Learn and Match Molecular Cations for Perovskites | The Journal of

Subscriber access provided by UNIV OF SOUTHERN INDIANA

A: New Tools and Methods in Experiment and Theory

Learn and Match Molecular Cations for Perovskites Heesoo Park, Raghvendra Mall, Fahhad H. Alharbi, Stefano Sanvito, Nouar Tabet, Halima Bensmail, and Fedwa El-Mellouhi J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b06208 • Publication Date (Web): 25 Jul 2019 Downloaded from pubs.acs.org on July 25, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Learn and Match Molecular Cations for Perovskites Heesoo Park,∗,† Raghvendra Mall,‡ Fahhad H Alharbi,† Stefano Sanvito,¶ Nouar Tabet,† Halima Bensmail,‡ and Fedwa El-Mellouhi∗,† †Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, PO BOX 34110, Doha, Qatar ‡Qatar Computing Research Institue, Hamad Bin Khalifa University, Doha,Qatar ¶School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland E-mail: [email protected]; [email protected]

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Forecasting the structural stability of hybrid organic/inorganic compounds, where polyatomic molecules replace atoms, is a challenging task; the composition space is vast and the reference structure for the organic molecules is ambiguously defined. In this work we use a range of machine-learning algorithms, constructed from state-of-the-art density functional theory data, to conduct a systematic analysis on the likelihood of a given cation to be housed in the perovskite structure. In particular, we consider both ABC3 chalcogenide (I-V-VI3 ) and halide (I-II-VII3 ) perovskites. We find that the effective atomic radius and the number of lone pairs residing on the A-site cation are sufficient features to describe the perovskite phase stability. Thus, the presented machine-learning approach provides an efficient way to map the phase stability of the vast class of compounds, including situations where a cation mixture replaces a single A-site cation. This work demonstrates that advanced electronic structure theory combined with machine-learning analysis can compose an efficient strategy superior to the conventional trial and error approach in materials design.

2


Page 2 of 36

Page 3 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Introduction The current world record for power-conversion efficiency in solar cells based on hybrid organic/inorganic lead-iodide perovskites is reaching beyond 24.2%. 1 This high efficiency has been achieved with a mixed formamidinium/methylammonium composition, FAx MA1-x PbI3 [FA = CH(NH2 )2 + , MA = CH3 NH3 + ], and comes at the end of half a decade of spectacular improvements. 2–4 In addition to the advantages related to the ease of growth, hybrid organic/inorganic perovskites (HOIPs) have the optimal bandgap for visible light absorption, low exciton binding energy, long exciton lifetime, and structure consisting of the 3D-network of corner-sharing [BC6 ]4 – octahedra that enable efficient carrier transport. 5–17 Unfortunately, these compounds also have some significant drawbacks, hampering the daily use of HOIP solar-cell. For example, MAPbI3 suffers from intrinsic phase instability, degrading spontaneously, especially under moisture or illumination conditions. 18–21 Together with a consistent effort to stabilize MAPbI3 and to encapsulate the compound in solar cells, many attempts have been devoted to find new stable and high-performing perovskites, in either hybrid or inorganic perovskite. 22–26 Cation substitution is a common strategy and perovskites with mixed-cation composition have shown some success in improving the intrinsic stability. 27,28 Concerning hybrid compounds, the use of molecular cations with different dipole moments have been suggested as a tool to avoid the hysteresis effects originating from ionic migration. 29 Compositional manipulations to overcome the perovskite instabilities due to external environmental factors must not be at the expense of the intrinsic phase stability of the perovskite crystal structure. In an ABC3 perovskite, halides and chalcogenides make the corner-sharing [BC6 ]4 – octahedra, while the voids accommodate the A-site cations. The Goldschmidt tolerance factor 30 (TF) provides a rough guideline for the phase stability of perovskite structure. This relevant √ descriptor writes TF = (rA + rC )/ 2(rB + rC ), where rα (α = A, B and C) is the ionic radius of the α species. The assessment of perovskite formability criterion assumes that the perovskites arrange their structure so that the number of anions surrounding a cation is as large 3



as possible. Generally, a compound with the value of TF between 0.9-1.0 has an ideal cubic perovskite structure adopting the Pm¯3m symmetry, while the value of TF between 0.8-0.9 results in a distorted quasi-cubic perovskite. Beyond these ranges, a compound would likely form a hexagonal phase when TF > 1 and orthorhombic phase when TF < 0.8 or even a non-perovskite structure. Depending on the ionic radii of the A, B and C constituents and the total equilibrium volume, some combination of atoms could be accommodated within the ideal cubic ABC3 perovskite structure; while in most cases the crystal structure relaxes away from the ideal cubic Pm¯3m symmetry via elongating, bending, and rotating the [BC6 ]4 – octahedra. Such a scenario could be observed, for example, when increasingly larger cations are incorporated in lead iodide perovskites leading to the monotonic increase of the octahedral deformation until the perovskite structure become no longer tolerable. In addition, the observed phase transitions of the perovskite structure 31 triggered by temperature and pressure are results of intrinsic factors such as the distortion and rotation in the [BC6 ]4 – octahedra. Due to the utilization of polyatomic cations in HOIPs, the definition of their TF has been extended by considering the effective radius of the organic cation. 32–34 TF, integrated with the octahedral factor, µ = rB /rC , could guide the mapping of the phase-stability boundary of HOIPs in high-throughput density functional theory (DFT) calculations. 35,36 This criterion limits the range of values that TF and µ can take and effectively provides a descriptor for screening the phase stability of existing and hypothetical compounds. 33,34 Recently, Filip and Giustino used the “no-rattling” principle to explore the perovskite phase stability limit for an extensive library of perovskites. 37 Coming back to the phase stabilization by mixing the cations, Li et al. successfully used the TF descriptor by averaging the radii of the mixed A-cations 38 enabling them to stabilize the perovskite phase by avoiding considerable internal distortions. Robinson et al. defined additional features to quantify and measure the [BC6 ]4 – octahedral distortion away from of the ideal cubic perovskites; they reported a linear correlation between the quadratic

4


Page 4 of 36

Page 5 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


elongation and angle variances. 39 Interestingly, all recent studies focused on the formability of HOIPs have used descriptors which are associated with geometrical quantities, but, to the best of our knowledge, few attempts have correlated it to the atomic chemical features. In the present work, we assess the impact of the ABC3 composition to estimate the phase stability of the perovskite structure by measuring the extent of octahedral deformation induced by a given A-cation. We thus perform extensive density functional theory (DFT) calculations for a range of chalcogenide and halide perovskites, both hybrid and inorganic, taking into account several geometrical orientations of the organic cations. Deviation from the ideal cubic perovskite structure is then measured by considering a descriptor consisting of the energy difference between the ideal cubic (Pm¯3m) and quasi-cubic (fully relaxed) perovskites. The resulting dataset is subsequently used for the construction of a Machine Learning (ML) model to estimate the phase stability of a given BC3 combination based on the ionic radius of the A-site cation and the number of lone pairs present. Our machine learning approach allows us to build a landscape of the perovskite phase stability enabling, for a given perovskite composition, to predict the best matching A-cation. Finally, our models are put to the test against some experimentally reported and hypothetical compounds with ternary cation mixtures in chalcogenide and halide perovskites.

Computational Methods DFT calculations and structural parameters We have studied 384 ABC3 chalcogenide (I−V−VI3 ) and halide (I−II−VII3 ) perovskites satisfying charge balance (see Figure 1). The elements are: B = V5+ , Nb5+ , Ta5+ , Ge2+ , Sn2+ and Pb2+ ; and C = O2− , S2− , Se2− , Te2− , F− , Cl− , Br− and I− . The A-site is instead occupied by a single atomic monovalent alkali metal, chosen among Li+ , Na+ , K+ , Rb+ and Cs+ , or a molecular cation. For this latter case we have explored the following molecules: Hydronium 5



