{{}^{2}}/(4\mu)}~{},$
(7)

which is the key to the DDH functional (where ${\bf{G}}$ is the reciprocal
lattice vector). The model dielectric function makes this construction quite
similar to that of the self-energy correction of $GW$, in particular when GGA
approximates $\sum_{COH}$ (Coulomb hole (COH)) and $\sum_{SEX}$ (screened
exchange (SEX)) by Fock term Cui et al. (2018).

It is readily apparent from Eq. LABEL:hy-eq6 that the macroscopic static
dielectric constant, $\epsilon_{\infty}$, is the key to DDH calculations. It
can be obtained using different procedures discussed in section III.2. Several
procedures are also available to determine the screening parameter $\mu$. In
particular, $\mu$ can be obtained (i) depending on the valance electron
density that participate to the screening Brawand et al. (2016); Skone et al.
(2016); Cui et al. (2018), (ii) from fitting with the accurate dielectric
function Chen et al. (2018), (iii) from empirical fitting Yang et al. (2023)
or (iv) from first principle way using linear-response TDDFT (LR-TDDFT)
approaches based on local density or local Seitz radius ($r_{s}$) Jana et al.
(2023). In particular, in this paper, we use procedure (iv) to determine
$\mu$, which is named as $\mu_{eff}^{fit}$ and obtained using the
compressibility sum rule together with LR-TDDFT Jana et al. (2023) having the
form,

$\mu=\mu_{eff}^{fit}=\frac{a_{1}}{\langle r_{s}\rangle}+\frac{a_{2}\langle
r_{s}\rangle}{1+a_{3}\langle r_{s}\rangle^{2}}~{},$ (8)

with $a_{1}=1.91718$, $a_{2}=-0.02817$, $a_{3}=0.14954$, and

$\langle
r_{s}\rangle=\frac{1}{V_{cell}}\int_{cell}\Big{(}\frac{3}{4\pi(n_{\uparrow}({\bf{r^{\prime}}})+n_{\downarrow}({\bf{r^{\prime}}}))}\Big{)}^{1/3}~{}d^{3}r^{\prime}~{}.$
(9)

The readers are referred to ref. Jana et al. (2023) for the details of this
formula and underlying derivations. It may be noted that the resultant
$\mu_{eff}^{fit}$ performs quite similarly to those obtained from the fitting
of the dielectric function Chen et al. (2018) as reported in ref. Jana et al.
(2023). On the other hand, the screening parameters determined from procedure
(i) are not always well-defined in some materials, especially where electrons
of different characters participate in the valence bands Lorin et al. (2021).
In this respect, determining screening parameters from method (iv) is quite
reasonable and well justified Jana et al. (2023).

In the subsequent discussion, we briefly overview the different levels within
the $GW$, which we have used whenever applicable. Various approximations exist
for $GW$, including the single-shot $G_{0}W_{0}$ calculation, which heavily
relies on the choice of the initial KS wavefunctions and orbital energies Gant
et al. (2022). On the other hand, $GW_{0}$ involves a self-consistent update
of the orbital energies in the Green’s function $G$ after the initial
$G_{0}W_{0}$ step Hybertsen and Louie (1986); Shishkin et al. (2007). One may
note the crucial distinctions between $GW$ and DDH. The $GW$ steps involve the
calculations of the frequency dependent dielectric function, including the
summations over both occupied and unoccupied states. This makes the self-
consistent of $GW$ computationally more expensive. In contrast, the DDH
calculations are performed only within the generalized KS (gKS) scheme.
However, it requires an additional calculation of dielectric constants
(details are provided in sec. III.3). Second, the outcomes of one-shot
$G_{0}W_{0}$ and partially self-consistent $GW_{0}$ are highly dependent on
the initial choice of the KS functional, while DDH, being a self-consistent
approach within gKS, yields outcomes independent of the initial state.

##  III Results and Discussions

###  III.1 Materials and calculation details

####  III.1.1 Materials

Table 1:  PBE optimized $a$ (in Å), $c$, $\eta$, $r_{A-X}$ (in Å), $r_{B-X}$ (in Å), and $u$ of all the chalcopyrite semiconductors used in this work. Solids | $a$ | c | $\eta$ | $r_{A-X}$ | $r_{B-X}$ | $u$
---|---|---|---|---|---|---
I-III-VI2
AgAlS2 | 5.740 | 5.251 | 0.915 | 2.574 | 2.277 | 0.294
AgAlSe2 | 6.029 | 5.562 | 0.922 | 2.677 | 2.425 | 0.285
AgAlTe2 | 6.409 | 6.122 | 0.955 | 2.812 | 2.661 | 0.270
AgGaS2 | 5.773 | 5.307 | 0.919 | 2.563 | 2.317 | 0.286
AgGaSe2 | 6.049 | 5.632 | 0.931 | 2.665 | 2.463 | 0.278
AgGaTe2 | 6.406 | 6.169 | 0.963 | 2.801 | 2.681 | 0.266
AgInS2 | 5.925 | 5.749 | 0.970 | 2.573 | 2.509 | 0.259
AgInSe2 | 6.195 | 6.039 | 0.975 | 2.673 | 2.647 | 0.254
AgInTe2 | 6.567 | 6.500 | 0.990 | 2.812 | 2.856 | 0.244
CuAlS2 | 5.336 | 5.274 | 0.988 | 2.324 | 2.28 | 0.257
CuAlSe2 | 5.651 | 5.576 | 0.987 | 2.443 | 2.430 | 0.252
CuAlTe2 | 6.094 | 6.055 | 0.994 | 2.603 | 2.664 | 0.241
CuGaS2 | 5.372 | 5.315 | 0.989 | 2.314 | 2.322 | 0.249
CuGaSe2 | 5.677 | 5.631 | 0.992 | 2.432 | 2.471 | 0.244
CuGaTe2 | 6.096 | 5.086 | 0.834 | 2.593 | 2.685 | 0.237
CuInS2 | 5.578 | 5.617 | 1.007 | 2.33 | 2.519 | 0.220
CuInSe2 | 5.871 | 5.908 | 1.006 | 2.447 | 2.657 | 0.219
CuInTe2 | 6.294 | 6.317 | 1.003 | 2.608 | 2.861 | 0.215
II-IV-V2
BeGeAs2 | 5.446 | 5.48 | 1.006 | 2.267 | 2.468 | 0.218
BeGeP2 | 5.207 | 5.229 | 1.004 | 2.177 | 2.347 | 0.222
BeSiAs2 | 5.372 | 5.368 | 0.999 | 2.266 | 2.388 | 0.230
BeSiP2 | 5.129 | 5.117 | 0.998 | 2.166 | 2.266 | 0.233
BeSnAs2 | 5.655 | 5.697 | 1.007 | 2.287 | 2.650 | 0.194
BeSnP2 | 5.428 | 6.675 | 1.230 | 2.198 | 2.538 | 0.195
CdGeAs2 | 6.052 | 5.746 | 0.949 | 2.676 | 2.485 | 0.277
CdGeP2 | 5.805 | 5.484 | 0.945 | 2.583 | 2.363 | 0.282
CdSiAs2 | 5.979 | 5.551 | 0.928 | 2.676 | 2.396 | 0.289
CdSiP2 | 5.728 | 5.289 | 0.923 | 2.584 | 2.271 | 0.296
CdSnAs2 | 6.218 | 6.103 | 0.981 | 2.688 | 2.664 | 0.253
CdSnP2 | 5.983 | 5.862 | 0.980 | 2.597 | 2.550 | 0.257
MgGeAs2 | 6.020 | 5.638 | 0.936 | 2.637 | 2.473 | 0.273
MgGeP2 | 5.796 | 5.365 | 0.926 | 2.551 | 2.355 | 0.279
MgSiAs2 | 5.968 | 5.410 | 0.906 | 2.638 | 2.385 | 0.286
MgSiP2 | 5.746 | 5.125 | 0.892 | 2.555 | 2.265 | 0.292
MgSnAs2 | 6.152 | 6.042 | 0.982 | 2.642 | 2.654 | 0.248
MgSnP2 | 5.935 | 5.799 | 0.977 | 2.558 | 2.543 | 0.252
ZnGeP2 | 5.500 | 5.420 | 0.985 | 2.383 | 2.357 | 0.254
ZnSiAs2 | 5.676 | 5.530 | 0.974 | 2.480 | 2.395 | 0.263
ZnSiP2 | 5.420 | 5.268 | 0.972 | 2.383 | 2.270 | 0.268
ZnSnAs2 | 5.944 | 5.961 | 1.003 | 2.498 | 2.659 | 0.226
ZnSiP2 | 5.703 | 5.718 | 1.003 | 2.402 | 2.545 | 0.228

