On-demand phase-field modeling: Three-dimensional Landau energy for HfO2 through machine learning

TL;DR

This work introduces a three-dimensional Landau–Devonshire energy for HfO$_2$ by learning a differentiable energy surface with a multilayer perceptron that takes nine mode amplitudes ($T_i$, $A_i$, $P_i$) and the strain tensor as input. By training on an extensive dataset generated with a universal ML potential, the model recovers the nonlinear mode couplings where an antipolar mode induces a polar distortion and enables TDGL-based phase-field simulations. The phase-field extension shows that thin films exhibit reduced polarization and a higher critical strain due to depolarization, with the antipolar mode acting as a trigger for polarization and the polar modes stabilized by the antipolar/polar coupling. This on-demand framework provides a scalable route to multiscale analysis of complex ferroelectric phenomena in HfO$_2$ and related materials.

Abstract

The unexpected emergence of ferroelectricity in HfO2 at reduced dimensions has attracted considerable attention, as it provides a pathway toward the realization of ultrasmall ferroelectric devices. Ab initio calculations suggest that this effect arises from a unique mode coupling, in which an antipolar displacement mode stabilizes a robust polar distortion. Based on these insights, Landau-Devonshire energy models have been proposed using such lattice modes as order parameters. However, most existing models are limited to a simplified one-dimensional model because of the computational cost of ab initio calculations and the limitations of conventional Landau polynomials. Here, we constructed a three-dimensional Landau-Devonshire potential for HfO2 by employing the tetragonal, antipolar, and polar modes as coupled order parameters, based on the latest machine-learning technologies. We generated a large-scale dataset of energies over a three-dimensional structural space, with the computational cost drastically reduced through the use of machine-learning interatomic potentials, and trained a multilayer perceptron (MLP) to learn the relationship between the order parameters and the energy. The energy predicted by the MLP successfully captures the characteristic coupling behavior whereby the antipolar modes induce the polar mode. Furthermore, by extending this MLP-based Landau potential to a position-dependent functional, that is, to a phase-field modeling framework, we revealed that the polarization magnitude in thin films decreases compared with the bulk state, while the critical strain required for the onset of spontaneous polarization increases due to surface effects. This study presents a new framework for the on-demand construction of Landau energy and phase-field modeling using the latest machine-learning techniques, enabling multiscale analysis of complex ferroelectric phenomena.

Figures (6)

• Figure 1: A schematic illustration of successive phase transitions in HfO$_2$. The arrows indicate the displacement direction in each mode.
• Figure 2: Schematic illustration of the dataset generation procedure combining (a) grid and (b) random sampling for training the MLP.
• Figure 3: (a) Comparison between the energies obtained from the machine learning potential (MatterSim) and the MLP-based Landau potential using 239,821 data points employed for training. (b) Potential energy surface for the transition from the tetragonal to orthorhombic phase by varying the order parameters. The strain $\varepsilon_{xx}$ was fixed at 0.05 and the tetragonal mode amplitude $T_x$ was set to 0.3 Å, while $A_x$ = $A_y$ = $A_z$ were varied simultaneously from 0 to 0.4 Å together with $P_z$. The solid lines represent predictions from the MLP-based Landau potential, while the dots indicate the energies obtained from MatterSim.
• Figure 4: Evolution of the order parameters during energy minimization.
• Figure 5: Schematic illustration of the simulation cells of HfO$_2$: (a) bulk and (b) thin film.