Nature of the octahedral tilting phase transitions in perovskites: A case study of ${\mathrm{CaMnO}}_3$

The temperature-induced antiferrodistortive (AFD) structural phase transitions in ${\mathrm{CaMnO}}_3$, a typical perovskite oxide, are studied using first-principles density functional theory calculations. These transitions are caused by tilting of the ${\mathrm{MnO}}_6$ octahedra that are related to unstable phonon modes in the high-symmetry cubic perovskite phase. Transitions due to octahedral tilting in perovskites normally are believed to fit into the standard soft-mode picture of displacive phase transitions. We calculate phonon-dispersion relations and potential-energy landscapes as functions of the unstable phonon modes and argue based on the results that the phase transitions are better described as being of order-disorder type. This means that the cubic phase emerges as a dynamical average when the system hops between local minima on the potential-energy surface. We then perform ab initio molecular dynamics simulations and find explicit evidence of the order-disorder dynamics in the system. Our conclusions are expected to be valid for other perovskite oxides, and we finally suggest how to predict the nature (displacive or order-disorder) of the AFD phase transitions in any perovskite system.

Figure 1

(a) The two unstable octahedral tilting modes in the cubic perovskite structure. (b) The orthorhombic ground-state structure of ${\mathrm{CaMnO}}_3$ (CMO) (space-group $Pnma$) where the octahedra are tilted $oop$ around the pseudocubic $a$ and $b$ axes and $ip$ around the $c$ axis. This corresponds to the tilt configuration $a^{-}{a}^{-}{c}^{+}$ in Glazer notation [7]. (c) The view along the $c$ axis, showing the definition of the tilt angle around this axis ${\varphi }_c$. Ca and O ions are represented by light blue and red spheres, respectively, whereas the Mn ions are in the center of the pink ${\mathrm{MnO}}_6$ octahedra.

Figure 2

Total and site-projected phonon density of states and dispersion relations along selected paths in the first BZ for the (a) cubic $Pm\overline{3}m$, (b) tetragonal $I4/mcm$, and (c) orthorhombic $Pnma$ phases of CMO.

Figure 3

Potential energy as a function of the amplitude of the unstable $M$-point $ip$ (the red circles) and $R$-point $oop$ (the blue squares) tilting modes in the cubic perovskite structure. The mode amplitude (tilts) are given as the offset distance of a single O ion in units of the lattice constant of the cubic perovskite. The lines are guides for the eye.

Figure 4

Potential-energy landscapes as functions of the amplitude of octahedral tilts in the cubic perovskite structure. The tilts are given as the offset distance of a single O ion in units of the lattice constant of the cubic perovskite. In (a) the $oop$ tilts around the pseudocubic $a$ and $b$ axes are fixed to be of the same magnitude, whereas in (b) the $oop$ tilt around the pseudocubic $a$ axis is fixed at its minimum values from (a) and the remaining $oop$ and $ip$ tilts are varied. Relaxations of both the cell vectors and the Ca ions are included.

Figure 5

GSSNEB minimum energy path between equivalent ($a^{-}{a}^{-}{c}^{+}$)-type tilt configurations. The reaction coordinate is the cumulative distance along the path divided by $\sqrt{N}$ where $N$ is the number of atoms.

Figure 6

Tilt angles ${\varphi }_a,{\varphi }_b$, and ${\varphi }_c$ around the three pseudocubic axes for one selected octahedron as functions of simulation time at temperatures 600, 900 and 1100 K. The black lines correspond to the running average.

Figure 7

Tilt angles ${\varphi }_a,{\varphi }_b$, and ${\varphi }_c$ for a chain of octahedra in the $a$ (lower panel), $b$ (middle panel), and $c$ (upper panel) pseudocubic directions. The data is from a simulation at $T=1100\mathrm{K}$. For clarity, tilt angles of different octahedra have been offset vertically.