We use $42$ chalcopyrite semiconductors having ABX2 structures (with space
group $I{\bar{42}}d$) grouped as I-III-VI2 (18 chalcopyrites) and I-III-VI2
(24 chalcopyrites), which are the iso-electronic analogous of the II$-$VI and
III$-$V ideal zinc blende structure (distorted), respectively Shaposhnikov et
al. (2012). Here, $(A,B)$ are the two cations tetrahedrally coordinated by
four anions ($X$), where each anion is again coordinated by two cations each
of which are $A$ and $B$ types Jaffe and Zunger (1983). Three structural
parameters, namely, ($i$) lattice constant $a$, ($ii$) tetragonal ratio
$\eta=c/2a$ where $c$ is the lattice constant along the $z$-direction, and
($iii$) the anion displacement parameter, $u$ are used to describe a
chalcopyrite structure. Note that $u$ is an important structural parameter
used to describe physics related to the interplay between structure and
electronic properties and is defined as Vidal et al. (2010),

$u=0.25+(r^{2}_{A-X}-r^{2}_{B-X})/a^{2}~{},$ (10)

where $r_{A-X}$ and $r_{B-X}$ are $A-X$ and $B-X$ bond lengths, respectively.

DDH, HSE06, and $GW_{0}$ band gap calculations are performed using the PBE
optimized geometries. The details of the PBE optimized geometries are given in
Table 1. In general, the computed lattice constants from PBE XC functionals
agree well with the experimental.

####  III.1.2 Calculation details

The density functional calculations are performed using the plane-wave
formalism as implemented in Vienna Ab initio Simulation Package (VASP) code
Kresse and Hafner (1993); Kresse and Furthmüller (1996); Kresse and Joubert
(1999); Kresse and Furthmüller (1996). A kinetic energy cutoff of $520$ eV is
used for all DFT calculations. We use Monkhorst-Pack (MP) like
$\Gamma-$centered $11\times 11\times 11$ k-points mesh to sample the Brillouin
Zone (BZ) for PBE calculations, whereas for DDH and HSE06, the k-points are
reduced to $8\times 8\times 8$. The electronic energies are allowed to
converge at $10^{-6}$ eV for all DFT methods to achieve self-consistency. The
relaxation of the structures is performed till the Hellmann-Feynman forces on
atoms are reduced to less than 0.01 eV/Å-1. The VASP-recommended PAW
pseudopotentials are used. Noteworthy, relatively deep Ga $3d$, Ge $3d$, and
In $4d$, states are treated as valence orbitals.

We also perform partially self-consistent quasi-particle $GW_{0}$ calculations
from the VASP code whenever required for bandgaps. The implementation of this
method in the VASP is described in ref. Shishkin et al. (2007). For the QP
calculations, the number of virtual orbitals is increased to $240$ (using
NBANDS=$240$), and $4$ iterations for the self-consistent $GW_{0}$ steps are
used (using NELMGW=$4$) after $G_{0}W_{0}$ calculation. The VASP-recommended
$GW$ pseudopotentials are used, where relatively deep Ga $3d$, Ge $3d$, and In
$4d$ states are treated as valence orbitals. $8\times 8\times 8$
$\Gamma-$centered $k-$ points are used to sample the Brillouin zone in
$GW_{0}$ calculations. For all our cases, the starting point of many-body
perturbation theory is the GGA PBE functional.

The optical absorption spectrum for DDH and HSE06 is also performed using the
VASP code with a $16\times 16\times 16$ MP-like $\Gamma-$centered
${\bf{k}}$-points with $72$ empty orbitals. We have performed the DDH and
HSE06 calculations in many shifted $4\times 4\times 4$ ${\bf{k}}$-points and
weight over the multiple grids as a straightforward calculation of hybrids
would be expensive. vas .

Figure 1:  Pictorial representation of the self-consistency of the screened
DDH functional used in this paper. Figure 2:  Convergence of
$\epsilon_{\infty}$ as obtained using RPA@DDH using the self-consistent cycle
of Fig. 1. See Fig. 3 for the experimental values. Figure 3:  The
orientationally averaged static dielectric constants (optical),
$\epsilon_{\infty}$ from experimental and RPA@DDH for nine Cu and Ag-based
chalcopyrite. The self-consistent cycle of Fig. 1 is used for RPA@DDH. The
experimental values are used 7.77 for CuInSe2 Neumann (1986), 9.30 for
CuInTe2Holah1981infrared, 6.15 for CuGaS2 Baars and Koschel (1972), 4.95 for
CuAlS2 Koschel et al. (1973), 7.18 for AgInSe2 Kanellis and Kampa (1977), 6.43
for AgInTe2 Kanellis and Kampa (1977), 6.87 for AgGaSe2 Madelung (1992), 5.70
for AgGaS2 van der Ziel et al. (1974), 11.43 for AgGaTe2 Madelung (1992).
Figure 4:  Comparison of model dielectric function as a function of ${\bf{G}}$
(in Bohr-1) for four chalcopyrite semiconductors. The black circle line
indicates the model function with parameters evaluated at RPA@DDH (see Table
2), whereas the red line corresponds to the $G_{0}W_{0}$ or RPA calculations.

