Energy levels and correlation crystal-field effects in ${\mathrm{Er}}^{3+}$-doped garnets

The crystal-field energy-level structures of three different ${\mathrm{Er}}^{3+}$-doped garnet systems are analyzed and compared in this study. The garnet hosts are ${\mathrm{Y}}_3$${\mathrm{Al}}_5$${\mathrm{O}}_{12}$ (YAG), ${\mathrm{Y}}_3$${\mathrm{Sc}}_2$${\mathrm{Al}}_3$${\mathrm{O}}_{12}$ (YSAG) doped with ${\mathrm{Tm}}^{3+}$ as a sensitizer ion, and ${\mathrm{Y}}_3$${\mathrm{Sc}}_2$${\mathrm{Ga}}_3$${\mathrm{O}}_{12}$ (YSGG) doped with ${\mathrm{Cr}}^{3+}$ as a sensitizer ion. The focus is on energy levels assigned to ${\mathrm{Er}}^{3+}$ ions substituted for ${\mathrm{Y}}^{3+}$ at dodecahedral (${\mathit{D}}_2$ symmetry) sites in the cubic garnet lattices. Analyses are carried out on experimental energy-level data that span up to 29 different ${}_{}{}^{2\mathit{S}+1\mathrm{}}{}_{}{}^{}$${\mathit{L}}_{\mathit{J}}$ multiplet manifolds (between 0 and 44 000 ${\mathrm{cm}}^{\mathrm{-}1}$) of the ${\mathrm{Er}}^{3+}$ 4${\mathit{f}}^{11}$ electronic configuration. These data include the locations of 117 crystal-field levels of ${\mathrm{Er}}^{3+}$ in YAG, 109 levels of ${\mathrm{Er}}^{3+}$ in YSAG, and 92 levels of ${\mathrm{Er}}^{3+}$ in YSGG.

The energy-level analyses are based on the use of a parametrized model Hamiltonian for the 4${\mathit{f}}^{11}$ electronic configuration of ${\mathrm{Er}}^{3+}$ in a crystal field of ${\mathit{D}}_2$ symmetry. The model Hamiltonian includes both atomic (‘‘free-ion’’) and crystal-field interactions, parametrized to fit calculated eigenvalues to experimentally observed energies. The crystal-field part of the Hamiltonian is defined to include the standard one-electron interaction operators, as well as additional operators that provide a partial, phenomenological consideration of electron-correlation effects in the 4f-electron–crystal-field interactions. The latter, correlation crystal-field (CCF) interactions, are introduced to address crystal-field splittings within several J-multiplet manifolds that are poorly represented by one-electron crystal-field interaction models. Inclusion of CCF terms in the model Hamiltonian leads to dramatic improvement in the fits between calculated and observed crystal-field splittings within the problematic multiplet manifolds. All of the energy-level analyses reported in this study were carried out within commensurate parametrization schemes, and the Hamiltonian parameters derived from these analyses provide a suitable basis for comparing the 4f-electron–crystal-field interaction properties of ${\mathrm{Er}}^{3+}$ in YAG, YSAG, and YSGG. These analyses are based entirely on experimental data that specify the locations of energy levels, but do not provide any explicit information about the angular momentum (${\mathit{JM}}_{\mathit{J}}$) compositions of the crystal-field wave functions.