Rules and mechanisms governing octahedral tilts in perovskites under pressure

The rotation of octahedra (octahedral tilting) is common in $AB{\mathrm{O}}_3$ perovskites and relevant to many physical phenomena, ranging from electronic and magnetic properties, metal-insulator transitions to improper ferroelectricity. Hydrostatic pressure is an efficient way to tune and control octahedral tiltings. However, the pressure behavior of such tiltings can dramatically differ from one material to another, with the origins of such differences remaining controversial. In this paper, we discover several new mechanisms and formulate a set of simple rules that allow us to understand how pressure affects oxygen octahedral tiltings via the use and analysis of first-principles results for a variety of compounds. Besides the known $A\text{−}\mathrm{O}$ interactions, we reveal that the interactions between specific $B$ ions and oxygen ions contribute to the tilting instability. We explain the previously reported trend that the derivative of the oxygen octahedral tilting with respect to pressure ($\mathit{dR}/\mathit{dP}$) usually decreases with both the tolerance factor and the ionization state of the $A$ ion by illustrating the key role of $A\text{−}\mathrm{O}$ interactions and their change under pressure. Furthermore, three new mechanisms/rules are discovered, namely that (i) the octahedral rotations in $AB{\mathrm{O}}_3$ perovskites with empty low-lying $d$ states on the $B$ site are greatly enhanced by pressure, in order to lower the electronic kinetic energy; (ii) $\mathit{dR}/\mathit{dP}$ is enhanced when the system possesses weak tilt instabilities, and (iii) for the most common phase exhibited by perovskites—the orthorhombic Pbnm state—the in-phase and antiphase octahedral rotations are not automatically both suppressed or both enhanced by the application of pressure because of a trilinear coupling between these two rotation types and an antipolar mode involving the $A$ ions. We further predict that the polarization associated with the so-called hybrid improper ferroelectricity could be manipulated by hydrostatic pressure by indirectly controlling the amplitude of octahedral rotations.

Figure 1

Derivative of (a) in-phase and (b) antiphase rotation with respect to pressure as a function of the tolerance factor. The results of several series ($RE\mathrm{Fe}{\mathrm{O}}_3,RE\mathrm{Al}{\mathrm{O}}_3,RE\mathrm{Cr}{\mathrm{O}}_3,\mathrm{CaB}{\mathrm{O}}_3,\mathrm{SrB}{\mathrm{O}}_3,\mathrm{LiB}{\mathrm{O}}_3$, and $\mathrm{NaNb}{\mathrm{O}}_3$) are shown.

Figure 2

Pressure effect on octahedral tiltings in Pbnm (a) $\mathrm{LaFe}{\mathrm{O}}_3$, (b) $\mathrm{LuFe}{\mathrm{O}}_3$, (c) $\mathrm{CaTi}{\mathrm{O}}_3$, and (d) $\mathrm{CaSi}{\mathrm{O}}_3$. The behaviors of both $R_5^{-}$ mode (antiphase tilt about the [110] axis) and $M_2^{+}$ mode (in-phase tilt about the [001] axis) are shown.

Figure 3

Dependence of the coefficients ($A_2,B_2,A_4,B_4$) of the energy model [Eq. (1)] on the lattice constant. The coefficients $A_2,A_4,B_2,B_4$ are in unit of eV/formula-unit. Four cases [(a) $\mathrm{LaFe}{\mathrm{O}}_3$, (b) $\mathrm{LuFe}{\mathrm{O}}_3$, (c) $\mathrm{CaTi}{\mathrm{O}}_3$, and (d) $\mathrm{CaSi}{\mathrm{O}}_3$] are shown.

Figure 4

Contributions ($B_2^{A\text{−}\mathrm{O}}$, $B_2^{B\text{−}\mathrm{O}}$, and $B_2^{\mathrm{O}\text{−}\mathrm{O}}$) from different interactions to $B_2$ of the energy model [Eq. (1)] as a function of lattice constant. The coefficients $B_2$ are in unit of eV/formula-unit. Four cases [(a) $\mathrm{LaFe}{\mathrm{O}}_3$, (b) $\mathrm{LuFe}{\mathrm{O}}_3$, (c) $\mathrm{CaTi}{\mathrm{O}}_3$, and (d) $\mathrm{CaSi}{\mathrm{O}}_3$] are shown. The local environment for an oxygen moving along the $x$ direction (due to a octahedral rotation about $z$) is shown in panel (e). ${\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_2$, and ${\mathrm{\Phi }}_3$ represent the $A\text{−}\mathrm{O}$, $B^{\prime }\text{−}\mathrm{O}$, and $B^{″}\text{−}\mathrm{O}$ force constants, respectively.

Figure 5

Origin of the pressure enhancement of octahedral rotations in $AB{\mathrm{O}}_3$ compounds with low-lying $d$ states of $B$ site. (a) Derivative of antiphase rotation with respect to pressure as a function of the shifted energy of oxygen $2p$ states. When the shift energy is more than $-2\mathrm{eV}$, $\mathrm{CaSi}{\mathrm{O}}_3$ becomes cubic with $R_{\mathrm{anti}}=0$ and $\frac{{\mathit{dR}}_{\mathrm{anti}}}{\mathit{dP}}=0$. (b) Contribution from the NNN $B\text{−}\mathrm{O}$ hybridization to $B_2$ of the energy model [Eq. (1)] from the TB calculation. One can see that the NNN $B\text{−}\mathrm{O}$ interaction makes $B_2$ much more negative when the lattice constant decreases in systems with low occupation of the $d$ orbitals.

Figure 6

Explanation for the pressure behavior of octahedral tilt in Pbnm $\mathrm{LuFe}{\mathrm{O}}_3$. (a) Parameters ($\lambda$ denotes the trilinear coupling strength, $A_4^M$ and $B_4^R$ are fourth-order coefficients of in-phase rotation and antiphase rotation, respectively) of the energy simple model incorporating the trilinear coupling, as a function of pressure. (b) Amplitude of the in-phase and antiphase rotations as a function of pressure from the DFT calculation and the energy model. Since the strain degree of freedom is not crucial for the pressure behavior of octahedral tilt (see Fig. S4 of the Supplemental Material [29]), the cell is kept cubic here.