###  III.2 Self-consistent screened-DDH calculation

The central quantity of the screened DDH is to determine high-frequency
macroscopic static dielectric constants or ion-clamped static (optical)
dielectric constant or electronic dielectric constant, $\epsilon_{\infty}$ and
screening parameter, $\mu$. As mentioned before, in our case, we use
$\mu=\mu_{eff}^{fit}$ of Eq. 8. The calculations of $\mu_{eff}^{fit}$ are done
using the LDA densities, and those are used further for all the chalcopyrite
systems. These values are supplied in Table 2. As shown in Table 2,
$\mu_{eff}^{fit}$’s remain almost constants to a typical average value $0.72$
bohr-1, although for a few systems, its value becomes $\sim$ $0.80$ Bohr-1.
These values are consistent with the earlier reported values for common
semiconductors and insulators Chen et al. (2018); Jana et al. (2023).
Noteworthy, in ref. Chen et al. (2018), $\mu$’s are calculated from the least-
squares fitting to the dielectric function in the long-wavelength limit
obtained from higher-level accurate calculations, such as random phase
approximation (RPA) calculated using PBE orbitals (RPA@PBE) or using
nanoquanta kernel and partially self-consistent GW calculations Shishkin et
al. (2007).

Next, we turn to the calculations of the static dielectric constant, which is
the central quantity for any DDH. As for our present case, we already fixed
the $\mu$ to $\mu_{eff}^{fit}$ from the scheme described above. Considering
$\epsilon_{\infty}$, it can be calculated using various schemes: (i) density
functional perturbation theory (DFPT) within the framework of the RPA Baroni
et al. (2001) or (ii) using the modern theory of polarization King-Smith and
Vanderbilt (1993); Resta (1994).

The fundamental equation that has been solved to calculate the dielectric
tensor for bulk solid is,

$\epsilon_{{\bf{G}},{\bf{G^{\prime}}}}({\bf{q}},\omega)=\delta_{{\bf{G}},{\bf{G^{\prime}}}}-\frac{4\pi}{|{\bf{G+q}}||{\bf{G^{\prime}+q}}|}\chi_{{\bf{G}},{\bf{G^{\prime}}}}({\bf{q}},\omega)~{}.$
(11)

Our interest lies on the inverse of the macroscopic dielectric matrix
$\epsilon^{-1}_{\infty}({\bf{q}},\omega)$, which is obtained from the head of
the inversion of the full microscopic dielectric tensor as,

$\epsilon^{-1}_{\infty}({\bf{q}},\omega)=\lim_{{\bf{q}}\to
0}\epsilon^{-1}_{0,0}({\bf{q}},\omega)$ (12)

In Eq. 11, the reducible polarizability
$\chi_{{\bf{G}},{\bf{G^{\prime}}}}({\bf{q}},\omega)$ is defined as,

$\chi_{{\bf{G}},{\bf{G^{\prime}}}}=\chi^{0}_{{\bf{G}},{\bf{G^{\prime}}}}+\chi^{0}_{{\bf{G}},{\bf{G^{\prime}}}}\Big{(}\frac{4\pi}{|{\bf{G+q}}||{\bf{G^{\prime}+q}}|}+f_{xc}({\bf{q}},\omega)\Big{)}\chi_{{\bf{G}},{\bf{G^{\prime}}}}~{},$
(13)

where $\chi^{0}_{{\bf{G}},{\bf{G^{\prime}}}}({\bf{q}},\omega)$ being the
irreducible polarizability matrix obtained from KS systems Gajdoš et al.
(2006); Nunes and Gonze (2001); Souza et al. (2002) and
$f_{xc}({\bf{q}},\omega)$ is the XC kernel, obtained from the derivative of
the exchange-correlation potential Baroni et al. (2001). However,
$f_{xc}({\bf{q}},\omega)$ can be neglected as its inclusion in the
polarizability calculation is observed to be negligible Paier et al. (2008).
Henceforth, in this work, the $\chi_{{\bf{G}},{\bf{G^{\prime}}}}$ and
dielectric constants are evaluated by neglecting $f_{xc}({\bf{q}},\omega)$,
i.e., within RPA approximations. Noteworthy, using Bloch notation, one can
write $\chi^{0}_{{\bf{G}},{\bf{G^{\prime}}}}({\bf{q}},\omega)$ in terms of the
KS orbitals and energies Adler (1962); Wiser (1963); Gajdoš et al. (2006).
Therefore, the evaluation of $\epsilon_{\infty}({\bf{q}},\omega)$ depends
strongly on the choices of the XC functionals. Here, we perform RPA
calculations of Eq. 11 with PBE orbitals (named RPA@PBE) and DDH orbitals
(named RPA@DDH).

Finally, the self-consistent cycle to calculate the static
$\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$ using screened-DDH is described
in Fig. 1. We conform to the following steps: (i) Firstly, we calculate
$\mu_{eff}^{fit}$ using the Eq. 8 with LDA orbitals, (ii) secondly, we start
with $\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$ as obtained from PBE
functional (RPA@PBE), and plug it in our DDH expression Eq. LABEL:hy-eq6 along
with previously calculated $\mu_{eff}^{fit}$, (iii) thirdly, we perform the
DDH calculation and update $\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$ as
obtained from RPA@DDH as long as the self-consistency in
$\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$ is reached. In Fig. 2 we
illustrate the self-consistency of $\epsilon_{\infty}({\bf{q}}\to 0,\omega\to
0)$ using RPA@DDH using the scheme of Fig. 1. The self-consistency of
$\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$ is achieved mostly within four
cycles. It’s worth mentioning that for certain chalcopyrites, the PBE predicts
a metallic nature with zero bandgaps, leading to
$\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)\to\infty$ at RPA@PBE. In such
cases, one needs to initiate the self-consistency process illustrated in Fig.
1 from a finite value of $\epsilon_{\infty}({\bf{q}}\to 0,\omega\to 0)$.
Noteworthy, the potential is solved in all materials using the generalized
Kohn-Sham (gKS) scheme Garrick et al. (2020).

###  III.3 High-frequency dielectric constants

In Table 2, we first calculate the orientationally-averaged
$\epsilon_{\infty}$ (i.e.,
$\epsilon_{\infty}=\frac{\epsilon^{xx}_{\infty}+\epsilon^{yy}_{\infty}+\epsilon^{zz}_{\infty}}{3}$)
with RPA using PBE and DDH XC approximations with $\mu$ using Eq. 8. The
analysis from Table 2 reveals that, for I-III-VI$2$, RPA@PBE tends to
overestimate $\epsilon\infty$ compared to RPA@DDH. Notably, for several cases
like AgGaTe2, AgInTe2, CuGaSe2, CuInS2, CuGaTe2, and CuInTe$2$, RPA@PBE
predicts significantly large $\epsilon\infty$, particularly in instances where
PBE incorrectly predicts a metallic structure. The dielectric constants
calculated with RPA@PBE show inaccuracies for such materials. A similar trend
is observed for II-IV-V2 chalcopyrite semiconductors. However, in cases like
ZnGeAs2 and ZnSnAs2, PBE calculations lead to a metallic outcome. Across all
instances, there is a noticeable magnitude difference of approximately $\sim
2$ when comparing RPA@DDH and RPA@PBE. Examining the $\epsilon_{\infty}$
values from RPA@DDH, we observe that all values are finite and fall within the
range expected for an ideal chalcopyrite semiconductor. This performance
highlights the inadequacy of RPA@PBE in accurately calculating
$\epsilon_{\infty}$ for these systems.

