Temperature-composition phase diagrams of Gd-doped EuO and EuS

We have computed the temperature-phase diagram of Eu$_{1-x}$Gd$_x$O alloys by combining density functional theory in the generalized-gradient approximation with Hubbard U correction on f-orbitals with the regular cluster expansion and Monte-Carlo approach. The cluster expansion fit has been performed with varying numbers of distinct cluster types until the formal cross-validation score is minimized. Our results indicate that (i) pair interactions are relatively stronger than other cluster types, (ii) the pair terms decay rapidly with distance up to 10 {\AA}, (iii) the pair terms are attractive for direct interactions between cations and repulsive for indirect interactions through anions, and (iv) the calculated convex hull is asymmetric about x=0.5, displaying more deep ground states in Eu-rich regions than in Gd-rich regions. The asymmetry of the convex hull may imply relative instability of Gd-rich compounds, as was shown by previously-reported experimental difficulties to make Gd-rich compounds. A comparison with a similar binary system - sulfur replacing oxygen - is made, showing that both oxides and sulphides are dominated by deformation interaction. The sulphides have a marginal tendency to phase-separate into pure compounds at low temperatures, whereas the oxides tend to order.