# First Principles Investigation of Polymorphism in Halide Perovskites

Jiaqi Yang<EMAIL_ADDRESS>[  Arun Mannodi-Kanakkithodi
<EMAIL_ADDRESS>[

###### Abstract

Halide perovskites have been extensively studied as materials of interest for
optoelectronic applications. There is a major emphasis on ways to tailor the
stability, defect behavior, electronic band structure, and optical absorption
in halide perovskites, by changing the composition or structure. In this work,
we present our contribution to this field in the form of a comprehensive
computational investigation of properties as a function of the perovskite
phase, different degrees of lattice strains and octahedral distortion and
rotation, and the ordering of cations in perovskite alloys. We performed first
principles-based density functional theory computations using multiple semi-
local and non-local hybrid functionals to calculate optimized lattice
parameters, energies of decomposition, electronic band gaps, and theoretical
photovoltaic efficiencies. Trends and critical observations from the high-
throughput dataset are discussed, especially in terms of the range of
optoelectronic properties achievable while keeping the material in a
(meta)stable phase or distorted, strained, or differently ordered polymorph.
All data is made openly available to the community and is currently being
utilized to train state-of-the-art machine learning models for accelerated
prediction and discovery, as well as to guide rational experimental discovery.

] School of Materials Engineering, Purdue University, West Lafayette, IN
47907, USA ] School of Materials Engineering, Purdue University, West
Lafayette, IN 47907, USA

## Introduction

Halide perovskites (HaPs) are very attractive materials for a variety of
electronic and optical applications, primarily owing to their massive chemical
space and engineerability in terms of composition, structure, alloying, and
doping 1, 2, 3, 4, 5. The canonical ABX3 perovskite family of materials are of
great interest as absorbers in solar cells 6, 7, with record efficiencies
achieved to date of 25.7% in single-junction solar cells 8 and very recently,
32.5% in Si-perovskite tandem solar cell 9, 10. While these efficiencies
firmly place HaPs in the fastest growing market for solar absorption, they are
also well below theoretical maximum values 11, 12, 13. The sheer size of the
possible HaP chemical space, including 3D and layered materials, purely
inorganic and hybrid organic-inorganic perovskites (HOIPs), and complex
alloys, makes it difficult to screen promising materials in a brute-force
manner, but also provides massive opportunities for discovery and
understanding via high-throughput computation.

The published literature contains several glittering examples of data-driven
and experimental efforts to optimize perovskite compositions for
optoelectronic performance 14, 15, 16, 17, 18. However, comprehensive
understanding of the effect of polymorphism on the properties of HaPs is still
an active area of research. The cubic phase is the standard prototype
structure any perovskite is simulated in, and there are established numerical
metrics such as the Goldschmidt tolerance and octahedral factors that
determine the stability of any ABX3 compound in the cubic phase 19. As shown
in our recent work, such factors are important but not sufficient conditions
for perovskite stability, as a thermodynamic evaluation based on first
principles reveals many materials that may decompose to alternative phases
despite suitable ionic radii 20. Perovskites could further adopt a series of
other prototype phases such as tetragonal, orthorhombic, or hexagonal, as well
as other corner-, edge-, or face-shared phases such as distorted orthorhombic
and needle-like 5, 21, 22, 23. Polymorphism may also manifest within the same
phase, in terms of energetically favorable (or metastable) distortions or
rotations in corner-shared BX6 octahedra or via uni-axial or multi-axial
lattice strains 24, 25, 26, 27, as well as in terms of re-optimization of the
compound in larger supercells with symmetry-breaking via small distortions 28.
The result is typically the existence of multiple competing phases that may
all contribute as an ensemble to experimentally measured band gaps and optical
absorption, instead of a sole ground state structure determining the
properties.

Doping and alloying are two of the most common ways to engineer the properties
of perovskites; the former typically involves a heterovalent ion of a suitable
size substituting an A or B cation 29, 30, 31, whereas the latter involves
multiple homovalent cations or anions mixed together at A/B or X sites 32, 33,
34, 35, respectively. The number of ways to dope HaPs or create mixed
compositions are practically infinite, introducing several degrees of freedom
in the HaP structure-composition-properties space 5. In multiple recent
studies, we comprehensively explored both B-site dopants 36, 37 and A/B/X-site
alloying 38, 20 in ABX3 compounds using density functional theory (DFT)
computations, and examined how the stability and optoelectronic properties
depend on the nature of ionic mixing. We used the special quasi-random
structures (SQS) approach 39 to simulate alloys in large supercells, as is the
norm in the literature, but for any given mixed composition, the properties
may depend heavily on ionic ordering as well, creating more polymorphs that
must be considered.

Figure 1: (a) 4 prototype ABX3 HaP phases, namely cubic, tetragonal,
orthorhombic, and hexagonal. (b) 4$\times$4$\times$4 cubic supercells showing
a MA(Pb-Sn-Ba-Sr-Ca)I3 quinary alloy with different ionic ordering. (c)
Octahedral distortion in the MAPbBr3 cubic lattice.

Additionally, the difficulties of brute-force experimentation within a massive
structure-composition space means that high-throughput DFT (HT-DFT)
computations are essential for systematically assessing trends and
correlations that may guide multi-objective optimization and rational
experimental synthesis and testing. DFT is extensively applied for determining
lattice parameters, heat of formation or decomposition ($\Delta$H), band gap
(Eg), optical absorption-derived spectroscopic limited maximum efficiency
(SLME) 40, 41, and defect formation energy (DFE) in perovskites, with mixed
accuracy compared to experiments 5, 42. While semi-local GGA-PBE and variants
such as PBEsol (improved PBE for solids) 43 and PBE-D3 (for weak dispersion
interactions) 44 reproduce bulk stability and structure well, they generally
under-predict Eg compared to non-local hybrid HSE06 functional or beyond-DFT
GW approximation 42, 45. For many Pb/Sn HOIPs, PBE Eg without including spin-
orbit coupling (SOC) is often as accurate as HSE Eg with SOC included 5, 38,
20, an effect that holds true for DFEs and corresponding defect charge
transition levels (CTLs) 37, 46. HSE+SOC is more expensive but generally more
accurate for electronic properties. Oftentimes, the Hartree Fock to semi-local
exchange parameter $\alpha$ needs to be tuned in HSE 47, which is highly
sensitive to material composition and not easy to perform over massive
chemical spaces. Thus, the DFT functional itself is an added factor that
determines perovskite properties: while general property-polymorph
relationship trends may be reliable from semi-local functionals, more advanced
theories would be necessary for quantitative estimates that could be compared
with experiments.

