Communication Cite This: J. Am. Chem. Soc. 2017, 139, 14905-14908
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Thermodynamic Stability Trend of Cubic Perovskites Qingde Sun†,‡ and Wan-Jian Yin*,†,‡ †
Soochow Institute for Energy and Materials Innovations (SIEMIS), College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, and ‡Key Laboratory of Advanced Carbon Materials and Wearable Energy Technologies of Jiangsu Province, Soochow University, Suzhou 215006, China
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S Supporting Information *
hybrid perovskites6−11 and exploration of emerging stable perovskites.12−15 Pellet et al.16 first observed improved stability and efficiency for (MA,FA)PbI3. Li et al.,6 Yi et al.,7 and Lee et al.8 stabilized photosensitive black perovskite by alloying the large-t FAPbI3 and small-t CsPbI3, in which the black phase is unstable at two end parts. Saliba et al.9,17 adopted the mixing of multiple cations, including much smaller Rb ions, to effectively tune tolerance factor t, and they obtained cell efficiency up to 21.6% with good stability. The crucial importance of tuning t is also reflected by the fact that, among the best six certified perovskite solar cell efficiencies reported by the National Renewable Energy Laboratory, four are based on mixed perovskite.10,18 For all the work above, intermixing of cations with different steric sizes (Cs, CH3NH3, NH2CHNH2) on A site or (I, Br, Cl) on X site19−21 leads to tunable effective ionic size and thus changes the tolerance factor t to stable range, explaining the improvement of stability. Although t or the (t, μ) map provides a good qualitative picture, the quantitative relationship between perovskite stability and t or μ has never been investigated, and the underlying fundamental issues are unclear. For example, is there any trend for perovskite stability? Is t a good quantitative descriptor for the stability? A good stability descriptor should not only provide quantitative guidance in experiments to stabilize the organic− inorganic hybrid perovskites but also be a key parameter for searching new emerging stable perovskites through highthroughput calculations and experimental synthesis. In this paper, thermodynamic stabilities of 138 cubic perovskite compounds (Figure 1), including 12 halide ABX3 compounds, 18 chalcogenide ABX3 compounds, 72 halide compounds A2B1B2X6, and 36 chalcogenide compounds A2B1B2X6, are investigated by calculating their decomposition energies (ΔHD) based on first-principles density functional theory (DFT) calculations. We have found that tolerance factor t is not a good descriptor for perovskite stability. Instead, a linear relationship between ΔHD and (t+μ)η was found, where η is the atomic packing fraction (APF). As a descriptor of thermodynamic stability, (t+μ)η predicts the relative stability among halide (chalcogenide) perovskites within the accuracy rate at 86% (90%). Such descriptor has then been adopted to predict decomposition energies (ΔHD) of another 69 perovskites, and excellent agreement among descriptor-predicted, DFT-calculated, and experimental results is found, clearly showing the generalization of the trend. Moreover, unlike t or the (t, μ) map,
ABSTRACT: Stability is of central importance in current perovskite solar cell research and applications. Goldschmidt tolerance factor (t) recently provided qualitative guidance for experimentalists to engineer stable ABX3 perovskite by tuning effective ionic size with mixing cations or anions and for theorists to search emerging perovskites. Through first-principles calculations, we have calculated decomposition energies of 138 perovskite compounds of potential solar cell applications. Instead of t, we have found that (μ + t)η, where μ and η are the octahedral factor and the atomic packing fraction, respectively, demonstrates a remarkably linear correlation with thermodynamic stability. As a stability descriptor, (μ + t)η is able to predict the relative stability among any two perovskites with an accuracy of ∼90%. This trend is then used to predict decomposition energies of another 69 perovskites, and the results are in excellent agreement with first-principles calculations, indicating the generalization of the trend. This thermodynamic stability trend may help the efficient high-throughput search for emerging stable perovskites and precise control of chemical compositions for stabilizing current perovskites.
