# Optoelectronic and Transport Properties of Vacancy Ordered Double Perovskite
Halides: A First-principles Study

Supriti Ghorui1, Jiban Kangsabanik1,2, M. Aslam1 and Aftab Alam1,∗ 1Department
of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
2Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby,
Denmark<EMAIL_ADDRESS>
###### Abstract

In the search for stable lead (Pb) free perovskites, Vacancy ordered double
perovskite (VODP), A2BX6 has emerged as a promising class of materials for
solar harvesting owing to their nontoxicity, better stability, and unique
optoelectronic properties. Recently, this class has been explored for a wide
range of applications such as photovoltaics, photodetectors, photocatalysis,
and light-emitting diodes. Here, we present the stability and the key physical
attributes of few selected compounds in a systematic manner using state-of-
the-art first-principle calculations. A careful structural and stability
analysis via simulating convex hull and compositional phase diagrams for
different structural prototypes discloses 14 stable and 1 metastable compounds
in this class. The electronic structure calculations using hybrid functional
reveals six compounds to acquire band gap in the ideal visible region. These
six compounds, namely Cs2SnI6, Cs2PdI6, Cs2TeI6, Cs2TiI6, Cs2PtI6, and
Cs2PdBr6, show high optical absorption ($\approx$ 105 cm -1) giving rise to
high spectroscopic limited maximum efficiency, SLME (15-23%) in the thin-film
thickness range. Close inspection of transport properties reveals polar
optical phonon scattering to be the dominant mechanism limiting the overall
mobility. Further analysis of the polaron excitations discloses the
possibility of large polaron formation at low to moderate defect
concentrations. At high defect concentrations, ionized impurity scattering
takes over. This suggests that, a simulation based guided control of defect
concentrations during synthesis can yield a desired candidate for promissing
device application. Additionally, few selected compounds show moderate to high
electron mobility values ($\sim$13-63 cm2V-1 s-1) at room temperature.
Overall, the present study paves an important path to help design VODP as Pb-
free potential candidates for future optoelectronic applications.

## I. Introduction

Lead halide perovskites(LHP) have reignited immense research interest in the
Photovoltaics (PV) community due to their remarkable power conversion
efficiency(PCE) of 25.6% jeong2021pseudo  (till date) and affordable device
processability. The rapid rise in PCE (3.8% to 25.6%) in a short period of
time (2009-2021) is attributed to its high absorption coefficient, high charge
carrier mobility, defect tolerance, and cost-effective flexible synthesis.
Because of their suitable optoelectronic properties, they have also been
explored as photodetectors (PD)li2019enhanced ; huang2021precursor ,
photocatalysts(PC) zhang2019stable ; yin2021controlling , and light emitting
diodes (LED) xu2019rational ; cui2021efficient . Yet, there remains two major
challenges in their large scale scalability: (1) Lead (Pb) toxicity and (2)
stability in the ambient environment. At present, major research efforts at
laboratory scale have been devoted to overcome these issues without losing
their original PV performance. lin2020piperidinium ;
stolterfoht2018visualization ; giustino2016toward ; shao2018highly ;
kangsabanik2020optoelectronic  This has led to a detailed exploration of the
diverse chemical space of halides perovskites (ABX3)stoumpos2013semiconducting
and their derivatives.zhao2017design ; xia2020unveiling ; cai2017computational
Among these perovskite derivatives, three major stoichiometric classes have
garnered immense research interest. One of the classes namely double
perovskites (DP) with stoichiometry A2BB′X6 is mainly generated via
transmutation of a combination of trivalent and monovalent elements at
B-sites.volonakis2016lead  For example, Cs2BiAgBr6,volonakis2016lead ;
slavney2016bismuth ; kangsabanik2018double  Cs2InAgCl6,volonakis2017cs2inagcl6
etc. belong to DP class which have been extensively explored for various
optoelectronic applications. Similarly, A3B2X9 (e.g. Cs3Bi2I9bai2018lead ,
Cs3Sb2I9singh2018photovoltaic  etc.) and A2BX6 (e.g. Cs2SnI6qiu2017unstable ,
Cs2TiI6ju2018earth  etc.) structures are constructed by replacing with
trivalent and tetravalent atoms respectively and leaving a vacant B-site.
Here, A2BX6 is also called vacancy ordered double perovskite where corner
shared alternate BX6 octahedras are removed along all three directions from
the unit cell as shown in Figure 1(a).

In the past few years, vacancy-ordered double perovskite family (A2BX6) has
gradually drawn ample attention in a wide range of optoelectronic applications
owing to their better environmental durability, tunable optical and electronic
properties. For example, Cs2SnI6 has been studied as a potential candidate in
PVlee2017solution , LED, PDshao2020high ; huang2021precursor , and PC
applications due to its direct band gap nature in the visible range( 1.1-1.62
eV), a high absorption coefficient ($\approx$105 cm-1)qiu2017unstable , a low
to high carrier mobility ($\approx$2-510 cm2 V-1 s-1)lee2014air ;
maughan2016defect ; saparov2016thin ; lopez2019optical ; guo2017two . The wide
range of measured mobilities of Cs2SnI6 can be attributed to variations
resulting from different synthesis and characterization methodologies.
Additionally, significant discrepancies have been observed between theoretical
and experimental results regarding the transport properties of this
material.maughan2018anharmonicity ; maughan2016defect ; cai2017computational ;
bhumla2022vacancy  The intrinsic limitations to mobility in Cs2SnI6 are still
not fully understood, and the underlying scattering mechanisms governing
carrier transport remain elusive. Therefore, a comprehensive and systematic
study encompassing both theoretical and experimental investigations is highly
desired to unravel the mobility ambiguity in Cs2SnI6 and shed light on its
transport characteristics. As of now, this compound exhibits a PCE of only
2.1%.lee2017solution  In contrast, substitutional alloying in Cs2SnCl6 yields
high photoluminescence quantum yield (PLQY) of 95.4% making it promising for
further exploration in LED applications.tan2020lead  Despite considerable
investigation into its structural, electronic, and optical properties, the
elucidation of charge-carrier dynamics in Cs2SnI6 still poses challenges that
hinder the optimization of conversion efficiencies.

