The nature of free-carrier transport in organometal halide perovskites

Organometal halide perovskites are attracting great attention as promising material for solar cells because of their high power conversion efficiency. The high performance has been attributed to the existence of free charge carriers and their large diffusion lengths, but the nature of carrier transport at the atomistic level remains elusive. Here, nonadiabatic quantum molecular dynamics simulations elucidate the mechanisms underlying the excellent free-carrier transport in CH3NH3PbI3. Pb and I sublattices act as disjunct pathways for rapid and balanced transport of photoexcited electrons and holes, respectively, while minimizing efficiency-degrading charge recombination. On the other hand, CH3NH3 sublattice quickly screens out electrostatic electron-hole attraction to generate free carriers within 1 ps. Together this nano-architecture lets photoexcited electrons and holes dissociate instantaneously and travel far away to be harvested before dissipated as heat. This work provides much needed structure-property relationships and time-resolved information that potentially lead to rational design of efficient solar cells.

atoms, we have implemented a series of techniques for efficiently calculating long-range exact exchange correction and excited-state forces 22,23 . Due to the use of excited-state forces, photoexcitation also modifies ground-state electronic structures. Details of our QMD and NAQMD simulation software are described in Ref. 23.

Bond-overlap population analysis:
To quantify the change in the bonding properties of atoms, we used bond-overlap population analysis 24 by expanding the electronic wave functions in an atomic-orbital basis set 25 . Based on the formulation generalized to the PAW method 26 , we obtained the gross population Z i (t) for the i th atom and the bond-overlap population O ij (t) for a pair of i th and j th atoms as a function of time t. From Z i (t), we estimate the charge of atoms, and O ij (t) gives a semi-quantitative estimate of the strength of covalent bonding between atoms. As the atomic-basis orbitals, we used numerical pseudo-atomic orbitals, which were obtained for a chosen atomic energy so that the first node occurs at the desired cutoff radius 27 . To increase the efficiency of the expansion, the numerical basis orbitals were augmented with the split-valence method 28 . The resulting charge spillage, which estimates the error in the expansion, was only 0.15 %, indicating the high quality of the basis orbitals. Figure S1 shows the characters of photoexcited electron and hole charge densities during NAQMD simulation, which are projected onto pseudoatomic orbitals of different angular momenta (i.e., s, p, and d) centered around different atoms. Finite-size effects: In the 2×2×2 system containing 96 atoms, time-averaged electron and hole participation numbers are 7.4 ± 3.1 and 9.8 ± 3.9, respectively. Namely, even in the smaller 2×2×2 system, a charge carrier occupies only 8~10 % of the total volume. The electron and hole wave functions are thus not percolating through the system and act as well defined wave packets, likely due to thermal disorder 29 . In order to quantify the effect of periodic boundary conditions (PBC), we have performed a PBC-compatible calculation of wave function's center of mass (COM). For the x direction, for example, it is given by where L is the simulation box length in the x direction and ψ(r) is the wave function. Figure S2 compares the hole mean square displacement (MSD) in the 2×2×2 system computed using Eq. (S1) with the original MSD calculation in the main text. Both calculations agree within 14% of the error bar of the new MSD calculation, demonstrating that finite-size effects are negligible. Figure S2. Hole mean square displacement in the 2×2×2 system. The original calculation (blue) is compared with that based on Eq. (S1) (magenta).

Radiative recombination time:
We calculated the spontaneous-emission contribution to the radiative recombination time from the oscillator strength as in Refs. 30,31, where we used the refractive index of 2.61 32 .

Electric polarizability:
We computed the dielectric constant ε using the fluctuation-dissipation theorem 33,34 : where k B is the Boltzmann constant, T is temperature, V is the volume of the simulation box, and M is the total dipole moment. 〈〉 denotes time average.

Movies of Carrier Transport.
QuickTime movie, S1.mov, animates isosurfaces of the quasielectron (red) and quasihole (blue) charge densities in a 3×3×3 unit-cell NAQMD simulation, where the threshold charge density is 1 ×10 -4 a.u. -3 . In the movie, H, C, N, I and Pb atoms are shown as white, cyan, blue, green and brown spheres, respectively. QuickTime movie, S2.mov, shows quasielectron and quasihole center-of-mass positions as yellow and black spheres, respectively, as well as the atoms occupied by quasielectron and quasihole charge densities, respectively, as red and blue transparent spheres.