Giant piezoelectric effects of topological structures in stretched ferroelectric membranes

<div style="color:black; background-color:#FFF3E9; border: 1px solid #FFE0C3; border-radius: 10px; margin-top:1rem; margin-bottom:1rem"> <p style="margin:1rem; padding-left: 1rem; line-height: 2.5;"> <a style="font-weight:bold"><em> ©️ <b> <i>Copyright 2024 @ Yihao Hu (胡逸豪)</i></b><br/></a></em> <i>Author： <b> <a href="mailto:huyihao@westlake.edu.cn"> Yihao Hu (胡逸豪) 📨 </a> <a href="mailto:yangjiyuan@westlake.edu.cn"> Jiyuan Yang (杨季元) 📨 </a> <a href="mailto:liushi@westlake.edu.cn"> Shi Liu (刘仕) ^†📨 </a> </b> </i> <br/> <i>Date：2024-01-11 (The last update was on 2024-07-29)</i><br/> <i>Lisence：This document is licensed under<a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"> Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) </a>license.<br/> 📖 <a style="font-weight:bold"> <b>Citing in your papers</b><br/></a> <i> We shall greatly appreciate if scientific work done using the published deep potential (<b>DP</b>) and/or the associated database and scripts for data analysis will contain an acknowledgment to the following references</i><br/> <i><a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.046802"> [1] Giant piezoelectric effects of topological structures in stretched ferroelectric membranes, Yihao Hu, Jiyuan Yang and Shi Liu*, Phys. Rev. Lett. 133, 046802 (2024) (Editors’ Suggestion)</a> </i><br/> <i><a href="https://liutheory.westlake.edu.cn/pdf/PhysRevB.107.144102.pdf"> [2] Modular development of deep potential for complex solid solutions, Jing Wu+, Jiyuan Yang+, Liyang Ma, Linfeng Zhang, and Shi Liu*, Phys. Rev. B 107, 144102 (2023)</a> </i><br/> <i><a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.143001"> [3] Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics, Linfeng Zhang, Jiequn Han, Han Wang*, Roberto Car, and Weinan E†, Phys. Rev. Lett. 120, 143001 (2018)</a> </i><br/> <i><a href="https://dl.acm.org/doi/10.5555/3327345.3327356"> [4] End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems, Linfeng Zhang, Jiequn Han, Han Wang*, Wissam A. Saidi†, Roberto Car, Weinan E‡, 32nd NeurIPS (2018)</a> </i><br/> </p> </div>

1. Introduction

We share the force field model, essential input files for density functional theory (DFT) calculations and molecular dynamics (MD) simulations, data analysis scripts, and selected original MD trajectories, as detailed in our paper [1]. The model file for PbTiO$_3$ , together with the complete training database and testing data, can be found in our <a href="https://github.com/huiihao/Spiral">GitHub repository</a>.

<!--The directory is organized as illustrated in the following diagram: - The `train` directory houses both the training dataset and the `input.json` file which holds the training metadata. - The `model` directory contains the force field file. - The `DFT` directory provides a sample `INCAR` file used for DFT scf calculations during the training database construction. - Within the `test` directory: - The `NEB` directory contains necessary files that compare the DFT and DP energy barriers for various polarization switching pathways in ferroelectric hafnia. See **Section 4.1**. - The `Energy barriers` directory contains necessary files that compare the DFT and DP energy barriers for oxygen vacancy diffusion. See **Section 4.2**. - The `Piezoelectric` directory is for piezoelectric coefficient $d_{33}$ calculations using finite-field MD simulations. See **Section 4.3**. - The `Mobility` directory contains selected MD trajectories and python scripts for oyxgen ion mobility calculations. See **Section 4.4**. This structure ensures clear categorization and easy navigation for users accessing the files. -->

The directory is organized as illustrated in the following diagram:

• train: Contains the training dataset and the input.json file which holds the training metadata. Refer to Section 2. Includes a representative INCAR file for DFT SCF calculations that were used to construct training database. Refer to Section 2.3.
• model: Stores the force field file compress01.pb.
• paper:
  • DFT_phase_diagram: We have shared the CONTCAR files and their energies for all ≈700 configurations. Refer to Section 3.1.
  • dipole_spiral: Offers selected MD trajectories and scripts to implement calculations (Refer to Section 3.2.) and demonstrate the robustness of dipole spiral Refer to (Section 3.3).
  • Piezoelectric: Dedicated to piezoelectric coefficient $d_{33}$ calculations via finite-field MD simulations. Refer to Section 3.4.
  • other_domain: Offers selected MD trajectories and scripts to implement calculations of various ferroelectric domains. Refer to Section 3.5.