Similarly, Cs2TiBr6, Cs2TeI6, and Cs2PtI6 are also studied experimentally for
PV absorbers with their band gaps in the ideal visible range: 1.8, 1.5, and
1.4 eV respectively, along with high absorption coefficients ($\sim$105
cm-1).vazquez2020vacancy ; schwartz2020air  Here, device efficiency for
Cs2TiBr6 as PV absorber is reported to be 3.3%.chen2018cesium  Indirect band
gap and material instability are reported to be responsible for poor PCE in
this case. In another report, PV device with Cs2PtI6 shows PCE of 13.88%,
which is a remarkable improvement on the reported efficiencies among all the
materials belonging to this class till date.schwartz2020air  Contribution of
larger carrier lifetimes along with direct band gap in ideal visible range and
robust stability help Cs2PtI6 to attain the high PCE. There are reports of
synthesizing Pdsakai2017solution , and Zrabfalterer2020colloidal  based
nanomaterials experimentally but not much has been explored in the direction
of optoelectronics. These background clearly indicates that A2BX6 class is
extremely interesting and fertile from the application perspective, yet a
detailed systematic study on their optoelectronic, carrier transport and
phonon properties connecting these observations is lacking. Moreover, it is
also noticed that substitutional alloying/doping of pure material is an
important strategy to improve optoelectronic properties, which again
necessitates an in-depth understanding of the pure materials themselves.

In this communication, we present a detailed and systematic study on the A2BX6
class of materials by using highly accurate ab-initio calculations. First, we
have performed a thorough stability analysis which includes choice of
different structural prototypes, thermodynamical stability via chemical
potential phase diagram and convex hull analysis, and lattice dynamics
simulation. Next, we have studied the electronic properties of the stable
compounds using hybrid (HSE06) functional, which is known to predict
reasonably accurate electronic structure information. Optical absroption and
PV device parameters are calculated on the promising set of systems showing
band gaps in the ideal visible region. Finally, carrier transport properties
of these compounds are studied by considering the important scattering
mechanisms. The importance of electron-phonon interactions, calculated within
the temperature-dependent Feynman polaron model, is also discussed in some
detail. We believe that such an in-depth study not only provides a solid
physical basis on these class of semiconductors but will also be immensely
beneficial for researchers working on their device application in the field of
PV, LED, PD, and PC.

Figure 1: (a) Crystal structure of A2BX6 compounds in Fm-3m space group. (b)
Convex energy hull (Ehull) for Cs2BX6 (B= Pd, Pt, Ti, Hf, Zr, Ge, Te, Se, Sn;
X=I, Br). Red circle denotes the top of the bar. Compositional chemical
potential phase diagram of (c) Cs2TiI6, (d) Cs2PdBr6, (e) Cs2PtI6, (f)
Cs2SnI6, and (g) Cs2TeI6 with respect to competitive secondary phases. The
green shaded regions show the stable regions of the corresponding materials.

## II. Structural Properties and Stability

Vacancy ordered double perovskite, A2BX6 is a class of compounds where
alternate BX6 octahedra are removed from the ABX3 unit cell as shown in Figure
1(a). In other words, 50% B cations are missing compared to the closed-packed
A2BB′X6 perovskite structure. Here, A possesses +1 oxidation state, B has +4
oxidation state and X is halide anion with -1 oxidation state. In general, for
perovskites, different crystal structures are possible depending on the ionic
radii of the constituent elements. These structures are roughly dictated by
few important geometrical factors as defined below,

  * •

Goldschmidt’s tolerance factor:
$t=\left(\text{r}_{\text{A}}+\text{r}_{\text{X}}\right)/\sqrt{2}\left(\text{r}_{\text{B}}+\text{r}_{\text{X}}\right)$

  * •

Octahedral factor : $\mu=\text{r}_{\text{B}}/\text{r}_{\text{x}}$

  * •

Radius ratio :
$\text{r}_{\text{A}}/\left(\text{D}_{\text{XX}}-\text{r}_{\text{X}}\right)$

In the above expressions, $\text{r}_{\text{A}}$, $\text{r}_{\text{B}}$,
$\text{r}_{\text{X}}$ and $\text{D}_{\text{XX}}$ are the empirical ionic radii
of the constituent elements A, B, X and the nearest neighbour X-X bond length,
respectively in the A2BX6 structure. All the calculated parameters are
tabulated in Table S1 of the supplementary information (SI).supplement  The
calculated Goldschmidt’s tolerance factor predicts formation of cubic
structures, which is also consistent with our stability analysis (discussed
later) and experimental observations for few of the compounds reported in the
literature.ju2018earth ; zhou2018all ; schwartz2020air ; maughan2016defect ;
saparov2016thin ; sakai2017solution