Based on the above ideas, and building upon our past work, we present here a
systematic investigation of the following types of polymorphism in a selected
chemical space of ABX3 HaPs, employing different types of PBE and HSE06
functionals within DFT:

  1. 1.

The effect of perovskite phase, as it changes from cubic to tetragonal to
orthorhombic to hexagonal. The four phases are pictured in Fig. 1(a).

  2. 2.

The effect of ionic ordering in B-site mixed compounds simulated in large
supercells; two example structures for a quinary alloys are pictured in Fig.
1(b).

  3. 3.

The effect of lattice strain and octahedral distortion/rotation within the
cubic perovskite lattice, as shown in Fig. 1(c).

This work provides an understanding of how the above factors may positively or
adversely affect the HaP stability and properties of interest for
optoelectronic applications, especially single-junction solar absorption.
Within a chemical space defined by A = FA, MA, or Cs (where FA and MA are
organic molecules, formamidinium and methylammonium, respectively), B = Pb,
Sn, Ge, Ba, Sr, and Ca, and X = I, Br, and Cl, we define easily-attainable and
generalizable descriptors that encode the composition, phase, elemental
properties, and ionic ordering, leading to important design rules such as the
favorability of Ba-Ba clustering in increasing the bulk stability and the
extent to which strain and distortions may keep the lattice stable while
tuning the band gap. We also obtain important insights on the accuracy of
different functionals and the inter-relationships between them. We believe the
datasets and understanding obtained from this work will be crucial for guiding
subsequent studies, both computational and experimental, as well as training
machine learning (ML) models for accelerated prediction and screening over
hundreds of thousands of possible structures and compositions. In the
following sections, we describe computational details and present a series of
plots and discussions unraveling the effects of different factors on the
properties of interest. All data is made openly available for the benefit of
the community.

## Computational Methodology

### DFT Details

All DFT computations were performed using VASP version 6.2 48, 49, 50,
employing Projector Augmented Wave (PAW) pseudopotentials. 51, 52 The Perdew,
Burke, and Ernzerhof (PBE) functional within the generalized gradient
approximation (GGA) 53 as well as the Heyd-Scuseria-Ernzerhof (HSE) functional
($\alpha$=0.25 and $\omega$=0.2) are used for the exchange-correlation energy.
The energy cutoff for the plane-wave basis is set to 500 eV. A Monkhorst-Pack
k-point mesh of 6$\times$6$\times$6 is used for cubic unit cells and a mesh of
4$\times$4$\times$3 is used for prototypical tetragonal, orthorhombic, and
hexagonal unit cells. For cubic supercell calculations, the k-point meshes are
reduced to 3$\times$3$\times$3 and gamma-point only for 2$\times$2$\times$2
and 4$\times$4$\times$4 supercells, respectively. The k-point meshes are
accordingly scaled down for the tetragonal, orthorhombic, and hexagonal
supercells as well. Starting from the PBE-optimized structures, full geometry
optimization is additionally performed using PBEsol, PBE-D3, and PBEsol-D3, by
adding relevant input tags. The force convergence threshold is set to be -0.05
eV/Å for all geometry optimization runs. Spin-orbit coupling (SOC) is
incorporated in HSE06 calculations using the LORBIT tag and the non-collinear
magnetic version of VASP 6.2 54.

The PBE-optimized structure is used as input for calculating the optical
absorption spectrum using the LOPTICS tag and setting the number of energy
bands to 1000, and the approach developed by Yu et al. 55 is then applied to
determine the spectroscopic limited maximum efficiency (SLME) as a function of
sample thickness. The SLME value at 5 $\mu$m thickness is taken as the
theoretical photovoltaic (PV) efficiency. The band gap is computed from the
PBE optimization runs and from static HSE calculations based on the PBE-
optimized structure, where k-point meshes of 2$\times$2$\times$2 and
2$\times$2$\times$1 are respectively used for cubic and
tetragonal/orthorhombic/hexagonal supercells. SLME values at the PBEsol,
PBE-D3, PBEsol-D3, and HSE levels are determined by shifting the PBE-computed
optical spectra by the difference between the PBE band gap and that computed
from the corresponding functional, and recalculating the SLME based on the
approach used in past work 20. We consider a series of pure and mixed-
composition ABX3 compounds in this work; all alloys are simulated using SQS
except for the 4$\times$4$\times$4 supercell structures, where we explicitly
examine the effect of ionic ordering by considering 20 to 25 randomly ordered
quaternary or quinary B-site mixed compounds.

Ultimately, multiple types of PBE (PBE-optimized, PBEsol-optimized,
PBE-D3-optimized, and PBEsol-D3-optimized) and HSE (static HSE+SOC on PBE-
optimized structure) functionals are applied for subsets of all compounds
being studied, and the effect of each is examined for the one or all of the
following properties: the effective lattice parameter (aeff), decomposition
energy ($\Delta$H), band gap (Eg), and SLME. While the a, b, c lattice
constants are computed for each cubic and non-cubic structure from every level
of theory, for efficient comparison, we define the “effective lattice
parameter” for any given pure or mixed composition material as aeff =
(Vsc/pfu)1/3, where Vsc is the supercell volume and pfu is the number of
perovskite formula units in the supercell. A negative $\Delta$H implies an
inherent resistance of any ABX3 compound to decompose to AX and BX2 phases. Eg
should typically be between 1 eV and 2 eV for suitable single-junction solar
absorption, whereas the SLME should be as high as possible. Equations for
calculating SLME can be found in the original publications 55 and in our past
work 20. $\Delta$H is calculated using equation (1), where Eopt(S) is the
total DFT energy per formula unit of any system S, kB is the Boltzmann
constant, T is the temperature fixed to be 300K here, and xi is the mixing
fraction of any species at A/B/X sites.

