Tile Hamiltonian for Decagonal AlCoCu

Ibrahim Al-Lehyani^{a,b} and Mike Widom^{a}

aDepartment of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

bDepartment of Physics, King AbdulAziz University, Jeddah, Saudi Arabia (April 24, 2002)

Abstract

A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions between and within tiles. We studied AlCoCu decagonal quasicrystals described as decorated Hexagon-Boat-Star (HBS) tiles using ab-initio methods. The dominant term in the TH counts the number of H, B and S tiles. Phason flips that replace an HS pair with a BB pair lower the energy. In Penrose tilings, quasiperiodicity is forced by arrow matching rules on rhombus edges. The edge orientation in AlCoCu is due to Co/Cu chemical ordering. Tile edges meet in vertices with 72^{∘} or 144^{∘} angles. We find strong interactions between edge orientations at 72^{∘} vertices that force a type of matching rules. Interactions at 144^{∘} vertices are somewhat weaker.

I. INTRODUCTION

Both quasicrystals and ordinary crystals are made of elementary building blocks. In crystals, copies of a single building block (known as a unit cell) are arranged side by side to cover the space periodically. In quasicrystals, building blocks are arranged to cover the space quasiperiodically. Two approaches to build quasilattices are proposed. One approach uses a single unit cell, but allows adjacent cells to overlap. Gummelt [1] proved that using a limited number of overlapping positions between decagons produces a quasicrystal structure. Steinhardt and Jeong [2] further proved that the overlapping conditions can be relaxed and supplemented by the maximizations of a specific cluster density to produce quasilattices. This is the “quasi-unit cell” approach.

In the other “tiling” approach, space is covered with building blocks called tiles. Two or more tile types are used [3]. No overlapping is allowed, and depending on the way the tiles are arranged, quasicrystal structures can be produced. Matching rules between tiles govern the local tile configurations by allowing a subset of all possible arrangement. Globally, the matching rules enforce quasiperiodicity [4,5]. Penrose proposed his famous matching rules before the discovery of quasicrystal materials. In 2D, Penrose tiles are fat and thin rhombi (Fig 1a). Edges are assigned arrow decorations which must match for common edges in adjacent tiles. In perfect quasicrystals these rules are obeyed everywhere. The very restrictive Penrose matching rules are sufficient, but not necessary, to force quasiperiodicity. Socolar [6] showed that weaker matching rules can still force quasilattices. The less restrictive set of rules are derived by allowing bounded fluctuations in perp space. Furthermore, matching rules can be abondened entirely and quasiperiodicity may arise spontaneously in the most probable random tiling.

A fundamental question is whether matching rules are enforced by energetics of real materials. Burkov [7] proposed matching rule enforcement by chemical ordering of CoCu pairs in Al-Co-Cu. However that model involved an unnatural symmetry linking Co sites to Cu sites. Cockayne and Widom [9] deduced a different, physically realistic, type of CoCu ordering. In their model, tile edges are assigned arrow direction based on their Co/Cu decorations (Fig.1b). The suggested physical origin of Co/Cu chemical ordering rests on the status of Cu as a Noble Metal with completely filled d orbitals, unlike normal transition metals such as Co.

In a tiling model of quasicrystals, the actual atomic interactions in the system Hamiltonian can be replaced with effective interactions between and within tiles [8]. The resulting tile Hamiltonian is a rearrangement of contributions to the actual total energy. In simple interaction pictures (pair potentials for example) the relation between the two is straight forward. It might be difficult to find the relations between them for more complicated descriptions (many body potentials, or full ab-initio energetics, for example) but it is theoretically possible. The tile Hamiltonian includes terms which depends only on the number of tiles and other terms for different interactions. The tile Hamiltonian greatly simplifies our understanding of the relationship between structure and energy, and it is a reasonable way to describe the tiles.

Space can be tiled in many different ways, even when holding the number of atoms or the number of similar tiles fixed. Figure 2 shows three tiling configurations of 132 atoms. The first two have the same number of tiles but arranged differently. The third has the same atoms but different tiles. These are called quasicrystal approximants (crystals that are very close to quasicrystals in structure and properties). One advantage of approximants is that they can be studied using the same conventional tools developed for ordinary crystals.

In a previous paper [14] we studied the matching rules in decagonal Al-Co-Cu using a limited group of quasicrystal approximants. Some specific interaction details in the tile Hamiltonian couldn’t be extracted from our limited data set. Here we study more thoroughly the set of rules controlling these compounds using different techniques and a much bigger set of approximants.

We describe the model of decagonal Al-Co-Cu in section II of this paper. Section III gives our detailed calculations using ab-initio methods. We extract a set of parameters that allow an excellent approximation to the total energy. Similar calculations done using pair potentials are described in section IV for comparison. In section V, we talk about various other effects that could be considered in a more accurate model. We analyze our findings and study their implications for Al-Co-Cu compounds in section VI.

