# Improved electronic structure prediction of chalcopyrite semiconductors from
a semilocal density functional based on Pauli kinetic energy enhancement
factor

Arghya Ghosh<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology, Hyderabad, India  Subrata Jana
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Physical Sciences,
National Institute of Science Education and Research, HBNI, Bhubaneswar
752050, India  Manish K Niranjan<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology, Hyderabad, India  Sushant Kumar Behera  School
of Physical Sciences, National Institute of Science Education and Research,
HBNI, Bhubaneswar 752050, India  Lucian A. Constantin  Istituto di
Nanoscienze, Consiglio Nazionale delle Ricerche CNR-NANO, 41125 Modena, Italy
Prasanjit Samal  School of Physical Sciences, National Institute of Science
Education and Research, HBNI, Bhubaneswar 752050, India

###### Abstract

The correct treatment of $d$ electrons is of prime importance in order to
predict the electronic properties of the prototype chalcopyrite
semiconductors. The effect of $d$ states is linked with the anion displacement
parameter $u$, which in turn influences the bandgap of these systems.
Semilocal exchange-correlation functionals which yield good structural
properties of semiconductors and insulators often fail to predict reasonable
$u$ because of the underestimation of the bandgaps arising from the strong
interplay between $d$ electrons. In the present study, we show that the meta-
generalized gradient approximation (meta-GGA) obtained from the cuspless
hydrogen density (MGGAC) [Phys. Rev. B 100, 155140 (2019)] performs in an
improved manner in apprehending the key features of the electronic properties
of chalcopyrites, and its bandgaps are comparative to that obtained using
state-of-art hybrid methods. Moreover, the present assessment also shows the
importance of the Pauli kinetic energy enhancement factor,
$\alpha=(\tau-\tau^{W})/\tau^{unif}$ in describing the $d$ electrons in
chalcopyrites. The present study strongly suggests that the MGGAC functional
within semilocal approximations can be a better and preferred choice to study
the chalcopyrites and other solid-state systems due to its superior
performance and significantly low computational cost.

##  I Introduction

In the last two decades, the $ABX_{2}$ (A=Cu,Ag,Be,Cd,Mg,Zn; B=Ga,Ge,In;
X=S/Se/Te) chalcopyrite semiconductors based on the zinc-blend structure have
emerged as highly promising advanced solar cells materials that allow the
tweaking of macroscopic physical properties as function of microscopic
chemical and structural degrees of freedoms 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, Cu-based ternary and quaternary
semiconductors and their alloys are now widely exploited as light absorbers in
thin film solar cells Birkmire and Eser (1997); Repins et al. (2008); Jackson
et al. (2011); Green et al. (2012); Rashkeev and Lambrecht (2001) and various
non liner optical devices Spiess et al. (1974); Parkes et al. (1973); Dittmer
et al. (2000). Unfortunately, the theoretical predictions of the structural
and electronic properties of Cu-based multinary semiconductors remain far from
being accurate because of the involvement of semicore $d$ or $f$ electrons,
which play a decisive role in the determination of the properties of these
materials Vidal et al. (2010); Siebentritt et al. (2010).

The straightforward applications of the Kohn-Sham (KS) density functional
theory (DFT) Kohn and Sham (1965) within the framework of the local density
approximation (LDA) Perdew and Wang (1992) or generalized gradient
approximation (GGA) Perdew et al. (1996) often result in wrong predictions of
bandgaps and band ordering as these suffer from the delocalization error Cohen
et al. (2008). Thus, the insufficiency and lack of appropriate theoretical
approaches hinder the proper understanding of the inherent electronic
properties of these systems. Although, some plausible attempts have been made
to correlate bandgap with the physical parameters in chalcopyrites such as
CuInSe2, CuInS2 in various literature Vidal et al. (2010); Jiang and Feng
(2007), theoretical studies of such correlations and structural anomalies in
relevant and important prototypes such as CuGaS2, CuAlS2, are still
fragmentary.

Nevertheless, using hybrid functional schemes Heyd et al. (2003); Heyd and
Scuseria (2004); Krukau et al. (2006); Paier et al. (2006); Jana et al.
(2018a); Jana and Samal (2019); Jana et al. (2020a, b), improved bandgaps of
chalcopyrites and other insulators have been reported. These schemes combine
Hartree-Fock exchange with LDA/GGA correlation. Further, the semilocal
exchange-correlation (XC) effect of strongly localized $d$-orbitals are
treated at the mean-field level Anisimov and Gunnarsson (1991); Anisimov et
al. (1997, 1991); Zhang et al. (2011). Although the hybrid density functionals
have been applied and studied for a variety of systems, these schemes are
often computationally expensive and prohibitive. On the other hand, the strong
electron correlations and self-interaction related problems within semilocal
approximation can be partially mitigated by the addition of separate Hubbard
$U$ for the $d$ or/and $f$ localized electrons. This scheme may be expected to
provide a qualitative assessment of the influence of the electronic
correlations on the physical properties of a system Anisimov and Gunnarsson
(1991); Anisimov et al. (1997). The DFT+U method attempts to fix hybridization
problems due to overly-delocalized orbitals and thereby has been recognized as
a potent tactic to treat the strongly localized $d$ electrons in chalcopyrites
Anisimov et al. (1991); Zhang et al. (2011).

In addition to hybrid and DFT+U schemes, the quasi-particle correction method
within the many-body GW approximation is even more well-founded among
different levels of theory which renders a relatively more accurate estimation
of bandgaps Anisimov and Gunnarsson (1991); Anisimov et al. (1997). However,
the calculations using the GW scheme are usually more computationally
expensive than those using hybrid functionals.

