First-principles prediction of oxygen octahedral rotations in perovskite-structure EuTiO${}_3$

We present a systematic first-principles study of the structural and vibrational properties of perovskite-structure EuTiO${}_3$. Our calculated phonon spectrum of the high-symmetry cubic structural prototype shows strong M- and R-point instabilities, indicating a tendency to symmetry-lowering structural deformations composed of rotations and tilts of the oxygen octahedra. Subsequent explicit study of 14 different octahedral tilt-patterns showed that the $I4/mcm$, $Imma$, and $R\overline{3}c$ structures, all with antiferrodistortive rotations of the octahedra, have significantly lower total energy than the prototype $Pm\overline{3}m$ structure. We discuss the dynamical stability of these structures, and the influence of the antiferrodistortive structural distortions on the vibrational, optical, and magnetic properties of EuTiO${}_3$, in the context of recent unexplained experimental observations.

Figure 1

Schematic diagrams[25] of the atomic displacements in the (a) slater mode in which the B-site cation moves in the opposite direction from the rigid oxygen octahedral cage, (b) Last mode in which the A-site cation moves opposite to the rigid BO${}_6$ cage, and (c) Axe mode in which the apical oxygens move opposite to the in-plane oxygens and the A- and B-site cations do not participate; a deformation of the oxygen cage results.

Figure 2

Phonon spectrum of ideal cubic perovskite EuTiO${}_3$, calculated at the experimental volume with lattice parameter $a=3.90$ Å. The imaginary wave numbers of the low-lying modes at the Brillouin zone boundary M and R points indicate structural instabilities. Ferromagnetic ordering of Eu magnetic moments was used for computational convenience.

Figure 3

A schematic diagram, indicating the high-symmetry cubic perovskite structure and the possible 14 subgroup structures, which are the superpositions of oxygen octahedra rotations.[42] Superpositions are indicated by Glazer's notations together with the corresponding crystalline symmetries. The structures indicated by rectangles relax to ideal cubic perovskite. The structures indicated by the rounded rectangles represent the families of tilted structures with energies lower than that of the ideal perovskite structure. Arrows indicate that a structure with lower symmetry relaxed to a higher-symmetry one. For example $P{2}_1/m$ and its high-symmetry neighbor $Pnma$ structure are both unstable and relax to $Imma$. Likewise the unstable $P\overline{1}$ and $C2/c$ structures relax to the $R\overline{3}c$ phase.

Figure 4

Full phonon spectra of EuTiO${}_3$ in $I4/mcm$ (upper panel), $Imma$ (middle panel), and $R\overline{3}c$ (lower panel) structures. G-type AFM ordering of Eu magnetic moments was used in each case. All three structures are dynamically stable. In units of the reciprocal lattice vectors, the high-symmetry points are as follows. $I4/mcm$: X $=$ (0,$\frac{1}{2},0$), M $=$ ($\frac{1}{2},\frac{1}{2},0$), Z $=$ ($\frac{1}{2},\frac{1}{2},\frac{1}{2}$). $Imma$: X $=$ ($\frac{1}{2},\frac{1}{2},-\frac{1}{2}$), S $=$ ($\frac{1}{2},0,0$), R $=$ ($0,\frac{1}{2},0$), T $=$ ($0,0,\frac{1}{2}$), W $=$ ($\frac{1}{4},\frac{1}{4},\frac{1}{4}$). $R\overline{3}c$: A $=$ ($\frac{1}{2},0,0$), D $=$ ($\frac{1}{2},\frac{1}{2},0$), Z $=$ ($\frac{1}{2},\frac{1}{2},\frac{1}{2}$).

Figure 5

Schematic representation of EuTiO${}_3$ in $I4/mcm$ space group structure with oxygen octahedra tilting. Left: view along the $z$ direction, where the out-of-phase tilting occurs, right: view along the $y$ direction without any tilting. The amount of tilting corresponds to the one obtained in calculations. Drawings were produced by vesta visualization software.[43]

Figure 6

Calculated and measured[24] mode-plasma frequencies, plotted as a function of the mode wave numbers. Our calculated values for four different structural symmetries are presented. Values originating from the TO1, TO2, and TO4 modes of the ideal perovskite are grouped together; the ungrouped symbols correspond to additional modes that appear due to the symmetry lowering.