In this work, we have investigated the following A2BX6 compounds: A= Cs; B=
Ge, Te, Se, Sn, Pd, Pt, Ti, Zr, Hf; X=I, Br. For each compound, we have
considered seven most common structural prototypes (as reported in
International Crystal Structure Database (ICSD))bergerhoff1983inorganic ; icsd
for A2BX6 class of compounds. Space group of these seven structures are Fm-3m
(cubic), I4$/$m (tetragonal), I4$/$mmm (tetragonal), P-3m1 (hexagonal), Pnma
(orthorhombic), P4$/$mnc (monoclinic), and P121$/$c1 (monoclinic). These
crystal structures are shown in Fig. S1 of SI.supplement  Most of these
structures are very similar in symmetry and differ in energy only within a few
meV (3-4 meV). Post structural optimization, the lowest energy structure for
most of the above set of compounds turns out to be cubic (Fm-3m). It has been
observed experimentally that several Cs based iodide and bromide compounds
indeed crystallize in the cubic space group.ju2018earth ; zhou2018all ;
schwartz2020air ; maughan2016defect ; saparov2016thin ; sakai2017solution

To further assess the chemical stability, we have calculated the convex hull
energies (E${}_{\text{hull}}$) of these compounds with respect to possible
secondary phases available in ICSD, open quantum materials database
(OQMD)kirklin2015open ; saal2013materials  and materials project (MP)
databaseWOS:000332272300003 . As evident from Fig. 1(b), most of the compounds
lie on the convex hull i.e. E${}_{\text{{hull}}}$ = 0, except Cs2GeI6,
Cs2SeI6, Cs2PdI6, and Cs2GeBr6, confirming the stability of the former. For
the remaining four compounds E${}_{\text{hull}}$ ranges between 5-60 meV/atom,
indicating likelihood of chemical (meta/in)stability.

Next, in order to explore the most probable secondary phases during synthesis
of Cs2BX6 materials, we have calculated the compositional phase diagrams
(chemical potential) for materials on the convex hull. More details about the
phase diagram calculations/analysis are given in the Sec. S1(A) of
SI.supplement  Figure 1(c-g) shows the phase diagrams for those materials
which can be potential for optoelectronic applications (based on their band
gap, optical and transport properties, as discussed later). The phase diagrams
for remaining compounds are displayed in Figure S2 of SI.supplement  The green
shaded portion shows the stability regions of these materials. The extent of
the stability region directly correlates with the ease/difficulty of
experimental synthesis.The theoretically optimized lattice parameters and bond
lenghts (B-X) of all the stable compounds in their cubic structure are
displayed in Table S2 of SI.supplement

Further, we have checked the dynamical stability of these compounds by
calculating phonon dispersions as shown in Figure S3 of SI.supplement  The
absence of any visible imaginary phonon modes indicates dynamical stability of
these compounds. For Cs2SnI6 and Cs2TiI6, one can observe small negative
phonon frequencies the magnitude of which decreases with increasing supercell
size. This is because the later captures the effect of higher order inter-
atomic force constants more accurately. This is also evident in previously
reported phonon dispersion for Cs2SnI6.bhui2022intrinsically ;
jong2019anharmonic  Nevertheless, these compounds are already experimentally
synthesized, and hence naturally stable.maughan2016defect ; liga2021colloidal

Following stability analysis, we have further studied the electronic
structures of 15 (14 stable and 1 metastable) compounds in the next section.

Figure 2: Band structures and the square of dipole transition matrix elements
(p2) between VBM and CBM, for (a) Cs2TiI6, (c) Cs2PdBr6(e) Cs2PtI6,
(g)Cs2SnI6, and (i) Cs2TeI6 respectively. (b), (d), (f), (h) and (j) show the
projected density of states (PDOS) for the same set of compounds respectively.
All the calculations are done using PBE functional including spin-orbit
coupling (soc) effect while band gap is scissor shifted to HSE+soc calculated
values. In band structure plots, VBM and CBM are indicated via green and red
circles respectively.

## III. Electronic structure

Band structure calculations for all the compounds are initially performed
using Perdew-Burke-Ernzerhof (PBE) exchange-correlation
functionalperdew1996generalized . As PBE functional is well-known to
underestimate the band gap, we also employ hybrid Heyd-Scuseria-Ernzerhof
(HSE06)krukau2006influence  functional which gives more accurate estimate of
band gap in comparison to experiment. Spin-orbit coupling (soc) effect in
included in all the calculations. Band structures for four potential compounds
calculated using PBE+soc functional (band gaps are scissor shifted to HSE+soc
values) are shown in Figure 2. In A2BX6 class of compounds, the topology of
band structure calculated using HSE+soc functional is very similar to that
calculated using PBE+soc functional except the enlargement of band gap in the
former (See Figure S4 of SI for few representative cases).

Figure 2 also shows the optical transition probability, (square of dipole
transition matrix elements, p2), and total/orbital projected density of states
(PDOS) of Cs2TiI6, Cs2PdBr6, Cs2PtI6, Cs2SnI6, and Cs2TeI6 respectively. The
HSE06+soc band gap values for the respective compounds are provided in Table
1. The band structure, PDOS and respective band gap values for other compounds
are provided in Figure S5-S7 and Table S3 and S4 of SI.supplement  In Fm-3m
phase, the estimated band gap values lie within 0.72 eV to 4.31 eV for
different compounds. Optical transitions at the fundamental direct gaps are
dipole forbidden for all the compounds, as confirmed by the calculated optical
transition probability (p2). Here the presence of inversion symmetry plays the
key role to induce parity forbidden transitions for these compounds,
effectively increasing the optical band gap.

