A data fusion approach to optimize compositional stability of halide perovskites

Abstract

•Physics-informed machine learning enables accelerated search of stable perovskites•Closed-loop Bayesian optimization takes the human out of the decision-making loop•>17-fold higher stability achieved within a combinatorial space of CsxMAyFA1−x−yPbI3•Improved perovskite thin-film stability translates into enhanced solar cell reliability Despite recent intensive efforts to improve the environmental stability of halide perovskite materials for energy harvesting and conversion, traditional trial-and-error explorations face bottlenecks in the navigation of vast chemical and compositional spaces. We develop a closed-loop optimization framework that seamlessly marries data from first-principle calculations and high-throughput experimentation into a single machine learning algorithm. This framework enables us to achieve rapid optimization of compositional stability for CsxMAyFA1−x−yPbI3 perovskites while taking the human out of the decision-making loop. We envision that this data fusion approach is generalizable to directly tackle challenges in designing multinary materials, and we hope that our successful showcase on perovskites will encourage researchers in other fields to incorporate knowledge of physics into the search algorithms, applying hybrid machine learning models to guide discovery of materials in high-dimensional spaces. Search for resource-efficient materials in vast compositional spaces is an outstanding challenge in creating environmentally stable perovskite semiconductors. We demonstrate a physics-constrained sequential learning framework to subsequently identify the most stable alloyed organic-inorganic perovskites. We fuse data from high-throughput degradation tests and first-principle calculations of phase thermodynamics into an end-to-end Bayesian optimization algorithm using probabilistic constraints. By sampling just 1.8% of the discretized CsxMAyFA1−x−yPbI3 (MA, methylammonium; FA, formamidinium) compositional space, perovskites centered at Cs0.17MA0.03FA0.80PbI3 show minimal optical change under increased temperature, moisture, and illumination with >17-fold stability improvement over MAPbI3. The thin films have 3-fold improved stability compared with state-of-the-art multi-halide Cs0.05(MA0.17FA0.83)0.95Pb(I0.83Br0.17)3, translating into enhanced solar cell stability without compromising conversion efficiency. Synchrotron-based X-ray scattering validates the suppression of chemical decomposition and minority phase formation achieved using fewer elements and a maximum of 8% MA. We anticipate that this data fusion approach can be extended to guide materials discovery for a wide range of multinary systems. Search for resource-efficient materials in vast compositional spaces is an outstanding challenge in creating environmentally stable perovskite semiconductors. We demonstrate a physics-constrained sequential learning framework to subsequently identify the most stable alloyed organic-inorganic perovskites. We fuse data from high-throughput degradation tests and first-principle calculations of phase thermodynamics into an end-to-end Bayesian optimization algorithm using probabilistic constraints. By sampling just 1.8% of the discretized CsxMAyFA1−x−yPbI3 (MA, methylammonium; FA, formamidinium) compositional space, perovskites centered at Cs0.17MA0.03FA0.80PbI3 show minimal optical change under increased temperature, moisture, and illumination with >17-fold stability improvement over MAPbI3. The thin films have 3-fold improved stability compared with state-of-the-art multi-halide Cs0.05(MA0.17FA0.83)0.95Pb(I0.83Br0.17)3, translating into enhanced solar cell stability without compromising conversion efficiency. Synchrotron-based X-ray scattering validates the suppression of chemical decomposition and minority phase formation achieved using fewer elements and a maximum of 8% MA. We anticipate that this data fusion approach can be extended to guide materials discovery for a wide range of multinary systems. The environmental instability of organic-inorganic halide perovskite materials limits their usage in optoelectronics, such as in solar cells, light emitters, lasers, and photodetectors.1Boyd C.C. Cheacharoen R. Leijtens T. McGehee M.D. Understanding degradation mechanisms and improving stability of perovskite photovoltaics.Chem. Rev. 2019; 119: 3418-3451https://doi.org/10.1021/acs.chemrev.8b00336Crossref PubMed Scopus (575) Google Scholar Compositional engineering is, to date, one of the most effective methods to improve the stability of perovskites in the presence of heat, humidity, and light without sacrificing optoelectronic performance.2Jeon N.J. Noh J.H. Yang W.S. Kim Y.C. Ryu S. Seo J. Seok S.I. Compositional engineering of perovskite materials for high-performance solar cells.Nature. 