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近年来, 软晶格被认为是钙钛矿材料实现缺陷容忍性的主要物理来源, 体模量则作为晶格“软硬度”的关键衡量指标. 本文针对立方钙钛矿体系, 基于SISSO (sure independence screening and sparsifying operator)与VS-SISSO (集成迭代变量选择机制的SISSO框架)方法, 构建了两类低维、物理可解释性强的体模量预测模型. 首先, 基于共价半径、熔点和体积等结构和热力学特征构建的热-结构耦合描述符模型, 在测试集上实现了均方根误差RMSE = 7.41 GPa, 决定系数R2 = 97.8% 的良好预测性能; 进一步引入电负性、原子价态与未配对电子数等电子层级特征后, 构建了电子-热-结构三重耦合描述符模型, 预测精度显著提升, 在测试集上RMSE降至5.34 GPa, R2提升至98.35%. 基于该模型, 本文对超过10000个卤族和硫族立方钙钛矿进行高通量预测, 筛选出约170种体模量位于10—20 GPa区间、与Pb-I钙钛矿相近的候选体系. 研究结果为软晶格机制在无铅体系中的适用性提供了初步支持, 并为高通量筛选具缺陷容忍潜力的稳定无铅钙钛矿材料提供了理论依据与数据支撑. 本文数据集可在
https://doi.org/10.57760/sciencedb.j00213.00161 中访问获取.In recent years, soft lattices have been considered a primary physical origin of defect tolerance in lead-halide perovskite materials, with bulk modulus serving as a key indicator of lattice “softness”. This work focuses on cubic perovskites and constructing a dataset of bulk moduli for 213 compounds based on density functional theory (DFT) calculations. A total of 138 features are compiled, including 132 statistical features extracted using the Matminer toolkit and 6 manually selected elemental descriptors. Four conventional machine learning regression models (RF, SVR, KRR, and EXR) are employed for prediction. Of them, the SVR model shows the best performance, achieving a test-set Root Mean Square Error (RMSE) of 7.35 GPa and Coefficient of Determination (R2) of 97.86%. Feature importance analysis reveals that thermodynamic-structural features such as melting point, covalent radius, and atomic volume play dominant roles in determining bulk modulus. Based on the 12 most important features, a thermodynamic-structural coupling descriptor is constructed using the SISSO method, yielding a test-set RMSE of 7.41 GPa and R2 of 97.80%. The resulting descriptor indicates that the bulk modulus is proportional to melting point and inversely proportional to atomic volume. Furthermore, the VS-SISSO method combined with a random subset selection and iterative variable screening strategy is used, enabling the selection of electronic-level features such as electronegativity, valence state, and number of unpaired electrons. The resulting electronic-thermodynamic-structural coupling descriptor further improves the prediction accuracy, reaching an RMSE of 5.34 GPa and R2 of 98.35% on the test set. Notably, due to the difference in valence states, this model effectively distinguishes between the bulk moduli of chalcogen-based (divalent) and halogen-based (monovalent) perovskites. Based on this model, high-throughput screening is performed on over 10000 cubic chalcogenides and halide perovskites, and approximately 170 lead-free candidates with bulk moduli in the range of 10–20 GPa are identified, which are comparable to Pb-I perovskites. These results provide preliminary evidence for supporting the applicability of the soft-lattice mechanism in lead-free systems and offer theoretical guidance and data support for the high-throughput discovery of stable, defect-tolerant, lead-free perovskite materials. All the data presented in this paper are openly available athttps://doi.org/10.57760/sciencedb. j00213.00161 .
