Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects

Abstract

The structural properties of free nanoclusters are reviewed. Special
attention is paid to the interplay of energetic, thermodynamic, and
kinetic factors in the explanation of cluster structures that are
actually observed in experiments. The review starts with a brief
summary of the experimental methods for the production of free nanoclusters
and then considers theoretical and simulation issues, always discussed
in close connection with the experimental results. The energetic
properties are treated first, along with methods for modeling elementary
constituent interactions and for global optimization on the cluster
potential-energy surface. After that, a section on cluster thermodynamics
follows. The discussion includes the analysis of solid-solid structural
transitions and of melting, with its size dependence. The last section
is devoted to the growth kinetics of free nanoclusters and treats
the growth of isolated clusters and their coalescence. Several specific
systems are analyzed.

Full text

Structural properties of nanoclusters: Energetic, thermodynamic,
and kinetic effects
Francesca Baletto
*
and Riccardo Ferrando
†
INFM and Dipartimento di Fisica, Università di Genova, via Dodecaneso 33, 16146
Genova, Italy
共Published 24 May 2005兲
The structural properties of free nanoclusters are reviewed. Special attention is paid to the interplay
of energetic, thermodynamic, and kinetic factors in the explanation of cluster structures that are
actually observed in experiments. The review starts with a brief summary of the experimental methods
for the production of free nanoclusters and then considers theoretical and simulation issues, always
discussed in close connection with the experimental results. The energetic properties are treated first,
along with methods for modeling elementary constituent interactions and for global optimization on
the cluster potential-energy surface. After that, a section on cluster thermodynamics follows. The
discussion includes the analysis of solid-solid structural transitions and of melting, with its size
dependence. The last section is devoted to the growth kinetics of free nanoclusters and treats the
growth of isolated clusters and their coalescence. Several specific systems are analyzed.
CONTENTS
I. Introduction 371
II. Experimental Methods for Free-Nanocluster
Production 373
III. Energetics of Free Nanoclusters 374
A. Geometric shells: Structural motifs and general
trends in energetics 375
B. Electronic shells 378
C. Calculation of the total energy of nanoclusters 378
D. Global optimization methods 380
E. Studies of selected systems 383
1. Noble-gas clusters 384
2. Alkali-metal clusters 385
3. Noble-metal and quasi-noble-metal clusters 385
a. Gold clusters 385
b. Silver clusters 387
c. Copper clusters 388
d. Platinum clusters 388
e. Palladium clusters 389
f. Nickel clusters 389
4. General structural properties of noble-metal
and quasi-noble-metal clusters 390
5. Silicon clusters 391
6. Clusters of fullerene molecules 393
IV. Thermodynamics of Free Nanoclusters 393
A. Entropic effects and solid-solid transitions 394
1. Structural transitions in the harmonic
approximation 395
2. Anharmonic and quantum corrections 396
B. Melting of nanoclusters 397
1. Experimental methods 397
2. Computational methods 398
3. Size dependence of the melting point 400
C. Studies of selected systems 403
1. Lennard-Jones clusters 403
2. Sodium clusters 403
3. Noble-metal and transition-metal clusters 404
4. Silicon clusters 405
V. Kinetic Effects in the Formation of Nanoclusters 405
A. Freezing of liquid nanodroplets 406
1. Lennard-Jones clusters 406
2. Silver clusters 406
3. Gold clusters 407
4. Entropic effects and kinetic trapping in Pt
55
408
5. Copper, nickel, and lead clusters 409
B. Solid-state growth 409
1. Universal mechanism of the solid-state
growth of large icosahedra 409
2. Growth of small noble-metal and
transition-metal clusters 410
3. Growth of intermediate- and large-size Ag
and Au
clusters 412
4. Aluminum clusters 413
5. Clusters of C
60
molecules 414
C. Coalescence of nanoclusters 415
VI. Conclusions and Perspectives 416
Acknowledgments 417
References 417
I. INTRODUCTION
In the last decade, we have seen the explosive devel-
opment of a new field, now commonly known as nano-
science 共Nalwa, 2004兲. This field extends through phys-
ics, chemistry, and engineering and addresses a huge
number of important issues, ranging from basic science
to a variety of technological applications 共in the latter
case, the word nanotechnology is often employed兲. The
purpose of nanoscience and nanotechnology is to under-
*
Present address: The Abdus Salam International Center for
Theoretical Physics, Strada Costiera 11, I34014 Trieste, Italy.
Electronic address: baletto@ictp.trieste.it
†
Electronic address: ferrando@fisica.unige.it
REVIEWS OF MODERN PHYSICS, VOLUME 77, JANUARY 2005
0034-6861/2005/77共1兲/371共53兲/$50.00 ©2005 The American Physical Society371

stand, control, and manipulate objects of a few nano-
meters in size 共say, 1–100 nm兲. These nano-objects are
thus intermediate between single atoms and molecules
and bulk matter. Their properties are often peculiar, be-
ing qualitatively different from those of their constituent
parts 共either atoms or molecules兲 and from those of mac-
roscopic pieces of matter. In particular, nano-objects can
present properties that vary dramatically with size. This
opens the possibility of controlling these properties by
controlling precisely their formation process.
Among nano-objects, nanoclusters occupy a very im-
portant place, since they are the building blocks of nano-
science. Nanoclusters are aggregates of atoms or mol-
ecules of nanometric size, containing a number of
constituent particles ranging from ⬃10 to 10
6
共Castle-
man and Bowen, 1996; Johnston, 2002; Wales, 2003兲.
In contrast to molecules, nanoclusters do not have a
fixed size or composition. For example, the water mol-
ecule contains one oxygen and two hydrogen atoms,
which are placed at a well-defined angle to each other.
On the other hand, silver, gold 共see Figs. 1 and 2兲,or
even water clusters may contain any number of constitu-
ent particles and, for a given size, present a variety of
morphologies. There are borderline cases that are not
easily classifiable unambiguously either as clusters or
molecules, the fullerene buckyball 共C
60
兲 being an ex-
ample. In the following, we shall classify C
60
as a mol-
ecule, since clusters and solids of buckyballs have been
produced in which each C
60
constituent preserves its in-
dividuality. Clusters can be homogeneous, that is, com-
posed of only one type of atom or molecule, or hetero-
geneous. They may be neutral or charged. They may be
held together by very different kinds of forces: strong
attraction between oppositely charged ions 共as in NaCl
clusters兲, van der Waals attraction 共as in He and Ar clus-
ters兲, covalent chemical bonds 共as in Si clusters兲,ora
metallic bond 共as in Na and Cu clusters兲. Small clusters
of metal atoms are held together by forces more like
those of covalent bonds than like the forces exerted by
the nearly free electrons of bulk metals.
Clusters containing no more than a few hundred par-
ticles 共diameters of 1–3 nm兲 are expected to have
strongly size-dependent properties 共for example, geo-
metric and electronic structure, binding energy, melting
temperature兲. Larger clusters, with many thousands of
atoms and diameters in the range of 10 nm and more,
have a smoothly varying behavior, which tends to the
bulk limit as size increases.
Nanoclusters have peculiar properties because they
are finite small objects. To finite objects, the constraint
of translational invariance on a lattice does not apply.
For this reason, clusters can present noncrystalline struc-
tures, icosahedra and decahedra being the best known.
Of course, it is possible also to build up crystalline clus-
ters, which are simply pieces of bulk matter. An impor-
tant issue in cluster science is to understand whether
crystalline or noncrystalline structures prevail for a
given size and composition. Being small objects, nano-
clusters have a very high surface/volume ratio. Thus the
surface energy contribution 共including terms from facets,
edges, and vertices兲 is not negligible and usually strongly
size dependent.
Nanoclusters are well suited for several applications,
whose number is rapidly increasing. There has been a
traditional interest in applications to catalysis 共see, for
FIG. 1. High-resolution electron microscopy image of Ag clus-
ters deposited on an inert substrate after being produced in an
inert-gas aggregation experiment. The clusters indicated by ar-
rows are identified as being icosahedra. Adapted from Rein-
hard, Hall, Ugarte, and Monot, 1997.
FIG. 2. High-resolution electron microscopy image of a 8.6
⫻6.3-nm
2
truncated decahedral gold particle deposited on
amorphous carbon. The particle was produced in an inert-gas
aggregation experiment and then deposited and observed.
Adapted from Koga and Sugawara, 2003.
372
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

example, Henry, 1998兲, because of the very favorable
surface/volume ratio of nanoclusters. More recently,
there have been developments towards biological uses.
For example, gold nanoparticles studded with short seg-
ments of DNA 共Alivisatos et al., 1996; Mirkin et al.,
1996兲 could form the basis of an easy-to-read test to
single out genetic sequences 共Alivisatos, 2001兲. Some ap-
plications of nanoclusters have a much older history. It
has been recently discovered 共Borgia et al., 2002;
Padeletti and Fermo, 2003兲 that the Renaissance masters
in Umbria, Italy, used nanoparticles in the decoration of
majolicas with lustre. Lustre consists of a thin film con-
taining silver and copper clusters with diameters up to a
few tens of nanometers, often of noncrystalline struc-
ture. Due to the inclusion of these nanoparticles, lustre
gives beautiful iridescent reflections of different colors.
The starting point for an understanding of cluster
properties is the study of their structure. With this goal
in mind, the first question to answer is the following:
Given size and composition, what is the most stable clus-
ter structure from an energetic point of view? To answer
to this question we need to find the global minimum on
the potential-energy surface 共PES兲 of the cluster 共Wales,
2003兲. The PES is the product of the elementary inter-
actions among the cluster constituents.
Once the answer to this first question is given, a sec-
ond one arises: What is the effect of raising the tempera-
ture on the structural properties of a cluster? The study
of this question introduces us to a fascinating field, the
thermodynamics of finite systems.
Finally, the experimental time scales of cluster pro-
duction are often short with respect to the time scales of
morphology transitions. This leads to a third important
question: What are the kinetic effects in the formation
of nanoclusters? To answer to this question, the cluster
growth process must be analyzed.
In this review, we try to summarize and discuss the
present knowledge about these three points and, to the
best of our ability, to answer the above questions con-
cerning the behavior of free clusters. Two general results
can, however, be anticipated.
First, what clearly emerges from the comparison of
experimental and theoretical results is that all three fac-
tors, energetics, thermodynamics, and growth kinetics of
nanoclusters, must be taken into account when analyz-
ing realistic situations. Second, interactions among the
constituent particles must be understood in terms of the
basic concepts of range and type. The interaction range
rules the behavior of systems with pair potentials 共see,
for example, Wales, 2003兲. In systems characterized by
interactions with strong many-body character, metals,
for example, other factors come into play together with
the interaction range, namely, the bond-order/bond-
length correlation and the directionality of the interac-
tions.