In the present study, we considered 9 different elements at the B site,
belonging to 4 distinct groups in the periodic table. Despite all elements
having a +4 oxidation state, their valence electron orbital configurations
differ, resulting in distinct electronic structures, including variations in
band structure and band gap types among the compounds. In the following, we
shall discuss the electronic structure of representative compounds from each
group and compare them with the electronic structures of other compounds
within the same group, including different halides.

For Cs2TiI6, band gap is indirect in nature with conduction band minimum (CBM)
at X and valence band maximum (VBM) at $\Gamma$ . But the direct band gap at
$\Gamma$ is very close to indirect band gap value ($\sim$50 meV) (Table 1).
From the orbital projected density of states (PDOS), we observe that CBM is
comprised of Ti-d and I-p i.e. B-d and X-p (see Figure 2)(a,b)). The
electronic band gap value is 1.77 eV which is overestimated by 0.75 eV with
respect to experimental value (1.02 eV)ju2018earth . The calculated optical
band gap lies within 100 meV from the fundamental direct gap. Apart from that,
the large difference between the calculated electronic band gap and optically
measured experimental band gap can be attributed to the excitonic effect (not
taken into account here) and the defects present in the measured sample, as
discussed by B.Cucco et.al.cucco2021electronic . All the electronic structure
information for the rest of the compounds can be found in Figure S5-S7 and
Table S3 and S4 of SI.supplement  It is clearly evident that the band gap
increases from Ti $\rightarrow$ Zr $\rightarrow$ Hf and also with I
$\rightarrow$ Br. In this group, only Cs2TiI6 shows band gap in the ideal
visible region.

Cs2PdI6 shows indirect band gap in both the space groups with CBM at X and VBM
at $\Gamma$ (see Fig. S5 of SI). The optically allowed direct band gap (0.88
eV) is very close to the indirect band gap values (0.72 eV) (shown in Table 1
). Experimentally, Cs2PdI6 nanocrystalszhou2018all  are synthesized and and a
band gap of 0.69 eV is reported. The reason behind the overestimation might be
similar to what is explained for the case of Cs2TiI6. In this case, the CBM is
comprised of Pd-d, I-p orbitals while VBM is composed of only I-p orbital (see
Fig. S5 of SI). Like Cs2PdI6, Cs2PtI6, Cs2PtBr6, and Cs2PdBr6 show similar
orbitals contribution at both CBM and VBM giving rise to indirect nature of
band gap. Their band gap values along with the formation energetics and
different between direct and indirect band gaps are presented in SI (see
Tables S3 and S4 and Fig. S5 and Fig. 2(c,e). For Cs2PtI6 and Cs2PdBr6, the
calculated band gap is close to experimentally reported values of Cs2PtI6
powder hamdan2020cs2pti6 ; yang2020novel  and Cs2PdBr6 nanocrystals
zhou2018all ; sakai2017solution  respectively. Here, we observe an increase in
band gap going from Pd $\rightarrow$ Pt and also from I $\rightarrow$ Br. In
this case, Cs2PdI6, Cs2PtI6, and Cs2PdBr6 compounds show band gaps within the
ideal visible range.

The band structure analysis of Cs2TeI6 reveals that it has indirect band gap
with a value of 1.85 eV, consistent with the study by Maughan
et.al.maughan2016defect . From PDOS analysis, we observe that the CBM is
comprised of Te-p and I-p orbitals whereas VBM is made up of I-p orbital (see
Figure 2(i) and (j)). The calculated electronic band gap value of 1.85 eV is
0.26 eV higher than the experimentally reported value maughan2016defect . For
Cs2TeBr6, the band gap nature and orbital contribution at both CBM and VBM is
similar to that of Cs2TeI6. The related electronic properties can be found in
SI (see Table. S4 and Fig. S6 ). All the electronic structure information for
Cs2SeBr6 can be found in SI (see Table S4 and Figure S6 (c,d) ), which shows
similar orbital characteristics.

For Cs2SnI6, the calculated band gap value is 0.85 eV which is direct in
nature with band edges (at $\Gamma$) in agreement with the values reported by
Maughan et.al.maughan2016defect  This is 0.38 eV higher than the
experimentally reported value maughan2016defect . From orbital analysis, we
observe that the CBM is made up of Sn-s and I-p orbitals, and VBM is comprised
of I-p orbital (see Fig. 2(g,h)). For Cs2SnBr6, the band gap nature and
orbital contribution at both CBM and VBM remains similar to that of Cs2SnI6.
The related electronic properties can be found in SI (see Table S4 and Fig.
S6(a,b)).

Figure 3:  (a) Absorption coefficients vs incident photon energy and (b)
spectroscopic limited maximum efficiency (SLME) (298 K) vs. film thickness for
Cs2TiI6, Cs2PdBr6, Cs2PtI6, Cs2SnI6, Cs2TeI6, and Cs2PdI6 compounds.

To summarize the electronic properties, one should note that the calculated
electronic band gaps are always overestimated as compared to the
experimentally reported optical band gap which is a well-known
fact.cucco2021electronic  Contrary to previous reports, the optical band gaps
are close to the lowest direct band gaps, confirmed by our calculation of
optical transition probability. We believe that the most probable reasons for
the theoretical overestimation of band gaps can be attributed to the excitonic
effects (not included in the present calculations) and defects present in the
experimental samples, as discussed by B.Cucco et.al.cucco2021electronic .

