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寻找尺寸小、稳定性高和易操控的纳米磁结构——磁斯格明子(magnetic skyrmion), 是发展下一代高密度、高速度和低能耗非易失性信息存储器件核心存储单元的关键. 磁性斯格明子根据其拓扑产生机制, 可以由非中心对称结构诱导的DMI (Dzyaloshinskii–Moriya interaction)作用项产生. 二维Janus结构具有两个不同面的原子层, 可以形成垂直内建电场, 打破中心空间反演对称性. 因此寻找具有本征磁性的二维Janus材料是研究新型磁存储的基础. 本文基于晶体材料数据库Materials Project中的1179种六角晶系ABC型Janus材料数据, 以其元素组分信息为特征描述符, 构建了随机森林、梯度提升决策树、极端梯度提升和极端随机树等四种机器学习模型, 基于上述模型对晶格常数、形成能和磁矩分类进行了预测, 并采用十折交叉验证法对模型进行了评估. 梯度提升决策树在磁矩分类预测显示出最高的精度和泛化能力. 最后, 基于上述模型对尚未发现的82018种二维Janus材料进行了预测, 筛选得到4024种具有热稳定性的高磁矩结构, 并基于第一性原理的方法对其中随机抽样的13种高磁矩结构进行了计算验证. 本研究为二维Janus材料磁矩分类和高通量筛选训练了有效的机器学习模型, 加速了二维Janus结构磁性的探索. 本文数据集可在
https://doi.org/10.57760/sciencedb.j00213.00072 中访问获取.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. The datasets provided in this work are available fromhttps://doi.org/10.57760/sciencedb.j00213.00072 .1. Introduction
Since 2004, the ultrathin two-dimensional (2D) material graphene [ 1 ] Transition metal sulfides (TMDs) [ 2 ] Transition metal carbides/nitrides (MXenes) [ 3 ] And two-dimensional metal halide perovskites [ 4 ] The successive discovery of [ 5 ] , for valleytronics, spintronics, and ferroelectric [ 6 – 8 ] It provides a wide range of platforms for the study of novel physical properties, and has attracted great attention in the fields of optoelectronics, sensing, nonvolatile ferroelectric and ferromagnetic memory. Among them, the design of multiferroic materials formed by ferroelectric and ferromagnetic coupling is one of the important ways to realize the next generation of new logic memory. Its memory principle is that the intrinsic topologically protected magnetic Sgemin is found in 2D materials, and the type of magnetic Sgemin is controlled by the reversal of ferroelectric polarization, so as to realize the logic state. [ 9 ] The upper and lower surfaces of the two-dimensional Janus structure are atomic layers composed of different elements. The difference in atomic electronegativity between the upper and lower layers makes it have a large built-in electric field in the vertical direction, which breaks the spatial inversion symmetry and shows asymmetry in physical and chemical properties. The two-dimensional Janus material with a thickness of several to more than ten atomic layers has a significant quantum confinement effect. As a typical quantum effect, it is closely related to the composition, structure, chemical bonds and electronic structure of the material. [ 10 – 13 ] Because of its unique structure, two-dimensional Janus materials play an important role in piezoelectric sensors, catalysts, [ 14 ] Biomedical fields and energy storage have shown broad application prospects. [ 15 , 16 ] In particular, the magnetic Janus system with large magnetocrystalline anisotropy, high magnetic moment and intrinsic magnetic topology is an important candidate for the development of a new generation of track-level magnetic storage. [ 2 , 6 , 17 – 20 ] .
Traditional trial-and-error methods are expensive and time-consuming in the exploration of new materials, while high-throughput screening based on first-principles calculations reduces the cycle and cost of experiments and research and development, but for large systems with complex structures, it faces a series of bottlenecks such as large computing power and high time cost. [ 21 , 22 ] In recent years, with the rise of machine learning in condensed matter physics and microelectronics research. [ 23 , 24 ] Researchers can quickly screen out systems that meet the required physical properties from a large number of unknown systems through the model, significantly reducing the cycle and cost, and then conduct theoretical research through first-principles calculations to verify the accuracy of model predictions. The combination of machine learning and first-principles calculations [ 25 – 27 ] It has greatly improved the speed and accuracy of materials research and promoted the rapid discovery and development of new materials.