II. DECAGONAL AL-CO-CU MODEL

Decagonal Al-Co-Cu quasicrystals have been studied by many authors, theoretically [7,9,11] and experimentally [12]. Cockayne and Widom [9] employed mock-ternary pair potentials to propose a model based on tiling of space by Hexagon, Boat and Star tiles (HBS) decorated deterministically with atoms (Fig. 2). The tile edge length is 6.4 Å. Tile vertices are occupied by 11-atom clusters. Each cluster consists of two pentagons of atoms stacked on top of each other at \frac{1}{2}c=2.07 Å and rotated by 36^{∘}. One layer contains only Al atoms. The other layer contains a mixture of transition metal (TM) atom species and can contain also Al atoms. The mixed Al/TM pentagon contains an additional Al atom at its center All TM atoms surrounding a vertex Al atom are parts of tile edges. Decagonal clusters meet along pairs of Co-Cu atoms.. It was shown in the original model [9] that TM atoms prefer to alternate in chemical species on tile edges. This was confirmed later by ab-initio calculations using the Locally Self-consistent Multiple Scattering (LSMS) method [13,14], and is confirmed again in this study.

We assign arrows to edges based on their TM atom decorations. By our definition, an arrow points from the Cu atom towards the Co atom. Tile edges meet in vertices of 72^{∘} or 144^{∘} angles. An angle is of type “i” if both edges point in towards their common vertex. Types “o” and “m” are out- and mixed-pointing, respectively.

Hexagon, boat and star tiles are the building blocks in the model. The HBS tiles are composed of Penrose rhombi with double arrow matching rules satisfied (by definition) inside the HBS tiles. Some of their properties are summarized in table I. Quasiperiodic tilings can then be constructed by only watching out for the single arrows (Fig. 1).

We choose to define a tile Hamiltonian for the system

H = ∑_{α} n_{t}^{α}E_{t}^{α} + λ_{72} ∑_{β} n_{72}^{β}E_{72}^{β} + λ_{144} ∑_{β} n_{144}^{β}E_{144}^{β}.  (1)

Here n_{t}^{α} is the number of specific tile type α which can be H, B or S; n_{72}^{β} and n_{144}^{β} are the numbers of 72^{∘} and 144^{∘} angles and β defines angle type which can be i, o or m. We fit energy parameters in the Hamiltonian (1) to achieve H ≈ E_{tot} for a wide class of structures. The tile energies are much bigger than the other terms, and it turns out that the 72^{∘} angle interactions are bigger than the 144^{∘}’s. It is desirable to check how each term can alter the system energies. We set λ_{72} and λ_{144} to 1 or 0 for the purpose of including and excluding the 72^{∘} and 144^{∘} angle interactions.

An H tile contains 25 atoms, counting those on the perimeter fractionally. All but 3 atoms (one Al and 2 Co) belong to vertex decagonal clusters. The Co atoms occupy symmetric sites and the Al atom breaks the symmetry by residing on one of two geometrically equivalent positions between the two Co atoms (Fig. 1b,left). Its interaction with tile edge TM atoms determines the preferred position. A B tile has 41 atoms. An internal Al atom breaks the symmetry(Fig. 1b, center) by residing in one of two equivalent sites. An S tile has 57 atoms. In a star, two internal Al atoms break the symmetry(Fig. 1b, right) by occupying any two of five equivalent sites as long they are 144^{∘} apart. A phason flip can switch a specially arranged star and hexagon into a pair of boats. This is shown in Fig. 2c outlined by a dashed line.

The present structure differs slightly from the original model [9]. In the original model, Cu atoms take certain symmetry-breaking positions inside the boat and the star which makes it difficult to parameterize their interactions. We choose to replace the Cu atoms in the tile interiors with Al atoms. In terms of the atomic surfaces [9], the atomic surface that is mainly Al with a thin ring of Cu becomes pure Al.

Many different quasicrystal approximants are exploited here to study different terms in tile Hamiltonian. All the approximants we used are listed in table II with some of their properties. The smallest structure is the monoclinic single-hexagon approximant H_{1} (Fig. 3). It contains one horizontal and two inclined tile edges. For the decoration shown, equation (1) becomes H = E_{t}^{H}+λ_{72}(E_{72}^{i}+E_{72}^{o})+λ_{144}(2E_{144}^{i}+2E_{144}^{o}). The next bigger approximant is a 41atom single-boat B_{1} (Fig. 4) for which H = E_{t}^{B} +λ_{72}(3E_{72}^{o})+λ_{144}(2E_{144}^{i} +E_{144}^{o}). Single stars do not cover the space, so a single-star unit cell approximant is not possible. Two-hexagon approximants can be constructed in orthorhombic cells, either by a genuine orthorhombic structure H′_{2} (Fig 5) with H = E_{t}^{H} +λ_{72}(2E_{72}^{i}+2E_{72}^{o})+λ_{144}(4E_{144}^{i}+4E_{144}^{o}) or by doubling the monoclinic H_{1} cell to create H_{2} shown in Fig. 3. The doubling process gives more freedom in controlling edge arrow orientations by performing Co/Cu swaps. Similarly, a two-boat unit cell can be constructed, either by doubling the single boat structure B_{1} or by tiling the two boats as shown in B_{2} (Fig 6) with H = E_{t}^{B} + λ_{72}(3E_{72}^{o} + 3E_{72}^{m}) + λ_{144}(3E_{144}^{i} + E_{144}^{m} + 2E_{144}^{o}).