O V S Nb x Se Ta Te

Li Na K Rb Cs H3O+ NH4+ H3S+ PH4+ NH3OH+ CH3NH3+ NH3NH2+ NH3COH+ +

CH(NH2)2 CH3CH2NH3

x

+

C(NH3)3+

+ F Ge Cl Sn x Br Pb I

x

6 different geometrical orientation

quasi-cubic

cubic

ΔH

σ2 λ

Figure 1: Structure of an ABC3 perovskite and the used constituent elements. A-site cations are in blue and comprise alkali metals and organic molecules, the rotated cations at the center of the [BC6 ]4 – octahedra are in green, and the anions (either halogens or chalcogens) are in yellow. ∆H, λ, and σ 2 are the parameters used to quantify the distortion out of the ideal cubic perovskites. Those represent the enthalpy change, quadratic elongation, and octahedral angle variance, respectively. (HY+ ), Ammonium (AM+ ), Sulfonium (SF+ ), Phosphonium (PH+ ), Hydroxylammonium (HA+ ), Methylammonium (MA+ ), Hydrazinium (HZ+ ), Formamidinium (FA+ ), Formamide (FO+ ), Ethylammonium (EA+ ) and Guanidinium (GA+ ). Since the organic cations are not spherical, in contrast to monovalent alkaline metals, their orientation, and interaction with the [BC6 ]4 – octahedra affect the compound’s electronic properties by many ways including the formation of hydrogen bond (H-bond) interactions with the anions 14,40 motivating us to consider various geometrical orientations. This nature of polyatomic ions encourages us to explore better the potential energy landscape to avoid trapping in local configurational minima. A randomly-generated set of 6 different structures have been constructed for each hybrid compound with differently oriented molecules. Then, we have conducted a full geometry optimization, which has delivered us a total of 1584 structures. We performed DFT calculations by using the projector-augmented wavefunction (PAW) method 41,42 as implemented in the VASP code. 43–45 We employ the Perdew–Burke–

6


Page 6 of 36

Page 7 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Ernzerhof (PBE) 46,47 parameterization of the generalized gradient approximation (GGA) of the DFT exchange and correlation energy. To compensate the lack of dispersion interactions in GGA, we include van der Walls corrections following the Tkatchenko-Scheffler procedure. 48 A plane-waves energy cutoff of 520 eV is employed alongside with a 6 × 6 × 6 k-points mesh generated according to the Monkhorst-Pack scheme. Energies and forces are converged within 10−7 eV/atom and 0.01 eV/Å, respectively. FireWorks 49 managed the calculation workflow while the Pymatgen 50–54 tools were used for pre- and post-processing the calculated data. The molecules are initially placed at the A-site voids of the Pm¯3m space group; they are rotated around a randomly-selected axis by a random angle followed by full structural relaxation. The full structural relaxation is conducted in three steps in the primitive unit cell: (1) the cell volume is optimized for the perovskite structure by constraining the Pm¯3m space group; (2) atoms are displaced in random directions by 0.1 Å to escape from the harmonic wells of possible local minima; (3) all the degrees of freedom, namely the atomic positions, the cell shape and the cell volume are optimized without any structural constraints. Another set of calculations is also conducted by constraining the cubic perovskite geometry adopting the Pm¯3m space group, and by iterating the steps (1) and (2) until the structure is converged. For each of the starting 1584 perovskite prototypes, differing in their geometry and composition, we define a descriptor ∆Hc that measures the phase stability of a fully relaxed ABC3 compound by comparing the DFT calculated total energies data in two configurations, namely, the fully relaxed one and the one in the cubic-symmetry within Pm¯3m space group. This energy difference defines the phase stability :

∆Hc = Ecubic − Erelax ,

(1)

where Ecubic and Erelax are the total energies of the cubic and fully relaxed structure, respec-

7



Page 8 of 36

tively. Depending on the ionic radii of the A, B and C constituents and the total equilibrium volume, the ABC3 perovskite structure relaxes away from the Pm¯3m symmetry via elongating, bending the [BC6 ]4 – octahedra unless the constituents fit perfectly in the ideal cubic crystal structure. The [BC6 ]4 – octahedra deform until the perovskite structure is no longer tolerable in the perovskite phase. Hence, ∆Hc metric measures the phase stability of a fully relaxed ABC3 compound compared to their ideal cubic structure counterpart. At this point, it should be noted that this energy change indicates the phase stability with respect to an ideal cubic perovskite. The octahedral deformation is measured by using the same descriptors introduced in our previous work. 39,55 An ideal cubic perovskite has a quadratic elongation λ = 1 and an angle variance σ 2 = 0. The quadratic elongation λ is defined as: 6

1X λ= 6 i=1

2 li , l0

(2)

where li is the distance between the central atom and i-th coordinated anion in an octahedron, and l0 is the same distance for the ideal cubic octahedron with the same volume. In contrast, the angle variance σ 2 is: 12

1 X σ = (φi − 90)2 , 11 i=1 2

(3)

where φi is the i-th ∠C-B-C0 angle. Among the calculated structures, 1339 entries have been used to analyze the octahedral deformation for our machine-learning analysis. We have then excluded non-perovskite compounds, for which the structure is so severely distorted that it cannot form [BC6 ]4 – octahedra resulting in ill-defined λ and σ 2 .

8


Page 9 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Machine-learning methods We have used five different machine-learning techniques to build the models. The most straightforward scheme is the generalized linear model (GLM), which extends ordinary linear regression and estimates the response variable as a linear combination of the input feature set. GLM allows the response variable to have an error distribution other than a normal one. Moreover, it also has a provision to penalize the coefficients of the weights obtained as a solution of the linear regression problem. We have then investigated supervised white-box decision-tree-based machine-learning techniques, including Random Forest and Gradient Boosting Machines, for our regression problems. Random Forest (RF) belong to the class of ensemble-based supervised-learning methods. The RF algorithm applies the general scheme of bagging or bootstrapped aggregating to decision tree learners. Given a dataset D = {Xi , Yi }N i=1 with response Yi ∈ R, the bagging algorithm selects repeatedly random samples with replacement from the dataset D and fits separate trees to these samples. The bootstrapped sampling procedures help to de-correlate the trees by showing them different parts of the dataset D. RF allows to automatically rank features based on their importance for a given task, by considering the average Information Gain for each feature across all the trees. Gradient-boosting machine (GBM) belongs to that family of predictive methods, which uses an iterative strategy such that the learning framework will consecutively fit new models so to have a more accurate estimate of the response variable after each iteration. 56,57 The primary notion behind this technique is to construct new tree-based learners to be as correlated as possible with the negative gradient of a given loss function, calculated by using all the training data. One here can use any arbitrary loss function. However, if the loss function is the most commonly used squared-loss one, the learning procedure will result in consecutive residual error-fitting. The advantage of the boosting process is that it works on decreasing the bias of the model, without increasing the variance. Learning uncorrelated base learners helps to reduce the bias of the final ensemble model. Moreover, features can be 9



ranked based on their ability to determine the split for the decision trees, thereby, providing feature importance. A more scalable and accurate version of GBM is XGBoost, also based on the principle of tree boosting. Both XGBoost and GBM follows the principle of gradient boosting. There are however differences in the details of modeling. Specifically, XGBoost uses a more regularized model formalization to control over-fitting, which gives it better predictive performance. XGBoost uses a scalable end-to-end tree boosting system by adopting an approximate algorithm, namely, weighted quantile sketch, in which we consider the only candidate split points in the dataset. More importantly, XGBoost can scale for a large number of samples using minimal computational resources and achieving state-of-the-art predictive performance. XGBoost allows parallel computations of many boosting machines utilizing all available computational resources, a feature not available in traditional gradient boosting machine. Finally, we have used a more complex and less interpretable machine-learning technique called Deep Learning (DL). DL is vaguely inspired by information processing and communication patterns in biological nervous systems. Of late, DL based models have been successfully applied in the complicated non-linear tasks. 58,59 The problem of the estimation of rA is a regression problem. In the case of DL, one learns a non-linear mapping function that takes as input the feature set, Xi , for the given perovskites and outputs a score ∈ R, namely t : Xi → Yi , where t is the mapping function. In this work, t is a Deep Fully Connected Feed-Forward Neural Network (DNN) that exploits the non-linear interactions between the input features to make its function estimation for rA . It has been stated in Ref. 60 , that a feed-forward neural network with a single hidden layer, containing a finite number of neurons, can approximate any continuous functions under certain mild assumptions on the activation function. This idea is commonly referred as the universal approximation theorem, which proves the advantage of deep learning over other white-box supervised learning techniques. However, most DL models are complex, and it is difficult to interpret these models to obtain a ranking of the input features. Moreover, they need a large number of samples to achieve

10


Page 10 of 36

Page 11 of 36

state-of-the-art predictive performance. We have used the ‘h2o’ package (Version 3.17.0.4195) to implement all the algorithms discussed above. These involve several hyper-parameters, whose optimal values are obtained by using a standard 5-fold cross-validation process.

