Computational Materials Science 17 (2000) 178±181
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Spatial con®gurations of monoatomic gold chains
N.V. Skorodumova *, S.I. Simak
Department of Applied Physics, Chalmers University of Technology and G
oteborg University, S-41296 Gothenburg, Sweden
Abstract
The equilibrium con®gurations for an in®nite gold chain depending upon stretching and compressing are investigated. We show that there are two structural rearrangements the chain may undergo. The stretched chain shows a
strong tendency to dimerization accompanied by a large energy gain whereas the compressed chain prefers a planar
zigzag con®guration. Electronic structure and its in¯uence on the stability of the chain are discussed. Ó 2000 Elsevier
Science B.V. All rights reserved.
1. Introduction
Much scienti®c and technological interest has
been recently focused on the properties of low-dimensional systems such as monoatomic nanowires
and clusters. Monoatomic gold bridges were obtained in scanning tunneling microscopy (STM) [1]
and mechanically controllable break-junction
(MCB) technique [2]. According to both methods
such chains have conductance close to 2e2 =h, i.e.,
to the theoretical limit for one-dimensional
monovalent metals, and demonstrate striking mechanical stability [3]. They can consist of as many
as seven atoms and the indirect estimations of the
stretched Au±Au bond length give the value up to
[3] that is much larger than the one in bulk
3.6 A
face-centered cubic (fcc) lattice of Au (2.88 A).
Though one-dimensional metals are wellknown as text-book examples studied, for instance, by the tight-binding approach there are
rather few ®rst-principles and molecular dynamics
calculations of such systems [4]. In particular de
*
Corresponding author. Tel.: +46-31-7728428; fax: +46-317728426.
E-mail address: [email protected] (N.V. Skorodumova).
Maria and Springborg [5] have shown that for one
atomic chain it is more preferable to dimerize than
to remain equally spaced when the bond length is
and longer. Searching for an explanation of
2.8 A
this dimerization the authors of Ref. [5] calculated
the electronic structure of an isolated in®nite gold
chain with even number of atoms per unit cell.
They found a half-®lled band for Au±Au distance
that should favor the Peierls distortion.
of 2.9 A
However, they allowed the chain to relax only in
one-dimension and did not discuss the situation
with an odd number of atoms in the unit cell which
could be di€erent.
In this paper, we present the results of our investigation of electronic structure and properties
of monoatomic chains with 1,2 and 3 atoms per
unit cell. We did not increase the size of the unit
cell further since it had been shown [5] that the
properties of the chains with 2, 4 and 6 atoms per
unit cell are much alike. We considered an in®nite
gold chain as a sort of a cluster trying to ®nd
con®guration with the minimal total energy. No
directional constraints for lattice relaxation were
applied. We used density-functional theory (DFT)
approach which was proven to be a rather accurate technique for metal clusters [10].
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N.V. Skorodumova, S.I. Simak / Computational Materials Science 17 (2000) 178±181
179
2. Details of calculations
We performed full-potential LMTO calculations [6±8] within local-density approximation
(LDA) for exchange-correlation term [9]. Within
this scheme the space was divided into nonoverlapping, so-called mun tin spheres (MTS), surrounding atomic sites and interstitial region. No
approximations were introduced for the shape of
the charge density and potential either inside MTS
or in the interstitial region. The basis set, charge
density and potential were expanded in spherical
harmonics within MTS and in Fourier series in the
interstitial region. The basis set of augmented linear mun±tin orbitals was used [11,12]. The tails
of the basis functions outside their parent spheres
were linear combinations of Hankel functions with
negative kinetic energies. The basis set consisted of
6s, 6p, 5d and 5f states. The spherical harmonic
expansions of the potential, basis functions and
charge density were carried out up to lmax ˆ 8. We
simulated one-dimensional chains by calculating
the three-dimensional periodic tetragonal lattice
with chains along z axis separated in x and y directions. These separations between the chains
were as large as 24 a.u. that was found to be suf®cient to neglect a contribution from their interaction to the total energy of the system under
consideration. The intergration over the Brillouin
zone was performed by the special point sampling
[13,14] with the Gaussian smearing of 10 mRy
using a mesh based on 3 3 15 k-points in kx ,
ky , and kz directions, respectively, resulting,
therefore, in 135 k-points in the full or equivalently
24 k-points in the irreducible part of the corresponding tetragonal Brillouin zone. According to
our convergence tests this was a relevant amount
for an accurate description of the energetics of the
considered system.
3. Results and discussion
The ®rst con®guration we studied is an equally
spaced linear chain as it is shown in Fig. 1(c).
Changing the interatomic distance we have got the
binding energy curve as shown in Fig. 2. The
equilibrium Au±Au bond length is about 2.54
Fig. 1. Scheme of the chain con®gurations with equilibrium
interatomic distances. (a) dimer; (b) planar zigzag chain with
(c) equally spaced chain
the lattice constant equal to 4.67 A;
with its equilibrium interatomic distance; (d) dimerized chain
with shorter and longer bonds; (e) con®guration with a dimer
and an atom along one line. d1 and d2 indicate interatomic
distances within the unit cell.