2015; 517: 476-480https://doi.org/10.1038/nature14133Crossref PubMed Scopus (4521) Google Scholar This fact has led to intensive research within combinatorial spaces, such as AxByC1−x−yPb(IzBr1−z)3.3Saliba M. Polyelemental, multicomponent perovskite semiconductor libraries through combinatorial screening.Adv. Energy Mater. 2019; 9: 1803754https://doi.org/10.1002/aenm.201803754Crossref Scopus (50) Google Scholar However, only a small fraction of this compositional space has been experimentally explored, in part due to the prohibitively expensive brute force synthesis. The paucity of resulting degradation data inhibits generalization of mechanisms across this diverse chemical and structural space, requiring each compositional search to begin with ab initio experimental investigations.4Khenkin M.V. Katz E.A. Abate A. Bardizza G. Berry J.J. Brabec C. Brunetti F. Bulović V. Burlingame Q. Di Carlo A. et al.Consensus statement for stability assessment and reporting for perovskite photovoltaics based on ISOS procedures.Nat. Energy. 2020; 5: 35-49https://doi.org/10.1038/s41560-019-0529-5Crossref Scopus (300) Google Scholar This challenge is similar to those faced by other materials communities, including the search for heterogeneous catalysts, alloyed battery electrodes, and high-entropy metal alloys for structural and magnetic materials.5Rohr B. Stein H.S. Guevarra D. Wang Y. Haber J.A. Aykol M. Suram S.K. Gregoire J.M. Benchmarking the acceleration of materials discovery by sequential learning.Chem. Sci. 2020; 11: 2696-2706https://doi.org/10.1039/c9sc05999gCrossref PubMed Scopus (26) Google Scholar, 6Zhang W. Mao J. Li S. Chen Z. Guo Z. Phosphorus-based alloy materials for advanced potassium-ion battery anode.J. Am. Chem. Soc. 2017; 139: 3316-3319https://doi.org/10.1021/jacs.6b12185Crossref PubMed Scopus (583) Google Scholar, 7George E.P. Raabe D. Ritchie R.O. High-entropy alloys.Nat. Rev. Mater. 2019; 4: 515-534https://doi.org/10.1038/s41578-019-0121-4Crossref Scopus (799) Google Scholar The halide perovskite field and several others require new tools to experimentally navigate these vast spaces efficiently to locate optima and to extract generalizable scientific insights.8Zhong M. Tran K. Min Y. Wang C. Wang Z. Dinh C.T. De Luna P. Yu Z. Rasouli A.S. Brodersen P. et al.Accelerated discovery of CO2 electrocatalysts using active machine learning.Nature. 2020; 581: 178-183https://doi.org/10.1038/s41586-020-2242-8Crossref PubMed Scopus (247) Google Scholar, 9Kumar G. Bossert H. McDonald D. Chatzidimitriou A. Ardagh M.A. Pang Y. Lee C. Tsapatsis M. Abdelrahman O.A. Dauenhauer P.J. Catalysis-in-a-Box: robotic screening of catalytic materials in the time of COVID-19 and beyond.Matter. 2020; 3: 805-823https://doi.org/10.1016/j.matt.2020.06.025Abstract Full Text Full Text PDF PubMed Scopus (8) Google Scholar, 10Burger B. Maffettone P.M. Gusev V.V. Aitchison C.M. Bai Y. Wang X. Li X. Alston B.M. Li B. Clowes R. et al.A mobile robotic chemist.Nature. 2020; 583: 237-241https://doi.org/10.1038/s41586-020-2442-2Crossref PubMed Scopus (191) Google Scholar, 11Cole J.M. A design-to-device pipeline for data-driven materials discovery.Acc. Chem. Res. 2020; 53: 599-610https://doi.org/10.1021/acs.accounts.9b00470Crossref PubMed Scopus (27) Google Scholar, 12Gómez-Bombarelli R. Aguilera-Iparraguirre J. Hirzel T.D. Duvenaud D. Maclaurin D. Blood-Forsythe M.A. Chae H.S. Einzinger M. Ha D.G. Wu T. et al.Design of efficient molecular organic light-emitting diodes by a high-throughput virtual screening and experimental approach.Nat. Mater. 