The focus of this review article is on theoretical and
simulation issues in free-cluster science. We remark that
cluster science is a field in which the interplay among
experiments, theory, and simulations is very active: in
fact, the analysis of an experiment is very often carried
out by some kind of simulation. In the words of Marks
共1994兲 “small particle structures cannot be understood
purely from experimental data, and it is necessary to
simultaneously use theoretical or other modeling.” The
discussion of theoretical and simulation results will
make frequent reference to the experiments.
We cannot pretend to be exhaustive on the subjects of
our review, since thousands of articles about nanoclus-
ters have been produced in the last decade. Some fasci-
nating fields, for example, the field of binary clusters and
nanoalloys, whose properties depend crucially not only
on size but also on composition, are completely left out
for space reasons. But even on the topics that we explic-
itly treat we do not claim to be exhaustive. If some con-
tributions have been left out, we apologize in advance.
The review is structured as follows. In Sec. II we give
a brief description of the methods for free-cluster pro-
duction. In Sec. III we deal with the energetics of free
nanoclusters, describing general trends, modeling the el-
ementary interactions, and finding the global minima of
the PES. In Sec. IV we consider the thermodynamics of
nanoclusters, focusing on the possibility of solid-solid
transitions at finite temperature and on the melting tran-
sition. Finally Sec. V treats the growth kinetics of nano-
clusters, which may take place either in the liquid or in
the solid state, and the coalescence of nanoclusters. In
each part, specific systems are treated in details.
II. EXPERIMENTAL METHODS FOR FREE-NANOCLUSTER
PRODUCTION
The experimental methods for producing free nano-
clusters have been reviewed by de Heer 共1993兲 and more
recently by Milani and Iannotta 共1999兲 and Binns 共2001兲.
Here we present a brief summary, focusing on those as-
pects 共such as the time scale of free-nanocluster growth兲
most closely related to the subjects treated below.
Sources producing free beams of nanoscale clusters
were made available about 30 years ago for the produc-
tion of noble-gas clusters 共Raoult and Farges, 1973兲 and
more than 20 years ago for producing metallic clusters
共Sattler et al., 1980; Dietz et al., 1981兲; more recent de-
velopments have made available sources that are ultra-
high vacuum compatible, that can produce binary clus-
ters 共Rousset et al., 1996; Cottancin et al., 2000兲, and that
incorporate extremely efficient mass-selection tech-
niques 共see Binns, 2001, and references therein兲.
In most cases, the heart of a cluster source is a region
where a supersaturated vapor of the material forming
the clusters is produced. The first step in cluster produc-
tion is the heating of the material to obtain a hot vapor.
This can be done in different ways, for example, by heat-
ing up a piece of bulk material in a crucible 共as in seeded
supersonic nozzle and inert-gas aggregation sources兲,or
by hitting a target with a laser pulse or an ion beam
共laser evaporation and ion sputtering sources兲. To obtain
supersaturation, the hot vapor must be cooled down.
There are essentially two ways to achieve this. The first
is by means of a supersonic expansion 共as in seeded su-
personic nozzle sources, where the material is mixed
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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with a high-pressure inert gas and then expanded兲,
which causes an adiabatic cooling 共Anderson and Fenn,
1985兲. Hot clusters are usually produced at temperatures
close to the evaporation limit 共Bjørnholm et al., 1991兲.In
the second way, the hot vapor is mixed with a cold inert-
gas flow, which acts as a collisional thermostat. This
method is used in inert-gas aggregation sources to pro-
duce very cold metallic clusters. Here, due to the low
temperature of the inert gas, cluster production pro-
ceeds mainly by the addition of single atoms, and re-
evaporation is negligible 共de Heer, 1993兲. When growth
ceases, the clusters can be further reheated or cooled
down by subsequent stages of contact with gas at differ-
ent temperatures.
What are the typical time scales of nanocluster pro-
duction? This question is extremely important, because,
as we see in the following, the finite lifetime of free clus-
ters in flight may cause their trapping into metastable
structures. Let us consider an inert-gas aggregation
source for noble-metal clusters 共Reinhard, Hall, Ugarte,
and Monot, 1997; Reinhard et al., 1998; Koga and Sug-
awara, 2003兲. There, the typical metal vapor tempera-
ture T
v
and pressure p
v
are of the order of 1000–1500 K
and 1–10 mbar, respectively. Consider now a growing
nanocluster, say with a radius R of 1 nm, which can cor-
respond to 100–200 atoms. If we assume a spherical clus-
ter, the kinetic theory of gases gives for the atomic flux
⌽
v
共i.e., the number of metal atoms per second hitting
the cluster surface兲
⌽
v
=
p
v
A
eff
冑
2
␲
m
v
k
B
T
v
, 共1兲
where A
eff
=4
␲
R
2
, and m
v
is the mass of the atoms. This
gives ⌽
v
⬃10
7
s
−1
, corresponding to an interval between
depositions
␶
dep
=⌽
v
−1
⬃10
2
ns. Thus a cluster of 10
3
at-
oms is grown on a time scale of a fraction of a millisec-
ond.
Once clusters are produced, they have to be detected
in some way, possibly while they are still in their beam.
Indeed, the detection of slow neutral clusters is a diffi-
cult task; however, the cluster structures can be investi-
gated by diffraction methods. They can then be ionized
for an efficient mass-selective detection and finally de-
posited and observed by several microscopy techniques
共Henry, 1998; José-Yacamán, Ascencio, et al., 2001兲.
In a diffraction experiment, a well-collimated electron
beam with an energy of 30 to 50 keV crosses a cluster
beam. The fast electrons are scattered from the cluster
atoms, and the diffraction pattern is recorded. A series
of diffraction rings around the position of the primary
electron beam is recorded. The interpretation of diffrac-
tion profiles is not straightforward 共Hall et al., 1991;
Reinhard, Hall, Berthoud, et al., 1997, 1998; Reinhard,
Hall, Ugarte, and Monot, 1997兲, being based on a fit to a
theoretical profile whose construction assumes the pres-
ence of some selected cluster structures. However, infor-
mation on the geometry, the average size, and the tem-
perature of the clusters in the beam can be extracted as
follows. Remembering that the scattered intensity is the
squared modulus of the Fourier transform of the scatter-
ers with respect to the momentum transfer q, one builds
a theoretical diffraction pattern by weighting contribu-
tions from clusters of different structures. In fact, one
can expect that, for example, an icosahedron has a dif-
ferent Fourier transform than a part of an fcc lattice.
The geometry is obtained directly by fitting the weights
of the different structures to the experimental data. On
the other hand, information on the size can be obtained
from the width of the diffraction rings, because the
larger the cluster, the narrower the diffraction rings. The
more scatterers add their contributions coherently, the
sharper the resulting pattern. Finally, information about
the cluster temperature is contained in the temperature
dependence of the scattered intensity
I共T兲 = I
共T=0兲
e
−2W
. 共2兲
Here W is the Debye-Waller factor,
W =
1
3
具u
2
典q
2
, 共3兲
where 具u
2
典 is the mean-square vibrational amplitude of
the cluster atoms. For harmonic vibrations, 具u
2
典⬀T.
III. ENERGETICS OF FREE NANOCLUSTERS
At low temperatures, the most favored structure of a
cluster of N particles is the one that minimizes its total
energy. For example, in atomic clusters, the most fa-
vored structure is the global minimum of the potential
energy as a function of the coordinates of the atomic
cores 共the potential-energy surface or PES兲. Tradition-
ally, great effort has been devoted to finding reliable
methods for calculating the total energy, and searching
for local and global minima. This task implies the fol-
lowing two steps:
共a兲 Construction of a model for the interactions be-
tween the elementary constituents of the cluster;
this can be accomplished either by trying to solve
directly the Schrödinger equation 共ab initio meth-
ods兲 or by constructing semiempirical interparticle
potentials 共see Sec. III.C兲.
共b兲 A search for the most favored isomers by some
global optimization algorithm 共see Sec. III.D兲.
Depending on material and size, both 共a兲 and 共b兲 may
present enormous difficulties. Therefore, as a prelimi-
nary step, it is extremely important to find general
trends that help to single out sequences of favorable
structures in different size ranges. This can be done ei-
ther on the basis of geometric considerations, as in Sec.
III.A 共where the construction of families of highly sym-
metric structures, the structural motifs, is treated兲,or
based on electronic shell effects, as in Sec. III.B. The
sizes of the most energetically stable structures are often
called magic sizes. Magic sizes may 共tentatively兲 corre-
spond either to the completion of a geometrically per-
fect structure 共geometric magic sizes兲 or to the closing of
an electronic shell 共electronic magic sizes兲.
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Section III.E is devoted to the study of selected sys-
tems of special interest.
A. Geometric shells: Structural motifs and general trends
in energetics
In general, the binding energy E
b
of a cluster of size N
with a given structure can be written in the form
1
E
b
= aN + bN
2/3
+ cN
1/3
+ d, 共4兲
where the first term corresponds to a volume contribu-
tion, while the others represent surface contributions
from facets, edges, and vertices. Volume and surface
contributions are in competition. Clusters with low sur-
face energy must have quasispherical shapes 共thus opti-
mizing the surface/volume ratio兲, and close-packed fac-
ets. On the other hand, it is not possible to build up
clusters of spherical shape without internal strain, which
gives a volume contribution.
A useful parameter for comparing the stability of clus-
ters in different size ranges is ⌬共N兲,
⌬共N兲 =
E
b
共N兲 − N␧
coh
N
2/3
, 共5兲
where ␧
coh
is the cohesive energy per particle in the bulk
solid and ⌬ is the excess energy 共that is, the energy in
excess with respect to N atoms in a perfect bulk crystal兲
divided approximately by the number of surface atoms.
Other indicators of structural stability are the binding
energy per atom, E
b
共N兲/N, and the first and second dif-
ferences ⌬
1
共N兲 and ⌬
2
共N兲 in the binding energy,
⌬
1
共N兲 = E
b
共N −1兲 − E
b
共N兲,
⌬
2
共N兲 = E
b
共N −1兲 + E
b
共N +1兲 −2E
b
共N兲. 共6兲
⌬
1
and ⌬
2
measure the relative stability of clusters of
nearby sizes. Peaks in ⌬
2
共N兲 were found to be well cor-
related to peaks in the mass spectra 共Clemenger, 1985兲.
Let us now build up structural motifs by trying to op-
timize either volume or surface energy contributions.
The easiest way to minimize volume contributions is to
cut a piece of bulk matter so that interparticle distances
inside the cluster are automatically optimized. For such
clusters of crystalline structure the parameter a in Eq. 共4兲
is simply ␧
coh
, so that lim
N→⬁
⌬=b. As we shall see in the
following, nanoclusters can be also of noncrystalline
structures; for these clusters a is larger than ␧
coh
, and ⌬
diverges at large sizes 共see Fig. 3兲.