At present, researchers at home and abroad have studied the physical and chemical properties of Janus materials and their heterojunctions by using machine learning and first-principles calculation methods. Huang et al. [ 28 ] Machine learning combined with density functional theory was used to study the significant effect of the local dipole moment of the active site of Janus materials on the catalytic activity of monatomic catalysts, and it was found that the local dipole moment of the active site was the key factor to regulate the catalytic activity, which provided a new strategy for the design of high-performance catalysts. Chaney et al. [ 29 ] The Janus heterojunction Mo/W is studied by using density functional theory and machine learning methods. XY ( X , Y = S, Se, Te), the effects of the structural asymmetry of two-dimensional Janus transition metal sulfides on the electronic properties were discussed, and the structural differences between Janus and conventional TMDs were compared, thus providing a mechanism explanation for the adsorption and diffusion mechanism of lithium. Yan et al. [ 30 ] Two-dimensional ferromagnetic Janus MXenes materials with different transition metals show the direct exchange of ground state ferromagnetic sources with d-orbital electrons of magnetic atoms, and the ferromagnetic materials are screened out based on machine learning methods, which provides a new strategy for the development of spintronic devices. Although the above studies have made important progress in the mining of Janus material properties based on machine learning methods, the magnetic prediction of hexagonal Janus structure remains to be explored. In addition, how to use the characteristic method.
In this paper, four machine learning models for the hexagonal ABC type Janus structure were trained based on the elemental composition information of the materials. They were used to predict the lattice constants, formation energies, and magnetic moment classifications of the Janus materials. Through high-throughput screening of the models, 4024 unexplored high-moment hexagonal Janus materials were found. Finally, based on first-principles calculations, the reliability of the model screening was further verified, which provided a theoretical basis for the experimental preparation of the high-moment Janus materials in the future. Fig. 1 Shown.
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Fig. 1. Steps for discovering high magnetic moment Janus materials by combining machine learning with density fun-ctional theory (DFT).2. Data and Research Methods
2.1 Data acquisition, analysis and pre-processing
This article is from the Materials Project database. [ 31 ] 1179 two-dimensional hexagonal ABC-type Janus materials were extracted as a data set. Fig. 2 (a) And Fig. 2 (B) The side and top views of the atomic structure of the two-dimensional hexagonal Janus material. The purple atoms in the middle layer are B atoms, and the upper and lower layers are green A and yellow C atomic layers with different radii. Because the electronegativity of A and C atoms is different, c The axis direction generates a built-in electric field that breaks the space inversion symmetry. The data set contains information on the chemical composition of the material (elements, stoichiometry), crystal space group, lattice constant, formation energy, and total magnetic moment.
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Fig. 2. (a) Side view and (b) top view of atomic structures of hexagonal ABC-type Janus materials.图3展示了数据集中二维Janus结构数量随晶格常数、形成能及总磁矩变化的分布情况. 图3(a)显示, 数据集中Janus结构的晶格常数a, b的分布主要在4 Å和8 Å附近, 具有两个显著峰值, 表明数据集中面内晶格较为集中. 图3(b)显示晶格常数c主要分布在3—10 Å的范围内, 在3—7 Å处具有显著峰值, 而少数二维Janus结构的c值大于10 Å, 这可能是模型中考虑了真空层的厚度所导致[32]. 图3(c)显示大部分Janus结构的形成能低于0 eV, 表明上述结构是热稳定的, 仅小部分是不稳定的. 图3(d)显示大部分材料的总磁矩主要集中在0—$ 0.25\mu_{\mathrm{B}} $的狭窄区间, 只有少数具有较高的磁矩. 上述特征表明, 数据集中的大部分Janus结构是热稳定的, 且具有较低的磁矩. 以上数据集中目标属性的特征对于机器学习算法的选择、超参数调试具有重要作用.
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Fig. 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.对数据集进行初步分析后, 接下来执行数据预处理步骤. 首先, 在六角晶系ABC型Janus材料中, 晶格常数a (Lattice a)和b (Lattice b)的数值始终一致, 因此在模型训练中, 将晶格常数a和b合并为一个变量(Lattice a=b), 以简化预测任务. 此外, 模型训练仅考虑晶格常数a和b小于10 Å的样本, 通过排除这些样本来减少噪声, 防止异常值对模型训练的干扰, 从而提高模型性能. 对于晶格常数c (Lattice c), 也采取了相同的操作.
