Calculated optical properties of Co in ZnO: internal and ionization transitions

Previous luminescence and absorption experiments in Co-doped ZnO revealed two ionization and one intrashell transition of $d(\textrm{Co}^{2+})$ electrons. Those optical properties are analyzed within the generalized gradient approximation to the density functional theory. The two ionization channels involve electron excitations from the two $\textrm{Co}^{2+}$ gap states, the $t_{2\uparrow}$ triplet and the $e_{2\downarrow}$ doublet, to the conduction band. The third possible ionization channel, in which an electron is excited from the valence band to the $\textrm{Co}^{2+}$ level, requires energy in excess of 4~eV, and cannot lead to absorption below the ZnO band gap, contrary to earlier suggestions. We also consider two recombination channels, the direct recombination and a two-step process, in which a photoelectron is captured by $\textrm{Co}^{3+}$ and then recombines via the internal transition. Finally, the observed increase the band gap with the Co concentration is well reproduced by theory. The accurate description of ZnO:Co is achieved after including $+U$ corrections to the relevant orbitals of Zn, O, and Co. The $+U(\textrm{Co})$ value was calculated by the linear response approach, and independently was obtained by fitting the calculated transition energies to the optical data. The respective values, 3.4 and 3.0~eV, agree well. Ionization of Co induces large energy shifts of the gap levels, driven by the varying Coulomb coupling between the $d(\textrm{Co})$ electrons, and by large lattice relaxations around Co ions. In turn, over $\sim 1$~eV changes of $\textrm{Co}^{2+}$ levels induced by the internal transition are mainly caused by the occupation-dependent $U(\textrm{Co})$ corrections.