To isolate the tile Hamiltonian parameters E_{t}^{α} and E_{θ}^{β}, larger approximants are needed. The three approximants in Fig. 2 each have 132 atoms per unit cell but different tile configurations. Two of them are related to each other by a single phason flip (S_{1}H_{3} and B_{2}H′_{2}). The phason flip lowers the energy, indicating that stars are disfavored in these compounds. Angle orientations are investigated by swapping a TM pair on tile edges, reversing the direction of the edge arrow.

Long range interactions and other small terms omitted from the tile Hamiltonian (1) can be estimated by swapping pairs surrounded by symmetric bonds on their sides. For example, the bonds on the sides of the horizontal tile edge of H_{2} can be specially arranged to cancel all angle interactions E_{θ}^{β} and leave only other effects.

III. AB-INITIO STUDY

For our calculations we employ ab-initio pseudopotential calculations utilizing the Vienna Ab-initio Simulation Package (VASP) program [15]. We use ultra soft Vanderbilt type pseudopotentials [16] as supplied by G. Kresse and J. Hafner [17]. Our calculations are carried out on the Cray T3E and on the newly installed Compaq TCS machines at the Pittsburgh Supercomputer Center. The k-space mesh size (among other parameters) determines the accuracy of the calculations. Bigger k-space grids are more accurate but more expensive in calculation time. One has to find a balance between the number of atoms in a unit cell and the size of the k-space grid in order to fit within the available computer resources. As explained before, we use many different approximants for our study. The k-space mesh is increased until a consistency of about 0.02 eV is reached in worst cases. But within a reasonable use of our allocated times we are able to get better convergence (0.002-0.01 eV) for most structures.

Convergence test calculations are summarized in table III for our B_{1} structures and in table IV for our B_{2}H_{2} structures. We used two slightly different structures per approximant, differing in orientation of a single arrow (Co/Cu pair circled in Fig. 6 and Fig. 2a). The tables suggest that at the chosen k-space grid used in our calculations (marked by a star in the tables), the accuracy is better than 0.02 eV. The smallest grid of 1x1x1 (single k-point in the middle of the k-space unit cell) for our B_{1} approximant takes about 8 minutes on the T3E machine (450 MHz alpha processors) using 8 processors. The largest grid we use for B_{1} of 5x5x15 (188 independent k-point) takes 4.8 hours on 64 processors. The B_{2}H_{2} structure has 132 atoms per unit cell. The largest grid of 2x2x10 (20 independent k-points) takes about 6 hours on 64 processors. The smallest 1x1x1 grid takes 2 hours on 8 processors.

We note that for a fixed number of processors, calculation time grows linearly with the number of independent k-points. For the same structure, calculation time decreases linearly with increasing number of processors only up to 16 processors. Beyond that, the total charging time (number of processors*elapsed time) increases notably. Large structures and big k-space grids require large numbers of processors because they need more memory. Note from tables III and IV that the more isotropic the distribution of the k-space grid points along k_{x}, k_{y} and k_{z}, the faster the convergence. For our B_{1} approximant, meshes that most isotropically distribute k-space points are 1x1x3 and its multiples, but for finer meshes 4x4x11 is slightly more isotropic than 4x4x12. For fixed numbers of k_{x} and k_{y} points, the total energy converges towards its limiting value as the number of k_{z} points approaches its isotropic value. In all our structures, we choose the most isotropic distribution of k-space points possible, consistent with an orthorhombic lattice.

Many rearrangements of arrows are performed by swapping TM atom pairs and the different structure energies are calculated. Most of these are done for the large approximants because they give more configurational freedom. In general, pure 72^{∘} angle interactions can be deduced if the middle bond has an equal number of 144^{∘} connections on each side (since we can arrange for 144^{∘} interaction to cancel) and either a single 72^{∘} connection which can be attached to any side or two 72^{∘} connections attached to the same side. An example is outlined by a dashed rectangle in Fig. 2a. The bond is surrounded by a single 144^{∘} on each side. Both point towards the middle bond making E_{144}^{i} on one side and E_{144}^{m} on the other. One extra 72^{∘} on one side makes E_{72}^{o}. If the middle pair is swapped, the 144^{∘} angle interactions will be conserved. The swap merely switches their sides. The E_{72}^{o} becomes E_{72}^{m}. The difference in energy E_{after} − E_{before} = 0.26 eV between the swapped and the basic structures yields the difference E_{72}^{m} − E_{72}^{o}. Reversing the 72^{∘} outer bounded then making the swap can give E_{72}^{i} − E_{72}^{m} = 0.35 eV. For pure 144^{∘} interactions, we use a middle bond surrounded by two 144^{∘} angle on one side and a single 144^{∘} angles on the other side. We arrange for any 72^{∘} angles to cancel. The pair surrounded by a dashed circle in Fig. 2a is one example that yields E_{144}^{m} − E_{144}^{o} = 0.06 eV.