2016; 15: 1120-1127https://doi.org/10.1038/nmat4717Crossref PubMed Scopus (465) Google Scholar, 13Kirman J. Johnston A. Kuntz D.A. Askerka M. Gao Y. Todorović P. Ma D. Privé G.G. Sargent E.H. Machine-learning-accelerated perovskite crystallization.Matter. 2020; 2: 938-947https://doi.org/10.1016/j.matt.2020.02.012Abstract Full Text Full Text PDF Scopus (38) Google Scholar, 14Li Z. Najeeb M.A. Alves L. Sherman A. Parrilla P.C. Pendleton I.M. Zeller M. Schrier J. Norquist A.J. Chan E. Robot-accelerated perovskite investigation and discovery (RAPID): 1. Inverse Temperature Crystallization.ChemRxiv. 2019; https://doi.org/10.26434/CHEMRXIV.10013090.V1Crossref Google Scholar Machine learning-based sequential learning approaches (e.g., Bayesian optimization [BO]) have emerged as efficient materials search tools that explore vast variable spaces in a “closed-loop” fashion, whereby the outcome of one experimental round informs the next without human intervention. BO, which has attracted increasing attention in the recent developments of self-driving laboratories in various fields of materials science, recently successfully directed experimentation in the search of organic hole-transport materials,15MacLeod B.P. Parlane F.G.L. Morrissey T.D. Häse F. Roch L.M. Dettelbach K.E. Moreira R. Yunker L.P.E. Rooney M.B. Deeth J.R. et al.Self-driving laboratory for accelerated discovery of thin-film materials.Sci. Adv. 2020; 6: eaaz8867https://doi.org/10.1126/sciadv.aaz8867Crossref PubMed Scopus (102) Google Scholar piezoelectric oxides,16Xue D. Balachandran P.V. Yuan R. Hu T. Qian X. Dougherty E.R. Lookman T. Accelerated search for BaTiO3-based piezoelectrics with vertical morphotropic phase boundary using Bayesian learning.Proc. Natl. Acad. Sci. U S A. 2016; 113: 13301-13306https://doi.org/10.1073/pnas.1607412113Crossref PubMed Scopus (84) Google Scholar and organic photocatalysts.10Burger B. Maffettone P.M. Gusev V.V. Aitchison C.M. Bai Y. Wang X. Li X. Alston B.M. Li B. Clowes R. et al.A mobile robotic chemist.Nature. 2020; 583: 237-241https://doi.org/10.1038/s41586-020-2442-2Crossref PubMed Scopus (191) Google Scholar Within the field of perovskite solar cells, machine learning has been combined with robotic liquid synthesis for microcrystal crystallization.17Epps R.W. Bowen M.S. Volk A.A. Abdel-Latif K. Han S. Reyes K.G. Amassian A. Abolhasani M. Artificial chemist: an autonomous quantum dot synthesis bot.Adv. Mater. 2020; 32: 2001626https://doi.org/10.1002/adma.202001626Crossref PubMed Scopus (70) Google Scholar, 18Gu E. Tang X. Langner S. Duchstein P. Zhao Y. Levchuk I. Kalancha V. Stubhan T. Hauch J. Egelhaaf H.J. et al.Robot-based high-throughput screening of antisolvents for lead halide perovskites.Joule. 2020; 4: 1806-1822https://doi.org/10.1016/j.joule.2020.06.013Abstract Full Text Full Text PDF Scopus (26) Google Scholar, 19Higgins K. Valleti S.M. Ziatdinov M. Kalinin S. Ahmadi M. Chemical robotics enabled exploration of stability and photoluminescent behavior in multicomponent hybrid perovskites via machine learning.ChemRxiv. 2020; https://doi.org/10.26434/CHEMRXIV.12436079.V1Crossref Google Scholar, 20Choubisa H. Askerka M. Ryczko K. Voznyy O. Mills K. Tamblyn I. Sargent E.H. Crystal site feature embedding enables exploration of large chemical spaces.Matter. 2020; 3: 433-448https://doi.org/10.1016/j.matt.2020.04.016Abstract Full Text Full Text PDF Scopus (15) Google Scholar However, such a model-free statistical approach shows limitations without principled guidance from domain expertise, because it has to learn everything from scratch. Recent in situ experiments and first-principle calculations independently revealed insights into the fundamental composition-dependent instability in organic-inorganic perovskites and their alloys; however, merging computational and experimental insights on selective compositions into a generalizable optimization policy over the entire chemical space remains a challenge.3Saliba M. Polyelemental, multicomponent perovskite semiconductor libraries through combinatorial screening.Adv. Energy Mater. 