FIG. 2.

(Left) The harmonic phonon dispersion for ${\text{CH}}_3{\text{NH}}_3{\text{PbI}}_3$ from a “pseudo-cubic” structure. The imaginary frequencies of acoustic modes at the M ($q=\frac{1}{2},\frac{1}{2},0$) and R ($q=\frac{1}{2},\frac{1}{2},\frac{1}{2}$) Brillouin zone boundary points correspond to an instability expressible in a supercell as alternate tilting of the octahedra. (Right) The imaginary acoustic mode at the R point in a $2\times 2\times 2$ supercell expansion shows a double-well potential in the DFT internal energy. The saddle point corresponds to a $1\times 1\times 1$ cubic structure, while the two local minima correspond to a distorted structure of lower symmetry. The energy barrier is small enough to allow both minima that can be accessed at room temperature, so the system is expected to exhibit dynamic disorder rather than static disorder. A similar behavior is found at the M point. The figure is adapted with permission from Refs. 32 and 33. The underlying phonon data are available from https://github.com/WMD-group/Phonons.