Caged clusters in Al 11 Ir 4 : Structural transition and insulating phase

Transcription

1 PHYSICAL REVIEW B 88, (2013) Caged clusters in Al 11 Ir 4 : Structural transition and insulating phase Marek Mihalkovič 1 and C. L. Henley 2 1 Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia 2 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York, , USA (Received 26 June 2012; revised manuscript received 1 July 2013; published 2 August 2013) Using pair potentials fitted to an ab initio database, combined with replica-exchange simulated annealing, we show that the complex, quasicrystal-related Al 11 Ir 4 compound contains a nonstandard version of the pseudo-mackay icosahedral cluster, with nonicosahedral inner Al 10 Ir and/or Al 9 Ir clusters that exist in various orientations and account for partial occupancies in the reported structure. We found two different compositions with different orientationally ordered grounds, each doubling the (cubic) unit cell and each reached by a first-order thermal transition. One of these is metallic and the other is insulating. DOI: /PhysRevB PACS number(s): Ks, Br, Lt, Cn I. INTRODUCTION Al 11 Ir 4 is a compound in one of the complex metallic alloy systems that resist ordinary approaches to determining crystallographic structure or phase diagrams because of two impediments to equilibration at low temperatures: (1) the alloy system has numerous competing complex phases of similar composition, and (2) in some of the phases, the intermediatetemperature structures have inherent entropy associated with block rearrangements of tiles or clusters. Despite the measurement of 457 independent reflections, the experimentally refined structure 1 of Al 11 Ir 4 has twice as many sites listed as atoms, many with a fitted occupancy of less than 0.5. In the absence of information on the occupancy correlations, structural data with this extent of partial occupancy is too ambiguous to allow computing electronic properties or total energies. Instead, in this paper we predict the structure, aided by its close relationship to Al-transition-metal (Al-TM) quasicrystals. These are described as networks of identically oriented icosahedral clusters linked along certain symmetry directions, 2 e.g., the Mackay icosahedron, 2 consisting of an empty center plus two concentric shells of full icosahedral symmetry Al 12 + Al 30 Mn 12. These are packed on a bcc lattice (lattice constant Å) to form the crystal structure of α-almnsi, or can be packed in a more general way to form the quasicrystal i-almnsi. It is generally accepted that the stable quasicrystals i-alcufe and i-alpdmn contain a modified cluster called the pseudo-mackay icosahedron (pmi) in which the inner Al 12 icosahedron is replaced by a cluster with fewer atoms and reduced symmetry, 3 the details of which are still unclear. We note that the Ca-Cd quasicrystals, another major class alongside the Al-TM quasicrystals, and complex alloys related to them, also contain icosahedral clusters with an inner shell of lower symmetry: 4 a tetrahedron, in that case. Thus, such inner rotatable clusters may be a common feature of well-ordered quasicrystal systems, and they present tractable examples of the above-mentioned block-rearrangement degrees of freedom. In the rest of this paper, we uncover the correct low-t structure of Al 11 Ir 4 using energy minimization, in several stages. We used both ab initio codes (VASP package 5 ) and empirically fitted pair potentials. 6 In Sec. II, we first determined the possible atom contents of one cluster to be Al 9 Ir or Al 10 Ir, and next determined each cluster s atom arrangement and preferred orientation within the cage of neighboring atoms. The partially occupied sites in the refined structure 1 are explained with an equal mixture of the two kinds of cluster. We went on (Sec. III) to find the collective ordering of orientations and locate the ordering temperature. This is applied to test these phases stability (at T = 0) relative to competing ones in Sec. IV. Section V addresses the electronic properties: the equal-mixture phase is metallic, as seen in experiment, 7 but the composition with Al 10 Ir clusters is predicted to be insulating. In the conclusion (Sec. VI), we relate this structure to other alloys containing the same cluster, and consider how this work can be extended to more detailed studies of the orientational ordering, including quasicrystal phases built from the same clusters. II. STRUCTURAL OPTIMIZATION AT SINGLE-CLUSTER LEVEL The reported crystal structure of Al 11 Ir 4 is a modification of the AuZn 3 structure in the space group Pm 3n. This is a bcc packing of Au icosahedral cages, distorted so as to share faces, and having icosahedral twofold axes aligned with the cubic fourfold axes. Inside each cage cluster is a AuZn 12 icosahedron, also aligned with the cubic fourfold axes but in the other way than its cage. (There are two symmetry-related ways of aligning an icosahedron with cubic axes.) The inner icosahedron sites are roughly aligned with the 12 pseudo-threefold directions of the cage icosahedron that are not along 111 directions. This structure is modified in Al 11 Ir 4 by the substitution Au Ir, Zn Al, and some partial occupancies. The result is a simple cubic crystal in which the body-center cages hold IrAl 12 icosahedra, but the cell-corner cages hold pmi clusters, each having an inner shell of 40 partially occupied sites with combined occupancy 10; these clusters will be the focus of this paper. The second shell surrounding each pmi cluster consists of the the Ir 12 icosahedron as well as 30 Al sites slightly outside that icosahedron s mid-edges, out of which the six sites along cubic twofold axes are only partially occupied. These second-shell Al sites are shared with the Al atoms from adjacent pmi and MI clusters. The key fact to determine by energy optimization is the atom configuration in the uncertain, pmi inner-shell sites /2013/88(6)/064201(10) American Physical Society

