Interplay of Couplings between Antiferrodistortive, Ferroelectric, and Strain Degrees of Freedom in Monodomain ${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ Superlattices

We report first-principles calculations on the coupling between epitaxial strain, polarization, and oxygen octahedra rotations in monodomain $({\mathrm{PbTiO}}_3{)}_n/({\mathrm{SrTiO}}_3{)}_n$ superlattices. We show how the interplay between (i) the epitaxial strain and (ii) the electrostatic conditions can be used to control the orientation of the main axis of the system. The electrostatic constrains at the interface facilitate the polarization rotation and, as a consequence, we predict large piezoelectric responses at epitaxial strains smaller than those required considering only strain effects. In addition, ferroelectric (FE) and antiferrodistortive (AFD) modes are strongly coupled. Usual steric arguments cannot explain this coupling and a covalent model is proposed to account for it. The energy gain due to the FE-AFD coupling decreases with the periodicity of the superlattice, becoming negligible for $n\ge 3$.

Figure 1

(a) Schematic representation of a ($2/2$) $\mathrm{PTO}/\mathrm{STO}$ superlattice. ${\mathrm{TiO}}_6$ octahedra are labeled according to the chemical identity of the first two neighbor $A\mathrm{O}$ planes and the direction of the polarization. (b) Definition of the angles of rotation around the $z$ axis, $\varphi$, and tilting around an axis in the ($x,y$) plane $\theta$ of octahedra.

Figure 2

(a) Polarization and (b) absolute value of the oxygen octahedra rotations and tiltings in monodomain ($2/2$) superlattices under different epitaxial strains. In (a), empty (patterned) circles correspond to bulk PTO (STO) under the same strain conditions. In (b), the rotation $\varphi$ and tilting $\theta$ angles of the O octahedra (labeled as in Fig. 1) are represented as up-pointing triangles and down-pointing triangles, respectively. As in (a), empty symbols refer to bulk values under strain. Top labels indicate strains induced by common substrates.

Figure 3

(a) Diagram showing the change in distances between Pb and Sr ions with O anions at the $\mathrm{PTO}/\mathrm{STO}$ interface under compressive strain for a ${\mathrm{P}}^{+}$ ${\mathrm{TiO}}_6$ octahedra. (b) Same as in (a) but for a ${\mathrm{P}}^{-}$ octahedra. Reduction of distance and reinforcement of the Pb-O bond is shown by full red lines while an increase in the Pb-O distance and weakening of the bond is shown by dotted lines. Blue dotted lines represent Sr-O distances. Green arrows on $A$ cations (Pb or Sr) represent out-of-plane displacement of the corresponding atoms, whose magnitude is written in green above the arrows. Yellow arrows on oxygen atoms represent their in-plane displacements.

Figure 4

Decomposition of the energy of the [001] phase into contributions from the polar modes, the AFD modes, and the coupling between them. The reference energy corresponds to a structure where neither polar nor AFD instabilities are allowed to condense.