School of Mathematics and Physics, University of Science and Technology
Beijing, Beijing 100083, China School of Mathematics and Physics, University
of Science and Technology Beijing, Beijing 100083, China Northwestern
University]Department of Materials Science and Engineering, Northwestern
University, Evanston, Illinois 60208, United States

# Thermal stabilities Landscape of A2BB′O6 compounds

Yateng Wang  Beijing Advanced Innovation Center for Materials Genome
Engineering, University of Science and Technology Beijing, Beijing, 100083
China  Bianca Baldassarri  Department of Materials Science and Engineering,
Northwestern University, Evanston, Illinois 60208, United States  Jiahong Shen
Department of Materials Science and Engineering, Northwestern University,
Evanston, Illinois 60208, United States  Jiangang He<EMAIL_ADDRESS>Beijing Advanced Innovation Center for Materials Genome Engineering,
University of Science and Technology Beijing, Beijing, 100083 China  Chris
Wolverton<EMAIL_ADDRESS>[

###### Abstract

Perovskite oxides have been extensively studied for their wide range of
compositions and structures, as well as their valuable properties for various
applications. Expanding from single perovskite ABO3 to double perovskite
$A_{2}BB^{\prime}$O6 significantly enhances the ability to tailor specific
physical and chemical properties. However, the vast number of potential
compositions of $A_{2}BB^{\prime}$O6 makes it impractical to explore them all
experimentally. In this study, we conducted high-throughput calculations to
systematically investigate the structures and stabilities of 4,900
$A_{2}BB^{\prime}$O6 compositions (with $A$ = Ca, Sr, Ba, and La; $B$ and
$B^{\prime}$ representing metal elements) through over 42,000 density
functional theory (DFT) calculations. Our analysis lead to the discovery of
more than 1,500 new synthesizable $A_{2}BB^{\prime}$O6 compounds, with over
1,100 of them exhibiting double perovskite structures, predominantly in the
$P2_{1}/c$ space group. By leveraging the high-throughput dataset, we
developed machine learning models that achieved mean absolute errors of 0.0444
and 0.0330 eV/atom for formation energy and decomposition energy,
respectively. Using these models, we identified 803 stable or metastable
compositions beyond the chemical space covered in our initial calculations,
with 612 of them having DFT-validated decomposition energies below 0.1
eV/atom, resulting in a success rate of 76.2 %. This study delineates the
stability landscape of $A_{2}BB^{\prime}$O6 compounds and offers new insights
for the exploration of these materials.

$\blacksquare$ Introduction Perovskite oxides $AB$O3 ($A$= alkaline-earth and
rare-earth elements; $B$ = transition metal elements) have attracted
significant research interest due to their unique properties,1, 2 such as
metal-insulator transition,3 ferroelectrics and multiferroic behavior,4, 5
half-metallic spin-polarized electrical conductivity,6 magnetocaloric effect,7
colossal magnetoresistance,8 superconductivity,9 among others. These
exceptional properties render perovskite oxides highly promising for a diverse
array of applications, encompassing magnetic refrigeration,7 magnetic random
access memory,6 spintronic devices, photocatalysis,10 solid oxide fuel
cells,11, 12 energy storage,13 solar thermal chemical water splitting,14
thermoelectricity,15 and biomedical applications.2 Additionally, the capacity
to manipulate and regulate these properties positions perovskite as a
captivating field of study with the potential for substantial technological
advancements. This exceptional diversity and adjustable properties originate
from a wide range of compositions, encompassing nearly all the elements in the
periodic table, and the flexible structures of the corner-sharing octahedral.
The perovskite structure was originally named after Russian geologist
Perovski.16 It is now extended to stand for a group of the compounds that
contain the chemical formula $AB$O3 and are characterized by corner-sharing
$B$O6 octahedral.

Although the number of potential $AB$O3 compounds and the derivations
($A_{2}BB^{\prime}$O6, $AA^{\prime}B_{2}$O6, and $AA^{\prime}BB^{\prime}$O6)
exceed 105,17 only around 1000 of them have been synthesized experimentally so
far.18 Exploring such an extensive chemical phase space through
experimentation is exceedingly challenging due to the time-consuming nature of
experimental synthesis and the unique synthesis conditions required for each
compound. Therefore, a revolution has transpired in materials design and
discovery, as the simulation of experimental synthesis and characterization is
now achievable through high-throughput density functional theory (DFT)
calculations.19, 20, 21, 22 Thermodynamic stability is a key parameter that
broadly governs whether a material is expected to be synthesizable (or whether
it may decompose under certain environment conditions) and can be computed
easily using DFT.23 As a result, computational materials discovery through
thermodynamic computation has been employed in various categories of inorganic
materials,24 such as Heusler compounds,19, 22 perovskites,21, 25, 26, 27, 28,
29, 30 garnets,31 delafossite,32 mixed anion compounds,33, 34 quaternary
chalcogenides,35 etc., with applications ranging from lithium-ion
batteries,36, 37, 38 to transparent conductors,39, 40, 28, 32 optoelectronic
devices,41, 42, 30 electrides,43 catalysis,17 photovoltaics,17, 44
thermoelectrics,35, 33 and to solar thermochemical water splitting.21, 45
Emery et al. computationally screened 5329 compositions of $AB$O3 compounds
considering four different perovskite structures (cubic, rhombohedral,
tetragonal, and orthorhombic) and found 383 stable $AB$O3 compounds.21 Schmidt
et al. screened 250,000 cubic perovskite and antiperovskite compositions by
employing DFT calculations and discovered 500 new stable compounds.46 Körbel
et al. performed high throughput DFT screening on $ABX_{3}$ compounds in a
chemical space containing 64 elements ($A$ and $B$ are 55 metal elements and
$X$ = nitrogen, oxygen, chalcogenides, and halides).30 They found 199 stable
and metastable $ABX_{3}$ compounds with various properties from 32,000
candidates. Zhao et al. investigated lead-free inorganic halide double
perovskites computationally toward photovoltaic application.47 Eleven
promising photovalitic materials with high thermodynamic stability, suitable
band gaps, small carrier effective masses, and low exciton binding energies
were discovered by screening 64 candidate compositions.