2 MAREK MIHALKOVIČ AND C. L. HENLEY PHYSICAL REVIEW B 88, (2013) TABLE I. Fitted parameters for Al-Ir EOPP potentials. C 1 η 1 C 2 η 2 k Al-Al Al-Ir Ir-Ir A. Fitted pair potentials Apart from a few ab initio calculations, our comparisons of alternative structures and our extensive molecular dynamics (MD) simulations used empirical oscillating pair potentials 6 (EOPP) of the form V (r) = C 1 + C 2 cos(k r η 1 r η r + φ ), (1) 2 where r is the distance between a pair of atoms. Therefore, the first prerequisite for our calculations was to generate these potentials valid for Al-Ir in this composition range, using the method of Ref. 6 to fit a database of ab initio forces, calculated with VASP. 5 The database contained snapshots from ab initio molecular dynamics (MD) simulations at high temperatures, specifically Al 11 Ir supercell samples (240 atoms/supercell) at 1700 K and fcc Al samples at T = 300 and 600 K (4 4 4 supercell, 256 atoms). It also contained relaxed T = 0 structures (i.e., vanishing forces on each atom) from various stable and unstable Al-Ir structures. These were as follows: (i) quenches from T = 1700 K of selected snapshots from the MD run of Al 11 Ir 4 in the supercell; (ii) the known binary Al-Ir structures Al 9 Ir 2 (Al 9 Co 2 prototype, Pearson symbol mp22), Al 3 Ir (hp8), AlIr (cp2), Al 5 Co 2 (hp28), and Al 45 Ir 13 (op236); (iii) Al-Ir in the Al 21 Pd 8 and B2 structures. The fitted coefficients we used are listed in Table I. Figure 1 gives a scatter plot demonstrating the goodness of fit and a plot of the potentials themselves. As is typical of Al-transitionmetal (Al-TM) potentials, with TM = Ir in this case: (a) the Al- Al potential has no nearest-neighbor well but only a shoulder, F [ev/a] FITTED F [ev/a] VASP E[eV] Al-Al (pot) Ir-Al (pot) Ir-Ir (pot) R[A] FIG. 1. (Color online) EOPP potentials fitted to ab initio (VASP) force and energy datapoints. Fit to forces (left panel) is shown with fitted forces (vertical axis) plotted against VASP forces (horizontal axis). The fitted pair potentials are shown in the right panel. Parameters are summarized in Table I. (b) the Al-Ir potential has a very deep well at the nearestneighbor distance, (c) the Ir-Ir potential is unfavorable for nearest neighbors but has a deep well at the second-neighbor distance, and (d) all the potentials have relatively strong Friedel oscillations. To capture data for fitting empirical pair potentials, and also in later calculations to evaluate phase stability (Sec. IV), we carried out first-principles total-energy calculations using the plane-wave density functional code VASP (Ref. 5) using the PW91 (Perdew-Wang) generalized gradient approximation, with projector augmented wave potentials, 8 at fixed energy cutoff ev (default for Ir atom). Our final total energies are converged to n K N at , where n K is the number of K points per Brillouin zone, and N at is the number of atoms per unit cell. B. Optimum arrangement and orientation of inner clusters The first question about the inner Al cluster is how many atoms it (optimally) contains. We addressed this directly by constructing initial configurations of the Al 11 Ir 4 unit cell with a chosen number m of inner Al atoms (assuming the periodicity of only one unit cell). The configuration was optimized, first by annealing with molecular dynamics (MD) using the fitted EOPP potentials at fixed volume; this was followed by ab initio relaxation using VASP, 5 optimizing all structural parameters. 9 The atoms within the inner shell rearrange rapidly under MD, whereas the surrounding cage atoms merely vibrate, so the end state approximates the optimum constrained to the chosen content m Al atoms in the cell-corner clusters and 12 Al atoms in the body-center clusters. We conducted two sets of trials. The first set of trials was to try different counts of Al atoms while assuming a periodicity of one unit cell of the framework (1 1 1 supercell in Table II). This identified the m = 8, m = 11, and m = 12 variants as having excessively high energy costs, 10 leaving only the m = 9 and 10 variants, which we call 9-cluster and 10-cluster from here on. In the second set of trials, we annealed in a larger supercell to accommodate possible alternations of the orientations, as well as mixtures of the two kinds of cluster. We found two nearly stable structures: the m = 10 filling ( 10-phase ) and an equal mixture of m = 9 and 10 fillings ( 9.5-phase ). From here on, in studies of the whole structure, we limit ourselves to these two fillings. Table III gives the symmetrized coordinates for both kinds of clusters. The 9-cluster is almost always a square antiprism, with one square enlarged and capped by an additional Al atom [see Fig. 2(a)]. (Equivalently, it could be considered a square pyramid with capping atoms on the three triangular faces.) In TABLE II. Relaxed energies (in mev/atom) as a function of Al per pmi inner cluster. Supercell used was 2 2 2, except for the 10 Al case. Last row is the mean lattice parameter. Inner cluster Al Supercell a cub (Å)