We compute energies for an over-complete set of structures and use a least square fit to determine average parameter values. First, we fit with λ_{72} = λ_{144} = 0 in equation (1), leaving only three adjustable parameters E_{t}^{h}, E_{t}^{b} and E_{t}^{s}. The fit finds the values of our parameters that minimize the root mean square of the difference between the calculated E_{tot} and model H values. The fitted values E_{t}^{B}, E_{t}^{H} and E_{t}^{S} are shown in the second colum of table VI. The values reported are the best fit for an ensemble of tiles with different angle orientations. The fitting is shown in Fig. 7. The five figures correspond to the different energy ranges spanned by our approximants. The first graph, showing data from 132 atom approximants, shows that a pair of boats is greatly favored over a star and a hexagon. Both have the same numbers of atoms of each type, but the S_{1}H_{3} has one more 72^{∘} interaction than B_{2}H_{2}. The graphs also show clearly that the three-parameter fit is not adequate because there is a significant variations of E_{tot} among different approximants.

Note that the values of H, B and S are individually meaningless unless compared with pure elemental energies. For example, the data quoted does not include arbitrary offsets EATOM [15] of each chemical species However, the difference 2E_{t}^{B} − E_{t}^{H} − E_{t}^{S} = 1.3 eV is meaningful. The EATOM is fixed for every element. Thus we define a tie-line energy E_{tie−line} to be the pure element fcc Al, fcc Cu and hcp Co energy per atom. The structure energies per atom lie above the tie-line and the difference is the cohesive energy per atom. We calculate E_{tot}[fccAl] = −4.17eV/atom, E_{tot}[fccCu] = −4.72eV/atom and E_{tot}[hpCo] = −7.88eV/atom. The fcc Co is stable only at high temperatures [18] and VASP predicts its energy to be E_{tot}[fccCo] = −7.87eV/atom, higher than the hcp phase by 0.01 eV/atom. The tile cohesive energies per atom for H, B and S are -0.35 , -0.34 and -0.33 eV/atom respectively.(Mike This is kind of misleading because 2B=S+H). The difference between two boats and a pair of hexagon and star is -0.24 eV.

The six-parameter fit values are shown in the third colum of table VI where we set λ_{72} = 1 to include E_{72}^{β} and λ_{144} = 0 to exclude E_{144}^{β}. The 72^{∘} angles are all internal in the tiles so their energies can’t be separated from the tile energies. Only differences in energies can be calculated such as E_{72}^{i} − E_{72}^{o} and E_{72}^{m} − E_{72}^{i}, so we set the lowest energy orientation E_{72}^{o} equal to zero without any loss of generality.

The fourth row of table VI shows the nine-parameter fit when we set λ_{72} = λ_{144} = 1. The 144 ^{∘} angles can come inside the tiles similar to the 72 ^{∘} angles or can come in what we call “hidden 144”, when two 72^{∘} angles combine together to form one 144^{∘} angle. The hidden 144 angles appear only when tiles are glued to each other to cover the space, and their number is configuration dependent. Our approximants have several of these. For example, the B_{1} structure (Fig. 4 has two hidden 144’s. The hidden 144 angles allow us to calculate the absolute energy cost (fourth column of table VI) of the 144^{∘} angle interactions E_{144}^{β}in contrast to the 72^{∘} angles. The fitting graphs are in Fig. 8. The large approximants spread over a large range of energies because of the freedom we have in performing different swaps. That’s why the range of the first graph is six times bigger than the range of the rest. The remaining deviation from the y=x line (s.d.=1.3×10^{−3} eV) is due to other effects not included in our model H (eq. 1), as well as incomplete convergence or other calculational inaccuracies.

The atoms in bonds meeting in a vertex are located at three different separations from each other. Concentrating on the smallest of all, the distance between atoms closed to the vertex. The are Co atoms in the in configuration, Cu atoms in the out configuration and a Co and a Cu atom in the mixed configuration. The mixed configuration is intermediate between out and in configurations which is true for all the other separations too. Knowing this fact and noting that mixed arrowing is very close to the average of the out and in configurations(table VI), we can conclude that the Co-Cu interaction is very close to the average of the Co-Co and Cu-Cu. In a previous paper [19], that averaging was assumed but not proven. Our results here shows that such an assumption is very close to reality.

IV. PAIR POTENTIALS

The electronic ground state of a system can be expanded in terms of a volume energy and potentials describing n-body (n=2,3,4,...) interactions [20]. The volume energy is the largest contribution to cohesive energy. It depends on the composition and density but not the specific structure. The n-body interactions distinguish between different crystal structures at the same composition and density. It is customary to truncate this n-body series at the pair potentials, because they are much easier to calculate and use, and higher order interactions are weak. Even for transition metals with their localized d-band, the decomposition to pair potentials proved to be practical [10,21].