Over the years, the search for efficient semilocal approaches which can
predict the accurate electronic structure of systems, has become an important
and fascinating research topic. The progress in semilocal density functionals
in recent years shows that the accuracy of the material properties can be
achieved from the advanced XC methods by satisfying more exact quantum
mechanical constraints. The advent of these accurate semilocal density
functionals Sun et al. (2015); Tao and Mo (2016); Patra et al. (2019a, b,
2020a); Furness et al. (2020) has also led to wide applicability of DFT in the
condensed matter physics. In particular, the meta-GGA functionals have been
found to be quite successful in overcoming the several challenges associated
with the lower rung of functionals such as GGA, LDA etc. Jana et al. (2019,
2018b, 2018c); Zhang et al. (2020a, b); Nokelainen et al. (2020); Patra et al.
(2021)

Though the chalcopyrite systems have been studied using different functionals,
the assessment, and performance of the advanced meta-GGA methods for these
systems have not been reported to the best of our knowledge. Therefore, in
this article, we investigate the performance of some advanced meta-GGA
functionals such as 1) meta-GGA functional obtained from the cuspless hydrogen
density (MGGAC) Patra et al. (2019a), 2) strongly constrained and
appropriately normed (SCAN) semilocal functional Sun et al. (2015) , and 3)
Tao-Mo (TM) Tao and Mo (2016) semilocal functional. We assess the performance
of these functionals in the study of structural and electronic properties,
bandgaps, and enthalpy formation energies of chalcopyrite systems. It may be
noted that both SCAN and MGGAC functionals have been found to be quite
successful in overcoming the several drawbacks of the GGA functional such as
severe underestimation of the bandgaps of the bulk and layered solids Patra et
al. (2020b). In particular, our study shows that the MGGAC functional is quite
advantageous to study the chalcopyrite semiconductors as it is computationally
inexpensive and exhibits good performance. The aforementioned meta-GGA schemes
span the third rung of the Jacob ladder of the XC approximation Perdew and
Schmidt (2001) owing to important features associated with them.

This paper is organized as follows. The methodologies used in this work are
discussed in section II. The structural properties, electronic properties, and
enthalpy formation energies are discussed in sections II, III, and IV
respectively. The results and the functional performances are further analyzed
in section VI. Finally, the conclusions are presented in section VI.

##  II Computational Methods

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). The core-valence electron interaction
is approximated using the projected augmented wave (PAW) method Blöchl (1994).
A kinetic energy cutoff of 600 eV is used to expand the Kohn-Sham (KS) single-
particle orbitals on a plane-wave basis. The Brillouin Zones (BZ) are sampled
using Monkhorst-Pack $9\times 9\times 9$ k-points grid for atomic relaxations
and 17$\times$17$\times$17 grid for the density of states (DOS) calculations.
The electronic energies are allowed to converge up to 10-6 eV (or less) to
achieve self-consistency in the calculations. The atomic relaxations are
performed till the Hellmann-Feynman forces on atoms are reduced to less than
$0.01$ eV/Å.

The exchange-correlation (XC) contributions are included using semilocal XC
functional schemes such as PBE Perdew et al. (1996), SCAN Sun et al. (2015),
MGGAC Patra et al. (2019a), and TM Tao and Mo (2016). Unlike GGAs, the meta-
GGA functionals SCAN, MGGAC, and TM are implemented in generalized KS (gKS)
Perdew et al. (2017); Yang et al. (2016) scheme because of their dependence on
KS orbitals. Usually, the meta-GGAs are quite reliable as compared to GGAs as
they can recognize and describe covalent, metallic, and non-covalent
interactions Sun et al. (2013); Della Sala et al. (2016).

In general, the meta-GGA XC functional which depends on the density ($\rho$),
gradient of density ($\nabla\rho$), and KS kinetic energy density
($\tau({\bf{r}})=\frac{1}{2}\sum_{i}|\nabla\psi_{i}|^{2}$) can be presented
as:

$E_{xc}[\rho_{\uparrow},\rho_{\downarrow}]=\int~{}d^{3}r~{}\rho({\bf{r}})\epsilon_{x}^{LDA}F_{xc}(\rho_{\uparrow},\rho_{\downarrow},\nabla\rho_{\uparrow},\nabla\rho_{\downarrow},\tau_{\uparrow},\tau_{\downarrow})~{},$
(1)

where $\epsilon_{x}^{LDA}$ is the exchange energy density in uniform electron
gas approximation and $F_{xc}$ is the XC enhancement factor. In case of
construction of SCAN Sun et al. (2015) and MGGAC Patra et al. (2019a)
functionals, one of the main ingredients in $F_{x}$ is the Pauli kinetic
enhancement factor $\alpha=(\tau-\tau^{W})/\tau^{unif}$, where
$\tau^{W}=|\nabla\rho|/[8\rho]$ is the von Weizsäcker kinetic energy density,
$\tau^{unif}=3/10(3\pi^{2})^{2/3}\rho^{5/3}$ is the Thomas-Fermi kinetic
energy density, and $k_{F}=(3\pi^{2}\rho)^{1/3}$ is the Fermi wave vector. It
may be noted that the exchange of the SCAN functional uses both the $\alpha$
and $p=s^{2}$, while the exchange in MGGAC uses only $\alpha$. This makes
MGGAC $F_{xc}$ in exchange-only or high-density limit ($r_{s}\to 0$)
independent of $s$. Importantly, both SCAN and MGGAC respect the tightened
bound of the exchange ($F_{x}\leq 1.174$ Perdew et al. (2014); Sun et al.
(2015)) and possess negative slope with respect to $\alpha$ ($\partial
F_{x}/\partial\alpha<0$), important for the bandgap improvement in semilocal
level Aschebrock and Kümmel (2019); Patra et al. (2020b, 2019a).