3 CAGED CLUSTERS IN Al 11 Ir 4 : STRUCTURAL... PHYSICAL REVIEW B 88, (2013) TABLE III. Atomic coordinates for a typical 10-cluster (labels C10-1 through C10-7) and a 9-cluster (labels C9-1 through C9-4) each in one of its optimal orientations with respect to the surrounding cage, obtained by symmetrizing coordinates found in the relaxed 9.5-phase [see Sec. III A2; this is structure oa236 in the Supplemental Material (Ref. 17)]. The radius R of each atom from the center is also shown (all in Å). Atoms C-10-1 and C-9-1 are the apex atoms. In the 10-cluster, the pseudo-threefold symmetry produces three approximately triangular orbits of atoms consisting C10-3/C10-7, C10-2/C10-6, and C10-2/C10-5, respectively, in order as one goes away from the apex. Similarly, the 9-cluster s pseudo-fourfold symmetry produces two approximately square orbits consisting of atoms C9-3/C9-4 and of atoms C9-2, respectively, going away from the apex. Notice that (i) all the atoms are at similar distances from the center, and (ii) this distance is slightly shorter in the case of the 9-cluster. Site μ x y z R C C C C C ± C ± C ± C C ±1.336 ± C ± C ± its optimum orientation (of multiplicity six), the fourfold axis is aligned with one of the cubic 100 axes, with the outer y x (a) (b) (c) FIG. 2. Inner clusters in Al 11 Ir 4 structures, shown as they order in the low-t 9.5-phase. The surrounding icosahedron of black atoms is the Ir 12 cage ; another Ir atom is at the center. The apex Al atoms are shown as gray. (The IrAl 12 icosahedra occupying the cubic body centers are not shown.) (a) Al 9 Ir clusters in the z = 0 plane of the 9.5-phase with their fourfold axes oriented up and down along the viewing z axis; there is a twofold axis normal to the page midway between the left two clusters, and another midway between the two right clusters. (b) Al 10 Ir clusters in the z = 0.5 plane,withtheir threefold axis almost parallel to the viewing z axis [notice how the foreground side of the lower cluster in (b) is a half icosahedron]. (c) Orientation relation of two Al 9 Ir clusters (left) with two Al 10 Ir clusters (right), viewed along the x direction. The layers of identicalcontent clusters are stacked normal to the page in (a) and (b), or left to right in (c), in a checkerboard pattern. In the 10-phase, the 10-cluster arrangement in (b) gets repeated in a supercell (with directions rotated such that xyz zxy). y z square of atoms in mirror planes. Each of the five outer atoms in the 9-cluster [Fig. 2(c)] is aligned approximately with a cubic twofold direction, where it simultaneously is counted as one of the Al in the pmi second shell. Thus, the only cage icosahedron edge not having a second-shell Al near its midpoint is the one near the apex atom of the 9-cluster. The 10-cluster is practically always a sort of trigonal prism, with one triangular end face enlarged and capped by an atom, and also capped on the three trapezoidal side faces. An alternate way to visualize the 10-cluster is as a reconstruction of a complete inner Al 12 icosahedron, rotated 90 with respect to the Ir 12 cage icosahedron (as found in the AuZn 3 structure, of which Al 11 Ir 4 is a modification). First choose any (fivefold) vertex of the cage icosahedron, which is approximately aligned with a certain threefold axis of the inner icosahedron: that becomes the threefold axis for the 10-cluster. Replace the triangle of three Al around that axis by a single Al, which becomes the apical capping atom; the three atoms next closest to that triangle move up towards the apex, and the three atoms after that move out radially a bit, to become the side-capping atoms of the trigonal prism. The effect is that the top hemisphere is approximately half a cube [see Fig. 2(b), lower cluster], the bottom hemisphere (opposite to the chosen direction) remains a half-icosahedron (upper cluster in the same figure), and they share a puckered ring of six atoms. By comparing the radii in Table III it can be seen that the 9-cluster is slightly shrunken compared to the 10-cluster. This is visible in Fig. 3(b), in which certain 9-cluster sites coincide with certain 10-cluster sites except for an inwards displacement. In its optimum orientation (of multiplicity 12), the 10- cluster is tilted rigidly so as to bring its threefold axis closer to (but not quite parallel to) the nearby 100 axis, as seen in Fig. 2(b). That brings the apical capping Al and all three of the side-capping Al atoms (which are farther out from the central Ir) close to cubic 100 axes, and each of these four simultaneously is counted as one of the Al atoms in the pmi second shell, on the mid-edges of the Ir 12 cage icosahedron. Note that pmi clusters overlap along certain 100 directions, such that some outer-shell atoms of one pmi can belong to the inner cluster of the neighboring pmi. 11 (In Sec. III A, we will (a) FIG. 3. (Color online) Partially occupied average structure of the inner pmi shell at the (0,0,0) node. (a) Diffraction fit of Ref. 1, (b) calculation based on pair-potential simulations, in which 9- clusters and 10-clusters are placed at random in their discrete optimal configurations as described in Sec. II. The shaded outer atoms (red online) are the Ir 12 icosahedral cage. In (b), sites derived from the 10-cluster and 9-cluster are shown, respectively, by small dark and light circles (blue and yellow online). (b)