From the electronic structure analysis, we notice that the band gaps of
Cs2TeI6, Cs2SnI6, Cs2PdI6 (Pnma), Cs2PtI6, Cs2TiI6, and Cs2PdBr6 lie in the
ideal visible range for photovoltaic application. Therefore, we shall now
focus on the optical properties of these six compounds along with the well-
known descriptor ‘Spectroscopic limited maximum efficiency’ (SLME) (as
proposed by Yu et.alyu2012identification ) to better understand their
potential as solar absorber.

Compound | $E_{g}^{(expt)}$ |  $E_{g}$(HSE+soc) |  $\Delta$$E_{g}^{da}$ |  JSC |  Jmax |  VOC |  Vmax | FF | SLME
---|---|---|---|---|---|---|---|---|---
| (eV) | (eV) | (meV) |  (mA cm-2) |  (mA cm-2) | (V) | (V) |  |  ($\eta\%$)
Cs2TiI6 |  1.02ju2018earth  | 1.77 (ID) | 72 | 16.64 | 16.34 | 1.50 | 1.39 | 0.91 | 22.78
Cs2PdBr6 |  1.6sakai2017solution , 1.69 zhou2018all  | 1.61 (ID) | 110 | 17.62 | 17.26 | 1.35 | 1.25 | 0.91 | 21.63
Cs2PtI6 |  1.25hamdan2020cs2pti6 , 1.37 yang2020novel , 1.4 schwartz2020air  | 1.31 (ID) | 149 | 22.23 | 21.66 | 1.08 | 0.98 | 0.89 | 21.26
Cs2SnI6 |  1.25 maughan2016defect , 1.62 saparov2016thin  | 0.87 (D) | 41 | 32.58 | 31.33 | 0.72 | 0.64 | 0.85 | 20.07
Cs2TeI6 |  1.59 maughan2016defect  | 1.85 (ID) | 190 | 12.95 | 12.72 | 1.55 | 1.45 | 0.92 | 18.44
Cs2PdI6 |  1.41 zhou2018all  | 0.72 (ID) | 166 | 43.20 | 40.94 | 0.54 | 0.46 | 0.81 | 18.97
Table 1: Simulated band gap ($E_{g}$), difference between electronic and
optically allowed direct band gap ($\Delta$$E_{g}^{da}$), short-circuit
current density (JSC ), open-circuit voltage (VOC ), current density (Jmax )
and voltage (Vmax ) at maximum power, spectroscopic limited maximum efficiency
(SLME), and fill factor (FF) for Cs2TiI6, Cs2PdBr6 Cs2PtI6 Cs2SnI6, Cs2TeI6,
and Cs2PdI6 compounds. ID and D indicates indirect and direct nature of band
gaps respectively. All the device-related parameters are shown for 500 nm
thickness at 298K. Experimental band gaps ($E_{g}^{(expt)}$) are also listed
for comparison.

## IV. Optical Properties

Figure 3(a) shows the absorption coefficients for the above mentioned six
promising compounds. All these compounds can act as potential solar absorber
as their absorption coefficients are as high as 104-105 cm-1 in the visible
range. The optical absorption is contributed by two factors: (1) optical joint
density of states (JDOS) and (2) optical transition strength. As we can see in
Figure S5(a-d),supplement  the square of dipole transition matrix elements
(aka transition strength) for Cs2PdI6 is pretty high contributing to better
optical absorption. This can be attributed to Cs-p, I-p to Pd-d transition.
Apart from that, the JDOS is also likely to be high as the bands near CBM and
VBM show flat nature. They are comprised of ‘p’ and ‘d’ orbitals, showing more
localized nature as compared to the other compounds. In addition to Cs2PdI6,
Cs2SnI6, and Cs2TeI6 also show similar absorption coefficient spectrum. The
absorption coefficient is directly related to the frequency dependent
dielectric function of a semiconductor via following equation

$\displaystyle\alpha(E)=\frac{2\omega}{c}\sqrt{\frac{\sqrt{\epsilon^{2}_{re}+\epsilon^{2}_{im}}-\epsilon_{re}}{2}}$
(1)

where $E$ is the incident photon energy, $\omega$ is the angular frequency
related to $E$ via $E=\hbar\omega$, $c$ is the velocity of light,
$\epsilon_{re}$ and $\epsilon_{im}$ are the real and imaginary part of the
dielectric function respectively. Figure 3(b) shows the thin-film thickness
dependance of spectroscopic limited maximum efficiency (SLME), which turn out
to be ¿15% for all six compounds. Interestingly, we can see a higher SLME for
the Cs2PdI6, Cs2PtI6 and Cs2TiI6 compounds as compared to $Cs_{2}SnI_{6}$.
Such increased SLME is essentially attributed to increased high absorption
spectra due to I-p to Pd/Pt/Ti-d orbital transition as well as suitable band
gaps. In Table 1, we present the simulated device parameter values for the six
compounds: short-circuit photo-current density (Jsc), open circuit voltage
(Voc), fill factor ($FF$), maximum current density (Jmax), and maximum voltage
(Vmax) obtained from the SLME calculation. The detailed description and method
of calculation of these parameters can be found in the SI.supplement  As
expected, materials with higher band gaps exhibit higher Voc values, while
materials with lower band gaps acquire higher Jsc values. The other compounds
do not have SLME as high as these six materials owing to higher band gaps.
Their absorption coefficients and SLME values are shown in Figure S8 and S9 of
SI respectively.supplement  Furthermore, it is worth noting that the band gaps
of other compounds are distributed within the visible range (larger than 1.8
eV), which makes them suitable for LED and photocatalytic water splitting
applications. Alloying at B and x sites are other avenues to tune the
optoelectronic properties of these systems and hence make them suitable for
different applications.

