Atomic dynamics in Al-rich Al-Co alloys near the composition of the decagonal quasicrystal
M. Mihalkovic
ˇ
, H. Elhor, and J.-B. Suck
Materials Research and Liquids, Institute of Physics, TU Chemnitz, D-09107 Chemnitz, Germany
Received 17 October 2000; published 14 May 2001
We used realistic Al-Co pair potentials R. Phillips, J. Zou, A. E. Carlsson, and M. Widom, Phys. Rev. B 49,
9322 1994; J. A. Moriarity and M. Widom, ibid. 56, 7905 1997 to study the atomic dynamics of Al
1-x
TM
x
crystalline structures with the fractional content x of the transition metal TM atom x 0.3. Our list comprises
rather simple structures of Al
3
Ni and Al
5
Co
2
alloys, complex structures related to the decagonal quasicrystal
(Al
9
Co
2
Ni, O-Al
13
Co
4
), and a model of the crystalline approximant of the decagonal quasicrystal d-AlNiCo.
Within the harmonic approximation, we assess the impact of the structural complexity on the phonon density
of states, sound velocity, Debye-Waller factor, and the character of the phonon states at low energies. In
complex structures related to the decagonal quasicrystal, a significant fraction of low-energy vibrations have
nonacoustic, strongly localized character. In a molecular-dynamics annealing of the decagonal approximant
model performed at elevated temperature, a fraction of aluminum atoms display signs of diffusive motion,
while the equilibrium positions of the cobalt atoms do not change.
DOI: 10.1103/PhysRevB.63.214301 PACS numbers: 61.44.Br, 63.20.-e
I. INTRODUCTION
One representative class of quasicrystals with both de-
cagonal and icosahedral phases are Al-rich aluminides, in
which icosahedral i-AlPdMn and decagonal d-AlNiCo form
at certain composition quasicrystalline structures, competing
in the perfection of the topological order with good-quality
periodic crystals. Understanding the energetic origins of qua-
sicrystal ordering requires comparison of the structural ener-
gies of quasicrystal models with energies of the crystalline
structures with similar composition. A promising systematic
approach is based on recent developments in modeling pair
interactions in Al-rich aluminides: semiempirical pair poten-
tials have been designed and tested for Al-Mn Ref. 3 and
Al-Co systems,
1
and ab initio generalized pseudopotential
theory GPT potentials for a range of Al-TM transition
metal systems.
2
The latter potentials were applied in a study
of Al-Ni and Al-Co binary phase diagrams.
4
In this paper, we report on a complementary exploration,
focusing on various aspects of the atomic dynamics in the
Al
1 -x
Co
x
system, with x ranging up to 0.3. Our primary
motivation is to obtain a comprehensive picture of the de-
pendence of the dynamical properties on the structure. The
stable phases occurring in the binary Al-Co and Al-Ni and
ternary Al-Ni-Co systems with the transition-metal content
ranging from x =0.2–0.3 provide particularly suitable
grounds for such a study: the rather simple crystalline struc-
tures of Al
3
Ni and Al
5
Co
2
stand in contrast to Al
9
Co
2
Ni and
O-Al
13
Co
4
structures with a close relationship to the decago-
nal quasicrystals near Al
70
(Ni,Co)
30
composition.
Although the details of the quasicrystalline structures re-
main uncertain, the amount of experimental and theoretical
work on their vibrational properties for a review, see Ref. 5
exceeds by far the available information on the crystalline
phases at similar compositions, the structures of which are
known accurately. In the long-wavelength limit, both icosa-
hedral ( i -AlPdMnRef. 6 and decagonal ( d -AlNiCoRef.
7 quasicrystals exhibit ordinary acoustic phonons. In
i-AlPdMn, the experiment found broad ( 4 meV optic
branches; the one with the lowest energy was at 8 meV. A
spring-model study applied to icosahedral tilings
8
predicted
the existence of optic branches with hierarchical structure,
with the lowest energy of an optic mode scaling inversely
with the size of the approximant unit cell; however, these
states had no impact on the phonon density of states that
scaled with
2
as expected from the Debye approximation,
and the low-energy modes had extended character. A study
using realistic pair potentials and atomic structure models of
Frank–Kasper icosahedral quasicrystals i-AlCuLi Ref. 9
and i-AlZnMg Ref. 10 revealed that besides the propagat-
ing excitations there exist other, spatially confined modes,
contributing appreciably to the density of states even in the
low-energy region. Recently, new experimental evidence of
nonacoustic low-energy modes became available: the low-
temperature heat capacity of i-AlPdMn Ref. 11 is larger
than the Debye value calculated from the sound velocities,
and low-energy tunneling states were found from low-
temperature variations of the sound velocity in i-AlPdMn,
12
i-AlCuFe,
13
and i-ZnMgY Ref. 14 quasicrystals. A similar
conclusion has been drawn from the comparison of the ex-
perimentally determined phonon density of states and model
calculations based on a simple structure model of d-AlNiCo
using the semi-empirical Al-Co pair potentials.
15
Our paper is organized as follows. In Sec. II, we discuss
the choice of the crystalline phases studied in this paper,
characterize their structures, and describe a pseudobinary
model of quasicrystal approximant in the d-AlNiCo sys-
tem. In Sec. II C we introduce semi-empirical
1
and GPT
2
pair potentials. While our selection of the crystalline phases
contrasts different structures, a parallel use of the two sets of
pair potentials tests the robustness of the conclusions we
draw. Section III provides the theoretical background for the
standard method we used to assess the dynamical properties:
the harmonic analysis via the direct diagonalization of the
complex dynamical matrices. Finally, our results are re-
viewed and discussed in Secs. IV and V.
II. ATOMIC STRUCTURES AND PAIR POTENTIALS
A. Model of quasicrystal approximant
One distinct feature of the decagonal quasicrystal struc-
tures near the Al
70
(CoNi)
30
composition are the decagonal
PHYSICAL REVIEW B, VOLUME 63, 214301
0163-1829/2001/6321/21430114/$20.00 ©2001 The American Physical Society 63 214301-1