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 $\mathbf{\text{94}}$, 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 $2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}2$ 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 $4\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}4\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}4$ 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.