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Discovering compact, stable, and easily controllable nanoscale non-trivial topological magnetic structures, such as magnetic skyrmions, is the key to developing next-generation high-density, high-speed, and low-energy non-volatile information storage devices. Based on the topological generation mechanism, magnetic skyrmions can be generated through the Dzyaloshinskii–Moriya interaction (DMI) caused by breaking space-reversal symmetry. Two-dimensional (2D) non-centrosymmetric Janus structurecan generate vertical built-in electric fields to break spatial inversion symmetry. Therefore, seeking for 2D Janus material with intrinsic magnetism is fundamental to develop the novel chiral magnetic storage technologies. In this work, we combine detailed machine learning techniques and first-principle calculations to investigate the magnetism of the unexplored 2D Janus material. We first collect 1179 2D hexagonal ABC-type Janus materials based on the Materials Project database, and use elemental composition as feature descriptors to construct four machine learning models: random forest (RF), gradient boosting decision trees (GBDT), extreme gradient boosting (XGB), and extra trees (ET). These algorithms and models are constructed to predict lattice constants, formation energy, and magnetic moment, via hyperparameter optimization and ten-fold cross-validation. The GBDT exhibits the highest accuracy and best prediction performance for magnetic moment classification. Subsequently, the collected data of 82018 yet-undiscovered 2D Janus materials, are input into the trained models to generate 4024 high magnetic moment 2D Janus materials with thermal stability. First-principles calculations are employed to validate random sample of 13 Janus materials with high magnetic moment. This study provides an effective machine learning framework for classifying the magnetic moments and screening highthroughput 2D Janus structures, thereby accelerating the exploration of their magnetic properties.
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Keywords:
- machine learning /
- two-dimensional Janus materials /
- magnetic moment /
- first-principles calculations
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图 1 机器学习结合基于密度泛函理论(DFT)发掘高磁矩Janus材料步骤
Figure 1. Steps for discovering high magnetic moment Janus materials by combining machine learning with density fun-ctional theory (DFT).
图 2 六角晶系ABC型Janus材料原子结构的(a)侧视图和(b)俯视图
Figure 2. (a) Side view and (b) top view of atomic structures of hexagonal ABC-type Janus materials.
图 3 数据集中二维 Janus 材料的(a)晶格常数 a 和b, (b)晶格常数 c, (c)形成能和(d)总磁矩的分布
Figure 3. The distribution of (a) lattice constants a and b, (b) lattice constant c, (c) formation energy and (d) total magnetic moment of the dataset of 2D Janus materials.
图 4 晶格常数预测: 最优模型在十折交叉验证中的散点图 (a) Lattice a = b预测任务最优模型: 极端随机树; (b) Lattice c预测任务最优模型: 极端梯度提升
Figure 4. Prediction of lattice constants: scatter plots for the optimal models in ten-fold cross-validation: (a) The optimal model for the lattice a = b prediction task: ET; (b) the optimal model for the lattice c prediction task: XGB.
图 5 形成能预测: 四种模型在十折交叉验证上的散点图 (a)随机森林; (b)梯度提升决策树; (c)极端梯度提升; (d)极端随机树
Figure 5. Prediction of formation energy: Scatter plots for four models in ten-fold cross-validation: (a) RF; (b) GBDT; (c) XGB; (d) ET
图 6 磁矩分类预测: 四种模型在十折交叉验证上的混淆矩阵 (a)随机森林; (b)梯度提升决策树; (c)极端梯度提升; (d)极端随机树
Figure 6. Prediction of magnetic moment classification: Confusion matrices for four models in ten-fold cross-validation: (a) RF; (b) GBDT; (c) XGB; (d) ET.
图 7 13种二维六角晶系Janus原子结构的侧视图
Figure 7. Side view of atomic structures of 13 two-dimensional hexagonal Janus materials.
表 1 不同训练任务中机器学习最优模型的超参数
Table 1. The hyperparameters of the optimal machine learning models in various training tasks.
模型 超参数 GBDT(磁矩分类) learning_rate = 0.01603011, max_depth = 5, n_estimators = 272, subsample = 0.69895067 GBDT(形成能) learning_rate = 0.02, max_depth = 6, n_estimators = 353, subsample = 0.93030056 ET(晶格常数a和b) max_depth = 10, max_features = 0.60, n_estimators = 100,
min_samples_leaf = 2, min_samples_split = 4XGB(晶格常数c) learning_rate = 0.02, n_estimators = 300, max_depth = 5,
subsample = 0.8, colsample_bytree = 0.49613519
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表 2 晶格常数预测
Table 2. Prediction of lattice constants.