Next, Fig.3 provides a comparison of $\epsilon_{\infty}$ for eleven Cu and Ag-
based chalcopyrites obtained from RPA@DDH and experimental data derived from
optical reflectivity experiments (refer to the references in the caption of
Fig.3). We observe a remarkable agreement between the calculated
$\epsilon_{\infty}^{RPA@DDH}$ and the experimental results. Conversely, for
most of these systems, PBE-predicted $\epsilon_{\infty}$ values are quite
large, with most systems exhibiting metallic behavior (as shown in Table 2).

Finally, Fig.4, we compare the model dielectric function described by Eq.7
(with $\alpha=1$, $\mu=\mu_{eff}^{fit}$, and self-consistent
$\epsilon_{\infty}^{-1}$ evaluated at RPA@DDH, as per Table 2) with the
dielectric function obtained through $G_{0}W_{0}$ or RPA calculations for
various chalcopyrite semiconductors. The details of the calculation procedure
can be found in ref.vas . This comparison illustrates that the model
dielectric function given by Eq.7 aligns well with dielectric functions
calculated through ab-initio methods.

Table 2:  High-frequency macroscopic static dielectric constants or ion-
clamped static (optical) dielectric constant or electronic dielectric constant
($\epsilon_{\infty}$), screening parameters ($\mu$ in bohr -1), and direct KS
bandgaps (in eV) using PBE, DDH, and HSE06. Here RPA@DDH is evaluated using
the scheme given in Fig. 1. The PBE, DDH, and HSE06 band gaps are corrected
for spin-orbit coupling (SOC). Error statistics in bandgaps with respect to
experimental values are also given. The experimental reference values of
I-III-VI2 and II-IV-V2 are taken from ref. Xiao et al. (2011) and ref.
Shaposhnikov et al. (2012), respectively. Here, the ‘$-$’ lines indicate that
those systems are treated as metals when using PBE. The bandgap values close
to the experiment one are in bold font. Total ME (in eV), MAE (in eV), and
MARE are also shown at the end of the Table.

Solids |  |  $\epsilon_{\infty}$ (RPA@PBE) |  $\epsilon_{\infty}$ (RPA@DDH) |  | $\mu=\mu_{eff}^{fit}$ |  | Eg (PBE) | Eg (DDH) | Eg (HSE06) | Eg ($GW_{0}$) | Eg (Expt.) | $\Delta_{SOC}^{a}$
---|---|---|---|---|---|---|---|---|---|---|---|---
I-III-VI2 |
AgAlS2 |  | 5.46 | 4.97 |  | 0.78 |  | 1.86 | 3.70 | 3.05 | 3.32b | 3.60 | 0.00
AgAlSe2 |  | 6.64 | 5.70 |  | 0.78 |  | 1.07 | 2.67 | 2.18 | 2.43b | 2.55 | 0.03
AgAlTe2 |  | 8.30 | 7.30 |  | 0.73 |  | 0.86 | 1.88 | 1.98 | 2.16b | 2.30 | 0.17
AgGaS2 |  | 6.65 | 5.38 |  | 0.82 |  | 1.11 | 2.86 | 2.30 | 2.16b | 2.73 | 0.01
AgGaSe2 |  | 8.85 | 6.18 |  | 0.82 |  | 0.51 | 2.06 | 1.62 | 1.29b | 1.83 | 0.03
AgGaTe2 |  | 15.39 | 8.78 |  | 0.79 |  | 0.13 | 1.19 | 1.23 | 1.17b | 1.36 | 0.17
AgInS2 |  | 7.56 | 5.21 |  | 0.74 |  | 0.49 | 2.23 | 1.57 | 1.32b | 1.87 | 0.01
AgInSe2 |  | $-$ | 6.09 |  | 0.74 |  | 0.02 | 1.56 | 1.06 | 0.73b | 1.24 | 0.04
AgInTe2 |  | 11.8 | 7.68 |  | 0.70 |  | 0.18 | 1.28 | 1.34 | 0.81b | 1.04 | 0.19
CuAlS2 |  | 6.23 | 5.20 |  | 0.78 |  | 1.66 | 3.87 | 3.20 | 2.96b | 3.46 | 0.01
CuAlSe2 |  | 7.73 | 6.05 |  | 0.78 |  | 0.84 | 2.76 | 2.32 | 2.13b | 2.65 | 0.02
CuAlTe2 |  | 9.42 | 7.85 |  | 0.75 |  | 0.93 | 1.96 | 2.01 | 1.85b | 2.06 | 0.09
CuGaS2 |  | 7.80 | 5.55 |  | 0.70 |  | 0.89 | 3.17 | 2.22 | 1.68d, 1.78b, 2.35e | 2.50 | 0.01
CuGaSe2 |  | 13.57 | 6.42 |  | 0.70 |  | 0.27 | 2.33 | 1.55 | 0.93d, 0.99b, 1.60e | 1.67 | 0.04
CuGaTe2 |  | 12.54 | 8.54 |  | 0.66 |  | 0.36 | 1.45 | 1.65 | 0.90b | 1.25 | 0.18
CuInS2 |  | $-$ | 5.98 |  | 0.73 |  | 0.03 | 1.72 | 1.03 | 0.77d, 0.69b, 1.41e | 1.55 | -0.03
CuInSe2 |  | 17.41 | 6.70 |  | 0.73 |  | 0.01 | 1.48 | 0.88 | 0.46b, 0.93e | 1.04 | 0 .01
CuInTe2 |  | $-$ | 9.05 |  | 0.71 |  | 0.01 | 1.14 | 1.17 | 0.70b | 1.00 | 0.17
II-IV-V2 |
BeGeAs2 |  | 11.30 | 10.47 |  | 0.81 |  | 0.55 | 1.31 | 1.30 | 1.07c | 1.68 | 0.07
BeGeP2 |  | 9.33 | 8.81 |  | 0.85 |  | 0.87 | 1.61 | 1.54 | 1.58c | 0.90 | -0.01
BeSiAs2 |  | 9.76 | 9.32 |  | 0.82 |  | 0.97 | 1.82 | 1.79 | 1.33c | 1.11 | 0.09
BeSiP2 |  | 8.63 | 8.37 |  | 0.86 |  | 1.18 | 1.95 | 1.90 | 1.75c | 1.30 | 0.03
BeSnAs2 |  | 11.52 | 10.79 |  | 0.79 |  | 0.56 | 1.32 | 1.34 | 1.25c | 1.15 | 0.11
BeSnP2 |  | 9.57 | 9.23 |  | 0.82 |  | 0.88 | 1.60 | 1.57 | 1.78c | 1.98 | 0.03
CdGeAs2 |  | 25.26 | 12.92 |  | 0.74 |  | 0.11 | 0.43 | 0.17 | 0.26c | 0.57 | 0.04
CdGeP2 |  | 11.45 | 9.55 |  | 0.78 |  | 0.63 | 1.39 | 1.38 | 1.61c | 1.72 | 0.00
CdSiAs2 |  | 15.30 | 10.64 |  | 0.76 |  | 0.33 | 1.12 | 1.19 | 1.29c | 1.55 | 0.04
CdSiP2 |  | 9.55 | 8.94 |  | 0.81 |  | 1.42 | 2.09 | 2.06 | 1.91c | 2.20 | 0.00
CdSnAs2 |  | 13.24 | 4.03 |  | 0.71 |  | 0.07 | 0.18 | 0.03 | 0.16c | 0.26 | 0.03
CdSnP2 |  | 18.53 | 9.34 |  | 0.74 |  | 0.24 | 0.97 | 0.98 | 1.10c | 1.17 | 0.02
MgGeAs2 |  | 11.54 | 9.31 |  | 0.72 |  | 0.49 | 1.34 | 1.26 | 1.27c | 1.60 | 0.05
MgGeP2 |  | 8.48 | 7.75 |  | 0.75 |  | 1.52 | 2.26 | 2.15 | 2.28c | 2.17 | -0.03
MgSiAs2 |  | 9.14 | 8.47 |  | 0.74 |  | 1.21 | 2.02 | 1.91 | 1.40c | 2.00 | 0.04
MgSiP2 |  | 7.79 | 7.42 |  | 0.77 |  | 1.37 | 2.18 | 2.03 | 1.83c | 2.26 | 0.00
MgSnAs2 |  | 12.13 | 9.02 |  | 0.69 |  | 0.31 | 1.17 | 1.10 | 1.07c | 1.20 | 0.07
MgSnP2 |  | 8.21 | 7.52 |  | 0.72 |  | 1.18 | 2.02 | 1.93 | 2.05c | 2.48 | 0.00
ZnGeAs2 |  | $-$ | 11.59 |  | 0.77 |  | 0.06 | 0.72 | 0.65 | 0.67c | 1.15 | 0.01
ZnGeP2 |  | 10.47 | 9.24 |  | 0.81 |  | 1.14 | 1.97 | 1.91 | 2.25c | 1.80 | 0.01
ZnSiAs2 |  | 11.59 | 10.16 |  | 0.79 |  | 0.81 | 1.76 | 1.79 | 1.94c | 1.60 | 0.05
ZnSiP2 |  | 9.34 | 8.68 |  | 0.83 |  | 1.35 | 2.09 | 2.04 | 1.92c | 2.30 | 0.01
ZnSnAs2 |  | $-$ | 11.69 |  | 0.74 |  | 0.00 | 0.46 | 0.44 | 0.44c | 0.75 | 0.00
ZnSiP2 |  | 10.45 | 8.68 |  | 0.83 |  | 1.35 | 2.09 | 1.48 | 1.92c | 2.30 | 0.01
TME |  |  |  |  |  |  | 0.95 | 0.02 | 0.15 | 0.17 |  |
TMAE |  |  |  |  |  |  | 0.95 | 0.24 | 0.27 | 0.28 |  |
TMARE |  |  |  |  |  |  | 0.57 | 0.17 | 0.20 | 0.19 |  |

