Defect tolerance of lead-halide perovskite (100) surface relative to bulk: band bending, surface states, and characteristics of vacancies

We characterized the formation of vacancies at a surface slab model and contrasted the results with the bulk of lead-halide perovskites using cubic and tetragonal CsPbI$_3$ as representative structures. The defect-free CsI-terminated (100) surface does not trap charge carriers. In the presence of defects (vacancies), the surface is expected to exhibit $p$-type behavior. The formation energy of cesium vacancies $V_\text{Cs}^{-}$ is lower at the surface than in the bulk, while iodine vacancies $V_\text{I}^{+}$ have a similar energy (around 0.25$-$0.4 eV) within the range of chemical potentials compatible with solution processing synthesis conditions. Lead-iodine divacancies ($V_\text{PbI}^{-}$) are expected to dominate over lead-only vacancies at the surfaces. Major surface vacancies create shallow host-like energy states with a small Franck-Condon shift, making them electronically harmless (same as in bulk). The spin-orbit coupling contributes to the defect tolerance of lead-halide perovskite surfaces by causing delocalization of electronic states associated with $n$-type defects and retraction of lowest unoccupied states from the surface due to a mixing of Pb-$p_{x,y,z}$ orbitals. These results explain a high optoelectronic performance of two-dimensional structures, nanoparticles, and polycrystalline thin films of lead-halide perovskites despite the abundance of interfaces in these materials.