First-principles study of phase stability of Gd-doped EuO and EuS

Phase diagrams of isoelectronic Eu${}_{1-x}$Gd${}_x$O and Eu${}_{1-x}$Gd${}_x$S quasibinary alloy systems are constructed using first-principles calculations combined with a standard cluster-expansion approach and Monte Carlo simulations. The oxide system has a wide miscibility gap on the Gd-rich side but forms ordered compounds on the Eu-rich side, exhibiting a deep asymmetric convex hull in the formation enthalpy diagram. The sulfide system has no stable compounds. The large difference in the formation enthalpies of the oxide and sulfide compounds is attributed to the contribution of local lattice relaxation, which is sensitive to the anion size. The solubility of Gd in both EuO and EuS is in the range of 10%–20% at room temperature and quickly increases at higher temperatures, indicating that highly doped disordered solid solutions can be produced without the precipitation of secondary phases. We also predict that rocksalt GdO can be stabilized under appropriate experimental conditions.

Figure 1

Electronic band structure of EuO calculated by (a) $\mathrm{GGA}+U$ and (b) QS$\mathit{GW}$, and of ferromagnetic phase GdO, calculated by (c) $\mathrm{GGA}+U$ and (d) QS$\mathit{GW}$. Black (red) curves correspond to the majority (minority) spin bands. Energy is referenced either from the valence band maximum (VBM) or from the Fermi level ($E$${}_F$). In GdO, states near $E_F$ at $\Gamma$ are of Gd $d$ character; the band whose value is $-$1.5 eV at $\Gamma$ is of $s$ character. If correlations were strong, the $d$ band would become narrow and possibly shift relative to the $s$ band. As can be seen, the GGA and QS$\mathit{GW}$ results are very similar for these bands: the GGA and QS$\mathit{GW}$ results differ mainly in the positions of the O $2p$ bands, at around $-$6 eV. That O $2p$ states shift downward relative to results in the GGA (or LDA) seems to be a universal property of oxide insulators.

Figure 2

Density of states (DOS) of FM EuO and EuS, AFM type II GdO, and AFM type II GdS calculated in the $\mathrm{GGA}+U$. For the AFM II phases, the solid line shows the total DOS including both the cation and the anion contributions, whereas the dashed lines show the partial majority- and minority-spin DOSs from the Gd cations. Energy is referenced from the Fermi level $E_F$.

Figure 3

Three different types of ECI parameters as functions of effective radius in units of the nearest-cation-neighbor distance $r_{\mathrm{NN}}$ in the cluster expansions for Eu${}_x$Gd${}_{1-x}$O, denoted by red crosses and Eu${}_x$Gd${}_{1-x}$S, denoted by blue circles.

Figure 4

Formation enthalpy per cation vs composition $x$ for all distinct cation orderings within rocksalt structure for (a) Eu${}_x$Gd${}_{1-x}$O with up to 30 atoms per unit cell and (b) Eu${}_x$Gd${}_{1-x}$S with up to 20 atoms per unit cell. The black open circles are the first-principles inputs, the red crosses are the fitted CE values for the input structures, and the gray dots are the predicted $\Delta H_{\mathrm{CE}}$ for all other structures.

Figure 5

Calculated phase diagrams obtained using Monte Carlo simulations from the cluster expansions for (a) Eu${}_x$Gd${}_{1-x}$O and(b) Eu${}_x$Gd${}_{1-x}$S alloys. The phase labels indicate the ordering within the cation sublattice. In (a), the phases in the region bounded by the dot-dashed line have not been identified. In (b), the solid lines correspond to the paramagnetic high-temperature phases, while the dotted lines show the prediction of the $T=0$ ferromagnetic cluster expansion.

Figure 6

Monte Carlo heating simulation of the $L$1${}_1$-to-disordered phase transition for Eu${}_{0.5}$Gd${}_{0.5}$O. Formation enthalpy and heat capacity are shown by black and red lines, respectively. The inset shows the temperature dependence of the composition $x$.