a) Spin-orbit coupling is defined as
$\Delta_{SOC}$=E${}_{g}^{PBE}-$E${}_{g}^{PBE-SOC}$ which has been added with
all SOC un-corrected bandgaps obtained from DFT calculations.

b) Present work calculated with $GW_{0}$ from VASP. See main text for details.

c) $GW_{0}$ values from ref. Shaposhnikov et al. (2012). These calculations
are performed using VASP code.

d) $G_{0}W_{0}$@PBE from ref. Zhang et al. (2013).

e) $G_{0}W_{0}$@PBE+U from ref. Zhang et al. (2013).

Figure 5:  The calculated versus experimental band gaps of $42$ chalcopyrite semiconductor using DDH, HSE06, and $GW_{0}$. The linear regressions formula is defined as, E${}_{g}^{calc}$ = $\kappa$E${}_{g}^{expt}$ \+ $\gamma$, where $\kappa$ is the slope, $\gamma$ is the intercept (in eV), and $r^{2}$ is the correlation coefficient. The dashed lines in every panel give E${}_{g}^{calc}$ = E${}_{g}^{expt}$. For $GW_{0}$, use the best value from the Table 2. Figure 6:  Bandstructure of CuGaS2 calculated using PBE, HSE06 and DDH. The dashed line indicates the Fermi energy. Figure 7:  Band gap vs the anion displacement $u$ for CuInS2 as obtained using different methods. The PBE, HSE06, sc-COHSEX, and sc-COHSEX+$G_{0}W_{0}$ values are taken from ref. Vidal et al. (2010). Figure 8:  Density of states (DOS) for CuInS2 calculated using DDH and HSE06. Figure 9:  Electronic charge density distribution contours of CuInS2 as obtained from DDH and HSE06. The positions of Cu, In, and S atoms are shown. The logarithmic scale is used for the better visualization of the isosurfaces. Table 3:  Mean positions of the occupied $d$ band (in eV) relative to the VBM for selective semiconductors. The theoretical values are calculated by averaging the $d$ state energies at the $\Gamma$ point. Solids |  |  | PBE |  |  | DDH |  |  | HSE06 |  |  | $GW_{0}$
---|---|---|---|---|---|---|---|---|---|---|---|---
AgGaS2 |  |  | -14.8 |  |  | -19.8 |  |  | -17.1 |  |  | -17.3
AgInS2 |  |  | -14.4 |  |  | -17.3 |  |  | -15.9 |  |  | -16.6
AgInTe2 |  |  | -14.8 |  |  | -17.8 |  |  | -16.4 |  |  | -16.9
AgGaTe2 |  |  | -15.3 |  |  | -20.4 |  |  | -17.9 |  |  | -16.7
AgGaSe2 |  |  | -14.9 |  |  | -20.0 |  |  | -17.3 |  |  | -16.5
CuGaS2 |  |  | -15.4 |  |  | -20.8 |  |  | -17.6 |  |  | -18.1
CuInS2 |  |  | -15.0 |  |  | -17.7 |  |  | -16.5 |  |  | -17.0
CuGaTe2 |  |  | -15.4 |  |  | -21.1 |  |  | -17.9 |  |  | -18.5

###  III.4 Analysis of band structures

####  III.4.1 Bandgaps and band structures

Now, let’s delve into the performance of DDH in estimating the band gaps.
Table 2 provides a comprehensive comparison by presenting (g)KS band gaps
obtained from PBE, HSE06, and $GW_{0}$. Band gap is defined as
$E_{g}=\epsilon_{LUMO}-\epsilon_{HOMO}$, where $\epsilon_{LUMO}$ and
$\epsilon_{HOMO}$ represent the corresponding lowest unoccupied molecular
orbital (LUMO) and highest occupied molecular orbital (HOMO) eigenvalues.

