Energetics of oxygen-octahedra rotations in perovskite oxides from first principles

We use first-principles methods to investigate the energetics of oxygen-octahedra rotations in $AB{\mathrm{O}}_3$ perovskite oxides. We focus on the short-period, perfectly antiphase or in-phase, tilt patterns that characterize the structure of most compounds and control their physical (e.g., conductive, magnetic) properties. Based on an analytical form of the relevant potential energy surface, we discuss the conditions for the stability of various polymorphs presenting different rotation patterns, and obtain numerical results for a collection of thirty-five representative materials. Our results reveal the mechanisms responsible for the frequent occurrence of a particular structure that combines antiphase and in-phase rotations, i.e., the orthorhombic $Pbnm$ phase displayed by about half of all perovskite oxides, as well as by many nonoxidic perovskites. In essence, the $Pbnm$ phase benefits from the simultaneous occurrence of antiphase and in-phase tilt patterns that compete with each other, but not as strongly as to be mutually exclusive. We also find that secondary antipolar modes, involving the $A$ cations, contribute to weaken the competition between tilts of different types, and thus play a key role in the stabilization of the $Pbnm$ structure. Our results thus confirm and better explain previous observations for particular compounds in the literature. Interestingly, we also find that strain effects, which are known to be a major factor governing phase competition in related (e.g., ferroelectric) perovskite oxides, play no essential role as regards the relative stability of different rotational polymorphs. Further, we discuss why the $Pbnm$ structure stops being the ground state in two opposite limits—namely, for large and small $A$ cations—showing that very different effects become relevant in each case. Our work thus provides a comprehensive discussion and reference data on these all-important and abundant materials, which will be useful to better understand existing compounds as well as to identify new strategies for materials engineering.

Figure 1

Sketch of the perovskite structure and the most important distortion modes discussed in this work. Blue and red circles represent $A$ and oxygen atoms, respectively, while we show the ${\mathrm{O}}_6$ octahedra centered on the $B$ atoms in green. (a) Pattern of antiphase ${\mathrm{O}}_6$ rotations, about the $z$ pseudocubic axis (perpendicular to the page). (b) In-phase rotations about the $z$ axis. (c) Antipolar $A$-cation displacements modulated according to the ${\mathbf{q}}_X=\pi /a(1,0,0)$ wave vector (horizontal direction in the figure). Here, $a$ is the lattice constant of the 5-atom elemental perovskite cell for the reference cubic phase. We show the ${\mathbf{q}}_X$ modulation in the figure for clarity; yet, the $a^{-}{a}^{-}{c}^{+}$ polymorph discussed in the text displays an equivalent antipolar distortion modulated by ${\mathbf{q}}_Z=\pi /a(0,0,1)$. (d) Antipolar $A$-cation displacements modulated according to the ${\mathbf{q}}_R=\pi /a(1,1,1)$ wave vector.

Figure 2

Summary of our first-principles results. (a) shows the energies of the different tilt phases considered, given in eV per formula unit (f.u.). We take the result for the $a^{-}{a}^{-}{a}^{-}$ structure as zero of energy. The insets display zooms of the results for the materials that do not present the $a^{-}{a}^{-}{c}^{+}$ ground state (energies in meV/f.u.). The case of ${\mathrm{SrGeO}}_3$ is not visible even in the inset; for this compound we obtain an $a^{-}{b}^0{b}^0$ ground state that is only 0.025 meV/f.u. below the $a^{-}{a}^{-}{a}^{-}$ structure. (b) shows the tolerance factor $t$ of the considered compounds. Note that the compounds are ordered from left to right as follows: we place together all the materials that share the same $A$ cation, and the ionic radius of $A$ grows as we move to the right. Compounds sharing the same $A$ cation are ordered so that the ionic radius of the $B$ cation decreases as we move to the right. All in all, the tolerance factor roughly grows when we move from left to right. (c)–(g) show the antiphase and in-phase rotation components (Å) corresponding to the relaxed structures. The chemical formulas are given following a color code, red corresponding to compounds with $A^{3+}{B}^{3+}{\mathrm{O}}_3^{2-}$ nominal ionizations, blue to $A^{2+}{B}^{4+}{\mathrm{O}}_3^{2-}$, and black to $A^{1+}{B}^{5+}{\mathrm{O}}_3^{2-}$.

Figure 3

Computed PES parameters. We use solid black squares for ${\kappa }_r,{\overline{\alpha }}_r$, and ${\overline{\gamma }}_r$, solid red circles for ${\kappa }_m,{\overline{\alpha }}_m$, and ${\overline{\gamma }}_m$, and solid blue squares for ${\overline{\alpha }}_{\mathrm{int}}$ and ${\overline{\beta }}_{\mathrm{int}}$; open black squares for ${\alpha }_r$ and ${\gamma }_r$; open red circles for ${\alpha }_m$ and ${\gamma }_m$; open blue squares for ${\overline{\alpha }}_{\mathrm{int}}^{\prime }$. The color code for the chemical formulas is as in Fig. 2.

Figure 4

Diagram showing the energies of different polymorphs, as well as some hypothetical structures, for representative compound ${\mathrm{GdFeO}}_3$ (see text). The reference cubic phase (denoted by $a^0{a}^0{a}^0)$ is taken as zero of energy. The interactions that dominate some energy variations are indicated for emphasis. See text for details.

Figure 5

Energy difference (meV/f.u.) between the $Pbnm$ and $R\overline{3}c$ phases, as obtained in usual (filled circles, $E^{\mathcal{O}}-E_r^{\mathrm{rho}})$ and frozen-$A$ (open squares, $E^{{\mathcal{O}}^{\prime }}-E_r^{\mathrm{rho}})$ conditions. A negative energy difference implies that the $Pbnm$ phase is more stable. The color code for the chemical formulas is as in Fig. 2.

Figure 6

Stereographic projection of the strain-renormalized PES of ${\mathrm{GdFeO}}_3$, as given by $\overline{E}(\mathbf{r},\mathbf{m})$. Panels (a) and (b) show the actual PES, as obtained from the usual fourth-order and corrected sixth-order models, respectively. Panel (c) displays the result obtained in frozen-$A$ conditions, and is derived from a model that includes a small sixth-order correction. Note that the directions corresponding to pure tilts are marked by their respective $r_{\alpha }$ or $m_{\alpha }$ symbols. We also mark the $Pbnm$ and $R\overline{3}c$ states. See text for details.