Structural phase transitions in $\mathrm{SrTi}{\mathrm{O}}_3$ from deep potential molecular dynamics

Strontium titanate ($\mathrm{SrTi}{\mathrm{O}}_3$) is regarded as an essential material for oxide electronics. One of its many remarkable features is the subtle structural phase transition, driven by the antiferrodistortive lattice mode, from a high-temperature cubic phase to a low-temperature tetragonal phase. Classical molecular dynamics (MD) simulation is an efficient technique to reveal atomistic features of phase transition, but its application is often limited by the accuracy of empirical interatomic potentials. Here, we develop an accurate deep potential (DP) model of $\mathrm{SrTi}{\mathrm{O}}_3$ based on a machine learning method using data from first-principles density functional theory (DFT) calculations. The DP model has DFT-level accuracy, capable of performing efficient MD simulations and accurate property predictions. Using the DP model, we investigate the temperature-driven cubic-to-tetragonal phase transition and construct the in-plane biaxial strain-temperature phase diagram of $\mathrm{SrTi}{\mathrm{O}}_3$. The simulations demonstrate that the strain-induced ferroelectric (FE) phase is characterized by two order parameters, FE distortion and antiferrodistortion, and the FE phase transition has both displacive and order-disorder characters. In this paper, we lay the foundation for the development of accurate DP models of other complex perovskite materials.

Figure 1

Schematic illustration of the deep potential generator (DP-GEN) process, including training, exploration, and labeling. (a) The training dataset contains hundreds of perturbed structures and corresponding density functional theory (DFT) energies and forces. (b) Starting with these DFT training data set, (c) four DP models are trained starting with different values of deep neural network parameters. (d) Performing one of the DP models is used for molecular dynamics (MD) simulations to explore the configuration space at different temperatures. (e) Labeling selected candidates by DFT calculation and then added to the training dataset of next iteration.

Figure 2

Comparison of (a) and (c) energies and (b) and (d) atomic forces calculated using the deep potential (DP) model and density functional theory (DFT) calculation for all configurations in the (a) and (b) final training dataset and (c) and (d) testing dataset, respectively.

Figure 3

The deep potential (DP) model and density functional theory (DFT) calculated energies for (a) hydrostatic pressure, (b) in plane strain, (c) Ti atom displacement of cubic phase, and (d) phase transition trajectory from cubic $Pm\overline{3}m$ to orthorhombic $Pnma$ phase. It should be noted that the $Pnma$ phases are not included in the training dataset of the DP model.

Figure 4

The phonon dispersion relations and density of states of cubic $Pm\overline{3}m$ phase of STO calculated by deep potential (DP) model and density functional theory (DFT).

Figure 5

Average lattice constants and rotation angle of the ${\mathrm{TiO}}_6$ octahedra around [001] (φ) for $\mathrm{SrTi}{\mathrm{O}}_3$ as a function of temperature determined from deep potential molecular dynamics (DPMD) simulations with a 5000-atom supercell.

Figure 6

Sketch of (a) the antiferrodistortive (AFD) order parameter φ and the distribution of φ within each unit cell in the $ab$ plane (in plane) at temperature of (b) 30 K, (c) 160 K, and (d) 200 K (d). The color illustrating the magnitude of φ.

Figure 7

Predicted phase diagram of bulk $\mathrm{SrTi}{\mathrm{O}}_3$ under biaxial in-plane strain by deep potential molecular dynamics (DPMD) simulations. Ferroelectric and antiferrodistortive (AFD) transition temperature are plotted by blue square dot and red dot at different strains, respectively.

Figure 8

Polarization ($P$) and rotation angle of the octahedra (φ) of 5000-atom supercell STO under −0.8% biaxial in-plane strain conditions. (a) $P$ and φ as a function of temperature. (b) The probability distribution of the local $P$ in unit cell along the [001] direction at different temperatures. (c) Typical snapshots of dipole configurations in the $bc$ plane (out of plane) at different temperatures. Each arrow represents the local electric dipole vector along out-of-plane within a pseudocubic unit cell.