Another reliable meta-GGA semilocal functional was proposed by Tao-Mo (TM) Tao
and Mo (2016) which has been quite successful at providing reasonably accurate
estimates of various quantum chemical and solid-state properties Mo et al.
(2017a, b); Tao and Mo (2016); Jana et al. (2019, 2020c). The TM and its
revised version revTMJana et al. (2019) functionals, use both the meta-GGA
ingredients i.e., $z=\tau^{W}/\tau$ and $\alpha$. However, these functionals
satisfy looser bound of exchange Lewin and Lieb (2015); Feinblum et al.
(2014); Constantin et al. (2015) and suffer from the order-of-limit problem
Patra et al. (2020a).

Table 1:  I-III-VI2 and I-IV-V2 chalcopyrite semiconductor systems considered in this paper. Group | Systems
---|---
|  CuInS2, CuInSe2, CuGaS2, CuGaSe2, CuAlS2, CuAlSe2,
I-III-VI2 |  CuAlTe2, CuGaTe2, CuInTe2, AgAlS2, AgAlSe2, AgAlTe2,
|  AgGaS2, AgGaSe2, AgGaTe2, AgInS2, AgInSe2, AgInTe2
|  BeSiP2, BeSiAs2, BeGeP2, BeGeAs2, BeSnP2, BeSnAs2
II-IV-V2 |  MgSiP2, MgSiAs2, MgGeP2, MgGeAs2, MgSnP2, MgSnAs2
|  ZnSiP2, ZnSiAs2, ZnGeP2, ZnGeAs2, ZnSnP2, ZnSnAs2
|  CdSiP2, CdSiAs2, CdGeP2, CdGeAs2, CdSnP2, CdSnAs2

In addition to aforementioned functionals, we have also used the hybrid
functional HSE06 scheme propounded by Hyde, Scuseria, and Ernzerhof Heyd et
al. (2003); Heyd and Scuseria (2004); Krukau et al. (2006); Paier et al.
(2006). The HSE06 scheme is based on the inclusion of fixed amount of Hartree-
Fock exchange. Briefly, the exchange-correlation energy of the HSE06
functional is given by Heyd et al. (2003); Heyd and Scuseria (2004); Krukau et
al. (2006); Paier et al. (2006),

$\displaystyle E_{xc}^{HSE}\left(\alpha,\omega\right)$ $\displaystyle=$
$\displaystyle\alpha
E_{x}^{HF,SR}\left(\omega\right)+\left(1-\alpha\right)E_{x}^{PBE,SR}\left(\omega\right)$
(2) $\displaystyle+$ $\displaystyle
E_{x}^{PBE,LR}\left(\omega\right)+E_{c}^{PBE}~{}.$

Here, the exchange-correlation energy $E_{xc}$ is range separated with the use
of a screening potential. The parameter $\alpha$ describes the fraction of the
Fock exchange while $\omega$ calibrates the range of the interaction. We have
used the conventional $\alpha=0.25$ and $\omega=0.11$ bohr-1 in computations
with HSE06 Krukau et al. (2006); Paier et al. (2006). It may be noted that
other variants of hybrid density funtionals Jana et al. (2018a); Jana and
Samal (2019); Jana et al. (2020a, b) have also been proposed.

In this work, we report the relative performance of HSE06, MGGAC, PBE, SCAN,
and TM functionals by comparing the computed estimates of the structural
parameters and properties of the Cu based chalcopyrite systems. In case of
other systems, the structural properties are calculated only using MGGAC, PBE,
SCAN, and TM functionals and the results are reported in supplementary
material Ghosh et al. (2021). Since, the main focus of the present work is the
study of Cu based chalcopyrites, we do not relax the structures of Ag, Be, Cd,
Mg, and Zn based chalcopyrites using HSE06 scheme, which is quite expensive
computationally. For these systems the structures are optimized and lattice
constants are computed using PBE (GGA) scheme.

In Table 1, we list the I-III-VI2 and II-IV-V2 group systems which are
considered in present work.

##  III structural properties

The chalcopyrite lattice structure adopts the $D_{2d}^{12}$ (No.122) space
group symmetry. The structure is a cation mutated super-structure of the cubic
zinc-blende structure ($T^{2}_{d}$), in which each of the two cations $(A,B)$
are tetrahedraly coordinated by four anions ($X$), and each anion is
coordinated by two $A$ and two type $B$ cations Jaffe and Zunger (1983). The
chalcopyrite structure can be described accurately by three structural
parameters: ($i$) lattice constant $a$, ($ii$) the tetragonal ratio
$\eta=c/2a$ $\neq$ 1, where $c$ is the lattice constant in the $z$-direction,
and ($iii$) the anion displacement parameter, $u$. In general, $A-X$ and $B-X$
bond lengths, denoted by $r_{A-X}$ and $r_{B-X}$, respectively, are not equal.
This unequal anion-cation bond length results in a tetragonal distortion vis-
a-vis an ideal zinc blende structure. The distortion parameter
$u=0.25+(r^{2}_{A-X}-r^{2}_{B-X})/a^{2}$, which describes the re-positioning
of the anions in the $x-y$ plane, sensitively influence the bulk electronic
bandgap of these materials Jaffe and Zunger (1983).