其次, 添加磁分类标签, 将目标属性总磁矩分为低磁矩($0 \mu_{\mathrm{B}} $—$ 0.25\mu_{\mathrm{B}} $)、中磁矩($0.25 \mu_{\mathrm{B}} $—$ 3.5\mu_{\mathrm{B}} $)、高磁矩材料(>$3.5 \mu_{\mathrm{B}} $), 标签分类为0/1/2, 每一类材料的数量分别为899, 214及66个. 通过将磁矩分为三个组别, 可以更好地捕捉不同磁矩类别之间的差异, 为进一步的材料发现和研究提供更详细的信息和指导.
本研究中, 模型训练的数据集按照9∶1的比例, 划分为训练集和测试集. 选择90%的数据作为训练集能够为模型提供充足的样本量进行学习, 确保模型充分捕捉数据中的模式和特征, 从而提高其拟合能力和预测性能. 此外, 为了在分类学习中保持训练集和测试集中各类别样本比例平衡, 采用分层抽样策略. 在模型训练过程中, 使用SMOTEENN (SMOTE and edited nearest neighbours)采样步骤[33]. SMOTEENN结合了SMOTE的过采样和ENN的欠采样, 这种方法首先对数据集应用 SMOTE技术, 对少数类样本进行过采样, 生成新的少数类样本, 然后对过采样后的数据集应用 ENN技术, 通过删除噪声和边界样本进行欠采样. 本研究提供一种综合方法来同时处理少数类样本不足和噪声样本问题. 这种方法提高了数据的质量和代表性, 使得模型能够更准确地学习和预测, 从而在实际应用中表现出更好的性能.
2.2 特征生成及改进
本文采用$\rm{Matminer}$[34]的元素数据和化学计量组成特征器—$\rm{ElementFraction}$, 生成了22组仅含材料组成信息的元素属性统计描述符(magpie). 每组元素属性统计描述符包括: 最小值(min)、最大值(max)、范围(range)、平均值(mean)、平均偏差(ave_dev)、众数(mode), 共计得到了包含元素组分信息的132个描述符.
对每种Janus结构构建一个$ 3 \times 3 $大小的矩阵, 将magpie统计的22个元素属性分别以主对角线为元素本身的属性值, 非主对角线则是原子i和j之间元素性质之和(如基态磁矩和与轨道电子数有关的属性)或其比值(其余属性)的形式构建材料的二维元素属性矩阵特征, 得到22组$ 3 \times 3 $型元素属性矩阵描述符[35]. 随后, 将每组元素属性矩阵按行展平并拼接, 最终获得198种特征描述符.
通过将特征数量从132个提高到198个, 不仅丰富了材料的描述信息, 还引入了元素间相互作用的特征, 从而提高了模型的学习能力和预测准确性. 这样的方法能够更全面地捕捉材料的复杂性质, 提高模型在实际应用中的表现. 补充材料图S1 (https://doi.org/10.7498/aps.73.20241278), 表明了各模型在训练过程中比较重要的特征.
2.3 机器学习模型、训练及超参数优化
本研究采用了四种机器学习算法: 基于Sklearn库[36]的随机森林(random forest, RF)、梯度提升决策树(gradient boosting decision tree, GBDT)、极端随机树(extremely randomized trees, ET), 以及基于专用的 XGBoost库[37]的极端梯度提升算法XGBoost, XGB). XGB是一种高效的梯度提升算法, 具有分块训练、自动处理缺失值、正则化防止过拟合等优化功能. 此外, 上述四种算法能有效处理高维度和复杂特征交互的数据, 具有快速计算和高精度预测的优势. 对于回归任务, 上述四种算法使用均方误差(mean squared error, MSE)作为损失函数, 并通过拟合优度(R2)、平均绝对误差(mean absolute error, MAE)和均方根误差(root mean squared error, RMSE)来评估其性能. 对于分类任务, RF和ET使用基尼不纯度(Gini impurity)作为评估准则, 而GBDT和XGB则采用对数损失(log-loss)函数作为评估准则. 分类任务的性能评估指标包括准确率(accuracy)、精确率(precision)、召回率(recall)和F1分数(F1 score). 上述评价指标的引入, 能够系统地比较和分析不同算法在分类和回归任务中的表现, 为模型选择提供了科学依据.