Results and discussion Comparisons of structures: high-throughput calculations (a)

800 r A (Å) Chalcogenide perovskite

(c)

0 .8 1 .6 2 .4 3 .2

1.3

300

1.2

200

0 1.0 800

1.1

0

1.0

TF < 1

500

Halide perovskite r A (Å)

0.8 1.6 2.4 3.2

0.9

400 2

600

1.1

100

1.3

1.2

400

1.4

TF ≥ 1

400

400 200

(b)

Chalcogenide + Halide

600 500

2

2

600

2

300

0.8

200

200

0.7

100 0

0 1.0

1.1

1.3

1.2

1.0

1.1

1.2

0.6 1.3

TF

Chalcogenide + Halide

(d) 600

0 .4

500

0 .3

400 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


300

0 .2

200

0 .1

100 0

1.0

1.1

1.2

1.3

0.0

ΔHc (eV/ion)

Figure 2: The angle variance, σ 2 , as a function of the quadratic elongation, λ, in all the calculated compounds. In panels (a) and (b) we show the relationship between σ 2 and λ, where the size of the circles is proportional to the radii of the A-site cations, rA . Panel (c) shows a color map plot using TF, while in panel (d) the color map uses ∆Hc (in eV/ion). For an ideal cubic perovskite λ = 1, σ 2 = 0 and ∆Hc = 0.0 eV/ion. We start by optimizing the structures of the 1704 perovskites constructed as described in the previous section. VASP calculations have been conducted in a high-throughput way by taking into account the different possible orientations of the organic cations. This workflow allows us to distinguish local minima against the ground state for each of the chemical compositions. At a high temperature, we may assume organic cations rotate freely in the 11



cavities, thus behaving like spheres by averaging the interactions. Meanwhile, at a lower temperature, their non-spherical structure and the presence of H-bonds promote symmetrybreaking distortions. 6,15–17 Figure S1 in the supplementary information (SI) illustrates the cases where two very different minima emerge from the full geometrical optimization of EAGeCl3 , depending on the Ethylammonium initial molecular orientation. In the figures and throughout this work, we take the effective radii of the various cations according to the method described in Ref. [ 32,33], and their values are listed in Table S1 in the SI. We use the ML model to extract solely the radius of Phosphonium since no Shannon’s ionic radius is available. The details are discussed in the SI. After full relaxation, when one excludes the non-perovskite structures, 1339 entries are left out of the 1704 ones. Figure 2 shows the angle variance, σ 2 as a function of the quadratic elongation, λ: panels (a) and (b) display the λ-σ 2 relationships illustrated using circles with sizes proportional the radii of the A-site cations for chalcogenide and halide perovskites respectively. One may observe that the small circles of chalcogenide perovskite populate in the low-low λ-σ 2 region near the origin indicating that chalcogenides tend to remain close to the cubic phase in the presence of small cations. (Figure 2a) In contrary, Figure 2b shows that the large circles of halide perovskite populate in the low-low λ-σ 2 region indicating that large cations can be accommodated in halide perovskites without generating substantial distortions. Figures 2c-d reveal some common trends to both chalcogenide and halide perovskites by the color-mapped TF and ∆Hc . One may notice that when TF < 1 the octahedron distorts by preserving a linear correlation between λ and σ 2 leading to a distorted quasi-cubic perovskite. In contrast, when TF >= 1 the perovskite structure will deform more severely by elongating its bond within the [BC6 ]4 – octahedra rather than by distorting the ∠C-B-C0 angle. In other words, λ is affected more strongly than σ 2 when TF >= 1; such bond elongation would lead to phase transition and in some cases reach a threshold after which the structure loses the characteristic of 3D perovskite indicating the severe intrinsic instability of cubic phase. One may

12


Page 12 of 36

Page 13 of 36

notice very similar tendency by using the color-mapped ∆Hc in in Figure 2d): The low-low linearly correlated λ-σ 2 region is dominated with black and dark green dots (∆Hc → 0) where the structure remains a quasi-cubic perovskite, whereas the higher λ-σ 2 values are dominated red to yellow color mapped dots for which ∆Hc 0. Therefore, one could use ∆Hc as a descriptor to measure/predict the phase stability of an ABC3 perovskite compound compared to their ideal cubic structure counterpart.

(b) Chalcogenide

0.4 0.3 0.2

A-site cations

H c (eV/ion)

(a) 0.5

0.1 0.0

H c (eV/ion)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Halide 0.4 0.3 0.2 0.1 0.0

0.8

1.0

TF

1.2

Li Na K Rb Cs HY AM SF PH HA MA HZ FA FO EA GA

Chalcogenide perovskite

Halide perovskite

0

0

92 139 164 172 188 140 146 184 181 216 217 217 253 268 274 278

0

100 200

r A (pm)

0.1 0.2 0.3 0.4 Hc (eV/ion)

0.1 0.2 0.3 0.4 Hc (eV/ion)

Figure 3: (a) ∆Hc of the chalcogenide and halide perovskites as a function of the tolerance factor. (b) Gaussian kernel density estimates of ∆Hc for each A-site cation. In addition to the prevailing trends, the subtle difference appears when we separate the two groups, namely chalcogenide and halide perovskites. Figure 3a illustrates the variation of ∆Hc as a function of TF for the chalcogenide and halide perovskites. One may notice that the considerable number of stable cubic perovskites (∆Hc ∼ 0) populate in the region where the TF ranges from 0.9-1.0; once TF deviates from this cubic perovskites stability range, ∆Hc increases differently between chalcogenide and halide perovskites. Plus, the increment of ∆Hc indicates that the compound would prefer to adopt either hexagonal or orthorhombic phase depending on the to the size of constituent ions. Figure S2 and Figure S3 in the SI showing the Gaussian kernel density of TF as a function of ∆Hc and octahedral deformation demonstrate that halide perovskites populate the less 13



deformed perovskite region than the chalcogenides. In other words, for a given value of the tolerance factor, the distribution of ∆Hc is much more narrowly concentrated around ∆Hc = 0 for halides than for chalcogenides. This fact suggests that the [BC6 ]4 – octahedra in a halide perovskite are relatively more rigid than those in chalcogenides. This trend could be explained based on the electronegativity difference between the B and C elements dictating the strength of the ionic and covalent bonds. Hence, the covalent bond character within halide perovskites appears to be stronger than that within the chalcogenides. The ultimate goal of this work is to train (learn) a machine learning model to be able to extract the phase stability as well as the most suitable A cation (match). While searching for the optimal A cation, we may consider the ionic radius of A-cation as the input feature. Thus, we can consider the structural deformation of octahedra, as the target feature for the phase stability. However, the rA is part of the definition of TF, which might lead to biased correlation and compromises our predictive abilities by assuming the ions are hard spheres. Besides, we observed that for some perovskites the quadratic elongation (λ) and octahedral angle variance (σ 2 ) behave non-linearly to each other. Interestingly, we showed that the DFT calculated energy difference ∆Hc , represents well the phase stability of the cubic perovskite structure in addition to being an independent feature from rA . For our learn-and-match exercise, we take ∆Hc as a target feature for building our machine learning model while avoiding any dependence on all ionic radii. Based on experimental and computational investigations, it is well accepted that the character of A-cation influence the phase stability of a perovskite structure for a given BC3 combination motivating the search for key features of hypothetical A-cation that match ideally into the voids of a [BC6 ]4 – networks. Furthermore, to explore the landscape of the octahedral deformation for each kind of A-cation, we construct the probability distribution of ∆Hc for each A-cation. Through this analysis, we can expect to discover additional features that influence the octahedral deformation besides to the ionic radius of the A-cation, rA . Figure 3b presents the distribution of ∆Hc over all the compounds for each A-site molec-

14


Page 14 of 36

Page 15 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ular/atomic cation, showing the influence of cation on the deformation of cubic perovskites. It is clear from the figure that the radius of the A-cation (rA ) correlates significantly with ∆Hc for both chalcogenide and halide perovskites. In the chalcogenide perovskites, sable compounds could form (∆Hc < 0.1 eV/ion) if the A-site is occupied by alkali metal ions such as Na, K, and Rb, and ammonium cation. Chalcogenides become highly unstable (∆Hc > 0.1 eV/ion) if the remaining organic cations occupy the A-site. For halide perovskites, it appears the BC3 framework is capable of housing cations over a broader range until it reaches a threshold where large organic cations such as EA and GA start to trigger phase instabilities (∆Hc > 0.1 eV/ion). A similar trend can be appreciated for λ in Figure S4 of the SI. In addition to rA , we now attempt to identify another physical/chemical feature to describe A-site cation. To perform this task, we compare perovskites where the A-site is occupied with different cations each having similar rA . For example, the radius of Hydronium (rA =140 pm) is almost the same as that of Ammonium (rA =146 pm), but the associated ∆Hc distributions are rather different, as shown in Figure 3 indicating that the phase stability depends strongly on the precise identity of the A-site cation. This disparity appears to be also the case while comparing results for Methylammonium (rA =217 pm) and Hydrazinium (rA =217 pm). This effect is much more pronounced in the case of organic cations than for inorganic cations suggesting that a closer look into the bonding of the molecular cation with the [BC6 ]4 – octahedra is necessary. We have thus measured all the relevant distances associated to the hydrogen bonds (Hbonds), namely d(DH − H) and d(AH · · · H) (see Figure S5 in the SI for both results and definitions), where DH is the H-bond donor in the organic cation and AH is the acceptor of the H-bond, with most of the C-site anions in the perovskite structure behaving as H-bond acceptor. As a general trend, d(DH − H) and d(AH · · · H) fall into the range expected from their chemical environment, 61 regardless of the degree of octahedral deformation. However, we have found a few cases where a proton is transferred from the molecular cation to the Bsite anion. This proton transfer during the relaxation occurs for HA, SF, and PH embedded