Nevertheless, high-throughput computational screening requires substantial
computational resources and has a relatively low success rate. With the
advancement of data science, the use of machine learning (ML) and machine-
learning-assisted high-throughput screening has become increasingly prevalent
in the field of materials science, leading to accelerated materials
discovery.48, 49 For example, Faber et al. screened 2 million elpasolite
($ABC_{2}D_{6}$ chemical formula with $Fm\bar{3}m$ space group) compounds
using a ML model trained with ten thousand DFT formation energies.50 The
accuracy of this model reaches 0.1 eV/atom and 90 stable compounds were
identified. Li et al. trained an ML model of thermodynamic stability using the
convex hull data of 1,929 perovskite oxides stored in the Materials Project.51
The RMSE (root mean squared error) of 28.5 meV/atom and MAE (mean absolute
error) of 16.7 meV/atom were accomplished by using the selected 70 features,
which are based on elemental properties. After screening 21,316 compositions
with the established ML model, 764 new perovskites were predicted and 98 of
them were validated to be stable by DFT calculations. Kim et al. developed a
regression model on formation energies of $A_{2}BB^{\prime}$O6 double
perovskites with R2 (coefficient of determination) of 0.98 and RMSE of 0.175
eV/atom and a classification model of the stability with F1-score of 0.771 by
means of training the formation energies of 1,052 double perovskites in the
Materials Project database.52, 53 These two models are employed to discover
synthesizable $A_{2}BB^{\prime}$O6 compounds from 11,763 candidates. The DFT
validated accuracy of these two models are 0.646 and 0.733 for the convex hull
energy and F1-score, respectively. A ML model based on crystal graph and
convolutional neural networks (CGCNN) established by Xie and Grossman was
trained with DFT-calculated data in the Materials Project to predict new
perovskites and 228 new compounds were predicted to be synthesisable.54 Li et
al. developed a deep transfer learning approach based on the formation
energies of spinel oxides and predicted 1,314 thermodynamically stable
perovskite oxides by training 5,329 spinel oxides and 855 perovskite oxides.55

Despite its advantages, machine learning necessitates a substantial volume of
data for model training, which has to be either carefully extracted from
experiments/literature or generated through massive DFT calculations when the
data is unavailable in established databases. Therefore, a nature marriage
between high-throughput DFT calculations and machine learning can
significantly accelerate material discovery procedure, allowing to access
materials in the compositional spaces that are too large even for high-
throughput DFT calculations. Schmidt et al. developed a ML model utilizing the
formation energies of 249,984 cubic $ABX_{3}$ perovskites computed by high-
throughput DFT calculations.46 Their ML model can reproduce the formation
energy data with errors around 0.1 eV/atom.

In this work, we performed a high throughput DFT calculations on
$A_{2}BB^{\prime}$O6 ($A$= Ca, Sr, Ba, and La; $B$ and $B^{\prime}$ are metal
elements) compositions with 10 common (5 perovskite and 5 non-perovskite)
crystal structures. We found 2022 stable/metastable compounds and 1785 of them
favor perovskite structures. Our statistic analysis based on ML model shows
that the formation energy of $A_{2}BB^{\prime}$O6 compounds is strongly
correlated to element properties such as the number of unfilled electrons in
the outermost layer of an atom, the number of valence electrons in an atom,
the thermodynamic properties of elements, and weakly correlated to the known
features such as tolerant factors. With the established ML model, we screened
the $A_{2}BB^{\prime}$O6 compositions out of our chemical space and discovered
612 stable/metastable $A_{2}BB^{\prime}$O6 compounds by verifying our
prediction using the OQMD database.

$\blacksquare$ Methodology

DFT parameters. In this work, all DFT calculations are performed using the
Vienna ab initio Simulation Package (VASP).56, 57 The Perdew-Burke-Ernzerhof
(PBE) version of the exchange-correlation functional,58 projector-augmented
wave (PAW) method,59, 60 and the plane wave basis sets with energy cutoff of
520 eV were employed for structure relaxation and static calculations. The
$\Gamma$-centered $k$-point grids with a density of 8000 $k$-points per
reciprocal atom (KPPRA) were used. The compounds containing the elements with
unfilled $d$-shells were carried out with spin-polarized calculations in a
ferromagnetic spin configuration, and the magnetic degree of freedom allowed
to relax to self-consistency within a primitive unit cell. Other parameters,
such as Hubbard U values,61 are consistent with the OQMD setting.62

Chemical space. In this work, we considered 4 commonly observed $A$-site
elements (Ca, Sr, Ba, and La) in perovskites and 50 metal elements for $B$ and
$B^{\prime}$ sites, see the element list in Figure 1. Perovskite is the most
common structure type of $A_{2}BB^{\prime}$O6 and its $A$-site has high
coordination number (e.g., 12 for cubic perovskite), which requires the cation
with large radius, i.e., Pauling’s first rule.63 We considered all the
combination of $B$ and $B^{\prime}$ elements, which generates 4,900 (4
$\times$ $C_{50}^{2}$) $A_{2}BB^{\prime}$O6 compositions. Rare earth elements
are excluded from $B$ and $B^{\prime}$ sites because they are not common at
the $B$-site of perovskite.18

Figure 1: $A$, $B$, and $B^{\prime}$-site elements used in the
$A_{2}BB{{}^{\prime}}X_{6}$ compounds of this work.

