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铁电材料广泛应用于功能器件中, 对铁电体进行方便、准确的理论建模, 是一个长期被关注的问题. 本文提出了一种基于图卷积神经网络的铁电相变模拟方法, 利用图卷积神经网络对铁电材料的势能面进行原子层面的建模, 并将得到的神经网络势函数作为计算器, 以驱动大体系的分子动力学模拟. 给定原子位置, 训练好的图卷积神经网络能够给出势能的高精度预测, 达到每原子1 meV级别, 与从头算(ab inito)精度基本相当, 同时在计算速度上相比从头算方法有数个数量级的提升. 得益于神经网络的高精度和快速预测能力, 结合分子动力学模拟, 本文对两种不同类型的铁电材料——GeTe和CsSnI3进行研究, 成功模拟了它们随温度发生的结构相变, 模拟结果和实验相符合. 这些结果说明了图卷积神经网络在铁电体建模和相变模拟应用中的准确性和可靠性, 为铁电体的理论探索提供了一个通用建模方法.Ferroelectric materials are widely used in functional devices, however, it has been a long-standing issue to achieve convenient and accurate theoretical modeling of them. Herein, a noval approach to modeling ferroelectric materials is proposed by using graph convolutional neural networks (GCNs). In this approach, the potential energy surface of ferroelectric materials is described by GCNs, which then serves as a calculator to conduct large-scale molecular dynamics simulations. Given atomic positions, the well-trained GCN model can provide accurate predictions of the potential energy and atomic forces, with an accuracy reaching up to 1 meV per atom. The accuracy of GCNs is comparable to that of ab inito calculations, while the computing speed is faster than that of ab inito calculations by a few orders. Benefiting from the high accuracy and fast prediction of the GCN model, we further combine it with molecular dynamics simulations to investigate two representative ferroelectric materials—bulk GeTe and CsSnI3, and successfully produce their temperature-dependent structural phase transitions, which are in good agreement with the experimental observations. For GeTe, we observe an unusual negative thermal expansion around the region of its ferroelectric phase transition, which has been reported in previous experiments. For CsSnI3, we correctly obtain the octahedron tilting patterns associated with its phase transition sequence. These results demonstrate the accuracy and reliability of GCNs in the modeling of potential energy surfaces for ferroelectric materials, thus providing a universal approach for investigating them theoretically.
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Keywords:
- phase transition /
- machien learning /
- potential energy surface
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图 1 方法流程图, 包括从头算分子动力学(ab inito molecular dynamics, AIMD)采样、图卷积神经网络(graph convolutional neural network, GCN)搭建和分子动力学模拟三个部分
Fig. 1. Workflow of this study, including ab inito molecular dynamics (AIMD) sampling, graph convolutional neural network (GCN) construction and MD simulations.
图 2 (a)图卷积神经网络框架, 以改进后的DimeNet++为例. 各个模块的具体结构和文献[22]一致; (b)相互作用模块, 包括消息传递$ f_{{\rm{inter}}} $和消息更新$ f_{{\rm{update}}} $两个过程
Fig. 2. (a) Architecture of the GCN model, a refined DimeNet++, where the design of blocks are inherited from Reference [22]; (b) interaction blocks, including message interaction and message update functions.
图 3 (a) GeTe和(b) CsSnI3的DFT数据集的能量分布柱状图, 以优化后的立方相的能量为能量零点
Fig. 3. The energy distribution for the data sets of (a) GeTe and (b) CsSnI3 relative to the energy of corresponding cubic phase.
图 4 图卷积神经网络模型(GCN)关于GeTe ((a), (b))和CsSnI3 ((c), (d))的测试集和验证集势阱预测效果, 以优化后的立方相结构为能量零点
Fig. 4. The test and validation results of the trained graph convolutional neural network (GCN) models for GeTe ((a), (b)) and CsSnI3 ((c), (d)) respectively, where the energy of the optimized cubic phase is set as the reference energy.
图 5 GeTe块体的相变模拟结果 (a)晶格常数随温度发生的变化, 红色虚线框表示铁电相变附近负的热膨胀效应; (b) Ge原子和Te原子两者沿$ x, y, z $三个方向的平均相对位移(虚线)及其模长(黑色实线)、自发极化(红色实线)随温度的变化情况; (c) Ge原子和Te原子的平均相对位移随体系大小的收敛性测试
Fig. 5. Phase transition simulations for bulk GeTe: (a) The temperature-dependence of simulated lattice parameters, where the red area indicates the negative volumetric thermal expansion of GeTe near the phase transition; (b) the average relative displacements between Ge and Te atoms during MD simulations and the spontaneous polarization; (c) the convergence of simulation with respect to the system size.
图 6 CsSnI3块体的相变模拟结果 (a)晶格常数随温度发生的变化; (b) CsSnI3在立方相(C)、四方相(T)和正交相(O)下的八面体转动情况, OOP (out-of-phase)和IP (in-phase)分别表示反相和同相转动
Fig. 6. Phase transition simulations for bulk CsSnI3: (a) The temperature-dependence of simulated lattice parameters; (b) the change of SnI6 octahedron tilting pattern during the cubic-tetragonal-orthorhombic (C-T-O) phase transition.
表 1 GeTe和CsSnI3的图卷积神经网络模型在各自测试集上的精度
Table 1. Prediction accuracy of the trained GCN models for GeTe and CsSnI3 on their test data sets
单位 能量 力 应力 /(meV·atom–1) /(meV·Å–1·atom–1) /(meV·Å–3) GeTe 0.197 1.016 2.371 CsSnI3 0.323 0.825 0.944 
下载: 导出CSV
表 2 图卷积神经网络(GCN)分别用于GeTe和CsSnI3的结构优化结果
Table 2. The structure optimization for GeTe and CsSnI3 using their corresponding graph convolutional neural network (GCN) models
Phases a/Å b/Å c/Å α/(°) β/(°) γ/(°) GeTe $ Fm\bar{3}m $ DFT 5.997 5.997 5.997 90 90 90 GCN 5.996 5.996 5.996 90 90 90 error 0.017% 0.017% 0.017% 0% 0% 0% $ R3 m $ DFT 6.076 6.076 6.076 88.04 88.04 88.04 GCN 6.061 6.061 6.061 88.37 88.37 88.37 error 0.244% 0.244% 0.244% 0.375% 0.375% 0.375% CsSnI3 $ Pm\bar{3}m $ DFT 6.270 6.270 6.270 90 90 90 GCN 6.270 6.270 6.270 90 90 90 error 0% 0% 0% 0% 0% 0% $ P4/mbm $ DFT 6.337 6.224 6.224 90 90 90 GCN 6.346 6.211 6.211 90 90 90 error 0.148% 0.195% 0.195% 0% 0% 0% $ Pnma $ DFT 6.243 6.243 6.254 90 90 89.63 GCN 6.225 6.225 6.235 90 90 89.72 error 0.295% 0.295% 0.311% 0% 0% 0.103%
下载: 导出CSV
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