15



in both chalcogenide and halide perovskites. Proton transfer has been previously reported in lead-free perovskites 62 as well as in inverse hybrid perovskites. 63 Hence, our 365 structures displaying proton transfer, together with other severely distorted ones, have been excluded from the training dataset used for the machine learning analysis. (b)

(c)

(d)

(e)

(a) HY

HZ

AM

MA

Figure 4: (a) Lewis structures of Hydronium (HY), Ammonium (AM), Hydrazinium (HZ) and Methylammonium (MA). In panels (b) through (e) we show the charge-density cross 3 sections, in the range of [0.01, 0.20] e/Å : (b) HYNbO3 (NLP = 1), (c) AMNbO3 (NLP = 0), (d) HZPbI3 (NLP = 1) and (e) MAPbI3 (NLP = 0).

This analysis allows us to further characterize any A-site cation by counting the number of lone pairs it can host, NLP . Visually one can identify lone pairs by looking at the crosssections of the charge densities associated with a given cation and these can be represented by drawing the Lewis structures. Such exercise is presented in Figure 4. Here, for example, HY and AM have an almost identical rA , 1.40 Å and 1.47 Å, but different NLP , 1 and 0, respectively. The same situation is encountered for HZ and MA, which have an identical rA = 2.17 Å, but have NLP = 1 and NLP = 0, respectively. It is notable that the lone pairs are repulsive to the C-anions and induces charge polarization. The net dipole is, however, the result of all the chemical and structural features of the entire cation (the bonds to the central atoms, the number of valence electrons, and the number of H).

16


Page 16 of 36

Page 17 of 36

(b)

(a) Li

2 600

Li K Na Rb Cs HY AM SF PH HA MA HZ FA FO EA GA VO3 300

K Na Rb Cs HY AM SF PH HA MA HZ FA FO EA GA

2

VO3

0

4

NbO3

VTe3 NbTe3 TaTe3 GeF3 SnF3 PbF3 GeCl3 SnCl3 PbCl3 GeBr3 SnBr3 PbBr3 GeI3 SnI3 PbI3

TaSe3

VSe3

TaS3

NbSe3

TaO3

NbS3

50

6

TaSe3 VTe3 NbTe3 TaTe3 GeF3 SnF3 PbF3 GeCl3 SnCl3 PbCl3 GeBr3 SnBr3 PbBr3 GeI3 SnI3 PbI3

100

10

0

NbSe3

150

VSe3

200

0

8

TaS3

250

VS3

Hc

Li

NbS3


MAD of

VS3

Li

VO3

VS3

NbO3

(d)

Hc

TaO3

(c)

100

TaSe3 VTe3 NbTe3 TaTe3 GeF3 SnF3 PbF3 GeCl3 SnCl3 PbCl3 GeBr3 SnBr3 PbBr3 GeI3 SnI3 PbI3

VTe3 NbTe3 TaTe3 GeF3 SnF3 PbF3 GeCl3 SnCl3 PbCl3 GeBr3 SnBr3 PbBr3 GeI3 SnI3 PbI3

TaSe3

VSe3

TaS3

NbSe3

TaO3

NbS3

VS3

NbO3

VO3

1.04

200

VSe3

1.12

300

NbSe3

1.20

400

NbS3

1.28

500

TaS3

1.36

TaO3


NbO3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5: Heat map for (a) λ, (b) σ 2 and (c) ∆Hc (meV/ion) for each chemical composition in the ground-state configuration. Panel (d) shows the mean absolute deviation (MAD) of ∆Hc (meV/ion) computed over all possible cation orientations. All plots map the spaces spanned by the A-site cation (the y-axis) and the [BC6 ]4 – octahedra (BC3 chemical composition; the x-axis).

The features of all the chemical compositions are presented in Figure 5, where one can notice that even cations of comparable size relax in structures showing different degree of distortion (Figure 5a-c). Intriguingly, when one looks at the MAD in the distribution of ∆Hc for all the molecular orientations investigated (Figure 5d), one may notice that groundstates with significant distortions correlate with large fluctuations in ∆Hc . By comparing the behavior of organic cations having the comparable ionic radius, such as AM and HY, 17



SF and PH, MA and HZ, we can relate the differences in the octahedral deformation to the presence of lone pair of electrons. To summarize this section, our high-throughput analysis has revealed that besides the molecular radius rA , number of lone pairs is an additional feature governing the interaction of an organic cation to its inorganic environment. Hence we will utilize both rA and NLP as input feature during the construction of our machine-learning models.

Regression on λ, σ 2 and ∆Hc We now move on to construct the ML models for the target features, namely, octahedral deformation parameters, λ and σ 2 , and the energy difference between the relaxed and ideal cubic structures, ∆Hc . To construct our ML models, we have to select carefully the input features describing the A-site cations. For inorganic perovskites, where the cation is just an atom, the input features merely relate to its chemical properties, namely the ionic radius, rA , the period and group numbers, PA and GA , the ionization energy, IEA and the electron affinity, EAA . The same chemical quantities also specify the elements at the B and C sites. However, one cannot use the same attributes for all the atoms defining the organic cations and thus, following the analysis of the previous section, we have replaced them with input features of A-cation: the effective radius, rA , and the number of lone pairs, NLP . Finally, we have also included in the feature dataset the octahedral factor, µ, and the TF, so that the full set consists of 12 input features: rA , rB , rC , GB , GC , PB , PC , EAC , IEB , NLP , µ and TF. We have constructed the ML models based on the DL, GBM, GLM, RF, XGBoost ML algorithms by randomly assigning 80 % of the entries to the training set, and by validating the models (the predicted values of the various quantities) over the remaining 20 % (see Figure 6). Among the various methods, DL, GBM, and XGboost achieve higher accuracy over all the quantities related to the octahedral deformation. (We present the accuracy and all the feature importance in Figure S6 and S7 in the SI, respectively.) These models are capable of reproducing efficiently the sophisticated features of our data by minimizing the errors 18


Page 18 of 36

Page 19 of 36

XGBoost

GBM

R^2=0.86 MSE=0.0016 1.0 1.1 1.2 1.3 True λ

R^2=0.87 MSE=0.0015 1.0 1.1 1.2 1.3 True λ

DL

R^2=0.87 MSE=0.0014 1.0 1.1 1.2 1.3 1.4 True λ

GBM

XGBoost

Predicted σ2 200 400

(b)

DL

600

Predicted λ 1.0 1.1 1.2 1.3 1.4

(a)

0

R^2=0.82 MSE=3749 0

200 400 True σ2

R^2=0.87 MSE=2566 0

200 400 True σ2

R^2=0.88 MSE=2481 0

200 400 True σ2

600

Predicted ∆H c 0.1 0.2 0.3 0.4

(c)

0.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


DL

GBM

R^2=0.99 R^2=0.98 MSE=0.0011 MSE=0.0019 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 True ∆H c True ∆H c

XGBoost

R^2=0.99 MSE=0.0011 0.0 0.1 0.2 0.3 0.4 True ∆H c

Figure 6: Predictions against the actual DFT data for (a) λ, (b) σ 2 and (c) ∆Hc (eV/ion). We consider three top-performing ML models, namely in DL, GBM and XGBoost. between the ML-predicted and the DFT-calculated data points. It appears that the ensemble of weak learners of GBM and XGBoost makes the models more robust when we compare them with RF. Meanwhile, the predicted values by RF ranges within a narrower width (see Figure S8 in the SI). It appears the RF method learn poorly from the variable orientations sticking to the mean value over the range. GLM predicts with the lowest accuracy since it is a linear regression scheme. The results of the validation of the models constructed with DL, GBM, XGboost are presented in Figure 6, in which we compare the predicted λ, σ 2 and ∆Hc for the test set. Interestingly, all the ML models tested provide a rather accurate description of λ, σ 2 and ∆Hc with ∆Hc being the most accurately predicted quantity. It appears that the structural parameters of the HOIPs are well reproduced by describing the A-site cation site with only rA and NLP despite ignoring its affinity and the ionization potential. This successful characterization might be because the A-cation plays mainly the

19



role of the charge balance within the perovskite structure. 64,65 Nonetheless, the steric effect exerted by the A-cation strains the perovskite and determines its structural deformation away from the cubic geometry.

