Screening for high-performance piezoelectrics using high-throughput density functional theory

We present a large-scale density functional theory (DFT) investigation of the $\mathit{AB}{\mathrm{O}}_3$ chemical space in the perovskite crystal structure, with the aim of identifying those that are relevant for forming piezoelectric materials. Screening criteria on the DFT results are used to select 49 compositions, which can be seen as the fundamental building blocks from which to create alloys with potentially good piezoelectric performance. This screening finds all the alloy end points used in three well-known high-performance piezoelectrics. The energy differences between different structural distortions, deformation, coupling between the displacement of the $A$ and $B$ sites, spontaneous polarization, Born effective charges, and stability is analyzed in each composition. We discuss the features that cause the high piezoelectric performance of the well-known piezoelectric lead zirconate titanate (PZT), and investigate to what extent these features occur in other compositions. We demonstrate how our results can be useful in the design of isovalent alloys with high piezoelectric performance.

Figure 1

(a) The prototypical ideal perovskite structure $\mathit{AB}{\mathrm{O}}_3$. The figure shows the 12-coordinated $A$-site cation (big, black) and the 6-coordinated $B$-site cations (big, pale shaded/blue) which are embedded in octahedra (blue) defined by oxygens (small, dark/red). (b) The tetragonally distorted phase. (c) The rhombohedrally distorted phase. (d) The structure where the oxygen octahedra have been rotated along the (1,1,1) direction. Note: the distortions are exaggerated for illustrative purposes as typical values of the rhombohedral tilt and tetragonal strain would not be visible. The figures were created using VESTA (Ref.33).

Figure 2

Key quantities in the energy landscape when the $A$- and $B$-site ions are displaced along the tetragonal (1,0,0) direction in the cubic ${\mathrm{BaTiO}}_3$. For definitions of the quantities, see Sec. 2.

Figure 3

Energy landscapes when the $A$- and $B$-site ions in cubic ${\mathrm{PbZrO}}_3$ and ${\mathrm{PbTiO}}_3$ are displaced various distances along the tetragonal (1,0,0) direction. For ${\mathrm{PbTiO}}_3$ when both of the ions are displaced simultaneously (the energy values in the center of the 2D plot), the energy is lower than when they are displaced individually (the energy values along the $x$ and $y$ axes). This means that the interaction between the ions (electrostatic and bond interaction across the oxygens) makes it energetically favorable for both ions to move together. Using the key quantities defined in Sec. 2 and Fig. 2, this is indicated by ${\alpha }_{\mathrm{AT}/\mathrm{BT}}^{\mathrm{cub}}\approx {45}^{°}$, rather than $0^{°}$ or ${90}^{°}$.

Figure 4

The 99 ternary perovskite oxides out of the chemical space of ${63}^2$ that are predicted as nonmetallic by our DFT calculations. The $x$ axis is the $A$-site species, and the $y$ axis is the $B$-site species, ordered by Mendeleev number which keeps chemically similar species together (Ref.60). The color shows (a) the band gap for the cubic structure and (b) the energy of the lowest distorted phase. Compositions with no Kohn-Sham gap are shown as black. Rows and columns that would have been completely black are excluded from the plots. In (b) compositions that stay cubic when they are allowed to distort are marked as gray.

Figure 5

(a) The span of phase energies (highest-lowest) $\Delta E^{\mathrm{P}}$ for the 49 nonmetallic ternary perovskite oxides from Fig. 4b with $\Delta E^{\mathrm{P}}<0.5\hspace{0.5em}\mathrm{eV}$. The color indicates which of the phase energies is the lowest. (b) $\Delta E^{\mathrm{P}}$ for structures at 10% expanded volume, using the same criteria. (c) $\Delta E^{\mathrm{P}}$ for structures at 10% compressed volume, using the same criteria. Compositions that stay cubic when they are allowed to distort are marked as gray.

Figure 6

Calculated properties of the selected 49 nonmetallic ternary perovskite oxides. The compositions are listed in order of Mendeleev number (Ref.60) on the $B$-site first, and on the $A$-site second. The quantities are divided into five categories defined and discussed in Sec. 2. A “$•$” symbol for “Stable” indicates that the composition is a formable perovskite according to the criteria of Ref.61. As indicated in the legends, some quantities have been rescaled to fit the scales used.

Figure 7

A closer look at the $A$-$B$ site interplay for those compositions where the energy landscape is on a small scale, $E_{\mathrm{AT}}^{\mathrm{cub}},E_{\mathrm{BT}}^{\mathrm{cub}}<50\hspace{0.5em}\mathrm{meV}$. The plotted quantities are defined and discussed in Sec. 2. The energy minimum used to calculate the angle ${\alpha }_{\mathrm{AT}/\mathrm{BT}}^{\mathrm{cub}}$ is taken on the fixed grid used for the energy landscapes, which means that only 5 different values are possible for this angle, $0^{°}$, ${27}^{°}$, ${45}^{°}$, ${63}^{°}$, or ${90}^{°}$. For these values to fit on the scale in the figure, they have been scaled to the range $-$50 to 50.

Figure 8

Energy landscape when the $A$- and $B$-site ions in cubic ${\mathrm{SnTiO}}_3$ are displaced various distances along the tetragonal (1,0,0) direction. This energy landscape has a strong resemblance to that of ${\mathrm{PbTiO}}_3$, shown in Fig. 3.