Thermodynamic stability at T=0 K. The thermodynamic stability of a compound at
0 K can be estimated by its energetic difference from the convex hull of the
formation energies in a given chemical space, which is the energy difference
between the formation energy of the compound and that of the lowest linear
combination of the phases at the same composition. Previous studies indicated
that the distance to the convex hull of formation energies (commonly referred
to as “convex hull distance” or “stability”) serves as a reliable metric for
assessing the likelihood of experimental synthesizability: a convex hull
distance of zero or close to zero is a strong indicator of successful
experimental synthesis.64, 65, 21, 33 All unique and ordered phases in the
OQMD (www.oqmd.org), a total of over 1,000,000 phases (as of June 2021),
including $\sim$40,000 known compounds from the Inorganic Crystal Structure
Database (ICSD)66, 67 and $\sim$600,000 hypothetical compounds based on common
binary, ternary, and quaternary structure types,68 were used for all convex
hull constructions. The hull distance ($\Delta{E_{\rm{h}}}$) of an
$A_{2}BB^{\prime}$O6 compound is defined as the energy difference between its
formation energy ($\Delta{E}$) and the lowest linear combination of the
formation energies of the phases in the $A$-$B$-$B^{\prime}$-O phase space
with an overall composition of $A_{2}BB^{\prime}$O6. These phases are referred
to as “competing phases” in this work.

Crystal structure determination. Crystal structure plays a crucial role for a
compound as it determines its properties. Although many compounds can have
several polymorphs at ambient condition, the lowest energy structure is
generally the most common form of a compound. Moreover, the lowest energy
structure is particularly useful to precisely evaluate thermal stability of a
composition. However, it is very computationally expensive to determine the
lowest-energy crystal structure of a composition containing multiple elements.
Although perovskite is the most common structure of $A_{2}BB^{\prime}$O6
chemical formula, many non-perovskite structures do exist and have lower
energy than perovskite. Furthermore, perovskite has many distorted structures,
which have lower energies than the cubic one. Owing to the high computational
cost of convention crystal structure prediction approaches, such as USPEX 69
and CALYPSO 70, 71, it is too expensive to perform convention crystal
structure search for so many compositions. Therefore, we performed high-
throughput structure search using the prototype structures approach. Since
double perovskite structures are the most commonly observed structures in
$A_{2}BB^{\prime}$O6 and the distortion of perovskite is very common and is
strongly correlated to $A$ and $B$-site cation size mismatch, we considered
several double perovskite structures with different octahedral distortion to
cover a wide range of radius mismatch between $A$ and $B$ ($B^{\prime}$)-site
cations, which is often characterized by the Goldschmidt tolerance factor
($\tau$).72 Here, we distinguish the $B$/$B^{\prime}$ ordered
$A_{2}BB^{\prime}$O6 perovskite from $B$/$B^{\prime}$ disordered one. If the
$B$-site cations are disordered, the compound is labeled as doped or
disordered single perovskite, instead of double perovskite. There are three
common pattern of forming $B$/$B^{\prime}$ ordered double perovskite: rock-
salt, columnar, and layered.73 The know $A_{2}BB^{\prime}$O6 compounds
summarized by Vasala and Karppinen18 indicates 58.1 % of $A_{2}BB^{\prime}$O6
perovskites are rock-salt ordered, while only 0.5% of them are layered. The
columnar ordering is even more uncommon. Our further analysis on rock-salt
ordered $A_{2}BB^{\prime}$O6 (also known as Elpasolite structure, after the
mineral K2NaAlF6) compounds show that 54.3 %, 30.2 %, 8.7 %, and 3.4 % of them
are monoclinic, cubic, tetragonal, and orthorhombic, respectively. Among
monoclinic, $P2_{1}/c$ is the dominant space group, which can be viewed as the
$B$/$B^{\prime}$ rock-salt ordered $Pnma$ of single perovskite. Similar to
$Pnma$, which is the most common structure of single perovskite $AB$O3,74
$P2_{1}/c$ is a distorted perovskite formed by octahedral rotation and titling
with respect to the cubic structure ($Fm\bar{3}m$).75 If only the rotation
($X_{3}^{+}$ in the language of irreducible representation, similar to
$R_{4}^{+}$ of perovskite) is involved, the subgroups of $Fm\bar{3}m$ are
$I4/m$ ($a^{0}a^{0}c^{-}$, Galzer notation76), $C2/m$ ($a^{0}b^{-}b^{-}$), and
$R\bar{3}$ ($a^{-}a^{-}a^{-}$). If only the tilting ($\Gamma_{4}^{+}$, similar
to $M_{3}^{+}$ of perovskite $AB$O3) is involved, the subgroups of
$Fm\bar{3}m$ are $Pn\bar{3}$ ($a^{+}a^{+}a^{+}$), $P4_{2}/nnm$
($a^{0}b^{+}b^{+}$), and $P4/mnc$ ($a^{0}a^{0}c^{+}$). If both rotation and
tilting are involved, 6 subgroups may form. But the most common structures are
$P2_{1}/c$ ($a^{+}b^{-}b^{-}$) and $C2/c$ ($a^{+}b^{+}c^{-}$). However,
rotation usually gains more energy than tilting,77, 78 therefore the most
commonly observed double perovskite structures are $P2_{1}/c$, $C2/m$,
$R\bar{3}$, $I4/m$, and $Fm\bar{3}m$. Also, the structure distortion is
strongly dependent on temperature and phase transitions are commonly observed
in many double perovskites. $Fm\bar{3}m$ is the high-temperature phase of many
double perovskite compounds, though the rotation and tilting distorted
structures are commonly adopted at low temperature. Moreover, Cs2Au2Cl6-type
($I4/mmm$) double perovskite structure is formed through charge disproportion
of 2Au2+ $\rightarrow$ Au+ \+ Au3+, which is similar to BaBiO3.78
Ba2InCuO6-type ($P\bar{3}m1$) and Ba2NiTeO6-type ($R\bar{3}m$) are the
commonly observed non-perovskite structures with large $A$-site radius, while
Li2SnTeO6-type ($Pnn2$) and Ni3TeO6-type ($R3$) are more common in the non-
perovskite $A_{2}BB^{\prime}$O6 compounds with small $A$-site radius.79

