Boxes within boxes: discovering the atomic structure of an Al-Co-Ni quasicrystal

Transcription

1 Boxes within boxes: discovering the atomic structure of an Al-Co-Ni quasicrystal Nan Gu, Marek Mihalkovič, and C.L. Henley, Cornell University LASSP Pizza talk, Tues July 11,

2 1. Quasicrystals Fourier transform (diffraction) has Bragg peaks Rotational symmetry (e.g. 5-fold) excludes periodicity Images of Al 72 Ni 20 Co 8 : (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. 2

3 2. Effective pair potentials Derived using Moriarty s Generalized Potential Theory [I. Al-Lehyani et al Phys. Rev. B 64, (2001)] Depend implicitly on electron density. Friedel oscillations [cos(2k F r + δ)/r 3 ] from F.T. of sharp Fermi surface. Calibrated from LDA results for (comparatively) simple crystals. 3

4 Al-Al potential: just hardcore 2.7Å. Al-TM (esp. Al-Co): deep well 2.5Å. TM-TM potential: deep 2nd well 4.5Å. Al-Al Al-Co Al-Ni Co-Co Co-Ni Ni-Ni Distance (A) Distance (A) 4

5 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)] 5

6 (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. 6

7 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. 7

8 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 a 0 = 2.45Å, period (2 layers) c = 4.08Å Experimental density and composition. 8

9 Here is the Penrose tiling of rhombi. 9

10 Another important tiling Hexagon-Boat-Star. Basis for structure model of Al 70 Ni 20 Co 10 by Mihalkovič, Widom, Henley, and collaborators, (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.) 10

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12 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 (a 0 = 2.45Å 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 Al 70 Ni 20 Co 10, [M. Mihalkovič et al, PRB 65, (2002)]. Generally? Try on Al 70 Co 20 Ni 10, Nan Gu et al, 2005: it works, but trickier than we thought! 12

13 5. Simulations: First stage rhombus edge a 0 = 2.45Å Two independent layers Tile flips and atom swaps. Second stage rhombus edge τa 0 = 4.0 Å One layer tiling // two-layer decoration of candidate sites 2 flavors of fat rhombus; Atom swaps only 13

14 Simulation image (first stage result) This breaks up into Hexagon-Boat-Star like the Ni-rich composition did, but there are also 8Å 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). 14

15 Dominant motif 13Å Decagon (really 12.8Å): ring 1 Cluster motifs Secondary motif Star cluster (fills spaces): ring 2 ring 2.5 ring 3 Al Co Ni Concentric rings... some irregularities on rings 2.5/3. Circled: are puckering channels when relaxation allowed Co/Ni Ni/Al Ring of five TM sites: only 30% TM (context dep.), otherwise Al. 15

16 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 10.4Å. Ni(s ) Co(1) Ni(3) Ni(s) Co(3) + - mark orientations σ i of TM s in centers. Other markings refer to a rule for TM decoration, which depends on the orientations. Composition around Al 70 Co 21 Ni 9. 16

17 Another version of idealized decoration This image emphasizes a decomposition into 2.45Å Hexagon, Boat, and Star tiles around 2.45Å 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. 17

18 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. 18

19 7. Relaxation and puckering Under relaxation, a subset of atoms puckers (deviates from layer plane). Also, the period doubled to c = 2c = 8Å, the real period for most d(alconi) subphases Why? There are channels of potential minima, running vertically, with just room to place three Al in every four layers, hence the period doubles to c = 2c. (a). TM Co τ a 0 4 A (b). z Al channel c 4 A ideal position U(z) 19

20 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! 20

21 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. 21

22 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. 22

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