Consider now fcc crystalline structures. Try to cut a
cluster from a bulk fcc crystal in such a way that its
surface has only close-packed facets. A possible result-
ing shape is the octahedron 共see Fig. 4兲, that is, two
square pyramids that share a basis. Even if the whole
surface of the octahedron is close packed, its shape does
not optimize the surface energy because of its high
surface/volume ratio. Clusters with more spherical
shapes are obtained by cutting the vertices, thus produc-
ing a truncated octahedron. Its surface has eight close-
packed 共111兲 and six square 共100兲 facets; the latter have
a higher surface energy in most materials. A deeper cut
1
See, for example, Hill 共1964兲, Northby et al. 共1989兲,Xieet al.
共1989兲, Cleveland and Landman 共1991兲, Jortner 共1992兲, Uppen-
brink and Wales 共1992兲, Baletto, Ferrando, et al. 共2002兲.
FIG. 3. Qualitative behavior of ⌬ 关Eq. 共5兲兴 for crystalline,
icosahedron, and decahedron clusters.
FIG. 4. Face-centered-cubic clusters: 共a兲 octahedron; 共b兲 trun-
cated octadedron; 共c兲 cuboctahedron. Each cluster is shown in
four views. 共a兲 An octahedron is made up of two square pyra-
mids sharing a common basis. Its surface consists of eight tri-
angular close-packed 共111兲 facets, but the structure has a high
surface/volume ratio. Polyhedra with a lower surface/volume
ratio, are obtained by truncating symmetrically the six vertices
of an octahedron, thus obtaining square and hexagonal 共or tri-
angular, see below兲 facets. A truncated octahedron can be
characterized by two indexes: n
l
is the length of the edges of
the complete octahedron; n
cut
is the number of layers cut at
each vertex. In the figure, for the octahedron in 共a兲共n
l
,N
cut
兲
=共7,0兲, the truncated octahedron in 共b兲共n
l
,N
cut
兲=共7,2兲, and
the cuboctahedron in 共c兲共n
l
,N
cut
兲=共7,3兲. A perfect truncated
octahedron has thus a number of atoms, N
TO
共n
l
,n
cut
兲=
1
3
共2n
l
3
+n
l
兲−2n
cut
3
−3n
cut
2
−n
cut
. This equation defines the series of
magic numbers for truncated octahedron structures. The
square facets have a 共100兲 symmetry and edges of n
cut
+1 at-
oms. The 共111兲 facets are not in general regular hexagons. In
fact, three edges of the hexagons are in common with square
facets, thus having n
cut
+1 atoms, while the remaining three
edges have n
l
−2n
cut
atoms. Regular hexagons are thus possible
if n
l
=3n
cut
+1; truncated octahedra with regular hexagonal fac-
ets are referred to as regular truncated octahedra. When n
l
=2n
cut
+1 the hexagonal facets degenerate to triangles and the
cuboctahedron is obtained, which is usually not energetically
favored because of its large 共100兲 facets.
375
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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gives a more compact shape having, however, larger
square facets. If size is sufficiently large, the optimal cut
is given by the Wulff construction. This was developed
to find the equilibrium shape of macroscopic crystals by
minimizing the surface energy at fixed volume 共see, for
example, Pimpinelli and Villain, 1998兲. From the Wulff
construction, the best truncated octahedron structure
should fulfill the condition
␥
共100兲
␥
共111兲
=
d
共100兲
d
共111兲
, 共7兲
where
␥
共100兲
and
␥
共111兲
are the 共100兲 and 共111兲 surface
energies, respectively, whereas d
共100兲
and d
共111兲
are the
distances of the facets from the center of the cluster. At
large size, the introduction of higher-order facets can
make fcc clusters more spherical 共see, for example,
Raoult et al., 1989a兲. Different groups 共Cleveland and
Landman, 1991; Valkealahti and Manninen, 1998; Bal-
etto, Ferrando, et al., 2002兲 have shown that the Wulff
construction is a reliable tool for identifying the best
crystalline clusters for nanometric sizes. In any case,
even the optimal Wulff shapes are quite far from being
spherical and are expected to be the most favorable clus-
ters at large sizes.
A better solution to the problem of building up com-
pact quasispherical shapes was found by Mackay 共1962;
see also Martin, 1996兲, who constructed the Mackay
icosahedron 共see Fig. 5兲. This is a noncrystalline struc-
ture, with fivefold rotational axes. Icosahedral clusters
are limited by 共111兲-like close-packed facets only, thus
optimizing the surface energy well. However, this is ob-
tained at the expense of a volume contribution, since
interatomic distances are not the ideal ones: radial 共in-
tershell兲 bonds are compressed, while intrashell bonds
are expanded. Therefore Mackay icosahedra are highly
strained structures, and their ⌬ is proportional to N
1/3
as
N→ ⬁. This indicates that icosahedra could be expected
to be the most favorable structures only at small sizes.
Icosahedra are not the only possible noncrystalline
structures. Another noncrystalline motif is represented
by the decahedra 共see Fig. 6兲. A decahedron is formed
by two pentagonal pyramids with a shared base; its sur-
face has only close-packed facets, but its shape is very
far from being spherical, so that truncations are advan-
tageous also in this case. Ino 共1969兲 proposed a trunca-
tion in which the five edges limiting the common basis of
the pyramids are cut to expose 共100兲-like facets. This
improves the surface/volume ratio, but usually does not
produce the best possible decahedra in a given size
range 共see, for example, Baletto, Ferrando, et al., 2002兲
FIG. 5. Icosahedral clusters. The Mackay icosahedron is a
noncrystalline structure organized in shells. An icosahedron
with k shells has N
Ih
共k兲=
10
3
k
3
−5k
2
+
1
3
k−1 atoms 共so that the
series of magic numbers is 1, 13, 55, 147,…兲. The icosahedron
in the figure has k=4 shells. An icosahedron with k shells has
the same number of particles as a cuboctahedron with n
cut
=k
−1. An icosahedron has 20 triangular facets of side k and 12
vertices. Each pair of opposite vertices lie along a fivefold sym-
metry axis. An icosahedron can be thought of as composed of
20 fcc tetrahedra sharing a common vertex 共in the central site兲.
When 20 regular tetrahedra are packed around a common ver-
tex, large interstices remain. To fill these spaces the tetrahedra
must be distorted, thus generating a huge strain on the struc-
ture. Intershell distances are compressed, while intrashell dis-
tances are expanded. The facets of an icosahedron are of dis-
torted 共111兲 symmetry. Adatoms deposited on the facets of a
Mackay icosahedron can be placed either on sites of fcc or hcp
stacking. Islands of fcc stacking are part of the next Mackay
shell, while islands of hcp stacking form a so-called anti-
Mackay overlayer.
FIG. 6. Decahedral clusters: 共a兲 regular decahedra; 共b兲 Ino
truncated decahedra 共Ino, 1969兲; 共c兲 Marks truncated decahe-
dra. A decahedron is made up of two pentagonal pyramids
sharing a common basis. It has a single fivefold axis and is
formed by five tetrahedra sharing a common edge along the
fivefold axis. When five regular tetrahedra are packed, gaps
remain, but they are smaller than in the case of the icosahedra.
These gaps are filled by distorting the tetrahedra, thus intro-
ducing some strain. Regular decahedra 共first row兲 are limited
by ten close-packed 共111兲-like facets, but have a large surface/
volume ratio, which can be lowered by truncating the edges
around the common basis, thus obtaining the Ino decahedron
with five 共100兲-like facets. An even better structure is the
Marks decahedron 共Marks, 1984兲, obtained by introducing re-
entrances that separate the 共100兲-like facets 共see the third row兲.
A decahedron is characterized by three integer indices
共m ,n,p兲, where m and n are the lengths of the sides of the
共100兲 facets, perpendicular and parallel to the axis, respec-
tively; p is the depth of the Marks reentrance. A regular deca-
hedron has indices of the form 共m,1,1兲关the 共5,1,1兲 decahedron
is shown in the top row兴; Ino decahedra have indices 共m ,n,1兲,
with n⬎1 关the 共4,2,1兲 Ino decahedron is shown in the second
row兴, and Marks decahedra have 共m,n ,p兲 with n ,p⬎1 关the
共2,2,2兲 Marks decahedron is shown in the third row兴. A Marks
decahedron has h=m +n +2p −3 atoms along its symmetry axis
and a total number of atoms given by N
M−Dh
=共30p
3
−135p
2
+207p−102兲/6+兵5m
3
+ 共30p −45兲m
2
+ 关60共 p
2
−3p兲+136兴m其/6
+兵n关15m
2
+共60p −75兲m+3共10p
2
−30p兲+66兴其/6−1. From this
formula it follows that a Ino decahedron has N
Ino
=关5m
3
−15m
2
+16m +n共15m
2
−15m +6兲兴/6−1 atoms. For n=m and p
=1 关square 共100兲-like facets兴 a decahadron has thus the same
number of atoms as an icosahedron of m shells and as a cub-
octahedron with n
cut
=m−1. Finally, a regular decahedron has
N
Dh
=共5m
3
+m兲/6 atoms.
376
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

because it creates large 共100兲-like facets. Marks 共1984,
1994兲 proposed a more efficient truncation scheme, with
reentrances exposing further close-packed facets which
separate neighboring 共100兲-like facets. Marks decahedra
can achieve a better optimization of the surface energy
than truncated octahedron structures. On the other
hand, decahedra are also strained structures, with a vol-
ume contribution to the excess energy giving ⌬⬀N
1/3
at
large N. The strain, however, is much smaller than for
icosahedra.
In summary, the icosahedral motif should be the most
favored at small sizes, while truncated octahedron clus-
ters are expected for large sizes; truncated decahedra
could be favored in intermediate ranges. This trend has
been verified in experiments. Farges et al. 共1986兲 found a
transition from icosahedron to close-packed 关not neces-
sarily fcc, see van de Waal et al. 共2000兲兴 structures at N
⬃750 in Ar clusters obtained in a free-jet expansion;
Reinhard, Hall, Berthoud, et al. 共1997, 1998兲 were able
to identify small icosahedra, intermediate-size decahe-
dra, and large fcc clusters in inert-gas aggregation ex-
periments on Cu. Several systems have been investi-
gated theoretically, showing that the icosahedral
→ decahedral and decahedral→ fcc crossover sizes are
strongly material dependent.
2
Doye et al. 共1995兲 proposed a quite simple principle
for a qualitative understanding of crossover sizes. This
principle states that soft interactions, with wide potential
wells, stabilize strained structures, while sticky interac-
tions, with narrow potential wells, cannot easily accom-
modate the strain and thus favor crystalline structures.
