Atomistic description for temperature-driven phase transitions in ${\mathrm{BaTiO}}_3$

Barium titanate (${\mathrm{BaTiO}}_3$) is a prototypical ferroelectric perovskite that undergoes the rhombohedral-orthorhombic-tetragonal-cubic phase transitions as the temperature increases. In this paper, we develop a classical interatomic potential for ${\mathrm{BaTiO}}_3$ within the framework of the bond-valence theory. The force field is parametrized from first-principles results, enabling accurate large-scale molecular dynamics (MD) simulations at finite temperatures. Our model potential for ${\mathrm{BaTiO}}_3$ reproduces the temperature-driven phase transitions in isobaric-isothermal ensemble ($NPT$) MD simulations. This potential allows for the analysis of ${\mathrm{BaTiO}}_3$ structures with atomic resolution. By analyzing the local displacements of Ti atoms, we demonstrate that the phase transitions of ${\mathrm{BaTiO}}_3$ exhibit a mix of order-disorder and displacive characters. Besides, from a detailed observation of structural dynamics during phase transition, we discover that the global phase transition is associated with changes in the equilibrium value and fluctuations of each polarization component, including the ones already averaging to zero, Contrary to the conventional understanding that temperature increase generally causes bond-softening transition, the $x$-polarization component (the one which is polar in both the orthorhombic and the tetragonal phases) exhibits a bond-hardening character during the orthorhombic-to-tetragonal transition. These results provide further insight about the temperature-driven phase transitions in ${\mathrm{BaTiO}}_3$.

Figure 1

Bond-valence vector sum and angle potential. (a) Tetragonal ${\mathrm{BaTiO}}_3$ with a nonzero BVVS; (b) cubic ${\mathrm{BaTiO}}_3$ with zero BVVS; (c) schematic of the angle potential. Ba, Ti, and O atoms are represented by green, blue, and red spheres, respectively.

Figure 2

Temperature dependence of the polarization, Ti displacement, and lattice constants in ${\mathrm{BaTiO}}_3$. Phase transitions among rhombohedral, orthorhombic, tetragonal, and cubic occur at 105, 115, and 160 K.

Figure 3

(a) The distribution of total Ti displacement magnitude at different temperatures. (b) Instantaneous compositions of different phases. Supercells at 30 K (rhombohedral), 70 K (rhombohedral), 110 K (orthorhombic), 150 K (tetragonal), and 190 K (cubic) are studied. Heights of the dark blue, light blue, orange, and red rectangles represent the percentages of rhombohedral, orthorhombic, tetragonal, and cubic unit cells, respectively. The phases of the unit cells are categorized by their Ti displacements $d$: for $d<0.1\AA$, the unit cell is considered as a nonpolar one; for a polar unit cell, if one component is larger than $d/\sqrt{6}$, then this component is considered a ferroelectric one. The ferroelectric phase (tetragonal, orthorhombic, and rhombohedral) is determined by the number of ferroelectric components.

Figure 4

The distributions of Ti displacement at different temperatures.

Figure 5

Schematic of the distributions of Ti displacement for displacive transition, order-disorder transition, and a mix of them.

Figure 6

Temperature dependence of Ti displacement distributions in three Cartesian directions. The horizontal axis shows the time. In these simulations, the temperature increases with time approximately linearly. The vertical axis represents the fraction of the Ti displacements, and the color scale represents the percentages of Ti displacement with a certain value. Note that, in the bottom center plot, the color showing the distribution becomes redder after the orthorhombic-to-tetragonal transition, indicating a narrower distribution around $d_z=0$ and a bond hardening in this direction.

Figure 7

The change in the average and standard deviations of the Ti displacement distribution. In the standard deviation plot of (b), the green and black lines increase with temperature and are parallel until the transition.

Figure 8

Schematics of bond-softening, bond-hardening, and displacive excitations. Two points worth mentioning: (1) For the $x$ component (first column), the minima of the energy profile for the tetragonal phase are further from the center and have higher curvatures, compared with those for the orthorhombic phase because the Ti displacement distribution has a larger average and smaller variance; (2) for the $z$ component (third column), compared with the energy profile for orthorhombic phase, the one for the tetragonal phase has a higher curvature at the center (Ti displacement more closely distributed around 0 as seen from Fig. 6) and smaller curvature for larger $z$-direction displacements [larger standard deviation, seen from Fig. 7].