模型 Lattice a = b Lattice c MAE RMSE $R^2$ MAE RMSE $R^2$ RF 0.5485 0.8104 0.7375 0.6491 1.0001 0.6872 GBDT 0.4477 0.7350 0.7829 0.6679 0.9924 0.6923 XGB 0.5427 0.7968 0.7462 0.5953 0.9474 0.7186 ET 0.3469 0.6808 0.8137 0.6534 1.0103 0.6817
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表 3 形成能预测: 四种机器学习模型的评价指标
Table 3. The prediction of formation energy: Evaluation metrics of four machine learning models.
模型 MAE RMSE $R^2$ RF 0.1054 0.1697 0.8671 GBDT 0.0798 0.1411 0.9070 XGB 0.0959 0.1533 0.8930 ET 0.1120 0.1701 0.8657
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表 4 磁矩分类预测: 四种机器学习模型的评价指标
Table 4. Prediction of magnetic moment classification: Evaluation metrics of four machine learning models.
模型 Accuracy Precision Recall F1 score RF 0.8770 0.8459 0.7636 0.7862 GBDT 0.8948 0.8498 0.8182 0.8263 XGB 0.8762 0.8398 0.7697 0.7883 ET 0.8795 0.8392 0.7778 0.7965
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表 5 13种结构优化后的六角晶系ABC型Janus材料的晶格常数、形成能和磁矩
Table 5. Optimized lattice constants, formation energies, and magnetic moments of 13 two-dimensional hexagonal ABC-type Janus materials.
Formula Lattice constants Formation energy/eV $ |\mu| / \mu_{\mathrm{B}} $ a = b/Å c/Å A B C ErFeTb 3.35 18.25 –2.02 2.51 3.03 6.24 FeNO 2.92 15.00 –11.87 1.17 0.08 0.47 HoRuSr 4.90 18.79 –6.66 3.79 0.02 0.05 DyOsSr 4.18 18.87 –6.89 4.89 0 0.13 EuSbSr 5.43 18.69 –5.53 6.85 0.01 0.05 HoIrSr 4.58 18.79 –7.24 3.72 0 0.05 LiUZn 2.89 18.13 –0.44 0 1.65 0.01 PuSZn 4.52 18.13 –6.75 5.61 0.10 0.01 GdKU 7.46 18.13 –2.39 7.33 0 2.96 LuNbTi 3.02 18.13 –1.76 0.02 0.28 1.67 GdHfSe 5.03 18.93 –8.46 7.33 0.34 0.02 NaTbZn 4.65 18.69 –1.87 0.02 6.00 0 HoNpSr 3.69 18.46 –1.80 3.81 4.38 0.08
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
Google Scholar
[2] Zhang Z W, Lang Y F, Zhu H P, Li B, Zhao Y Q, Wei B, Zhou W X 2024 Phys. Rev. Appl. 21 064012
Google Scholar
[3] Liu B, Feng X X, Long M Q, Cai M Q, Yang J L 2022 Phys. Rev. Appl. 18 054036
Google Scholar
[4] 熊祥杰, 钟防, 张资文, 陈芳, 罗婧澜, 赵宇清, 朱慧平, 蒋绍龙 2024 物理学报 73 137101
Google Scholar
Xiong X J, Zhong F, Zhang Z W, Chen F, Luo J L, Zhao Y Q, Zhu H P, Jiang S L 2024 Acta Phys. Sin. 73 137101
Google Scholar
[5] Zhao Y Q, Liu Z S, Nie G Z, Zhu Z H, Chai Y F, Wang J N, Cai M Q, Jiang S L 2021 Appl. Phys. Lett. 118 173104
Google Scholar
[6] Lang Y F, Zou D F, Xu Y, Jiang S L, Zhao Y Q, Ang Y S 2024 Appl. Phys. Lett. 124 052903
Google Scholar
[7] Liao C S, Ding Y F, Zhao Y Q, Cai M Q 2021 Appl. Phys. Lett. 119 182903
Google Scholar
[8] Tan W, Zhang Z W, Zhou X Y, Yu Z L, Zhao Y Q, Jiang S L, Ang Y S 2024 Phys. Rev. Mater. 8 094414
Google Scholar
[9] Liang J H, Wang W W, Du H F, Hallal A, Garcia K, Chshiev M, Fert A, Yang H X 2020 Phys. Rev. B 101 184401
Google Scholar
[10] Zhang S Q, Xu R Z, Luo N N, Zou X L 2021 Nanoscale 13 1398
Google Scholar
[11] Dai C Y, He P, Luo L X, Zhan P X, Guan B, Zheng J 2023 Sci. China Mater. 66 859
Google Scholar
