Atomic structure of metal-halide perovskites from first principles: The chicken-and-egg paradox of the organic-inorganic interaction

We have studied the prototype hybrid organic-inorganic perovskite ${\text{CH}}_3{\text{NH}}_3{\text{PbI}}_3$ and its three close relatives, ${\text{CH}}_3{\text{NH}}_3{\text{SnI}}_3,{\text{CH}}_3{\text{NH}}_3{\text{PbCl}}_3$, and ${\text{CsPbI}}_3$, using relativistic density function theory. The long-range van der Waals (vdW) interactions were incorporated into the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional using the Tkatchenko-Scheffler pairwise scheme. Our results reveal that hydrogen bonding, which is well described by the PBE functional, plays a decisive role for the structural parameters of these systems, including the position and orientation of the organic cation as well as the deformation of the inorganic framework. The magnitude of the inorganic-framework deformation depends sensitively on the orientation of the organic cation, and directly influences the stability of the hybrid perovskites. Our results suggest that the organic and the inorganic components complement each other; the low symmetry of the organic cation is the origin of the inorganic-framework deformation, which then aids the overall stabilization of the hybrid perovskite structure. This stabilization is indirectly affected by vdW interactions, which lead to smaller unit-cell volumes than in PBE and therefore modulate the interaction between the organic cation and the inorganic framework. The vdW-induced lattice-constant corrections are system dependent and lead to PBE+vdW lattice constants in good agreement with experiment. Further insight is gained by analyzing the vdW contributions. In all iodide-based hybrid perovskites, the interaction between the organic cation and the iodide anions provides the largest lattice-constant change, followed by iodine-iodine and the organic cation—heavy-metal cation interaction. These corrections follow an almost linear dependence on the lattice constant within the range considered in our study and are therefore approximately additive.

Figure 1

Cubic primitive cell of the perovskite structure used to model the investigated compounds ${\mathrm{ABX}}_3$ in this paper. The monovalent cation ${\mathrm{A}}^{+}$, the divalent cation ${\mathrm{B}}^{2+}$, and the halide anions ${\mathrm{X}}^{-}$ are represented by the green, blue and red spheres, respectively.

Figure 2

Cubic unit cell of ${\text{MAPbI}}_3$ optimized with PBE+vdW: (a) structure a, with the colinearity of $\text{Pb}$ (blue), $\text{C}$ (green), and $\text{N}$ (orange) indicated by the gray dotted line; and (b) structure b, indicating the $\left(020\right)$ plane where the $\text{C}$–$\text{N}$ bond is located in gray. The hydrogen bonds between ${\text{I}}^{-}$ anions (red) and $\text{H}$ atoms (gray) in the ${\text{-NH}}_3^{+}$ group are highlighted by black dashed lines.

Figure 3

Graphical representation of observables defined to characterize the atomic structure of hybrid perovskites: (a) $\mathrm{\Delta }x:\mathrm{\Delta }y:\mathrm{\Delta }z\left(\text{C–N}\right)$, (b) $u_{\text{NC}}$, and (c) $\Delta \left(\text{X}\right)$. The carbon, nitrogen, halide and divalent metal atoms are colored in green, orange, red and blue, respectively. In (a), $\mathrm{\Delta }x,\mathrm{\Delta }y$, and $\mathrm{\Delta }z$ are the projections of the $\text{C}$–$\text{N}$ bond onto the lattice-vector directions $x,y$, and $z$, respectively. In (b), $M$ denotes the unit-cell center, and $\overline{\text{N}M}$ and $\overline{\text{C}M}$ are the distances between it and the carbon and nitrogen atoms, respectively. $u_{\text{NC}}$ is defined by their ratio. In (c), ${\text{X}}^{\prime }$ denotes the Wyckoff position for the ${\text{X}}^{-}$ anion, and $\Delta \left(\text{X}\right)$ is the distance between it and the position of ${\text{X}}^{-}$ in the real system.

Figure 4

PBE+vdW results of $u_{\text{NC}}$ and $\Delta \left(\text{X}\right)$ for ${\text{MAPbI}}_3$-a (black), ${\text{MASnI}}_3$-a (red), and ${\text{MAPbCl}}_3$-a (blue).

Figure 5

PBE+vdW results of $\mathrm{\Delta }x/\mathrm{\Delta }z\left(\text{C–N}\right)$ and the $\Delta \left(\text{X}\right)$ range for ${\text{MAPbI}}_3$-b (black), ${\text{MASnI}}_3$-b (red), and ${\text{MAPbCl}}_3$-b (blue).

Figure 6

DFT results of ${\text{MAPbI}}_3$-a and b in models I, II, and III. Shown are (a) the $\Delta \left(\text{I}\right)$ values (or ranges for b-II and b-III) which define the models, (b) the PBE+vdW total energies, (c) $u_{\text{NC}}$, and (d) $\mathrm{\Delta }x/\mathrm{\Delta }z\left(\text{C–N}\right)$ for structure b. Results calculated at both PBE and PBE+vdW lattice constants ($6.49$ and $6.36\AA$, respectively) are shown. In (c) and (d), PBE and PBE+vdW results are given in dashed and solid lines, respectively.

Figure 7

TS-vdW contributions for different interatomic pairs $\mathrm{\Delta }{E}_{\text{total}}^{\text{TS-vdW}}$ for several perovskites: (a) ${\text{MAPbI}}_3$-a, (b) ${\text{MAPbI}}_3$-b, (c) ${\text{MASnI}}_3$-a, (d) ${\text{MAPbCl}}_3$-a, and (e) ${\text{CsPbI}}_3$ [note the different scale for (e)]. Also shown are the PBE total energy curves vertically shifted so that the minimal energy lies at 0 (black curves, note the different scale of $E_{\text{total}}^{\text{PBE}}$ axes). The lattice constants optimized with PBE+vdW are indicated by the vertical gray dashed lines.