Many-electron multiplet effects in the spectra of3dimpurities in heteropolar semiconductors

The excitation energies of impurities in semiconductors, as well as their donor and acceptor ionization energies, represent a combination of one-electron and many-electron multiplet effects, where the latter contribution becomes increasingly significant as localized states are formed. Analysis of the absorption and ionization data for $3d$ impurities is often obscured by the inability of contemporary multiplet theories (e.g., the Tanabe-Sugano approach) to separate these two contributions and by the inadequacy of mean-field, one-electron theories that neglect multiplet effects altogether. We present a novel theory of the multiplet structure of localized impurities in semiconductors that circumvents the major shortcomings of the classical Tanabe-Sugano approach and at the same time separates many-electron from mean-field effects. Excitation and ionization energies are given as a sum of mean-field (MF) and multiplet corrections (MC): $\Delta E={\Delta E}_{\mathrm{MF}}+{\Delta E}_{\mathrm{MC}}$. We determine ${\Delta E}_{\mathrm{MC}}$ from the analysis of the experimental data. This provides a way to compare experimentally deduced mean-field excitation and ionization energies ${\Delta E}_{\mathrm{MF}}=\Delta E-{\Delta E}_{\mathrm{MC}}$ with the results of electronic-structure calculations. The three central quantities of the theory—the $e$- and $t_2$- orbital deformation parameters and the effective crystal-field splitting—can be obtained from mean-field electronic-structure calculations, or, alternatively, can be deduced from experiment. In this paper, we analyze the absorption spectra of $3d$ impurities in ZnO, ZnS, ZnSe, and GaP, as well as those of the bulk Mott insulators NiO, CoO, and MnO, in light of the new approach to multiplet effects. These mean-field parameters are shown to display simple chemical regularities with the impurity atomic number and the covalency of the host crystal; they combine, however, to produce interesting non-monotonic trends in the many-electron correction terms ${\Delta E}_{\mathrm{MC}}$. These trends explain many of the hitherto puzzling discrepancies between one-electron (${\Delta E}_{\mathrm{MF}}$) theory and experiment ($\Delta E$). This approach unravels the chemical trends underlying the excitation and donor or acceptor spectra, provides predictions for unobserved excitations and donor or acceptor energies, and distinguishes the regime where one-electron theory is applicable (${\Delta E}_{\mathrm{MC}}$ small) from the region where it is not (${\Delta E}_{\mathrm{MC}}\sim \Delta E$).