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Ab-initio energetics of transition metal ordering in decagonal
Al-Co-Cu
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Michael Widom\footnote{Corresponding author. Tel.: 412-268-7645; fax 412-681-0648. E-mail address: widom@andrew.cmu.edu} and Ibrahim Al-Lehyani
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Department of Physics, Carnegie Mellon University, Pittsburgh, PA  15213 USA
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Yang Wang
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Pittsburgh Supercomputer Center, Carnegie Mellon University,
Pittsburgh, PA 15213 USA
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Eric Cockayne
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Ceramics Division, 
Materials Science and Engineering Laboratory, NIST, Gaithersburg, MD 20899-8520
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\today
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{\bf Abstract}\vspace{2ex}\\
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We perform ab-initio total energy calculations for approximants to a
model of decagonal AlCoCu. These calculations support previously
proposed chemical ordering of Co and Cu atoms along tile edges.
Preferred arrow orientations are identified and shown to obey a subset
of the single-arrow Penrose matching rules.
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{\it Keywords:}\/
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Decagonal Quasicrystal, AlCoCu, ab-initio
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\section{Introduction}
\label{intro}
A fundamental question in the field of quasicrystals is the origin of
thermodynamic stability of these non-periodic structures. Competing
explanations range from {\em energetic stabilization}~\cite{energy}
utilizing matching rules such as those that force quasiperiodicity in
the Penrose tiling~\cite{gardner}, to {\em entropic
stabilization}~\cite{entropy} focusing on the configurational entropy
available in random tiling models~\cite{RTM}. Experimental evidence
fails to unambiguously settle the matter, and the true situation is
certainly more complex than either of the two extremes just described.
Considerable progress has been made in two areas that, together, pave
the way for additional theoretical insight on the problem.  First,
highly reliable structural models identify the positions and chemical
identity of the majority of atoms in a variety of decagonal and
icosahedral phases. Second, ab-initio calculations based upon
electronic density-functional theory now yield meaningful total
energies for small quasicrystal approximants.

Quasicrystal structures may be effectively described as coverings of
space by shapes known as tiles, each tile being decorated in some
manner by atoms. The preferred positions and chemical identities of
atoms within a tile may be coupled to the environment in which the
tile sits. Conversely, the environment is influenced by the presence
of the tile and its atomic decoration. This paper explores the
coupling between tile decoration and local tiling structure in
decagonal AlCoCu. We focus especially on the chemical ordering
between Co and Cu atoms along tile edges~\cite{CW}. We demonstrate
that chemical ordering exists, assigning arrows along tile edges.
Given the existence of arrows along tile edges, we investigate the
relative energies of different arrow configurations and show these
energies can be modeled by interactions between arrows. We find that
the favored arrowing of tile edges sharing 72$^{\circ}$ vertices is
consistent with a subset of the Penrose matching rules.

\section{Model and calculational method}
\label{model}
We recently introduced a model for decagonal AlCoCu based upon tilings
of space by hexagon, boat and star-shaped tiles (HBS tilings)
decorated deterministically with atoms~\cite{CW}. A key feature of the
model is the zigzag chains of alternating Cu and Co atoms extending
along the periodic axis. Within the zigzag, Cu and Co atoms are
separated by a near-neighbor distance of 2.55~\AA~ consisting of a
1.52~\AA~ separation within the quasiperiodic plane and a
$c/2$=2.04~\AA~ separation along the periodic ($c$) axis. We may
describe the Cu-Co chain geometry, when projected onto the
quasiperiodic plane, as an 'arrow' pointing from the Cu atom towards
the Co atom. The directions of arrows in our deterministic model is in
perfect agreement with the Penrose 'single arrow' matching
rules~\cite{rules}. It is not known if this special arrangement of
arrows is favored energetically.  If there were a unique ground state
exhibiting the arrowing of the Penrose tiling, then it is conceivable
that quasiperiodicity in this model would be forced energetically.