<!--The directory structure is as shown in the following diagram. The training dataset and `input.json` are located in the *train* directory. The force field file is located in the *model* directory. The *DFT* directory provides an `INCAR` file for SCF calculations. The *NEB*, *strain*, and *Mobility* directories in the *test* directory correspond to the respective tests for polarization switching pathways in ferroelectric hafnia, Energy barriers of oxygen vacancy diffusion, strain vs. Electric Field along the z-axis, and Mobility of oxygen ions.-->

2. Database Construction

2.1. Training database

The force field of PbTiO$_3$ utilized in this work is a deep neural network-based model potential, referred to as deep potential (DP).

Details regarding the construction of the training database, DFT calculations, and metadata of the DP model were documented in our previous work [2]. Specifically, we adopted the DP-GEN, a concurrent learning procedure, to construct the training database (see details in Section 2.1). The initial training database contains DFT energies and atomic forces for structures derived from random perturbations of ground-state structures of $P4mm$ (tetragonal) and $Pm3m$ (cubic) phases of PbTiO$_3$ . The final training database comprises 13021 PbTiO$_3$ configurations. You can access the training database in Spiral/train/PSTO-data.zip.

2.2. DP-GEN

We employ the Deep Potential Generator (DP-GEN) to construct the training database. DP-GEN is a concurrent learning procedure consisting of three stages: labeling, training, and exploration, which together form a closed loop. Starting with an initial training database that contains DFT energies and forces for a few configurations, four DP models with distinct random initializations of neural networks are trained. In the exploration phase, one of these models is employed for MD simulations to explore the configuration space. Predictions (energies and atomic forces) are generated using all four models for each new configuration sampled from MD. For configurations that are well represented by the current training database, these four models should display nearly identical predictive accuracy. However, for those not well-represented, we expect the four models to produce scattered predictions with significant deviations. The maximum standard deviation of predictions from the four models serves as a criterion for labeling: configurations from MD with significant model deviation are labeled. The energies and atomic forces of these labeled configurations, as computed using DFT, are subsequently integrated into the training database for the next training cycle. Here, the maximum atomic force standard deviation, denoted as ε, is used as the labeling criterion. We introduce two thresholds, ε_lo and ε_hi ; only configurations for which ε_lo < ε < ε_hi are labeled for DFT calculations. We set ε_lo = 0.12 and ε_hi = 0.25. The introduction of ε_hi is to handle the exceptions due to highly distorted configurations resulting from low-quality DP models (especially in the first few cycles of DP-GEN) or unconverged DFT calculations. The iteration stops when all configurations sampled from MD simulations satisfy a predefined accuracy across all four models.  A primary advantage of the DP-GEN approach is its streamlined and largely autonomous data generation, minimizing human intervention.

<div align=center> <img src="./picture/DP-GEN.png" width="80%" height="auto"> </div>

2.3. DFT calculations

We employ 2x2x2 supercells of 40 atoms for first-principles DFT calculations using the Vienna Ab initio Simulation (VASP) package. The projected augmented wave method is employed, and the generalized gradient approximation of the Perdew-Burke-Ernzerhof (PBE) type is chosen as the exchange-correlation functional. The energy cutoff is set at 800 eV, and the k-spacing is set at 0.3 Å^-1 . A sample INCAR file for the self-consistent field (SCF) calculations can be found in the Spiral/train/ directory.

2.4. Deep Potential

The DP model, based on a deep neural network with the number of learnable parameters on the order of 10$^{6}$, offers a robust mathematical structure to represent highly nonlinear and complex interatomic interactions while bypassing the need to handcraft descriptors that represent local atomic environments. Specifically, the DP model features a symmetry-preserving embedding network that maps an atom's local environment to inputs for a fitting neural network which then outputs the atomic energy; the sum of atomic energies yields the total energy. The original references to the DP model can be found in <a href