2019; 9: 1803754https://doi.org/10.1002/aenm.201803754Crossref Scopus (50) Google Scholar State-of-the-art two-step approaches of directly applying theoretical screening as a hard constraint before shortlisted synthesis are limited by inefficiencies arising from: (1) high-performing theoretical calculations for organic-inorganic systems are often too sparse to guide experimentation, and (2) the discrepancies between the calculation assumptions and the experiments at non-thermodynamic equilibria decreases search accuracy.21Curtarolo S. Hart G.L. Nardelli M.B. Mingo N. Sanvito S. Levy O. The high-throughput highway to computational materials design.Nat. Mater. 2013; 12: 191-201https://doi.org/10.1038/nmat3568Crossref PubMed Scopus (1110) Google Scholar,22Jesper Jacobsson T. Correa-Baena J.-P. Pazoki M. Saliba M. Schenk K. Grätzel M. Hagfeldt A. Exploration of the compositional space for mixed lead halogen perovskites for high efficiency solar cells.Energy Environ. Sci. 2016; 9: 1706-1724https://doi.org/10.1039/C6EE00030DCrossref Google Scholar The lack of physics-informed and iterative materials search hinders the ultimate goal of designing perovskite compositions for enhanced environmental stability. Here, we introduce a data fusion approach to incorporate both Gibbs free energy of mixing (ΔGmix) from density functional theory (DFT) calculation23Schelhas L.T. Li Z. Christians J.A. Goyal A. Kairys P. Harvey S.P. Kim D.H. Stone K.H. Luther J.M. Zhu K. et al.Insights into operational stability and processing of halide perovskite active layers.Energy Environ. Sci. 2019; 12: 1341-1348https://doi.org/10.1039/c8ee03051kCrossref Scopus (77) Google Scholar and experimentally quantified degradation from accelerated aging tests to every decision that the BO algorithm is making. We apply this closed-loop machine learning framework to optimize lead iodide perovskites that suffer from severe heat and moisture-induced degradation within the five-element space of CsxMAyFA1−x−yPbI3. Under multiplex environmental stress tests with increased temperature, humidity, and illumination in air, we identified compositions overperforming the MAPbI3 starting point by 17-fold and our state-of-the-art reference composition of (Cs0.05(MA0.17FA0.83)0.95Pb(I0.83Br0.17)3) by 3-fold within three optimization rounds, and the results are found transferable to device stability. DFT here serves as principled guidance within the decision-making algorithm to constrain the search space to not only chemically, but also the structurally stable α-perovskite alloys. To efficiently guide the compositional search, we constructed a physics-informed batch BO framework (Figure 1). In BO, promising compositions for the next experimental round are suggested by an acquisition function, such as expected improvement, EI(Θ), which balances the exploitation of the most stable regions and the exploration of high-uncertainty regions within the compositional space. As a key algorithm contribution, we fuse ΔGmix as a probabilistic constraint of the BO acquisition function in the “composition selection” step, providing additional information on phase stability to effectively identify multi-cation perovskites that are thermodynamically stable relative to their single-cation counterparts (Figures 1A and 1B). We define “instability index” (Ic), a figure of merit for optimizing stability. The goal of each optimization round, which consists of three steps of composition selection, “film synthesis,” and “instability quantification,” is to minimize this value. Our batch BO algorithm makes use of a surrogate ML model, Gaussian process (GP) regression,24SheffieldML/GPyOpt: Gaussian Process Optimization Using GPy n.d..https://github.com/SheffieldML/GPyOptGoogle Scholar to estimate the value and uncertainty of Ic in non-explored regions of the compositional space (see the Experimental procedures). Within each optimization round (one batch in BO), 28 spin-coated thin-film samples (Figure 1C) are examined in situ in parallel using an environmental chamber under 85% relative humidity (RH) and 85°C in the air (Figure S1). 0.15 Sun visible only illumination is applied to enable automatic image capture every 5 min using an RGB camera (~200 μm resolution). Photoactive α-perovskite phases within CsxMAyFA1−x−yPbI3 exhibit a band gap of ~1.5 eV, whereas their main degradation products under hot and humid conditions, PbI2 (2.27 eV)25Zhu X.H. Wei Z.R. Jin Y.R. Xiang A.P. Growth and characterization of a PbI2 single crystal used for gamma ray detectors.Cryst. Res. Technol. 