$\displaystyle\Delta
H=E_{opt}(ABX_{3})-\sum_{i}x_{i}E_{opt}(AX)-\sum_{i}x_{i}E_{opt}(BX_{2})+k_{B}T(\sum_{i}x_{i}ln(x_{i}))$
(1)

###  0.1 Simulating Polymorphs using Perovskite Supercells

We first study 27 pure ABX3 compounds, defined by A = FA, MA, or Cs, B = Pb,
Sn, or Ge, and X = I, Br, or Cl, in 4 different phases, namely cubic,
tetragonal, orthorhombic, and hexagonal, as pictured in Fig. 1(a). All 27*4 =
108 structures are optimized using PBE, PBEsol, PBE-D3, and PBEsol-D3. We then
perform HSE+SOC computations on all 108 structures using the PBE-optimized
structures as input. Next, we consider 19 random alloyed compositions each in
two (most likely) phases each and perform PBE as well as HSE+SOC computations
on them, which leads to a total dataset of 146 points. We thus have the
ability to compare the aeff, $\Delta$H, Eg, and SLME calculated from multiple
functionals for 146 compounds with any known experimental values, as well as
to visualize entire datasets of different properties plotted against each
other.

After studying the effect of DFT functional and perovskite phase, we turn our
attentions to ionic ordering in mixed compounds: this is accomplished by
simulating cubic MA-(Pb/Sn/Ba/Sr/Ca)-I3 4$\times$4$\times$4 supercells in
quaternary (4 species mixed at B) and quinary (5 species mixed at B)
compositions, with 20 possible structures each considered for the 5
quaternaries and 25 structures for the quinary. This leads to a dataset of 125
compounds, and the PBE-computed $\Delta$H and Eg are visualized in terms of
clustering of different B-site cations. Finally, we consider 6 compounds in
the cubic phase, namely CsPbI3, CsPbBr3, CsPbCl3, MAPbI3, MAPbBr3 and MAPbCl3,
and induce a series of lattice strains and octahedral distortions/rotations
starting from the PBE-optimized ground state structures. First, we apply
systematic compression and elongation in the lattice by changing the lattice
constants and running volume-fixed geometry optimization; the changes in
$\Delta$H and Eg for the newly optimized structures are visualized against the
amount of strain. Further, we distort corner-shared octahedra by changing
positions of the bridging X atoms, perform geometry optimization on the new
structures containing different amounts of distortion and rotation, and
finally visualize the computed $\Delta$H and Eg.

###  0.2 Compiled Datasets

Table 1 presents a description of all the DFT datasets generated as part of
this work, as well as the experimental data collected for comparison.
Ultimately, we compute lattice constants, stability, band gap, and PV
efficiency for 146 (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 compounds (both pure and
mixed composition), from PBE, PBEsol, PBE-D3, PBEsol-D3, and HSE-PBE+SOC
(henceforth referred to as HSE). 125 quinaries and quaternaries are simulated
in 4$\times$4$\times$4 cubic supercells of MA-(Pb/Sn/Ba/Sr/Ca)-I3 which yields
lattice constants, stability, and band gaps from PBE. Lattice strain and
octahedral distortion/rotation applied on (MA-Cs)(Pb)(I-Br-Cl)3 results in
datasets of 65 and 677 points respectively, with lattice constants, stability,
and band gaps computed from PBE. Experimental data is collected from across
several publications 56, 57, 58, 59, 60, leading to lattice constants of 32
compounds, band gaps of 31 compounds, and power conversion efficiency (PCE)
values of 19 compounds, which are compared against corresponding values from
the multi-phase HaP datasets of 146 compounds.

Dataset Chemical Space Functional Data Points Properties Multi-phase HaPs (MA-
FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 PBE 146 a, b, c, $\Delta$H, Eg, SLME Multi-phase
HaPs (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 PBEsol 146 a, b, c, $\Delta$H, Eg, SLME
Multi-phase HaPs (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 PBE-D3 146 a, b, c, $\Delta$H,
Eg, SLME Multi-phase HaPs (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 PBEsol-D3 146 a, b,
c, $\Delta$H, Eg, SLME Multi-phase HaPs (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 HSE-
PBE+SOC 146 $\Delta$H, Eg, SLME Quaternaries & Quinaries MA(Pb-Sn-Ba-Sr-Ca)I3
PBE 125 a, b, c, $\Delta$H, Eg Lattice strain (MA-Cs)(Pb)(I-Br-Cl)3 PBE 65 a,
b, c, $\Delta$H, Eg Octahedral distortion (MA-Cs)(Pb)(I-Br-Cl)3 PBE 677 a, b,
c, $\Delta$H, Eg Experimental Lattice Constants (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3
- 32 a, b, c Experimental Band Gaps (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 - 31 Eg
Experimental PV Efficiencies (MA-FA-Cs)(Pb-Sn-Ge)(I-Br-Cl)3 - 19 PCE

Table 1:  Description of all the datasets used in this work, in terms of the
chemical space, DFT functionals used, number of data points, and computed
properties. a, b, and c refer to the optimized or experimental lattice
constants. Experimental data is collected from across several publications 56,
57, 58, 59, 60. PCE stands for power conversion efficiency. Figure 2:
Visualization of the DFT dataset of multi-phase HaPs: (a) PBE $\Delta$H vs Eg,
(b) PBE Eg vs SLME (PV efficiency), (c) HSE-PBE+SOC $\Delta$H vs Eg, and (d)
HSE-PBE+SOC Eg vs SLME. Different shapes of the scatter points represent
hybrid organic-inorganic HaPs and purely inorganic HaPs, and the different
colors represent different perovskite phases.