Therefore, we use the $B$/$B^{\prime}$ rock-salt-ordered cubic double
perovskite ($Fm\bar{3}m$) as the first prototype structure to perform the
screening of 4,900 compositions because this structure (high symmetry and less
free parameters, only contains 10 atom per primitive unit cell) requires a
smaller amount of computational resources. Then, for the 2815 compositions
that are within 200 meV/atom from the convex hull, we perform further
prototype structure screening with the other 9 structures, including 5
perovskite with different distortion ($P2_{1}/c$, $C2/m$, $I4/m$, $R\bar{3}$,
$I4/mmm$) and 4 non-perovskite (Ba2InCuO6-type, $P\bar{3}m1$; Ba2NiTeO6-type,
$R\bar{3}m$; Li2SnTeO6-type, $Pnn2$; Ni3TeO6-type, $R3$) structures, see
Figure S1 of the supporting information. As mentioned above, these structures
cover a wide range of radius mismatch between $A$ and $B$-site cations. This
procedure can significantly reduce the computational cost but may miss the
compounds with the ground state structure that is more than 200 meV/atom lower
than $Fm\bar{3}m$. However, this chance should be quite low since 200 meV/atom
is a pretty large number and $Fm\bar{3}m$ is a reasonable guess for most
compounds.

Machine learning model. Our DFT computed formation energies and decomposition
energies of these compounds are used to train machine learning modes to unveil
the underlying correlation between formation energy/stability and the
fundamental properties of constituent elements and to accelerate new
$A_{2}BB^{\prime}$O6 compounds discovery. 443 element-based features are
generated using Matminer,80 including element information such as Mendeleev
number, occupation of $d$-orbitals, electronegativity, the properties of
elemental bulk (melting temperature, covalent radius, etc.), and the
properties of forming common compounds (oxidation state). Additionally, we
adopted 146 features of perovskite from literature.81

Since both formation energy and decomposition energy are continuous numbers,
23 regression models are tested in this study. All the algorithm except
XGBoost, which is from XGBoost package,82 are adopted from scikit-learn
package.83 The hyper-parameters are determined using a 5-fold cross-validation
with a view to maximize accuracy while minimize the standard deviation on
unseen data. We use recursive feature elimination (RFE) and Spearman rank to
perform feature selection, we use greedy search to find the optimal
parameters. Please see Table S1 of the supporting information for more
details. The data is randomly split into 80% for training and 20 % for cross-
validation for 10 times. The averaged R2 score, RMSE, and MAE are used to
evaluate the performance of models.

$\blacksquare$ Results and Discussion
Thermodynamic stability at 0 K. Thermodynamic stability forms the basis of
material properties. It is particularly critical when predicting as-yet
synthesized materials, i.e., computational materials discovery. Ideally, the
thermodynamic stability should be evaluated at room temperature or the
conditions of service. However, the entropy contribution to free energy,
especially the vibrational entropy of solids, is computationally costly for a
large set of materials. Also, the ambient pressure (10-4 GPa) is approximately
to zero for solids, therefore the contribution of work (product of pressure
and volume) is negligible to free energy at ambient condition. Therefore, we
ignore the contribution from entropy and work terms in the Gibbs free energy,
and only use zero-kelvin energy to estimate thermodynamic stability of
solids.33 Following the DFT screening for 4,900 compositions with $Fm\bar{3}m$
structure, our DFT screening with 9 prototype structures for the compositions
that are within 200 meV/atom above the convex hull generate 420 stable and
1,602 metastable (convex hull distance is less than 100 meV/atom) compounds,
which is much larger than the known $A_{2}BB^{\prime}$O6 compounds reported in
the ICSD and literature, see Table 1.18, 66, 67 Note the compounds reported in
the ICSD and literature include both $B$/$B^{\prime}$ ordered double
perovskites, doped perovskite, and non-perovskite, whereas in this work we
only studied $B$/$B^{\prime}$ ordered $A_{2}BB{{}^{\prime}}$O6 compounds. The
configurational entropy of $B$/$B^{\prime}$ disorder can lower the free energy
of the $B$/$B^{\prime}$ disordered compounds at elevated temperatures, and
therefore the observed $A_{2}BB{{}^{\prime}}$O6 structure may be
$B$/$B^{\prime}$ disordered at finite temperature if the energy gained by
ordering $B$/$B^{\prime}$ can be canceled by configurational entropy at
certain temperatures. The distribution of the convex hull distance of all
these $A_{2}BB{{}^{\prime}}$O6 compounds are depicted in Figure 2 and 3 for
Ba/Sr (Ba: upper triangle; Sr: lower triangle) and Ca/La (Ca: upper triangle;
La: lower triangle), respectively. The compounds that have been experimentally
reported are indicated by smaller open squares within the color-coded circles.
We can see that most of the reported compounds exhibit green color, i.e., on
the convex hull or very close to the hull. This is consistent with previous
observations in many other compounds.21, 33, 84, 85

Figure 2: Heat map of the convex hull distance of Ba${}_{2}BB^{\prime}$O6
(upper triangle) and Sa${}_{2}BB^{\prime}$O6 (lower triangle) compounds. White
unfilled circles indicate that the calculations of the corresponding compound
fails. Small empty squares inside the larger circle indicate that the compound
has been reported in the ICSD or literature. Figure 3: Heat map of the convex
hull distance of La${}_{2}BB^{\prime}$O6 (upper triangle) and
Ca${}_{2}BB^{\prime}$O6 (lower triangle) compounds. White unfilled circles
indicate that the calculations of the corresponding compound fails. Small
empty squares inside the larger circle indicate that the compound has been
reported in the ICSD or literature.