2007; 42: 456-459https://doi.org/10.1002/crat.200610847Crossref Scopus (71) Google Scholar δ-CsPbI3 (2.82 eV),26Hu Y. Bai F. Liu X. Ji Q. Miao X. Qiu T. Zhang S. Bismuth incorporation stabilized α-CsPbI3 for fully inorganic perovskite solar cells.ACS Energy Lett. 2017; 2: 2219-2227https://doi.org/10.1021/acsenergylett.7b00508Crossref Scopus (349) Google Scholar or δ-FAPbI3 (2.43 eV)27Masi S. Echeverría-Arrondo C. Salim K.M.M. Ngo T.T. Mendez P.F. López-Fraguas E. Macias-Pinilla D.F. Planelles J. Climente J.I. Mora-Sero I. Chemi-structural stabilization of formamidinium lead iodide perovskite by using embedded quantum dots.ACS Energy Lett. 2020; : 418-427https://doi.org/10.1021/acsenergylett.9b02450Crossref Scopus (44) Google Scholar show deteriorated photophysical properties (Figure S2). As shown in Figure 1D, we hence used a color-based metric as a proxy to capture the macroscopic evolution of the high-band-gap, non-perovskite phases (see Videos S1 and S2).28Hashmi S.G. Tiihonen A. Martineau D. Ozkan M. Vivo P. Kaunisto K. Ulla V. Zakeeruddin S.M. Grätzel M. Long term stability of air processed inkjet infiltrated carbon-based printed perovskite solar cells under intense ultra-violet light soaking.J. Mater. Chem. A. 2017; 5: 4797-4802https://doi.org/10.1039/c6ta10605fCrossref Scopus (60) Google Scholar, 29Hashmi S.G. Martineau D. Li X. Ozkan M. Tiihonen A. Dar M.I. Sarikka T. Zakeeruddin S.M. Paltakari J. Lund P.D. Grätzel M. Air processed inkjet infiltrated carbon based printed perovskite solar cells with high stability and reproducibility.Adv. Mater. Technol. 2017; 2: 1600183https://doi.org/10.1002/admt.201600183Crossref Scopus (104) Google Scholar, 30Stoddard R.J. Dunlap-Shohl W.A. Qiao H. Meng Y. Kau W.F. Hillhouse H.W. Forecasting the decay of hybrid perovskite performance using optical transmittance or reflected dark-field imaging.ACS Energy Lett. 2020; 5: 946-954https://doi.org/10.1021/acsenergylett.0c00164Crossref Scopus (8) Google Scholar We define the instability index (Ic) as the integrated color change of an unencapsulated perovskite film over accelerated degradation test duration T. Complementary direct band-gap measurements before and after the degradation tests using UV-vis spectroscopy are listed in Figure S12.Ic(Θ)=∑c={R,G,B}∫0minT|c(t,Θ)−c(0,Θ)|dt,(Equation 1) where composition Θ=(x,y,1−x−y), t is time, and c are area-averaged, color-calibrated red, green, and blue pixel values of the sample. The cutoff time was set to T = 7,000 min based on the observed divergence between the most- and least-stable compositions (Figure S3). Our closed-loop and iterative workflow enable the systematic optimization of multi-cation perovskites against degradation by varying the nominal compositions, Θ, within CsxMAyFA1−x−yPbI3 (x, y limit to two decimal places) (Tables S1 and S2). https://www.cell.com/cms/asset/b8d56803-5bb1-4c11-8a89-20768dc0abcd/mmc3.mp4Loading ... Download .mp4 (0.15 MB) Help with .mp4 files Video S1. Optical changes of perovskite thin films in initialization Round 0 https://www.cell.com/cms/asset/74c8d777-8b82-4d42-bed7-e64ea6fcb67d/mmc4.mp4Loading ... Download .mp4 (0.16 MB) Help with .mp4 files Video S2. Optical changes of perovskite thin films in optimization Round 3 Due to their polymorphic nature, identical perovskite compositions crystallized into different phases can exhibit diverse degradation behaviors, making it essential to evaluate phase stabilities in any perovskite composition optimization.1Boyd C.C. Cheacharoen R. Leijtens T. McGehee M.D. Understanding degradation mechanisms and improving stability of perovskite photovoltaics.Chem. Rev. 2019; 119: 3418-3451https://doi.org/10.1021/acs.chemrev.8b00336Crossref PubMed Scopus (575) Google Scholar The end members of the compositional space in this study consist of the cubic α-FA/MAPbI3 perovskites and the non-perovskite δ-CsPbI3 at the synthesis temperature.31Kim S. Eom T. Ha Y.-S. Hong K.-H. Kim H. Thermodynamics of multicomponent perovskites: a guide to highly efficient and stable solar cell materials.Chem. Mater. 