ENERGETICS AND DYNAMICS OFCAGED Zn4 IN i-ScZn

Marek Mihalkovic and Christopher L. Henley

[Support: U.S. Dept. of Energy]

ICQ11 Poster P02-3, June 14, 2010, Sapporo Japan

1

Introduction

Recall: cluster in CaCd-type structure.

(a). Zn4 tetrahedron (focus of this poster)

(b). 20 Zn dodecahedron (very deformable!)

(c). 30Zn + 12 Sc (2nd shell)

At higher T , the tetrahedron can reorient; what is its dynamics,

and what determines the optimum orientation(s)?

2

Outline:

• Formulate eff. potential for orientations of Zn4 tetrahedron

• Find optimal orientations

• Theory: model the dynamics (as seen in neutron diffraction)

Based on molecular dynamics (MD) simulations using pair

potentials.

3

Framework for tetrahedron eff. Ham.

Degrees of freedom are labeled by rigid-body rotation matrix

(orientation) Ωi. (Actually: tetrahedra deform, but we assume this

just follows the orientation.)

One-body terms:

— an icosahedrally symmetric part

— a term with cubic symmetry of the 1/1 approximant.

Pair terms:

vij(Ωi,Ωj) orientations of tetrahedra in neighboring clusters.

Determine structural orderings (seen in CaCd case)

Probably mediated elastically.

4

Our “single Zn4” construct

This sample: all but one Zn4 → one (large) Sc atom.

(Out of 16 clusters in our 2 × 2 × 2 supercell.)

Important: constrained lattice const. same as in all-Zn4 sample.

Purpose: separate single-body and pair interactions.

potential is acting.

• The 1/1 approximant with Zn4 → Sc is stable against

competing phases over a wide range of Sc fraction substituted.

(Explains compositional variability in expts?)

• Behavior in single-Zn4 and all-Zn4 samples similar

⇒ we infer pair interactions negligible.

(From now on, only consider single-body terms.)

5

Pair potentials used

Our MD simulation based on these. (Ab-initio-based:

computationally demanding, in progress [Euchner, Stuttgart])

Salient features of pair potentials:

φZnSc(R) strongly attractive well at RSc ≈ 3.0A;

φZnZn(R) optimal distance RZn ≈ 2.8A.

6

2 3 4 5 6 7 8-0.1

-0.05

0

0.05

0.1

E[e

V]

Sc-ScZn-ScZn-Zn

7

Zn potential

Handy: “Zn potential” Φ(r) (analog of electrical potential ...)

ΦZn(r) =∑

i

φZn,i(r− ri),

i runs over neighbors; φZn,i(R) are the pair potentials

(Note: “Al potential” was used previously for analogous

(fluctuating) atoms

in d-AlCuCo [Gu, Mihalkovic, and Henley 2006]).

Result: A5 = lowest energy (best distance for Zn-Sc potential to

5-fold Sc; also good Zn-Zn distance to 3-fold Zn in cage).

8

Each curve runs from one symmetry direction to another:

— Am = an m-fold symmetry axis)

— A2c = cubic (100) axis (6 directions),

— A2 = other 2-fold (24 directions)

— A3c = cubic (111) (8 directions)

— A3c = other 3-fold (12 directions).

0 10 20 30 40 50-25

-24.5

-24

-23.5

-23

EN

ER

GY

[eV

]

A2->A3cA2c->A2A2c->A3A2c->A5A3->A2A5 -> A2A5-A3cA5->A3

A3c almost as low E (Zn along 〈111〉 far away),

but A3 is highest E (other 3 fold Zn are close, sterically bad)

9

Optimal orientations

ΦZn(r) ⇒ want to place all four Zn close to a 5-efold axis, but

possible for at most three:. Angle between tetrahedral directions is

109, angles between 5-efold directions are 63 and (best) 117.

We found two minimum orientations (evidence below).

(1) optimal orientation: 2mm symmetry around (say) (001) axis.

Zn(1,2) close to (±τ, 0, 1) [117 apart] whereas Zn(3,4) go with

(0,±1,−τ) directions [63 apart] – this pulls the latter pair closer

by ∼ 0.05A.

(2) secondary minimum: place Zn(1′,2′,3′) near to 5-fold axes

(−τ, 0, 1) and cyclic permutations, which are 117 apart. Put

Zn(4′) near (1, 1, 1) This has no symmetry (all four Zn atoms are

inequivalent, point symmetry “1”).

10

Evidence 1: movies

3 frames (352, 630, 705) from simulation of Zn4 in 20-Zn cage.

First and last are 2mm states; middle is a secondary min. state

Conclude: Transition path is via secondary minimum.

11

Evidence 2: probability density

"all Zn4"

"single Zn4"

T=210K, 2x2x2 supercell

z

x x

y y

z

z

x x

y y

z

Probability density for Zn atoms, projected in the (001) direction.

12

Vibrational density of states (VDOS)Dyn.; structure factor S(q, ω) from neutron scatt ⇒ VDOS

G(ω) ≡ limq→0

ω2

|q|2S(q, ω)

0 5 10 15 20 25 30 35E [meV]

0

0.02

0.04

0.06

0.08

VD

OS

partial Zn4, T=100Kpartial Zn4, T=210KVDOS total, 200 ps

Red/blue = partial Zn4 contributions at T = 100K/210K. Black

curve = whole spectrum. (called VDOS since it agrees with phonon

spectrum – if phonons are the only slow dynamics.

13

Previous plot matches calculated harmonic phonons – except at low

ω end)

0 0.1 0.2 0.3 0.4 0.50

0.005

0.01

0.015

0.02

Very low ω (T = 100K) showing (dominant) Zn4 contributions –

dynamics of occasional reorientations.

In both plots the axis is hω in meV

14

Theory: reorientation dynamics and VDOS

Can be shown

G(ω) = Fourier transform(Gv(t))

Gv(t)= velocity autocorrelation.

Our situation: a constant state for a long time, then randomly

has abrupt transitions to another discrete state. Described by

discrete master equation with transition rates Γαβ from state

α → β.

In our case, where position xi(t) is confined, it can be shown that

Gv(t) =d2〈xi(0) · xi(t)〉

dt2.

Thus, don’t need to know how large the velocities are during the

short and rare transitions!

15

For the simplest case of a double well we obtain

G(ω) =2

π

x20

τ

[

1 −1

1 + ω2τ2

]

.

This fits our data well for the 1/1 approximant (red line in

low-frequency plot of VDOS).

Note it does not fit simulation results for the 2/1 approximant.

Interpretation: the local environment is less symmetric than cubic

⇒ various transition barriers ⇒ spectrum of exponential relaxation

rates (not just the one term in fitting form.)

16

Discussion

1. Comparison to CaCd 1/1 approximant

Cd4 cluster diffraction study by C. P. Gomez et al, PRB 2003.

Our similar simulations for CaCd show that (in contrast to ScZn)

the optimal configuration is the asymmetric one, while the 2mm

confiuration is secondary.

2. Compare central cluster in i-AlPdMn

i-AlPdMn and i-AlCuFe built from “pseudo-Mackay” icosahedron

clusters with an irregular Mn Al7 core. The orientational dynamics

of this core calls for analysis like Zn4, but much more complicated.

17