Note the $A_{2}BB{{}^{\prime}}$O6 compounds reported in literature haven’t
been included in the OQMD prior to our high throughput DFT calculations, the
“re-discovery” of these compounds confirms our metric of stability for these
compounds, which is more reliable than formation energy (all the formation
energies of the studied compounds are negative but the majority of them are
unstable) because it includes phase competition, which is more important in
multi-component systems. Comparing with experiments, there are only 7
exceptions, as indicated in Figure 4: Ba2VBiO6, Sr2MoSnO6, Ca2PdWO6,
Ca2AsBiO6, La2FeRhO6, La2VCuO6, La2ZnTaO6. The lowest-energy structure of
Ba2BiVO6 in our calculations is $R3$, which is 131 meV/atom above the hull,
while the experimentally observed structure is pretty complex: a non-
perovskite structure with $P2_{1}/a$ space group, large unit cell (a=24.640 Å,
b=7.73 Å, c=5.63 Å), and octahedrally coordinated Bi but tetrahedrally
coordinated V.86 The lowest-energy structure of Ca2VBiO6 we found is
$P2_{1}/c$, which is 168 meV/atom above the convex hull, while the
experimental structure is $Cmc2_{1}$, which is a non-perovskite structure with
VO4 tetrahedral and (BiO2)- chain.87 This is presumably because V5+ is too
small for octahedral and therefore the tetrahedral coordination is more
favorable than octahedral. Our calculated energy of Ca2VBiO6 with $Cmc2_{1}$
space group is 201 meV/atom lower than our $P2_{1}/c$ phase, which is more
than enough to bring it to the convex hull. Ca2AsBiO6 is isostructural with
Ca2VBiO6 and is expected to have a smaller energy difference because As5+ is
bigger than V5+ and the octahedral coordinated As5+ is less unfavorited. Our
calculated $Cmc2_{1}$ Ca2AsBiO6 is 159 meV/atom lower in energy than
$P2_{1}/c$, which is the lowest energy phase of our prototype search and is
159 meV/atom above the convex hull. Therefore, $Cmc2_{1}$ Ca2AsBiO6 is just on
the convex hull of formation energy. Our relaxed structure of Ca2PdWO6
($C2/m$), which is heavily distorted from $P2_{1}/c$ prototype structure
because of the strong preference of Pd2+ to square planer coordination rather
than octahedral,88 is 192 meV/atom lower in energy than the experimentally
reported structure ($Pmm2$), where both W6+ and Pd2+ cations are octahedrally
coordinated with six oxygen anions.89 However, our $C2/m$ structure is still
115 meV/atom above the convex hull, which means the experimental structure is
307 meV/atom above the convex hull. Therefore, it is plausible that the
experimental structure is incorrect or the composition is not correct.
Although La2ZnTaO6 is collected in the ICSD, the corresponding reference
reported is actually La2ZnTaO5N.90 The experimentally synthesized La2FeReO6
and Sr2MoZrO6 are the $B$/$B^{\prime}$ disordered perovskite with space group
$Pnma$.91, 92 The lowest-energy structure of our calculated Sr2MoZrO6 and
La2FeReO6 double ($B$/$B^{\prime}$) perovskite is $P2_{1}/a$, which is 125 and
167 meV/atom above the convex hull, respectively. The configurational entropy
induced by $B$/$B^{\prime}$ disorder is ln2 $K_{\rm B}$T ($K_{\rm B}$ is the
Boltzmann constant and T is the temperature) and the free energy gain at 300 K
is 17 meV/atom. Although these two compounds are metal and La2FeReO6 has
magnetic moments and the electric and magnetic entropies have certain
contribution to free energy as well, the gained free energy due to entropy is
unlike to overcome such large energy (more than 100 meV/atom) at room
temperature. Therefore, these two compounds are metastable. In summary,
$\Delta{\rm E_{\rm stable}}$ = 0.1 eV/atom is a reasonable threadhold for
metastable $A_{2}BB{{}^{\prime}}$O6 compounds.

Since $A_{2}BB^{\prime}$O6 are typical ionic compounds, the compounds that
satisfy charge balance are indicated with open squares in the circles in
Figure S2 and S3 of the supporting information. The nominal charges of the
elements studied in this work are from Pymatgen, see Table S2 of the
supporting information. We can see that all compounds that are on the hull are
charge-balanced, while most unstable compounds (i.e., $\Delta{E}_{\rm stab}$
$>$ 100 meV/atom) are not charge-balanced, which highlights the importance of
charge balance in identifying stable/metastable $A_{2}BB{{}^{\prime}}$O6
compounds. Therefore, charge balance can be used as a pre-screening criteria
for perovskite discovery in the future. The full list of the stable and
metastable compounds are tabulated in the supporting information csv table. As
shown in Figure 2 and 3, the $B$/$B^{\rm\prime}$ pairs of the
stable/metastable Ba${}_{2}BB^{\prime}$O6 compounds are mainly from the
combinations of 4$d$ and 3$d$ as well as 5$d$ and 3$d$ transition metals, but
fewer pairs are from 4$d$ and 5$d$ elements. This is presumably due to the
diverse oxidation states and smaller ionic radius of 3$d$ transition metal
elements, which are easy to satisfy charge balance and favor octahedral
coordination. Since Ba, Sr, and Ca are divalent cations, the elements with
oxidation states from +1 to +7 can be $B$ and $B^{\prime}$, more charge-
balanced compositions are expected in these divalent cations than trivalent
La, where only the elements with oxidation states from +1 to +5 can be $B$ and
$B^{\prime}$. This is inline with our observation of charge-balance in Figures
S2 and S3 of the supporting information, as well as a previous study.18
Lastly, many Technetium-containing compounds are predicted to be stable in
this work, but they are not yet observed in experiments, which is likely due
to its radioactivity.