Our original model was based on mock-ternary pair potentials. Pair
potentials provide only an approximation to the true total energy
because they neglect many-body interactions that might be important,
especially for transition metal atoms with two or more transition
metal neighbors as indeed occur along tile edges. Here we consider Cu
as a transition metal atom due to the proximity of its d-orbits to the
Fermi level. An added uncertainty in our previous calculations involved
our treatment of the Cu interactions. In the absence of genuine Cu potentials
we treated Cu as a 50\%-50\% mixture of Al and Co.


In the present paper we employ a more rigorous all-electron ab-initio
approach utilizing the Locally Self-consistent Multiple
Scattering~\cite{LSMS} (LSMS) method based upon density functional
theory. This is an order-N method that is particularly well adapted to
ab-initio calculations on alloy systems with complex structures. The
major advantage of order-N methods is their computational effort
scales linearly with the number of atoms in the unit cell. We employ a
muffin-tin approximation for the potential and von Barth-Hedin
exchange-correlation for the local density approximation. Each
processor in a parallel supercomputer treats the electron scattering
problem for an individual atom, including multiple scattering effects
associated with neighboring atoms within a "local interaction zone".
Local charge density and one-electron potentials are iterated until
self-consistency is achieved. Each self-consistent iteration takes
approximately 1 minute on a Cray-T3E system, regardless of the number
of atoms in the structure under consideration.  Typically 50-100
iterations are required to achieve convergence for a 50 atom structure
starting from pure elementary potentials.

The muffin-tin approximation for potentials can be dangerous when
applied to the structures in which different atoms have different
muffin-tin radii. This may lead to non-physical energy shifts of their
semi-core levels, and invalidate the energy change of atomic swap
between inequivalent sites. All atoms involved in our swap lie on
geometrically equivalent tile edge sites and therefore share equal
muffin-tin radii. Under this condition, energy change calculated by
LSMS are comparable to those calculated by VASP~\cite{VASP,Nassrin},
an electronic structure calculation package using a plane wave
pseudopotential method.

\begin{figure}[htb]
\epsfxsize=3.5in
\hspace{-0.25in}
\epsfbox{fig.ps}
%\vspace{0.25in}
\caption{(a) Monoclinic and (b) orthorhombic tilings, both utilizing
orthorhombic cells as indicated with dashed lines. Open circles denote
Al, Black disks denote Co and gray disks denote Cu. Atoms labeled a-l
are discussed in the text.}
\label{structs}
\end{figure}

Because our method demands periodic boundary conditions we must carry
out our calculations for quasicrystal ``approximants'', periodic
crystalline structures whose complex unit cells contain fragments of
the infinite quasicrystal structure.  The calculations reported in
this paper treat two distinct 50 atom structures containing only hexagonal
tiles, two tiles per unit cell. The atomic formula for both structures
is the same, Al$_{34}$Co$_{10}$Cu$_{6}$. These structures are
illustrated in figure~\ref{structs}a and b. Space group and lattice
constants are, respectively, symmetry mC with a cell $23.3 \times 7.6
\times 4.09$~\AA$^3$ and oP with a cell $14.4 \times 12.3
\times 4.09$~\AA$^3$. All atoms occupy idealized positions defined by the model
of Cockayne and Widom - no relaxation has been performed. The atoms
are all at height $c/4$ or $3c/4$ except for the cell vertex and
center atoms that are at height 0.

\section{Results}
We employ swaps of atoms among the labeled sites in
figure~\ref{structs} to convert directed (CoCu) bonds to pure (CoCo
and CuCu) bonds and also to reverse bond directions. All energy
changes are reported ``per cell''. Because of the periodic boundary
condition, the energy changes are calculated as if the swaps are
preformed simultaneously among all periodic images of the calculation
cell. Approximate energies for truly localized swaps may be inferred:
the $a$ and $b$ lattice constants are sufficiently large that
interaction of a swap bond with its periodic images in the $x-y$ plane
should be quite weak; Interaction with images displaced by $\pm c$ are
quite strong because they involve nearest-neighbor atoms, but may be
estimated because they create pure (CoCo and CuCu) bonds whose
energies we will determine.