The ternary ABC2 [A = I; II = Cu, Ag; Be, Mg, Zn, Cd, B = III; IV = In, Ga,
Al; Si, Ge, Sn, and C = VI; V = S, Se, Te; P, As] chalcopyrite compounds of
composition I$-$III$-$VI2 or II$-$IV$-$V2 are iso-electronic analogous of the
II$-$VI or III$-$V binary semiconductors, respectively Shaposhnikov et al.
(2012). The crystal structure of these chalcopyrites is described by
$D^{12}_{2d}$ ($I42d$) space group symmetry, and can be regarded as a super-
lattice of the zinc-blende structure with small structural distortions Feng et
al. (2011). The lattice constants ($a$ and $c$), distortion parameter ($u$),
and tetragonal ratio ($\eta$) of Cu based chalcopyrites computed using
different XC schemes are listed in Table S1 of the supporting information
Ghosh et al. (2021). The trend seen in the results obtained from semilocal
functionals may be explained from the perspective of computed electronic
properties.

In general, the computed lattice constants of solids obtained from PBE-like
GGA XC functionals are reasonably accurate. However, the PBE-GGA computed
lattice constants are generally found slightly overestimated because of the
lack of slowly varying density correction, which is taken care of in its
solid-state version PBEsol Perdew et al. (2008); Terentjev et al. (2018).
Furthermore, the higher rung of functionals i.e., meta-GGAs are generally
better than PBE-GGA functional for estimating the lattice constants. For
instance, the SCAN meta-GGA performs reasonablly well in predicting the
lattice constants of different solids.

Figure 1: Relative error (%) in (a) in-plane lattice constants, (b) lattice
constants in $z$ direction, (c) distortion parameter ($u$), and (d) tetragonal
ratio ($\eta$) of the Cu based chalcopyrite systems reported in Table S1 of
the supporting information Ghosh et al. (2021). The levels along $x-$ axis is
same as the solids presented in Table S1 of the supporting information Ghosh
et al. (2021) and shown in the caption.

In Fig 1, we show the relative error (in %) in parameters $a$, $c$, $u$, and
$\eta$ of the Cu based chalcopyrites as obtained from different methods. As
can be seen, the HSE06 performs better than other methods. However, within
semilocal methods, the SCAN performs better than MGGAC and PBE for $a$ and
$c$, while MGGAC performs better than SCAN for $u$. Further, in most cases,
the MGGAC underestimates $a$ and $c$ parameters as compared to experimental
and SCAN values. It may also be noted that for the tetragonal ratio ($\eta$),
the MGGAC performs in a more balanced way than SCAN due to its tendency to
underestimate both $a$ and $c$. Physically, the improvements in $u$ obtained
using MGGAC as compared to those obtained using SCAN comes from bond lengths
$r_{A/B-X}$, which are also significantly improved by much better description
of the Cu $d$-states. It may also be seen in Fig 1 that the $u$ parameters
obtained from MGGAC are very close to those obtained using HSE06. This
indicates that the performance of MGGAC is quite close to that of HSE06 in
treating Cu-$d$ states. We will further discuss this point in more detail in
section IV.

##  IV electronic properties

The phenomenological treatment of $p$-$d$ interaction in chalcopyrite systems
depends on how good are the various approximation of XC for the $d$-electron
of Cu. For instance, the LDA (or GGA) based methods tend to delocalize
$d$-electrons which results in underestimation of $p-d$ hybridization Zhang et
al. (2011); Vidal et al. (2010).

Further, a significant shortcoming in standard DFT based methods is the
presence of inherent self-interaction error (SIE) Vidal et al. (2010); Kim et
al. (2016). In the case of chalcopyrites, this shortcoming results in shallow
$d$ states description for the cations, and in general, the absence of
derivative discontinuity in the KS bandgap which is defined as the difference
between eigenvalues of the CBM and VBM. As a consequence, the theoretical
bandgap is severely underestimated as compared to the experimental bandgap
(the difference between the ionization potential and electron affinity). The
absence of the LDA/GGA computed bandgaps in some of the chalcopyrites and
other material systems has been attributed to large ratio of SIE in exchange
functionals. Hybrid functionals eliminate some amount of SIE and improve the
performance and estimates of bandgaps Vidal et al. (2010); Kim et al. (2016).
The performance of the meta-GGA functionals which are implemented in gKS
scheme (such as SCAN and MGGAC), in providing estimates of the electronic
properties of chalcopyrites has not been reported. We discuss these in next
subsection.

To evaluate the relative performances of different functionals, we first
discuss the band structures as obtained from HSE06, MGGAC, PBE, and SCAN
functionals. Following this, the density of states (DOS) and charge density
will be analyzed and discussed. We have not included the performance of the TM
functional here as its performance is known to lie in between that of PBE and
SCAN functionals. The bandgap values computed using TM functional are shown in
Table S1 of the supporting information Ghosh et al. (2021). Overall, the
performance of TM functional is found to qualitatively similar to that of the
PBE functional.

In order to analyze the performance of different functionals, the band
structures, charge density, and DOS of CuInS2 are computed using the HSE06,
MGGAC, PBE, and SCAN functionals. In supporting information Ghosh et al.
(2021), we also present and analyze the electronic properties of CuInSe2.
These two systems are particularly chosen since their electronic band
structures, density of states (DOS), and charge densities are quite distinct
from one another. Further, both these systems are known to exhibit strong
interplay between structure and electronic properties Vidal et al. (2010).

###  IV.1  Band structures

Figure 2: Band structures of CuInS2 as obtained from different methods.