2020; 32: 4265-4272https://doi.org/10.1021/acs.chemmater.0c00893Crossref Scopus (9) Google Scholar Phase de-mixing during synthesis leads to minority phases within thin-film samples before degradation tests and are, therefore, not captured in Ic. Nevertheless, phase de-mixing during film formation or soon after is not desirable because it causes deterioration of the electronic properties of the perovskite.32Knight A.J. Herz L.M. Preventing phase segregation in mixed-halide perovskites: a perspective.Energy Environ. Sci. 2020; 13: 2024https://doi.org/10.1039/d0ee00788aCrossref Scopus (84) Google Scholar Schelhas et al.23Schelhas L.T. Li Z. Christians J.A. Goyal A. Kairys P. Harvey S.P. Kim D.H. Stone K.H. Luther J.M. Zhu K. et al.Insights into operational stability and processing of halide perovskite active layers.Energy Environ. Sci. 2019; 12: 1341-1348https://doi.org/10.1039/c8ee03051kCrossref Scopus (77) Google Scholar recently demonstrated the use of DFT calculations to predict the phase de-mixing tendency between α-CsxMAyA1−x−yPbI3 (Gmix) and their single-cation perovskite polymorphs APbI3 (A = Cs, MA, or FA) (G0) at a given temperature. Here, we fuse the composition-dependent change in Gibbs free energy of mixing, ΔGmix as a constraint into the experimental optimization loop (Figure 2A). This approach allows the α- and δ-phase relative stability in the non-degraded perovskite samples to be considered in the composition selection, thus enabling us to reduce sampling in regions with high probability of minority phase formation. Data fusion refers to a set of techniques where ML is used to map two or more datasets coming from related but distinct distributions. In our case, we relate the theoretical ΔGmix(Θ) and the experimental Ic(Θ). The two data streams account for distinct mechanisms of modeled thermodynamic phase instability and measured macroscopic thermal-moisture instability, respectively. Hence, it is inadequate to combine both datasets as equivalent or include DFT directly as a prior following state-of-the-art model-free BO.33Herbol H.C. Poloczek M. Clancy P. Cost-effective materials discovery: Bayesian optimization across multiple information sources.Mater. Horiz. 2020; 7: 2113https://doi.org/10.1039/D0MH00062KCrossref Google Scholar,34Doan H.A. Agarwal G. Qian H. Counihan M.J. Rodríguez-López J. Moore J.S. Assary R.S. Quantum chemistry-informed active learning to accelerate the design and discovery of sustainable energy storage materials.Chem. Mater. 2020; 32: 6338https://doi.org/10.1021/acs.chemmater.0c00768Crossref Scopus (14) Google Scholar Here, we define a data-fused probabilistic constraint approach according to Equation 2:P(ΔGmix(Θ),βDFT)=11+e−ΔGmix(Θ)/βDFT,(Equation 2) where P(ΔGmix(Θ),βDFT) is a logistic cumulative distribution function modeling the phase mixing probability and βDFT is a data fusion parameter calibrated according to ΔGmix calculations to control the smoothness of the boundaries from stable to unstable compositions, forming a soft compositional boundary as shown in Figure 2A (see the Experimental procedures for algorithm details). Given the computational cost and complexity of DFT calculations on organic-inorganic hybrid systems, we first regress 85 DFT-modeled ΔGmix values on 47 single-cation and binary alloyed compositions (29 MAFA and CsFA compositions from Schelhas et al.23Schelhas L.T. Li Z. Christians J.A. Goyal A. Kairys P. Harvey S.P. Kim D.H. Stone K.H. Luther J.M. Zhu K. et al.Insights into operational stability and processing of halide perovskite active layers.Energy Environ. Sci. 2019; 12: 1341-1348https://doi.org/10.1039/c8ee03051kCrossref Scopus (77) Google Scholar and 12 CsMA compositions computed for the present work using the same methods) over the quasi-ternary CsxMAyA1−x−yPbI3 phase space using an auxiliary GP model that defines ΔGmix(Θ). Figure 2A visualizes the probability of phase mixing P(ΔGmix(Θ),βDFT)∈[0,1], where low values suggest phase instability (ΔGmix>> 0) and high values suggest phase stability (ΔGmix << 0). Our work is inspired by the unknown constraint BO proposed by Gelbart et al.35Gelbart M.A. Snoek J. Adams R.P. Bayesian optimization with unknown constraints.Uncertain. Artif. Intell. Proc. 30th Conf. UAI. 2014; 2014: 250-259Google Scholar By developing a probabilistic constraint model P(ΔGmix(Θ),βDFT) instead of applying a hard