4 MAREK MIHALKOVIČ AND C. L. HENLEY PHYSICAL REVIEW B 88, (2013) explain how clusters are oriented relative to their neighbors in specific ordered states.) At T = 600 K, just 100 K above the orientational T c (see Sec. III), an inner Al 10 Ir cluster spends 95% of the time close to one of the 12 ideal orientations described above (as shown by quenching a crystal using Al 10 filling in each cage). At higher temperatures, the threefold axis of the 10-cluster may instead align along a cubic 111 axis, or a pseudo-twofold axis (of the Ir 12 cage). (We have not examined the high-t orientations of the 9-cluster.) These discrete additional orientations are presumably local minima of higher energy. One other form of disorder, seen only at high temperatures, is that one inner-cluster Al atom may wander into a neighboring pmi shell, so the respective clusters are Al 9 Ir and Al 11 Ir, like a vacancy-interstitial pair. Such exchanges, as well as the easier transfer of an Al atom from a 10-cluster to a 9-cluster, could form a mechanism for fast Al self-diffusion. Note that the body-center MI are never involved in this disorder: the body-center MI inner clusters continue to have 12 Al atoms, in contrast with the cell-corner pmi inner clusters, even at high T. C. Agreement with crystallographic data (high-t phase) The Al partial occupancies refined in Ref. 1 areinclose agreement with our model s predictions, provided that close to 50% of the clusters are 9-clusters and the rest are 10- clusters, and that each is oriented independently in one of the (respectively) six or 12 optimum orientations. 12 The predicted and observed sites are compared in Fig. 3.It will be seen that the sites outline the edges of a nearly regular dodecahedron. Indeed, in the time-averaged atom density from a high-t simulation [Fig. 4(a)], the lines of closely spaced alternative sites are smeared into what appears to be a diffuse atom density outlining the dodecahedron edges. Presumably, these lines are troughs of the potential energy function felt by an Al atom due the surrounding Ir. In Ref. 1, this spread-out density is represented by the refined Al(3) and Al(4) sites. However, according to our model calculation, each of those actually represents four or five distinct sites due to both 9- and 10-clusters in various orientations. The reported Al(2) site comes only from one site of the 9-cluster, while the Al(5) site comes only from two sites of the 10-cluster in our model, showing that both kinds of cluster must be present. As detailed in the Appendix, the rms discrepancy between predicted and reported occupancies of each site is 12% of the occupancy, and the estimated fraction of 9-clusters is 53 ± 12%. Further details of the structural comparison are given in the Appendix. III. ORIENTATIONAL ORDERING We next investigated the collective behavior of systems with many interacting clusters, using molecular dynamics simulations. The pair-potential interaction radius was cut off at r c = 10 Å. 13 We used a supercell with the composition of the 10-phase (thus, = 1920 atoms); the initial state was our best single-cell model, repeated in all 4 3 cells. To properly equilibrate the system, we set up a replicaexchange 14 simulation, as we have previously described. 6,15 This method involves annealing multiple samples, each at a different temperature; in our case, 16 samples were used, spaced at 10-K intervals from T = 310 to 460 K. The simulation lasted 4000 cycles, each consisting of 1000 molecular dynamics (MD) steps with time increment t = 5 fs, for a total simulation time 20 ns. Each cycle is an opportunity to exchange the temperatures of two samples. Additional single-t simulations were done at higher temperatures. The simulated alloy remained solid beyond 1700 K. [The experimental melting point is about 1900 C(Ref.16).] A. Ground-state structures The low-temperature optimal structures of the 9.5-phase and 10-phase were determined by rapidly quenching configurations from the lowest replica (T = 310 K) to a T = 0 energy minimum, in 50 iteration steps, and selecting those with the lowest relaxed energies. Their crystal structures are P(E) K 480 K 495 K 500 K 510 K (a) 100 (b) E [ev/atom] FIG. 4. (Color online) Simulation results from a supercell of the basic (30-atom) unit cell with the 10-phase composition, using replica-exchange molecular dynamics. (a) Disordered and ordered states of inner clusters: panel shows time-averaged occupancy distributions for a slice of thickness 6 Å. Left side is at T = 470 K, below the ordering transition, while right side is above it at T = 510 K. The time-averaging restores the ordered structure s cubic symmetry. (b) Phase transition in the 10-phase: plot shows the probability distribution P (E) for total energy (normalized per atom) at five temperatures from 470 to 510 K. The double peak at 480 K T 500 K is diagnostic of coexistence and a first-order phase transition. According to statistical mechanics, the variance of this distribution is proportional to the heat capacity. The two peaks are separated by about 3 mev/atom, the relative weight of which shifts with T