P
erovskite solar cells have attracted substantial attention in the past few years and have been viewed as ideal candidates for next-generation photovoltaics. The power conversion efficiency increased from 3.8% in 2009 to 22.1% in 2016,1 which is comparable with commercialized thin-film solar cells, such as a-Si, CdTe, and Cu(In,Ga)Se2. The remaining main challenge for industrialization is the long-term stability of perovskite materials.2 Despite the proposal by Goldschmidt a century ago,3 tolerance factor t, defined as t = (rA + rX)/√2(rB + rX), where rA, rB, and rX are ionic radius of ions A, B, and X, respectively, has recently played an important role for engineering stable halide perovskite solar cells. As shown in Figure S1, in an ideal cubic perovskite, t is equal to 1, which is based on the assumption that the bond lengths of A−X and B−X are the sum of ionic radii of rA, rX, and rB, rX, respectively. In general, perovskite can be formed in the range of 0.8 < t < 1.0. To further clarify the formability of ABX3 perovskites, Li et al.4,5 introduced a (t, μ) map, where μ is octahedral factor, that is, the ratio of the radii of the B-site cation (rB) and the X-site anion (rX). Based on the existing perovskites, they have concluded that the stable region for halide perovskite is 0.813 < t < 1.107 and 0.377 < μ < 0.895. Those empirical rules have recently guided the stabilization of organic−inorganic © 2017 American Chemical Society
Received: September 2, 2017 Published: October 6, 2017 14905
DOI: 10.1021/jacs.7b09379 J. Am. Chem. Soc. 2017, 139, 14905−14908
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Journal of the American Chemical Society
energy of organic−inorganic hybrid perovskite depends on the orientation of molecule. The joint effect could lead to uncertainties of results when organic−inorganic hybrid perovskites are included. The decomposition energies, which are directly correlated with thermodynamic stability, of perovskites are calculated following the standard procedure. Although the main results presented in this paper are based on decomposition pathway to binary compounds, the results involving ternary products (Table S3) are also investigated in Supporting Information. The (t, μ) map is constructed, and all 138 perovskite compounds are represented by dots in Figure 2, with red dots indicating stable
Figure 1. Schematic crystal structures of (a) cubic perovskite (Pm3m ̅ symmetry) and (b) double perovskite (Fm3̅m symmetry). (c,d) Typical halide and chalcogenide perovskites that are considered in this study. The 138 perovskite compounds could be considered as chemical mutations from CsPbI3 (c) and CaZrS3 (d) with left-hand side cubic perovskites and right-hand side double perovskites. The Roman numerals in (c) and (d) refer to the charge states.
Figure 2. Map of (t, μ) for 138 perovskite compounds considered in our work. Red dots mean ΔHD > 0 (predicted to be stable), and black dots mean ΔHD < 0 (predicted to be unstable). The available experimental results (see Table S2 for more details) are marked, with red (black) text fonts indicating cubic phase can (not) be found.
this trend is proven to be independent of the choice of different sets of ionic radii. Perovskites usually have octahedral distortion from cubic phases to several kinds of symmetry-reduced phases. For example, the CsPbBr3 crystal transforms from cubic perovskite structure (Pm3̅m) to tetragonal (P4/mbm) at 130 °C and further to orthorhombic (Pmbn) at 88 °C.22 Different materials have different phase transitions at different temperatures, which makes complete comparisons among 138 compounds impossible. For fair comparison, we choose cubic Pm3̅m phase (Figure 1a) because it is the prototype phase of perovskite, where tolerance factor t and octahedral factor μ are originally defined. For double perovskites, chemical mutation based on cubic phase leads to the Fm3̅m phase23 (Figure 1b), which has been observed in recent experiments.24−26 Even for the cubic phase, it has recently been shown to have dynamic tilting instabilities.27 These dynamic contributions to the total energy are ignored in static DFT calculations and thus not considered in the present calculation. As shown in Figure 1, typical systems with potential for photovoltaic applications are chosen in our studies, including inorganic counterparts of CH3NH3PbI3 such as Rb- and Csrelated perovskites and their derived double perovskites, which hold the potential to overcome the toxicity and instability issues of hybrid halide perovskites. Chalcogenide perovskites are also considered as they are recently proposed to be an alternative avenue to achieve Pb-free stable perovskite photovoltaics.28 Inclusion of both halide and chalcogenide perovskites with diverse chemical compositions would test the generalization of our proposed stability trend. Only inorganic perovskites are considered in our work due to two facts: (i) the ionic radii of organic molecules are not well-defined; (ii) the calculated total