##  1 Results and Discussion

###  1.1 Tracking Computed Properties Against Phase and Functional

Fig. 2 shows $\Delta$H vs Eg and Eg vs SLME plots for the multi-phase HaP
dataset of 146 points, from PBE and HSE-PBE+SOC (henceforth referred to as
HSE). Corresponding plots for PBEsol, PBE-D3, and PBEsol-D3 are presented in
Fig. S1. The general property distributions are very similar from all 5
functionals. We note here that while the PBE-based functionals are clearly not
intended to estimate band gaps and PV efficiencies, our purpose is to
establish a baseline of computation accuracy from these cheaper semi-local
functionals. Across the plots in Fig. 2 and Fig. S1, the scatter points are
distinguished in terms of the perovskite phase and whether a compound is a
hybrid organic-inorganic perovskite (MA- or FA-based) or a purely inorganic
perovskite (Cs-based). As seen from Fig. 2(a) and (c), a majority of the
compounds have $\Delta$H < 0 eV p.f.u. from both PBE and HSE, indicating a
strong resistance to decomposition for nearly all chemistries and phases,
except for a few inorganic compounds where $\Delta$H $\sim$ 1.5 eV p.f.u. The
unstable compounds are primarily Cs-based chlorides whereas the most stable
compounds (with very negative $\Delta$H) are usually FA-based iodides. There
is no clear phase-specific stability preference, with all phases seemingly
distributed across the range of $\Delta$H values from $\sim$ -1.5 eV p.f.u. to
1.5 eV p.f.u.

Figure 3: Multi-phase property visualization for 16 MA-based hybrid HaPs: (a)
PBE $\Delta$H, (b) PBE Eg, (c) PBE SLME, (d) HSE $\Delta$H, (e) HSE Eg, and
(f) HSE SLME.

Eg ranges from a low of $\sim$ 0.5 eV (0.2 eV) from PBE (HSE) for cubic CsSnI3
to a high of $\sim$ 3.5 eV (4 eV) from PBE (HSE) for hexagonal MAGeCl3. SLME
(labeled as PV efficiency in % in the plots) values range from 0 for the
highest band gap compounds to a high of 16% from both PBE and HSE for
compounds with Eg in the vicinity of 1 eV. The SLME vs Eg plot shows the
characteristic shape that has been explored in past works 61, 62, 20.
Interestingly, many of the highest SLME values are shown by cubic, tetragonal,
or orthorhombic phases of Cs-based compounds, which also show some of the
lowest Eg values—with the caveat that some of these compounds also have very
large $\Delta$H and are thus unstable. The outer boundary of the SLME vs Eg
plots is dominated by hexagonal phase hybrid perovskites because of their
tendency to show larger Eg. Very similar ranges of values for all three
properties are observed in the PBEsol, PBE-D3, and PBEsol-D3 datasets as well,
as shown in Fig. S1, with PBEsol-D3 showing a tendency for broadening both the
$\Delta$H and Eg ranges. Fig. S2 shows the PBE-computed $\Delta$H, Eg, and PV
efficiency plotted against the corresponding values from the remaining four
functionals, namely PBEsol, PBE-D3, PBEsol-D3, and HSE. We find that the
$\Delta$H values track very linearly except for some compounds which are
predicted to be more stable from HSE than from PBE. Eg and PV efficiency also
track linearly except for when D3 corrections are used; many of the inorganic
compounds show strange behavior, seemingly arising from the fact that vdW
corrections are unnecessary for Cs-based compounds although they should be
incoporated for FA- and MA-based compounds.

###  1.2 Benchmarking Computed Optoelectronic Properties

To further examine the phase-dependent properties of different hybrid and
inorganic HaPs, we plotted the PBE and HSE properties of 16 MA-based compounds
in Fig. 3, using individual plots with compound labels on the x-axis.
Corresponding PBE and HSE data for 12 FA-based compounds and 17 Cs-based
compounds are presented in Figs. S3 and S4. Plots for the PBEsol, PBE-D3, and
PBEsol-D3 data are presented across Figs. S5, S6, and S7. Known experimental
values of Eg and PCE are also shown in all such plots. We find that the
tetragonal phase is largely the most stable one for MA compounds with the
hexagonal phase also showing low energies in many cases, from both PBE and
HSE. Hexagonal phase compounds show the largest Eg and the lowest SLME,
whereas the orthorhombic phase displays the opposite behavior. There is also
an impressive match between the PBE-computed Eg of tetragonal phase compounds
and the corresponding measured Eg values, which comes from the aforementioned
cancelation of errors and accidental accuracy of GGA-PBE when SOC is not
included 20. The HSE Eg for the same compounds end up being under-estimated
because of SOC reducing the band gap to a larger extent than desired, and
likely because of a need for tuning the mixing parameter $\alpha$ in the HSE
calculation. The measured PCE values on average do not match well with PBE or
HSE computed SLME values, although some qualitative trends are captured.

Figure 4: Accuracy of different DFT functionals compared to experiments: PBE
and HSE-PBE+SOC Eg for (a) 31 MA-, FA-, and Cs-based compounds, (b) 17 hybrid
HaPs (MA- and FA-based), and (c) 14 inorganic Cs-based HaPs, and (d) pseudo-
cubic lattice constants of 32 MA-, FA-, and Cs-based compounds from PBE,
PBEsol, PBE-D3, and PBEsol-D3.

Similarly, Fig. S3 shows that the hexagonal or cubic phase are most preferred
for FA-based compounds, there is a smaller range of Eg values across the four
phases and both PBE and HSE match very well with experiments, and the PV
efficiency values from both functionals also match reasonably well with
experiments. The best match with experiments for Eg and SLME of FA compounds
comes from the hexagonal phase. Fig. S4 shows that the orthorhombic or cubic
phase are most preffered for Cs compounds from both PBE and HSE, and there is
little match between computed Eg and SLME and corresponding measured values
across the 17 compounds. PBE-computed Eg and SLME show pretty good accuracy
for CsPbX3 compounds (where X is some combination of I, Br, and Cl) but falter
for most of the Sn-containing compounds. Finally, the DFT-computed Eg from
different functionals are plotted against experimental values for 31 known
compounds in Fig. S8(a). Surprisingly, PBE shows the lowest root mean square
error (RMSE, DFT vs experiment) of 0.57 eV, with the remaining functionals,
including HSE, showing RMSE values between 0.81 eV and 1.06 eV. These errors
are highly dependent on perovskite type and can certainly be improved using
advanced functionals, including tuning the HSE mixing parameter.