Doye et al. 共1995兲 considered a pair potential and de-
composed the cluster energy into three parts,
E
b
=−n
NN
␧
NN
+ E
strain
+ E
NNN
, 共8兲
where n
NN
is the number of nearest-neighbor pairs, ␧
NN
is their bond strength at the optimal distance, E
strain
is
the strain contribution due to the fact that some nearest-
neighbor pairs can be at nonoptimal distances, and E
NNN
is the contribution from further neighbors, which is, to a
first approximation, negligible. The usual competition is
between the first and the second term in Eq. 共8兲; icosa-
hedral structures, which have the largest number of
nearest-neighbor bonds, optimize the first term at the
expense of the second, while the opposite holds for fcc
clusters. The relative weights of these two terms depend
on the range of the potential. Decreasing the range has
the effect of destabilizing strained structures, because
the potential wells narrow so that the distortion of the
nearest-neighbor distance with respect to its ideal value
becomes more costly. In the case of a Morse 共1929兲 in-
teraction potential, Doye et al. 共1995兲 were able to con-
struct a structural phase diagram. The Morse potential
U
M
can be written as
U
M
=
⑀
兺
i⬍j
e
␳
0
共1−r
ij
/r
0
兲
关e
␳
0
共1−r
ij
/r
0
兲
−2兴, 共9兲
where the r
ij
are the interatomic distances, r
0
is the equi-
librium separation, and
⑀
is the well depth. Thus by vary-
ing the parameter
␳
0
one can adjust the width of the
potential well without changing the position or depth of
the minimum. Large values of
␳
0
give short-ranged at-
tractions with a steep repulsive part, that is, a narrow
well and a sticky interaction 共see Fig. 7兲. Small
␳
0
corre-
spond to soft potentials. As can be seen in Fig. 8, the
most favored structure changes from icosahedral at
small
␳
0
to decahedral in the intermediate range, and
finally to close-packed clusters at high values of
␳
0
.A
qualitative idea of the trends among different materials
can be obtained by fitting the parameters of the Morse
potential. This fit gives large values of
␳
0
for the interac-
tion potentials between fullerene molecules 关
␳
0
=13.62
and 11.92 for Girifalco 共1992兲 and Pacheco and Prates-
2
See Raoult et al. 共1989b兲, Cleveland and Landman 共1991兲,
Uppenbrink and Wales 共1992兲, Turner et al. 共2000兲, Doye et al.
共2001兲, Baletto, Ferrando, et al. 共2002兲, Doye and Hendy 共2003兲
and Sec. III.E for details on some specific systems.
FIG. 7. 共Color in online edition兲 Morse potential for
␳
0
values:
dash-dotted line, sodium atoms; dashed line, Lennard-Jones
molecules; and solid line, C
60
molecules. The last is a very
sticky interaction.
FIG. 8. Phase diagram for Morse clusters: The lines separate
domains pertaining to different structural motifs. Figure cour-
tesy of Jonathan Doye.
377
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

Ramalho 共1997兲 potentials, respectively兴,
␳
0
=6 for a
Lennard-Jones crystal,
␳
0
=3.96 for Ni 共Stave and De-
Pristo, 1992兲, and even smaller values for alkali metals
关
␳
0
=3.15 and 3.17 for sodium and potassium 共Girifalco
and Weizer, 1959兲兴.
Baletto, Ferrando, et al. 共2002兲 applied a similar crite-
rion, based on the stickiness of the interactions, to dis-
cuss crossover sizes in noble-metal and quasi-noble-
metal clusters modeled by many-body semiempirical
potentials. This point is discussed in detail in Sec.
III.E.3, where it is shown that there are other factors,
besides the interaction range, that determine the struc-
ture of metallic clusters 共Soler et al., 2000兲. These factors
arise from the many-body character of the metallic in-
teractions, which causes a strong correlation between
bond order and bond length, whose effects tend to make
icosahedral structures less stable than what follows from
the analysis of the interaction range. Moreover, bond
directionality effects can be important. Finally, the pres-
ence of multiple minima or of secondary maxima in the
two-body part of the interaction 共Doye and Wales, 2001;
Doye, 2003兲 can have strong effects on the preferred
cluster structures.
B. Electronic shells
Electronic shell closing has been extremely successful
in explaining the observed experimental abundances of
alkali-metal clusters, like those found in the seminal ex-
periments of Knight et al. 共1984兲 on sodium clusters. In
this framework, a cluster is modeled as a super-atom;
valence electrons are delocalized in the cluster volume
and fill discrete energy levels. There are several degrees
of sophistication of this model. Detailed accounts can be
found in the reviews by Brack 共1993兲 and de Heer
共1993兲; a simple and very clear discussion is found in
Johnston 共2002兲. Here we briefly sketch only the spheri-
cal jellium model.
The spherical jellium model assumes a uniform back-
ground of positive charge, in which electrons move and
are subjected to an external potential. The simplest
forms of the potential are the infinitely deep spherical
well and the harmonic well. The solution of the single-
electron Schrödinger equation for the spherical well
gives the following series of magic numbers: 2, 8, 18, 20,
34, 40, 58,…, etc. On the other hand, the harmonic well
gives the series: 2, 8, 20, 40, 70,…, etc. The experimental
spectra of Knight et al. 共1984兲 reveal high peaks at 8, 20,
40 in agreement with both models. There are, however,
less evident peaks at 18 and 58, which appear only in the
spherical well. Experiments on larger sodium clusters
have revealed electronic shells in these clusters, up to
about 2000 atoms 共Martin et al., 1991a; Martin, 1996,
2000兲, and in other metals; see, for example, Johnston
共2002兲.
Candidates for observing electronic shell effects are
the metals with weakly bound valence electrons, prima-
rily the alkali and then the noble metals. It seems also
that the temperature T, which determines whether the
cluster is solid or liquid depending on N 共see Sec.
IV.B.3兲, plays a crucial role. The major evidence for elec-
tronic shell closing is for small alkali-metal clusters
共Knight et al., 1984; Bjørnholm et al., 1990; Nishioka et
al., 1990兲. For these systems, several calculations 共Röth-
lisberger and Andreoni, 1991; Spiegelman et al., 1998;
Solov’yov et al., 2002兲 indicate that electronic shell clos-
ing is a better criterion than atomic packing for deter-
mining the most stable clusters. For larger alkali clusters
the situation is more complicated. In fact, Martin et al.
共1991a兲 and Martin 共1996, 2000兲 have shown that Na
clusters, after presenting electronic magic numbers up to
2000 atoms, reveal a series of geometric magic numbers
at larger sizes 共see Fig. 9兲. This would correspond to a
transition from liquid to solid clusters at increasing size.
Moreover, when solid Na clusters are heated and
melted, geometric magic numbers disappear to the ad-
vantage of electronic magic numbers. The same kind of
behavior was also found for aluminum clusters 共Martin
et al., 1992; Baguenard et al., 1994兲. Therefore the indi-
cation is that electronic shells are seen when hot liquid
clusters are produced, while geometric shells are exhib-
ited by cold, solid clusters 共Johnston, 2002兲. In small
clusters, both electronic and geometric effects can play
important roles, as Zhao et al. 共2001兲 have shown in their
tight-binding global optimization study of Ag clusters at
N艋20. In conclusion, the interplay of electronic and
geometric shell effects, depending on material and size,
remains to be fully understood.
C. Calculation of the total energy of nanoclusters
A key point in the theoretical study of clusters is the
choice of an appropriate energetic model. This depends
on the material and the size of the cluster, as well as on
FIG. 9. Mass spectrum of Na
n
clusters of size n, photoionized
with 3.02-eV photons. Closed-shell clusters are more difficult
to ionize, so that they correspond to minima in the spectrum.
Two sequences of minima appear in the spectrum. These se-
quences are at equally spaced n
1/3
intervals on the size scale
and correspond to an electronic shell sequence and a structural
shell sequence. Adapted from Martin, 2000.
378
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

the physical and chemical properties one wishes to in-
vestigate. To cover and illustrate in detail all the meth-
ods used in theoretical cluster science would require an
entire book, and it is well beyond the scope of our re-
view. Here we mention only the main methods, trying to
give an idea of their underlying philosophy and range of
applicability.
As pointed out in the review by Bonacic
´
-Koutecký et
al. 共1991兲, even clusters of a few atoms are very compli-
cated systems. Indeed, the complexity of quantum me-
chanics forces one to employ approximate methods.
The ab initio methods of quantum chemistry 关Hartree-
Fock and post Hartree-Fock; see Bonacic
´
-Koutecký et
al. 共1991兲, and references therein兴 were extensively ap-
plied to the study of small clusters about 20 years ago.
For example, small-size 共2艋N艋9兲 Li and Na clusters
contain relatively few electrons, so that all-electron cal-
culations were possible 共Boustani et al., 1987兲. When ei-
ther the size of the cluster or the nuclearity of the atoms
increase, these methods become cumbersome, and at
present they are less commonly employed than in the
past.
Methods based on density-functional theory 共Hohen-
berg and Kohn, 1964; Kohn and Sham, 1965兲, when ad-
equately tested, can be of very high accuracy and less
cumbersome from the computational point of view,
making it possible to treat a wide variety of systems and
somewhat larger sizes. Calculations on metals up to a
few hundred atoms are present in the literature, even for
difficult systems such as the transition and noble metals
共see, for example, Häberlen et al., 1996; Jennison et al.,
1997; Garzón, Michaelian, et al., 1998; Fortunelli and
Aprà, 2003; Nava et al., 2003兲. The weak point in
density-functional calculations is often the exchange and
correlation term, which is treated in an approximate
way. The validity of the treatment depends on the sys-
tems and has to be checked each time. The simplest ap-
proximation is the local-density approximation; more so-
phisticated approaches include gradient corrections
共very important in transition and noble metals兲, which
are often called the generalized gradient approximation.
At this level, different exchange and correlation func-
tionals are available 共see Perdew and Wang, 1986;
Becke, 1988, 1996; Perdew et al., 1992, 1996兲. In order to
check the validity of gradient-corrected density-
functional calculations, Mitás et al. 共2000兲 performed
quantum Monte Carlo simulations 共Ceperley, 1994兲 on
small-size silicon clusters. In quantum Monte Carlo,
which is computationally very demanding, many-body
correlations are directly taken into account by an ex-
plicit correlation in the trial wave function. It turned out
that there were significant discrepancies between the
density-functional 共with different types of exchange and
correlation functionals兲 and the quantum Monte Carlo
results. Mitás et al. 共2000兲 confirmed the already known
bias of density-functional calculations towards compact
structures; this bias is very strong at the level of the
local-density approximation, and considerably reduced
if gradient corrections are included. There are some
functionals that give the same qualitative results as the
quantum Monte Carlo calculations in ordering the dif-
ferent isomers of the Si
20
cluster, even though there are
still some quantitative differences.
Density-functional calculations can be inserted into a
molecular-dynamics procedure to give the ab initio mo-
lecular dynamics. The best known example is the Car-
Parrinello method 共Car and Parrinello, 1985兲, which
could be well suited to the investigation of thermody-
namic and kinetic properties at T⬎ 0K.
