Interplay of octahedral rotations and breathing distortions in charge-ordering perovskite oxides

We investigate the structure-property relationships in $AB$O${}_3$ perovskites exhibiting octahedral rotations and cooperative octahedral breathing distortions (CBD) using group theoretical methods. Rotations of octahedra are ubiquitous in the perovskite family, while the appearance of breathing distortions—oxygen displacement patterns that lead to approximately uniform dilation and contraction of the $B$O${}_6$ octahedra—are rarer in compositions with a single, chemically unique $B$ site. The presence of a CBD relies on electronic instabilities of the $B$-site cations, either orbital degeneracies or valence-state fluctuations, and often appear concomitant with charge order metal-insulator transitions or $B$-site cation ordering. We enumerate the structural variants obtained from rotational and breathing lattice modes and formulate a general Landau functional describing their interaction. We use this information and combine it with statistical correlation techniques to evaluate the role of atomic scale distortions on the critical temperatures in representative charge ordering nickelate and bismuthate perovskites. Our results provide microscopic insights into the underlying structure-property interactions across electronic and magnetic phase boundaries, suggesting plausible routes to tailor the behavior of functional oxides by design.

Figure 1

Illustration of the effect of charge disproportionation (CDP) on the local $B$O${}_6$ octahedral site equivalence. Before the CDP transition, all octahedra are equivalent (left); afterwards, the $B$O${}_6$ octahedra disproportionate into nonequivalent sites with the simplest two-site case shown here: charge transfer between $B$ sites causes one octahedron to dilate (site 1) and the other octahedron to contract (site 2), resembling a “breathing” mode of the $B$O${}_6$ perovskite building blocks.

Figure 2

Schematic octahedral representations of the two cooperative structural breathing distortions that coexists with charge ordering in $AB$O${}_3$ perovskites. In (a), the breathing distortion is two-dimensional, irrep $M_1^{+}$, and produces a columnar ordering of the two $B$ cations reducing the symmetry to $P4/mmm$. In (b), a three-dimensional ordering produces a checkerboard arrangement of the $B$ cations and is described by the $R_1^{+}$ irrep ($Fm\overline{3}m$ symmetry).

Figure 3

Representation of the rotation patterns that are compatible with each (or both, given by the intersection of the) CBD without requiring a further symmetry reduction.

Figure 4

Group-subgroup relationship between $Pm\overline{3}m$ and $P{2}_1/c$. Irreps within $Pm\overline{3}m$ responsible for transitions are shown. In addition, we note that $M_3^{+}$$\oplus$$R_4^{+}$$\oplus$$R_3^{+}$ and $M_3^{+}$$\oplus$$R_4^{+}$$\oplus$$M_5^{+}$ would also transform $Pm\overline{3}m$ $\to$ $P{2}_1/c$. Dotted line indicates that the phase transition is not allowed to be continuous within the confines of Landau theory.

Figure 5

Illustration of the symmetry-adapted orthonormal distortion modes found in the low-symmetry $R$NiO${}_3$ and Ba${}_{1-x}$K${}_x$BiO${}_3$ perovskites. The undistorted octahedra are shown in (a) for comparison. Octahedral tilting distortions refer to the rotation of the octahedral units (we use “tilting” and “rotation” interchangeably). The out-of-phase distortions are differentiated using arrows (lines and dashes) indicating the direction of cooperative atomic displacements. (b) $M_2^{+}$ describes the $d$-type Jahn-Teller mode, where two bonds shrink and two elongate; (c) $M_3^{+}$ describes an in-phase rotation mode; (d) $M_5^{+}$ describes an out-of-phase tilting mode, which could also act as an in-phase tilting mode depending on the order parameter direction (e) $R_3^{+}$ describes the $a$-type Jahn-Teller mode, where four bonds contract and two expand, and vice versa; (f) $R_4^{+}$ describes an out-of-phase rotation mode; (g) $R_5^{+}$ describes an out-of-phase bending mode; and (h) $X_5^{+}$ describes an in-phase tilting mode accompanied by $A$ cation displacements, which are not shown for clarity.

Figure 6

Correlation heat map capturing the degree of linear relationship between unit cell level structural distortion modes and macroscopic transition temperatures, $T_{\mathrm{MI}}$, and $T_N$ in $R$NiO${}_3$ perovskites. Dark blue (red) indicates a strong positive (negative) correlation and white indicates no statistical correlation between between the two variables. Units for the irreps and temperatures are given in Å and in degrees Kelvin (K), respectively.

Figure 7

A strong linear relationship is seen between (a) $R_3^{+}$ and the Néel temperature ($T_N$) with a correlation coefficient of 0.957 and (b) $M_5^{+}$ and $T_N$ with a correlation coefficient of 0.916. In (c), we plot the average $\langle$Ni–O–Ni$\rangle$ bond angle and $T_N$ to illustrate that the $R_3^{+}$ and $M_5^{+}$ irreps capture the main variation in $T_N$ for both $T_{\mathrm{MI}}=T_N$ and $T_{\mathrm{MI}}>T_N$ nickelates compared to the $\langle$Ni–O–Ni$\rangle$ bond angle. Experimental data are from Ref. 34.

Figure 8

Relationship between crystal symmetry and structural distortions in Ba${}_{1-x}$K${}_x$BiO${}_3$ as a function of $A$-site doping. The number of symmetry unique structural modes decreases with increasing K concentration. Pure BaBiO${}_3$ is described by four modes: $R_1^{+}$, $R_3^{+}$, $R_4^{+}$, and $R_5^{+}$. As the concentration of K increases, the Jahn-Teller $R_3^{+}$ mode disappears rapidly, followed by loss of the three-dimensional CBD $R_1^{+}$. The bar graph (inset) shows quantitatively in units of angstroms the change in amplitude of the structural modes observed in each space group.