Inspecting the band gaps of I-II-VI2, one can readily observe that, as usual,
PBE underestimates the band gaps for all systems and becomes zero for AgGaTe2,
AgInTe2, CuGaSe2, CuGaTe2, and CuInS2. Although HSE06 offers an improvement
over PBE, yet important improvement is observed in DDH calculations. Using
DDH, the band gap increases to $\sim~{}0.8$ eV for Ag-based chalcopyrites
compared to HSE06, making DDH band gaps close to the experimental values.
Similarly, for Cu-based chalcopyrites, the gaps obtained using DDH are also in
good agreement. We observe good performance for CuAlSe2, CuGaTe2, and CuInTe2
using DDH, which brings it closer to experimental values. One may note that
for Cu-based chalcopyrites, the interplay between Cu $d$ and anions $p$ is
identified as a crucial factor Zhang et al. (2013); Vidal et al. (2010);
Aguilera et al. (2011). Interestingly, for CuGaS2, CuGaSe2, CuInS2, and
CuInSe2, the underestimation in the band gap from $G_{0}W_{0}$ and $GW_{0}$ is
noticeable. As mentioned in ref. Shaposhnikov et al. (2012), the screening is
not treated correctly by $G_{0}W_{0}$ and the PBE+U method may be the better
starting point for quasiparticle $GW$ calculations Zhang et al. (2013); Vidal
et al. (2010); Aguilera et al. (2011). For example, using the PBE+U as a
starting point for $G_{0}W_{0}$, the band gaps of CuInS2 improve to 1.41 eV,
where $G_{0}W_{0}$@PBE value was $0.77$ eV (in our present calculation it is
$0.69$ for $GW_{0}$ which is close to $G_{0}W_{0}$@PBE value of ref. Zhang et
al. (2013)). A similar improvement in results is also noticed when employing
PBE+U as a starting point for Cu-based chalcopyrite semiconductors. In
accordance with the insights from ref. Vidal et al. (2010), these systems call
for careful treatment of $GW$, requiring fully self-consistent (sc)COHSEX or
$GW$ for precise band gap predictions. Nevertheless, the remarkable
performance of DDH in these semiconductor materials hints at its potential to
be the preferred method, offering a commendable balance between accuracy and
relatively lower computational cost compared to the self-consistent $GW$.
Similarly, DDH also delivers accurate results for II-IV-V2 semiconductors and
is closely aligned with the experimental values.

Fig. 5 shows the experimental versus theoretical band gaps obtained using DDH,
HSE06, and $GW_{0}$. We use linear regression analysis to understand better
the errors coming from different methods. We calculate slope ($\kappa$),
interception ($\gamma$), and correlation coefficient ($r^{2}$) for comparison.
For DDH, the $\kappa$, $\gamma$, and correlation coefficient $r^{2}$ are found
to be about $0.98$, $0.08$, and $0.81$ which are slightly better than HSE06.
Considering the $GW_{0}$, its error statistics are similar to HSE06. The most
relevant parameter here is the $r^{2}$, which is marginally better for DDH
functional than HSE06 and GW0. Also, as shown in Table 2, in terms of total
mean absolute error (MAE) and mean absolute relative error (MARE), DDH
performs marginally better than HSE06 as well as $GW_{0}$.

In Fig. 6, we plot the band structures from PBE, HSE06, and DDH for CuGaS2, a
direct band gap (located at $\Gamma-$ point) semiconductor. We observe an
identical band structure and curvature apart from the shift in the conduction
band for different methods. Specifically, DDH shows a noticeable $\sim 0.97$
eV shift in the conduction band at the $\Gamma$-point compared to HSE06.

####  III.4.2 Variation of bandgaps with $u$

To understand the variation of the band gaps with the distortion parameter
‘$u$’ (defined in Eq. 10), , we show the band gap variation of CuInS2, one of
the prototype chalcopyrite semiconductor, using various methods in Fig 7. This
is important because, experimentally, one can observe the stability of
chalcopyrite with the variation of $u$. Typically, PBE predicts the system to
be metallic for $0.20\leq u\leq 0.215$. HSE06 improved over PBE but
underestimated bandgap by $\sim 0.2$ eV. A very similar performance is also
observed from various sc-COHSEX+$G_{0}W_{0}$, while fully self-consistent sc-
COHSEX improves over single shot sc-COHSEX, i.e., sc-COHSEX+$G_{0}W_{0}$ Vidal
et al. (2010). In contrast, the improvements in the bandgaps for DDH are quite
noticeable over the ranges of $u$. Although DDH overestimates the bandgaps
only by $\sim 0.2$ eV compared to the experimental values, it is well within
the shaded part of Fig. 7. For each $u$, the DDH uses different Fock mixing by
$\epsilon_{\infty}^{-1}$. The seemingly different band gap values from
different methods occur because of the different screening of these systems
with various $u$, which is expected due to interplay between the $u$ and the
hybridization of the $p-d$ orbitals Vidal et al. (2010).

####  III.4.3 Analysis of DOS and charge density

More analysis can also be drawn from the density of states (DOS) for CuInS2 in
Fig 8. Typically, the main contribution comes from metal Cu$-3d$, anion
S$-3p$, and their (non-bonding/anti-bonding) hybridization. For DDH, there is
a downward shift of non-bonding $3d$ states because of the stronger
hybridization, which reduces the $p-d$ repulsion, hence the enlargement of
bandwidth.

The differences in the performances of DDH and HSE06 can also be drawn from
the charge density contour plot of CuInS2 as shown in Fig 9. As known, the
In$-$S bond is ionic, whereas Cu$-$S bond is covalent. The nature of covalency
changes due to the repulsive $p-d$ non-bonding nature, which is depicted
through the change density iso-surface plot. The reduction in the isosurface
value between Cu$-$S indicates a decrease of $p-d$ repulsion in DDH; hence,
the atomic distance becomes slightly lower than HSE06.

####  III.4.4 Positions of valance $d-$ bands

Finally, in Table 3, we show the mean positions of the occupied $d$ band (in
eV) (relative to the VBM) for several chalcopyrite semiconductors. As PBE
functional suffers from the known de-localization, its occupied $d$ band is
quite underestimated. Although HSE06 improves over PBE, DDH generally recovers
the mean positions of the occupied $d$ band quite remarkably compared to both
HSE06 Jana et al. (2023). However, our results suggest non-empirical DDH
obtains a bit of deep occupied $d-$ band positions compared with $GW_{0}$.
This is because, for these systems, the band gaps are overestimated from DDH.
Unfortunately, no experimental results are available to compare with.

The analysis of band gaps, distortion parameter versus band gap variations,
the density of states, and charge density indicate the screening effect as
determined using DDH, which is important for determining accurate properties
from hybrid functionals. Importantly, a judicious choice of the percentage of
the Fock screening is required for a more accurate prediction of the
properties. In that case, the functional becomes empirical or tuned DDH Ohad
et al. (2022). However, looking at the nonempirical settings and superiority
of the obtained properties, the present DDH can be considered as one of the
useful methods, especially when higher-level accurate methods like $GW$ are
necessary but computationally unaffordable.

Figure 10: The band gap values for CuxAg1-xGaSe2 alloy are calculated at
compositions $x=0.0,0.25,0.50,0.75,\text{ and }1.0$, with DDH and HSE06. The
curves are fitted using the Eq. 14.

