Electronic structure and bonding properties of cobalt oxide in the spinel structure

The spinel cobalt oxide Co${}_{3}$O${}_{4}$ is a magnetic semiconductor containing cobalt ions in Co${}^{2+}$ and Co${}^{3+}$ oxidation states. We have studied the electronic, magnetic, and bonding properties of Co${}_{3}$O${}_{4}$ using density functional theory (DFT) at the Generalized Gradient Approximation (GGA), GGA$+$U, and PBE0 hybrid functional levels. The GGA correctly predicts Co${}_{3}$O${}_{4}$ to be a semiconductor but severely underestimates the band gap. The GGA$+$U band gap (1.96 eV) agrees well with the available experimental value (1.6 eV), whereas the band gap obtained using the PBE0 hybrid functional (3.42 eV) is strongly overestimated. All the employed exchange-correlation functionals predict three unpaired $d$ electrons on the Co${}^{2+}$ ions, in agreement with crystal field theory, but the values of the magnetic moments given by GGA$+$U and PBE0 are in closer agreement with the experiment than the GGA value, indicating a better description of the cobalt localized $d$ states. Bonding properties are studied by means of maximally localized Wannier functions (MLWFs). We find $d$-type MLWFs on the cobalt ions, as well as Wannier functions with the character of ${\mathit{sp}}^{3}d$ bonds between cobalt and oxygen ions. Such hybridized bonding states indicate the presence of a small covalent component in the primarily ionic bonding mechanism of this compound.