Spin-state ordering and intermediate states in the mixed-valence cobalt oxyborate Co$_3$O$_2$BO$_3$ with spin crossover

Spin-state ordering - a periodic pattern of ions with different spin-state configurations along a crystal lattice - is a rare phenomenon, and its possible interrelation with other electronic degrees of freedom remains little explored. Here we perform a structural investigation of the mixed-valence Co homometallic ludwigite Co$_2^{2+}$Co$^{3+}$O$_2$BO$_3$. A superstructure consistent with a long-range Co$^{3+}$ spin-state ordering is observed between $T_{4}=580$ K and $T_{3}=510$ K. Intermediate states with mesoscopic correlations are detected below $T_{3}$ down to $T_{1}=480$ K with a change of dimensionality at $T_{2}=495$ K. The spin-state correlations are connected to the charge sector as revealed by the abrupt changes in the electrical resistance at $T_1$ and $T_2$. The evolution of the structural parameters below $T_{1}$ indicate that the spin crossover is ignited by a moderate degree of thermally-induced Co$^{2+}$/Co$^{3+}$ charge disorder. Charge and spin-state degrees of freedom can be interrelated in mixed-valence spin-crossover materials, leading to sharp transitions involving intermediate spin-state/charge correlated states at the mesoscale.