To improve the DFT-experiment correspondence, we applied some corrections to
the PBE and HSE Eg values and observed a marked reduction in RMSE. Fig. 4(a)
shows PBE Eg shifted up by 0.4 eV and HSE Eg shifted up by 0.8 eV plotted
against measured Eg values for 31 compounds, showing RMSE values of 0.47 eV
and 0.35 eV respectively for PBE and HSE. This data is plotted for 17 hybrid
HaPs in Fig. 4(b) and for 14 inorganic HaPs in Fig. 4(c). As observed earlier,
PBE Eg without any correction shows a very low RMSE of 0.23 eV for hybrid
HaPs, while HSE Eg shifted up by 0.5 eV also shows an RMSE of 0.23 eV. PBE Eg
shifted up by 0.7 eV improves the inorganic RMSE to 0.44 eV and the HSE Eg
shifted up by 1 eV for inorganic compounds shows an RMSE of 0.30 eV. Thus,
based on the observations so far, we posit here that (a) PBE Eg can be used
with reasonable accuracy for screening across MA- and FA-based HaPs, (b)
HSE+SOC Eg (using a default mixing parameter of $\alpha$=0.25) shifted up by 1
eV can serve as an accurate estimate for Cs-based HaPs, and (c) it would very
likely be possible to accurately learn the experiment-level Eg of all
important HaP compositions by combining PBE, HSE, and experimental data and
performing multi-fidelity learning 63, 64, as we plan to do in future work.
Table 2 lists the RMSE values for Eg from all DFT functionals compared against
experiments, with and without corrections, for different datasets.

###  1.3 Benchmarking Computed Lattice Constants

Finally, we examine the accuracy and inter-dependence of lattice constant
values from the four PBE-based functionals. Fig. S9 shows the PBE-computed
effective lattice parameter (aeff) plotted against the corresponding values
from PBEsol, PBE-D3, and PBEsol-D3, for the entire multi-phase HaP dataset of
146 compounds. In general, aeff values decrease from PBE to PBE-D3 to PBEsol
to PBEsol-D3, showing how the lattice becomes more closely packed upon the
inclusion of vdW interactions and the use of PBEsol. There is a clear linear
correlation between aeff values from different functionals, and equations
listed in the plots in Fig. S9 could be trivially applied to calculate lattice
parameters for any functional using the PBE values, with 99% accuracy. It is
noted here that during geometry optimization using any functional, the
prototype cubic/tetragonal/orthorhombic/hexagonal structure is used as the
starting point, with necessary SQS-based ionic mixing, but the cell size and
shape are allowed to change a small amount for lowering the energy, meaning a
lot of the compounds have slight deviations from ideal prototype structures.
Lattice constants will further change if full geometry optimization were to be
performed with HSE06, but as shown in our past work 20, HSE06-relaxation is
often unnecessary for HaPs.

Functional Dataset Property RMSE PBE 32 HaPs Lattice Constant (Å) 0.16 PBEsol
32 HaPs Lattice Constant (Å) 0.06 PBE-D3 32 HaPs Lattice Constant (Å) 0.05
PBEsol-D3 32 HaPs Lattice Constant (Å) 0.14 PBE 45 HaPs Band Gap (eV) 0.57
PBEsol 45 HaPs Band Gap (eV) 0.86 PBE-D3 45 HaPs Band Gap (eV) 0.81 PBEsol-D3
45 HaPs Band Gap (eV) 1.06 HSE-PBE+SOC 45 HaPs Band Gap (eV) 0.83 PBE-
corrected 45 HaPs Band Gap (eV) 0.47 HSE-corrected 45 HaPs Band Gap (eV) 0.35
PBE 28 Hybrid HaPs Band Gap (eV) 0.23 HSE-corrected 28 Hybrid HaPs Band Gap
(eV) 0.23 PBE-corrected 17 Inorganic HaPs Band Gap (eV) 0.44 HSE-corrected 17
Inorganic HaPs Band Gap (eV) 0.30

Table 2:  Root mean square errors between DFT-computed lattice constants and
band gaps and corresponding experimental values collected from the literature.
Figure 5: Pearson correlation coefficients between DFT-computed properties
($\Delta$H, Eg, and SLME) and 49-dimensional input descriptors, for (a) PBE,
and (b) HSE-PBE+SOC.

Fig. 4(d) shows the computed lattice constants for 32 selected compounds
plotted against their corresponding experimentally measured values. For better
visibility, the longer a/b/c lattice parameters from tetra/ortho/hex phases
are moved to a separate plot in Fig. S8(b). We find that all 4 functionals
match reasonably well with experiments, but PBEsol and PBE-D3 show the lowest
RMSE values of 0.06 Å and 0.05 Å respectively whereas PBE and PBEsol-D3 show
higher RMSE values of 0.16 Å and 0.14 Å respectively. It tracks that the
PBEsol corrections are desired for inorganic compounds and the D3 corrections
are suitable for hybrid compounds, but both together may be unnecessary and
lead to under-predicted lattice constants. PBE alone clearly over-predicts the
lattice constants which motivates the inclusion of the necessary functional
modifications. Table 2 further lists the lattice constant RMSE values for all
PBE-related functionals compared against experiments.

###  1.4 Correlation Between Properties and Material Descriptors

Next, the DFT dataset is mined to obtain some qualitative insights into the
physical and chemical factors that contribute to the HaP properties of
interest, namely $\Delta$H, Eg, and SLME, computed at all levels of theory.
For this purpose, we utilize a strategy applied by us in multiple previous
studies, of converting every ABX3 compound into unique composition-based
vectorial representations 38, 20. The “descriptor” for any compound is a
49-dimension vector, where the first 9 dimensions encode the fraction of any
species (Cs, MA, FA, Ge, Sn, Pb, Cl, Br, I) in the compound using a value
between 0 and 1, the next 4 dimensions provide a 1 or 0 score based on the
phase of the compound, and the next 36 dimensions represent weight-averaged
elemental properties of species at the A, B, and X sites, using well-known
properties such as ionic radii, electronegativity, and electron affinity. Fig.
5(a) and (b) show heatmaps capturing the Pearson coefficients of linear
correlation 65 between all 49 descriptor dimensions and the three properties
from PBE and HSE, respectively; corresponding plots for PBEsol, PBE-D3, and
PBEsol-D3 are pictured in Fig. S10.

