Co-rich Decagonal Al-Co-Ni
predicting structure, orientatio
order, and puckering
Nan Gu, Marek Mihalkovič, and C.L. Henley,
Cornell University
ICQ9 poster, Monday May 23, 2005
Supported by DOE grant DE-FG02-89ER-45405
1
Problem: structure prediction for d(Al
A fundamental problem: Given we can perfectly compute
energy of a complex compound, what’s the atomic arrange
Brute-force Monte Carlo (MC) / Molecular Dynamics (MD
typically get stuck in glassy states.
Try this on d(AlCoNi) which has many subphases [S. Rits
Phil. Mag. Lett. 78: 67 (1998).] in particular, ‘basic Ni’ (
Al70 Co10 Ni20 ) and ‘basic Co’ (∼ Al70 Co20 Ni10 ).
Input information:
Quasilattice constant a0 = 2.45Å, period (2 layers) c =
Pair potentials
2
Pair potentials
Derived using Moriarty’s ‘Generalized Potential Theory” [
Al-Lehyani et al Phys. Rev. B 64, 075109 (2001)]
Al-Al potential:
just hardcore ∼ 2.7Å.
Al-TM (esp. Al-Co):
deep well ∼ 2.5Å.
TM-TM potential:
deep 2nd well ∼
0.5
0.5
0.4
0.4
0.3
0.3
Al-Al
Co-Co
Al-Co
Co-Ni
0.2
Al-Ni
Pair potential (eV)
Pair potential (eV)
0.2
0.1
0.0
0.1
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
Ni-Ni
-0.3
0.0
2.0
4.0
6.0
8.0
10.0
0.0
2.0
Distance (A)
4.0
6.0
Distance (A)
3
8.0
10.0
Our recipe – Multiscale Procedur
Represent structures as decorations of Penrose rhombi
1 MC simulations with atoms as lattice gas on a fixed li
sites, allowing atom swaps and ’tile-flips’ with a0 = 2.
rhombi. Lowest energy configuration in each run is vie
identify common motifs.
2 Promote the observations [1] to rules for a larger scale
with fewer degrees of freedom.
3 New MC simulations (fixed sites) on larger-scale tiling
4 Relaxation (and MD simulations) to find atoms’ true
equilibrium positions.
This method worked once for “basic Ni” [M. Mihalkovič e
Phys. Rev. B 65, 104205 (2002). Will it work in general?
4
Simulations:
First stage
rhombus edge a0 = 2.45Å
Two independent layers
Tile flips and atom swaps.
Second stage
rhombus edge τ a0 =
One layer tiling //
two-layer decor
of candidate sit
2 flavors of fat rhom
Atom swaps only
We used tilings that maximize number of (nonoverlapping
decagons. In effect, we really have a 10.4Å edge Binary
broken down for decoration purposes into a 4.0Å edge De
Hexagon-Boat-Star tiling
5
Simulation image
Left side – 2.45Å tiles
...
Right side – 4.0Å ti
Large and small circles indicate top and second layer.
Gray = Al
Black = Ni
Blue= Co
6
Cluster motifs
Secondary motif “Star c
(fills spaces):
Dominant motif “13Å Decagon”
(really 12.8Å):
ring 1 ring 2 ring 2.5
ring 3
Al
Co
# $#
$
Co/Ni
Ni
Concentric rings... some
irregularities on rings 2.5/3.
Circled: are puckering channels
when relaxation allowed
"!"! "!"!
Ni/Al
Ring of five “TM” site
∼30% TM (context dep
erwise Al.
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Decoration on 10.4A Binary tiling
These clusters are centered respectively on rhombus vertic
“Binary Tiling” with edge τ 3 a0 = 10.4Å.
Ni(s )
Co(1)
Ni(3)
Co(3)
Ni(s)
“+” “-” mark orientations σi of TM’s in centers. Other
ings refer to a rule for TM decoration, which depends o
orientations. Composition around Al70 Co21 Ni9 .
8
Another version of idealized decorat
This image emphasizes a decomposition into 2.45Å He
Boat, and Star tiles around 2.45Å Decagon tiles. The di
possible tilings of the HBS tiles correspond to options for
in rings 2.5/3, all of which maximize the number of Al-Co b
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Orientational order
Define σi = ±1 on each cluster, like an “Ising model” s
expresses the center’s orientation (also controls in which lay
TM’s of ring 1 sit.) What is the pattern of orientations?
For the (artificial!) case of fixed site list, the energy diff
is delicately balanced. We often find a “ferromagnetic” ar
ment, which means the point symmetry is pentagonal – i
pentagonal symmetry occurs in part of the Al-Co-Ni phas
gram. This can be understood by matching the edges of th
Decagon cluster: Al atoms alternate in which way they
correlated with center orientation.
At low and high densities, or in a approximant with
Star clusters,“antiferromagnetic” arrangment is preferred
can also be rationalized, because it avoids adjacent Ni-N
forming a 72◦ angle in the Star cluster.
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Relaxation and puckering
Under relaxation, we found rearrangements of atoms in
2.5/3. Also, the period doubled to c0 = 2c = 8Å, the re
riod for most d(AlCoNi) subphases [when simulation was t
a sample cell of that size]. Why?
(a).
(b).
τ a0
TM
4A
z
Co
channel
c
Al
4A
ideal
position
U(z)
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Explanation of Al in channels...
We can define “Al potential”
X
X
0
UAl (r) ≡
VAlCo (r − r ) +
VAlNi (r − r00 )
r0
r00
given fixed positions of TM’s. Isolated wells of this functi
filled by a single Al, but there are also “channels” of po
minima, running between columns of Co in the c direction.
is just room to place three Al in every four layers, hence
c0 = 2c. The placement of the atoms is determined by (
timum single-atom potential [plot (b) on previous sheet]
in the atom layer with more TM’s in distant locations (ii)
to have maximum transverse displacement to satisfy Al-A
core, since 2c/3 ≈ 2.67Å is a little too close.
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Al potential map
Left plot shows Al potential map in the TM poor layer (at
TM rich layer (at right). In middle is atom configuration
layers (actual sites after relax - MD - relax). TM poor layer
that will pucker) is larger circles. At right is a slice of the
map along the c axis. Double lines show intersection of sli
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Interaction of channels
Adjacent channels pucker in opposite sense
as shown, to avoid Al-Al contacts in mirror layers. The actual top view (rihgt) is a
“crooked cross” of 2Al+2Al (in projection)
around a column of Co.
puckered
mirror
puckered
mirror
puckered
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Global puckering pattern
Puckering pattern in the 23×32×8 Å cell. Key: black/u
shows sign and size shows magnitude of displacements.
rings of puckered atoms.
15
Simulated vs. actual W(AlCoNi)
The top figure is a
mirror layer; bottom
is a puckered layer.
Empty circles are
experimentally determined sites: teeth
= TM, smooth=Al,
large
=
mixed
Al/TM. Filled circles
are
simulations:
Dark gray = TM,
light gray = Al.
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Conclusions
Methods of Mihalkovic et al (2002) for ‘basic Ni’ d(Al70
successfully led to a structural understanding (at T = 0) o
Co’ d(Al70 Co20 Ni10 : a completely different tiling (domina
decagon clusters) despite a very similar nearest-neighbor o
We have described the origin of puckering and orientati
order. The fixed-site list simulations can lead to the wron
conclusion for low-energy effects, unless relaxations are ap
intelligently.
When applied to a unit cell the same size as crystalline
approximant W(AlCoNi) [Sugiyama K. Sugiyama et al J.
Comp. 342: 65 (2002)] many details of our prediction agre
not all.
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