Application of Machine Learning Force Fields for Modeling Ferroelectric Materials

doi:10.11900/0412.1961.2024.00177

Acta Metall Sin

2024, Vol. 60

Issue (10): 1312-1328 DOI: 10.11900/0412.1961.2024.00177

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Application of Machine Learning Force Fields for Modeling Ferroelectric Materials

LIU Shi¹(

), HUANG Jiawei², WU Jing¹

1 Department of Physics, School of Science, Westlake University, Hangzhou 310030, China
2 Department of Mechanical Engineering, The University of Hong Kong, Hong Kong 999077, China

Cite this article:

LIU Shi, HUANG Jiawei, WU Jing. Application of Machine Learning Force Fields for Modeling Ferroelectric Materials. Acta Metall Sin, 2024, 60(10): 1312-1328.

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Abstract

Ferroelectric materials, which are characterized by tunable spontaneous polarization, show remarkable application potential for nonvolatile information storage; however they present various challenges. The performance of these materials is strongly influenced by their dynamic polarization behavior under multiple external fields. Due to the limited precision of experimental observations, precise atomic-level material simulations are crucial. Although molecular dynamics (MD) offers an ideal method for investigating material dynamics over a wide spatiotemporal range, its application to new materials is often limited by challenges such as low accuracy, complex development, and limited portability of conventional classical force fields. Advances in machine learning have provided new possibilities for developing force fields. Among different machine learning potentials, deep potential (DP) based on deep neural networks stands out. DP offers accuracy comparable to that of density functional theory while providing computational efficiency similar to that of conventional classical force fields. This review primarily focused on the development and application of DP in ferroelectric materials, specifically examining the phase transition mechanisms and polarization reversal processes at the atomic scale. Considerable efforts have been made to develop and evaluate DP for crucial ferroelectric materials such as hafnium dioxide (HfO₂) and classic perovskite ferroelectric materials. Furthermore, this review explores the high oxygen-ion migration kinetics in HfO₂ using DP and investigates the flexoelectricity induced by polar domain boundaries and the bulk photovoltaic effects in strontium titanate. By highlighting the use of DP molecular dynamics approaches in ferroelectric materials, this review emphasizes the role of machine learning approaches in optimizing and accelerating material simulations to facilitate further breakthroughs and discoveries.

Key words: ferroelectric material molecular dynamics machine learning deep potential

Received: 23 May 2024

ZTFLH:

TM22

Fund: National Natural Science Foundation of China(92370104)

Corresponding Authors: LIU Shi, professor, Tel: (0571)85273989, E-mail: liushi@westlake.edu.cn

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Fig.1 Schematic structures of perovskite ferroelectric material represented by PbTiO₃ (Polarization (denote as P) directions are marked with blue arrows. Pb atoms in dark gray, Ti atoms in blue, and O atoms in red) (a) and ferroelectric phase HfO₂ (Hf atoms in light gray and O atoms in red. O^np and O^p are non-polar and polar O atoms, respectively) (b)

Fig.2 Schematic diagram of the deep potential (DP) energy model, amplified by a deep neural network^[98] ( R_i is the global coordinates of the atom i, and R_ij = R_i - R_j describes the neighboring atoms. D_ij is the local coordinate information and serves as neural network input. E_i is the “atomic energy” of atom i, while E is the total energy of the whole system)

Fig.3 Deep potential generator (DP-GEN) framework diagram^[86]
(a) the exploration, labeling, and training steps in the iterative procedure; the deep potential molecular dynamics (DPMD)-based explora-tion strategy is used here as an example. The molecular dynamics simulation is driven by an ensemble of DP models given an initial structure, and a series of conforma-tions are sampled
(b) for each configuration, the maximum deviation from the atomic force, defined as the error metric ϵ, is predicted by the DP model ensemble. σ_hi and σ_lo are the upper and lower bounds of the trust levels, respectively. F_w,_i (R_t) denotes the force on the atom with index i predicted by the model E_ω, and N_m denotes the number of atoms
(c) labeling is performed, and first-principles cal-culations are employed to obtain the energies and forces of the candidate structures,

E ˜

and

F ˜

are energies and forces computed through first-principles calculations, respectively
(d) based on the accumulated training set, a new DP model ensemble is generated and passed to the next iteration

Fig.4 DP model predictions of energies (a) and atomic forces along different directions (f_x, f_y, and f_z ) (b-d) for all structures in the final training dataset vs density functional theory (DFT) results (Insets represent absolute error distributions)^[103]

Fig.5 Equation of state for different crystalline phases of HfO₂ (V is the volume of the cell. Solid lines and crosses mark the results of DFT calculations and DP model predictions, respectively)^[103] (a), phonon spectra of HfO₂P2₁/c, Pbca, and Pca2₁phases^[103] (b), energy barrier calculations between multiple crystalline phases of HfO₂ (Solid lines are DFT calculations, and hollow circles are DP predictions)^[103] (c), probability distributions of the local displacement of oxygen atoms along the [010] direction (d_[010]) at different temperatures for a pressure of 0 GPa (Inset shows the distribution of the displacements of oxygen atoms along the [100], [010], and [001] directions at 400 K) (d), and temperature increase-driven changes in the lattice constant and in the average value of d_[010], indicating the occurrence of a phase transition (e)