(ΔHD > 0) and black ones unstable (ΔHD < 0). It is clear that the stable and unstable perovskites are well separated by blue dashed lines, which divided the (t, μ) map into two regions. Eighty-four percent of compounds (48 out of 57) in the stable region are predicted to be stable, and 90% of compounds (73 out of 81) in unstable region are unstable. The materials with the discrepancies are mainly close to the borderlines. The theoretical results are in great agreement with available experiments. We have extensively searched the available experimental results for all 138 compounds and classified them into three types: (i) compounds which have been reported and cubic perovskites are found; (ii) compounds which have been reported and cubic perovskites are not found; (iii) compounds without experimental report by far. All 22 compounds of types (i) and (ii) are marked in Figure 2 (also see Table S2). The theoretical results successfully predict the (in)stability of those compounds in cubic phases, and the only exceptions are CsPbI3 and BaZrS3, both of which are close to the borderline and may be within the error range. Despite separating stable and unstable regions, the (t, μ) map cannot give a quantitative description of stability, which is more meaningful for stabilizing current perovskites-in-use and searching more stable perovskites. To quantitatively understand the stability, the (ΔHD, t) and (ΔHD, u) relations are plotted in Figure S2. Unfortunately, there is no obvious correlation between ΔHD and t/μ, which is in agreement with Zhao’s calculations29 (Figure S2 in ref 29). Those results mean that, although t or μ can provide qualitative range for the formability of 14906
DOI: 10.1021/jacs.7b09379 J. Am. Chem. Soc. 2017, 139, 14905−14908
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Journal of the American Chemical Society
further optimize the descriptor, we introduce APF (η), which is defined as the fraction of volume in a crystal structure that is occupied by constituent atoms treated by a rigid sphere (see Figure S1). In a rigid sphere model, higher APF usually leads to more stable structures. Therefore, we modified μ+t by adding η as the power index and plotted ΔHD as the function of (μ+t)η, as shown in Figure 3b. Remarkably, two linear trends are improved between ΔHD and (μ+t)η, especially for halides. This trend does not have significant change when ternary secondary products are considered in pathways (Figure S3). It predicts the decreasing stability from Cs2AgBiCl6 (stable), Cs2AgBiBr6 (stable), to Cs2AgBiI6 (unstable), which is consistent with recent experimental observations.24 As a descriptor, (μ+t)η predicts the relative stability for halides (chalcogenides) with an accuracy of 86% (90%), which is much better than that of μ+t. As an example, μ+t predicts that KPbCl3 is more stable than CsPbI3, which is contradictory to the results by first-principles calculations. The (μ+t)η corrects this error by considering that CsPbI3 has higher APF. The results show that APF is an important factor that is not considered in previous studies for the formability of perovskites. It is noted that all of the structural parameters μ, t, and η are the functions of the ionic radii, which are not well-defined quantities and depend on the ionicity/covalency of chemical bonds and the coordination number.30 The choice of ionic radius is critical to predict the formability of perovskite in the (t, μ) map approach. Recently, Travis et al.13 found that tolerance factor t based on widely used Shannon radii31 fails to accurately predict the stability of 32 known inorganic iodide perovskites, partially because Shannon radius derived from ionic compounds may not work well for more covalent crystals. Meanwhile, coordination number is another factor that affects the ionic radii.30 A good stability trend should not be significantly affected by the choice of different sets of ionic radii. We considered Shannon crystal radii31 and Pauling’s covalent radii32 and also coordinationdependent radii. The results (Figures S4 and S5) show that the stability trend of perovskites will not change by the choice of atomic radii, which has proven the robustness of our found trend. This stability trend is identified based on the sample group including 138 inorganic perovskite compounds of current photovoltaic interest. It would be interesting to see whether the trend can be generalized to other independent perovskite systems. To check the generalization ability of the trend, we chose a testing group that included 42 potassium perovskite compounds (with potassium atom at A