###  III.5 Optical bowing parameters

The band gap of ternary chalcopyrites varies from CuBX2 to AgBX2 due to the
different sizes of Cu and Ag atoms. This variation is evident in Table 2 for
all XC approximations. The band gap varies as the Cu/Ag ratio changes Shaukat
(1990). These size-dependent variations are further intensified by band
bowing, known as optical bowing Mourad and Czycholl (2012). This effect arises
from atomic-level fluctuations in the lattice structure of (Cu, Ag)BX2
compounds Mudryi et al. (2003). Due to level repulsion between chalcopyrite
energy levels in the alloy, the band gap of the alloy experiences a downward
shift from the linear average, as described by the following equation:

$E_{g}(A_{x}B_{1-x})=xE_{g}(A)+(1-x)E_{g}(B)-bx(1-x)~{},$ (14)

where $E_{g}(A)$ and $E_{g}(B)$ are the band gaps of A and B for the compound
$A_{x}B_{1-x}$. Considering the quaternary chalcopyrite semiconductor alloy
CuxAg1-xGaSe2, the common Ga$-$X bond length remains almost unchanged with
concentration $x$, while the Ag$-$Se and Cu$-$Se bond lengths associated with
Ag and Cu increase with $x$ Wei and Zunger (1995). This is consistent with the
nearly identical Ga-Se bond lengths in CuGaSe2 and AgGaSe2 because the local
environment of Ga does not change with the alloy concentration $x$. In the
case of DDH, the band gap varies from 2.33 to 2.06 eV as we go from CuGaSe2 to
AgGaSe2, and in the case of HSE06, it varies from 2.55 to 2.62, as shown in
Table 2. The optical bowing parameter ’$b$’ can be obtained using the band gap
difference. The band gaps for CuxAg1-xGaSe2 with concentrations of ($x=$0.0,
0.25, 0.50, 0.75, 1.0) are calculated using PBE, DDH and HSE06 XC
approximations. The band gap value using PBE for $x=$0.0 and 0.25 is 0 eV, so
PBE band gap errors are largely canceled in the calculation. The optical
bowing parameter ’$b$’ can be calculated at different $x$ along with pure
AgGaSe2 and CuGaSe2 according to Equation 14.

Figure 10 is plotted by curve-fitting the calculated band gap using HSE06 and
DDH using Eq. 14. The experimental value of ’$b$’ for CuxAg1-xGaSe2 is 0.280
eV Choi et al. (2000). The value of ’$b$’ using HSE06 is 0.251 eV, whereas for
DDH, it is 0.286 eV. Our study shows that the value of ’$b$’ using DDH is in
very good agreement with the experimental optical bowing parameter compared to
HSE06.

Figure 11:  Imaginary part of absorption spectra ($\epsilon_{2}(\omega)$) of
CuInSe2 (upper panel) for light polarized along the $c$ axis (upper panel) and
for light polarized perpendicular to the $c$ axis (lower panel) calculated
with DDH (non-empirical) and DDHfit (with empirical tuning of parameter
$\gamma$) and HSE06. The experimental and BSE@$GW$ results are taken from
Alonso et al. Alonso et al. (2001) and Korbel et al. ref. Körbel et al.
(2015), respectively. Figure 12:  Same as Fig. 11 but calculated for CuGaS2.
The experimental values are taken from Alonso et al. Alonso et al. (2001) and
Levcenko et al. Levcenko et al. (2007). The BSE@$GW$ results are taken from
Aguilera et. al. Aguilera et al. (2011).

###  III.6 Description of absorption spectra

Optical absorption properties of solids in DFT are mostly calculated using the
linear-response TDDFT (LR-TDDFT) by solving the Casida equation using KS or
gKS orbitals CASIDA . However, to get an accurate optical spectrum for a
solid, it is necessary to get the correct band gap and accurate treatment of
the electrons and holes. State-of-the art BSE@$GW$ includes the physics
correctly but comes with huge computational expenses. Typically, in VASP, the
excitation spectra for both the TDDFT and BSE@$GW$ methods are calculated by
solving the Casida/Bethe-Salpeter equation CASIDA ; Onida et al. (2002);
Sander et al. (2015); Sander and Kresse (2017); Tal et al. (2020).

Within the Tamm-Dancoff approximation, the TDDFT spectrum is calculated by
solving the matrix element of direct transition from occupied to unoccupied
states and the electron-hole ($e-h$) interaction as CASIDA ; Sander et al.
(2015); Sander and Kresse (2017); Tal et al. (2020),

$\displaystyle A_{ij;ab}=\omega_{ia}\delta_{ij;ab}+\langle
ib|K_{Hxc}|aj\rangle~{},$ (15)

where $\omega_{ia}=\varepsilon_{i}-\varepsilon_{a}$. The occupied states are
noted as $i$, $j$, unoccupied states as $a$, $b$, and corresponding KS or gKS
eigenvalues are denoted by $\varepsilon$. The matrix element $\langle
ib|K_{Hxc}|aj\rangle$ is evaluated using the Hartree plus XC kernel,
$K_{Hxc}$, which includes the $e-h$ interaction. Because the semilocal
functionals underestimate the band gaps, typically, the first term of the
right-hand side of Eq. 15, depending on the KS eigenvalues, is not well
treated. The $e-h$ interaction term is further simplified using the framework
of the gKS and screened-DDH as Hartree, screened exchange, and the term
involves the XC kernel as Tal et al. (2020),

$\displaystyle\langle ib|K_{Hxc}|aj\rangle$ $\displaystyle=$ $\displaystyle
2\langle ib|V(|{\bf{q+G}}|)|aj\rangle$ $\displaystyle-$ $\displaystyle\langle
ib|f_{xc}^{non-local}|ja\rangle+\langle ib|f_{xc}^{local}|aj\rangle$

The first term on the right side of the above matrix element involves the
Hartree term common for both TDDFT and BSE@$GW$. The term involving $\langle
ib|f_{xc}^{non-local}|ja\rangle=\langle
ib|\epsilon_{\infty}^{-1}(|{\bf{q+G}}|)V(|{\bf{q+G}}|)|ja\rangle$ does not
appear when considering only the semilocal XC functional. The local XC kernel,
$f_{xc}^{local}$, is the functional derivative of semilocal XC approximations
with respect to the density and does not include any excitonic effect.
However, some specially designed low-cost XC kernels are also developed and
include necessary features Trevisanutto et al. (2013); Terentjev et al.
(2018); Sharma et al. (2011); Rigamonti et al. (2015); Van Faassen et al.
(2002); Cavo et al. (2020); Byun and Ullrich (2017); Byun et al. (2020).
Importantly, for a short-range screened hybrid like HSE06,
$\epsilon_{\infty}^{-1}(|{\bf{q+G}}|)=\beta(1-e^{-|{\bf{G}}|{{}^{2}}/(4\mu)})$
and the resultant screened exchange varies as $\sim\beta$, a constant with
$\beta=0.25$ at $q\to 0$. However, for bulk systems, the correct behavior of
screened exchange must be
$~{}\sim\frac{\epsilon_{\infty}^{-1}(|{\bf{G}}|)}{q^{2}}$ at $q\to 0$, which
is the key to improving the optical absorption spectra from DDH Ullrich
(2011); Paier et al. (2008); Yang et al. (2015); Wing et al. (2019b); Städele
et al. (1999); Petersilka et al. (1996); Kim and Görling (2002); Sun and
Ullrich (2020); Sun et al. (2020); Kootstra et al. (2000), where the high-
frequency dielectric constant of the material replaces dielectric function Tal
et al. (2020).

On the other hand, in BSE, the same set of equations are solved Onida et al.
(2002); Tal et al. (2020). However, the orbitals are related to the previous
$GW$ calculations Onida et al. (2002); Tal et al. (2020), typically obtained
from different levels of approximations, where the screened exchange is
frequency-dependent (through the dielectric function) Tal et al. (2020) and
the inclusion of the “nanoquanta” vertex correction may also required for
accurate calculations Shishkin et al. (2007). For the corresponding VASP
implementation of the BSE equation and details differences of BSE and TDDFT
using DDH, the readers are referred to ref. Tal et al. (2020). While BSE
calculations neglect dynamical effects, it’s worth noting that determining the
screened exchange in prior $GW$ steps adds complexity to the overall problem
Tal et al. (2020).