Figure 3: Experimental versus theoretical bandgaps of I-III-VI2 (upper panel)
and II-IV-V2 (lower panel) semiconductors as obtained from different methods.
The values within dotted lines indicate the bandgaps within the error of $\pm
0.5$ eV of the experimental values. The experimental bandgaps of I-III-VI2 are
taken from ref. Xiao et al. (2011) and II-IV-V2 from ref. Shaposhnikov et al.
(2012). Figure 4: Variation of bandgaps of CuInS2 and CuInSe2 as a function of
distortion parameter ($u$). Figure 5: Electronic charge density distribution
contours of CuInS2 in the plane contain the Cu, In, and S atoms according to
different methods. The logarithmic scale is used for the clear visualization
of the isosurfaces.

Figure 6: Density of states (DOS) for CuInS2 calculated from different
methods. Right panel shows the DOS in energy range
$-1~{}eV<(E-E_{F})<2.5~{}eV$. The Fermi energy (EF) is set to $0$ eV and is
same as EVBM.

The bandgaps of different Cu-based chalcopyrite systems as obtained from
different methods are shown in Table S1 of the supporting information Ghosh et
al. (2021). As can be seen, the bandgaps computed using hybrid HSE06 method,
are in close agreement with the experimental values. Further, among the
semilocal functionals, MGGAC performs in a significantly better way than
others, as it opens the bandgaps in CuInS2, CuInSe2, and CuInTe2. The bandgaps
computed using SCAN, TM and PBE are found to be severely underestimated and
thus these methods perform poorly.

Fig 2 shows the band structure of CuInS2 along high-symmetry directions in BZ.
As can be seen, the bandgap computed using MGGAC is improved significantly and
is quite close to that obtained using HSE06. The SCAN method though opens the
bandgap for CuInS2, its magnitude comes out to be very small. The use of PBE
results in zero bandgap.

It may be noted that the HSE06 functional has more often been used to study
chalcopyrites as compared to other global hybrid functionals such as PBE0,
because of the presence of screened Coulomb interaction, which reduces the
computational cost significantly.

As can be seen in Table S1 of the supporting information Ghosh et al. (2021),
excellent estimates of electronic properties are obtained from hybrid
functional HSE06 with its default value of $25\%$ fraction of Fock exchange.

Interestingly, among considered semilocal approximations, the MGGAC functional
performs quite closely to HSE06 for most semiconductors. (see supplemental
material). A comparative plot of theoretical and experimental bandgaps is
shown in Fig 3 which clearly indicates that bandgap estimates obtained using
MGGAC are more accurate than those obtained using other semilocal functionals.

Based on the comparision of the results presented in this work with those in
Ref. Braga et al. (2011); Shaposhnikov et al. (2012) for I-III-VI2 and II-
IV-V2 chalcopyrite semiconductors, the performances of different functionals
are twofold:

  * •

I-III-VI2 semiconductors: The direct bandgaps are found to be most severely
underestimated when computed using semilocal functionals like GGA, SCAN, and
TM. For Ag based chalcopyrites too, a similar trend is observed. A consistent
improvement in the bandgap magnitudes is observed, when computed using the
MGGAC functional. This is expected due to the improvement in the description
of $p-d$ hybridization discussed in the previous section. In the case of
CuAlTe2, the bandgap computed using MGGAC and HSE06 is found to be
overestimated whereas PBE-GGA estimate of the bandgap is closer to the
experiment. Further, for all the Te based systems, MGGAC overestimates the
bandgaps as compared to HSE06. In particular, the MGGAC overestimates the
bandgaps of CuAlTe2 and AgInTe2 by a large margin. Nevertheless, for other Te
based systems, the performance of MGGAC is reasonably good with bandgap
estimates closer to the experiments as compared to other semilocal methods.

  * •

II-IV-V2 semiconductors: The computed GGA bandgaps are found to be in
excellent agreement with the experiments Shaposhnikov et al. (2012) and
earlier reported theoretical results Shaposhnikov et al. (2012). The MGGAC
computed bandgaps are found to be comparable with HSE06 computed bandgaps.
However, both MGGAC and HSE06 overestimate the bandgaps in case of BeSiAs2,
BeSiP2, CdGeAs2, CdSnP2. In case of these semiconductors, the SCAN functional
performs well as compared to other semilocal (MGGAC, TM, GGA) methods (see
ref. Ghosh et al. (2021) for details). As can be seen in Fig 3, the MGGAC
computed bandgaps are overestimated by more than 0.5 eV in case of CdSnAs2,
CdGeAs2, ZnSnAs2, BeGeP2, BeSiAs2, and BeSiP2.

It may be noted that for chalcopyrite semiconductors containing Be and Cd, the
HSE06 bandgaps are comparable to the GW0 bandgaps as reported in Ref.
Shaposhnikov et al. (2012). Further, the semilocal functionals SCAN, TM, GGA
underestimate the bandgaps. However, MGGAC overestimates the bandgap for
BeSiAs2, BeSiP2, CdGeAs2, CdSnP2. In case of BeGeAs2 and BeSnP2, MGGAC
bandgaps are comparable to the experimental values and are in better agreement
than that for HSE06 functional.

For semiconductors containing Mg and Zn, all functionals estimate bandgaps
close to the experimental values except for some semiconductors containing Zn.

Overall, it may be concluded from the calculations and above discussions that
the bandgaps estimated by semilocal MGGAC are comparable to those estimated by
hybrid functional HSE06 to a great extent with a minimal deviation (see Fig
3).

