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Role of Polar Phonons in the Photo Excited State of Metal Halide Perovskites

The development of high efficiency perovskite solar cells has sparked a multitude of measurements on the optical properties of these materials. For the most studied methylammonium(MA)PbI3 perovskite, a large range (6–55 meV) of exciton binding energies has been reported by various experiments. The existence of excitons at room temperature is unclear. For the MAPbX3 perovskites we report on relativistic Bethe-Salpeter Equation calculations (GW-BSE). This method is capable to directly calculate excitonic properties from first-principles. At low temperatures it predicts exciton binding energies in agreement with the reported ‘large’ values. For MAPbI3, phonon modes present in this frequency range have a negligible contribution to the ionic screening. By calculating the polarization in time from finite temperature molecular dynamics, we show that at room temperature this does not change. We therefore exclude ionic screening as an explanation for the experimentally observed reduction of the exciton binding energy at room temperature and argue in favor of the formation of polarons.

The exciton binding energies presented in the paper have been calculated using 'semi-cubic' unit cells. These structures have been obtained by allowing all atoms to relax (keeping the experimental lattice shape and volume fixed) using damped relaxation algorithms with a 4 × 4 × 4 k-point grid. The bromine and chlorine structures were constructed from the resulting high-symmetry iodine unit cells.
For these structures, in the first step all degrees of freedom were relaxed (including cell shape and volume) at a high energy cut off of 600 eV. Hereafter, a simulated annealing was performed, linearly cooling the structure from 800K to 500K in 50000 steps. Approximately every ∼1000 steps a snapshot was taken from the trajectory. For these structures the cell shape as well as the internal coordinates were relaxed and the lowest global energy structure was determined.
To confirm that the lowest energy structures for the primitive unit cells were correctly determined, a second molecular dynamics run was started from the first lowest energy structure. The same annealing procedure was used with the volume and cell shapes fixed to those of the yet best structures. More than 15 structures were picked from the MD and fully relaxed (including volume and cell shape). In only two cases, an even lower energy structure than in the first cycle was found, however, these two structures were only 15 meV lower in energy than in the first cycle. We are therefore confident that we have determined the lowest energy structures for the primitive unit cells reliably. Furthermore, all structures were carefully checked for instabilities in the vibrational frequencies and no instabilities were found, whereas virtually all initial structures constructed from experimental data alone exhibited such instabilities, even after careful relaxation. The resulting structures have been attached to this document.

Model dielectric function
A local model dielectric function (ε m ) [1] has been used in the BSE calculations to converge the exciton binding energies on dense k-point grids.
A local function makes the screened Coulomb kernel diagonal (G = G ) in the screened Coulomb potential and allows us to calculate the high resolution plots of Fig. 1 (right) in the paper. The parameter ε m comes from DFPT calculations on a shifted 8×8×8 k-point grid and the screening length parameter (λ) was fitted to match the diagonal (G = G ) part of dielectric function from the GW 0 calculations on the shifted 6 × 6 × 6 k-point grid, see Table 1. This approximation is also used in Ref. [2], however in that work a different functional was used. This approximation works very well, in particularly in the low energy part as is illustrated in Fig. 1. The imaginary part of the dielectric function calculated with the 'normal' GW-BSE matches with the one calculated with the model BSE (mBSE) method. Both calculations were performed using a 4 × 4 × 4 k-point grid with the same KS orbital basis and GW 0 quasiparticle energies.  Figure 2: Eigenvalue of the first exciton with e-h interactions (circles) and in the independent particle picture (squares). The exciton binding energy is the difference between these two levels. The inset shows a zoom-in for the levels calculated at dense k-meshes. The green and blue lines are linear fits to the last three data points. (Example: FASnI 3 )

K-point convergence
The calculated exciton binding energies have been carefully converged with respect to the k-point grid density. Coarse meshes result in exciton binding energies up to an order too large. The exciton binding energies presented in the paper are the result of a linear extrapolation to an infinite k-point grid [2] as is illustrated in Figure 2.
All unit cell structures have been added and formatted as Crystallographic Information File (CIF) files. The CIF formatted files can, for instance, be viewed with the freely available atomic structure visualization package VESTA (http://jp-minerals.org/vesta/).