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Keywords:
- bulk modulus /
- defect tolerance /
- soft lattice /
- perovskite /
- interpretable machine learning
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图 1 (a) 卤族立方钙钛矿包括典型的单钙钛矿ABX3 (B = M2+)和双钙钛矿A2B1B2X6 (B1 = M+, B2 = M3+); (b) 硫族立方钙钛矿包括ABY3 (B = M4+)型单钙钛矿和A2B1B2Y6 (B1 = M3+, B2 = M5+)型双钙钛矿; 图中展示了数据集中213个钙钛矿的元素组成分布, 其中双色半圆表示该元素既可作为A位也可作为B位
Fig. 1. (a) Halide cubic perovskites include typical ABX3-type single perovskites (B = M2+) and A2B1B2X6-type double perovskites (B1 = M+, B2 = M3+); (b) chalcogenide cubic perovskites include ABY3-type single perovskites (B = M4+) and A2B1B2Y6-type double perovskites (B1 = M3+, B2 = M5+); the figure summarizes the elemental composition of 213 cubic perovskites in the dataset. Dual-colored semicircles indicate elements that can occupy both A and B sites.
图 2 各机器学习模型预测值与真实值的对比图 (a) 随机森林; (b) 支持向量回归; (c) 核岭回归; (d) 极端随机树; 红色点和蓝色三角形分别表示训练集和测试集数据, 虚线表示理想预测线y = x
Fig. 2. Comparison between predicted and calculated bulk modulus for different machine learning models: (a) Random forest (RF); (b) support vector regression (SVR); (c) kernel ridge regression (KRR); (d) extremely random trees (EXT); red dots and blue triangles correspond to the training and testing datasets, respectively, while the dashed line represents the ideal prediction (y = x).
图 3 (a) 基于支持向量回归模型的特征重要性排序; (b) 特征之间的皮尔逊相关系数热图
Fig. 3. (a) Feature importance ranking based on the support vector regression (SVR) model; (b) Pearson correlation coefficients between features.
图 4 (a) VS-SISSO (三轮独立训练)与$ d_{{\text{B-X}}}^{ - {4}} $, SISSO所构建体模量描述符的预测RMSE对比; (b)—(d) 3种不同模型的预测结果对比图, 其中(b)为使用$ d_{{\text{B-X}}}^{ - {4}} $描述符的拟合结果, (c)为使用SISSO构建的热-结构耦合描述符的拟合结果, (d)为使用VS-SISSO构建的电子-热-结构三重耦合描述符的拟合结果; 横轴表示DFT计算所得体模量, 纵轴为模型预测值; 蓝色点表示卤族钙钛矿, 红色点表示硫族钙钛矿
Fig. 4. (a) Comparison of prediction RMSE among the VS-SISSO (three independent trainings), the $ d_{{\text{B-X}}}^{ - {4}} $ descriptor, and the SISSO-derived descriptor; (b)–(d)comparison of predicted vs. DFT-calculated bulk modulus using three different models, (b) the $ d_{{\text{B-X}}}^{ - {4}} $ descriptor, (c) the SISSO-derived thermo-structural descriptor, (d) the VS-SISSO-derived electronic-thermo-structural descriptor; the x-axis represents the bulk modulus obtained from DFT, and the y-axis indicates model predictions; blue dots denote halide perovskites, while red dots represent chalcogenide perovskites.
表 1 各类机器学习模型参数设置
Table 1. Machine learning model parameters.
模型 参数设置 RF n_estimators=200, max_features=“auto”, bootstrap=True SVR kernel=“rbf”, C=497, gamma=0.01 KRR kernel=“poly”, alpha=0.01 EXT n_estimators=110, max_features=“auto” 
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表 2 SISSO与VS-SISSO参数设置
Table 2. Parameters for SISSO and VS-SISSO.
参数名称 设置值 说明 desc_dim 1 描述符维度 ptype 1 回归任务 opset 11 运算符: (+)(–)(*)(/)(exp)(exp–)
(^(–1))(^2)(sqrt)(log)(|–|)ntask 1 任务数(无任务划分) nsample 213 样本数 nsf 12 特征总数上限 fcomplexity 6 最大选中特征数 subs_sis 10000 SIS阶段筛选特征子集 method “L0” 稀疏回归方法 metric “RMSE” 评价指标 nm_output 100 最终模型输出数量 Sa 4 随机子集维数 (VS-SISSO) max_iter 200 最大迭代次数 (VS-SISSO)
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表 3 各模型性能评估结果
Table 3. Evaluation results of different models.