To visualize the variation of the bandgap with the distortion parameter ($u$),
we plot in Fig. 4 the bandgaps of CuInS2 and CuInSe2 computed using different
functionals as a function of $u$. As can be seen, the bandgap increases almost
linearly with $u$ in range 0.2 $<$ $u$ $<$ 0.25. For both the chalcopyrite
systems, HSE06 bandgaps are improved over MGGAC bandgaps, which in turn are
improved over SCAN bandgaps. For CuInS2 and CuInSe2, the SCAN bandgap opens
for $u>0.2$ and $u>0.22$, respectively. As can be seen, the MGGAC and HSE06
bandgaps are finite for all values of $u$ (in range shown) for both the
chalcopyrites. The difference in the bandgaps obtained from different methods
is expected due to strong interplay between the structure parameter $u$ and
the hybridization of the $p-d$ orbitals Vidal et al. (2010). We will revisit
this point later in this paper.

In supporting information Ghosh et al. (2021), we also discuss and analysis
another chalcopyrite system CuInSe2. As can be seen in supporting information
Ghosh et al. (2021), the MGGAC and HSE06 open the bandgap of this system,
whereas it remains closed for PBE and SCAN functionals.

###  IV.2  Charge densities

The charge density contours for CuInS2 are shown in Fig.5, which reveals the
the presence of both ionic and covalent bonds. The mixed bonding is seen
irrespective of the XC functionals used. As can be seen, the electron density
is more enhanced towards the side of the S-atom than that of the In-atom,
which indicates that the In$-$S bond is more ionic. Further, the even
distribution of the electron density in the case of the Cu$-$S bond indicates
that the bonding is more covalent. The arrow labeled in the Cu$-$S bond (see
Fig. 5) indicates the isosurface value. The size of the arrow is larger in the
case of PBE and SCAN than the one for MGGAC and HSE06, which follows the
bandgap enlargement rule as well as the differences in the $u$ parameter as
the arrow size is related to the corresponding bond length and varying degrees
of $p-d$ hybridization for these chalcopyrites. From Table S1 of the
supporting information Ghosh et al. (2021), it can be seen that the calculated
$u$ parameter, which depends on the bond length of Cu$-$S, is less for PBE and
SCAN (0.218) than that for MGGAC (0.224) and HSE (0.226). The greater the
isosurface arrow, the lower the atomic distance is, which indicates greater
$p-d$ repulsion and thereby smaller bandgap. It is generally understood that
methods (within the DFT framework) that do not take into account the self-
energy correction, underestimate the bandgaps. However, in the case of MGGAC,
the presence of internal screened coulomb energy $U$ results in bandgap
estimates which are comparable to bandgaps obtained using hybrid functionals
as well as state-of-the-art many-body perturbation technique. We will revisit
and elaborate more on this point in a later section. The MGGAC computed
bandgaps also exhibit general trend which is similar to that seen for bandgaps
obtained using other functionals. For instance, the computed bandgaps of
semiconductors containing $P$-atoms are always larger than bandgaps of
semiconductors containing As-atoms.

###  IV.3  Density of states

The electronic structures of all chalcopyrite semiconductors are qualitatively
similar. In the following, we present and discuss the electronic structure of
CuInS2. The electronic structure of CuInSe2 is presented in the supporting
information Ghosh et al. (2021). The total density of states (DOS) and partial
DOS as computed with different XC functionals are shown in Fig 6 and Fig. S3
of supporting information Ghosh et al. (2021) for CuInS2 and CuInSe2,
respectively. As can be seen, the contribution from the valence Cu-$4s$ and
Cu-$3p$ states near the VBM is negligible as compared to Cu-$3d$ states. The
valence band region exhibits two main groups: up to around $-2$ eV from the
VBM, the region is mainly composed of $p$ and $d$ states with primary
contribution from Cu$-3d$ states. The deeper valence band in the energy range
-$2$ eV to $0$ eV results from strong hybridization of Cu$-3d$ and S$-3p$
states. The conduction band region is contributed by In$-5p$ and S$-3p$ states
along with a minor contribution of Cu$-3d$ states. The dispersion of the
valence and conduction bands are found to be almost the same for all semilocal
functionals (GGA, SCAN, and MGGAC) with an exception that the band width
(energy between the highest and lowest allowed levels) for MGGAC is wider than
that for PBE and SCAN.

To encapsulate the behavior of the VBM and CBM, we show the DOS plot for the
respective functionals in the right panel of Fig 6, in the energy range
$-1~{}eV<(E-E_{F})<2.5~{}eV$. The enhancement in the bandwidth is evident in
case of MGGAC as compared to that for SCAN and PBE semilocal functionals. In
case of MGGAC, the CBM shifts to the higher energy level due to slightly
enhanced hybridization between Cu$-3d$ and S$-3p$ states at VBM, resulting in
the opening of the bandgap. However, in case of PBE and SCAN, the $3d$ non-
bonding region lies a little bit of the higher side on the energy scale, which
results in the reduction of the bandgap as compared to that in case of MGGAC
and HSE06.

In the case of HSE06, the relative bandgap enhancement compared to that for
semilocal functionals happens because of the inclusion of the non-local HF
exchange, which reduces the repulsion between Cu$-3d$ and S$-4p$ states and
thereby results in the bandgap opening Kim et al. (2016). Importantly, the
bandgap enlargement in the case of HSE06 as compared to other semilocal
functionals (MGGAC, SCAN, TM, and PBE) may be attributed to ($i$) enlargement
of bandwidth and the orbital spacing which directly influences the bandgap and
($ii$) the shift of non-bonding $3d$ states downward which reduce the $p-d$
repulsion.