To present our data we need a notation for different structures
considered.  Those based upon the monoclinic and orthorhombic
structures depicted in figures~\ref{structs}a and b are denoted,
respectively, M and O. Atomic swaps are denoted with suffixes of the
form Sxy, where x and y are the labels of the sites whose atoms have
been swapped. For example, MSab reverses a directed bond, and MSdg
replaces the pair of horizontal directed bonds, in
figure~\ref{structs}a with a pair of pure bonds.  Energy differences
are reported in table~\ref{data} between pairs of structures, always
subtracting the lower energy structure from the higher, resulting in
positive energy changes.


\begin{table}[htb]
\caption{Energy differences among structures. Units eV/cell.}
\begin{tabular}{||l|r||l|r||l|r||}
\hline
  Comparison  & Energy    &  Comparison   & Energy    \\
\hline
MSdg-M        & 0.168     &  OSab-O       & 0.130  \\
MSabScd-MSab  & 0.011     &  OSgh-O       & 0.099  \\
M-MSabScd     & 0.029     &  OSabSgh-O    & 0.230  \\
\hline
\end{tabular}
\label{data}
\end{table}


From our total energy calculations for the structures M and MSdg we
determine the energy difference MSdg-M=0.168 eV/cell. Apparently there
is a substantial energetic preference, 0.168 eV/cell, for directed
bonds as opposed to pure bonds. To gauge the strength of this
preference, we consider dimensionless energy units by dividing the
strength of a preference in eV by k$_{\rm B}$T=0.0862 eV at T=1000K, a
typical temperature at which quasicrystals form and are clearly
equilibrium phases. We must take care of the number of bonds involved
in each swap, recognizing the periodic boundary conditions.  For
example, in the swap MSdg, 4 directed bonds are lost and 4 pure bonds
created. Thus the energy change for swapping one pure CoCo and one
pure CuCu bond with a pair of directed CoCu bonds is 0.168eV/2=0.084eV
and the dimensionless energy change is 0.102eV/k$_{\rm B}$T = 0.97.

When interpreting energy differences in terms of bond strength we must
recognize assumptions underlying the interpretation. Of particular
importance are the spatial range and many-body nature of interatomic
interactions. Attributing the energy difference between structures to
the presence or absence of certain bonds implies a degree of spatial
localization of the interatomic interactions. Pair potentials used to
model interatomic interactions in similar compounds~\cite{Moriarty}
exhibit long-ranged decaying oscillations that are especially strong
between transition metal atoms. Many-body interactions, also, are
especially strong when one transition metal atom has multiple
transition metal near neighbors. Consequently, the bond strength
calculated here applies rigorously only to the case where entire
zigzags of alternating Co and Cu are replaced with entire zigzags of
pure Co and pure Cu.

Presumably the energetics of certain other bonding configurations
will be consistent in sign and rough magnitude with the calculation
given above. Consider stacking disorder, for example. Double the unit
cell of the monoclinic structure along the $c$ axis, and compare the
energy of the structure MSdgM (MSdg placed above the the structure M)
with the energy of the structure MM (M placed above itself). Counting
near-neighbor transition metal bonds we find 4 directed bonds are
replaced with 4 pure bonds in MSdgM compared with MM. It seems likely
that the energy difference MSdgM-MM will be positive and similar in
magnitude, but not {\em exactly} equal, to MSdg-M.

We extracted the energy for replacing directed bonds with pure bonds
using the monoclinic structure in figure~\ref{structs}a because the
bonds involved in this particular swap do not participate in any
72$^{\circ}$ vertices, allowing a clean separation of the energy
associated with chemical ordering from the energies associated with
arrow patterns at 72$^{\circ}$ vertices. There should be some energy
associated with 144$^{\circ}$ vertices, and some even longer-ranged
interactions with tile edges that do not share a vertex in common with
the edges involved in the swap. We presume that these interactions are
weak compared with the 0.168 eV energy found for this swap. Our later
discussion confirms this.