RF SVR KRR EXT MAE/GPa 3.79 3.54 4.89 4.27 RMSE/GPa 8.42 7.35 10.88 11.00 R2/% 97.20 97.86 95.31 95.22 CV-MAE/GPa 3.09 3.08 3.96 2.70 CV-RMSE/GPa 9.04 8.37 11.36 13.47 CV-R2/% 95.36 96.86 95.40 95.39
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表 4 SISSO模型的输入特征及其含义
Table 4. Input features and their meanings in the SISSO model.
特征名称 含义 特征名称 含义 meanMT 平均熔点 meanGSV 平均基态高斯体积 modeMT 众数熔点 meanSGN 平均空间群数 modeAW 众数原子质量 avgAW 原子质量的平均偏差 minGSV 最小基态
晶胞体积minN 最小原子序数 maxCR 最大共价半径 ranN 原子序数极差 ranCR 共价半径极差 meanMN 平均门捷列夫序数
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[1] NREL. Best research-cell efficiency chart. https://www.nrel.gov/pv/cell-efficiency.html [2025-05-18]
[2] Yin W, Shi T, Yan Y 2014 Adv. Mater. 26 4653
Google Scholar
[3] Huang J, Yuan Y, Shao Y, Yan Y 2017 Nat Rev Mater 2 17042
Google Scholar
[4] Xian Y M, Wang X M, Yan Y F 2024 Chin. Phys. B 33 096803
Google Scholar
[5] Yin W J, Shi T, Yan Y 2014 Appl. Phys. Lett. 104 063903
Google Scholar
[6] Ming C, Wang H, West D, Zhang S, Sun Y Y 2022 J. Mater. Chem. A 10 3018
Google Scholar
[7] Miyata K, Meggiolaro D, Trinh M T, Joshi P P, Mosconi E, Jones S C, De Angelis F, Zhu X Y 2017 Sci. Adv. 3 e1701217
Google Scholar
[8] Zhu X Y, Podzorov V 2015 J. Phys. Chem. Lett. 6 4758
Google Scholar
[9] Bonn M, Miyata K, Hendry E, Zhu X Y 2017 ACS Energy Lett. 2 2555
Google Scholar
[10] Yang J F, Wen X M, Xia H Z, Sheng R, Ma Q S, Kim J, Tapping P, Harada T, Kee T W, Huang F Z, Cheng Y B, Green M, Ho-Baillie A, Huang S J, Shrestha S, Patterson R, Conibeer G 2017 Nat. Commun. 8 14120
Google Scholar
[11] Chu W B, Zheng Q J, Prezhdo O V, Zhao J, Saidi W A 2020 Sci. Adv. 6 eaaw7453
Google Scholar
[12] Chu W B, Saidi W A, Zhao J, Prezhdo O V 2020 Angew Chem Int Ed 59 6435
Google Scholar
[13] Wu X W, Ming C, Shi J, Wang H, West D, Zhang S B, Sun Y Y 2022 Chin. Phys. Lett. 39 046101
Google Scholar
[14] Zhao X G, Yang J H, Fu Y, Yang D, Xu Q, Yu L, Wei S H, Zhang L 2017 J. Am. Chem. Soc. 139 2630
Google Scholar
[15] Sun Q D, Wang J, Yin W J, Yan Y F 2018 Adv. Mater. 30 1705901
Google Scholar
[16] Sun Q, Yin W J, Wei S H 2020 J. Mater. Chem. C 8 12012
Google Scholar
[17] Ghorpade U V, Suryawanshi M P, Green M A, Wu T, Hao X, Ryan K M 2023 Chem. Rev. 123 327
Google Scholar
[18] 余毅, 安治东, 蔡晓艺, 郭明磊, 敬承斌, 李艳青 2021 物理学报 70 048503
Google Scholar
Yu Y, An Z D, Cai X Y, Guo M L, Jing C B, Li Y Q 2021 Acta Phys. Sin. 70 048503
Google Scholar
[19] Dunn A, Wang Q, Ganose A, Dopp D, Jain A 2020 npj Comput. Mater. 6 138
Google Scholar
