Microscopic picture of paraelectric perovskites from structural prototypes

We highlight with first-principles molecular dynamics the persistence of intrinsic $\langle 111\rangle$ Ti off-centerings for ${\mathrm{BaTiO}}_3$ in its cubic paraelectric phase. Intriguingly, these are inconsistent with the $Pm\overline{3}m$ space group often used to atomistically model this phase using density-functional theory or similar methods. Therefore, we deploy a systematic symmetry analysis to construct representative structural models in the form of supercells that satisfy a desired point symmetry but are built from the combination of lower-symmetry primitive cells. We define as structural prototypes the smallest of these that are both energetically and dynamically stable. Remarkably, two 40-atom prototypes can be identified for paraelectric ${\mathrm{BaTiO}}_3$; these are also common to many other ${AB\mathrm{O}}_3$ perovskites. These prototypes can offer structural models of paraelectric phases that can be used for the computational engineering of functional materials. Last, we show that the emergence of $B$-cation off-centerings and the primitive-cell phonon instabilities is controlled by the equilibrium volume, in turn, dictated by the filler $A$ cation.

Figure 1

CPMD simulation of ${\mathrm{BaTiO}}_3$. Histograms of the $xy$-plane projection of the displacement of Ti atoms with respect to the barycenter of the surrounding ${\mathrm{TiO}}_6$ octahedron at (a) equilibrium volume (lattice parameter: 4.00 Å) and (b) under a 2.8% compressive hydrostatic strain (lattice parameter: 3.89 Å), integrated over 13 ps of fixed-volume NVE CPMD simulations and on all Ti atoms of a $4\times 4\times 4$ cubic supercell. The average temperature of the simulation is 315 K. The inset: graphical three-dimensional representation depicting the $xy$-plane projection of the Ti cation position in a representative ${\mathrm{TiO}}_6$ octahedron.

Figure 2

Crystal structures (top) and phonon dispersions (bottom) for various ${\mathrm{BaTiO}}_3$ cells with Ba, Ti, and O atoms shown in green, blue, and yellow, respectively. (a) Five-atom primitive cubic cell (standard crystallographic labels from Ref. [54]), displaying unstable phonon modes. The high-symmetry ${\mathrm{\Gamma }}_4^{-}$, $X_5^{+}$, and $M_2^{-}$ instabilities are marked in red with a cross, a filled circle, and an empty square, respectively. (b) Forty-atom undistorted supercell; $X^{\prime }$, $M^{\prime }$ and $R^{\prime }$ labels indicate the $X$, $M$, and $R$ points of the $2\times 2\times 2$ supercell, respectively [the same labels are used also in panels (c) and (d)]. All three instabilities marked in panel a fold at $\mathrm{\Gamma }$ in this supercell. Forty-atom supercells of the (c) 4+4 and (d) 2+6 displacement patterns. In panels (c) and (d) (top), red arrows indicate the direction of atomic displacements (only shown for $B$ cations for clarity). We refer the reader to the SM Sec. 5 [36] for the displacement pattern associated with each unstable mode and the other modes that contribute to the 4 $+$ 4 and 2 $+$ 6 displacement patterns, which can also be visualized with the interactive phonon visualizer tool on the Materials Cloud [38].

Figure 3

(a) Magnitude of the Ti-atom displacements for the 4 $+$ 4 and 2 $+$ 6 patterns in ${\mathrm{BaTiO}}_3$ as a function of the lattice parameter. Displacements are along the $\langle 111\rangle$ directions, and we plot the displacement in angstrom along one Cartesian coordinate. The DFT (PBEsol) equilibrium lattice parameter in its lowest-energy cubic configuration is indicated by the blue arrow. (b) Plot of the unstable phonon modes of ${\mathrm{BaTiO}}_3$ with irrep ${\mathrm{\Gamma }}_4^{-}$, $X_5^{+}$, and $M_2^{-}$ in the five-atom undistorted cubic cell as a function of the lattice parameter. The gray arrows indicate the lattice parameters at which the modes at $X$ and $M$ become unstable; they are in agreement with the corresponding displacement onsets in panel (a).

Figure 4

Magnitude of the Ti-atom displacements for the 4 $+$ 4 pattern in (Ba, Pb, Sr, Ca)${\mathrm{TiO}}_3$ (labeled as BTO, PTO, STO, and CTO) as a function of the lattice parameter. Displacements are along the $\langle 111\rangle$ directions, and we plot the displacement in angstrom along one Cartesian coordinate. The DFT (PBEsol) equilibrium lattice parameter is indicated by the arrow of the corresponding color. A clear universal trend across the titanates is demonstrated; the 4 $+$ 4 pattern is stable only in unstrained ${\mathrm{BaTiO}}_3$ because of the larger $A$ cation and, thus, larger lattice parameter.