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The search for ferromagnetic materials with high Curie temperature (Tc) is a hot issue in condensed matter physics. In this work, an effective machine learning model of Curie temperature based on material component information is established to predict a variety of ferromagnetic materials with high Curie temperature. Based on the collected data of 1568 ferromagnetic materials, and taking the component information of ferromagnetic materials as descriptors, in this work four efficient machine learning models are constructed, namely support vector regression, kernel ridge regression, random forest and extremely randomized trees, through hyperparameter optimization and ten-break cross-validation. Of them, extremely randomized tree model has the best prediction performance, and its cross-validation R2 score can reach 81.48%. At the same time, the extremely randomized tree model is also used to predict 36949 materials in the materials project database, and 338 ferromagnetic materials with Tc greater than 600 K are found in this work. The method proposed in this paper can help obtain ferromagnetic materials with high Curie temperature and accelerate the process of ferromagnetic material design.
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Keywords:
- machine learning /
- ferromagnetic materials /
- material component /
- Curie temperature
[1] Sanvito S, Oses C, Xue J, Tiwari A, Zic M, Archer T, Tozman P, Venkatesan M, Coey M, Curtarolo S 2017 Sci. Adv. 3 e1602241
Google Scholar
[2] Jiang Z, Wang P, Jiang X, Zhao J 2018 Nanoscale Horiz. 3 335
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Google Scholar
[15] Coey J M D 2011 IEEE Trans. Magn. 47 4671
Google Scholar
[16] Buschow K J 2003 Handbook of Magnetic Materials (Amsterdam: Elsevier) pp293–456
[17] Connolly T F 2012 Bibliography of Magnetic Materials and Tabulation of Magnetic Transition Temperatures (New York: Springer Science & Business Media) pp1–30
[18] Long T, Fortunato N M, Zhang Y, Gutfleisch O 2021 Mater. Res. Lett. 9 169
Google Scholar
[19] Zhai X, Chen M, Lu W 2018 Comput. Mater. Sci. 151 41
Google Scholar
[20] 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
[21] Mirjalili 2019 Enetic Algorithm (Berlin: Springer) pp43–55
[22] Syarif I, Prugel A, Wills G 2016 Telecommun. Comput. Electron. Control 14 1502
Google Scholar
[23] 杨自欣, 高章然, 孙晓帆, 蔡宏灵, 张凤鸣, 吴小山 2019 68 210502
Google Scholar
Yang Z X, Gao Z R, Sun X F, Cai H L, Zhang F M, Wu X S 2019 Acta Phys. Sin. 68 210502
Google Scholar
[24] Benesty J, Chen J, Huang Y, Cohen I 2009 Pearson Correlation Coefficient (Berlin: Springer) pp1–4
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Google Scholar
[26] Popoel E, Tuszynski M, Zarek W, Rendecki T 1989 J. Less-Common Met. 146 127
Google Scholar
[27] Li J G, Pan M X, Sun J R, Zhang F X, Li S P, Zhao D Q, Yu X H, Zhao S X, Chen X C 1996 Solid State Commun. 97 pp1047
Google Scholar
[28] Dang M Z, Rancourt D G 1996 Phys. Rev. B 53 2291
Google Scholar
[29] Miyata N, Kamimori T, Goto M 1986 J. Phys. Soc. Jpn. 55 2037
Google Scholar
[30] Trumpy G, Both E, Djéga-Mariadassou C, Lecocq P 1970 Phys. Rev. B 2 3477
Google Scholar
[31] Meinert M 2016 J. Phys. Condens. Matter 28 056006
Google Scholar
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图 1 1568个铁磁材料数据集Tc的分布情况
Figure 1. Distribution of Tc in 1568 ferromagnetic material data sets.
图 2 数据集中铁磁材料的元素分布情况 (a) Tc大于300 K时元素分布; (b) Tc大于600 K时元素分布
Figure 2. Element distribution of ferromagnetic materials in data set: (a) Element distribution when Tc is greater than 300 K; (b) element distribution when Tc is greater than 600 K.
图 3 均匀网格搜索 (a) 随机森林参数优化图; (b) 极端随机树参数优化图
Figure 3. Uniform grid search: (a) Random forest parameter optimization map; (b) extreme random tree parameter optimization map
图 4 四种机器学习模型实验值和预测值对比的二维散点图 (a) 核岭回归; (b) 支持向量机; (c) 随机森林; (d) 极端随机树
Figure 4. Two-dimensional scatter plots comparing experimental and predicted values of four machine learning models: (a) Kernel ridge regression; (b) support vector machine; (c) random forests; (d) extremely random tree.
图 5 基于极端随机数模型的特征重要性排序图
Figure 5. Feature importance ranking graph based on extreme random number model.
图 6 预测集中铁磁材料元素分布情况 (a) 2531个Tc大于300 K数据的元素分布图; (b) 338个Tc大于600 K数据的元素分布图
Figure 6. Element distribution of ferromagnetic materials in prediction set: (a) Element distribution of 2531 data with Tc greater than 300 K; (b) element distribution of 338 data with Tc greater than 600 K.
图 7 预测集中338个Tc > 600 K的铁磁材料的Tc分布情况
Figure 7. Curie temperature distribution of 338 ferromagnetic materials with Tc >600 K in prediction set.
表 1 本研究中四种机器学习模型的超参数
Table 1. Hyperparameters of four machine learning models in this study.
模型 超参数 KRR alpha = 0.00567165, kernel = “rbf” SVR kernel = “rbf”, C = 181.8797945, gamma = 0.18646131 RF n_estimators = 170, max_features = 0.30, min_samples_leaf = 0.001 EXT n_estimators = 180, max_features = 0.58, min_samples_leaf = 0.001 
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表 2 基于特征筛选获得的化学参数描述符
Table 2. Chemical parameter descriptors obtained based on feature screening.