在机器学习的分类与回归任务中, 虽然RF, GBDT, XGB和ET使用的基本算法结构和超参数类型相同, 但这些超参数的调整重点因任务性质的不同而有所区别. 分类任务侧重于提高算法的泛化能力和准确预测类别, 而回归任务则强调预测值的精确度和误差的控制. 对于上述四种算法, 树的数量($ n\_{\mathrm{estimators}} $)、最大深度($ {\mathrm{max}}\_{\mathrm{depth}} $)以及最大特征数($ {\mathrm{max}}\_{\mathrm{features}} $)是共通的关键参数, 影响模型的学习能力和复杂性. 在分类任务中, RF算法与ET算法的叶节点上所需的最小样本数 ($ {\mathrm{min\_samples\_leaf }}$)及内部节点再划分所需的最小样本数(${\mathrm{ min\_samples}}\_{\mathrm{split }}$)是特别重要的; 而对于GBDT和XGB, 样本子集比例($ {\mathrm{subsample}} $)和最小化损失的增益(γ)的作用亦如此, 这些参数降低模型对训练数据的噪声敏感性, 减少模型的深度, 提高对数据变化的适应能力, 防止过拟合、欠拟合及提升泛化能力. 另一方面, 对于回归任务, RF和ET算法所需的最大特征数(${\mathrm{ max\_features}} $)及XGB与GBDT算法所需的每棵树所采样的特征比例($ {\mathrm{colsample\_bytree}} $)和学习率($ {\mathrm{learning\_rate}} $)是特别重要的, 这些超参数的精确调整可以帮助模型找到更好的局部最优解, 加速模型训练, 减少对训练数据的依赖, 从而降低过拟合的可能性. 因此, 虽然共通参数为模型提供了基础结构, 但是特定任务相关的参数调整是优化性能、提高精确度和防止过拟合的关键.
本研究采用贝叶斯超参数搜索技术[38], 在超参数优化过程中, 每次迭代都基于贝叶斯统计预测各参数组合的潜在性能, 优先评估那些最有可能提升模型表现的参数组合. 这种策略通过持续更新概率模型, 并根据获得的性能数据优化搜索策略, 有效地识别出最佳参数配置. 通过在十折交叉验证中综合评估各参数组合的表现, 在最终迭代中选出表现最优的超参数配置, 显著提升了模型的预测精度和泛化能力. 不同学习任务中最优算法模型的超参数见表1.
表 1 不同训练任务中机器学习最优模型的超参数Table 1. The hyperparameters of the optimal machine learning models in various training tasks.模型 超参数 GBDT(磁矩分类) learning_rate = 0.01603011, max_depth = 5, n_estimators = 272, subsample = 0.69895067 GBDT(形成能) learning_rate = 0.02, max_depth = 6, n_estimators = 353, subsample = 0.93030056 ET(晶格常数a和b) max_depth = 10, max_features = 0.60, n_estimators = 100,
min_samples_leaf = 2, min_samples_split = 4XGB(晶格常数c) learning_rate = 0.02, n_estimators = 300, max_depth = 5,
subsample = 0.8, colsample_bytree = 0.496135192.4 第一性原理计算
本文采用基于DFT的第一性原理计算方法[39,40], 通过维也纳从头计算模拟包(Vienna ab initio simulation package, VASP)进行结构优化和静态计算. 电子交换关联能由Perdu-Burke-Ernzerhof(PBE)泛函描述. 布里渊区中采用 5$ \times $5$ \times $5 k点网格进行优化和自洽, 平面波截断能量设置为 500 eV. 每个原子上所受力的收敛阈值设置为–0.1 eV/Å, 能量收敛标准为 $ 1 \times 10^{-6} $ eV. 为了描述局域化电子之间的库仑排斥力, 对含有d, f轨道电子的强关联体系设置了有效场库仑相互作用参数Hubbard U. 所有的计算中, 原子的初始磁矩设置为零, 通过原子尺度的结构优化和静态计算, 提取二维Janus结构中的原子磁矩并与模型预测结果对比, 验证模型预测可靠性.