Finally, in TDDFT, the frequency-dependent and small wave vectors limit of the
imaginary ($\epsilon_{2}(\omega)$) part of the macroscopic dielectric function
(optical), $\epsilon^{M}$ is calculated via

$\displaystyle\epsilon_{2}(\omega)$ $\displaystyle=$
$\displaystyle\Im\\{\lim_{q\rightarrow 0}\epsilon^{M}(q,\omega)\\}~{},$ (17)

where $\lim_{q\rightarrow 0}\epsilon^{M}(q,\omega)$ is given by the Eq. (48)
of ref. Sander and Kresse (2017), which is also related to the calculated
matrix elements of Eq. (15) and how the kernels are evaluated. It is quite
apparent that the results obtained from TDDFT using the semilocal only and
hybrid DFT functionals give drastically different results.

In the following, we choose CuInSe2 and CuGaS2 to calculate the TDDFT spectrum
for which high-level calculations and experimental values are available. For
CuInSe2, we consider $a=5.780$ Å, $c=11.618$ Å, and $u=0.230$ according to the
experimental values in Table I of the ref Kim et al. (2016). Similarly, for
CuGaS2 we choose $a=5.351$ Å, $c=10.478$ Å, and $u=0.259$ according to the
experimental values as supplied in Table II of the ref Han et al. (2017). To
make a meaningful comparison, we compare the results from the KS (as obtained
using DDH and HSE06) applying TDDFT, BSE solving the $GW$ (BSE@$GW$), and the
experimental dielectric functions. Fig. 11 presents the absorption spectra for
CuInSe2 and Fig. 12 for CuGaS2. In both cases, we show the spectra for the
light-polarized perpendicularly to the $c$ axis
(($\epsilon_{2}^{xx}(\omega)+\epsilon_{2}^{yy}(\omega)$)/2) or along the $c$
axis ($\epsilon_{2}^{zz}(\omega)$).

Fig. 11 compares the absorption onset of TDDDH, TDHSE06, BSE@$GW$, and
experimental in the case of CuInSe2. As evident from the figure, a fairly good
agreement of TDDDH with BSE@GW (BSE@$GW$ spectrum is taken from ref. Körbel et
al. (2015)) and experimental spectrum from Alonso et al. Alonso et al. (2001)
is observed. Notably, excitonic peak positions obtained from TDDDH slightly at
higher energies or right-shifted when compared with the experimental and
BSE@$GW$. This is because of the overestimations in the orbital energies and
hence the band gap values from DDH (connected to the Casida Eq. 15). On the
other hand, in the case of TDHSE06, we also observe good agreements with
TDDDH. Note that the HSE06 absorption spectrum for these semiconductors is
also reasonably good compared to the experimental. On the other hand, the
advantage of TDDDH is that very accurate spectra can be obtained for both the
semiconductor and insulators Jana et al. (2023).

Very similar tendencies are obtained from TDDDH when compared for CuGaS2. As
referred to the ref. Aguilera et al. (2011), for CuGaS2, two absorption
spectra are available from Alonso et. al. Alonso et al. (2001) and Levcenko
et. al. Levcenko et al. (2007). We consider the experimental spectra of Alonso
et al. Alonso et al. (2001), shown in Fig. 12, and those have better agreement
with BSE spectra Aguilera et al. (2011). Considering TDDDH spectra, the first
peak is higher energy than the experimental and BSE@$GW$, as shown in Fig. 12.
Regarding the TDHSE06, the excitonic peaks also agree with the experimental
one, similar to the previous studies.

One can obtain a good absorption spectrum from TDDDH by empirically tuning the
parameter $\gamma$ of Eq. 4. Several other works have adopted this strategy
Wing et al. (2019b); Ohad et al. (2022); Gant et al. (2022); Camarasa-Gómez et
al. (2023). The following strategies can be adopted to obtain a reasonable
spectrum from TDDDH: (i) Tune parameter $\gamma$ of Eq. 4 to match with the
band gaps of $G_{0}W_{0}$ or $GW_{0}$ when no experimental band gap is
available, or (ii) if experimental band gaps are available then the tuning of
the $\gamma$ of Eq. 4 can be done to match with the experimental band gap of
the system, keeping screening parameter $\mu$ fixed. Here, we consider the
second strategy. After tuning $\gamma$ from experimental band gaps, we compare
the excitation spectrum with the BSE@$GW$ or experimental. We obtain
$\gamma=0.08$ for CuGaS2 and $\gamma=0.06$ for CuInSe2 to match the
experimental band gaps. Consistent agreement is noted in absorption onsets,
excitonic peak positions, and the higher energy spectrum when compared to the
spectrum for light polarized in both directions. Notably, the tuning procedure
is usually not needed for systems where DDH adequately describes band gaps.

##  IV Conclusions

The comparative assessment of the screened-range-separated hybrids (such as
HSE06), screened DDH, and methods based on the many-body perturbation theory
are assessed for various properties of chalcopyrite semiconductors such as
band gaps, optical bowing parameters, and optical absorption spectrum. It is
demonstrated that the screened DDH approach is promising not only as a cheaper
alternative to many-body perturbation theory based approaches (such as
$G_{0}W_{0}$ or $GW$ and BSE@$GW$) but more flexible and physically sound than
HSE06. The important fact is that in the screened DDH, the amount of screening
correlation is determined from the static dielectric constant instead of a
fixed screening used in HSE06. This makes the screened DDH more flexible,
especially for the band gaps of chalcopyrites, where the amount of $p-d$
hybridization is determined from screening correlation. Though the overall
mean absolute error suggests that the band gap performance of HSE06, screened
DDH, and $GW_{0}$ (or $G_{0}W_{0}$) are quite similar, the screened DDH has a
better overall slope, intercept, and correlation coefficient when compared
with the experimental band gaps. We also observe that in Cu-based
chalcopyrites, the accuracy of screened DDH for band gaps is slightly better
than $G_{0}W_{0}$@PBE, which strongly depends on the initial starting point.

Also, a notable success of the TDDDH in the case of calculating the optical
absorption spectrum is demonstrated, which is both cost-efficient and free
from empiricism when compared with the BSE@$GW$. We hope that screened DDH can
be the method of choice for evaluating the optical properties of chalcopyrite
systems and different heterostructures where the BSE calculations are not
feasible.

Finally, the overall quality performances of screened-DDH are encouraging as
they are quite close to many-body perturbation calculations in providing the
estimates of various properties. Importantly, the screened DDH is free of
adjustable parameters (as opposed to HSE06). All the results are obtained by
solving the generalized Kohn-Sham equation, and unlike the $GW$ method, it
involves no virtual orbitals. This makes the functional easy to use with
minimum computational cost. The present study shows this can be a method of
choice for other chalcopyrite semiconductors, especially for Cu-based
multinary semiconductors.

## Acknowledgments

SJ would like to thank Dr. Lucian A. Constantin for valuable comments,
suggestions, and technical details.

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