Even though density-functional methods are nowa-
days reliable and efficient for a large variety of systems,
there is still a great interest in developing methods re-
quiring a smaller computational effort. In fact, global
optimization 共see Sec. III.D兲 of clusters in the frame-
work of ab initio calculations is not feasible at present
except for a few systems and at very small sizes. More-
over, ab initio molecular dynamics is limited to small
systems on short time scales, so that both the accurate
sampling of thermodynamic properties and the simula-
tion of kinetic processes 共diffusion, structural transfor-
mations, growth兲 are far beyond present capabilities.
Therefore several approximate energetic models for
clusters have been developed, often on semiempirical
grounds. At an intermediate degree of computational
effort, there is the tight-binding model for semiconduc-
tors 共Ho et al., 1998兲 and metals 共Barreteau, Guirado-
Lopez, et al., 2000; Bobadova-Parvanova et al., 2002兲.
This allows global optimization searches for clusters of a
few tens of atoms, and molecular-dynamics simulations
even for clusters of 10
2
atoms 共Yu et al., 2002兲, even
though on rather short time scales.
Larger sizes and longer time scales can be now treated
by classical atom-atom 共or even molecule-molecule兲 po-
tentials, which are built up on the basis of approximate
quantum models. These potentials contain parameters
fitted to experimental material properties 共semiempirical
potentials兲 or to density-functional calculations 关ab-
initio-based potentials 共Kallinteris et al., 1997; Garzón,
Kaplan, et al., 1998兲兴. For metallic systems, several atom-
atom potentials have been developed, such as
embedded-atom 共Daw and Baskes, 1984; Voter, 1993兲,
glue 共Ercolessi et al., 1988兲, second-moment tight-
binding 共Gupta, 1981; Rosato et al., 1989兲, Sutton-Chen
共Sutton and Chen, 1990兲, and effective-medium 共Jacob-
sen et al., 1987兲 potentials. A discussion of the criteria
for fitting the parameters in Gupta 共1981兲 potentials is
found in López and Jellinek 共1999兲. For metals, these
potentials must contain a many-body term, which is re-
sponsible for the correct surface relaxations 共Desjon-
queres and Spanjaard, 1998兲. The advantage of this ap-
proach is that it allows full global optimization to sizes
of the order of ⬃200 atoms, local relaxation of clusters
of ⬃10
5
atoms, and molecular-dynamics simulations on
time scales of 10
␮
s and more for clusters of 10
2
atoms,
quite close to the growth time scales of free clusters in
inert-gas aggregation sources. The disadvantage is that
even the qualitative accuracy of these potentials is often
questionable, and they must be tested carefully before
being used 共see, for example, Ala-Nissila et al., 2002兲.As
we shall see in the following, there are several systems
379
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

for which reliable potentials have been developed. In
any case, the use of semiempirical potentials is in prac-
tice a necessary tool for the study of medium- or large-
size clusters. In contrast to bulk materials and crystal
surfaces, where one usually knows where to place the
atoms, the best structures of clusters are not known in
principle, so that a semiempirical modeling is the start-
ing point for more sophisticated approaches. For ex-
ample, in the work of Garzón, Michaelian, et al. 共1998兲,
global optimization by a semiempirical potential selects
the most promising candidates for a further relaxation
study by ab initio methods.
Semiempirical interaction potentials for molecular
clusters have been built up too. Here we note the Giri-
falco 共1992兲 and Pacheco and Prates-Ramalho 共1997兲 po-
tentials for fullerene molecules. Finally we mention the
oldest semiempirical potentials, Lennard-Jones and
Morse potentials, which are the usual benchmarks for
testing new theoretical tools. The Lennard-Jones poten-
tial is also a popular model for noble gases.
D. Global optimization methods
Given the potential-energy surface of the cluster, that
is, the potential energy U共兵r其兲 关where 兵r其=共r
1
,r
2
,...,r
N
兲兴
as obtained by methods like those of the previous sec-
tion, one is confronted with the formidable task of find-
ing its deepest minimum. Indeed, Wille and Vennik
共1985兲 demonstrated that this problem is NP-hard by a
mapping to the traveling-salesman problem.
3
The num-
ber of minima increases more than polynomially with
the size 关there are indications of a proportionality to
exp共N兲兴: a Lennard-Jones cluster of 13 atoms has about
10
3
local minima 共Hoare and McInnes, 1976; Tsai and
Jordan, 1993兲, but this number is at least 10
12
for a 55-
atom cluster 共Doye and Wales, 1995兲. Clearly, a com-
plete sampling of all these minima would be simply im-
possible, and the ability of a given system to reach its
energy global minimum 共or at least one of the usually
few good local minima兲 should reside in some special
features of its PES. This point has also been extensively
debated in the field of protein folding 共Wales, 2003兲,
since proteins efficiently fold to their native state 共which
may not coincide with the lowest minimum in the PES兲
even if their PES presents a huge number of local
minima. A good search algorithm should exploit the fea-
tures of the PES to ensure a fast convergence to low-
lying minima. However, as we show below, depending on
the interaction potential, size, and composition of the
cluster, there are easy potential-energy surfaces where
most algorithms converge fast to some good putative
global minima, and others where there are low-lying
minima 共often of very high symmetry兲 that are ex-
tremely difficult to reach, so that most algorithms get
stuck in some less favorable configuration. A remark is
necessary at this stage: no global optimization technique
can warrant that the lowest minimum is really reached;
the only way to reach the global minimum with prob-
ability one is to sample all minima, compare them, and
choose the lowest one. This can be done only in essen-
tially infinite time in cases of practical interest. For ex-
ample, while good putative global minima for Lennard-
Jones clusters have been obtained at sizes well above
100 atoms, only for much smaller sizes have these
minima proven to be global 共Maranas and Floudas,
1992兲.
Let us now analyze the good features of a PES which
allows fast convergence to its global minimum and cor-
respondingly how an efficient global optimization algo-
rithm must be constructed to exploit these features
共Doye, 2004兲. To this end, a few definitions are needed
in the framework of what was proposed by Stillinger and
Weber 共1982兲 introducing the inherent structure of liq-
uids. Here below we follow the very clear exposition of
Becker and Karplus 共1997兲. A thorough account is given
in the excellent book by Wales 共2003兲.
We introduce a mapping from the continuum configu-
rational space of the cluster into the discrete set of its
local minima. The mapping associates each point 兵r其
with its closest minimum, i.e., the one reached by a
steepest-descent 共or a quenching兲 minimization starting
at 兵r其. This amounts effectively to the following transfor-
mation of the PES 共see Fig. 10兲:
U
˜
共兵r其兲 = min关U共兵r其兲兴, 共10兲
where min means that the minimization is started from
兵r其. U
˜
共兵r其兲 is a multidimensional staircase potential 共Li
and Scheraga, 1987; Doye and Wales, 1998a兲.
The set of all points associated with a minimum s con-
stitutes its basin. All points of a given basin are con-
nected by definition. This mapping is a partition of the
共3N−6兲-dimensional space of the internal coordinates of
the cluster into disjoint sets, which are indeed the attrac-
tion basins of the different minima. The boundaries be-
tween the basins constitute a network of 3N−7 dimen-
sions, where the mapping is not defined. Nearby basins
共usually of the same structural family兲 can be grouped
into metabasins, at different level of complexity. A very
3
NP stands for nonpolynomial. The computational complex-
ity of this optimization increases more than polynomially with
cluster size.
FIG. 10. 共Color in online edition兲 Transformation of the
potential-energy surface 共PES兲 to a staircase. Figure courtesy
of Giulia Rossi.
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convenient representation of the connections between
basins, metabasins, etc. is given by the disconnectivity
diagram 共see Fig. 11兲, which allows one to understand
pictorially which basins are connected at different values
of the total energy of the cluster 共two basins are con-
nected when the highest point on the minimum-energy
path between them lies below that total energy兲.
Portions of the PES can be classified into three differ-
ent types according to the features of the connections
among the basins that they contain: rough PES, single-
minimum PES with weak noise, and funnel-like PES
共see Fig. 11兲. It can be easily understood that the abso-
lute minimum is reached quickly in both a single-
minimum or a steep funnel-like portion of the PES,
while a rough PES will resist global minimization.
Therefore we expect that in systems whose entire PES is
given by a single minimum with weak noise or by a
single funnel the global minimum can be found quickly.
Correspondingly, such systems are much easier to attack
by good global optimization algorithms than those with
rough PES or with multiple-funnel PES. As an example,
for a cluster of given size, icosahedral structures are usu-
ally grouped in a wide metabasin with a funnel structure,
which is easily accessible from liquidlike configurations
共Doye and Wales, 1998a兲, because the latter have in
common with the icosahedra a pronounced polytetrahe-
dral character 共Nelson and Spaepen, 1989; Doye and
Wales, 1996a兲, being formed by tetrahedral units packed
together. In contrast, decahedral and truncated octahe-
dral funnels are much narrower, so that global minima
pertaining to these motifs are not easily reached from
liquidlike configurations. The best-known example of
this kind is the Lennard-Jones cluster of 38 atoms, which
presents a double-funnel PES 共see Fig. 11兲, with a wide
icosahedral funnel and a narrower but deeper close-
packed funnel, so that the trapping of search algorithms
in the former is very likely. Moreover, one can expect
that systems with short-range 共sticky兲 potentials will
present a larger number of minima and a rougher PES
than systems with soft potentials 共Wales et al., 2000;
Doye, 2004兲. Multicomponent systems will be harder to
optimize because of their much smaller number of
equivalent permutational isomers with respect to pure
systems 共Darby et al., 2002兲.
From the above considerations, it is clear that an effi-
cient search algorithm must be able to perform and in-
tegrate the following tasks:
共a兲 From any given point on the PES, to find the local
minimum with which it is associated; this is simply
what is needed to construct the above-described
mapping, or equivalently to transform the PES into
a multidimensional staircase.
共b兲 To make transitions possible from a given basin to
another and, more important, from one metabasin
to another. In this way, the algorithm would also be
able to explore a multiple-funnel PES.
The search by global optimization algorithms can be
unbiased when the starting configuration is randomly
chosen, or seeded when a set of 共supposedly兲 good struc-
tures is used to begin the optimization procedure.
Seeded searches are often faster, since they use prior
knowledge about the system under study, but they have
the disadvantage of making difficult the finding of unex-
pected low-lying minima. Almost every algorithm men-
tioned below can be used either for unbiased or for
seeded searches; obviously, when comparing different al-
gorithms, the unbiased search is more significant.
