Boxes within boxes: discovering the

atomic structure of an Al-Co-Ni

quasicrystal

Nan Gu, Marek Mihalkovic, and C.L. Henley,

Cornell University

LASSP Pizza talk, Tues July 11, 2006

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1. Quasicrystals

Fourier transform (diffraction) has Bragg peaks

Rotational symmetry (e.g. 5-fold) excludes periodicity

Images of Al72Ni20Co8:

(L) electron microscope; (R) electron diffraction.

But this talk on how microscopic interactions relate to the atomic

structure; won’t focus on long range order/diffraction.

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2. Effective pair potentials

Derived using Moriarty’s ‘Generalized Potential Theory” [I.

Al-Lehyani et al Phys. Rev. B 64, 075109 (2001)]

Depend implicitly on electron density.

Friedel oscillations [cos(2kF r + δ)/r3] from F.T. of sharp Fermi

surface.

Calibrated from LDA results for (comparatively) simple crystals.

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Al-Al potential:

just hardcore ∼ 2.7A.

Al-TM (esp. Al-Co):

deep well ∼ 2.5A.

TM-TM potential:

deep 2nd well ∼ 4.5A.

Al-Al

Al-Co

Al-Ni

Pair

poten

tial (e

V)

Distance (A)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.0 2.0 4.0 6.0 8.0 10.0

Co-Co

Co-Ni

Ni-Ni

Pair

poten

tial (e

V)

Distance (A)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.0 2.0 4.0 6.0 8.0 10.0

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3. Where are (or should be) the atoms?

Fundamental problem of crystal chemistry: Given we can perfectly

compute total energy of a complex compound, what’s the atomic

arrangement?

Why?

(i) Diffraction is insufficient in quasicrystal case. Most

structure fits include some impossible distances; if you pluged into

an ab-initio code, the high energies from these rare (and spurious)

environments would swamp the small differences that decide the

equilibrium phase diagram.

Also, usually partial occupations – which combinations are

simultaneously occupied?

ternary, but X-rays (or electron diffraction) won’t separate

transition metals with similar atomic numbers [Co (Z=27) vs Ni

(Z=28)]

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(ii) Check agreement with experiment to see how far we can

trust the pair potentials.

(iii) Do interactions favor an essentially unique ground state

(matching-rule scenario) or does order emerge from an

entropy-dominated state (random tiling scenario)? Modeling may

be the fastest way to answer this old question.

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why is it so hard?

Implicitly demands comparing with infinity of possible structures!

What people do:

i. compare a few known candidate structures

Usually, too few!

ii. Brute-force Monte Carlo / Molecular Dynamics?

Typically gets stuck in glassy states.

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4. The solution......

An empirical fact: quasicrystal atomic structure is well described

by tilings. (That is, atoms are close to tile vertices.) The same tile

appears with different surroundings.

Input information:

Quasilattice constant a0 = 2.45A, period (2 layers) c = 4.08A

Experimental density and composition.

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Here is the Penrose tiling of rhombi.

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Another important tiling – Hexagon-Boat-Star.

Basis for structure model of Al70Ni20Co10 by Mihalkovic, Widom,

Henley, and collaborators, 2002. (Figure: Al=gray, Ni=black,

Co=blue; size is top or bottom layer.)

Note: you can describe the same configuration by tiles of different

size scales. (Note supertiling here which is also an H-B-S tiling.)

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4. Our recipe – Multiscale Procedure

Represent structures as decorations of Penrose rhombi

1 MC simulations, atoms as lattice gas on fixed list of ideal sites.

Allow atom swaps and ‘tile-flips’ (a0 = 2.45A rhombi). View

low E configuration from each run, identify common motifs.

2 Promote observations [1] to rules for larger tiles: fewer degrees

of freedom.

3 New MC simulations (fixed sites) on larger-scale tiling; iterate.

4 Relaxation/MD simulations to find true equilibrium positions.

Thus “boxes within boxes.”

It worked for “best case” Al70Ni20Co10, [M. Mihalkovic et al, PRB

65, 104205 (2002)]. Generally? Try on Al70Co20Ni10, Nan Gu et al,

2005: it works, but trickier than we thought!

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5. Simulations:

First stage

rhombus edge a0 = 2.45A

Two independent layers

Tile flips and atom swaps.

