Modeling ferroelectric phase transitions with graph convolutional neural networks

Ouyang Xin-Jian; Zhang Yan-Xing; Wang Zhi-Long; Zhang Feng; Chen Wei-Jia; Zhuang Yuan; Jie Xiao; Liu Lai-Jun; Wang Da-Wei

doi:10.7498/aps.73.20240156

Abstract

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.

Keywords:

Authors and contacts

1.
School of Microelectronics, Faculty of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
2.
School of Physics, Henan Normal University, Xinxiang 453007, China
3.
College of Materials Science and Engineering, Guilin University of Technology, Guilin 541004, China

Corresponding author: Wang Da-Wei, dawei.wang@xjtu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11974268, 12111530061).

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Cited By

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

Figure 1. Workflow of this study, including ab inito molecular dynamics (AIMD) sampling, graph convolutional neural network (GCN) construction and MD simulations.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 3. The energy distribution for the data sets of (a) GeTe and (b) CsSnI₃ relative to the energy of corresponding cubic phase.

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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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

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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

Figure 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.

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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
单位	/(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

DownLoad: 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%

DownLoad: CSV

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[21]	Gasteiger J, Groß J, Günnemann S 2020 International Conference on Learning Representations Virtual, April 26–May 1, 2020
[22]	Gasteiger J, Giri S, Margraf J T, Günnemann S 2020 Machine Learning for Molecules Workshop, NeurIPS Virtual, December 6–12, 2020
[23]	Chattopadhyay T, Boucherle J X, vonSchnering H G 1987 J. Phys. C: Solid State Phys. 20 1431 Google Scholar
[24]	Dangić D, Murphy A R, Murray E D, Fahy S, Savić I 2018 Phys. Rev. B 97 224106 Google Scholar
[25]	Yamada K, Funabiki S, Horimoto H, Matsui T, Okuda T, Ichiba S 1991 Chem. Lett. 20 801 Google Scholar
[26]	da Silva E L, Skelton J M, Parker S C, Walsh A 2015 Phys. Rev. B 91 144107 Google Scholar
[27]	Schütt K, Unke O, Gastegger M 2021 Proceedings of the 38th International Conference on Machine Learning Virtual, July 18–24, 2021 p9377
[28]	He K, Zhang X, Ren S, Sun J 2016 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Las Vegas, USA, June 27–30, 2016 p770
[29]	Musaelian A, Batzner S, Johansson A, Sun L, Owen C J, Kornbluth M, Kozinsky B 2023 Nat. Commun. 14 579 Google Scholar
[30]	Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, Kopf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S 2019 33rd Conference on Neural Information Processing Systems Vancouver, Canada, December 8–14, 2019 p8026
[31]	Batzner S, Musaelian A, Sun L, Geiger M, Mailoa J P, Kornbluth M, Molinari N, Smidt T E, Kozinsky B 2022 Nat. Commun. 13 2453 Google Scholar
[32]	Frenkel D, Smit B 2002 Understanding Molecular Simulation: from Algorithms to Applications (Amsterdam: Elsevier
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[34]	Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169 Google Scholar
[35]	Enkovaara J, Rostgaard C, Mortensen J J, et al. 2010 J. Phys.: Condens. Matter 22 253202 Google Scholar
[36]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[37]	Melchionna S, Ciccotti G, Lee Holian B 1993 Mol. Phys. 78 533 Google Scholar
[38]	Melchionna S 2000 Phys. Rev. E 61 6165 Google Scholar
[39]	Larsen A H, Mortensen J J, Blomqvist J, et al. 2017 J. Phys.: Condens. Matter 29 273002 Google Scholar
[40]	Smith J S, Nebgen B, Lubbers N, Isayev O, Roitberg A E 2018 J. Chem. Phys. 148 241733 Google Scholar
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[49]	Savory C N, Walsh A, Scanlon D O 2016 ACS Energy Lett. 1 949 Google Scholar
[50]	Quan L N, García de Arquer F P, Sabatini R P, Sargent E H 2018 Adv. Mater. 30 1801996 Google Scholar
[51]	Heidari Gourji F, Velauthapillai D 2021 Molecules 26 2010 Google Scholar
[52]	Glazer A M 1972 Acta Crystallogr., Sect. B Struct. Crystallogr. Cryst. Chem. 28 3384 Google Scholar
[53]	Xie N, Zhang J, Raza S, Zhang N, Chen X, Wang D 2020 J. Phys.: Condens. Matter. 32 315901 Google Scholar
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[55]	Resta R, Vanderbilt D 2007 Theory of Polarization: A Modern Approach (Berlin: Springer) p31
[56]	Schubert K, Fricke H 1951 Zeitschrift für Naturforschung A 6 781
[57]	Wdowik U D, Parlinski K, Rols S, Chatterji T 2014 Phys. Rev. B 89 224306 Google Scholar
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