Regulatory tolerance and octahedral factors by using vacancy in APbI3 perovskites

The Goldschmidt tolerance factor and the octahedral factor are widely used to predict stable crystal structures of perovskite materials. A t value between 0.8 and 1.0, and a u value larger than 0.41 usually result in stable perovskite structures. In this study, we set the effective radii of ions of the vacancy at both Pb sites and I sites as zero, decreased the weighted average effective ion radius, and accurately controlled the Goldschmidt tolerance factor and octahedral factor by the content of Pb-sites or I-sites vacancy. Considered by ionization, the results suggest the potential existence of the undiscovered perovskites as APb(l-x)I(3) with the single cation vacancies, CsPbI3-y with y = 0-1.00 with the single anions vacancies, and double vacancies at both the Pb-sites and I-sites perovskites APb(1-z)X(3-2z) (A = Na, K, Rb, and Cs). Finally, a simulation study was performed based on first principle, and the results indicated the structural stability and the effect of the vacancy on the physical properties.