## V. Transport properties

Figure 4:  (a,b,c) Hole mobility ($\mu_{h}$) and (d,e,f) electron mobility
($\mu_{e}$) for Cs2TiI6, Cs2PdBr6, Cs2PtI6, Cs2SnI6, Cs2PdI6, and Cs2TeI6
compounds as a function of defect concentrations at three different
temperatures, T= 100K, T=300 K and T= 500K respectively.  Figure 5:  (a,b,c)
Hole mobility ($\mu_{h}$) and (d,e,f) electron mobility ($\mu_{e}$) for
Cs2TiI6, Cs2PdBr6, Cs2PtI6, Cs2SnI6, Cs2PdI6, and Cs2TeI6 compounds as a
function of temperatures at three different defect concentrations, ND=1010
cm-3, 1015 cm-3, and 1020 cm-3 respectively.

Detailed analysis of optoelectronic properties and calculation of solar
efficiency (SLME) reveals six compounds to be promising. This can be
attributed to their optimal band gaps falling within the ideal visible region
of the solar spectrum coupled with their excellent absorption coefficients.
However, in a practical photovoltaic device, extraction of charge carriers is
one of the key component determining its power conversion efficiency. As such,
mobility of the charge carriers is an integral quantity dictating the promise
of a semiconductor for solar harvesting. Most of the past theoretical studies
on photovoltaic materials rely on calculation of transport properties based on
constant relaxation time approximation (RTA). Within this approximation, all
the scattering mechanisms are averaged out via a single relaxation time
(chosen to be approximately 10 fs). This practice, however, can be misleading
as the carrier relaxation time is a complex parameter which sensitively
depends on a number of physical properties and can be significantly different
for different materials belonging to the same class (as illustrated in this
study). In this section, we perform a thorough analysis of the carrier
mobilities of these compounds considering three relevant scattering
mechanisms, namely, acoustic phonons (ADP), ionized impurities (IMP), and
polar optical phonons (POP) scattering. We have excluded piezoelectric
scattering due to the inherent centro-symmetry present in these compounds. In
Figure 4 and 5, we show the temperature and defect concentration dependence of
electron and hole mobilities ($\mu_{e}$ and $\mu_{h}$) for these compounds.
Contribution of individual scattering mechansims on these mobilities for the
six compounds are provided in Figure S10 to S15 of SI.supplement

Figure S16 of SIsupplement  displays the total relaxation times of six
compounds at varying defect concentrations, ranging from 1010 cm-3 to 1020
cm-3 at three different representative temperatures (100 K, 300 K, and 500 K)
for both hole and electron transport. For defect concentrations in the low to
moderate range, the relaxation times remains almost constant. However, as the
defect concentration increases, relaxation times vary in an irregular manner.
To comprehend the cause of this anomalous behavior, a more in-depth analysis
was conducted. The relaxation times for all three scattering mechanisms were
calculated for each compound, and plotted in Figures S17 to S22.supplement  A
close inspection of these data confirms that in the low to moderate
concentration range, the primary scattering mechanism is due to POP
scattering. In contrast, as the concentration increases into the higher range,
the dominant scattering mechanism shifts to IMP scattering, resulting in the
emergence of anomalous behavior. Such unusual behavior is also reflected in
the mobility shown in Fig. 4.

Speaking about the behavior of mobilities for each individual compounds, one
can notice that at low temperature (100 K), the hole mobility ($\mu_{h}$) is
highest for Cs2TiI6 ($\sim$20.9 cm2V-1 s-1). With increasing temperature,
$\mu_{h}$ decreases slowly to reach a shallow minimum and increases again with
increasing defect concentration. At higher temperature, Cs2PdI6 shows the
highest hole mobility ($\sim$5.9 cm2V-1 s-1 @300 K and $\sim$4.2 cm2V-1 s-1
@500 K ) among all the compounds. This compound also shows the highest
electron mobility ($\sim$183 cm2V-1 s-1 @ 100 K, $\sim$51 cm2V-1 s-1 @ 300 K,
and $\sim$32 cm2V-1 s-1 @ 500 K) throughout the temperature range. At room
temperature, the hole mobilities remain relatively low but except Cs2TiI6,
electron mobilities show moderate to high values ($\sim$13-63 cm2V-1 s-1).
This is commensurate with the electronic band structures of these compounds
where the VBM shows flat bands whereas the CBM is more dispersive.
Consequently, n-type doping could prove advantageous for efficient charge
carrier collection in photovoltaic devices, aligning with the experimental
findings for this class of compounds.maughan2018tolerance ; lee2014air ;
saparov2016thin  A closer look at the individual contributions from the
different scattering mechanisms show that at low to moderate defect
concentrations ($<10^{18}$ cm-3), POP scattering is the dominant scattering
mechanism limiting the mobilities. With increasing temperatures, the number of
activated polar optical phonons increase, and as a result we see a decrease in
overall mobility going from 100 K $\rightarrow$ 300 K $\rightarrow$ 500 K. At
higher defect concentrations( 1018-1020 cm-3), we see ionized impurity
scattering begins to dominate as can be seen from Figures
S10(a,b,c)-S15(a,b,c) of SI.supplement  At these concentrations, there is one
more mechanism that starts to impact the carrier mobility, which is the
screening of polar optical phonons by free carriers. This in effect reduces
the POP scattering, effectively increasing the overall mobility in some cases.
Now, the temperature has also an effect on this screening mechanism. At higher
temperatures, there are more activate polar optical phonons, which require a
higher density of free carriers to effectively screen the Coulomb field
created by these phonons. This is clearly evident from our SI plots (see
Figures S10(a,b,c)-S15(a,b,c)).supplement  In all the cases, ADP scattering
remains low which is common in hybrid perovskites arising out of small
deformation potentials.ganose2021efficient ; irvine2021quantifying