3. 结果与讨论
3.1 晶格常数
本研究对晶格常数的模型训练分为两类: 一类是针对晶格常数a和b的训练, 另一类是针对晶格常数c的训练. 为了全面评估这两个部分的模型性能, 使用十折交叉验证评估模型. 此方法不仅能够评估模型的稳定性, 还能有效防止过拟合. 表2展示了四种模型分别对上述两类任务在测试集上的平均评估指标.
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 对于晶格常数a和b的预测, ET模型表现最佳, 其MAE为0.3469, RMSE为0.6808及R2评分达到了0.8137. 这表明ET模型在完成训练晶格常数a和b的任务时具有较高的准确性和稳定性. 在晶格常数c的预测中, XGB模型表现最佳, 其MAE为0.5953, RMSE为0.9474及R2评分为0.7186. 图4(a)和图4(b)分别展示了最优模型ET和XGB在晶格常数a, b 和c的预测任务里, 在十折交叉验证过程中样本集上所有真实值与预测值拟合程度的散点图.
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Fig. 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.从上述数据可以看出, 晶格常数c预测的性能评估低于晶格常数a和b. 这主要是由于Materials Project数据库中晶格常数c的标准不一致, 并未阐明是否包含真空层厚度. 如果考虑了真空层厚度, 其晶格会比体相的实际晶格c大很多, 而往往在具有大的晶格常数c的Janus体系中, 真空层可能被包含在其中. 这就导致数据集中的晶格c比实际晶格要大得多. 特别是在高晶格常数c的材料中, 大多增加了真空层, 导致c值显著偏大. 这种处理方式增加了数据分布的极端性和复杂性, 使得数据的统计特性存在显著差异和额外的变异性, 进一步加剧了数据的复杂性, 从而增加了模型的学习难度, 导致晶格常数c的预测效果略低于晶格常数a和b.
3.2 形成能
本文分别采用了RF, GBDT, XGB和ET四种算法训练二维Janus材料形成能的预测模型, 并对模型性能进行了评估. 表3展示了十折交叉验证过程中各模型在测试集上的平均评估指标MAE, RMSE及$R^2$. 图5(a)—(d)展示了各模型在十折交叉验证过程中样本集上所有真实值与预测值拟合程度的散点图.
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 
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Fig. 5. Prediction of formation energy: Scatter plots for four models in ten-fold cross-validation: (a) RF; (b) GBDT; (c) XGB; (d) ET评价指标结果表明, GBDT模型表现最为优异, 具有最低的MAE和RMSE, 以及最高的R2评分, 其MAE为0.0798, RMSE为0.1411, 且R2评分达到了0.9070. 尽管XGB在MAE, RMSE, R2指标上也表现良好, 但总体上仍稍逊于GBDT. 散点图直观地展示了GBDT模型在十折交叉验证过程中样本集上所有真实值与预测值之间良好的拟合程度, 大多数样本点都分布在对角线上. 综上所述, 基于GBDT训练的形成能模型在各项性能评估指标上表现更优异, 能有效建立二维Janus材料的元素组分信息与形成能之间的映射关系, 保证了预测结果的准确性.
3.3 磁矩分类
为了评估磁矩分类模型的性能, 对四种模型进行了十折交叉验证, 并使用宏平均(macro-averaging)方式对每个类别单独计算评价指标, 然后对所有类别的结果取平均, 以确保每个类别的表现都得到公平的评估. 这种方法在类别不平衡的情况下, 能够对模型性能提供更加全面和真实的评价. 其在十折交叉验证过程中各模型在测试集上的平均评估指标如表4所列. 图6(a)—(d)直观展示了各模型在十折交叉验证过程中所有预测值和真实值的混淆矩阵图.
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 
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Fig. 6. Prediction of magnetic moment classification: Confusion matrices for four models in ten-fold cross-validation: (a) RF; (b) GBDT; (c) XGB; (d) ET.上述评价指标结果显示, GBDT模型在准确率、精确率、召回率和F1分数上均展现出优越性, 表明其能更有效地识别和分类复杂数据集中的类别. 相比之下, RF和ET虽然在处理高维数据时稳健, 但在某些类别的识别上存在误判, 这可能会影响在实际应用中对未知样本的预测效果. XGB虽然处理速度快, 适合大规模数据, 但其对异常数据的敏感性可能在未探测的数据集上造成性能波动. 因此, GBDT由于较强的泛化能力和稳定的性能表现, 成为寻找高磁矩Janus材料最具潜力的模型.