Genetic algorithms have applications in a large variety
of fields; they are based on the analogy of evolution
through natural 共fitness-based兲 selection. The fitness is
the parameter to be optimized, here the potential en-
ergy. The coordinates of each cluster are encoded in a
string of bits, called the chromosome. At each step, from
the present generation of clusters, a new generation is
built up. Sons are built up by mixing the chromosomes
of parent clusters, or simply by inserting some mutation
in the present chromosomes. The individuals of the old
FIG. 11. 共Color in online edition兲 Disconnectivity diagrams:
Top panel, schematic of a single-minimum potential-energy
surface 共PES兲 with weak noise; center panel, a single-funnel
PES. The disconnectivity diagrams of these two panels show at
which energetic level the different local minima of a PES can
be considered connected. Adapted from Becker and Karplus,
1997. The lowest panel gives the disconnectivity diagram for
the complete double-funnel PES of the Lennard-Jones cluster
of size 38. Figure courtesy of Jonathan Doye.
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generation are compared to the sons via their fitness,
and a new generation is formed from the old individuals
and the sons, by some rule, which always includes the
best-fit individual in the new generation. Very often, sev-
eral populations are evolved in parallel, and individuals
are exchanged between them from time to time. In a
genetic algorithm, task 共b兲 is accomplished by chromo-
some mixing and mutations and by exchanges of indi-
viduals between subpopulations, while task 共a兲 corre-
sponds to comparing the fitness of the individuals after a
local minimization on them, i.e., comparing U
˜
共sons兲 to
U
˜
共parents兲. Recent developments include similarity
checking among cluster structures to keep the diversity
of the population as the genetic optimization goes on, as
in Cheng et al. 共2004兲. The use of genetic algorithms in
cluster optimization was pioneered by Hartke 共1993兲 and
by Xiao and Williams 共1993兲, who made applications to
Si
4
and various molecular clusters, respectively. Their
work was followed by applications to a wide variety of
systems. Michaelian et al. 共1999兲 optimized transition
and noble-metal clusters by a symbiotic variant of a ge-
netic algorithm 共Michaelian, 1998兲. Hartke 共2000, 2003兲
also treated water clusters; Deaven and Ho 共1995兲,
Deaven et al. 共1996兲, and Ho et al. 共1998兲 optimized
Lennard-Jones and silicon clusters; Rata et al. 共2000兲 and
Bobadova-Parvanova et al. 共2002兲 considered silicon and
iron clusters; Darby et al. 共2002兲, Massen et al. 共2002兲,
Bailey et al. 共2003兲, and Lloyd et al. 共2004兲 optimized
Pd-Pt, Au-Cu, and Al-Ni nanoalloy clusters; finally
Rossi et al. 共2004兲 searched for the global minima of
Ag-Cu, Ag-Ni, and Ag-Pd clusters. Recent reviews are
those of Hartke 共2002兲 and Johnston 共2003兲.
The basin-hopping algorithm 共Li and Scheraga, 1987;
Doye and Wales, 1998a; Doye, 2004兲 differs from genetic
algorithms because it accomplishes task 共b兲 simply by a
canonical Monte Carlo simulation at constant T on the
transformed PES. In this framework, the transformation
of the PES amounts to lowering the barriers between
basins to the maximum possible extent, while keeping
the levels of the minima unchanged. Starting from a
given basin, a move with random displacements is tried,
and the energy difference between the new and the old
position ⌬U
˜
is calculated. If ⌬U
˜
⬍0 the move is ac-
cepted with probability 1, otherwise with probability
exp关−⌬U
˜
/共k
B
T兲兴.OnU
˜
, transitions can occur in any di-
rection, not only through saddle points, and transitions
to lower-lying minima are always accepted. Basin hop-
ping has been applied successfully to the optimization of
a large variety of systems, from Lennard-Jones clusters
共Wales and Doye, 1997兲 to clusters of fullerene mol-
ecules 共Doye et al., 2001兲, of transition and noble metals
共Doye and Wales, 1998b兲, and of aluminum 共Doye,
2003兲. Basin hopping was able to find the difficult puta-
tive global minima in Lennard-Jones clusters at sizes 38,
75, 76, 77, and 98, starting from random initial condi-
tions. Recently, Lai et al. 共2002兲 optimized sodium clus-
ters by both genetic and basin-hopping algorithms, find-
ing the same set of minima. This could indicate that both
algorithms have comparable efficiency.
Another popular algorithm is simulated annealing
共Kirkpatrick et al., 1983; Biswas and Hamann, 1986;
Freeman and Doll, 1996兲. In thermal simulated anneal-
ing, the system is evolved at constant high T on the un-
transformed PES by either Monte Carlo or molecular
dynamics, then slowly cooled down. Simulated anneal-
ing has the advantage of being pretty much physical: one
tries to mimic the procedure of cooling a sample which
hopefully will reach its most stable configuration if cool-
ing is sufficiently slow. This algorithm is easily incorpo-
rated into standard Monte Carlo and molecular-
dynamics codes 共including ab initio molecular dynamics兲,
and the analysis of its output is easy and intuitive. For
these reasons, this algorithm has been used for a large
variety of systems.
4
There is, however, a major drawback
to simulated annealing: it utilizes effectively a single lo-
cal optimization, and if the system fails to be confined to
the basin of attraction of the global minimum as the
temperature is decreased, the algorithm may fail, even if
it passed through the basin of the global minimum when
T was high. As a result, simulated annealing is less effi-
cient than genetic and basin-hopping algorithms
共Hartke, 1993; Zeiri, 1995兲. To avoid trapping in a single
basin, one can make long high-T Monte Carlo or
molecular-dynamics runs, collecting a large quantity of
different snapshots and quenching them down 共Sebetci
and Güvenc, 2003; Baletto et al., 2004兲, monitoring the
short-term average of kinetic energy 共Jellinek and
Garzón, 1991; Garzón and Posada-Amarillas, 1996兲. T
must be chosen carefully. If it is too high, the probability
of catching the global minimum is very small; if T is too
low, the cluster is likely to remain trapped in the basin it
starts from.
Another family of annealing algorithms is known as
quantum annealing 共Amara et al., 1993; Finnila et al.,
1994; Freeman and Doll, 1996兲. In quantum annealing,
the system is first collapsed into its quantum ground
state by using diffusion or Green’s-function Monte Carlo
techniques 共Ceperley, 1994兲 and then quantum mechan-
ics is slowly turned off. This algorithm utilizes delocal-
ization and tunneling as the primary means for avoiding
trapping in metastable states. Amara et al. 共1993兲 and
Finnila et al. 共1994兲 applied quantum annealing to
Lennard-Jones clusters up to N=19. Lee and Berne
共2000兲 coupled quantum and thermal annealing, being
thus able to find the fcc global minimum of the Lennard-
Jones cluster of size 38, but this approach failed for
other difficult cases at larger size, such as the Marks
decahedron at N =75. Recently, Liu and Berne 共2003兲
have developed a new quantum annealing procedure,
based on quantum staging of path-integral Monte Carlo
sampling and local minimization of individual imaginary
time slices. This method is able to locate all Lennard-
4
To cite but a few, Jones 共1991兲, Kumar and Car 共1991兲, Röth-
lisberger and Andreoni 共1991兲, Jones and Seifert 共1992兲, Stave
and DePristo 共1992兲, Vlachos et al. 共1992兲, Cheng et al. 共1993兲,
Christensen and Cohen 共1993兲, Jones 共1993兲, Ma and Straub
共1994兲, Ahlrichs and Elliott 共1999兲, Bacelo et al. 共1999兲.
382
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Jones minima for N艋 100 except at N =76,77,98, being
thus comparable to basin-hopping and genetic algo-
rithms. This form of quantum annealing uses local mini-
mization to accomplish task 共a兲, and mutations 共in which
coordinates of randomly selected atoms are reset兲 to fa-
cilitate task 共b兲.
An algorithm that tries to combine the advantages of
simulated-annealing, genetic, and basin-hopping ap-
proaches has been recently developed by Lee et al.
共2003兲. Their algorithm was able to optimize Lennard-
Jones clusters up to size 201.
Finally, a very recent proposal is to search for global
minima by means of a growth procedure. Solov’yov et al.
共2003, 2004兲 applied this method to Lennard-Jones clus-
ters, starting from a tetrahedron, and adding one atom
at a time according to different procedures 共adding an
atom on the surface, inside the surface, or at the center
of mass of the cluster兲. After adding a new atom, the
cluster is quenched down to search for the closest local
minimum, and the results of the different addition pro-
cedures are compared to select the best minimum. This
procedure exploits the fact that clusters of similar sizes
are likely to have global minima with some resemblance,
and in this respect it is a seeded algorithm. Solov’yov et
al. 共2004兲 were able to find all global minima up to N
=150. Again, their procedure works on U
˜
, since clusters
are compared for selection after local minimization.
This last observation lead us to a summarizing com-
ment. The most successful algorithms make extensive
use of local minimization, comparing the structures for
selection after having performed that minimization. This
amounts effectively to working on U
˜
. To use the words
of Doye 共2004兲, “the method for searching U
˜
is of sec-
ondary importance to the use of the transformation it-
self.” The searching method may be based on thermal
Monte Carlo, quantum tunneling, or genetic operations,
but in the end the efficiencies are comparable, given that
the algorithms are well constructed.
Doye and Wales 共1998a兲 have shown that the transfor-
mation to U
˜
modifies the thermodynamics of the system
in such a way that the temperature range in which tran-
sitions among different funnels are possible is enlarged,
and the probability of occupying the basin of the global
minimum is increased. This can be understood by ana-
lyzing the occupation probability of a minimum s. For an
untransformed PES within the harmonic approximation
共see Sec. IV兲 p
s
⬀exp共−
␤
E
s
兲/⍀
s
3N−6
, where E
s
is the
potential energy of minimum s, ⍀
s
is the geometric
mean vibrational frequency, and
␤
=共k
B
T兲
−1
, with k
B
the
Boltzmann constant. For a transformed PES, p
s
⬀A
s
exp共−
␤
E
s
兲, where A
s
is the hyperarea of the basin
of attraction of minimum s. The differences between
these expressions, the vibrational frequency and hyper-
area terms, have opposite effects on the thermodynam-
ics. The higher energy minima are generally less rigid
and have a favorable vibrational entropy, so that the
transitions are pushed down to lower temperature and
sharpened. By contrast, the hyperarea of the minima de-
creases with increasing potential energy, thus stabilizing
the lower energy states and broadening the thermody-
namics. The transformation to a staircase PES is not the
only useful transformation. Other possibilities have been
suggested by Pillardy and Piela 共1995兲, Locatelli and
Schoen 共2003兲, and Shao et al. 共2004兲.
What are the maximum sizes presently tractable by
global optimization? This answer depends on the system
under study and on the degree of sophistication of the
interaction potential. In the Cambridge Cluster Data-
base 共see www-wales.ch.cam.ac.uk/CCD.html兲, putative
global minima for several systems are reported, ranging
from Lennard-Jones to Morse, metallic, and water clus-
ters. Even for the simplest potentials, the largest sizes
are up to N=190 关for Al 共Doye, 2003兲 and Pb clusters
共Doye and Hendy, 2003兲 modeled by the glue potential
共Lim et al., 1992兲兴, and sizes decrease for more complex
interactions. Very recent seeded optimizations of
Lennard-Jones clusters 共Shao et al., 2004兲 reached N
=330. In order to treat large sizes, Krivov 共2002兲 has
proposed a hierarchical method for optimizing qua-
siseparable systems. In these systems, distant parts are
independent, so that a perturbation to the coordinates of
a given part has very little influence on distant atoms.
