Tolerance Factors Revisited: Geometrically Designing the Ideal Environment for Perovskite Dopants

Focusing on case studies relevant to solar energy conversion, the replacement of lead with a layer of germanium or isolated germanium atoms in CsPbCl3 and CsPbBr3, we develop a novel geometric approach to design optimal environments for perovskite dopants. In doing so, we extend the sphere-packing arguments that motivate Goldschmidt tolerance factors beyond bulk ABX3 perovskite compounds to doped and substituted perovskite superstructures. To assess the stability of our proposed superstructures relative to competing phases and structural distortions, we compute total energies and phonon frequencies using density functional theory (DFT)-based methods. We extend these ideas toward the formulation of a generalized tolerance factor that applies to perovskite dopant environments and identify superstructures in which the stability of these dopants is significantly improved relative to the bulk parent compounds. This approach holds promise in uncovering general design rules for stable doped perovskites.