Figure 6: Effect of ionic ordering and clustering on alloy properties: PBE-
computed $\Delta$H plotted against Eg for 20 structures each of 4 quaternary
compositions and 25 structures of a quinary composition. Figure 7: Pearson
correlation coefficients between PBE-computed properties ($\Delta$H and Eg)
and nearest neighbor pairs of B-site cations across 125 quinary and quaternary
alloy structures.

It is seen that from both PBE and HSE, the Cs-content, A-site boiling point,
and A-site electron affinity have a strong positive correlation with
$\Delta$H, while the FA-content, A-site ionic radius, and A-site atomic number
(approximated for MA and FA using an artificial continuation of Group I, as
described in past work 38) are strongly negatively correlated with $\Delta$H.
This is consistent with the earlier observation that many Cs compounds have
large $\Delta$H and are thus unstable, while the most stable compound with low
$\Delta$H are FA-based, with MA lying somewhere in the middle. Further, the
X-site properties such as ionic radius, melting point, electron affinity, and
electronegativity, have the strongest correlation with both Eg and SLME.
Increasing the size, the boiling/melting point, and the heat of
fusion/vaporization of the X-site species helps decrease the gap and increase
the PV efficiency, while increasing the electron affinity, ionization energy,
and electronegativity has the opposite effect. This is consistent the iodides
or iodide-bromide compounds having the most desirable optoelectronic
properties and the chlorides lying on the other end of the spectrum. The Cl-
content (I-content) also shows strong negative (positive) correlation with the
SLME, as shown in Fig. 5(a) and (b). Although many B-site and A-site
properties also show notable correlation with Eg and SLME, the contributions
are dominated by X-site species. Most of the qualitative correlations remain
the same from the PBEsol, PBE-D3, and PBEsol-D3 datasets as well, with the
inclusion of D3 corrections leading to some interesting changes in the
stability trends.

###  1.5 Effect of Ionic Ordering on HaP Alloy Properties

Every alloy composition investigated so far was simulated using the SQS
approach in a 2$\times$2$\times$2 or 2$\times$2$\times$1 cubic or non-cubic
supercell. While this is a fine representation of what the alloy would look
like on average, larger supercell sizes provide the opportunity to explore
different types of ordered and disordered arrangements. For this purpose, we
performed additional PBE computations on a series of “high-entropy” HaP alloys
belonging to the chemical space MA(Pb-Sn-Ba-Sr-Ca)I3, as shown in Table 1 and
discussed in the Methodology section. Equimolar quinary (5 ions mixed in 20%
fractions each at the B-site) or quaternary (4 ions mixed in 25% fractions
each at the B-site) compositions would exhibit the largest mixing entropy
contributions to the decomposition energy (kBT($\sum$ixiln(xi)) in Eqn. 1),
and are thus referred to as high-entropy perovskite alloys here. These
compositions are chosen because of the general interest in MA-based iodides
and in partially or completing replacing Pb in such compounds, via alloying at
the B-site. We consider 4$\times$4$\times$4 cubic supercells, starting from
the optimized MAPbI3 geometry, and perform random mixing of ions to obtain 20
structures each for the five possible quaternaries and 25 structures for the
quinary.

Species | Pb | Sn | Ba | Sr | Ca
---|---|---|---|---|---
Pb | Pb-Pb | Pb-Sn | Pb-Ba | Pb-Sr | Pb-Ca
Sn |  | Sn-Sn | Sn-Ba | Sn-Sr | Sn-Ca
Ba |  |  | Ba-Ba | Ba-Sr | Ba-Ca
Sr |  |  |  | Sr-Sr | Sr-Ca
Ca |  |  |  |  | Ca-Ca
Table 3:  A matrix of possible B1-B2 pairs in all MA(Pb-Sn-Ba-Sr-Ca)I3 alloys
simulated in 4$\times$4$\times$4 cubic supercells.

Fig. 6 shows the PBE Eg plotted against the PBE $\Delta$H for all 125 systems,
with different colors and shapes representing different compositions. It can
be seen that while the Sn-free composition MAPb0.25Ba0.25Sr0.25Ca0.25I3 shows
the largest Eg between 2.6 eV and 2.8 eV, all other compositions show Eg
spread between $\sim$ 1.75 eV and $\sim$ 2.5 eV. The Eg values in
MAPb0.25Sn0.25Sr0.25Ca0.25I3, for instance, range from 1.75 eV to 2.2 eV, with
$\Delta$H between -30 meV p.f.u. to -18 meV p.f.u., meaning that that band gap
could be changed by nearly 0.5 eV while keeping the material robustly stable
against decomposition, simply by altering the ionic ordering in the system. It
stands to reason that the properties displayed by the polymorphs of this
material would emerge from some kind of an ensemble average over all these
configurations. MAPb0.25Sn0.25Sr0.25Ba0.25I3 also shows negative $\Delta$H
across its 20 structures as Eg ranges from $\sim$ 1.8 eV to $\sim$ 2.1 eV,
implying that Pb-Sn-Sr combinations mixed with either Ba or Ca are good for
achieving stability and lower gaps.

Figure 8: Effect of lattice strain and octahedral distortion/rotation for 6
selected compounds: (a) PBE $\Delta$H and (b) Eg plotted against average
lattice strain, and (c) PBE $\Delta$H and (d) Eg plotted against average
octahedral distortion. CsPbCl3 is not pictured in some plots due to very large
$\Delta$H values.