[20] Geurts P, Ernst D, Wehenkel L 2006 Mach Learn 63 3
Google Scholar
[21] Chen T, Guestrin C 2016 Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining San Francisco California USA pp785–794
[22] Hancock J T, Khoshgoftaar T M 2020 J. Big Data 7 94
Google Scholar
[23] De Breuck P P, Hautier G, Rignanese G M 2021 npj Comput. Mater. 7 83
Google Scholar
[24] Wang A Y T, Kauwe S K, Murdock R J, Sparks T D 2021 npj Comput. Mater. 7 77
Google Scholar
[25] Xie T, Grossman J C 2018 Phys. Rev. Lett. 120 145301
Google Scholar
[26] Schütt K T, Sauceda H E, Kindermans P J, Tkatchenko A, Müller K R 2018 J. Chem. Phys. 148 241722
Google Scholar
[27] Chen C, Ye W, Zuo Y, Zheng C, Ong S P 2019 Chem. Mater. 31 3564
Google Scholar
[28] Choudhary K, DeCost B 2021 npj Comput. Mater. 7 185
Google Scholar
[29] Gasteiger J, Giri S, Margraf J T, Günnemann S 2020 arXiv: 2206.13578 [cs. LG]
[30] Guerrero P, Hašan M, Sunkavalli K, Měch R, Boubekeur T, Mitra N J 2022 ACM Trans. Graph. 41 1
[31] Ruff R, Reiser P, Stühmer J, Friederich P 2024 Digit. Discov. 3 594
Google Scholar
[32] Ong S P, Cholia S, Jain A, Brafman M, Gunter D, Ceder G, Persson K A 2015 Comput. Mater. Sci. 97 209
Google Scholar
[33] Akinpelu S B, Abolade S A, Okafor E, Obada D O, Ukpong A M, Kumar R. S, Healy J, Akande A 2024 Res. Phys. 65 107978
[34] Ouyang R, Curtarolo S, Ahmetcik E, Scheffler M, Ghiringhelli L M 2018 Phys. Rev. Mater. 2 083802
Google Scholar
[35] Guo Z, Hu S, Han Z K, Ouyang R 2022 J. Chem. Theory Comput. 18 4945
Google Scholar
[36] Roy P B, Roy S B 2005 J. Phys.: Condens. Matter 17 6193
Google Scholar
[37] Heyd J, Peralta J E, Scuseria G E, Martin R L 2005 J. Chem. Phys. 123 174101
Google Scholar
[38] Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169
Google Scholar
[39] Ward L, Dunn A, Faghaninia A, Zimmermann N E R, Bajaj S, Wang Q, Montoya J, Chen J, Bystrom K, Dylla M, Chard K, Asta M, Persson K A, Snyder G J, Foster I, Jain A 2018 Comput. Mater. Sci. 152 60
Google Scholar
[40] Fabian P, Gaël V, Alexandre G, Vincent M, Bertrand T, Olivier G, Mathieu B, Peter P, Ron W, Vincent D, Jake V, Alexandre P, David C, Matthieu B, Matthieu P, Duchesnay É 2011 J. Mach. Learn. Res. 12 2825
[41] Benesty J, Chen J, Huang Y, Cohen I 2009 Pearson Correlation Coefficient (Berlin: Springer) pp1–4
[42] Cohen M L 1993 Science 261 307
Google Scholar
[43] Guo Z, Wang J, Yin W J 2022 Energy Environ. Sci. 15 660
Google Scholar
[44] Verma A S, Kumar A 2012 J. Alloy. Comp. 541 210
Google Scholar
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