According to Krivov 共2002兲, one can exploit this prop-
erty to build up a hierarchical procedure that can treat
Lennard-Jones clusters of several hundreds of atoms.
More complex systems such as water clusters have been
optimized up to N ⯝30 共Wales and Hodges, 1998;
Hartke, 2003兲. Ahlrichs and Elliott 共1999兲 optimized alu-
minum clusters within a density-functional approach up
to N=15 by simulated annealing. Tekin and Hartke
共2004兲 optimized Si clusters by a combination of an em-
pirical global search and density-functional local optimi-
zation up to N =16. In the case of tight-binding modeling
of the interactions, the sizes amenable by global optimi-
zation approaches extend to a few tens of atoms. See,
for example, Zhao et al. 共2001兲, who optimized Ag clus-
ters up to N =20 by a genetic algorithm.
How do the best algorithms scale with cluster size?
This might be an ill-posed question, since there is no
demonstration that the minima found by the different
algorithms are really global. However, Liu and Berne
共2003兲 find that their quantum annealing algorithm
scales as N
3.2
for Lennard-Jones clusters in the size
range 11–55, and Hartke 共1999兲 finds that his genetic
algorithm scales approximately as N
3
, again for
Lennard-Jones clusters up to N =150.
E. Studies of selected systems
In this section we review the energetics of atomic clus-
ters of some selected materials, chosen both for their
importance from the point of view of basic science 共for
example, noble-gas and alkali-metal clusters, which have
been the benchmark for testing theoretical models兲 and
for their practical applications 共silicon clusters, transition
and noble-metal clusters兲. Finally, we treat a specific
kind of molecular cluster, those of fullerene molecules,
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whose interpretation in energetic terms has been a long-
standing puzzle.
1. Noble-gas clusters
Noble-gas clusters have been thoroughly studied since
the early 1980s. Excellent accounts of their properties
are those of Haberland 共1994兲 and Johnston 共2002兲.
Here we briefly review their energetics, focusing on the
transition sizes among structural motifs and comparing
theory with experiment. A special place among noble-
gas clusters is occupied by He clusters, which are the
best known example of quantum clusters. Recently
there has been a noticeable experimental interest in He
clusters, which can be formed by
4
He,
3
He, or a mixture
of the two.
5
In the following, we treat “classical” clusters
first and then He clusters.
In a classical series of electron-diffraction experi-
ments, Farges et al. 共1983, 1986兲 investigated the struc-
ture of neutral Ar clusters produced in a free-jet expan-
sion. They compared the experimental diffraction
patterns with those obtained by freezing Lennard-Jones
droplets of different sizes in molecular-dynamics simula-
tions and were thus able to identify the experimental
products up to N ⬃10
3
atoms. Harris et al. 共1984兲 mea-
sured the mass spectra of charged Ar clusters at N
⬍100. At 20⬍N ⬍ 50, both experiments found polyi-
cosahedral structures, composed by interpenetrating 13-
atom icosahedra. From N =50 to N ⬃750 multishell
structures based on the Mackay icosahedron were
found. Farges et al. 共1986兲 were also able to single out
features from diffraction by fcc planes in their spectra of
clusters with N⬎ 600 atoms. Thus they came to the con-
clusion that the icosahedron→ fcc transition is placed at
N⬃750. However, in a more recent analysis, van de
Waal et al. 共2000兲 have shown that the transition is to a
mixture of fcc, hcp, and random close-packed regions,
with no significant preference for the fcc bulk structure.
Let us see how the energetics calculations compare
with these experimental results. Most of the calculations
are based on the use of the Lennard-Jones potential,
which is a very popular model for noble-gas systems
even though not completely accurate. Several global op-
timization studies have shown that the icosahedron mo-
tif is dominant at small sizes,
6
with few exceptions. Poly-
icosahedral clusters are formed at sizes 19, 23, and 26
共Farges et al., 1988; Ikeshoji et al., 1996兲. Indeed, Wales
and Doye 共1997兲 found that there are only seven excep-
tions in the range 13艋N 艋 150: at N =38 the global mini-
mum is a truncated octahedron; at N=75, 76, 77 and N
=102, 103, 104 the best structures are based on the
共2,2,2兲 and 共2,3,2兲 Marks decahedra, respectively. Later
on, Leary and Doye 共1999兲 discovered that the global
minimum at N =98 is tetrathedral. Romero et al. 共1999兲
optimized clusters in the range 148艋N 艋 309 on icosa-
hedral and decahedral lattices, finding again a preva-
lence of icosahedron putative global minima with only
11 decahedron exceptions.
At larger sizes, Raoult et al. 共1989b兲 compared perfect
icosahedron and decahedron structures and concluded
that Marks decahedra finally prevail over icosahedra at
N⬎1600. Concerning the transition to fcc structures,
Xie et al. 共1989兲 compared icosahedral and cuboctahe-
dral clusters, finding a crossover at N ⬃10
4
. However,
cuboctahedra are not the most favorable fcc clusters.
Therefore the crossover should be at much lower sizes
共Raoult et al., 1989a; van de Waal, 1989兲, comparable to
the crossover size with Marks decahedra.
The combination of these energy-minimization results
gives a reasonable interpretation of the experiments,
even though there are some points that probably cannot
be explained in terms of the energetics alone. At small
sizes, the agreement between the global optimization
calculations and the experiments is good, even though
the decahedron global minima are not yet observed.
This may indicate the presence of entropic effects, of the
kind discussed in Sec. IV.A, or that even at small sizes
some kind of kinetic trapping is taking place 共see Sec.
V. A 兲. At larger sizes, the comparison of theory and ex-
periment is much more difficult, and there has been con-
siderable debate in the interpretation of the diffraction
data in terms of known structures 共van de Waal, 1996兲.
For example, fcc clusters with twin faults give electron
diffraction patterns very similar to those of Marks deca-
hedra. In any case, a coarse-grained picture is in agree-
ment with the general trend of a transition from the
icosahedron motif to other structures. A more detailed
analysis reveals that several kinds of clusters can be
present in the same size range, so that structural transi-
tions are not really sharp. Quantitative agreement on
the transition size between the Lennard-Jones calcula-
tions and the experiments has not been reached, prob-
ably due to the limited accuracy of the Lennard-Jones
potential for Ar. At present, it is very difficult to ascer-
tain whether the diffraction data from large clusters sup-
port the existence of either entropic or kinetic effects.
Some interesting results addressing this point 共Ikeshoji
et al., 2001兲 are discussed in Sec. V.A.
Let us now consider
4
He clusters. While the dimer is
stable but very weakly bound 共Luo et al., 1993; Grisenti
et al., 2000兲, the trimer 关which should have a noticeable
contribution from linear configurations according to the
calculations by Lewerenz 共1996兲兴 and larger clusters are
much more stable 共Toennies and Vilesov, 1998兲. Chin
and Krotscheck 共1992兲 calculated the ground-state prop-
erties of various
4
He clusters modeled by the Aziz et al.
共1987兲 potential, considering several sizes up to N=112,
by diffusion Monte Carlo simulations. They found evi-
dence of density oscillations indicating a possible shell
structure. Chin and Krotscheck 共1995兲 confirmed the ex-
istence of these oscillations for larger clusters, as well,
5
Recent discussions of He clusters are those of Whaley
共1994兲, Toennies and Vilesov 共1998兲, Scoles and Lehmann
共2000兲, Callegari et al. 共2001兲, Northby 共2001兲, Johnston 共2002兲,
Jortner 共2003兲.
6
See Hoare and Pal 共1975兲, Farges et al. 共1985兲, Freeman and
Doll 共1985兲, Northby 共1987兲, Wille 共1987兲, Coleman and Shal-
loway 共1994兲,Xue共1994兲, Pillardy and Piela 共1995兲, Deaven et
al. 共1996兲, Wales and Doye 共1997兲, Romero et al. 共1999兲.
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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and were also able to recover the bulk limit of the exci-
tation spectrum. In a very recent experiment Brühl et al.
共2004兲 observed magic sizes at N =10, 11, 14, 22, 26, 27,
and 44 atoms, in
4
He clusters produced in a free-jet ex-
pansion. By comparing the experimental results with dif-
fusion Monte Carlo calculations, Brühl et al. 共2004兲
showed that these magic sizes are not related to en-
hanced binding energies at specific values of N, but to
the sizes at which excited levels cross the chemical-
potential curve and become stabilized.
Clusters of
3
He are much less stable, because the con-
stituent atoms are fermions, so that the minimum num-
ber of atoms N
min
needed to form a stable cluster is
relatively large. A configuration-interaction calculation
based on a phenomenological density functional gave
N
min
=29 共Barranco et al., 1997兲, while subsequent varia-
tional Monte Carlo calculations with the Aziz potential
gave 34–35 as an upper bound to N
min
共Guardiola, 2000;
Guardiola and Navarro, 2000兲. The same kind of calcu-
lations 共Guardiola and Navarro, 2002兲 have recently
been applied to mixed
4
He-
3
He clusters, finding that
their stability has a nontrivial dependence on size and
composition.
2. Alkali-metal clusters
The structural properties of small Li and Na clusters
have been the subject of extensive theoretical activity,
following the seminal experiments of Knight et al.
共1984兲. Bonacic
´
-Koutecký et al. 共1991兲 gave an excellent
account of the earlier developments, which are very
briefly summarized here. Systematic ab initio
configuration-interaction studies on Li and Na clusters
共Boustani et al., 1987; Bonacic
´
-Koutecký et al., 1988兲 for
N艋9 have shown a complete analogy between the two
elements. The lowest isomers are planar up to the pen-
tamer; at N =6 a pentagonal pyramid and a triangular
planar structure are in very close competition; larger
clusters are 3D. In agreement with the experimental
magic numbers and with the electronic shell closing ar-
guments, the cluster of 8 atoms, of tetrahedral symmetry
共Jellinek et al., 1994兲, is especially stable compared to
neighboring sizes. Röthlisberger and Andreoni 共1991兲
and Röthlisberger et al. 共1992兲 performed Car-Parrinello
calculations on Na at slightly larger sizes. At N =13, they
found that the most stable isomer is neither an icosahe-
dron nor a cuboctahedron, but a capped pentagonal bi-
pyramid. At N=18 the most stable isomer was the
double icosahedron minus one vertex, and at N=20 the
isomers based on pentagonal symmetries 共which are
closely related to the double icosahedron兲 were found to
be more stable than structures with tetrahedral symme-
try. On the other hand, Bonacic
´
-Koutecký et al. 共1991兲
found that two tetrahedron structures are the lowest in
energy for Li
20
and Na
20
共Bonacic
´
-Koutecký, Fantucci, et
al., 1993兲. Spiegelman et al. 共1998兲 found results in rea-
sonable agreement with the previous calculations by a
tight-binding approach.
