Low-temperature structure of ${\xi }^{\prime }$-Al-Pd-Mn optimized by ab initio methods

We have studied and resolved occupancy correlations in the existing average structure model of the complex metallic alloy ${\xi }^{\prime }$-Al-Pd-Mn [Boudard et al., Philos. Mag. A 74, 939 (1996)], which has approximately 320 atoms in the unit cell and many fractionally occupied sites. Model variants were constructed systematically in a tiling-decoration approach and subjected to simulated annealing by use of both density functional theory and molecular dynamics with empirical potentials. To obtain a measure for thermodynamic stability, we reproduce the Al-Pd-Mn phase diagram at $T=0$ K, and derive an enthalpy of formation for each structure. Our optimal structure resolves a cloud of fractionally occupied sites in pseudo-Mackay clusters. In particular, we demonstrate the presence of rotational degrees of freedom of an Al${}_9$ inner shell, which is caged within two icosahedrally symmetric outer shells Al${}_{30}$ and Pd${}_{12}$. Outside these clusters, the chemical ordering on a chain of three nearby sites surprisingly breaks the inversion symmetry of the surrounding structure, and couples to an Al/vacancy site nearby. Our refined tiling-decoration model applies to any structure within the $\varepsilon$-phases family, including the metastable decagonal quasicrystalline phase.

Figure 1

Atomic clusters in ${\xi }^{\prime }$. (a) PMI cluster consisting of a centered Mn atom, a symmetry-broken inner Al shell, a Pd icosahedron, and an Al icosidodecahedron. (b) LBPP cluster composed of two Mn-centered Al icosahedra which are capped within a Pd pentagonal prism, and an Al icosidodecahedron-type shell. Atomic coordinates and labeling is taken from Ref. 4. Note that in these pictures all atomic sites are fully occupied.

Figure 2

Alignment of the atomic clusters in ${\xi }^{\prime }$. (a) Projection of the structure along the stacking axis of the PMI clusters. Black lines connect adjacent PMI-cluster columns. Dashed rectangle outlines the unit cell, dotted circles indicate the positions of the LBPP clusters. (b) Side view of one flattened hexagon. For the sake of simplicity, only one PMI column is shown. The Pd${}_{10}$ pentagonal prisms are omitted from the LBPP clusters in this picture. The “free” Al atoms not covered by any cluster are encircled by black-dashed lines.

Figure 3

Top: HBS tiling models for $\xi$ and ${\xi }^{\prime }$. Red (thick) lines connect adjacent PMI-cluster columns. Unit cells are highlighted with dash-dotted lines. Bottom: Tiling-decoration rule for the boat tiles reproducing the Boudard model. Larger circles indicate atoms sitting in lower layers.

Figure 4

The Al-Pd-Mn energy-phase diagram in the Al-rich corner at $T=0$ K. Circles label known stable binary phases, diamonds indicate known metastable phases, squares are either unreported or unknown structures, and triangles correspond to high-pressure phases. Tie lines connect low-enthalpy phases, constituting the convex hull vertices. Enthalpies $\Delta H$ and energy differences $\Delta E$ to the convex hull are given in parentheses in meV/atom. The red spot outlines the approximate compositions of the $\varepsilon$ phases.

Figure 5

PMI clusters in $\xi$ after relaxation. The vertical axis corresponds to the PMI stacking axis. The outer Al${}_{30}$ shells are shown only for the 4$\times$Al${}_8$ occupation. The triangles of the trigonal Al${}_9$ shell and the antiprisms of the Al${}_{10}$ shells are shaded in blue to outline their threefold and twofold axes, respectively.

Figure 6

(a)–(c) Icosahedra of the LBPP clusters with different chemical orderings. The structure in (a) is unstable by 0.2 meV/atom. The structures in (b) and (c) are stable with respect to all competing phases. All three structures contain Al${}_9$ inner shells in their PMI clusters. (d) Alignment of the LBPP cluster from (b) along the PMI stacking axis. Only atoms of the flat and adjacent puckered layers are displayed in the right picture. Larger circles indicate atoms sitting in lower layers. PMI clusters are outlined with dashed circles. The Al decagon located on the flat layer is outlined with a dash-dotted circle. The unoccupied sites in (b) and (c) correspond to Al(21) in Ref. 4. Vac means vacancy.

Figure 7

Comparison of the electronic density of states (DOS) of $\xi$ with different inner PMI shells. The structure with Al${}_9$ inner shells is stable with respect to the competing phases, whereas the structures with Al${}_8$ and Al${}_{10}$ shells are unstable by 28.0 and 4.7 meV/atom, respectively.

Figure 8

Density plot of the atoms in the $\xi$ phase at 1200 K, time averaged after 50 ps. The density plot is projected along the PMI stacking axis. Dotted lines outline the unit cell, dashed circles outline the inner PMI shells. The upper part shows a single PMI cluster and its inner Al${}_9$ shell: only atoms within the dashed lines are shown in the density plot. For the sake of clarity, one Al${}_9$ shell is superimposed in the density plot.

Figure 9

Snapshots of an inner PMI shell at different times after relaxation. The vertical fivefold axis is the stacking axis of the PMI clusters. The triangles of the trigonal prisms are highlighted in green to outline their threefold axis.

Figure 10

Optimized Al${}_3$Pd structure. (a) Side view of PMI clusters with central Pd atoms, inner Al${}_9$ and outer Pd${}_{11}$ shells. (b) Top view along the stacking axis of PMI clusters. Only atoms on the flat and its adjacent puckered layers are shown. PMIs are encircled with black-dotted lines. Color coding same as in previous figures.

Figure 11

HBS-tiling models for the $H$, $H^{\prime }$, and ${\xi }_1^{\prime }$ phases. The skinny rhombi are outlined to illustrate the different tile-tile interactions existing in each phase. See text for more details.

Figure 12

Optimized structure model for ${\xi }_1^{\prime }$ with 398 atoms in the primitive unit cell. (a) Projection of the structure along the PMI stacking axis. Dashed circled outline the locations of the LBPP clusters. (b) Side view of one nonagon tile. The Pd${}_{10}$ pentagonal prisms and the PMI-cluster columns are omitted for the sake of simplicity. The nonagon contains five LBPP clusters per unit-cell height.

Figure 13

Low-temperature energy-phase diagrams of the Al-Pd-Mn system in the Al-rich corner with our optimized structures. The structures are labeled according to Tables and . Left: The two ${\xi }^{\prime }$ phases are stable down to $T=0$ K with respect to all competing phases. The recently discovered Al${}_{16}$Pd${}_4$Mn phase is not included in the stability evaluation. Right: If we include the Al${}_{16}$Pd${}_4$Mn phase in our stability evaluations, both ${\xi }^{\prime }$ phases lie slightly above the convex hull.