Lattice stability of aluminum-rare earth binary systems: A first-principles approach

The thermodynamics of over 330 compounds in 15 Al-RE ($\mathrm{RE}=\text{rare}$ earth elements) binary systems is studied via first-principles density-functional theory at low-temperature limit. The calculated phase stabilities at $T=0\mathrm{K}$ are in very good agreement with experimentally reported ones for the majority of the systems. For example, we show the ${\mathrm{Al}}_2\mathrm{RE}.\mathrm{cF}24$ structure is the most stable compound phase in each binary and it indeed has the highest (congruent) melting point in each system. In some other cases, we obtain results previously unknown experimentally. For example, we suggest that the structure of the observed compound $\mathrm{Al}{\mathrm{Tm}}_2$ is isostructural with ${\mathrm{Co}}_2\mathrm{Si}.\mathrm{oP}12$ (prototype ${\mathrm{Co}}_2\mathrm{Si}$, Pearson symbol oP12), we confirm the stability of AlEu.oP20 rather than AlEu.oP18 by examining the energetics of vacancy substitution, and we predict the unreported Al-Pm phase diagram. Relative accuracy of different potentials and calculational details are addressed. This study predicts that the Al-RE phase diagrams evolve systematically across the entire RE series, interrupted by anomalies at elements Ce and especially Eu and Yb. Trends in lattice stability across the RE series are explained on the basis of interatomic bonding and strain. This study demonstrates that first-principles calculations can be employed to (1) further examine and improve the experimentally established binary-alloy phase diagrams, and (2) provide accurate enthalpy data for stable and hypothetical compounds and structures.

Figure 1

(a) The enthalpy of various (stable and hypothetical) lattice structures of RE elements in bcc (cI2), hcp (hP2), dhcp (hP4), and hR3 with respect to fcc (cF4) lattice at the ground state at $0\mathrm{K}$ calculated in this study. (b) The calculated atomic volume of RE elements in hcp and dhcp structures. The experimental data are the volume of the stable structure at $1\mathrm{atm}$ and $298\mathrm{K}$, calculated based on the lattice parameters presented in Ref. 20.

Figure 2

(Color online) The convex hull plots of the enthalpies of formation of Al-early RE (i.e., La, Ce, Pr, Nd, Pm, Sm, and Eu) binary systems calculated at $0\mathrm{K}$. See the text for details about the choice of potentials. The plotting symbol notations are (black or blue) heavy circles for known stable binary phases, (red or blue) light circles for known high-temperature phases, (blue) triangles for known high-pressure phases, and (red, blue, or black) squares for imperfectly known, unknown, or hypothetical structures. Tie-lines run along convex hull edges, joining low enthalpy structures at the vertices of the convex hull.

Figure 3

(Color online) The convex hull plots of the enthalpies of formation of Al-late RE (i.e., Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) binary systems calculated at $0\mathrm{K}$. Plotting symbols are as in Fig. 2.

Figure 4

The trend of the enthalpies of formation of several groups of competing compounds across the Periodic Table: (a) ${\mathrm{Al}}_4\mathrm{RE}$ family, (b) ${\mathrm{Al}}_3\mathrm{RE}$ family, (c) ${\mathrm{Al}}_2\mathrm{RE}$ family, (d) AlRE family, (e) ${\mathrm{Al}}_2{\mathrm{RE}}_3$ family, (f) $\mathrm{Al}{\mathrm{RE}}_2$ family, and (g) $\mathrm{Al}{\mathrm{RE}}_3$ family. The volume per atom plot is shown in (h) for certain selected compounds.

Figure 5

(a) Comparison of the lattice parameter of ${\mathrm{Al}}_2\mathrm{RE}.\mathrm{cF}24$ between experiments and the present FP calculations. (b) The melting point of RE and ${\mathrm{Al}}_2\mathrm{RE}.\mathrm{cF}24$.

Figure 6

(Color online) (a) The atomic structure of ${\mathrm{Al}}_{11}{\mathrm{La}}_3.\mathrm{oI}28$. The viewing direction is indicated on top of each figure together with the height of repeating unit cell along that direction. The size of atoms indicates their height in the structure along the viewing direction. The caption is the same for Figs. 6, 7, 8, 9, 10. (b) Comparison in the pair distribution of ${\mathrm{Al}}_{11}{\mathrm{La}}_3.\mathrm{oI}28$ vs ${\mathrm{Al}}_{11}{\mathrm{Lu}}_3.\mathrm{oI}28$.

Figure 7

(Color online) (a) The atomic structure of ${\mathrm{Al}}_3\mathrm{La}.\mathrm{hP}8$. (b) Comparison in the pair distribution of ${\mathrm{Al}}_3\mathrm{La}.\mathrm{hP}8$ vs ${\mathrm{Al}}_3\mathrm{Lu}.\mathrm{hP}8$.

Figure 8

(Color online) (a) The atomic structure of ${\mathrm{Al}}_2\mathrm{La}.\mathrm{cF}24$. (b) Comparison in the pair distribution of ${\mathrm{Al}}_2\mathrm{La}.\mathrm{cF}24$ vs ${\mathrm{Al}}_2\mathrm{Lu}.\mathrm{cF}24$.

Figure 9

(Color online) (a) The atomic structure of ${\mathrm{Al}}_2\mathrm{La}.\mathrm{hP}3$. (b) Comparison in the pair distribution of ${\mathrm{Al}}_2\mathrm{La}.\mathrm{hP}3$ vs ${\mathrm{Al}}_2\mathrm{Lu}.\mathrm{hP}3$.

Figure 10

(Color online) (a) The atomic structure of ${\mathrm{Al}}_3\mathrm{La}.\mathrm{cP}4$. (b) Comparison in the pair distribution of ${\mathrm{Al}}_3\mathrm{La}.\mathrm{cP}4$ vs ${\mathrm{Al}}_3\mathrm{Lu}.\mathrm{cP}4$.