Many different pair potentials have been used to study this and other quasicrystals. Cockayne and Widom [9] proposed mock-ternary potentials extracted from Al-Co pair potentials. Noting that, in Al-Co-Cu, Cu substitutes for an equal combination of Al and Co, they approximated Cu interactions by the average interactions of Al and Co. They adopted Al-Co pair potentials calculated by Phillips et al. [22]. Their discovery of alternation of TM atoms on tile edges, and the effect of edge decorations on the central Al, are all consistent with our VASP results.

Later, more rigorous pair potentials in the context of Generalized Pseudopotential Theory (GPT) were used for Al-Co-Ni [10]. The original GPT pair potentials suffered from TM over-binding which is an unphysical attraction between TM atoms at small separations. The strongest over-binding appears in Co-Co pair potentials. We modified the TM pair potentials at short distances by adding a repulsive term using VASP to get the energy and length scale [10]. The resulting potentials behave really well in simulations [23]. Originally Co-Cu pair potentials were defined as equal to the Ni-Ni pair potentials [19] so in the modified potential the corrected Ni-Ni pair potentials played the same rule for Co-Cu. The other AlCoCu pair potentials were found to be well behaved up to large Cu composition [19].

We calculated the energies of the approximants using both mock-ternary and modified GPT pair potentials to check how pair potential results compare to VASP.

The central position for the Al atom is preferred by at least 0.11 eV/cell and 0.01 eV/cell respectively.

Results of the fitting are summarized in table VI for all the methods used. Aside from a difference in energy scale between the modified GPT and the mock-ternary pair potential calculations, they are qualitatively close to each other and to VASP. The order of 144^{∘} angle interaction is reversed compared to VASP, but these interactions are very weak

V. OTHER EFFECTS

Chemical ordering of TM atoms on tile edges creates matching rules in our model. We study chemical ordering here using our H_{2} approximant (The orthorhombic unit cell in Fig. 3) which contains two horizontal tile edges. When a Cu atom from one of them is swapped with the Co atom on the other, the resulting edges contain pairs of similar species (one CoCo and one CuCu). The process raises the energy by 0.68 eV/cell. The same swap were studied before with LSMS [14] and gave same conclusion with different value (0.17 eV/cell). Using the pair potentials, TM atoms favor alternation on the tile edges by 0.022 eV/cell for mGPT and 0.079 eV/cell for mock-Ternary.

In Al-Co-Ni, CoCo and NiNi pairs are slightly preferred over CoNi pairs. As a result Al-Co-Ni has no arrow decorations at low temperatures. Cu and Ni are adjacent in the periodic table, but they are notably different in their properties. In an isolated Ni atom, the 3d shell has six electrons and the 4s shell is filled. The partial filling of the d-band strongly influence atomic interactions. The 3d shell in Cu is filled with electrons and the 4s has one electron which makes Cu act more like a simple metal. The d-band is buried and doesn’t participate strongly in interactions.

An important issue is the position of the symmetry-breaking Al atom inside a hexagon we mentioned in section II. There are two symmetrically related positions between the two internal Co atoms, and we force the Al atom to take one of these positions. If the horizontal edge arrows are parallel to each other, the Al atom prefers to reside in the side where Co atoms are by about 0.03 eV. In the smaller tiling representation (τ^{2} deflated [23]), the TM pairs on edges becomes the interior of the small hexagon(see Fig. 2a), and the area inside the hexagon is decomposed into a boat and a star. The position of the internal Al atom specifies the decomposition into boat and star, and its hopping from one site to the other is a phason flip between the small star and the small boat. This decomposition is lost by placing the Al atom exactly at the center of the hexagon. However, this position is lower in energy by 0.2 eV. In the pair potential picture, the central position for the Al atom is preferred by at least 0.11 eV/cell using mGPT and 0.01 eV/cell using mock-ternary potentials. Depending on the edge decoration, this Al may relax very slightly from the central positions, but this effect is minimal and does not significantly influence the energy.

With the off-center Al, we define the decomposition of the hexagon such that the symmetry-breaking Al is placed as in Fig. 1. The position of the internal Al atom inside a hexagon, together with the horizontal tile edge arrows, define a “direction” for the hexagon. We noticed that generally hexagons like to align in our H_{2} structure. The effect is very small (≈ 0.01 eV) but enough to account for some of the discrepancies between E_{tot} and H in our calculations.

One more small effect appears in the hidden 144^{∘} angles, where two 72^{∘} angles share one edge making an extra 144^{∘}. The shared edge orientation affects the angle interaction E_{144}^{β} of the outer edges. We calculate the difference E_{144}^{o}-E_{144}^{m} with the shared edge pointing inward and outward. With outward pointing, the difference E_{144}^{o}-E_{144}^{m} equals -0.075 eV. An inward middle edge raises the difference by 0.015 eV, so that E_{144}^{o} − E_{144}^{m} equals -0.060 eV.