(b) A(0)PbI3

300 250

DL XGboost

200 150 100

300 250

150 100 50

0

0

1.5

2.0

2.5

r A (Å)

A (0)PbI3 A (1)PbI3 A (2)PbI3 A (0)PbBr 3 A (1)PbBr 3 A (2)PbBr 3

350

200

50

1.0

(c)

A (0)NbO3 A (1)NbO3 A (2)NbO3 A (0)NbS3 A (1)NbS3 A (2)NbS3

350

H c, prd (m eV/ion)

350

H c, prd (m eV/ion)

(a) H c, prd (m eV/ion)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 36

300 250 200 150

GA

100 50

1.0

1.5

2.0

2.5

r A (Å)

0

1.0

1.5

2.0

2.5

3.0

r A (Å)

Figure 7: ML predictions of ∆Hc,prd (eV/ion) as a function of rA for different NLP . (a) ∆Hc for perovskites APbI3 (NLP = 0) is predicted by using DL and XGBoost. In panel (b) and (c) we present the same curves, respectively for chalcogenide and halide perovskites and different NLP . Here a cation with NLP = n is labeled as A(n) . The shadowed light-yellow area marks the phase stability range, defined as ∆Hc < 50 eV/ion. All results in panels (b) and (c) have been obtained with XGBoost. As mentioned before, ∆Hc represents a straightforward measure of phase stability with respect to the ideal cubic phase stability, for which we built several rather accurate ML models that can be used to extrapolate predictions in parameters regions not covered by our DFT calculations. In particular, ∆Hc is predicted at a high level of accuracy, with XGBoost presenting the impressive variance ( R2 ≥ 0.99) meaning that our model is very accurate in the parameter region 0.8 < rA < 3.0 and 0 ≤ NLP ≤ 2. Subsequently, we run our ML models in order to identify regions of the parameters space promote the formation of stable, cubic perovskites allowing us to evaluate compounds containing a proper mixture of different A-site cations also. 66–68 For example, in Figure 7a, we show predicted ∆Hc by continuously varying the value of rA in the interval [0.8, 3.0] Å. We employ models generated by DL and XGBoost to plot the stability curve of the APbI3 cubic perovskite when a hypothetical cation has a radius 0.8 < rA < 3.0 and no lone-pair (NLP = 0) is accommodated within the [BC6 ]4 – octahedra. The 20


Page 21 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


DL-predicted curve is smooth, in contrast to the one predicted by XGBoost, which shows the stepwise prediction of ∆Hc as a function of rA . This stepwise prediction is due to the nature of the XGBoost learning algorithm based on the decision trees concept. Despite these technical differences, both methods return similar stability curves, that has its minimum (high-stability point) at rA ∼ 2Å (1.94±0.03 Å). Interestingly this value lays between the ionic radii of the Cs (1.88 Å) and Methylammonium (2.17 Å), which experimentally form a distorted perovskite structure at low temperature, but turn into cubic at room temperature. 69,70 In Figure 7b-c, we compare ∆Hc as a function of rA computed with the XGBoost model, respectively for Nb-based chalcogenide and Pb-based halide perovskites. In the figure, we consider values on NLP going from 0 to 2 (we denote A(n) for an A-site cation with NLP = n). For both chalcogenide and halide perovskites, it appears that the structure becomes more stable with increasing the number of lone-pairs. The phase is stabilized by an order of 10 meV/ion when going from NLP = 0 to NLP = 1 and a less dramatically when going from NLP = 1 and NLP = 2. This fact confirm that the lone pair of electrons residing on the A-site cation and interacting with the B-site are important drivers of the stability. To continue our analysis, we now establish a phase stability criterion; namely, we determine a value for ∆Hc below which a given structure would be the closest to the cubic perovskite phase together with the corresponding A-site cation radius. The largest organic cation we considered is Guanidinium, whose ionic radius is 2.78 Å. It is experimentally shown that it can be incorporated into MAPbI3 , forming a mixed-cation perovskite with higher stability than the pure MAPbI3 one. 66 Our ground state DFT-calculated ∆Hc for GAPbI3 is 43 meV/ion, while local configurations emerging from molecular orientations may trigger additional strain raising the energy by about 10 meV/ion (see Ref. 71 and panel (d) of Figure 5). This contribution of local structure variation suggests us that mixed-ion perovskites can form in the close to cubic phase given that ∆Hc is less than 50 meV/ion, a value below the utmost limit of crystalline metastability (∼70 meV/ion). 72 By using the ∆Hc < 50 meV/ion criterion of stability, we predict that among all our compounds, 32 % (61

21



chalcogenide and 452 halide perovskites) are possibly stable, this stability region highlighted with a light-yellow shadowed in all panels of Figure 7. For every B and C constituents of the perovskite structure, we could extract the optimal (rA,opt ) as well as the lower and upper bounds for rA , namely rA,min < rA < rA,max where the cubic phase is predicted to be stable. r A, m ax

r A, m in

r A, opt

Halide Perovskite

r A, m ax

r A, opt

Chalcogenide Perovskite r A, m in

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 36

AVO3

n/a n/a n/a

AGeF3

n/a n/a n/a

AVS3

n/a n/a n/a

AGeCl 3

116 145 193

AVSe 3

n/a n/a n/a

AGeBr 3

108 138 241

AVTe 3

n/a n/a n/a

AGeI3

99 135 255

ANbO3

114 147 172

ASnF3

146 166 187

ANbS3 132 161 186

ASnCl 3

127 168 224

ANbSe 3 146 164 181

ASnBr 3

124 168 248

ANbTe 3

n/a n/a n/a

ASnI3

120 179 276

ATaO3

101 147 174

APbF3

164 191 222

ATaS3

122 157 185

APbCl 3 155 194 243

ATaSe 3

142 161 177

APbBr 3

ATaTe 3

189 196 205

50

100

155 197 252

APbI3 148 195 259

200 150 Radius (pm )

250

300

Figure 8: XGBoost machine-learning estimates of the minimum, rA,min , maximum, rA,max and optimal, rA,opt , A-site cation radius defining the stability of the cubic perovskites. Here we consider the case of NLP = 0 and both chalcogenide (left-hand side panels) and halide (right-hand side panels) perovskites. "n/a" marks that the ML model cannot provide the prediction of rA due to the lack of input feature. We then apply our machine-learning models to compounds with all possible BC3 combinations, exploring their stability for the choice of the A-site cation. An example of our results is presented in Figure 8, where we list the predicted rA range, namely [rA,min , rA,max ] and the optimal value, rA,opt , as computed by XGBoost for NLP = 0. We notice that, when NLP = 0, for halide perovskites rA,min decrease and rA,max increases as one moves from flu22


Page 23 of 36

oride to iodide. This trend implies that the range of stability of halide perovskites gets larger as heavier halide elements are involved in the structure. The opposite is observed for chalcogenide perovskites, where both boundaries of the [rA,min , rA,max ] interval increase as the chalcogenide ion gets heavier, producing an overall reduction of the stability range. The difference in behavior between chalcogenide and halide perovskites (for NLP = 0) can be appreciated further by comparing all the ∆Hc (rA ) curves in Figure S9 of the SI. For NLP = 1, rA,min increases marginally for chalcogenide and remains approximately constant for halide perovskites, while rA,max increases for both as one move down the period of the anions. This tendency results in an increase (decrease) of the stability range for halide (chalcogenide) perovskites, although smaller than in the NLP = 0 cases. A similar trend is also found for NLP = 2. Then, we can conclude that overall halide perovskites with larger anions such as I− appear more tolerant of the A-site cation choice. As such, these seem to offer a better prospect to form mixed-cation complexes.