Now we evaluate the energetics of edge arrowing patterns at
72$^{\circ}$ vertices. We denote a vertex as outward, mixed or inward
according to whether both arrows point in the outward direction, one
is out and the other in, or both point inwards. The Penrose rules
demand arrows point outwards, but periodic boundary conditions force
at least two matching rule violations. In figure~\ref{structs}b the
arrow rules are satisfied (arrows point outwards) at two vertices
(efij and ghkl) but violated (arrows are mixed) at (abkl) and (cdef).
We believe this arrowing pattern represents the lowest possible energy
for the tiling of figure~\ref{structs}b.

Our numerical results supports this claim. By reversing bonds we may
vary the numbers of inwards, mixed and outwards vertices. for example,
interchanging atoms a and b converts the 72$^{\circ}$ vertex (abkl)
from mixed to inwards. Such a swap raises the total energy by 0.130
eV/cell. Similarly interchange of atoms g and h converts the
72$^{\circ}$ vertex (ghkl) from outwards to mixed. Such a swap raises
the total energy by 0.099 eV/cell.

The energy changes just presented include contributions from the
interaction of edge arrowing within a 72$^{\circ}$ vertex and also
interactions within 144$^{\circ}$ vertices. We can isolate certain
linear combinations of 72$^{\circ}$ and 144$^{\circ}$ interactions.
Comparing MSabScd with MSab isolates the combination
(2*mixed-in-out=0.011 eV/cell) of 144$^{\circ}$ vertex interactions
without any contribution from 72$^{\circ}$ vertex inteactions. In
contrast, comparing M with MSabScd isolates the combination
(2*mixed-in-out=-0.029 eV/cell) of 72$^{\circ}$ interactions without
any contribution from 144$^{\circ}$ vertices. It seems that the
72$^{\circ}$ interactions are indeed stronger than the 144$^{\circ}$
interactions, roughly by a factor of 3. Unfortunately, even comparing
all possible edge arrowing patterns in figures~\ref{structs}a and b it
is impossible to determine individually all 72$^{\circ}$ and
144$^{\circ}$ vertex interactions.

To gauge the strength of interactions among arrows that do not share a
vertex in common, we compare the energy change for two simultaneous
swaps with the sum of the single swap energy changes. For example, consider
the interchange of atoms a and b. We find that the resulting energy change
is independent of the orientation of the bond (gh) to the accuracy of
our calculations (about 0.001 eV/cell). Note that the bond (gh) is
located across the short axis of the hexagon from the bond (ab). The
Penrose rules demand that these bonds be parallel.

\section{Discussion}

Our conclusion, based upon these calculations, is that in the
decagonal phase of AlCoCu chemical ordering along tile edges is
favored and that the directionality of these bonds at
72$^{\circ}$ vertices tends to follow the sense of the Penrose arrowing.

The energy scale associated with chemical ordering and vertex
interactions is roughly comparable to k$_{\rm B}$T at T=1000K. At this
temperature there will be an evident preference for directed bonds
rather than pure bonds.  Among the directed bonds there will be an
evident preference for arrowing to follow the Penrose rules at
72$^{\circ}$ vertices. But the energies involved are sufficiently
small that a substantial minority of bonds will lack arrows, and among
the tile vertices with two arrowed bonds a substantial minority will
violate the Penrose rules. Monte Carlo simulations that distribute a
set of atoms among a set of allowed sites, weighted according to the
total energy of a given configuration, could shed considerable light
on the statistics and possible phase transitions of the model.

Several additional questions must be examined before the true ground
state may be identified. The HBS tile set is overcomplete, so
quasiperiodicity does not fix the relative numbers of hexagon, boat
and star tiles. Phason flips can transform hexagon-star pairs into
pairs of boats. The energetics of this transformation define a
``chemical potential'' for the tile numbers. Currently we do not know
the energy of this phason flip.

Another consideration is the role of atomic relaxation. Holding atoms
fixed on the ideal sites shown in figure~\ref{structs}, we found the
main interaction among edge arrows occurred at 72$^{\circ}$ and
144$^{\circ}$ vertices. This finding is consistent with interatomic
interactions that are not too long-ranged. However, elastic
interactions in solids typically fall off slowly raising the
possibility of energetic effects not yet accounted for in which the
edge arrowing couples to the elastic strain field.

