Machine learning interatomic potential for molecular dynamics simulation of the ferroelectric $\mathrm{KNb}{\mathrm{O}}_3$ perovskite

Ferroelectric perovskites have been ubiquitously applied in piezoelectric devices for decades, among which ecofriendly lead-free $\left(\mathrm{K},\mathrm{Na}\right)\mathrm{Nb}{\mathrm{O}}_3\text{−}\mathrm{based}$ materials have been recently demonstrated to be an excellent candidate for sustainable development. Molecular dynamics is a versatile theoretical calculation approach for the investigation of the dynamical properties of ferroelectric perovskites. However, molecular dynamics simulation of ferroelectric perovskites has been limited to simple systems, since the conventional construction of interatomic potential is rather difficult and inefficient. In the present study, we construct a machine-learning interatomic potential of $\mathrm{KNb}{\mathrm{O}}_3$ [as a representative system of $\left(\mathrm{K},\mathrm{Na}\right)\mathrm{Nb}{\mathrm{O}}_3$] by using a deep neural network model. Including first-principles calculation data into the training data set ensures the quantum-mechanics accuracy of the interatomic potential. The molecular dynamics based on machine-learning interatomic potential shows good agreement with the first-principles calculations, which can accurately predict multiple fundamental properties, e.g., atomic force, energy, elastic properties, and phonon dispersion. In addition, the interatomic potential exhibits satisfactory performance in the simulation of domain wall and temperature-dependent phase transition. The construction of interatomic potential based on machine learning could potentially be transferred to other ferroelectric perovskites and consequently benefit the theoretical study of ferroelectrics.

Figure 1

(a) Temperature-dependent phase transitions of $\mathrm{KNb}{\mathrm{O}}_3$. The purple, green, and red atoms correspond to K, Nb, and O. Comparison of (b) energies and (c) atomic forces between the prediction of the DP model and reference DFT calculations. The insets show the distributions of absolute errors.

Figure 2

(a) Equation of states of polymorphic $\mathrm{KNb}{\mathrm{O}}_3$ as a function of volume. Intrinsic polarization switching in (b) tetragonal phase, (c) orthorhombic phase, and (d) rhombohedral phase.

Figure 3

(a) Independent elastic stiffness constants and (b) elastic moduli of cubic, tetragonal, orthorhombic, and rhombohedral phases. Detailed values can be found in Tables SI–SV of the Supplemental Material [39].

Figure 4

Phonon dispersion spectra of (a) rhombohedral, (b) orthorhombic, (c) tetragonal, and (d) cubic phases at 0 K. Phonon dispersion spectra of (e) tetragonal and (f) cubic phases at 500 and 800 K respectively.

Figure 5

Simulation of 180DW and 90DW in orthorhombic $\mathrm{KNb}{\mathrm{O}}_3$. (a) Illustration of domain-wall configurations. (b) Formation energies of domain walls. Illustration of domain-wall motion of (c) 180DW and (d) 90DW. Calculated transition paths correspond to the domain-wall motions of (e) 180DW and (f) 90DW. Please note that $8\times 1\times 1$ supercells were illustrated here only for better demonstration, but $10\times 1\times 1$ supercells were actually calculated.

Figure 6

Phase transition in $\mathrm{KNb}{\mathrm{O}}_3$ simulated by using the variable-cell c-NEB method. Three independent transition paths, which correspond to cubic-tetragonal, tetragonal-orthorhombic, and orthorhombic-rhombohedral phase transitions, were merged and normalized into a continuous transition path.

Figure 7

Simulation of temperature-dependent phase transition in $\mathrm{KNb}{\mathrm{O}}_3$ within $NPT$ ensemble from 815 to 15 K. (a) Lattice volume, (b) lattice parameters including $a,b,c$, and (c) lattice parameters including $\alpha ,\beta ,\gamma$, normalized (via averaging) from a $13\times 13\times 13$ supercell. (d) Variation of normalized total energy consists of potential and kinetic energies. (e) Time-averaged RDF at different temperatures. (f) Ideal RDF of four major phases at 0 K.

Figure 8

Evolution of the effective polarization vectors in the simulated supercell at (a) 800 K, (b) 600 K, (c) 200 K, and (d) 15 K, which correspond to cubic, tetragonal, orthorhombic, and rhombohedral respectively. Each box component corresponds to an effective polarization vector within a perovskite unit cell. The boxes are colored according to the color map displayed in the right panel.