Nevertheless, the performance of the MGGAC within semilocal XC functionals is
quite encouraging and the same trend of results also follows for all I-III-VI2
semiconductors as the valence band maximum is influenced by the $4d$ states of
I-group atoms. On the other hand, in the case of II-IV-V2 semiconductors, the
bandgap character is not influenced by $d$ states of II-group atoms, as these
are highly localized and contribute to valence bands far from VBM Shaposhnikov
et al. (2012).

In supporting information Ghosh et al. (2021), we show the DOS and PDOS for
CuInSe2. In this case also, the opening of the bandgap can be explained
sililarly as discussed for CuInS2. The reduction in repulsion between Cu$-3d$
and Se$-4p$ states is responsible for the opening of bandgap. For MGGAC, this
repulsion reduces at the VBM and the CBM is shifted towards the higher energy.
This results in the opening of the bandgap more than that in case of SCAN.

##  V Formation enthalpies

Table 2:  Enthalpy formation energies (eV/f.u.) of CuInSe2 and AgInSe2 as obtained from different methods. PBE and HSE06 values are taken from ref. Kim et al. (2016). Numbers in the parentheses represent the percentage deviation. Solids | PBE | SCAN | TM | MGGAC | HSE06 | Expt. Berger et al. (1973)
---|---|---|---|---|---|---
CuInSe2 | -1.78 | -2.02 | -1.39 | -2.35 | -2.40 | -2.77
| [-35.6] | [-27.1] | [-49.5] | [-15.1] | [-13.4] |
AgInSe2 | -1.71 | -1.90 | -1.26 | -2.25 | -2.22 | –2.51
| [-31.8] | [-24.3] | [-49.5] | [-10.3] | [-11.4] |
Figure 7: Enthalpy formation energies (meV/atom) of Cu-Se binaries as obtained
from different methods

We now discuss the enthalpy formation energies of chalcopyrites and competing
binary phases of Cu-Se, computed using different meta-GGA functionals. In the
thermodynamical framework, the enthalpy formalism energy per atom for a
chalcopyrite system is defined as:

$\Delta H=[\epsilon_{system}-\sum_{i}n_{i}\epsilon_{i}]/m~{},$ (3)

where $\epsilon_{system}$ is the energy of the chalcopyrite system, $n_{i}$
and $\epsilon_{i}$ are the number and energy of the $i^{th}$ constituent in
the system, and $m$ is the total number of atoms in the system. All energies
are calculated in their stable solid form. A negative (positive) enthalpy
formation energy indicates that the system is stable (unstable).

The Table 2 shows the formation enthalpies of CuInSe2 and AgInSe2 computed
using different functionals. For these solids, the experimental formation
enthalpies have been reported in ref. Berger et al. (1973). The PBE and HSE06
results in Table 2 are taken from ref. Kim et al. (2016). As can be seen, the
PBE underestimates the formation energies of both the chalcopyrites by more
than $30\%$. The SCAN method improves the enthalpy estimates as compared to
those obtained using PBE. The SCAN values are underestimated by less than
$30\%$. Further, Table 2 shows that the formation enthalpy estimates are
improved significantly with the use of MGGAC functional, which performs as
accurately as hybrid HSE06 functional and with computational cost much lesser
than that of HSE06.

It may be noted that the PBE functional does not properly describe different
bondings such as metallic, covalent, and/or ionic in different solids
Stevanović et al. (2012); Jain et al. (2011); Hautier et al. (2012). The
bonding nature is much better described by meta-GGASun et al. (2013)
functionals, which is reflected in the performance of SCAN and MGGAC. This is
because the meta-GGA functionals are generally better than GGAs in describing
the covalent, metallic, and non-covalent interactions Sun et al. (2013); Della
Sala et al. (2016).

Incidentally, the TM functional fails drastically in providing reasonable
estimates of formation energies as it exhibits error more than PBE. This
failure of the TM may be attributed to its order-of-limit problems, which is
important for estimating the formation of enthalpies Patra et al. (2020a). In
case of the HSE06, the good performance is due to reasonably correct
desprition of the $d$-states of Cu and Ag atoms.

The formation enthalpy of CuInSe2 computed using MGGAC and HSE06 is in close
agreement with the experiments, which is expected due to improved estimates of
energy eigenvalues, In-4$d$ states, band structure and DOS in it (see
supporting material Ghosh et al. (2021)).

Next, we assess the functionals performance for the several competing
candidate binary phases of Cu-Se. The formation enthalpies per atom for
different Cu$-$Se phases are plotted in Fig. 7. As expected, PBE underbinds
the formation enthalpies, with a positive $\Delta H$ for Cu2Se compound. The
HSE06 estimated formation energies are found to be in closest agreement with
the experimental values. The SCAN and MGGAC perform closely, albeit SCAN
performance slightly better than MGGAC in all cases. This may be partially
attributed to the metallic character of the CuSe, CuSe2, and Cu3Se2, which is
better described by the SCAN functional as it takes into account the slowly
varying second/fourth order terms in the gradient expansion of the exchange.
Furthermore, the SCAN correlation is free from spurious one-electron self-
interaction which tends to improve the formation enthalpy estimates.
Nevertheless, both SCAN and MGGAC formation energies are negative (including
semiconductor Cu2Se) and therefore are much better estimated as compared to
those by PBE-GGA. Fig. 7 also shows that the performance of the TM functional
is quite poor for all the binary phases of Cu-Se. As can be seen, the TM
functional incorrectly predicts CuSe2 and Cu3Se2 as unstable. This
significantly poor performance of the TM functional in providing formation
enthalpy estimates may be attributed to its order-of-limit problem Patra et
al. (2020a).