No. Meanings Features 1 Mean atomic number MAN 2 Mean IonicRadius MIR 3 Mean Modulus of Elasticity MME 4 Mean melting point MMP 5 Mean GSmagmom MGSM 6 Mean covalentradius MCR 7 Mean IonizationEnergy MIE 8 Mean ElectronAffinity MEA 9 Mean AtomicVolume MAV 10 Mean MendeleevNumber MMN 11 Composition of Mn CMn 12 Composition of Fe CFe 13 Composition of Co CCo 14 Composition of Ni CNi 15 range AtomicRadius RAR 16 range Electronegativity REE 17 avg p valence electrons ApV 18 avg d valence electrons AdV 19 frac f valence electrons FfV 20 transition metal fraction TMF 21 2-norm 2N
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表 3 本研究中四种机器学习模型的最终评估结果
Table 3. Final evaluation results of four machine learning models in this study.
KRR SVR RF EXT MAE 89.43 83.21 76.52 70.86 RMSE 125.50 125.77 114.71 109.32 R2/% 80.75 80.67 83.92 85.39 CS MAE 95.41 88.64 81.18 74.04 CS RMSE 141.21 137.62 124.13 117.98 CS R2/% 73.70 74.92 79.45 81.48
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[1] Sanvito S, Oses C, Xue J, Tiwari A, Zic M, Archer T, Tozman P, Venkatesan M, Coey M, Curtarolo S 2017 Sci. Adv. 3 e1602241
Google Scholar
[2] Jiang Z, Wang P, Jiang X, Zhao J 2018 Nanoscale Horiz. 3 335
Google Scholar
[3] Lu X, Fei R, Yang L 2019 Phys. Rev. B 100 205409
Google Scholar
[4] Claussen N, Bernevig B A 2020 Phys. Rev. B 101 245117
Google Scholar
[5] Jiang Z, Wang P, Xing J, Jiang X, Zhao J 2018 ACS Appl. Mater. Interfaces 10 39032
Google Scholar
[6] Kabiraj A, Kumar M, Mahapatra S 2020 npj Comput. Mater. 6 35
Google Scholar
[7] Lu S H, Zhou Q H, Guo Y, Wang J L 2022 Chem 8 769
Google Scholar
[8] Lu S H, Zhou Q H, Guo Y L, Zhang Y H, Wu Y L, Wang J L 2020 Adv. Mater. 32 2002658.
Google Scholar
[9] Nelson J, Sanvito S 2019 Phys. Rev. Mater. 3 104405
Google Scholar
[10] Xue Y F, Shen Z, Wu Z B, Song C S 2022 J. Appl. Phys. 132 053901
Google Scholar
[11] Zhang B, Zheng X Q, Zhao T Y, Hu F X, Sun J R, Shen B G 2018 Chin. Phys. B 27 067503
Google Scholar
[12] Vishina A, Vekilova O Y, BJörkman T, Bergman A, Herper H C, Eriksson O 2020 Phys. Rev. B 101 094407
Google Scholar
[13] Kwon H Y, Kim N J, Lee C K, Won C 2019 Phys. Rev. B 99 024423
Google Scholar
[14] Choudhary K, Garrity K F, Ghimire N J, Anand N, Tavazza F 2021 Phys. Rev. B 103 155131
Google Scholar
[15] Coey J M D 2011 IEEE Trans. Magn. 47 4671
Google Scholar
[16] Buschow K J 2003 Handbook of Magnetic Materials (Amsterdam: Elsevier) pp293–456
[17] Connolly T F 2012 Bibliography of Magnetic Materials and Tabulation of Magnetic Transition Temperatures (New York: Springer Science & Business Media) pp1–30
[18] Long T, Fortunato N M, Zhang Y, Gutfleisch O 2021 Mater. Res. Lett. 9 169
Google Scholar
[19] Zhai X, Chen M, Lu W 2018 Comput. Mater. Sci. 151 41
Google Scholar
[20] 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
[21] Mirjalili 2019 Enetic Algorithm (Berlin: Springer) pp43–55
[22] Syarif I, Prugel A, Wills G 2016 Telecommun. Comput. Electron. Control 14 1502
Google Scholar
[23] 杨自欣, 高章然, 孙晓帆, 蔡宏灵, 张凤鸣, 吴小山 2019 68 210502
Google Scholar
Yang Z X, Gao Z R, Sun X F, Cai H L, Zhang F M, Wu X S 2019 Acta Phys. Sin. 68 210502
Google Scholar
[24] Benesty J, Chen J, Huang Y, Cohen I 2009 Pearson Correlation Coefficient (Berlin: Springer) pp1–4
[25] Jain A, Ong S P, Hautie G, Chen W, Richards W D, Dacek S, Cholia S, Gunter D, Skinner D, Ceder G 2013 APL Mater. 1 011002
Google Scholar
[26] Popoel E, Tuszynski M, Zarek W, Rendecki T 1989 J. Less-Common Met. 146 127
Google Scholar
[27] Li J G, Pan M X, Sun J R, Zhang F X, Li S P, Zhao D Q, Yu X H, Zhao S X, Chen X C 1996 Solid State Commun. 97 pp1047
Google Scholar
[28] Dang M Z, Rancourt D G 1996 Phys. Rev. B 53 2291
Google Scholar
[29] Miyata N, Kamimori T, Goto M 1986 J. Phys. Soc. Jpn. 55 2037
Google Scholar
[30] Trumpy G, Both E, Djéga-Mariadassou C, Lecocq P 1970 Phys. Rev. B 2 3477
Google Scholar
[31] Meinert M 2016 J. Phys. Condens. Matter 28 056006
Google Scholar
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