Competition of defect ordering and site disproportionation in strained LaCoO$_{3}$ on SrTiO$_3$(001)

The origin of the $3 \times 1$ reconstruction observed in epitaxial LaCoO$_{3}$ films on SrTiO$_3(001)$ is assessed by using first-principles calculations including a Coulomb repulsion term. We compile a phase diagram as a function of the oxygen pressure, which shows that ($3 \times 1$)-ordered oxygen vacancies (LaCoO$_{2.67}$) are favored under commonly used growth conditions, while stoichiometric films emerge under oxygen-rich conditions. Growth of further reduced LaCoO$_{2.5}$ brownmillerite films is impeded by phase separation. We report two competing ground-state candidates for stoichiometric films: a semimetallic phase with $3 \times 1$ low-spin/intermediate-spin/intermediate-spin magnetic order and a semiconducting phase with intermediate-spin magnetic order. This demonstrates that tensile strain induces ferromagnetism even in the absence of oxygen vacancies. Both phases exhibit an intriguing ($3 \times 1$)-reconstructed octahedral rotation pattern and accordingly modulated La-La distances. In particular, charge and bond disproportionation and concomitant orbital order of the $t_{2g}$ hole emerge at the Co sites that are also observed for unstrained bulk LaCoO$_3$ in the intermediate-spin state and explain structural data obtained by x-ray diffraction at elevated temperature. Site disproportionation drives a metal-to-semiconductor transition that reconciles the intermediate-spin state with the experimentally observed low conductivity during spin-state crossover without Jahn-Teller distortions.