Cs

x

0.6

0.8

1

MA

20

0

Cs

10

ΔH c of APbI3

0 .4

30

0.2

0.4

x

0.6

0.8

1

MA

r A,mix (Å) 1.5 2.0 2.5

0 25 0 20 0 15 0 10

NLP

0 Rb

0

3.0 0 1 2

50

0 .8

0 .8 0 .4

0.4

1.0

0

0

0

0.2

y 0 .2

0 .2

0 .2

0 .6

0 .4

Na

0

1

1

1 0 .8

1

0 .6

0.8

40

1

x

0.6

1

0.4

z

(d)

50

0 .8

0 .8

0.2

y

0 .4

0 .6

0 .6 1

Li

0

0 .4

z

0 .2

0 .2

0 .2

y

ΔH c (m eV/ion)

FA 0

0

0 0 .4

z

(c)

EA

0 .6

(b)

K

0 .6

(a)

0 .8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 meV/ion

FA

Cs MA

MA0.75EA0.25

Figure 9: Diagrams illustrating ∆Hc for the specific choice of A-site mixture, for ternary mixtures as computed with the XGBoost model. Results are shown for (a) Lix Nay Kz NbO3 , (b) Csx MAy EAz PbI3 and (c) Csx MAy FAz PbI3 . Experimentally reported mixed-cation compounds are marked by green-filled circles in panels (a) and (b). (d) The predicted ∆Hc (meV/ion) of APbI3 as a function of rA,mix in Å. The radii of the representative cations and mixture are marked in the axis of rA,mix . The possibility to form a mixed-cation compound is explored in Figure 9, where we present ∆Hc of Ax A0y A00z BC3 , having a ternary mixture of A-site cation. Typically, the volume of a mixed-ion perovskite is linearly related to the concentration of the ions. 38,73 So, we define the average cation radius of the mixture as the composition-weighted average of the individual radii, accordingly, rA,mix = xrA + yrA0 + zrA00 , where x + y + z = 1 (x, y, z ≥ 0). 23



Subsequently, we make prediction based on our ML models using this approximation. For this exercise, we select a chalcogenide compound, ANbO3 , with possible cations Li– Na–K, and a halide , APbI3 , with two possible cation mixtures, Cs–MA–EA and Cs–MA–FA. It must be noted that some of the mixed-cation perovskites will be able to accommodate a given composition only by incorporating some level of local strain and probably by involving non-equilibrium growth. Figure 9 illustrates the predicted ∆Hc over the ternary phase space of the cations obtained by using the XGBoost model. In the panels (a) and (b) we mark with green dots ternary compositions for which a mixed-cation perovskite has been experimentally synthesized before. For ANbO3 , we find that any known ternary cation mixture 67,74–78 presents the value of ∆Hc within 50 meV/ion, laying within the stability criterion. The criterion of APbI3 is consistent with the representative and composition-weighted radii (Figure 9d). Therefore, we may expect a Cs–MA–EA mixed perovskite to be stable with low-concentration of EA. To our knowledge, no ternary cation Cs–MA–EA mixture has been reported to date for APbI3 . At the same time, we find that all compositions in the Cs–MA–FA diagram of APbI3 appear to be potentially synthesized (∆Hc < 50 meV/ion; 0.85 < TF < 0.99), meaning that ternarycation Cs–MA–FA perovskites should form across the entire compositional range. 79–81 This successful application gives us confidence that our perovskite stability predictor performs well. Finally, in our previous work, searching for mixed-ion perovskites, we have shown how to use the t-SNE method to visualize compounds presenting structural similarity. 55 The same exercise is conducted here for the much more extended materials dataset investigated in the present work providing a more complete picture. (see Figure S10 in SI)

Conclusions In the search for novel perovskites for photovoltaic applications, there have been several computational studies using large-scale inorganic databases of electronic structure proper-

24


Page 24 of 36

Page 25 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ties. These, however, are limited to only inorganic compounds, since the data for hybrid organic/inorganic compounds are limited, and thus the construction of the convex hull stability diagrams is problematic. To overcome this issue, we constructed a number of machine learning models estimating the likelihood of incorporating a given A-site cation in an ABC3 perovskite structure. This investigation is done by assessing the impact of the composition and more precisely the A-site cation on the phase stability of the perovskite structure by determining the degree of octahedral deformation a given cation induces once embedded within an [BC6 ]4− and how severely it deforms far from the ideal perovskite structure. We have found that the perovskite octahedral deformation and its related stabilization energy can be estimated with high accuracy by machine learning models, whose dominant features are the effective cationic radii and the number of lone pairs. This knowledge has allowed us to evaluate the cubic perovskite phase stability of hypothetical compounds across a wide cation range and to predict the range of effective A-cation radius offering the minimum deviation from the ideal perovskite structure in learning-and-matching exercises. Our proposed triple cation phase diagrams enable to emphasize the concept of effective cation radius based on a mixture of two, three or even four cations, which is a lab synthesis practice nowadays recognized to enhance the perovskite phase stability and shall not be left behind due to the myriad of possible combinations while preparing precursor components. Our work thus provides a powerful map for exploring a vast compositional space to guide the future synthesis of novel mixed cation hybrid organic/inorganic perovskites.

Acknowledgement This work is sponsored by the Qatar National Research Fund (QNRF) through the National Priorities Research Program (NPRP8-090-2-047) and by the Qatar Environment and Energy Research Institute (FE, FHA and NT). Computational resources have been provided by the research computing group at Texas A&M University at Qatar.

25



Supporting Information Available The supplementary information file includes: Details of the used organic cation; Visualization of DFT calculations; Comparisons of ML prediction; Pearson correlation coefficient of input features; t-SNE clustering.

This material is available free of charge via the Internet at

http://pubs.acs.org/.

References (1) NREL. PV efficiency chart. https://www.nrel.gov/pv/assets/pdfs/best-reserch-cellefficiencies.20190411.pdf. (2) Kim, H.-S.; Lee, C.-R.; Im, J.-H.; Lee, K.-B.; Moehl, T.; Marchioro, A.; Moon, S.-J.; Humphry-Baker, R.; Yum, J.-H.; Moser, J. E. et al. Lead Iodide Perovskite Sensitized All-Solid-State Submicron Thin Film Mesoscopic Solar Cell with Efficiency Exceeding 9%. Sci. Rep. 2012, 2, 591. (3) Green, M. A.; Ho-Baillie, A.; Snaith, H. J. The emergence of perovskite solar cells. Nat. Photonics 2014, 8, 506–514. (4) Zhang, M.; Lyu, M.; Yu, H.; Yun, J.-H.; Wang, Q.; Wang, L. Stable and Low-Cost Mesoscopic CH3 NH3 PbI2 Br Perovskite Solar Cells by using a Thin Poly(3-hexylthiophene) Layer as a Hole Transporter. Chem. Eur. J. 2015, 21, 434–439. (5) Frost, J. M.; Butler, K. T.; Brivio, F.; Hendon, C. H.; van Schilfgaarde, M.; Walsh, A. Atomistic Origins of High-Performance in Hybrid Halide Perovskite Solar Cells. Nano Lett. 2014, 14, 2584–2590. (6) Zhang, Z.; Long, R.; Tokina, M. V.; Prezhdo, O. V. Interplay between Localized and Free Charge Carriers Can Explain Hot Fluorescence in the CH3 NH3 PbBr3 Perovskite: Time-Domain Ab Initio Analysis. J. Am. Chem. Soc. 2017, 139, 17327–17333. 26


Page 26 of 36

Page 27 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(7) Katayama, T.; Suenaga, H.; Okuhata, T.; Masuo, S.; Tamai, N. Unraveling the Ultrafast Exciton Relaxation and Hidden Energy State in CH3 NH3 PbBr3 Nanoparticles. J. Phys. Chem. C 2018, 122, 5209–5214. (8) Lee, J.-H.; Bristowe, N. C.; Bristowe, P. D.; Cheetham, A. K. Role of hydrogen-bonding and its interplay with octahedral tilting in CH3NH3PbI3. Chem. Commun. 2015, 51, 6434–6437. (9) Maughan, A. E.; Ganose, A. M.; Candia, A. M.; Granger, J. T.; Scanlon, D. O.; Neilson, J. R. Anharmonicity and Octahedral Tilting in Hybrid Vacancy-Ordered Double Perovskites. Chem. Mater. 2018, 30, 472–483. (10) Yang, R. X.; Skelton, J. M.; da Silva, E. L.; Frost, J. M.; Walsh, A. Spontaneous Octahedral Tilting in the Cubic Inorganic Cesium Halide Perovskites CsSnX3 and CsPbX3 (X = F, Cl, Br, I). J. Phys. Chem. Lett. 2017, 8, 4720–4726. (11) Zhou, L.; Neukirch, A. J.; Vogel, D. J.; Kilin, D. S.; Pedesseau, L.; Carignano, M. A.; Mohite, A. D.; Even, J.; Katan, C.; Tretiak, S. Density of States Broadening in CH3 NH3 PbI3 Hybrid Perovskites Understood from ab Initio Molecular Dynamics Simulations. ACS Energy Lett. 2018, 3, 787–793. (12) Rashkeev, S. N.; El-Mellouhi, F.; Kais, S.; Alharbi, F. H. Domain Walls Conductivity in Hybrid Organometallic Perovskites and Their Essential Role in CH3 NH3 PbI3 Solar Cell High Performance. Sci. Rep. 2015, 5, 11467. (13) Berdiyorov, G. R.; El-Mellouhi, F.; Madjet, M. E.; Alharbi, F. H.; Rashkeev, S. N. Electronic transport in organometallic perovskite CH3 NH3 PbI3 : The role of organic cation orientations. Appl. Phys. Lett. 2016, 108, 053901. (14) Motta, C.; El-Mellouhi, F.; Sanvito, S. Charge carrier mobility in hybrid halide perovskites. Sci. Rep. 2015, 5, 12746.