Figure 4: DFT-calculated formation energy vs convex hull distance of all the reported $A_{2}BB^{\prime}$O6 compounds in the chemical space defined in this work. A hull distance Estable = 0 (on the convex hull) indicates the compound is thermodynamically stable at 0 K. Table 1: Summary of the A${}_{2}BB{{}^{\prime}}$O6 compounds studied in this work. Stable (S), metastble (M), and unstable (U) are the compounds with convex hull distance of 0, 0 - 100, and $>$ 100 meV/atom, respectively. The experimentally observed compounds are under “ICSD” and “Literature”. Here, “Literature” means the compounds are reported in literature but are not included in the ICSD. $A$ site | S | M | ICSD | Literature
---|---|---|---|---
|  |  | S | M | U | S | M | U
Ba | 158 | 394 | 48 | 21 | 0 | 33 | 32 | 1
Sr | 140 | 416 | 58 | 34 | 0 | 21 | 25 | 1
Ca | 79 | 421 | 12 | 27 | 3 | 24 | 31 | 0
La | 43 | 371 | 10 | 40 | 1 | 10 | 37 | 1

Low-energy structure. As mentioned above, 10 prototype structures are employed
in our high throughput DFT calculations to filter out the low-lying
$A_{2}BB{{}^{\prime}}$O6 compounds. Our DFT screening yields 381 stable double
perovskites, 1404 metastable double perovskites, 38 stable non-perovskites,
and 198 metastable non-perovskite compounds, which cover nearly all the
compositions observed in experiments (see Table 1). The full lists of the
perovskite and non-perovskite compounds are in single csv file of the
supporting information.

Goldschmidt’s tolerance factor ($\tau=\frac{r_{A}+r_{\rm
O}}{\sqrt{2}(r_{B}+r_{\rm O})}$, where $r_{A}$, $r_{B}$, and $r_{\rm O}$ are
the ionic radii of $A$, $B$, and O ions, respectively), which is an indicator
of the mismatch between $A$-O and $B$-O layers, is widely used to determine
the structural stability and distortion of perovskite.72 The distribution of
stable and metastable $A_{2}BB{{}^{\prime}}$O6 perovskites with respect to
Goldschmidt’s tolerance factor is shown in Figure 5. The span of $\tau$ for
stable double perovskite is from 0.82 to 1.10 and most of the stable double
perovskites have $\tau$ around 0.97. The metastable double perovskite have a
wider spread (from 0.72 to 1.11). This is expected because both small ($<$
0.9) and large $\tau$ ($>$ 1.0) suggest large mismatch between $A$-O layer
$B$-O layers, which result in large lattice strain and therefore low
stability. Therefore, high-pressure and high-temperature methods are commonly
used to synthesize perovskites and related compounds with low $\tau$.93, 79
Our finding is similar to the analysis for the perovskites observed in
experiment.18

Figure 5: The number of stable and metastable $A_{2}BB{{}^{\prime}}$O6 double
perovskite (a) and non-perovskite (b) compounds with Goldschmidt’s tolerance
factor.

The correlation between Goldschmidt’s tolerance factor and space group of the
stable and metastable perovskites is shown in Figure 6. Overall, the compounds
with lower symmetry (lower space group number) has smaller $\tau$. Note there
are more space groups than our prototype structures because of symmetry
breaking during structure relaxation. For instance, $P1$ and $P\bar{1}$ are
mainly from $P2_{1}/c$ and $C2/m$, respectively. As expected, $Fm\bar{3}m$ has
$\tau$ $\sim$ 1.0 with a much narrow spread. This is slightly different from
experiments because our calculations are performed at zero kelvin and the
structure distortions are strongly dependent on temperature, i.e., the higher
the temperature, the less the distortion.

Figure 6: Distribution of the Goldschmidt’s tolerance factor for stable (a)
and metastable (b) $A_{2}BB{{}^{\prime}}$O6 double perovskites with respect to
space group.

The space group distribution of the stable and metastable double perovskites
are shown in Figure 7. Most of compounds adopt $P2_{1}/c$ space group, which
is the combination of octahedral rotation and tilting, consisting with the
experiment observation.18 The second ($P\bar{1}$) and seventh ($P1$) favorable
structures are not our prototype structures and these structures are due to
the distortion during relaxation. Similarly, three space groups ($R\bar{3}$,
$Pc$, $P2_{1}$) that are only observed in metastable compounds are due to
symmetry breaking during structure relaxation. As previously mentioned,
perovskites usually undergo structural phase transition associated with
octahedral distortion and, the lower the temperature, the large octahedral
distortion. Therefore, our 0 K calculations find more distorted perovskite
than experiments.

Recently, Filip and Giustino revisited a rigid-sphere model of the perovskite
geometry and proposed a generalised tolerance factor by taking into account
electronegativity.94 However, our results show that this tolerance factor has
very similar results as Goldschmidt’s tolerance factor, see Figure S4 and
Figure S5 of the supporting information.

Figure 7: Distribution of stable and metastable compounds for stable (a) and
metastable (b) $A_{2}BB{{}^{\prime}}$O6 double perovskites with respect to
space group.

Statistic analysis based on machine learning methods. In this section, we use
machine learning models to analysis formation energies and decomposition
energies of our high throughput DFT calculated 4,855 compositions to unveil
the hidden connection between formation energy (and decomposition energy) and
composition. 431 features based on atomic characters of constituent elements
are generated using Matminer80 and are adopted from literature. Twenty one
most common machine learning algorithms are examined here, including Random
Forest,95 Gradient Boosting Regressor,96 and XGBoost. In order to verify the
performance of these models, the DFT datasets were shuffled and split into 80%
and 20% randomly for training and cross-validation, respectively. Note our
training data is different from previous studies,46, 50 we used the lowest
energy of the 10 prototype structures screened from our high throughput DFT
calculations for each composition, which takes into account different
structure distortion and structural variability, and therefore is more
challenging to build this model than the formation energies (and decomposition
energies) of the compounds with the same structure, where the structural
information is implicitly utilized in fact.