5 CAGED CLUSTERS IN Al 11 Ir 4 : STRUCTURAL... PHYSICAL REVIEW B 88, (2013) TABLE IV. Crystal data for Al 11 Ir 4 structures. The first two rows are from the diffraction refinement of Grin et al. (Ref. 1) and our prediction for the disordered high-t 9.5-state (Sec. IIC), in which each pmi is equally likely to hold a 9-cluster or a 10-cluster, in any of their optimum orientations (Sec. IIB). The last three rows describe the ordered states of the 9-phase, 9.5-phase, and 10-phase [details in the Supplemental Material (Ref. 17)]. The columns include lattice constants and the counts of Al and Ir atoms per cell. Pearson Space Parameters (Å) Structure symbol group a b c Al Ir Ref. 1 cp60 P (cubic) 22 8 High-T 9.5 cp178 Pm (cubic) phase oi232 Ibam phase oa236 Abm phase op60 Pma summarized in Table IV. In the rest of this section, we describe the cluster arrangements in each phase; their crystallographic coordinates are available in Ref phase ground state In the 10-phase, the 10-clusters pair from Fig. 2(b) is repeated in a supercell doubled along the y direction (with respect to the simple cubic framework). Remember that adjacent pmi clusters overlap along all three 100 directions, such that their inner clusters touch; the orientational relations of the inner clusters, in the 10-phase, are described next: (i) Along the [001] direction, one 10-cluster shows its apex atom while the other shows a pair of Al atoms, each aligned roughly in an 011 direction, so as to form a triangle of Al atoms. (ii) Along the [100] direction, a capping atom and an Al pair similarly form a triangle. (iii) Along the [010] direction, there is just one contact between two capping Al s from the respective clusters. It should be noted that telling the 100 axis that the apex points at is incomplete, since each 10-cluster s axis is tilted one way or the other from that axis. To fully specify their orientations, note the 10-clusters of +z and z alignment are always related by a mirror reflection in the xy plane phase ground state The ordered arrangement of the 9.5-phase contains alternating (001) layers of 9-clusters and 10-clusters, stacked along the z direction. The layers are, respectively, made by copying Figs. 2(a) or 2(b). Their main (fourfold or threefold) apex orientations alternate within each layer between +z and z directions, so as to form a 2 2 checkerboard pattern in the xy plane. [Thus, a layer of 10-clusters within the 9.5-phase is different from the corresponding layer in the 10-phase, but the pairing in Fig. 2(b) occurs in both phases.] The layers are stacked such that overlaying clusters (related by a [001] lattice vector) have the same orientation of main axes; thus, a chain of clusters related by [001] separation [normal to the page in Fig. 2(c)] alternates 9-clusters with 10-clusters, but all of them have the same apex direction. The relation between adjoining clusters is shown in Fig. 2(c). B. Cluster ordering phase transition We pinpointed the ordering temperature of the 10-phase in three ways. First, we can examine the atom density distribution, averaged over moderate times [Fig. 4(a)]; for T>T c this has the full symmetry of the unit cell, but for T<T c this shows (despite some fluctuations) a clear symmetry breaking (to the cell-doubled structure just described). A second evidence for a phase transition is available within the replica-exchange simulation: a larger energy spacing Ē i = E(T i ) E(T i 1 ) between the replicas at consecutive temperatures, suggesting a latent heat. The third and most convincing signature is that, for a finite system, the ensemble at temperatures close to T c is a mixture of the two phases, weighted according to the difference in their free energies. This is evident in Fig. 4(b), from which we can read off T c 495 K. The peak separation in Fig. 4(b) shows that the ordered and disordered states in the 10-phase differ by (a latent heat of) 3 mev per atom, requiring an entropy difference at T c of (30) (3 mev)/t c ln(9) per 30-atom cell. That is, we have effectively N cs 9 states per cluster, comparable with N cs = 12 ideal orientations for each 10-cluster. We tentatively concluded there are no other phase transitions (in the 10-phase) at any higher temperature by performing the MD simulation at temperatures T = K, and examining the symmetry of the time-averaged densities visible in Figs. 4(a) and 4(b). With increasing T, the streaks in Fig. 4(b) get stronger, but they do not change in shape. Although we did not simulate the ordering transition of the 9.5-phase, we will briefly speculate on the expected nature of that transition. The 9.5-phase has two kinds of order, the 9Al/10Al alternation in cluster content and the alignment of the orientations, which might appear in separate transitions. The first ordering requires vastly longer equilibration times (for intercluster Al diffusion) and we were unable to identify any sharp transitions. Also, we can estimate the entropy difference (and hence the latent heat) by the same approach as for the 10-phase transition. At high T the effective N cs = 2 (6)(12) 17. (This includes a factor of 2 per cluster for deciding which centers get the 9.5 or 10 filling, then each 9-cluster has six possible orientations.) Thus, the latent heat of the 9.5-phase might be 10% 20% higher than for the 10-phase. Recently, Oishi et al. 7 reported that Al 11 Ir 4, annealed at/above 1173 K, adopts a structure with supercell (a = Å) and space group Pa 3, based on indexing electron diffraction patterns (the structure itself was not refined). Our own studies did not explore the structures of the simulated high-temperature phases (which might be based on the favored orientations of the 9- and 10-clusters but with partial disordering). Therefore, our work provides no clues to interpret the results of Ref. 7. IV. PHASE STABILITY IN Al-Ir SYSTEM We now turn to the T = 0 phase stability of Al 11 Ir 4 and other Al-Ir compounds. The currently accepted Al-Ir phase diagram 16 with 0 <x Ir 0.5 shows six compounds as stable: Al 9 Ir 2 (in the Al 9 Co 2 mp 22 structure); AlIr; and around x Ir 1 4, there is Al 3Ir plus the complex phases Al 11 Ir 4,Al 28 Ir 9 and orthorhombic Al 45 Ir

