In fact, it is better to ask this question differently. First, why perovskites are usually not cubic at room temperature? Second, why they become cubic at high temperatures?

Answer to the first question is that for the perfect cubic structure two main conditions must be fulfilled. The ionic sizes should match geometry of the ideal perovskite cell (what is usually described as tolerance factor) and no additional electronic effects should exist. If ionic sizes do not fit the cubic cell, it distorts in such way to accomodate wrong sizes (the most typical situation is too small A that leads to the GdFeO3-like distortion) . Electronic effects can be very different, for example Jahn-Teller distortion of the oxygen octahedra, or off-center shift of the B ion that leads to ferroelectric phases, or stereochemical activity of the lone pair of Bi or Pb.

As for the second question, it should be taken into account that while distortion of the structure is almost inevitable, there are always several symmetrically equivalent ways of such distortion. Thus, several states with the same energy exist that are separated by certain energy barrier. At some temperature thermal motion of the ions becomes strong enough to pass through this barrier. At such temperature the phase will spend equal time at all possible configurations. Observed structure will be average of all these configurations. For such average opposite distortions will negate each other and cubic symmetry will be observed.

Temperature is not the only parameter which may influence the phase which may be observed by XRD. Of course also pressure will have an influence, but also the quality of the crstal (or crystallite). The smaller a crystallite is, the more defects inside a crystallite exist, the more the crystal will tend to show a higher symmetry. Eample: A single crystal of high quality of BaTiO3 will switch from the tetragonal to the cubic state about at 126°C. A polycrystalline sample will show the same transition about at 120°C and nanocrystalline powder with a crystallite size below 20nm will show the cubic structure even at room temperature.

The thermal vibration (high temperature) and imperfect crystals will cause a decrease of the distortion or even prevent prom forming the structure of lower symmetry.

This is valid if XRD is used for determination of the structure. Using Raman spectroscopy the influence of the local environment instead of an average structure will be seen and things may look a little bit different.

Hello everyone,

As a biologist, I find these issues and discussions interesting. I wonder whether these may also apply to biological systems, e.g. muscle. Active muscle force increases with temperature and could this be due to structural changes discussed here????

KW

May be due to conformational changes in the protein molecules. But I cannot say surely. A biochemist can answer it. So, dear Ranatunga, please create a separate question in RG so that people who work in Biochemistry or related field can see and answer you. 

The stability and the distortion of the perovskite structure depend on the ratio of the (M-X) distance to the (A-X) distance, called the Goldschmidt tolerance factor (t) for AMX3 structure,

t = (RM + RX)/ (RA+RX)

where RA, RM, and RX are ionic radii of A, M and X, respectively for AMX3b .

Ideal perovskites have a cubic geometry if t = 1. Perovskites have a pseudocubic or distorted cubic structure for stability when t deviates from 1. Such a distortion will affect the electronic, optical, and dielectric properties of perovskite materials. An additional consideration for perovskite formability is the octahedral factor (μ) , μ = RM/RX. In the case of the alkali metal halide perovskite, formability is determined from the t-μ mapping where the perovskite is stabilized for a tolerance factor ranging between 0.813 and 1.107 and an octahedral factor ranging from 0.442 and 0.895.

MAPbX3 perovskite has 4 phases,  α, β, and γ and the non-perovskite δ-phase. α is the high-temperature phase for temperatures T > 327 K and has a pseudocubical crystal structure. according to Poglitsch et al., the position of the CH3NH3+ cation in CH3NH3PbI3 perovskite is only fixed in the orthorhombic phase at low temperature and could not be fixed at room temperature due to the cubic symmetry requiring eight identical positions for the cations. As a result, the tetrahedral-coordinated C and N atoms exhibit a disordered state inside the eight tetrahedral of the cuboctahedron around the normal A positions (1/2, 1/2, 1/2) of the AMX3 perovskite. Therefore, the CH3NH3PbI3 perovskites have tetragonal β-phase (non-centrosymmetric, space group I4cm with lattice parameters a = 8.855 Å and c = 12.659 Å) [27], with a slightly distorted PbI6 octahedral around the c axis at room temperature. The exact parameters depend on molecular orientation. In the α- and β-phase, the methylammonium cations are disordered. Ferroelectric response like capacitance and non-ohmic behavior, which might be responsible for hysteretic behavior in the current/voltage curves, could be attributed to the reorientation of the methylammonium cations in an external field and the resistance of the inorganic lead-iodide lattice. Due to the tilting of the octahedra during the phase transition from α- to β-phase, the unit cell doubles its length, thus octuples its volume and forms a super-cell.

For temperatures below 162 K, the perovskite undergoes a phase transition to orthorhombic γ -phase (space group Pmc21), where the methylammonium cations are ordered. During the phase transitions from α to β and γ, the octahedra are tilted and deform from the ideal octahedron with respect to cubic phase. For decreasing temperatures, tilting and deformation effects increase. Phase transitions between the α-, β-, and γ -phases occur in the solid phase, while the transition to the non-perovskite δ-phase happens in the presence of solvents.

A reversible phase transition has found from the tetragonal → cubic transition occurring at 57.3 °C upon heating and a cubic → tetragonal transition at 56°C upon cooling from a crystal chemistry study of CH3NH3PbI3.

Dear Alexander B. Missyul’

Thank you for your detailed explanations.

But, can you please explain me in a simple manner, why cubic phase is only the stable phase at high temperatures ?