Importance of exchange anisotropy and superexchange for the spin-state transitions in $R$CoO$_3$ ($R=$ rare earth) cobaltates

Spin-state transitions are the hallmark of rare-earth cobaltates. In order to understand them, it is essential to identify all relevant parameters which shift the energy balance between spin states and determine their trends. We find that Δ, the eg-t2g crystal-field splitting, increases by ∼250 meV when increasing pressure to 8 GPa and by about 150 meV when cooling from 1000 K to 5 K. It changes, however, by less than 100 meV when La is substituted with another rare earth. Moreover, the Hund's rule coupling Javg is about the same in systems with very different spin-state transition temperature, like LaCoO3 and EuCoO3. Consequently, in addition to Δ and Javg, the Coulomb-exchange anisotropy δJavg and the superexchange energy gain δESE play a crucial role and are comparable with spin-state-dependent relaxation effects due to covalency. We show that in the LnCoO3 series, with Ln=Y or another rare earth (RE), superexchange progressively stabilizes a low-spin ground state as the Ln3+ ionic radius decreases. We give a simple model to describe spin-state transitions and show that, at low temperature, the formation of isolated high-spin/low-spin pairs is favored, while in the high-temperature phase, the most likely homogeneous state is high spin rather than intermediate spin. An orbital-selective Mott state could be a fingerprint of such a state.