Machine learning model of formation energy. As shown in Figure S6 of the
supporting information, formation energies of these compounds are continuous
and spread from -3.7 to -1.2 eV/atom with the maximum number of compounds at
-2.5 eV/atom, which is appropriate to use regression models. Therefore,
several commonly used regression models in material science are examined. The
performance of all these models are presented in Figure S7 of the supporting
information. Among them, XGBoost, Random Forest, and Gradient Boosting show
the best performance. Therefore, we focus on these three models and carried
out feature filter to avoid overfitting. We reduce the number of features from
589 to $\sim$ 30 by gradually removing the features that have less impact on
model’s performance, i.e., the feature has smallest effect on R2 and MAE. All
the generated features are employed in these 21 models and their performances
are evaluated using R2 and MAE. As we can see in Figure S8 of supporting
information, the R2 of Random Forest is not improved when the number of
features is over 20, but it decreases rapidly when the number of features is
less than 20. Therefore, we keep 20 features for the subsequent training in
the Random Forest model. Similarly, we find 30 features are sufficient for
Gradient Boosting. XGBoost, provides three feature sorting methods, namely
gain, weight, and cover. The gain metric signifies the contribution of each
feature during node splitting, and its calculation incorporates the reduction
in loss resulting from feature splitting. If a feature can significantly
reduce its loss when split, its gain will be higher. Weight represents the
number of times the feature is used to split the node. During the construction
of the tree, each feature is assigned importance based on the frequency with
which it is used to split the dataset; the more times it is used, the higher
its weight. Cover refers to the number of observations covered by a feature in
all splits, and the coverage reflects the importance of the feature in the
entire dataset. Since gain can quantify the contribution of each feature to
loss reduction in each split, it directly measures the actual impact of each
feature on model performance, thereby aiding in the understanding of the
importance the model assigns to features. Therefore, the gain method is more
robust compared to frequent splits and can help to reduce the risk of model
overfitting. Then, we subsequently select the features obtained by gain as the
input features of our XGBoost model. Results from weight and cover are
presented as well, see Figure S9 of supporting information. Therefore, we
choose 20 features with feature importance sorted by gain (labeled as XGBoost
with gain), 30 features with feature importance sorted by weight (labeled as
XGBoost with weight), and 35 features by cover (labeled as XGBoost with
cover). Then, we conducted 20 training and cross-validation with randomly
dataset splitting for each of these three models and ranked the feature
importance after each training to obtain the most important features.

Figure 8: Machine learning prediction vis DFT calculated formation energy and
decomposition energy. (a)-(c) performances of XGBoost, GBR, and random forest
(RF) algorithm on formation energy, respectively. (d)-(f) performances of
XGBoost, GBR, and Random Forest algorithm on convex hull distance,
respectively.

Then, we used Spearman’s rank correlation to further filter the redundant and
correlated features. For the top features from 20 training iterations for each
model, we compute the Spearman’s rank correlation coefficient between each
pair of features, see Figure S10 of the supporting information. For any pair
of features with a correlation coefficient greater than 0.8, we calculate the
impact of the feature on the formation energy and remove the feature with
lower correlation coefficient. This approach allows us to identify and
eliminate non-independent features in a data-driven and statistically sound
manner. After this feature filtering step, the number of features is further
reduced for all these models. The final number of features for each model are:
14 features for Random Forest, 24 for Gradient Boosting, 20 for XGBoost with
gain, 22 for XGBoost with weight, and 35 for XGBoost with cover. It is evident
that Random Forest requires much less features than other algorithms. These
reduced feature sets allow more efficient and accurate modeling formation
energy while minimizing the risk of overfitting. Secondly, we can see that
Random Forest, Gradient Boosting, and XGBoost models outperform others.
XGBoost has the highest R2 (0.9799) and smallest MAE (0.0458 eV/atom), which
is followed by Random Forest (R2=0.9751, MAE=0.0500 eV/atom) and Gradient
Boosting (R2=0.9557, MAE=0.0747 eV/atom), see Figure 8. However, the
differences among these models are very small, indicating the robustness of
our models. The results from other methods are presented in Table S3 and Table
S4 of the supporting information. Our MAE is smaller than previous studies on
perovskite oxides,50, 55 especially considering we have more diverse crystal
structures than these previous studies.

To unveil the underlying physics of our models, we conduct feature analysis
for the three best models utilizing the SHAP algorithm,97 which comprises two
parts: the feature importance derived from the tree models themselves and the
importance analysis generated by SHAP. The feature importance analysis shows
that 6 of the top 10 features ranked by SHAP are common for all ML models,
namely MagpieData mean Nunfilled (average number of unfilled electrons in the
outermost shell of atoms in a compound), frac d valence electrons (Number of
valence electrons in d orbital), avg ionic char (computational features in
matminer for quantifying ionic properties in materials80), MagpieData maximum
Nvalence (Number of valence electrons), DemlData mean electronegativity, and
DemlData maximum heat_fusion (heat of fusion). Five features are common in at
least two models, namely DemlData std_dev atom_radius (atomic radius),
MagpieData maximum NdValence (the maximum number of valence electrons in the
$d$ orbital of an atom in a compound), HOMO (the highest occupied molecular
orbital) energy, frac $p$ valence electrons (the number of valence electrons
in the $p$ orbital of an atom in a compound), and MagpieData range NdValence
(the range of valence electrons in the $d$ orbital of an atom in a compound).
Additionally, tree models also can obtain their own feature importance during
training. We compare the SHAP feature importance analysis with that directly
obtained from these three tree models and find that the top-ranked features
exhibit high similarity, i.e., the features related to the distribution of
electrons outside the nucleus (MagpieData mean Nunfilled, frac d valence
electrons, etc.) have the highest rank. Secondly, properties such as
electronegativity, thermodynamic properties (DemlData maximum heat_fusion),
atomic radius, and Mendeleev number have relatively lower rank. Although the
ranks of these properties vary among different models, overall they are the
properties correlated to electron distribution. Surprisingly, the tolerance
factor, which plays important roles in structural distortion of perovskite,
has low rank in formation energies of $A_{2}BB^{\prime}$O6 compounds. More
details of the feature rank can be found in Figure S11 of the supporting
information.

