Exploring the cation dynamics in lead-bromide hybrid perovskites

Density functional theory including a many-body treatment of dispersive forces is used to describe the interplay between structure and electronic properties of two prototypical Br-based hybrid perovskites, namely, ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbBr}}_3$ and $\mathrm{HC}{\left({\mathrm{NH}}_2\right)}_2{\mathrm{PbBr}}_3$. We find that, like for some of their iodine-based counterparts, the molecules' orientation plays a crucial role in determining the shape of both the conduction and valence bands around the band edges. This is mostly evident in the case of ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbBr}}_3$, which is a direct band-gap semiconductor when the ${\mathrm{CH}}_3{\mathrm{NH}}_3$ group is oriented along the (111) direction but turns indirect when the orientation is (100). We have constructed a simple dipole model, with parameters all evaluated from ab initio calculations, to describe the molecules' depolarization dynamics. We find that, once the molecules are initially orientated along a given high-symmetry direction, their room-temperature depolarization depends on the specific material investigated. In particular we find that the ratio between the polarization decay constant of ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbBr}}_3$ and that of $\mathrm{HC}{\left({\mathrm{NH}}_2\right)}_2{\mathrm{PbBr}}_3$ is about 2 at room temperature. With these results at hand we suggest a simple luminescence decay experiment to prove our findings and establish a correlation between optical activity and the molecules' dynamics in these materials.

Figure 1

Optimized structures for the Pm3m phase of (a) ${\mathrm{FAPbBr}}_3$ and (b) ${\mathrm{MAPbBr}}_3$. The dotted lines indicate the hydrogen bonds between H and Br, and the bond lengths are in angstroms. The noncovalent interaction density isosurfaces, shown in green, provide a useful representation of the hydrogen bonds occurring between the H and Br atoms.

Figure 2

Electronic band structure of (a) ${\mathrm{FAPbBr}}_3$ and (b) ${\mathrm{MAPbBr}}_3$ calculated at the PBE level and including spin-orbit interaction. In the case of ${\mathrm{MAPbBr}}_3$ we present results for the two inequivalent orientations of the methylammonium cation. The energy zero is taken at the valence band maxima.

Figure 3

Phonon density of states at the $\mathrm{\Gamma }$ point for ${\mathrm{FAPbBr}}_3$ and ${\mathrm{MAPbBr}}_3$. The arrows indicate the quasirigid rotational modes of the molecules within the inorganic cage. The shaded area represents the projection of the DOS on the organic molecule on a percentage scale relative to the $y$-axis range.

Figure 4

Calculated potential energy surface associated with a molecular rotation between two stable symmetrically equivalent configurations. The results are shown for FA (blue), MA(001) (cyan), and MA(111) (brown). Note that during the rotation between two (111) directions MA passes through (001) for an angle of ${45}^{\circ }$.

Figure 5

Total polarization as a function of the rescaled MC time for ${\mathrm{FAPbBr}}_3$ (cyan) and ${\mathrm{MAPbBr}}_3$ (blue) resulting from a Monte Carlo simulation at 300 K of a $20\times 20\times 20$ periodic lattice in an initial ferroelectric configuration. Actual data (circles) and the fit function (solid) are shown. In the inset, the ratio ${\tau }_{\mathrm{FA}}/{\tau }_{\mathrm{MA}}$ is plotted as a function of the temperature (red) and electric field (orange). The data are averaged over 24 simulations, and error bars are shown.

Figure 6

Dependence of the ${\tau }_{\mathrm{FA}}/{\tau }_{\mathrm{MA}}$ ratio on some of the model parameters. We present data for a rescaled anisotropy barrier $K^{\prime }$ and for different permittivity ${\epsilon }_{\mathrm{r}}$. Note that in both cases the ratio does not change significantly and remains in between 2 and 3. The data are averaged over 24 simulations, and error bars are shown. In both cases simulations are conducted at room temperature.

Figure 7

Dipole distribution over a cross section of the simulation cell after a MC equilibration time of $5\times {10}^5$ MC steps for (a) MA and (b) FA. In the picture the angle in the $x\text{−}y$ plane is represented by the direction of the arrow, while the one along $z$ is represented by the color code.