Multi-scale model for disordered hybrid perovskites: the concept of organic cation pair modes

We have studied the properties of the prototype hybrid organic-inorganic perovskite $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ using relativistic density functional theory (DFT). For our analysis we introduce the concept of $\text{CH}_3^{}\text{NH}_3^+$ "pair modes", that is, characteristic relative orientations of two neighboring $\text{CH}_3^{}\text{NH}_3^+$ cations. In our previous work [Phys. Rev. B \textbf{94}, 045201 (2016)] we identified two preferential orientations that a single $\text{CH}_3^{}\text{NH}_3^+$ cation adopts in a unit cell. The total number of relevant pairs can be reduced from the resulting 196 combinations to only 25 by applying symmetry operations. DFT results of several $2\!\times\!2\!\times\!2$ supercell models reveal the dependence of the total energy, band gap and band structure on the distribution of $\text{CH}_3^{}\text{NH}_3^+$ cations and the pair modes. We have then analyzed the pair-mode distribution of a series of $4\!\times\!4\!\times\!4$ supercell models with disordered $\text{CH}_3^{}\text{NH}_3^+$ cations. Our results show that diagonally-oriented $\text{CH}_3^{}\text{NH}_3^+$ cations are rare in optimized $\text{CH}_3^{}\text{NH}_3^{}\text{PbI}_3^{}$ supercell structures. In the prevailing pair modes, the $\text{C--N}$ bonds of the two neighboring $\text{CH}_3^{}\text{NH}_3^+$ cations are aligned approximately vertically. Furthermore, we fit the coefficients of a pair-mode expansion to our supercell DFT reference structures. The pair-mode model can then be used to quickly estimate the energies of disordered perovskite structures. Our pair-mode concept provides combined atomistic-statistical insight into disordered structures in bulk hybrid perovskite materials.