Fig. 2. Binding energy curves with respect to the total energy of
Au atom. By solid diamonds we show the total energy for the
isolated dimer; solid circles stand for the simple chain; open
diamond is the energy of the planar zigzag chain with the lattice
open circles are for the dimerized chain.
parameter 4.76 A;
Dotted line indicates the energy of the dimer in equilibrium.
(Fig. 1(c)) that is slightly larger than the calcuA
lated interatomic distance in Au dimer 2.48
(Fig. 1(a)) (exp. 2.47 A
[15]). The binding enA
ergy for the isolated dimer is also shown in Fig. 2.
The binding energy of the chain in the minimum is
180
N.V. Skorodumova, S.I. Simak / Computational Materials Science 17 (2000) 178±181
1.7 eV. The energy for fcc Au from our calculations is 4.2 eV that shows usual LDA overestimation of the binding energy (exp. 3.8 eV [16]).
If the chain is allowed to relax it shows di€erent
equilibrium con®gurations and interatomic distances. Let us introduce an average distance in the
chain d
dˆ
d1 ‡ ‡ dn
;
n
where d1 ; dn are interatomic distances in the chain
as it is indicated in Fig. 1. For the chains having d
the equilibrium con®guration
smaller than 2.54 A
is a planar zigzag chain. Here we report the result
for one zigzag con®guration only which is shown
Though this is not a
in Fig. 1(b) (d ˆ 2.38 A).
completely relaxed con®guration one can see the
energy gain taken place in this case (Fig. 2). For
the equilibrium conchains with d around 2.54 A
®guration is a simple linear chain (Fig. 1(c)) which
remains energy favourable up to d 2:86A.
Above this d the system shows a tendency to dimerization. The equally spaced nanowire breaks
into a structure of lined up dimers (Fig. 1(d)). The
Au±Au bond length in the dimers varies from
for the average length 2.96 A
to the
2:56 A
Fig. 2 shows
value of the isolated dimer (2.48 A).
that the energy of the dimerized chain increases
upon stretching approaching the energy of the
isolated dimer. The displacement and energy gain
upon this rearrangement are essential as it is
shown in Table 1.
To check how an odd number of atoms in the
unit cell can change the tendency to dimerization
we have calculated three atomic unit cell. We have
found that the con®guration with a dimer and an
atom in the row (Fig. 1(e)) still brings the energy
gain of 6.1 mRy/atom, though, it is not the most
stable of possible con®gurations.
Thus our results show that equally spaced
chains are unfavourable almost in all intervals of
stretching and distorted con®gurations are preferable for the system. In particular dimerization is
rather favourable that can be explained by the
Peierls distortion mechanism. We should also note
that there is a tendency to trimerization for chains
with three atoms per unit cell. The complete
analysis of all possible con®gurations is in progress.
The described structural changes are determined by the band term of the energy. In Figs. 3
and 4 one can see the band structures for the
and 2.96 A.
The
chains with d equal to 2.38 A
in
electronic structure of the chain with d ˆ 2:38 A
the linear con®guration Fig. 1(c) contains a ¯at
band lying at the Fermi level (Fig. 3(a)) that
should lead to an instability of this con®guration.
It turns out that dimerization does not improve the
Table 1
Displacement and energy gain (DE) upon dimerization
Average
distance, A
2.96
3.18
3.18
Displacement,%
DE, mRy
12
20
20
3.8
17.8
6.1
a
3.70
3.97
32
38
46.1
58.1
a
These values are calculated for the three atomic unit cell for
the con®guration shown in Fig. 1(e).
Fig. 3. Band structure of the compressed chain with d ˆ 2:38
(a) linear chain; (b) zigzag chain. Points X and Z indicate the
A.
Brillouin zone boundary in kx and kz directions. Right panels
show the corresponding density of states (DOS).
N.V. Skorodumova, S.I. Simak / Computational Materials Science 17 (2000) 178±181
181
4. Conclusion
We have considered an in®nite gold chain as a
cluster searching for an equilibrium state at every
step of stretching. We have found that there are
three di€erent equilibrium con®gurations depending on stretching. The situation in real experiments, however, is far from equilibrium and there
are many factors we did not take into account,
such as, for example, ®niteness of the chain, supporting surfaces, applied bias, dynamic stretching.
However, since the time necessary for atomic relaxation is much smaller than the time steps of the
applied force, a static picture can give one a clue
for preferable structural rearrangements in
monoatomic chains.
References
Fig. 4. Band structure of the stretched chain with d ˆ 2:96 A.
(a) linear chain; (b) dimerized chain. Points X and Z indicate
the Brillouin zone boundary in kx and kz directions. Right
panels show the corresponding density of states (DOS).
situation, however, the rearrangement into the
zigzag chain removes this band from the Fermi
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Dimerization removes states from the
d ˆ 2:96 A.
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still can be mechanically stable. The ¯at bands
along CX line show the absence of interaction
between chains in the unit cell (Figs. 3 and 4) as it
should be for well-modeled one-dimensional systems.
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