Machine learning model of stability. As mentioned above, convex hull distance
is defined as the energy difference between the formation energy of a compound
with respect to the lowest formation energy of the linear combination of the
competing phases with the same composition. Therefore, if the energy
difference is less than zero, the convex hull distance is mathematically zero.
However, regression models tend to yield more accurate results when the
dataset exhibits a relatively even distribution across the range of values. In
order to obtain a better ML model, decomposition energy, which measures the
energy difference between the formation energy of a compound with respect to
the lowest formation energy of the linear combination of the competing phases
with the same composition, is adopted as the target here for stability
prediction because it has a more even distribution than convex hull distance.
As shown in Figure S6 of the supporting information, decomposition energy has
a narrower spread (from -0.2 to 0.9 eV/atom) than the formation energy (from
-3.7 to -1.2 eV/atom). Although it is expected that decomposition energy model
would be worse than the formation energy one for regression with the same
amount of data, the decomposition energy model is more useful in material
discovery than the formation energy model. Our decomposition energy models are
shown in Figure 8. As expected, all these models have lower R2 and larger MAE
than the formation energy models. Similar to the results of formation energy
models, XGBoost has the best performance (high R2 and small MAE) among these
three models, which is followed by Random Forest and gradient boosting
regression. Overall, the performance of our model surpasses or is on par with
that of previous studies.46, 51, 98

Figure 9: Comparison of XGBoost prediction results for formation energy with
OQMD database results.

Application of machine learning model for new material discovery. The power of
machine learning lies in its ability to expedite material discovery by
directly predicting the thermal stabilities of a composition without prior
knowledge of its structure. Subsequent computations only are needed for the
compositions that are predicted to be stable, which can save a lot of CPU
hours. Since we have calculated all possible $A_{2}BB^{\prime}$O6 compositions
within the chemical space $A$= Ca, Sr, Ba, and La, we then use our ML models
to predict the stability (decomposition energy) of the $A_{2}BB^{\prime}$O6
compositions that are not included in our dataset, namely the
$A_{2}BB^{\prime}$O6 compounds with $A$ = Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho,
Er, Tm, Yb, and Lu and same $B$ and $B^{\prime}$. We compare our predicted
stabilities of these compounds with those that already exist in the OQMD.
Although these elements are not included in our training set, our ML models
are based on the features of constituent elements and should be extendable to
these elements. We screen 2071 compositions and predict their formation energy
and decomposition energy using our XGBoost model. The prediction results of
the formation energy are shown in Figure 9, the results predicted by XGBoost
are positively correlated with the results of the OQMD. For decomposition
energy, 803 compositions are predicted to be stable or metastable (with
decomposition energy $\leqslant$ 0.1 eV/atom) and 612 of them have $\Delta{\rm
stable}$ less than 0.1 eV in the OQMD, indicating a 76.2 % success rate.

$\blacksquare$ CONCLUSIONS
In summary, we explored the thermal stabilities of $A_{2}BB^{\prime}$O6
compounds ($A$=Ca, Sr, Ba, and La; $B/B^{\prime}$ are metal elements), using
high throughput DFT calculations. We not only rediscovered 470 compounds that
have been reported in the ICSD and literature, but also discovered 1,552 new
synthesizable $A_{2}BB^{\prime}$O6 compounds. Our analysis shows that most
$A_{2}BB^{\prime}$O6 compounds adopt $P2_{1}/c$ structure at 0 K, which is
consistent with tolerance factor analysis. Using 4,855 formation energies and
decomposition energies from our high throughput DFT calculations, we build
several machine learning models based on the element features and ionic radius
of constitution elements. The MAE of formation energy and stability are as low
as 0.0444 and 0.0330 eV/atom, respectively. Formation energy and stability are
strongly correlated with the number of empty electron of constitution
elements. We then used our stability model to predict new stable/metastable
$A_{2}BB^{\prime}$O6 compounds that are outside the chemical space explored in
the high-throughput study. A success rate of 76.2 % is achieved by comparing
our ML prediction with those that already exist in the OQMD. Our work
delineates the stability map of $A_{2}BB^{\prime}$O6 compounds and offers new
insights in $A_{2}BB^{\prime}$O6 compounds discovery.

The Supporting Information is available free of charge on the ACS Publications
website at DOI:
The prototype structures; the convex hull distance of the charge-balanced
compounds; the tolerance factor of $A_{2}BB^{\prime}$O6; the performance of
all the machine learning models used in this work; feature importance of the
selected models; the Spearman correlation coefficient of the selected
features; models and parameters used in this work.

$\blacksquare$ AUTHOR INFORMATION
Corresponding Author
*(C.W.) E-mail<EMAIL_ADDRESS>
*(J.H.) E-mail<EMAIL_ADDRESS>

Notes
The authors declare no competing financial interest.

$\blacksquare$ ACKNOWLEDGMENT Y.W. and J.H. acknowledge the support of the
Fundamental Research Funds for the Central Universities China (USTB). C.W.
J.S., and B.B. acknowledge primary support via National Science Foundation
through the MRSEC program (NSF-DMR 1720139) at the Materials Research Center.

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