Assessing Density Functionals Using Many Body Theory for Hybrid Perovskites

Which density functional is the “best” for structure simulations of a particular material? A concise, first principles, approach to answer this question is presented. The random phase approximation (RPA)—an accurate many body theory—is used to evaluate various density functionals. To demonstrate and verify the method, we apply it to the hybrid perovskite ${\mathrm{MAPbI}}_3$, a promising new solar cell material. The evaluation is done by first creating finite temperature ensembles for small supercells using RPA molecular dynamics, and then evaluating the variance between the RPA and various approximate density functionals for these ensembles. We find that, contrary to recent suggestions, van der Waals functionals do not improve the description of the material, whereas hybrid functionals and the strongly constrained appropriately normed (SCAN) density functional yield very good agreement with the RPA. Finally, our study shows that in the room temperature tetragonal phase of ${\mathrm{MAPbI}}_3$, the molecules are preferentially parallel to the shorter lattice vectors but reorientation on ps time scales is still possible.

Figure 1

Polar distribution of the orientation $\{\varphi ,\theta \}$ of the (a) molecules and (b) PbI octahedra (represented by C-N and I-I connecting vectors, respectively) in a $2\times 2\times 2$ supercell as function of temperature and the XC functional. For the molecules, cubic symmetry is applied to downfold the full polar distribution into a single octant. The molecular relaxation times are shown (in ps) by the white numbers. The legends below illustrate the orientations (a) of the MA molecules in the cage (principle axes, green; face diagonal, red; and room diagonal, blue) and (b) of the I-I connecting vectors (note the different values for the range of the angles).

Figure 2

(a) Snapshot of the RPA-MD simulation of ${\mathrm{MAPbI}}_3$, the $\sqrt{2}\times \sqrt{2}$ cell is indicated by the $a$ and $b$ lattice vectors. The histogram presents the energy difference between the RPA and selected density functionals for the RPA ensemble. In order to center the histograms the mean differences have been subtracted. The $x$ axis shows the energy difference (in eV), whereas the $y$ axis indicates the probability of finding a certain energy difference. (b) Same as in (a) but for the SCAN ensemble buildup with the larger $2\times 2\times 2$ cell. Additionally, the difference between the PBE-MBD and PBE energies [$\mathrm{MBD}-\mathrm{PBE}$] is shown in (a).

Figure 3

(a) Stress difference in the tetragonal system as a function of $c/a$ ratio and temperature. (b) Polar distributions of the MA molecules and PbI octahedra orientations obtained with $c/a=1.01$. Only I-I connecting vectors in the $ab$-plane are included. Fraction of the molecules aligned along the $a$, $b$, and $c$ axes are indicated by white numbers.