So far we have examined interactions within the quasiperiodic plane. Now consider “vertical” interactions. Pairs of TM atoms on tile edges are 1.51 Åapart within the quasiperiodic plane and 2.07 Åapart along the periodic direction. The net bond length is 2.56 Å. The lines connecting them makes a zigzag of alternating TM atoms along the periodic axes. We turn our attention to atomic order in this direction. Approximant B_{2}H_{2} (Fig. 2a) has a glide plane parallel to the long side of its unit cell that can be exploited for this purpose. We swap one pair contained within τ^{2} deflated hexagon in Fig. 2a (call this pair a) and call the structure (A). Another structure (B) is made from B_{2}H_{2} by swapping the glide-equivalent image of pair a (pair b). These two structures have equal energies by symmetry. Further, we build a 264 atom unit cell by stacking two 132 atom unit cells. it is built once by stacking an A layer over an A layer and another time by stacking a B layer over an A layer. In AA, TM alternation along the vertical zigzag is conserved. In AB the zigzag sequence is violated along pair a and along pair b. Along each pair the alternation defect includes a CoCo pair and a CuCu pair. The AA and AB structures are calculated with a k-point mesh of 2x2x5. The difference E_{AB} − E_{AA} equals 0.392 eV/264-atom cell.

VI. DISCUSSION

We discuss here the implications of our findings on the structure of decagonal AlCoCu. The main result is that now energy can be calculated quickly and accurately for these compounds by adding the relevant terms in the tile Hamiltonian H (eq. 1) using parameters obtained in table VI.

Adjacent 72^{∘} angles either share the same vertices or point in opposite directions. The first is the hidden 144 ^{∘} we discussed above and the second form a zigzag of 72^{∘} angle. This can be seen in H_{2} structure where an infinite zigzag forms perpendicular to the hexagons. Due to the fact that E_{72}^{m} is very close to the average of E_{72}^{o} and E_{72}^{i}, the middle edge decoration an isolated zigzag with any direction is degenerate. Similar argument apply to the case of 144^{∘} zigzag but the effect is much smaller.

The last point leads us to a bigger question about the importance of the 144^{∘} angle interactions. Most of the times bonds participate in combinations of 144^{∘} and 72^{∘} angles. The stronger interactions determine bond arrowing. When a bond is surrounded by a total of four 144^{∘} angles and no 72^{∘} the middle bond is a part of 144^{∘} zigzag and its decoration doesn’t matter. An example of this is circled in Fig. 6. There is only one configuration where decorations is determined by 144^{∘} interactions. This is the configuration we used to get pure 144^{∘} angle effects (see sec. III). These configurations occur occasionally (one is in Fig. 2b) and as shown in table V including the 144^{∘} angle terms in the fitting did improve it.

However, at high temperatures the situation is different. Quasicrystals may be stabilized by entropic contributions at high temperatures. Transition from crystal to quasicrystal phases are reported at about 1000K [24]. That correspond to about k_{B}T=0.1 eV. At such temperatures the 144^{∘} angle interactions are irrelevant because they are small compared to energy fluctuations, and the structure is determined primarily by its 72^{∘} angle interactions

Our model expectations are in agreement with calculated energies. Figure 8 shows that for most of the structures the model energies fall near the y=x line. The worst deviation is about 0.1 eV. Out of that we account for 0.03-0.05 eV from the internal Al atom effects on tile edges. The rest can be a collection of long range interactions. We do see these long range effects in some instances. For example, when calculating pure 72^{∘} angle interactions using the bond surrounded by a rectangle in Fig. 2a in two different approximants (B_{2}H_{2} and S_{1}H_{3}). The environments are identical up to about 7 Å, but a difference of about 0.02 eV in E_{72}^{m}-E_{72}^{o} between the two cases shows up.

Pair potential calculations show that they are capable of catching the dominant 72^{∘} interactions we are investigating with a lot less calculation times.

In our previous paper [14], we reported several results related to edge arrowing calculated using LSMS. Approximants H_{2} and H′_{2}were used, with internal Al atoms centered. The swap for chemical ordering agrees in sign but VASP’s is four times bigger that LSMS. Other swaps that give 2E_{72}^{m}-E_{72}^{i}-E_{72}^{o} agree in sign with the same factor disagreement in magnitudes.

Further studies might include the effect of TMAl arrows on tile edges. The difficulty comes from the fact that such arrowing exist not only on tile edges but also inside the tiles. At high temperatures, only 72^{∘} angles interactions survive, some freedom is left to the system. Further investigations for the different system configurations can shed some light on the favored structures imposed by our Hamiltonian. Phason disorder along the periodic axes is important as we showed and might be studied more thoroughly. Finally, the system’s behavior under relaxation and the preferred relaxed atomic positions are wide areas to explore.

ACKNOWLEDGMENTS

The authors like to thank C. Henley and Y. Wang for useful discussions. IA wishes to thank King Abdul Aziz University (Saudi Arabia) for supporting his study and MW acknowledges support by the National Science Foundation under grant DMR-0111198. We thank the Pittsburgh Supercomputer Center for the computer time for this study.