6 MAREK MIHALKOVIČ AND C. L. HENLEY PHYSICAL REVIEW B 88, (2013) We additionally tried several complex phases not reported in Al-Ir, but found in related alloys. First, we note that a closely related variant of the Al 11 Ir 4 structure occurs in Al-Rh according to the refinement of Ref. 1; thus, since Al 5 Rh 2 occurs in the Al 5 Co 2 structure, 18 we tried Al 5 Ir 2 in that structure. Second, we tried Al 21 Ir 8 in the Al 21 Pd 8 structure, 19 and Al 4 Ir in Al 4 Pt structure, 20 both packings of our 10-clusters without any icosahedral cages (and sharing a few Al atoms). Third, we tried Al 41 Ir 23 in the Al 41 Cu 8 Ir 15 structure, 21 equivalent to Al 68 Pd 20 Ru 12, 22 AlCuRuSi, 23 or Al 64.5 Ir 22 Pd In this last structure, the cell-corner inner clusters alternate between an Al 10 Ir cluster (simulations 24 show it is the same trigonal 10-cluster described in Sec. II B) and an AlCu 8 cube cluster (capped by six more Al). The trigonal clusters appear in several orientations 24 producing a fractional occupancy pattern reminiscent of Al 11 Ir 4,givinga2 2 2 supercell with the space group Fm 3 structure. 21 Of these, the available structures of Al 11 Ir 4 and Al 28 Ir 9 include many fractional sites, so in total-energy calculations we had to try versions of these structures with various ways of realizing the occupations. The relaxed total energies of all these compounds (and the pure elements) were computed using VASP. 5 We found AlIr and Al 9 Ir 2 to be stable, 25 as expected; the only stable phases around x Ir 1 4 were Al 45Ir 13 and (surprisingly) Al 21 Ir 8. Relative to a corrected tie line including Al 21 Ir 8, we found Al 3 Ir to be unstable by 23 mev/atom; this is reduced to only 8 mev/atom in a variant (Al 17 Ir 6 ) with a tripled cell in which one Al is removed. Thus, in contradiction to Ref. 16, we believe Al 3 Ir is a high-temperature phase only. The phases Al 41 Ir 23,Al 28 Ir 9, and Al 11 Ir 4 in the 9.5-phase were unstable by the small amounts of 6, 7, and 9 mev/ atom, respectively; in the latter two, the site disorder contributes entropy which may explain their stability at higher temperatures. We can apply the estimate from Sec. IIIB of the configurational entropy due to cluster placement in the 9.5-phase: the entropy is ln(17) per 30 atoms (the contents of one cell); thus at the annealing temperature T 1160 K 100 mev, the free energy is decreased by 9 mev/atom, exactly enough to stabilize the 9.5-phase. The heat of formation for Al 11 Ir 4 is large: ev/atom; for Al 21 Ir 8 it is ev/atom. We predicted that Al 5 Ir 2, in the Al 5 Co 2 structure also has a large heat of formation, ev/atom. Nevertheless, this (Al5Co2) structure is unstable by 3 mev/atom relative to a tie line between the Al 21 Ir 8 and the Al 45 Ir 13 phases. Table V gives the data used to find the predicted binary phase diagram of the Al-Ir system, by the usual convex-hull construction. Despite the structure names, a binary Al-Ir composition is used in all cases, with Ir replacing any transition metals. All energies are ab initio calculations with the VASP code. 5 V. ELECTRONIC DENSITY OF STATES A specific interest in Al-Ir and certain other Al-TM systems is the possibility of an insulating alloy, the elemental constituents of which are all good metals, e.g., Al 2 Ru in the TiSi 2 structure 29 and Al 2 Fe in the MoSi 2 type structure. 29,30 Quasicrystal i-alpdre (built from pmi clusters) is long TABLE V. Data for phase diagram of Al-Ir system at T = 0K. Column E is energy in mev/atom by which a structure is unstable relative to a mixture of competing stable compounds. Stable compounds forming a convex hull of energy-composition scatter plot have E = 0. The last column cites a reference for each structure. E H x(al) x(ir) Structure (mev/at.) (mev/at.) (%) (%) Ref. Al 9 Co 2.mP Al 45 Ir 13.oP a Al 21 Pd 8.tI AlIr.cP Al 5 Co 2.hP Al 11 Ir 4.oP60 b Al 4 Pt.hP102 c a Al 41 Ir 23.hR d Al 28 Ir 9.hP a Al 3 Ir.hP24 e Al 11 Ir 4.oA236 f Al 3 Ir.hP Al 11 Ir 4.oI a Lowest-energy realization of fractional occupancies. b 10-phase. c Structure has 102 nominal sites, 12 Al sites in two Wyckoff orbits are vacant at T 0K. d In the experimental structure, 10-clusters are centered on a fcc lattice with random orientations giving space group Fm 3, but in our calculated T = 0 structure all 10-clusters are oriented along a threefold axis (the optimal pattern), reducing the symmetry to rhombohedral. e Triple supercell of Al 3 Ir, with one less Al atom (site vacant). f 9.5-phase. claimed to be a semiconductor. 31,32 Predicting gap formation in these alloys depends critically on the accurate relaxation of atomic positions. 31 We find the Al 11 Ir 4 electronic density of states (DOS) in the ordered (T = 0) 10-phase has a significant gap ( 0.4 ev) as shown in Fig. 5(a), so this alloy should be a semiconductor. DOS [states/ev/atom] phase 9.5-phase E [ev] FIG. 5. Electronic density of states (DOS) of T = 0 ordered structures, where E = 0 is the Fermi energy; these curves are smoothed using the Methfessel-Paxton smearing method. The inset is a blowup of the curves near the Fermi energy (these curves calculated using the tetrahedron method). Note that the 10-phase has a gap of 40 mev, whereas in the 9.5-phase, the Fermi level is 0.5eVshifted from a pseudogap