Even focusing on short-ranged effects, atomic displacement may play a
role. For example, we choose to occupy the symmetric site at the
center of a hexagon (see the center of the unit cell in
figure~\ref{structs}b) with an Al atom. Additional, unoccupied, ideal
sites lie nearby, displaced slightly from the center along the long
axis of the hexagon. Possibly, the symmetry breaking of the hexagon
due to the orientation of the nearby bond (ab) could force the Al atom
to one of these nearby sites. This asymmetric Al position could, in
turn, influence the orientation of the nearby bond (gh), creating an
effective interaction between (ab) and (gh).

In conclusion, we find that chemical ordering on tile edges is
strongly favored in AlCoCu decagonal quasicrystals. Interactions among
arrow directions at 72$^{\circ}$ vertices favor arrowing patterns that
obey a subset of the Penrose matching rules. We note, however, that
this subset of the Penrose rules is insufficient to force
quasiperiodicity~\cite{private}. Furthermore, we expect significant
interactions associated with 144$^{\circ}$ vertices, which have no
counterpart in the Penrose rules.  Further calculations on larger
approximants are needed to identify this complete set of arrow
interactions and identify the ground state of the model.

\section{Acknowledgments}
We thank Nassrin Moghadam, G. Malcom Stocks, Chris
Henley and Marek Mihalkovic for useful discussions. We also thank
Chris Henley for his hospitality at Cornell University at the time
this project was begun.  This research is supported in part by the
National Science Foundation under grant DMR-9732567 and by the
Saudi-Arabian government. We thank the Pittsburgh Supercomputer Center
for grants of computing time.

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\begin{footnotesize}
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\begin{thebibliography}{99}
\vspace{3ex}
\itemsep 0.2ex
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%  REFERENCES
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\bibitem{energy} For a review of matching rules see K. Ingersent, 
in {\em Quasicrystals: the State of the Art}, eds. D.P. diVincenzo and
P.J. Steinhardt (World Scientific, Singapore, 1991)
\bibitem{gardner} M. Gardner, {\em Scientific American} {\bf 236} (1977) 110
\bibitem{entropy} M. Widom, K.J. Strandburg and R.H. Swendsen, 
{\em Phys. Rev. Lett.} {\bf 58} (1987) 706
\bibitem{RTM} V. Elser, {\em Phys. Rev. Lett.} {\bf 54} (1985) 1730;
For a review of random tiling models see C.L. Henley, in {\em
Quasicrystals: the State of the Art} eds. D.P. DiVincenzo and P.J.
Steinhardt (World Scientific, Singapore, 1991)
\bibitem{CW} E. Cockayne and M. Widom, {\em Phys. Rev. Lett.}
{\bf 81} (1998) 598
\bibitem{rules} Penrose matching rules employ two types of rhombus
edges: ``single-arrow'' and ``double-arrow''. When double arrow edges
are removed from a Penrose tiling, a tiling of hexagons, boats and
stars remains whose single-arrow edges force
quasiperiodicity~\cite{CW}. In this article, ``arrows'' refers to
single arrows on the edges of an HBS tiling.
\bibitem{LSMS} Y. Wang, G.M. Stocks, D.M.C. Nicholson, W.A. Shelton, 
Z. Szotek, and W.M.  Temmerman, {\em Phys. Rev. Lett.}, {\bf 75} (1995) 2867
\bibitem{VASP} G. Kresse and J. Furthuller, {\em Comput. Mater. Sci.}
{\bf 6} (1996) 15; G. Kresse and J.  Furthuller, {\em Phys. Rev. B} 
{\bf 54}, (1996) 11169
\bibitem{Nassrin} N. Moghadam performed VASP calculations for swaps
in AlCoNi alloys that compare reasonably with LSMS values.
\bibitem{Moriarty} J.A. Moriarty and M. Widom, {\em Phys. Rev. B}
{\bf 56} (1997) 7905; M. Widom and J.A. Moriarty, {\em Phys. Rev. B}
{\bf 58} (1997) 8967
\bibitem{private} We are indebted to M. Mihalkovic and C.L. Henley for
discussions on this point.
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