3.4 高通量材料生成及第一性原理计算
以数据集中二维六角晶系的ABC型Janus结构为基础, 在元素周期表中对A, B和C包含的所有元素随机替换, 生成了82018 种尚未发现的ABC型二维Janus材料, 上述结构不属于原数据集.
将上述各部分评价性能最好的模型用于预测82018种二维六角晶系Janus材料的晶格常数、形成能和磁矩. 模型共预测出4204种包含晶格信息并具有热稳定和高磁矩的结构. 为了进一步验证其可靠性, 本文从高通量筛选结果中随机挑选了13种涵盖了不同元素的二维Janus结构并结合第一性原理的研究, 计算其晶格常数、形成能和磁矩. 图7分别为ErFeTb, FeNO, HoRuSr, DyOsSr, EuSbSr, HoIrSr, LiUZn, PuSZn, GdKU, LuNbTi, GdHfSe, NaTbZn及HoNpSr共13种二维Janus原子结构的侧视图. 优化后的晶格常数、形成能和磁矩如表5所列. 补充材料表S1 (https://doi.org/10.7498/aps.73.20241278)给出了13种未优化结构的形成能和磁矩.
图 7 13种二维六角晶系Janus原子结构的侧视图Fig. 7. Side view of atomic structures of 13 two-dimensional hexagonal Janus materials.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 根据上述结果, 优化后的13种二维六角晶系Janus结构的形成能均为负值, 表明上述Janus体系是热力学稳定的. 此外, 基于第一性原理的研究表明, 所有13种Janus结构均具有磁性, 理论计算与机器学习模型的预测值一致. 其中ErFeTb, HoRuSr, DyOsSr, EuSbSr, HoIrSr, PuSZn, GdKU, GdHfSe, NaTbZn和HoNpSr的单原子磁矩超过$3 \mu_{\mathrm{B }}$, 如表5所列. 上述具有高磁矩的原子主要属于锕系金属(如Pu, Np)和稀土元素(如Tb, Gd, Dy). 上述元素因为未填满的f轨道电子的净磁矩贡献诱导了磁性. 这表明含有锕系金属和稀土元素的化合物是设计高磁矩六角晶系ABC型Janus材料的有力候选. 补充材料S1 (https://doi.org/10.7498/aps.73.20241278)为基于机器学习模型预测的13种未优化的Janus结构, 结合第一性原理计算的形成能和磁矩.
4. 结 论
本研究基于材料的元素组分信息, 构建了包含22组元素属性的矩阵特征, 每组特征通过行展平, 生成了198种特征描述符. 针对六角晶系ABC型Janus材料的晶格常数、形成能和磁矩分类, 对四种机器学习模型(RF, GBDT, XGB, ET)采用了贝叶斯超参数优化和十折交叉验证方法进行训练, 并得到了在各自任务中表现最佳的模型(晶格常数a = b : ET; 晶格常数c: XGB; 形成能: GBDT及磁矩分类: GBDT). 通过对数据集中六角晶系结构进行元素替换, 生成了82018种尚未发现的六角晶系ABC型Janus材料. 应用上述训练的机器学习模型, 筛选出了4024种结构稳定且具有高磁矩的六角晶系Janus材料. 鉴于算力的限制, 随机选择了13种结构结合第一性原理进行计算验证, 得到了10种优化后热力学稳定且单原子磁矩超过$ 3\mu_{\mathrm{B}} $的Janus材料, 研究表明, 含有稀土和锕系金属元素的化合物是设计高磁矩Janus材料的有力候选, 进一步验证了机器学习模型预测二维六角晶系Janus材料磁性的可靠性.
数据可用性说明
支撑本研究成果的数据集可在科学数据银行https://doi.org/10.57760/sciencedb.j00213.00072中访问获取.
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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
Fig. 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
Fig. 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
Fig. 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
Fig. 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
Fig. 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
Fig. 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
Fig. 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 = 4 XGB(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
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