In Figure 5(a-f), we show average hole and electron mobilities with respect to
temperatures ranging from 100 K to 500 K for three different defect
concentrations, low (1010 cm-3), moderate (1015 cm-3) and high (1020 cm-3).
Due to weak dependence on IMP scattering in low to moderate defect
concentrations, we see the carrier mobility remains similar in these two
defect concentrations. But we can see that at higher concentrations, IMP
starts to dominate. As such, controlling the defect concentrations can impact
device efficiencies, not only because at higher defect concentrations, IMP
becomes the dominant scattering mechanism, but also because the prevalence of
free carriers will start to screen the POP scattering effect. As expected, the
overall mobility has a strong temperature dependence for most of the compounds
and remains high to moderate for the electrons whereas the hole mobility
values remain consistently low.

The above analysis reveals that in A2BX6 class, polar optical phonons play a
dominant role at the realistic defect concentrations relevant for photovoltaic
application. As such, next we study the properties of the polaronic states via
calculating the Fr$\ddot{o}$hlich interactions under the temperature-dependent
Feynman polaron model.feynman1955slow [Reference] In polar semiconductors, for
example, halide perovskite and its derivatives, the interaction between charge
carriers and the macroscopic electric field generated by longitudinal optical
phonon (LO) is well known to be the dominant scattering mechanism near room
temperature which is expected to be the case for our studied materials as
well.hellwarth1999mobility ; manna2020lattice ; maughan2018anharmonicity ;
maughan2018tolerance ; ganose2021efficient  To investigate the same, we
studied the influence of changing B-site in A2BX6 on the electron-phonon
coupling (EPC). Within the Fr$\ddot{o}$hlich interaction model, the
interaction strength ($\alpha$) is defined as

$\displaystyle\alpha=\frac{1}{4\pi\epsilon_{0}}\frac{1}{2}\left(\frac{1}{\epsilon_{\infty}}-\frac{1}{\epsilon_{static}}\right)\frac{e^{2}}{\hbar\omega_{LO}}\left(\frac{2m^{*}\omega_{LO}}{\hbar}\right)^{1/2}$
(2)

were, $\epsilon_{0}$ is dielectric constant of vacuum, $\epsilon_{\infty}$ and
$\epsilon_{static}$ are high frequency and static dielectric constants of the
semiconductor, $\hbar$ is the reduced Plank constant, $\omega_{LO}$ is the
characteristic angular LO frequency where all the infrared active optical
phonon branches are taken into account via a spectral
average,hellwarth1999mobility  $m^{*}$ is the carrier effective mass. Table 2
display all the associated values related to Fr$\ddot{o}$hlich interaction for
electrons for the six compounds. The corresponding list of parameters for the
holes for these six compounds are reported in Table S5 of SM.supplement  To
validate our simulation, we compare the calculated values of $\alpha$ for
Cs2SnI6, with recent literature and observe a fair
agreement.maughan2018anharmonicity  In case of A2BX6 class, calculated
$\alpha$-values lie in the moderate range (1$<\alpha<$ 6). Estimated values of
polaron radius (lp) indicates formation of large polarons, similar to what is
observed for hybrid halide perovskites and double
perovskites.hellwarth1999mobility ; manna2020lattice ; saxena2020contrasting
$\alpha_{e}$ value is highest for Cs2TiI6 mainly due to higher electron
effective mass compared to the other compounds. Additionally, taking an
inference from the electronic structure of these materials, we see that CBM in
Cs2TiI6 has a contribution from Ti-d and I-p orbitals whereas for Cs2SnI6, it
is Sn-s and I-p orbitals. Now, Ti-d orbitals are more localized arising out of
the flat band and hence higher effective mass. For other compounds, we see
more dispersive bands (see Figure 2) at CBM, and the corresponding $\alpha$
values are in the range close to that of Cs2SnI6. Interestingly, the hole
mobility turn out to be significantly lower than the electron mobility. To
conclude, large polaron is the main carrier related to moderate mobility for
our studied compounds. These crucial observations clearly indicates the
importance of studying charge carrier behavior in A2BX6 class of compounds and
its implications in future applications.

Parameters |  Cs2TiI6 |  Cs2PdBr6 |  Cs2PtI6 |  Cs2SnI6 |  Cs2TeI6 |  Cs2PdI6
---|---|---|---|---|---|---
me∗(m0) | 1.66 | 0.66 | 0.49 | 0.31 | 0.35 | 0.61
$\epsilon_{\infty}$ | 4.91 | 4.15 | 4.82 | 5.04 | 4.89 | 7.58
$\epsilon_{static}$ | 10.81 | 7.21 | 8.40 | 10.41 | 11.21 | 11.68
$\nu_{eff}$(THz) | 3.77 | 3.16 | 2.16 | 2.60 | 2.71 | 2.25
$\alpha_{e}$ | 4.23 | 2.66 | 2.44 | 2.02 | 2.38 | 1.38
$\mu_{e}$ (cm2V-1 s-1) | 1.5 | 11.9 | 22.7 | 31.4 | 27 | 51.3
l${}_{p}(e)$(Å) | 29.40 | 32.75 | 25.88 | 32.48 | 30.92 | 35.33