site) and 27 oxide perovskite compounds. We can see good agreement between predicted and DFT-calculated values, and their mean square errors are 20 and 59 meV/atom for potassium and oxide perovskites, respectively (Figure S6). The descriptor (μ+t)η successfully predicts the unstable nature of most potassium perovskites and high stability of most oxide perovskites (Table S4), indicating its strong generalization for other perovskite systems. In conclusion, through first-principles calculations on decomposition energies of 138 perovskite compounds, we have identified a thermodynamic stability trend, a linear correlation between decomposition energies and descriptor (μ+t)η, where t, μ, and η are the tolerance factor, the octahedral factor, and the atomic packing fraction, respectively. As a descriptor, (μ+t)η can predict the relative stability between any two perovskites within the accuracy rate at ∼90%, which is much better than μ, t, or η alone. Utilizing this trend, we have successfully predicted the thermodynamic stability of another 69 perovskite compounds. Our work opens a way for quantitatively describing perovskite
perovskite, it is not a good stability descriptor, that is, quantitative correlation with stability. Enlightened by the borderline in Figure 2, which is approximate as μ+t = C, where C is a constant, we reorganized the decomposition energies of 138 perovskites as the function of μ+t in Figure 3a. Two distinct trends between ΔHD and μ + t are
Figure 3. Decomposition energies of 138 perovskite compounds dependent on (a) μ+t and (b) (μ+t)η. Halides and chalcogenides are marked as blue and orange.
observed for halides and chalcogenides, respectively. The steep slope for chalcogenides could be explained by the strong Madelung energy for 2− chalcogens instead of 1− halogens. From the ionic view, the compounds are a stack of cations and anions through Coulomb interactions (Madelung energy in real crystal). Therefore, the change of decomposition energy directly correlated with the change of Coulomb energy. In chalcogenide perovskite, the valence charges on both cations and anions are higher than those on halides. Therefore, their Coulomb energies are more sensitive to structural change. The linear correlation between ΔHD and μ+t means that μ+t could act as a descriptor for thermodynamic stability. Its prediction accuracy rate for the thermodynamic stability between any two halide (chalcogenide) perovskites is 74% (85%), whereas that for μ and t are 56% (65%) and 71% (81%), respectively (Figure 4). It is interesting to see, as a stability descriptor, μ+t performs better than μ or t alone. To
Figure 4. Accuracy rate for μ, t, η, (μ+t), and (μ+t)η as a descriptor to predict the relative stabilities among two perovskites. See Supporting Information for how we calculate the prediction accuracy rate. 14907
DOI: 10.1021/jacs.7b09379 J. Am. Chem. Soc. 2017, 139, 14905−14908
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Journal of the American Chemical Society stability instead of qualitative tolerance factor t or (t, μ) map that has been used for decades in the field. The stability descriptor of (μ+t)η may provide direct guidance for precise control of chemical compositions toward stable perovskites and enable efficient high-throughput searching of emerging stable perovskites, in particular, to photovoltaic applications.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.7b09379. Computational methodologies, ΔHD dependence on tolerance factor t, octahedral factor μ, and atomic packing ratio η, linear correlation between ΔHD and (μ + t)η under Shannon’s crystal radii, Pauling’s covalent radii, and various coordination environments, ionic radii we used, available experimental results in comparison with descriptor-predicted and DFT-calculated results (PDF)
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AUTHOR INFORMATION
Corresponding Author
*
[email protected] ORCID
Wan-Jian Yin: 0000-0003-0932-2789 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS W.J.Y. thanks Prof. Xingao Gong and Prof. Su-Huai Wei for their insightful discussion. The authors acknowledge the funding support from National Key R&D Program of China Grant No. 2016YFB0700700, National Natural Science Foundation of China (Grant Nos. 51602211 and 11674237), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20160299), and Suzhou Key Laboratory for Advanced Carbon Materials and Wearable Energy Technologies, China. The work was carried out at National Supercomputer Center in Tianjin, Lvliang, and Guangzhou, China, and the calculations were performed on TianHe-1(A) and TianHe-II.
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REFERENCES
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DOI: 10.1021/jacs.7b09379 J. Am. Chem. Soc. 2017, 139, 14905−14908