7 CAGED CLUSTERS IN Al 11 Ir 4 : STRUCTURAL... PHYSICAL REVIEW B 88, (2013) By contrast, the ordered 9.5-phase is predicted to be robustly metallic [Fig. 5(b)]. We call attention to the fact that these very similar structures have radically different electronic properties, all due to details of the placement of certain Al atoms that link adjacent clusters. In the disordered high-temperature form of 10-phase (not showninfig.5), the gap tends to get filled in. Hence, that phase is likely to be metallic, too. However, the difference between the electronic DOS of the high-t 10-phase and the low-t 10-phase is small compared to the large difference between the 9.5-phase and the 10-phase. VI. DISCUSSION In conclusion, starting by reanalyzing a variant of the known AuZn 3 structure 1 as a network of icosahedral clusters, we combined molecular dynamics simulations with pair potentials to discover a well-defined, asymmetric inner Al 9 Ir and Al 10 Ir clusters, with variable orientations, and an ordering transition to new orthorhombic phases. We suggest an experimental search for the ordered phase having all clusters of Al 10 Ir type, as it has a gap exactly at the Fermi energy, and hence should be a semiconductor. We also found that, contrary to the accepted phase diagram, the stable low-temperature phase at x Ir 1 4 is Al 21 Ir 8. A. Other structures containing similar clusters Several other known structures are made by placing pmi clusters and/or variants on the same same simple cubic lattice with a 7.7 Å, alternating with highly distorted icosahedral inner clusters at the body centers. These include not only Al 41 Cu 8 Ir 15, 21 Al 68 Pd 20 Ru 12, 22 and AlCuRuSi, 23 which we mentioned, but Al 70 Pd 10 Fe 20, 33,34 which is supposed to have the same structure as Al 41 Cu 8 Ir The ɛ Ag 7 Mg 26 structure is also of this class, with Al Mg. Not only do the body centers contain distorted AgMg 12 icosahedra, but here half of the cell-corner clusters are undistorted AgMg 12 icosahedra too; the latter clusters alternate regularly on the cell-corner sites with AgMg 8 cubes (thus the mean content is Mg 10 ). Furthermore, similar inner clusters are a feature of phases related to decagonal AlPdMn. In particular, decagonal-related ξ (AlPdMn) phase contains pmi clusters with Al 9 Mn and Al 10 Mn inner clusters, not quite identical to those we presented here. 35 In these decagonal-type phases, the Al 9 Mn inner cluster is trigonal, like our Al 10 Ir with the end cap removed; the Al 10 Mn is square, like our Al 9 Ir but capped at both ends. Finally, we point out three Al-TM phases which can be constructed from nothing but Al 10 Ir clusters like the pmi inner cluster (with varying degrees of Al sharing): Al 21 Ir 8 phase, Al 6 Mn.oC28, and Al 4 Pt.hP102. Thus, this coordination shell is a stable motif that does not depend on the second icosahedral shell. Strikingly, after shared atoms are taken into account, Al 21 Ir 8 has practically the same composition as Al 11 Ir 4, while Al 4 Pt.hP102 is only slightly unstable, according to our phase stability evaluation in Table V. B. Relevance to icosahedral quasicrystals When the energetic optimization of the i-alpdmn quasicrystal structure 36 was further pursued, the lowest-energy version yet found 37 (using the 2/1 approximant) consists of exactly the same pmi clusters described above. In particular, (i) they have a short ( 8 Å) linkage along the icosahedral twofold direction, rather than the 12.5 Å linkage known from α-almnsi 2, and (ii) they have an Al 10 Mn inner cluster in the same shape as the one described here, which appears in various orientations with only small energy differences. (The previous best energy version for i-alpdmn, the Quandt-Elser model, 38 as improved by Zijlstra, 39 was also built from small pmis containing the same 9-cluster that we proposed here for Al Ir.) Very recently, a refinement was made 40 of a large cubic approximant with composition about Al 70 Pd 22 Cr 2 Fe 6, demonstrating it is built from a network of clusters connected like canonical cells 41 using the short kind of linkage, with pmi clusters on the even nodes, just like our hypothetical AlPdMn structure. Based on the above, it is reasonable to conjecture that i-alpdmn consists of a small canonical-cell tiling, with pmis on the even nodes, each having as its inner shell a 10-cluster (or possibly a 9-cluster in some environments). An icosahedral cage containing an inner cluster that breaks icosahedral symmetry is known in a quite different family of quasicrystals and related compounds: CaCd 6 or ScZn 6, with an inner Cd 4 or Zn 4 tetrahedron. 4,42 Those, too, show orientational orderings at low temperatures We observe that such cluster orientations are highly pertinent to the longstanding problem of whether realistic atomic-level interactions can implement local matching rules in icosahedral quasicrystals that would stabilize an ideal quasiperiodic ground state, in the spirit of Penrose s matching rules. Just as the Penrose arrows spoil the equivalence of different edges in a rhombus tiling, more generally one needs a marking to spoil the high symmetry of the clusters that form the structure s framework. The likeliest candidate for this marking is the orientation of the inner clusters. It should be noted that, apriori, such a set of interactions that split the energies of different tilings might just as well stabilize a crystal structure. C. Possible future directions An interesting topic for future systematic research is the path by which a 10-cluster converts from one discrete orientation to another. Movies from our molecular dynamics indicate that, whereas Zn tetrahedra in ScZn 6 undergo quasirigid rotations, 42 that is not so for the 10-clusters. Instead, a large motion of two atoms combines with small adjustments of the others reconstitutes the cluster in a different orientation, a mode of reorientation called pseudorotation. 