Descriptors for the interpretability of cubic perovskite bulk modulus oriented towards soft lattice screening

JIANG Jinming; SUN Qingde; ZHANG Weibing

doi:10.7498/aps.74.20250652

Abstract

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 at https://doi.org/10.57760/sciencedb. j00213.00161.

Keywords:

Authors and contacts

Hunan Provincial Key Laboratory of Flexible Electronic Materials Genome Engineering, School of Physics & Electronic Science, Changsha University of Science and Technology, Changsha 410114, China

Corresponding author: SUN Qingde, qingdesun@csust.edu.cn ; ZHANG Weibing, zhangwb@csust.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12474219) and the Natural Science Foundation of Hunan Province, China (Grant No. 2023JJ40041).

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References

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Cited By

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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).

DownLoad: Full-Size Img PowerPoint

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

Figure 3. (a) Feature importance ranking based on the support vector regression (SVR) model; (b) Pearson correlation coefficients between features.

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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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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”

DownLoad: CSV

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)

DownLoad: CSV

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

DownLoad: CSV

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

DownLoad: CSV

[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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