The MAPb0.2Sn0.2Ba0.2Sr0.2Ca0.2I3 quinary also shows a wide range of Eg
between $\sim$ 1.8 eV and $\sim$ 2.3 eV, but $\Delta$H values become slightly
positive. The same is true for the MAPb0.25Ba0.25Sn0.25Ca0.25I3 and
MASn0.25Ba0.25Sr0.25Ca0.25I3 quaternaries, implying that Sn-Ba-Ca combinations
are less desirable when it comes to the HaP stability. To further understand
the effect of specific types of ionic clustering on the computed properties,
we generated descriptors for all 125 structures based on the number of Pb-Pb,
Pb-Sn, Sn-Sn, etc. pairs that occur in them; any B1-B2 pair is defined based
on the existence of B1-I-B2 combinations with B1 and B2 connected via a
bridging I anion; example structures are pictured in Fig. 1(b). The matrix in
Table 3 covers all possible B1-B2 pairs, resulting in 15 possible
combinations. Fig. 7 shows a heatmap of Pearson correlation coefficients
between the 15 types of B-cation pairs and the PBE computed $\Delta$H and Eg,
across the dataset of 125 quinaries and quaternaries. Interestingly, we find
that Pb-Pb, Pb-Sr, and Sr-Sr pairs are least desirable for improving the
perovskite stability, whereas Ba-Ba, Sn-Ba, and Ba-Sr pairs are helpful for
making $\Delta$H more negative. Furthermore, Sn-Sn and Pb-Sn pairs are most
responsible for reducing Eg, which is consistent with most Sn-based compounds
showing lower gaps than Sn-free compounds, and pairs such as Ba-Ba, Ca-Ca, Ba-
Sr, and Sr-Ca help increase the Eg. Overall, this analysis reveals that (a) by
changing the ionic ordering in a given HaP composition, Eg could be
drastically reduced or increased while keeping the material stable, and (b)
certain B-site cations clustering together may be helpful or harmful to the
stability and desired Eg.

###  1.6 Effect of Lattice Strain and Octahedral Distortion and Rotation on
HaP Properties

Finally, we investigate how intentional distortions applied upon the
perovskite lattice and/or on the corner-shared BX6 octahedra can change the
stability and band gap. For this analysis, 6 compounds given by the chemical
space (MA-Cs)(Pb)(I-Br-Cl)3 are considered, as listed in Table 1 and described
in the Methodology section. Fig. 8(a) and (b) show the PBE $\Delta$H and PBE
Eg respectively plotted against the average lattice strain in Å (over x, y,
and z dimensions), for 5 compounds. CsPbCl3 structures are missing in these
plots because of their much higher $\Delta$H. It can be seen from Fig. 8(a)
that $\Delta$H shows an expected trend, becoming more positive for both
negative (compression) and positive (elongation) lattice strain, with the
lowest energy shown by the no-strain structures as expected. Interestingly, Eg
generally shows a monotonically increasing behavior all the way from large
negative to large positive lattice strain. Once again, it should be noted that
compounds such as CsPbBr3 and MAPbBr3 can be kept metastable with small
amounts of strain, as shown by the negative $\Delta$H values, while changing
the Eg by nearly 0.3 eV.

Fig. 8(c) and (d) show the PBE $\Delta$H and PBE Eg respectively plotted
against the average octahedral distortion in Å. It can be seen that there are
many configurations of CsPbBr3, MAPbBr3, and MAPbCl3 with small amounts of
octahedral distortion that maintain $\Delta$H below 0 eV p.f.u. and change Eg
by as much as 0.5 eV in some cases. Distorted CsPbCl3 has a high Eg $>$ 2.5 eV
which does not change by a great amount as the average distortion increases
from $\sim$ 0.15 Å to 0.3 Å, and also shows large positive $\Delta$H.
Comparing plots in Fig. 8(c) and (d) to plots in Fig. 8(a) and (b), it is
clear that is a non-trivial correlation between octahedral distortion and the
properties as compared to lattice strain vs the properties. Ideally, one would
express the octahedral distortion in terms of how every atom in a BX6
octahedral unit is displaced relative to all other BX6 units it is connected
to, and correlations would be sought between an “octahedral distortion vector”
and the properties of interest. With the current representation, a range of
property values are often observed for the same average distortion, which
arises from different types of distortions resulting in lowering or raising
$\Delta$H or Egap. To wrap up this discussion, we plotted the entire dataset
of lattice/octahedra strained/distorted structures for all compounds in Fig.
9, in terms of the PBE Eg against the PBE $\Delta$H. By drawing a cut-off at a
low enough $\Delta$H value, dozens of possible structures could be obtained
for the same composition, showing a range of Eg values that may be suitable
for different applications.

Figure 9: PBE-computed Eg plotted against $\Delta$H for the dataset of
strained and octahedrally manioulated HaPs.

##  2 Perspective and Future Work

Our systematic first principles investigation reveals that the lattice
parameters, energetic stability, and optoelectronic properties of ABX3 halide
perovskites depend heavily on the identity and mixing of atoms at the A, B, or
X sites, the perovskite phase, the level of theory being used, the type of
ionic ordering in an alloy, and on possible lattice strains and octahedral
distortion or rotation. Ideally, a framework for predicting the properties of
HaPs, and designing novel HaP compositions/structures with multiple targeted
properties, must take each of these factors into account, potentially in
addition to important experimental conditions and past experimental
measurements. Generating large DFT datasets as in the present work is
invaluable for meaningful correlations and inter-dependencies between
different properties and different levels of theory, as well as for
establishing a reliable benchmark for computations against experiments. An
important question that will be raised is in the usefulness of different PBE
and HSE functionals going forward, when applying them to other related HaP
compositions and structures. Other than applying simple linear corrections
like pictured in Fig. 4, the best way to improve DFT-level predictions is to
continue applying higher levels of theory and tuning necessary parameters in a
compound-by-compound case, which is clearly not conducive to a high-throughput
treatment.

Figure 10: Band gaps computed from HSE-PBE+SOC for 5 compounds, plotted as a
function of the HSE06 mixing parameter ($\alpha$). The dotted horizontal lines
represent known experimental values.