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[10] I. Al-Lehyani, M. Widom, Y. Wang, N. Moghadam, G.M. Stocks and J.A. Moriarty, Phys Rev. B64, 075109 (2001)

[11] See for example: A. Yamamoto, K. Kato, T. Shibuya and S. Takeuchi, Phys. Rev. Lett. 65, 1603 (1990); S.E. Burkov, Phys. Rev. B 67, 614 (1991); M. Torres, G. Pastor, I. Jimenez, J.L. Aragon and D. Romeu, Scripta Metallurgica et Materialia 27, 83 (1992); G. Trambly de Laissardiere and T. Fujiwara, Phys. Rev. B 50, 9843 (1994) and G. Zeger and H.-R. Trebin, Phys. Rev. B 54, R720 (1996)

[12] See for example: L.X. He, Y.K. Wu and K.H. Kuo, J. Mater. Sci. Lett. 7, 1284 (1988); A.P. Tsai, A. Inoue and T. Masumoto, JIM Mater. Trans. 30, 300 (1989); H. Chen, S. E. Burkov, Y. He, S.J. Poon and G.J. Shiflet, Phys. Rev. Lett. 65, 72 (1990); A.R.

Kortan, R.S. Becker, F.A. Thiel and H.S. Chen, Phys. Rev. Lett., 64, 200 (1990); W. Steurer and K.H. Kuo, Acta Cryst. B46, 703 (1990); K. Hiraga Sci. Rep. RITU A, 36, 115 (1991); B. Grushko, R. Wittmann and K. Urban, J. Mater. Res. 9, 2899 (1994); K. Saitoh, K. Tsuda, M. Tanaka, A.P. Tsai, A. Inoue and T. Masumoto. Phil. Mag. A 73, 387 (1996) and K. Edagawa, K. Suzuki and S. Takeuchi, Phys. Rev. Lett. 85, 1674 (2000)

[13] For LSMS method see: Y. Wang, G.M. Stocks, D.M.C. Nicholson, W.A. Shelton, Z. Szotek, and W.M. Temmerman, Phys. Rev. Lett., 75 2867 (1995)

[14] M. Widom, I. Al-Lehyani, Y. Wang and E. Cockayne, Matter. Sci. Eng. A295, 8 (2000)

[15] G. Kresse and J. Hafner, Phys. Rev. B 47, RC558 (1993); G. Kresse, Thesis, Technische Universität Wien 1993. G. Kresse and J. Furthmüller, Comput. Mat. Sci. 6, 15-50 (1996). G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).

[16] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).

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[18] P. Villars and L.D. Calvert,Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed. (American Society of Metals, Metals Park, Ohio, 1991)

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[20] J. Hafner, From Hamiltonians to Phase Diagrams (Springer-Verlag, Berlin Heidelberg, 1987).

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Y. Wang and M. Widom, Phys. Rev. B 65, 104205 (2002)

[24] P.A. Bancel, Phys. Rev. Lett. 63, 2741 (1989)

TABLES

TABLE I. Basic Tiles in HBS model and their compositions

tile  Composition  Penrose tiles
H  Al_{17}Co_{5}Cu_{3}  2T+F
B  Al_{29}Co_{8}Cu_{4}  T+3F
S  Al_{41}Co_{11}Cu_{5}  5F

TABLE II. The approximant tilings we use for our study , their compositions and the number of different decorations (Nd) of each of them. The unit cells are either orthorhombic (a, b are given) or monoclinic (a, b and θ are given). All of them have c=4.14 Å in the z-direction. The number of independent k-points is the number on which most of the structures are calculated. To investigate the convergence we go higher for a few structures (see tables III and IV).

Tiling  figure  Composition  a, b (θ)  Indep. K-points  Nd
H_{1}  3  Al_{17}Co_{5}Cu_{3}  12.14, 7.50 (72^{∘})  132  3
B_{1}  4  Al_{29}Co_{8}Cu_{4}  12.13,12.13 (108^{∘})  88  6
H_{2}  3  Al_{34}Co_{10}Cu_{6}  23.08, 7.56  66  5
H′_{2}  5  Al_{34}Co_{10}Cu_{6}  14.27,12.13  66  4
B_{2}  6  Al_{58}Co_{16}Cu_{8}  12.13,30.30 (134.5^{∘})  55  3
B_{2}H_{2}  2a  Al_{92}Co_{26}Cu_{14}  19.63,23.08  20  27
B_{2}H′_{2}  2b  Al_{92}Co_{26}Cu_{14}  19.63,23.08  20  6
S_{1}H_{3}  2c  Al_{92}Co_{26}Cu_{14}  19.63,23.08  20  12

TABLE III. Energies of approximant B_{1} and one of its variants (single swap). Our convergence investigation goes through several k-point grids. Isotropic k-point distributions are its 1x1x3 mesh and its multiples. For finer grids 4x4x11 is more isotropic than 4x4x12.

