We propose a model for the Co charge disproportionation in $\mathrm{L}{\mathrm{a}}_{2-x}\mathrm{C}{\mathrm{a}}_x\mathrm{Co}{\mathrm{O}}_{4\pm \delta }$ ($0.4\le x\le 0.7$) and $\mathrm{L}{\mathrm{a}}_{1.5}\mathrm{S}{\mathrm{r}}_{0.5}\mathrm{Co}{\mathrm{O}}_4$ layered cobaltites based on the photon energy, azimuthal angle, and polarization dependence of the scattered intensity of selected reflections in a resonant x-ray scattering experiment. Tetragonal superlattice $(h/2,h/2,l)$-type reflections were detected at room temperature for all the samples independently from the Ca (or Sr) doping rate in agreement with a checkerboard ordering of two different Co sites. The corresponding average charge disproportion is accounted for by a semiempirical model from which the imaginary part of the resonant atomic scattering factors of each of the two Co atoms is obtained resulting in about $0.5\pm 0.1$ electron. Moreover, no forbidden $(h/4,h/4,l)$-type reflections were observed at room temperature indicating the lack of any local anisotropy ordering in contrast with the behavior found in the related layered manganites. Finally, we found the pattern of small distortions $(\sim 0.05\AA )$ around the two Co sites compatible with the resonant x-ray scattering results. The symmetry of this displacement pattern is consistent with the $A2mm$ (or Ammm) orthorhombic structure of the ordered phase and the structural transition is accounted for by the condensation of two soft modes—${\mathrm{X}}_1{}^{+}\left({\mathrm{B}}_{2\mathrm{u}}\right)$ and ${\mathrm{X}}_1{}^{+}\left({\mathrm{A}}_1\right)$—acting on the oxygen atoms.

• Received 4 December 2017
• Revised 26 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.085111