[12] Wang P, Zong Y X, Wen H Y, Xia J B, Wei Z M 2021 Acta Phys. Sin. 70 026801 [王盼, 宗易昕, 文宏玉, 夏建白, 魏钟鸣 2021 物理学报 70 026801]
Google Scholar
Wang P, Zong Y X, Wen H Y, Xia J B, Wei Z M 2021 Acta Phys. Sin. 70 026801
Google Scholar
[13] Ren K, Wang K, Zhang G 2022 ACS Appl. Electron. Mater. 4 4507
Google Scholar
[14] Peng Z L, Huang J X, Guo Z G 2021 Nanoscale 13 18839
Google Scholar
[15] Zhang L, Yang Z J F, Gong T, Pan R K, Wang H D, Guo Z N, Zhang H, Fu X 2020 J. Mater. Chem. A 8 8813
Google Scholar
[16] Vafaeezadeh M, Thiel W R 2022 Angew. Chem. Int. Edit. 61 e202206403
Google Scholar
[17] Mukherjee T, Kar S, Ray S 2022 J. Mater. Res. 37 3418
Google Scholar
[18] Li C Q, An Y K 2022 Phys. Rev. B 106 115417
Google Scholar
[19] Zhang L, Zhao Y, Liu Y Q, Gao G Y 2023 Nanoscale 15 18910
Google Scholar
[20] Xu L J, Wan W H, Peng Y R, Ge Y F, Liu Y 2024 Ann. Phys. 536 2300388
Google Scholar
[21] Gao Z Y, Mao G Y, Chen S Y, Bai Y, Gao P, Wu C C, Gates I D, Yang W J, Ding X L, Yao J X 2022 Phys. Chem. Chem. Phys. 24 3460
Google Scholar
[22] Liu H, Sun J T, Liu M, Meng S 2018 J. Phys. Chem. Lett. 9 6709
Google Scholar
[23] Nelson J, Sanvito S 2019 Phys. Rev. Mater. 3 104405
Google Scholar
[24] Belot J F, Taufour V, Sanvito S, Hart G L 2023 Appl. Phys. Lett. 123 042405
Google Scholar
[25] Miyazato I, Tanaka Y, Takahashi K 2018 J. Phys.: Condens. Matter 30 06L
[26] 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
[27] Ma X Y, Lyu H Y, Hao K R, Zhao Y M, Qian X F, Yan Q B, Su G 2021 Sci. Bull. 66 233
Google Scholar
[28] Huang T, Yang Z X, Li L, Wan H, Leng C, Huang G F, Hu W Y, Huang W Q 2024 J. Phys. chem. Lett. 15 2428
Google Scholar
[29] Chaney G, Ibrahim A, Ersan F, Çakır D, Ataca C 2021 ACS Appl. Mater. Interfaces 13 36388
Google Scholar
[30] Yan X H, Zheng J M, Zhao X, Zhao P J, Guo P, Jiang Z Y 2024 Phys. Status Solidi Rapid Res. Lett. 18 2300468
Google Scholar
[31] Jain A, Ong S P, Hautier G, Chen W, Richards W D, Dacek S, Cholia S, Gunter D, Skinner D, Ceder G, Persson K A 2013 APL Mater. 1 011002
Google Scholar
[32] Chen P Y, Lam C H, Edmondson B, Posadas A B, Demkov A A, Ekerdt J G 2019 J. Vac. Sci. Technol. A 37 050902
Google Scholar
[33] Khushi M, Shaukat K, Alam T M, Hameed I A, Uddin S, Luo S, Yang X, Reyes M C 2021 IEEE Access 9 109960
Google Scholar
[34] Ward L, Dunn A, Faghaninia A, Zimmermann N E, 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 Comp. Mater. Sci. 152 60
Google Scholar
[35] Chen J, Song Y Y, Li S Z, Que Z X, Zhang W B 2023 Sci. China Technol. Sci. 1 011002
[36] Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay E 2011 J. Mach. Learn. Res. 12 2825
[37] Ester M, Kriegel H P, Xu X 2023 Geogr. Anal. 55 207
Google Scholar
[38] Wu J, Chen X Y, Zhang H, Xiong L D, Lei H, Deng S H 2019 J. Electron. Sci. Technol. 17 26
[39] Ma Q Y, Wan W H, Ge Y F, Li Y M, Liu Y 2022 J. Magn. Magn. Mater. 605 172314
[40] Yin W J, Tan H J, Ding P J, Wen B, Li X B, Teobaldi G, Liu L M 2021 Mater. Adv. 2 7543
Google Scholar
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