Dissolution of rare-earth clusters in ${\mathrm{SiO}}_2$ by Al codoping: A microscopic model

A microscopic model for the incorporation of ${\mathrm{Er}}_2{\mathrm{O}}_3$ units in silica codoped with ${\mathrm{Al}}_2{\mathrm{O}}_3$ is presented. The model assumes that Er clustering is counteracted by the formation of Er-Al complexes in which each Er ion provides valence compensation for three substitutional Al ions. These complexes are investigated by theoretical calculations within the framework of density functional theory. Bond lengths and coordination numbers for Er and Al are in good agreement with results from extended x-ray absorption fine structure spectroscopy. The total energy of the Er-Al complexes is slightly higher than that of the phase-separated state, but thermodynamic arguments show that they are favored by entropy considerations and will prevail for sufficiently high values of the Al/Er ratio and a fictitious temperature parameter controlling the entropy contribution to the free energy of the glass. An analysis of the Kohn-Sham eigenvalue spectrum and the electrostatic potential around the dissolved Er ion provides only limited support for the approximations commonly made in the discussion of ligand-field effects.