27



(15) Mozur, E. M.; Maughan, A. E.; Cheng, Y.; Huq, A.; Jalarvo, N.; Daemen, L. L.; Neilson, J. R. Orientational Glass Formation in Substituted Hybrid Perovskites. Chem. Mater. 2017, 29, 10168–10177. (16) Ambrosio, F.; Wiktor, J.; De Angelis, F.; Pasquarello, A. Origin of low electron–hole recombination rate in metal halide perovskites. Energy Environ. Sci. 2018, 11, 101–105. (17) Ghosh, D.; Smith, A. R.; Walker, A. B.; Islam, M. S. Mixed A-Cation Perovskites for Solar Cells: Atomic-Scale Insights Into Structural Distortion, Hydrogen Bonding, and Electronic Properties. Chem. Mater. 2018, 30, 5194–5204. (18) Berhe, T. A.; Su, W.-N.; Chen, C.-H.; Pan, C.-J.; Cheng, J.-H.; Chen, H.-M.; Tsai, M.C.; Chen, L.-Y.; Dubale, A. A.; Hwang, B.-J. Organometal halide perovskite solar cells: degradation and stability. Energy Environ. Sci. 2016, 9, 323–356. (19) Latini, A.; Gigli, G.; Ciccioli, A. A study on the nature of the thermal decomposition of methylammonium lead iodide perovskite, CH 3NH 3PbI 3: an attempt to rationalise contradictory experimental results. Sustain. Energy Fuels 2017, 1, 1351–1357. (20) Juarez-Perez, E. J.; Ono, L. K.; Maeda, M.; Jiang, Y.; Hawash, Z.; Qi, Y. Photodecomposition and thermal decomposition in methylammonium halide lead perovskites and inferred design principles to increase photovoltaic device stability. J. Mater. Chem. A 2018, 6, 9604–9612. (21) Saliba, M. Perovskite solar cells must come of age. Science 2018, 359, 388–389. (22) Lu, S.; Zhou, Q.; Ouyang, Y.; Guo, Y.; Li, Q.; Wang, J. Accelerated discovery of stable lead-free hybrid organic-inorganic perovskites via machine learning. Nat. Commun. 2018, 9, 3405. (23) Balachandran, P. V.; Emery, A. A.; Gubernatis, J. E.; Lookman, T.; Wolverton, C.;

28


Page 28 of 36

Page 29 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Zunger, A. Predictions of new ABO3perovskite compounds by combining machine learning and density functional theory. Phys. Rev. Mater. 2018, 2, 2825. (24) Mao, X.; Sun, L.; Wu, T.; Chu, T.; Deng, W.; Han, K. First-Principles Screening of All-Inorganic Lead-Free ABX3 Perovskites. J. Phys. Chem. C 2018, 122, 7670–7675. (25) Körbel, S.;

Marques, M. A. L.;

Botti, S. Stable hybrid organic–inorganic

halide perovskites for photovoltaics from ab initiohigh-throughput calculations. J. Mater. Chem. A 2018, 6, 6463–6475. (26) Nakajima, T.; Sawada, K. Discovery of Pb-Free Perovskite Solar Cells via HighThroughput Simulation on the K Computer. J. Phys. Chem. Lett. 2017, 8, 4826–4831. (27) Binek, A.; Hanusch, F. C.; Docampo, P.; Bein, T. Stabilization of the Trigonal HighTemperature Phase of Formamidinium Lead Iodide. J. Phys. Chem. Lett. 2015, 6, 1249–1253. (28) Giorgi, G.; Fujisawa, J.-I.; Segawa, H.; Yamashita, K. Organic–Inorganic Hybrid Lead Iodide Perovskite Featuring Zero Dipole Moment Guanidinium Cations: A Theoretical Analysis. J. Phys. Chem. C 2015, 119, 4694–4701. (29) Tong, C.-J.; Geng, W.; Prezhdo, O. V.; Liu, L.-M. Role of Methylammonium Orientation in Ion Diffusion and Current–Voltage Hysteresis in the CH3 NH3 PbI3 Perovskite. ACS Energy Lett. 2017, 2, 1997–2004. (30) Goldschmidt, V. M. Die Gesetze der Krystallochemie. Die Naturwissenschaften 1926, 14, 477–485. (31) Swainson, I. P.; Hammond, R. P.; Soullière, C.; Knop, O.; Massa, W. Phase transitions in the perovskite methylammonium lead bromide, CH3ND3PbBr3. J. Solid State Chem. 2003, 176, 97–104.

29



(32) Glazer, A. M. The classification of tilted octahedra in perovskites. Acta Cryst. B 1972, 28, 3384–3392. (33) Kieslich, G.; Sun, S.; Cheetham, A. K. Solid-state principles applied to organic–inorganic perovskites: new tricks for an old dog. Chem. Sci. 2014, 5, 4712–4715. (34) Burger, S.; Ehrenreich, M. G.; Kieslich, G. Tolerance factors of hybrid organic–inorganic perovskites: recent improvements and current state of research. J. Mater. Chem. A 2018, 6, 21785–21793. (35) Becker, M.; Klüner, T.; Wark, M. Formation of hybrid ABX3 perovskite compounds for solar cell application: first-principles calculations of effective ionic radii and determination of tolerance factors. Dalton Trans. 2017, 46, 3500–3509. (36) Pilania, G.; Balachandran, P. V.; Kim, C.; Lookman, T. Finding New Perovskite Halides via Machine Learning. Front. Mater. 2016, 3, 13285. (37) Filip, M. R.; Giustino, F. The geometric blueprint of perovskites. Proc. Natl. Acad. Sci. 2018, 115, 5397–5402. (38) Li, Z.; Yang, M.; Park, J.-S.; Wei, S.-H.; Berry, J. J.; Zhu, K. Stabilizing Perovskite Structures by Tuning Tolerance Factor: Formation of Formamidinium and Cesium Lead Iodide Solid-State Alloys. Chem. Mater. 2016, 28, 284–292. (39) Robinson, K.; Gibbs, G. V.; Ribbe, P. H. Quadratic Elongation: A Quantitative Measure of Distortion in Coordination Polyhedra. Science 1971, 172, 567–570. (40) Lee, J.-H.; Bristowe, N. C.; Lee, J. H.; Lee, S.-H.; Bristowe, P. D.; Cheetham, A. K.; Jang, H. M. Resolving the Physical Origin of Octahedral Tilting in Halide Perovskites. Chem. Mater. 2016, 28, 4259–4266. (41) Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953–17979.

30


Page 30 of 36

Page 31 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(42) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 1999, 59, 1758–1775. (43) Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558–561. (44) Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 1996, 54, 11169–11186. (45) Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6, 15–50. (46) Perdew, J. P.; Burke, K.; Wang, Y. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Rev. B 1996, 54, 16533– 16539. (47) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (48) Tkatchenko, A.; Scheffler, M. Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data. Phys. Rev. Lett. 2009, 102, 073005. (49) Jain, A.; Ong, S. P.; Chen, W.; Medasani, B.; Qu, X.; Kocher, M.; Brafman, M.; Petretto, G.; Rignanese, G.-M.; Hautier, G. et al. FireWorks: a dynamic workflow system designed for high-throughput applications. Concur. Comp.-Pract. E. 2015, 27, 5037–5059. (50) Ong, S. P.; Richards, W. D.; Jain, A.; Hautier, G.; Kocher, M.; Cholia, S.; Gunter, D.; Chevrier, V. L.; Persson, K. A.; Ceder, G. Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis. Comput. Mater. Sci. 2013, 68, 314–319. 31