##  VI Relation between performance and mathematical features of functionals

Figure 8: The exchange enhancement factor of SCAN and MGGAC as a function of
$\alpha$ for different values of reduced density gradient $s$. The MGGAC
exchange is independent of $s$.

Next, we discuss the plausible reasons behind improvement in performance of
MGGAC functional over GGA or SCAN methods. To analyze the results, let us
consider the behavior of the exchange enhancement factor ($F_{x}$) of SCAN and
MGGAC functionals, the gKS potential of the meta-GGA functionals, and its
relation to the derivative discontinuity ($\Delta_{x}$), which is connected
with the localized $d$\- or $f$ -states, often encountered in the chalcopyrite
systems. In Fig. 8, we plot the $F_{x}$ as a function of meta-GGA ingredient
$\alpha$ for different values of reduced density gradient $s$. The MGGAC
$F_{x}$ does not depend on $s$ and is much smoother than that for SCAN and its
slope $\partial F_{x}/\partial\alpha$ is more negative in the energetically
relevant regions of solids ($0<s\leq 1$) resulting in inclusion of more
$\Delta_{x}(\propto\partial F_{x}/\partial\alpha$). In fact the negative slope
$\partial F_{x}/\partial\alpha<0$ is also connected to the ultranonlocality
effect which results in polarization effect in solids Aschebrock and Kümmel
(2019). Furthermore, the $\alpha$ is also interpreted as the measure of the
well-known electron localization factor via the exchange energy, similar to
the EXX Aschebrock and Kümmel (2019). Importantly, because of the more
enhanced $\alpha$ dependent and $\partial F_{x}/\partial\alpha<0$, some amount
of inherent on-site Coulomb interaction ($U$) is included in the orbital
energies and eigenvalues corresponding to MGGAC functional. The on-site
Coulomb interaction ($U$) is important to treat the highly delocalized band
states of rare-earth elements and strongly localized late transition metal
elements. It may be noted that the SCAN functional also includes some amount
of inherent $U$ Peng and Perdew (2017) which results in improved performances
as compared to PBE+U and SCAN-no-U Sai Gautam and Carter (2018); Varignon et
al. (2019); Kirchner-Hall et al. (2021); Wexler et al. (2020); Long et al.
(2020). However, because of the more realistic and enhanced $\partial
F_{x}/\partial\alpha<0$, the MGGAC is a slightly better in treating the $d$\-
and $f$\- orbitals and its results for chalcopyrite systems are often close to
the HSE06, where the $25\%$ HF exact exchange is used to treat the strongly
localized $d$\- bands. It may be noted that along with the localized $d$\-
states, the $s$\- and $p$-states are also well described by the MGGAC. From
the structure of the DOS plot (see Fig 6 and Fig. S3 of supporting information
Ghosh et al. (2021)) obtained using the MGGAC functional, one can readily
observe that it treats the non-bonded Cu$-3d$ and hybridized $3d-4p$ (of Cu
and Se) states in a bit better way than PBE and SCAN functionals. This results
in the reduction of strong localization effect of $3d$ states, which in turn
opens a sizable bandgap between occupied and unoccupied sub-bands.

In general, the success and overall improvement of MGGAC functional vis-a-vis
LDA, GGA, and SCAN functioanls are retained for other chalcopyrite systems
discussed in this article.

##  VII Concluding remarks

The comparative assessment of the structural properties, electronic
properties, and formation energies of Cu-based multinary semiconductors is
performed using GGA, meta-GGA, and hybrid density functionals within the
framework of density functional theory. In particular, we assess the
performances of SCAN, TM, and MGGAC meta-GGA functionals, which are very
recent and reasonably successful exchange-correlation (XC) functionals in
predicting various solid-state properties. To assess the structural
properties, the tetragonal distortion ($\eta$) and/or anion displacement ($u$)
are first computed from different functionals. The PBE-GGA functional is found
to be inadequate to estimate accurately the electronic structures and
associated properties of Cu-based chalcopyrite systems due to its inability to
describe the intrinsic localization and/or strong correlation of the $d$
electrons.

Our results for PBE and HSE06 are in agreement with earlier reported studies,
wherein it was shown that the HSE06 generally improves over PBE because of
less delocalization error, which in turn improves the bandgap, and density of
state (DOS) estimates.

The estimates of various quantities computed using MGGAC and SCAN are found to
be significantly improved as compared to those obtained using GGA. Further,
the performances of MGGAC is found to be encouraging among meta-GGA
functionals and quite close to that of the HSE06 in providing the estimates of
$\eta$, $u$, band structures, and enthalpy formation energies of Cu-based
multinary semiconductors. The reasonably improved and better performance of
MGGAC is attributed to its a better description of strongly localized $d$
electrons in these systems.

In addition, the performance of the aforementioned functionals is also
explored for several competing candidate binary phases of Cu-Se. For these
phases also, the MGGAC is found to be quite accurate. However, the SCAN
functional performs slightly better than MGGAC for binary phases. The
performance of TM functional is found to be poor in comparison to other meta-
GGA functionals possibly due to its order of limit problem. Overall, our
results strongly suggest that MGGAC (meta-GGA) functional can be highly useful
for studying Cu-based multinary semiconductors as well as other systems with
$d$-electrons, due to its good performance and low computational cost.

##  VIII Acknowledgemets

A.G. would like to thank INSPIRE fellowship, DST, India for financial support.
S.J. would like to thank NISER, Bhubaneswar for partial financial support.
Calculations are performed high performance computing (HPC) clusters of IITH,
Hyderabad. Part of calculations are also performed in the KALINGA and NISERDFT
high performance computing (HPC) clusters of NISER, Bhubaneswar.

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