Low-energy paths for octahedral tilting in inorganic halide perovskites

Instabilities relating to cooperative octahedral tilting is common in materials with perovskite structures, in particular in the subclass of halide perovskites. In this work the energetics of octahedral tilting in the inorganic metal halide perovskites ${\mathrm{CsPbI}}_3$ and ${\mathrm{CsSnI}}_3$ are investigated using first-principles density functional theory calculations. Several low-energy paths between symmetry equivalent variants of the stable orthorhombic $\left(Pnma\right)$ perovskite variant are identified and investigated. The results are in favor of the presence of dynamic disorder in the octahedral tilting phase transitions of inorganic halide perovskites. In particular, one specific type of path, corresponding to an out-of-phase “tilt switch,” is found to have significantly lower energy barrier than the others, which indicates the existence of a temperature range where the dynamic fluctuations of the octahedra follow only this type of path. This could produce a time averaged structure corresponding to the intermediate tetragonal $(P4/mbm)$ structure observed in experiments. Deficiencies of the commonly employed simple one-dimensional “double-well” potentials in describing the dynamics of the octahedra are pointed out and discussed.

Figure 1

Illustration of the in-phase (a) and out-of-phase (b) octahedral tilting modes in perovskites and the orthorhombic $\left(Pnma\right)$ structure with Glazer tilt pattern $a^{-}{a}^{-}{c}^{+}$, in the psuedocubic setting (c). Cs and I ions are represented by green and purple spheres, respectively, and the Pb/Sn ions are the black spheres in the center of the octahedra.

Figure 2

Potential energy as a function of tilt amplitude for in-phase and out-of-phase tilts in ${\mathrm{CsPbI}}_3$ and ${\mathrm{CsSnI}}_3$. Circles and squares in all four panels represent fixed and tetragonally relaxed unit cells, respectively.

Figure 3

$a^{-}{a}^{-}{c}^{+}$ PES of (a) ${\mathrm{CsPbI}}_3$ and (b) ${\mathrm{CsSnI}}_3$. The $x$ coordinate axis gives the magnitude of the in-phase tilt around the pseudocubic $c$ axis, while the $y$ coordinate axis gives the magnitude of the out-of-phase tilts around the $a$ and $b$ pseudocubic axes. The tilt amplitude is given as the offset of one I ion in units of the lattice constant of the ideal cubic perovskite. Relaxation of Cs ions and lattice parameters are allowed in the symmetry given by the tilt pattern. The energy reference is the cubic phase (origin). The energy contours are spaced by 10 and 5 meV/f.u. in (a) and (b), respectively. Note that the energy scales differ in (a) and (b).

Figure 4

2D octahedral tilting PES in ${\mathrm{CsPbI}}_3$ with the out-of-phase tilt around the $a$ pseudocubic axis fixed to its value in the fully relaxed $a^{-}{a}^{-}{c}^{+}$ structure. The $x$ coordinate axis gives the magnitude of the in-phase tilt around the pseudocubic $c$ axis, while the $y$ coordinate axis gives the magnitude of the out-of-phase tilt around $b$ pseudocubic axes. Relaxation of Cs ions and lattice parameters are allowed in the symmetry given by the tilt pattern. The energy reference is an $a^{-}{b}^0{b}^0$ structure, where the magnitude of the $a^{-}$ tilt is the same as in the fully relaxed $a^{-}{a}^{-}{c}^{+}$ structure. The energy contours are spaced by 10 meV/f.u.

Figure 5

(Top row) NEB energy profiles for ${\mathrm{CsPbI}}_3$ on a set of distinct paths between symmetry equivalent variants of the $a^{-}{a}^{-}{c}^{+}$ structure. (Bottom row) Maximum value of the octahedral distortion parameter $\mathrm{\Delta }$ [Eq. (1)], along the NEB paths. Blue and red markers correspond to a varying and fixed lattice vectors along the NEB path, respectively.