Changes in charge density vs changes in formal oxidation states: The case of Sn halide perovskites and their ordered vacancy analogues

Shifting the Fermi energy in solids by doping, defect formation, or gating generally results in changes in the charge density distribution, which reflect the ability of the bonding pattern in solids to adjust to such external perturbations. In the traditional chemistry textbook, such changes are often described by the formal oxidation states (FOS) whereby a single atom type is presumed to absorb the full burden of the perturbation (change in charge) of the whole compound. In the present paper, we analyze the changes in the position-dependence charge density due to shifts of the Fermi energy on a general physical basis, comparing with the view of the FOS picture. We use the halide perovskites CsSnX₃ (X=F, Cl, Br, I) as examples for studying the general principle. When the solar absorber CsSnI₃ (termed 113) loses $50%$ of its Sn atoms, thereby forming the ordered vacancy compound Cs₂SnI₆ (termed 216), the Sn is said in the FOS picture to change from Sn(II) to Sn(IV). To understand the electronic properties of these two groups we studied the 113 and 216 compound pairs CsSnCl₃ and Cs₂SnCl₆, CsSnBr₃ and Cs₂SnBr₆, and CsSnI₃ and Cs₂SnI₆, complementing them by CsSnF₃ and Cs₂SnF₆ in the hypothetical cubic structure for completing the chemical trends. These materials were also synthesized by chemical routes and characterized by x-ray diffraction, ¹¹⁹Sn-M\"ossbauer spectroscopy, and K-edge x-ray absorption spectroscopy. We find that indeed in going from 113 to 216 (equivalent to the introduction of two holes per unit) there is a decrease in the s charge on Sn, in agreement with the FOS picture. However, at the same time, we observe an increase of the p charge via downshift of the otherwise unoccupied p level, an effect that tends to replenish much of the lost s charge. At the end, the change in the charge on the Sn site as a result of adding two holes to the unit cell is rather small. This effect is theoretically explained as a ``self-regulating response'' [Raebiger, Lany, and Zunger, Nature (London) 453, 763 (2008)] whereby the system rehybridizes to minimize the effect of the charge perturbation created by vacancy formation. Rather than having a single preselected atom (here Sn) absorb the full brunt of the perturbation producing two holes, we find that the holes are distributed in a complex pattern throughout the octahedral systems of X₆ ligands, forming hole orbitals with some specific symmetries. This clarifies the relation between FOS and charge transfer that can be applied to a wide variety of materials.