Fig.6 Schematic diagram of modular development of deep potential (ModDP) strategy (As an example, DP-GEN is used to obtain DP models for HfO₂ and ZrO₂, respectively, using Hf _x Zr_{1 -}_x O₂ (HZO). The initial training dataset consists of the converged datasets of HfO₂and ZrO₂ with the random structures of Hf _x Zr_{1 -}_x O₂ (x = 0.25, 0.50, and 0.75), and the converged DP model of Hf _x Zr_{1 -}_x O₂ is finally obtained by DP-GEN. MD—molecular dynamics, MC—Monte Carlo) (a)^[105] and schematic diagram of the UniPero process (The workflow for building a generic force field for chalcogenide oxides follows the strategy of ModDP) (b)^[107]

Fig.7 Correlation of temperature with lattice constant based on generalized interatomic potential simulations (T—tetragonal, C—cubic, R—rhombohedral, O—orthorhombic)
(a) PbTiO₃ (b) SrTiO₃ (c) BaTiO₃ (d) Pb_0.5Sr_0.5TiO₃ (e) KNbO₃
(f) 0.29PIN-0.45PMN-0.26PT (PIN—Pb(In_1/2Nb_1/2)O₃, PMN—Pb(Mg_1/3Nb_2/3)O₃, PT—PbTiO₃)
(g) PbZr_0.5Ti_0.5O₃ (h) K_0.5Na_0.5NbO₃ (i) 0.36PIN-0.36PMN-0.28PT

Fig.8 Schematic representation of the inverse ferroelectric distortion octahedral turning angle (φ) (φ_n is defined as the antiferrodistortive (AFD) order parameter, where the index n is the sequence number of unit cell, and θ is the rotation angle of TiO₆ octahedra in each unit cell) (a), phase diagram of the SrTiO₃ (STO) block under biaxial in-plane strain simulated by deep potential molecular dynamics (The ferroelectric (FE) and AFD transition temperatures at different strains are indicated by blue square dots and red dots, respectively. T represents temperature) (b), and curves of polarization and octahedral turning angle (ϕ) versus temperature for STO supercells (5000 atoms) under biaxial in-plane strain (-0.8%) (c)^[116]

Fig.9 Schematic of the domain lattice point model in SrTiO₃ (STO) films (Where the domains generate spontaneous concave and convex undulations in the (001) plane, with the convex angle corresponding to the domains with upward polarization and the concave angle corresponding to the domains with downward polarization, and the domains can be oriented to rotate in order to flip the polarization. a and c are lattice parameters, and α is the angel between a-c domain) (a), schematic diagram of a 90° rotation of polarized domain boundaries by bending deformation (P_macro is the macroscopic polarization, head-to-tail and head-to-head domain configurations are denoted as HT and HH, respectively) (b), and hysteresis-like features of the STO film under strain, and after the stress is removed, the STO can still maintain the bent state with residual polarization (c)^[117]

Fig.10 Polarization reversal paths in the ferroelectric phase HfO₂^[128]
(a) vibrational modes of

X 2 -

crystals in a single cell of Pca2₁, with outwardly and inwardly shifted oxygen atoms indicated by purple and brown-orange spheres, respectively, and polarization along the z-axis
(b) Pca2₁ has alternating arrangements of O^np and O^p, with gray shaded regions marking the polar regions
(c) schematic representation of the shift in (SI) and shift across (SA) polarization reversal paths driven by an external electric field (ε). In the SA path, the O^np ion

X 2 -

sign is reversed (indicated in grey during the transition). The SI and SA paths can start from the same structure, represented by P vectors in opposite directions (green arrows) to ensure compatibility with classical electrodynamics
(d) calculated results of the minimum energy paths for different polarization reversal paths, lines indicate the results of the DFT, and scatters indicate the results of the deep neural network-based force field prediction (λ represents the polariza-tion state (polar-nonpolar-polar))
(e) polarization reversal energy barrier versus electric field strength (|ε |)

Fig.11 Oxygen ion transport is coupled to the nucleation and growth mechanism of the ferroelectric polarization reversal^[128]
(a) polar-antipolar phase cycling induced by successive SI and SA ferroelectric transitions. The highlighted marker O^np in the initial structure, δ = δ₀ (defined as displacement of the O^p ion relative to the top Hf plane), becomes O^p after SI-2, and δ = -δ₀, becomes O^p after SA
(b) schematic representation of stochastic nucleation in a three-dimensional bulk. The nucleus is two-dimensional in the x-z plane and its thickness corresponds to half the cell length along the y-axis of the Pca2₁ phase of HfO₂
(c) 2D-dimensional nucleus extracted from a DPMD simulation trace of a supercell containing 28800 atoms. The profile of the nucleus was determined based on the displacement δ-value of the O^p ion
(d) δ profiles along the z and x directions labeled in Fig.11c

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