First-principles study of epitaxial strain in perovskites

Using an extension of a first-principles method developed by King-Smith and Vanderbilt [Phys. Rev. B, 49, 5828 (1994)], we investigate the effects of in-plane epitaxial strain on the ground-state structure and polarization of eight perovskite oxides: $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$, $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$, $\mathrm{Ca}\mathrm{Ti}{\mathrm{O}}_3$, $\mathrm{K}\mathrm{Nb}{\mathrm{O}}_3$, $\mathrm{Na}\mathrm{Nb}{\mathrm{O}}_3$, $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$, $\mathrm{Pb}\mathrm{Zr}{\mathrm{O}}_3$, and $\mathrm{Ba}\mathrm{Zr}{\mathrm{O}}_3$. In addition, we investigate the effects of a nonzero normal stress. The results are shown to be useful in predicting the structure and polarization of perovskite oxide thin films and superlattices.

Figure 1

(a) Ideal cubic perovskite structure for an $\mathrm{A}\mathrm{B}{\mathrm{O}}_3$ compound; the $z$-polarized soft-mode atomic displacements are indicated by arrows. (b) Epitaxial paraelectric (or $p$) phase, in which the atoms are constrained in the $xy$ plane due to the presence of the substrate.

Figure 2

(Color online) Sketches of different behaviors found for the elastic enthalpy curves of the most stable phases. Solid circles show where two parabolas meet. Empty circles show the points where a tie line meets a parabola. The curves that meet at the circles do so with equal first derivatives. Panels (a) and (b) are representative of $c\text{−}r\text{−}aa$ and $c\text{−}p\text{−}aa$ or $c\text{−}p\text{−}a$ sequences of phase transitions. Panels (c) and (d) illustrate the cases where formation of domains occurs in going from the $c$ to the $aa$ or to the $a$ phases.

Figure 3

(Color online) Comparison of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ energy curves for the six epitaxial phases studied, as obtained from direct DFT calculations (Ref. 5) (left), and from the KSV theory (right). Energies are relative to the paraelectric structure at zero misfit strain. The lines in the left panel and the symbols in the right panel are provided as guides to the eye.

Figure 4

(Color online) Comparison of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ atomic displacements for the most stable phase at each given value of strain, as obtained from direct DFT calculations (Ref. 5) (left), and from our KSV model (right). ${\Delta }_z\left(\mathrm{Ti}\right)$ indicates the displacement of the Ti atom along the $z$ direction, etc. Symmetry implies that ${\Delta }_y\left(\mathrm{Ti}\right)={\Delta }_x\left(\mathrm{Ti}\right)$, ${\Delta }_x\left({\mathrm{O}}_2\right)={\Delta }_y\left({\mathrm{O}}_1\right)$, ${\Delta }_y\left({\mathrm{O}}_2\right)={\Delta }_x\left({\mathrm{O}}_1\right)$, ${\Delta }_z\left({\mathrm{O}}_2\right)={\Delta }_z\left({\mathrm{O}}_1\right)$, and ${\Delta }_y\left({\mathrm{O}}_3\right)={\Delta }_x\left({\mathrm{O}}_3\right)$. The lines in the left panel are guides to the eye.

Figure 5

(Color online) Value of the polarization along the $z$ axis for different perovskites. The continuous thick part of each curve indicates easily achievable misfit strain conditions, with misfit strain values between $-20\times {10}^{-3}$ and $20\times {10}^{-3}$, while the thin dashed part of each curve corresponds to larger strains. The curves for $\mathrm{Ca}\mathrm{Ti}{\mathrm{O}}_3$ and $\mathrm{Pb}\mathrm{Zr}{\mathrm{O}}_3$ fall outside the plotted region.

Figure 6

External stress versus misfit strain phase diagrams for eight epitaxially strained perovskites. Straight lines represent second-order phase transitions. Shadowed areas indicate the presence of $c$ and $aa$ (or $a$) domains.