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.