As an example, we performed new HSE+SOC computations for 5 compounds by tuning
the mixing parameter $\alpha$ all the way from 0.20 to 0.50; these results are
shown in Fig. 10, along with known experimental values. We find that
$\alpha$=0.50 reproduces the experimental Eg perfectly for cubic FAPbI3, and
$\alpha$=0.48 work best for orthorhombic CsPbI3 and CsPbBr3. The same values
of $\alpha$ might work for cubic CsPbI3 and CsPbBr3 as well. Such an analysis
may be enormously useful for these particular compounds, as HSE+SOC
computations could now be performed using ideal $\alpha$ values for optical
absorption or defect calculations. Of course, it must be noted that the input
structure used for these static HSE+SOC computations is itself a huge factor;
for Fig. 10, we utilized the FAPbI3 and CsPb(I/Br)3 structures that matched
best with experiments, from across the different PBE-based functionals
discussed earlier. Once again, it should be noted that HSE+SOC with default
$\alpha$ provides means for uniform evaluation of properties across a large
chemical space, whereas tuning would need to be performed independently for
every compound. Machine learning (ML) models could potentially be trained in
the future that combine data from different PBE and HSE functionals to yield
the ideal $\alpha$ values for any given HaP composition/structure.

Currently, all the DFT data presented in this work is being utilized for
training a myriad of ML predictive models, using both composition-based
descriptors 38 explained earlier in the article and used extensively in the
past, as well as using state-of-the-art crystal graph-based neural network
(GNN) models. While the former are elegant and simple models, they cannot
typically be applied for multiple polymorphs of the same composition, whereas
GNN models can appropriately represent entire crystal structures as graphs and
use them as input to train predictive NN models that yield multiple properties
as output 66, 67, 68. A crystal graph representation automatically takes into
account the identity, mixing, and bond-lengths between atoms, any lattice or
octahedral distortion, ionic ordering, perovskite phase, etc. Furthermore, in
addition to material $\rightarrow$ property forward predictive models, inverse
design models could also be trained using techniques such as genetic algorithm
69, Bayesian optimization 70, generative neural networks 71, and variational
autoencoders 72, to design novel HaP atom-composition-structure combinations
that show the desired mix of negative $\Delta$H, PV-suitable Eg, and highest
possible PV efficiency.

##  3 Conclusion

In conclusion, we performed a series of first principles-based DFT
computations to investigate polymorphism in halide perovskites, in terms of
changing the perovskite composition and phase, ionic ordering in alloys, and
strain/distortions applied to the lattice and interconnected octehdral units.
Our work shows that the stability, band gap, and PV efficiency can vary quite
a lot for the same composition when the phase or ionic ordering is changed, or
when distortions are applied. Different semi-local and non-local DFT
functionals are explored, revealing that PBEsol or PBE-D3 may be necessary for
estimating correct lattice parameters, whereas PBE and/or HSE+SOC computed
band gaps can match very well with experiments if some corrections are
applied. We further find that different types or ordering or distortion could
be used to keep materials stable while drastically altering their band gaps.
Our linear correlation analyses further show the positive or negative effect
of different cations/anions, their elemental properties, and their clustering,
on the bulk properties of interest. All computational data is made available
to the community and is currently being utilized for trained multiple machine
learning models and for guiding collaborative experimental synthesis and
characterization.

## Conflicts of Interest

There are no conflicts to declare.

## Data Availability

All tabulated data and scripts used to analyze the DFT computed properties can
be accessed from
https://github.com/mannodiarun/perovs$\\_$mfml$\\_$ga/tree/polymorph$\\_$data.
Crystal structure files for all datasets discussed in this article are
attached with the Supplementary Information.

## Acknowledgements

This work was performed at Purdue University, under startup account
F.10023800.05.002 from the Materials Engineering department. This research
used resources of the National Energy Research Scientific Computing Center
(NERSC), the Laboratory Computing Resource Center (LCRC) at Argonne National
Laboratory, and the Rosen Center for Advanced Computing (RCAC) clusters at
Purdue.

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Supplemental material to "First Principles Investigation of Polymorphism in
Halide Perovskites"

Jiaqi Yanga and Arun Mannodi-Kanakkithodia

aSchool of Materials Engineering, Purdue University, West Lafayette, IN 47907,
USA; E-mail<EMAIL_ADDRESS>
111<EMAIL_ADDRESS>Figure S1: Visualization of the PBEsol, PBE-D3, and
PBEsol-D3 datasets in terms of $\Delta$H vs Eg and Eg vs SLME plots. Figure
S2: PBE-computed $\Delta$H, Eg, and SLME plotted against corresponding values
from PBEsol, PBE-D3, PBEsol-D3, and HSE-PBE+SOC. Figure S3: Multi-phase
property visualization for 12 FA-based hybrid HaPs, showing $\Delta$H, Eg, and
SLME from PBE and HSE-PBE+SOC. Figure S4: Multi-phase property visualization
for 27 Cs-based inorganic HaPs, showing $\Delta$H, Eg, and SLME from PBE and
HSE-PBE+SOC. Figure S5: Multi-phase property visualization for 16 MA-based
hybrid HaPs, showing $\Delta$H, Eg, and SLME from PBEsol, PBE-D3, and
PBEsol-D3. Figure S6: Multi-phase property visualization for 12 FA-based
hybrid HaPs, showing $\Delta$H, Eg, and SLME from PBEsol, PBE-D3, and
PBEsol-D3. Figure S7: Multi-phase property visualization for 27 Cs-based
inorganic HaPs, showing $\Delta$H, Eg, and SLME from PBEsol, PBE-D3, and
PBEsol-D3. Figure S8: (a) DFT-computed band gaps plotted against measured
values for 45 compounds, showing values from PBE, PBEsol, PBE-D3, PBEsol-D3,
and HSE-PBE+SOC. (b) DFT-computed non-cubic lattice constants plotted against
measured values for 6 compounds, showing values from PBE, PBEsol, PBE-D3, and
PBEsol-D3. Figure S9: Average lattice parameter, defined as the cube root of
the supercell volume per formula unit in any perovskite structure, plotted
from PBE against corresponding values from PBEsol, PBE-D3, and PBEsol-D3.
Figure S10: Pearson correlation coefficients between DFT-computed properties
($\Delta$H, Eg, and SLME) and 49-dimensional input descriptors, for (a)
PBEsol, (b) PBE-D3, and (b) PBEsol-D3 datasets of 146 compounds.