K-point grid  Indep. K-points  E_{b1}  E_{b1−sw}  ΔE=E_{b1−sw}-E_{b1}
1x1x1  1  -186.83238  -186.78158  0.05080
1x1x2  1  -217.02636  -216.68522  0.34114
1x1x3  2  -215.95009  -215.63608  0.31401
2x2x2  4  -218.18586  -217.90238  0.28348
2x2x4  8  -216.75311  -216.51774  0.23537
2x2x6  12  -216.74436  -216.49251  0.25185
3x3x3  14  -216.59210  -216.32712  0.26498
3x3x9  41  -216.74029  -216.48619  0.25410
4x4x4  32  -216.77625  -216.53784  0.23841
4x4x11*  88  -216.74116  -216.48698  0.25418
4x4x12  96  -216.74323  -216.48616  0.25707
5x5x15  188  -216.74291  -216.48615  0.25676

TABLE IV. Similar results for approximant B_{2}H_{2}.

K-point grid  Indep. K-points  E_{B_{2}H_{2}}  E_{B_{2}H_{2}−sw}  ΔE=E_{B_{2}H_{2}−sw}-E_{B_{2}H_{2}}
1x1x1  1  -606.30421  -606.22773  .07648
1x1x2  1  -704.05048  -703.77620  .27428
1x1x4  2  -700.48922  -700.24548  .24374
2x2x2  4  -704.02670  -703.77286  .25384
1x2x8  8  -700.38011  -700.12094  .25917
2x2x4  8  -700.73700  -700.49654  .24046
1x2x10  10  -700.42044  -700.15409  .26635
2x2x8  16  -700.44533  -700.18672  .25861
2x2x10*  20  -700.45159  -700.19033  .26126

‘

TABLE V. Fitting our data with different number of parameters. The standard deviation for each set is reported in the last row. Including the 144^{∘} angle interactions improved the fit as shown

Parameters  λ_{72} = λ_{144} = 0  λ_{72} = 1, λ_{144} = 0  λ_{72} = λ_{144} = 19-para
E_{t}^{H}  -133.17  -133.15  -133.15
E_{t}^{B}  -216.42  -216.76  -216.77
E_{t}^{S}  -298.32  -300.08  -300.15
2E_{t}^{B}-E_{t}^{H}-E_{t}^{S}  -1.35  -0.29  -0.24
E_{72}^{i}  -  0.55  0.55
E_{72}^{m}  -  0.23  0.22
E_{72}^{o}  -  0  0
E_{144}^{i}  -  -  0.037
E_{144}^{m}  -  -  -0.003
E_{144}^{o}  -  -  -0.034
l-sym  -  -  -
s.d.  0.37  0.0026  0.0013

TABLE VI. The energy costs of significant parameters in our model. All are 9-parameter fit

Parameters  Energy (VASP)  Energy (mGPT)  Energy (mock-T)
(eV)  (eV)  (eV)
2E_{t}^{B}-E_{t}^{H}-E_{t}^{S}  -0.24  -0.59  -0.13
E_{72}^{i}  0.55  0.60  0.12
E_{72}^{m}  0.23  0.31  0.05
E_{72}^{o}  0  0  0
E_{144}^{i}  0.037  -0.042  -0.0079
E_{144}^{m}  -0.003  0.010  0.0001
E_{144}^{o}  -0.034  0.032  0.0080

FIGURES

FIG. 1. HBS tiles and their decompositions to Penrose tiles (a) and atoms (b). In (b), only TM and Symmetry breaking Al atoms are shown.

25

FIG. 2. Space can be tiled in many ways using HBS tiles. All these approximants (B_{2}H_{2}, B_{2}H′_{2} and S_{1}H_{3}) have 132 atoms per unit cell. Structures in (b) and (c) differ by a phason flip outlined in (c) with a small dashed line. Bonds in dashed shapes in (a) are bonds that can give information about pure angle interactions. Thin lines outline τ^{2} deflated hexagon and also indicate bond used in vertical stacking fault.

FIG. 3. Filling space with hexagons. This approximant has one hexagon per monoclinic cell (fine dashing, H_{1} structure in the text). The unit cell has 25 atoms. The cell can be doubled to get 50-atom orthorhombic unit cell (coarse dashing, H_{2} structure in the text).

FIG. 4. Single boat (B_{1}) approximant. It contains 41 atoms per monoclinic cell. Swapping the pair in the small rectangle changes the distribution of angle decorations. The two 144^{∘}’s surrounding the middle bond are images of each other. So total cancellation of 144^{∘} interactions is not possible in this unit cell. Enlarging the unit cell gives the freedom of assigning different decorations to the outer bonds to cancel that.

FIG. 5. Orthorhombic two hexagon approximant (H′_{2}). Different arrangements of hexagons from fig 3. Even with a single tile, space can be covered in different ways.

FIG. 6. Two boat approximant (B_{2}). One of the keel bonds (circled) in the boats participates only in 144^{∘} angles.

30

FIG. 7. plots of calculated structure energies vs. our model expectations using only tile energies (turning off angle interactions). The spread of energies vertically is due to angle contribution not accounted for in the model energies. The graphs in the set are on different scales to keep it consistent with the coming set of graphs.

FIG. 8. Turning the angle interactions back on certainly improves the fitting. All the approximants in the first graph spread over the range shown for better comparison