constraint boundary, we are able to discount regions predicted by DFT to go through phase de-mixing rather than completely exclude any unfavorable regions. This approach accounts for the inherent uncertainty in DFT predictions, chemical accuracy, and data scarcity through the use of the soft compositional boundary to model the stability threshold (see the Experimental procedures for βDFT calibration). The proposed algorithm allows us to seamlessly adapt DFT into the experimental optimizations loop, thereby achieving a physics-informed and sample-efficient search without being limited by the unknown exact phase boundaries across a vast compositional space (Figures S4 and S5). To integrate the probabilistic constraint into the BO formulation, we weigh the acquisition function with the value of P(ΔGmix(Θ),βDFT) and obtain a DFT-weighted BO acquisition function, EIC(Θ),as illustrated in Figure 2A. Traditional EI(Θ) utilizes the Ic results of our first experimental round without DFT and indicates two potential optima in Cs-poor and Cs-rich regions, respectively. The DFT-weighted EIC(Θ) effectively reduces sampling in energetically unfavorable Cs-rich regions despite low Ic: the subsequent optimization rounds converge to stable nominal compositions with a high probability of stable α-perovskite films among Cs-poor regions (Figures 2A, S6, and S7). Comparisons of optimization with and without DFT weighting using a teacher-student model are shown in Figures S8 and S9, which validates that, without data fusion, the model-free BO algorithm continues to suggest sampling in Cs-rich regions despite their phase instability. Figure 2B demonstrates that batch BO sequentially identifies the most stable regions over one initialization and three optimization rounds of synthesis and degradation tests. Iterative evolution of the landscape (posterior mean of Ic, Ic(Θ), with uncertainty) is presented in Figures S3 and S4. Figure 2C reveals a rapid decrease in experimentally quantified Ic from Rounds 0–3. The search converges after three optimization rounds (see Figure S5 for convergence conditions) to an optimal composition region centered at Cs0.17MA0.03FA0.80PbI3 and bounded by 8%–29% Cs, <14% MA, and 68%–92% FA. The identification of the global optimum lying within an FA-rich, and Cs- and MA-poor region is consistent with the reports that FA-rich perovskites show superior environmental stability compared with their MA-rich counterparts and the less volatile Cs is expected to enhance the heat and moisture resistance.36Saliba M. Matsui T. Seo J.Y. Domanski K. Correa-Baena J.P. Nazeeruddin M.K. Zakeeruddin S.M. Tress W. Abate A. Hagfeldt A. Grätzel M. Cesium-containing triple cation perovskite solar cells: improved stability, reproducibility and high efficiency.Energy Environ. Sci. 2016; 9: 1989-1997https://doi.org/10.1039/C5EE03874JCrossref PubMed Google Scholar Interestingly, we found a local optimum near Cs0.26MA0.36FA0.38PbI3, which emerged in Round 1. We sampled four additional compositions in Round 3 and validated that the non-intuitive local optima suggested by the algorithm is reproducible. The ability to rapidly identify non-intuitive regions of success is a major advantage of using an automated closed-loop optimization algorithm over materials search strategies leveraging human intuition alone. Further experimental validation and mechanisms study of the identified compositional regions of interest are discussed in the next subsection. We define the compositional space as the discretized quasi-ternary-phase space subdivided by the minimum achievable experimental resolution (1% composition). This yields 5,151 possible singular, binary, and ternary cation compositions, 1.8% of which were sampled experimentally while converging to the optimal region (i.e., 94 unique compositions and 112 samples within Round 0–3, see the Supplemental information for more details). Three additional degradation rounds of seven representative compositions were performed to validate the instability trend, with structural and optical characterization shown in Table S3 and Figures S10–S13. We find the overall stability landscape within the CsxMAyFA1−x−yPbI3 compositional space to be non-linear. To quantify the divergence in de