Table 2:  Calculated electron effective mass (me∗ in units of electron rest
mass (m0)), high frequency and static dielectric constants
($\epsilon_{\infty}$ and $\epsilon_{static}$), effective phonon frequency
($\nu_{eff}$), Fr$\ddot{o}$hlich coupling constants for electrons
($\alpha_{e}$), electron mobility ($\mu_{e}$), and polaron radius (lp) for
Cs2TiI6, Cs2PdBr6, Cs2PtI6 Cs2SnI6, Cs2TeI6 and Cs2PdI6 compounds. The
corresponding set of parameters for holes are displayed in Table S5 of
SI.supplement

## VI. Conclusion

In summary, we performed an accurate and systematic investigation of Pb-free
vacancy ordered double perovskites (A2BX6) from the optoelectronic application
perspective. We carried out a thorough stability analysis considering
different structural prototypes and carefully simulating the convex hull
energy diagram including all possible secondary phases. We found 14 compounds
to be stable and 1 in the metastable phase. For stable compounds, we further
simulated the compositional phase diagrams to assist the experimentalists
identifying the most probable secondary phases which might emerge during
synthesis. Next, a carefull electronic structure analysis reveals six
compounds, namely Cs2TeI6, Cs2SnI6, Cs2PdI6, Cs2PtI6, Cs2TiI6, and Cs2PdBr6 to
possess optically allowed band gaps in the ideal visible range (0.8-1.85 eV).
The detailed investigation of optical properties confirms that few of these
compounds possess favorable optoelectronic properties facilitating better
efficiency than some of the existing ones. A close inspection of transport
properties reveals that Cs2PdBr6, Cs2PtI6, Cs2SnI6, Cs2PdI6, and Cs2TeI6
compounds acquire moderate to high electron mobilities ($\sim$13 - 63 cm2V-1
s-1). In all the cases, polar optical phonons (POP) remain the dominant
scattering mechanism at low to moderate defect concentrations. At high defect
concentrations, ionized impurity scattering starts to dominate while
accumulation of free carriers shows a screening effect on the POP scattering.
This study is expected to facilitate the necessary base and guidance for
future experimental synthesis of some of these compounds to achieve desired
features for promising device applications.

## VII. Computational Details

First-principles calculations are carried out using density functional theory
(DFT)PhysRev.140.A1133  with projector augmented wave (PAW)blochl1994projector
basis set as implemented in Vienna Ab-Initio Simulation Package
(VASP).PhysRevB.59.1758 ; kresse1993ab ; kresse1994ab ; kresse1996efficient ;
KRESSE199615  A plane wave energy cutoff of 520 eV, $\Gamma$-centered
4$\times$ 4$\times$4 k-mesh, and Perdew-Burke-Ernzerhof (PBE) exchange-
correlation functionalperdew1996generalized  were employed to perform the
geometry optimization. The crystal structure was relaxed with force tolerance
criteria of 0.001 eVÅ-1. The spin-orbit coupling (soc) effect is included
while simulating the electronic and optical properties. Hybrid(HSE06)
functionalkrukau2006influence  is used to calculate the band gap and band
edges which are known to provide a more accurate estimate for the same.
Optical absorption spectra are simulated within the independent particle
approximation and then the absorption onset value is scissor shifted to HSE06
band gap values. This method makes it possible to accurately assess the SLME
for the materials under consideration. The chemical phase diagrams are drawn
using Chesta software package.chesta  Phonon dispersion is calculated using
the density functional perturbation theory (DFPT) using $\Gamma$-centered
4$\times$4$\times$4 k-mesh under the supercell method. The 2nd order force
constant is calculated using 2$\times$2$\times$2 supercells of primitive cells
for cubic structures and in similar proportion for other structures. Next, the
rotational sum rule is applied using the hiphive packageeriksson2019hiphive
to renormalize the phonon frequencies. Transport calculations are performed
using the AMSET code,ganose2021efficient , where we have considered three
different scattering mechanisms, namely scattering due to acoustic phonons
(ADP), ionized impurities (IMP), and polar optical phonons (POP).
Piezoelectric scattering is not included due to the centro-symmetric crystal
structure of A2BX6, whereas screening due to free carriers at high defect
concentrations is included. This program uses the Boltzmann transport
equation’s (BTE) momentum relaxation time approximation (MRTA) to determine
scattering rates and carrier mobilities. Polaron related parameters were
simulated via implementing a temperature dependent Feynman polaron
model.hellwarth1999mobility ; frost2017calculating  Born effective charges and
static and high-frequency dielectric tensors were calculated using density
functional perturbation theory (DFPT) as implemented in VASP. The effective
mass has been calculated using the following equation:

$m^{*}=3\left[\frac{1}{m_{xx}^{*}}+\frac{1}{m_{yy}^{*}}+\frac{1}{m_{zz}^{*}}\right]$
(3)

where, $m_{ii}^{*}$ is the effective mass in the $i$-th direction
($i$=x,y,z).mortensen2005real ; enkovaara2010electronic ; larsen2017atomic ;
kangsabanik2022indirect

## VIII. Acknowledgments

SG acknowledges financial support from IIT Bombay for research fellowship. AA
and MA acknowledges National Center for Photovoltaic Research and Education
(NCPRE) funded by Ministry of new renewable energy (MNRE), Government of
India, and IIT Bombay for possible funding to support this research.

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