(51) Ong, S. P.; Wang, L.; Kang, B.; Ceder, G. Li–Fe–P–O2 Phase Diagram from First Principles Calculations. Chem. Mater. 2008, 20, 1798–1807. (52) Ong, S. P.; Jain, A.; Hautier, G.; Kang, B.; Ceder, G. Thermal stabilities of delithiated olivine MPO4 (M = Fe, Mn) cathodes investigated using first principles calculations. Electrochem. Commun. 2010, 12, 427–430. (53) Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.; Cholia, S.; Gunter, D.; Skinner, D.; Ceder, G. et al. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Mater. 2013, 1, 011002. (54) Jain, A.; Hautier, G.; Moore, C. J.; Ong, S. P.; Fischer, C. C.; Mueller, T.; Persson, K. A.; Ceder, G. A high-throughput infrastructure for density functional theory calculations. Comput. Mater. Sci. 2011, 50, 2295–2310. (55) Park, H.; Mall, R.; Alharbi, F. H.; Sanvito, S.; Tabet, N.; Bensmail, H.; El-Mellouhi, F. Exploring new approaches towards the formability of mixed-ion perovskites by DFT and machine learning. Phys. Chem. Chem. Phys. 2019, 21, 1078–1088. (56) Mall, R.; Cerulo, L.; Garofano, L.; Frattini, V.; Kunji, K.; Bensmail, H.; Sabedot, T. S.; Noushmehr, H.; Lasorella, A.; Iavarone, A. et al. RGBM: regularized gradient boosting machines for identification of the transcriptional regulators of discrete glioma subtypes. Nucleic Acids Res. 2018, 46, e39–e39. (57) Rawi, R.; Mall, R.; Kunji, K.; Shen, C.-H.; Kwong, P. D.; Chuang, G.-Y. PaRSnIP: sequence-based protein solubility prediction using gradient boosting machine. Bioinformatics 2017, 34, 1092–1098. (58) Khurana, S.; Rawi, R.; Kunji, K.; Chuang, G.-Y.; Bensmail, H.; Mall, R. DeepSol: a deep learning framework for sequence-based protein solubility prediction. Bioinformatics 2018, 32


Page 32 of 36

Page 33 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(59) Elbasir, A.; Moovarkumudalvan, B.; Kunji, K.; Kolatkar, P. R.; Mall, R.; Bensmail, H. DeepCrystal: A Deep Learning Framework for Sequence-based Protein Crystallization Prediction. Bioinformatics 2018, (60) Csáji, B. C. Approximation with artificial neural networks. Faculty of Sciences, Etvs Lornd University, Hungary 2001, 24, 48. (61) Arunan, E.; Desiraju, G. R.; Klein, R. A.; Sadlej, J.; Scheiner, S.; Alkorta, I.; Clary, D. C.; Crabtree, R. H.; Dannenberg, J. J.; Hobza, P. et al. Definition of the hydrogen bond (IUPAC Recommendations 2011). Pure Appl. Chem. 83, 1637–1641. (62) El-Mellouhi, F.; Akande, A.; Motta, C.; Rashkeev, S. N.; Berdiyorov, G.; Madjet, M. E.-A.; Marzouk, A.; Bentria, E. T.; Sanvito, S.; Kais, S. et al. Solar Cell Materials by Design: Hybrid Pyroxene Corner-Sharing VO4 Tetrahedral Chains. ChemSusChem 2017, 10, 1931–1942. (63) Gebhardt, J.; Rappe, A. M. Transition metal inverse-hybrid perovskites. J. Mater. Chem. A 2018, 6, 14560–14565. (64) Xiao, Z.; Meng, W.; Wang, J.; Mitzi, D. B.; Yan, Y. Searching for promising new perovskite-based photovoltaic absorbers: the importance of electronic dimensionality. Mater. Horiz. 2017, 4, 206–216. (65) Filip, M. R.; Eperon, G. E.; Snaith, H. J.; Giustino, F. Steric engineering of metal-halide perovskites with tunable optical band gaps. Nat. Commun. 2014, 5, 5757. (66) Jodlowski, A. D.; Roldán-Carmona, C.; Grancini, G.; Salado, M.; Ralaiarisoa, M.; Ahmad, S.; Koch, N.; Camacho, L.; de Miguel, G.; Nazeeruddin, M. K. Large guanidinium cation mixed with methylammonium in lead iodide perovskites for 19% efficient solar cells. Nat. Energy 2017, 2, 972–979.

33



(67) Gholipour, S.; Ali, A. M.; Correa-Baena, J.-P.; Turren-Cruz, S.-H.; Tajabadi, F.; Tress, W.; Taghavinia, N.; Grätzel, M.; Abate, A.; De Angelis, F. et al. GlobularitySelected Large Molecules for a New Generation of Multication Perovskites. Adv. Mater. 2017, 29, 1702005. (68) Gholipour, S.; Saliba, M. From Exceptional Properties to Stability Challenges of Perovskite Solar Cells. Small 2018, 14, 1802385. (69) Xiang, S.; Fu, Z.; Li, W.; Wei, Y.; Liu, J.; Liu, H.; Zhu, L.; Zhang, R.; Chen, H. Highly Air-Stable Carbon-Based α-CsPbI3 Perovskite Solar Cells with a Broadened Optical Spectrum. ACS Energy Lett. 2018, 3, 1824–1831. (70) Weller, M. T.; Weber, O. J.; Henry, P. F.; Di Pumpo, A. M.; Hansen, T. C. Complete structure and cation orientation in the perovskite photovoltaic methylammonium lead iodide between 100 and 352 K. Chem. Comm. 2015, 51, 4180–4183. (71) Woods-Robinson, R.; Broberg, D.; Faghaninia, A.; Jain, A.; Dwaraknath, S. S.; Persson, K. A. Assessing High-Throughput Descriptors for Prediction of Transparent Conductors. Chem. Mater. 2018, 30, 8375–8389. (72) Sun, W.; Dacek, S. T.; Ong, S. P.; Hautier, G.; Jain, A.; Richards, W. D.; Gamst, A. C.; Persson, K. A.; Ceder, G. The thermodynamic scale of inorganic crystalline metastability. Sci. Adv. 2016, 2, e1600225–e1600225. (73) Rehman, W.; McMeekin, D. P.; Patel, J. B.; Milot, R. L.; Johnston, M. B.; Snaith, H. J.; Herz, L. M. Photovoltaic mixed-cation lead mixed-halide perovskites: links between crystallinity, photo-stability and electronic properties. Energy Environ. Sci. 2017, 10, 361–369. (74) Isaza-Zapata, V.; Arias, A.; Maya, C.; Martínez, W.; Agudelo, A.; Álvarez, B.; Gómez, A.; Izquierdo, J. L. Ferroelectric response of Na0.9 Li0.1 NbO3 at room temperature. J. Phys.: Conf. Ser. 2017, 850, 012009. 34


Page 34 of 36

Page 35 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(75) Jin, B. M.; Bhalla, A. S.; Kim, J. N. Dielectric properties of Li0.4 K0.6 NbO3 crystal. Ferroelectrics 1994, 157, 221–226. (76) Madaro, F.; Tolchard, J. R.; Yu, Y.; Einarsrud, M.-A.; Grande, T. Synthesis of anisometric KNbO3 and K0.5 Na0.5 NbO3 single crystals by chemical conversion of nonperovskite templates. CrystEngComm 2011, 13, 1350–1359. (77) Kusumoto, K. Dielectric and Piezoelectric Properties of KNbO3 -NaNbO3 -LiNbO3 SrTiO3 Ceramics. Jpn. J. Appl. Phys. 2006, 45, 7440. (78) Choi, H.; Jeong, J.; Kim, H.-B.; Kim, S.; Walker, B.; Kim, G.-H.; Kim, J. Y. Cesiumdoped methylammonium lead iodide perovskite light absorber for hybrid solar cells. Nano Energy 2014, 7, 80–85. (79) Ono, L. K.; Juarez-Perez, E. J.; Qi, Y. Progress on Perovskite Materials and Solar Cells with Mixed Cations and Halide Anions. ACS Appl. Mater. Interfaces 2017, 9, 30197–30246. (80) Charles, B.; Dillon, J.; Weber, O. J.; Islam, M. S.; Weller, M. T. Understanding the stability of mixed A-cation lead iodide perovskites. J. Mater. Chem. A 2017, 5, 22495– 22499. (81) Saliba, M.; Matsui, T.; Seo, J.-Y.; Domanski, K.; Correa-Baena, J.-P.; Nazeeruddin, M. K.; Zakeeruddin, S. M.; Tress, W.; Abate, A.; Hagfeldt, A. et al. Cesiumcontaining triple cation perovskite solar cells: improved stability, reproducibility and high efficiency. Energy Environ. Sci. 2016, 9, 1989–1997.

35



Graphical TOC Entry

36


Page 36 of 36

Learn and Match Molecular Cations for Perovskites | The Journal of

Recommend Documents