Second stage

rhombus edge τa0 = 4.0 A

One layer tiling //

two-layer decoration

of candidate sites

2 flavors of fat rhombus;

Atom swaps only

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Simulation image (first stage result)

This breaks up into Hexagon-Boat-Star like the Ni-rich composition

did, but there are also 8A Decagons with the same edge – in fact

5-fold symmetry extends farther out. But many defects present.

Idealizing clusters lowers the energy (so we make it a rule in the

next stage).

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Cluster motifsDominant motif “13A Decagon”

(really 12.8A):

Secondary motif “Star cluster”

(fills spaces):

Al Co Ni

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ring 2

ring 1

ring 2.5

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Concentric rings... some

irregularities on rings 2.5/3.

Circled: are puckering channels

when relaxation allowed

Ring of five “TM” sites: only

∼30% TM (context dep.), oth-

erwise Al.

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6. Decoration on 10.4A Binary tiling

At this composition, Decagon motif always appears. How are the

clusters placed relative to each other? We identified a tiling on a

larger scale – the “binary rhombus tiling”– with large/small cluster

centers on the vertices. The edge length is τ 3 times larger than the

original small rhombi – 10.4A.

Ni(s )Co(1)

Co(3)

Ni(3)

Ni(s)

“+” “-” mark orientations σi of TM’s in centers. Other mark-

ings refer to a rule for TM decoration, which depends on the

orientations. Composition around Al70Co21Ni9.

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Another version of idealized decoration

This image emphasizes a decomposition into 2.45A Hexagon,

Boat, and Star tiles around 2.45A Decagon tiles. The different

possible tilings of the HBS tiles correspond to options for the Al

in rings 2.5/3, all of which maximize the number of Al-Co bonds.

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Orientational order

How did we know the relative orientation of Decagons? (Since

each actually has 5-fold symmetry.)

Laborious to answer. Turns out best if the transition-metal atoms

(from the central Co ring) are in the same layers; this creates a

deeper modulation of the potential energy of an Al atom (at

second-neighbor distances), allowing certain variable Al atoms to

lower their energies. This happens if all Decagons oriented the

same – quasicrystal is pentagonal, not decagonal!

Note there are competing contributions: answer is affected by

composition. Get wrong if we stick with the discrete-site

approximation – need MD/relaxation.

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7. Relaxation and puckeringUnder relaxation, a subset of atoms “puckers” (deviates from

layer plane). Also, the period doubled to c′ = 2c = 8A, the real

period for most d(AlCoNi) subphases

Why? There are “channels” of potential minima, running verti-

cally, with just room to place three Al in every four layers, hence

the period doubles to c′ = 2c.

A4c

aτ 0 A4

idealposition

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U(z)

(b).

Alz

(a).

TM Co

channel

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Puckering pattern...

An obvious result (related to the orientation order of clusters) is

that every other layer becomes singled out as a mirror layer (flat by

symmetry and richer in transition metal atoms), the other layers

being puckered alternately. Nearby channels interacts: adjacent

ones want to have opposite puckering sense, and more globally the

interaction is frustrated. Again, we have several important length

scales – boxes within boxes again!

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Conclusion

Our approach “works” in that prominent features match

experiment (NiNi pairs and decagons with triangle feature in

center, in previous Ni-rich case; decagons and overall 5-fold

symmetry, in present Co-rich case; puckering). The inherent bias in

starting with fixed-site simulations doesn’t necessarily prevent us

from finding the correct answers for relaxed structures.

But it’s an art, not a science. We missed some things by not trying

a broad enough range of densities, compositions, and system sizes

in the initial stages, and by taking too big a step in devising a

constrained model. To do better on some of these points requires

technical improvements on the code which are underway. Further

simulations are being done by undergrad Sejoon Lim, which (so

far) explored toy models in which matching rules do emerge from

potentials, also the Al-Ni-Fe decagonal phase.

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Oops!

One run of the more constrained simulation, using a somewhat

enlarged site list, showed emergence of a larger decagon unit. The

local order is very similar, but at large scales it has a different set

of supertiles. Quite possibly, both structures are “correct” in that

they occur for some nearby composition – the Al-Co-Ni decagonal

phase is known to be fragmented into modifications with various

larger scale modulations.

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