48 This is feasible because one side of the 10-cluster is a fragment of an icosahedron and these sites are partly shared between the different orientations. Simulations of the inner-cluster dynamics can be compared to neutron measurements of the dynamic structure factor, analogous to the Zn 4 tetrahedron in ScZn 6. 42,45 One other direction in which we have already begun to extend this work 49 is to fit an orientational interaction between adjacent clusters, as was done for the Cd 4 -Cd 4 interaction 46,47 for Monte Carlo estimation of the orientational T c. A final possible extension is to model a hypothetical Al-Ir quasicrystal based on these clusters

8 MAREK MIHALKOVIČ AND C. L. HENLEY PHYSICAL REVIEW B 88, (2013) ACKNOWLEDGMENTS We thank J. Dshemuchadse, P. Kuczera, and W. Steurer for sharing structural data on Al 41 Cu 8 Ir 15 phase prior to publication, and R. Tamura and Y. Grin for fruitful discussions. This work was supported by DOE Grant No. DE-FG02-89ER (M.M., C.L.H.), and Slovak Research and Development Agency funding under Contracts No. VEGA 2/0111/11 and No. APVV (M.M.). APPENDIX: Al 10 Ir CLUSTERS AND AVERAGE STRUCTURE In this appendix, we define symmetry-averaged 9-cluster and 10-cluster positions, and relate them to the experimentally refined average-structure sites of Ref. 1. The upper part of Table VI shows the predicted Al positions with a particular orientation of an inner cluster (taken from Table III) within the fundamental cubic cell (a cub = Å) : the 10-cluster (first seven sites) or the 9-cluster (next four sites). The bottom rows of Table VI give the coordinates of all atoms in the experimentally refined (average) structure. 1 We included all sites including the small Al 12 Ir icosahedron centered at the body-center positions (part of MI cluster) as well as Ir cage sites. In those MI positioned at body centers, the small icosahedron composed of Al(1) atoms is rotated by an angle of 90 relative to the large icosahedron of Ir(3) atoms, as in the AuZn 4 prototype structure. Table VII addresses the effects averaging the inner cluster coordinates from Table VI over all possible orientations. The predicted and refined structures are compared quantitatively in two ways. First, the right-hand column shows the spatial displacement between the predicted and refined site positions TABLE VII. Average structure of the 9.5-phase, compared to the diffraction-data refined average structure. Rows in the table label the 9- and 10-cluster Al sites, while columns show which inner-shell refined site they contribute to [of Al(2), Al(3), Al(4), or Al(5)]. One kind of model site (C10-4) has been split between the refined Al(3) and Al(4) sites. Site μ Al2 Al3 Al4 Al5 dr (Å) C C C C ,0.48 a C b C C C10-sum C C C C C9-sum phase Expt a For Al3 or Al4, respectively. b 0.74 Å from Al4 site. (that were shown in Table VI). Second, the totals at the bottom give the net count (per cluster) of Al atoms assigned to each site type defined in the refinement. (The actual site occupancy is obtained when these are divided by the multiplicity of that site according to the space group.) Evidently, Grin et al. constrained the total Al count to be exactly 10 atoms; we would TABLE VI. Coordinates of all atoms predicted from the simulation are given in a form convenient for comparison with Grin s refinement (Ref. 1). All coordinates are given as multiples of the lattice constant. The numbering of Ir and Al Wyckoff sites is taken from Ref. 1. The 9-cluster or 10-cluster is centered at (0,0,0). In the last column, sites are tagged according to their structural role ( inner-pmi, etc). Site μ x y z Expt. label Comment C Al3 Inner pmi C Al4 C Al4 C Al4 C ± Al5 C ± Al4 C ± Al5 C Al3 C ± ± Al2 C ± Al4 C ± Al3 Occ. Ir1 1b 1/2 1/2 1/2 1 Cube B.C. Ir2 1a pmi center Ir3 12j Large ico. Al1 12j Small ico. Al Inner pmi Al Inner pmi Al Inner pmi Al Inner pmi

9 CAGED CLUSTERS IN Al 11 Ir 4 : STRUCTURAL... PHYSICAL REVIEW B 88, (2013) thus multiply all the experimental numbers by 0.95 to give the best comparison to our predictions. It is interesting to allow the fraction of 9-clusters and 10- clusters to be x and 1 x, respectively, with variable x. Then, we can obtain an independent estimate of x from the atom count in each of the four Al sites. When averaged, this gives x = 0.4 ± 0.2. In general, the rather large separation of some actual 9- and 10-cluster sites from the diffraction-refined structure indicates that four crystallographic sites are not sufficient to accurately describe the full set of equilibrium positions: those of our sites having largest displacements from the experimentally refined sites (namely, sites C10-4, C10-5, and C9-4) ought to be included in future refinements to more accurately represent the data. 50 NoticeinTableVI that in the refined structure, 1 the Ir(3) atoms get split into half-occupied sites that are 0.3 Å away from each other. We propose that this is an indirect consequence of the fact that the 9.5-phase contains two kinds of inner shells, so that the surrounding large icosahedron adjusts its shape in two inequivalent ways. These displacements of the cage Ir atoms are noticeable in Fig. 2(c). 1 Y. Grin and K. Peters, Z. Kristallogr. 212, 439 (1997). 2 V. Elser and C. L. Henley, Phys.Rev.Lett.55, 2883 (1985). 3 A. Katz and D. 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