First principles investigation on how site preference and entropy affect the stability of (Eu_xM_1-x)₂Ge₂Pb (M = Ca, Sr, Ba) polar intermetallics

Density functional theory calculations have been carried out to analyze the factors contributing to the stabilities of a set of recently synthesized quaternary polar intermetallic compounds, (Eu_xM_1-x)₂Ge₂Pb with M = Ca, Sr, and Ba. Experiments showed that these preferentially crystallized with Pbam (M = Ca) or Cmmm (M = Sr, Ba) symmetry. We systematically explored how the electronic energies of these structures depended on how they were "colored" by the europium/M ions for a wide composition range. It was found that whereas there was very little site preference in the Cmmm structure, the "B" site in the Pbam structure strongly preferred smaller cations. The configurational entropy was also found to play a role in determining which structures might be preferred. However, the experimentally obtained product ratios could only be fully rationalized by the Gibbs free energies of structures containing M:Eu ratios resembling those that were synthesized experimentally. Our results highlight the importance of calculating vibrational contributions to the entropy for realistic structure models (in terms of coloring and composition) to explain product ratios for syntheses carried out at high temperatures.