Recent developments are discussed by Ishikawa et al.
共2001兲, Solov’yov et al. 共2002兲, and Matveentsev et al.
共2003兲. Ishikawa et al. 共2001兲 reanalyze the problem of
Li
6
, in which planar and 3D structures were found to be
in competition, by a global optimization procedure em-
ploying the replica exchange method 共Swendsen and
Wang, 1986兲 applied to ab initio calculations. They find a
3D structure of D
4h
symmetry group, which is slightly
lower in energy than the planar triangle and the pen-
tagonal pyramid. Poteau and Spiegelmann 共1993兲 per-
formed a search for the best Na isomers up to N =34 by
a growth Monte Carlo algorithm in the framework of a
distance-dependent tight-binding 共Hückel兲 model. They
found that at N =34 the best isomer is the double icosa-
hedron surrounded by a ring of 15 atoms. Sung et al.
共1994兲 optimized Li clusters up to N =147 by simulated
annealing within the local spin-density approximation to
the density-functional theory, finding that for 55艋N
艋147 the structures based on the Mackay icosahedron
are the lowest in energy. This has been recently con-
firmed by Reyes-Nava et al. 共2002兲, who performed a
global genetic optimization of sodium modeled by
Gupta 共1981兲 potentials. A thorough global optimization
study 共by both genetic and basin-hopping algorithms兲 of
Na, K, Rb, and Cs clusters in the framework of the
Gupta potential 共as developed by Li et al., 1998兲 has
been performed by Lai et al. 共2002兲 up to N =56. They
found a sequence based on icosahedron clusters except
for N =36 and N =38. At N =38 the global minimum is a
truncated octahedron, with distorted facets in the cases
of Na and Cs; at N =36 the structure is a distorted in-
complete truncated octahedron.
The transition from electronic to geometric magic
numbers in Na clusters has already been discussed in
Sec. III.B. Here we simply recall that for 2000⬍N
⬍20 000, there is experimental evidence 共Martin et al.,
1991a; Martin, 1996, 2000兲 of peaks in the abundances of
Na clusters at sizes corresponding to the completion of
perfect icosahedron, Ino decahedron 共Ino, 1969兲, or cub-
octahedron structures, which have exactly the same
magic numbers. From these magic numbers, one cannot
directly infer the actual cluster structures. However,
there is at least one indirect argument in favor of icosa-
hedral structures: in fact, while complete icosahedra are
surely more favorable than incomplete ones, Ino deca-
hedra are not usually the best decahedral structures, nor
are cuboctahedra the best among the possible truncated
octahedron clusters.
3. Noble-metal and quasi-noble-metal clusters
Here we treat the energetics of pure clusters of the
elements of the last two columns of the transition series:
Ni, Cu, Pd, Ag, Pt, and Au. We restrict our treatment to
neutral clusters unless otherwise specified.
a. Gold clusters
Small Au clusters 共N 艋10兲 have been thoroughly
treated by ab initio methods. Bravo-Pérez et al. 共1999a兲
studied sizes 3艋N艋6, finding that the best structures
are planar 共Bravo-Pérez et al., 1999b兲. These results have
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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been confirmed and extended by others.
7
Au clusters are
planar up to N=10 at least, according to Bonacic
´
-
Koutecký et al. 共2002; see Fig. 12兲, and only to N =6
according to Wang et al. 共2002兲, who however found that
flat 共but nonplanar兲 structures are the lowest in energy
up to N =14. In the experiments on Au cations by Gilb et
al. 共2002兲 there is evidence for planar clusters at least up
to N=7. The physical origin of the preference for planar
structures is discussed in Sec. III.E.4. On the other hand,
Li et al. 共2003兲 found experimental and computational
evidence that Au
20
is a tetrahedron, well separated from
higher isomers. This finding has been confirmed by the
calculations of Wang, Wang, and Zhao 共2003兲.
Global optimization methods 共Wilson and Johnston,
2000兲 have been applied to larger clusters modeled by
semiempirical potentials. Garzón and co-workers
共Garzón, Michaelian, et al., 1998; Garzón, Beltrán, et al.,
2003兲 and Michaelian et al. 共1999兲 modeled Au by the
Gupta 共1981兲 potential and searched the best isomers by
a genetic algorithm, to use them as the starting point of
a local relaxation by the density-functional method.
They found that Au presents low-symmetry structures at
some geometric magic numbers, such as N=38 and 55,
which therefore are not true magic numbers for Au clus-
ters. These structures are often called “amorphous” in
the literature, even though the most appropriate termi-
nology is low-symmetry structures. The low-symmetry
cluster at N=55 is a strongly rearranged and distorted
icosahedron, which conserves some fivefold vertexes
共see Fig. 13兲. Low-symmetry structures are found in sev-
eral semiempirical calculations,
8
the only exception be-
ing the Murrell-Mottram potential 共Cox et al., 1999兲,
which gives high-symmetry clusters. All these results in-
dicate that high- and low-symmetry structures are in
close competition, and the global minimum is sensitive
to the fine details of the potential parametrization. An
argument in favor of the low-symmetry structures fol-
lows from the density-functional relaxation at N=75
seen by Michaelian et al. 共1999兲, who found that a low-
symmetry structure is lower in energy than the 共2,2,2兲
Marks decahedron, in contrast with all semiempirical re-
sults. The physical origin of low-symmetry structures is
discussed in Sec. III.E.4.
For larger sizes, Cleveland et al. 共1997兲 and Baletto,
Ferrando, et al. 共2002兲 performed semiempirical calcula-
tions to compare structures of the different motifs, find-
ing that Marks decahedra are the most favorable, the
crossover with fcc structures being at N ⬃500.
The comparison of these theoretical predictions with
the experiments is complicated by the fact that Au clus-
ters are very often passivated during or after the forma-
tion process 共Whetten et al., 1996; Schaaff and Whetten,
2000兲, and it is difficult to determine to what extent the
action of the passivating agents can modify the struc-
tures 共Alvarez et al., 1997兲. With this in mind, we try,
however, to determine how the calculated crossover
sizes compare with the experimental results. To this pur-
pose, we must single out those experiments in which the
observed clusters were able to reach their equilibrium
structure. This was very likely the case in the work of
Patil et al. 共1993兲, who produced free Au clusters in
inert-gas aggregation sources. The clusters were subse-
quently slowly heated up above their melting tempera-
ture and then slowly cooled down 共for about one sec-
ond兲. Finally, the clusters were deposited and observed.
These clusters were identified as being fcc, even at the
smallest sizes 共N ⯝400兲, in good agreement with the
crossover sizes calculated by Baletto, Ferrando, et al.
共2002; see Table I and Fig. 14兲. On the other hand, there
have been several observations of large icosahedral
structures,
9
at N Ⰷ400. For example, Ascencio et al.
共1998, 2000兲 have observed by high-resolution electron
microscopy a variety of structures 共decahedron, trun-
cated octahedron, icosahedron, and amorphous兲 for pas-
7
See Grönbeck and Andreoni 共2000兲, Häkkinen and Land-
man 共2000兲, Bonacic
´
-Koutecký et al. 共2002兲, Gilb et al. 共2002兲,
Häkkinen et al. 共2002兲, Wang et al. 共2002兲, Zhao et al. 共2003兲.
8
See, for example, Doye and Wales 共1998b兲, Li, Cao, and
Jiang, 共2000兲, Darby et al. 共2002兲, Garzón et al. 共2002兲.
9
See Buffat et al. 共1991兲, Marks 共1994兲, Martin 共1996兲, Ascen-
cio et al. 共1998兲, Ascencio et al. 共2000兲, Koga and Sugawara
共2003兲.
FIG. 12. Lowest-energy isomers of Au clusters according to
Bonacic
´
-Koutecký et al. 共2002兲. All clusters are planar up to
N=10 at least. The binding energy per atom E
b
/N 共in eV兲 is
also shown. For isomers with energy difference smaller than
0.1 eV, both competing structures are shown 共see sizes 3, 4,
and 7兲. From Bonacic
´
-Koutecký et al., 2002.
FIG. 13. 共Color in online edition兲 Two views of the lowest-
energy isomer of Au at N =55 as found by Garzón, Michaelian,
et al. 共1998兲. This cluster is a strongly rearranged and distorted
55-atom icosahedron, similar to the double-rosette structure
discussed in the case of Pt
55
in Sec. V.A.4. Figure courtesy of
Ignacio Garzón.
386
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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sivated particles of a few nanometer’s diameter, with a
prevalence, however, of 共possibly defected兲 Marks and
Ino decahedra. Defected clusters 2–4 nm in diameter
have been detected by x-ray diffraction 共Zanchet et al.,
2000兲. Moreover, in a recent experiment, Koga and Sug-
awara 共2003兲 have analyzed a large sample of free Au
clusters obtained in an inert-gas aggregation source,
with diameters of 3–18 nm and found that icosahedra
were the most frequent and decahedra second-most fre-
quent, fcc clusters being absent. These results are again
an indication in favor of kinetic effects and are discussed
in Sec. V.B.3.
b. Silver clusters
As in the case of Au, the energetics of Ag clusters
have been the subject of intensive study in all size
ranges. For N艋 9, Bonacic
´
-Koutecký, Cespiva, et al.
共1993, 1994兲 searched for the lowest-energy clusters by a
self-consistent Hartree-Fock procedure which was able
to produce structures in good agreement with the ex-
periments 共Ganteför et al., 1990; Ho et al., 1990;
Alameddin et al., 1992; Jackschath et al., 1992兲. They
found that the best isomers are planar up to the pen-
tamer, as confirmed by Matulis et al. 共2003兲 by density-
functional calculations. Santamaria et al. 共1994兲 found a
planar structure also at N =6. Recent density-functional
studies have considered N 艋 12 共Fournier, 2001兲, and N
=13 共Oviedo and Palmer, 2002兲. The latter study sug-
gested that the best isomer of 13 atoms is of low sym-
metry, and that even the cuboctahedron is more favor-
able than the icosahedron. By contrast, previous density-
functional studies on larger Ag clusters 共Jennison et al.,
1997兲 found that at N=55 the icosahedron is lower in
energy than the cuboctahedron.
Larger clusters have been studied mainly by semi-
empirical interatomic potentials. This approach has al-
lowed global-minimum searches for N up to 100 atoms
共Doye and Wales, 1998b; Erkoç and Yilmaz, 1999兲 and
the comparison of selected magic structures belonging to
the icosahedral, decahedral, and truncated octahedral
motifs for N up to 40 000 atoms 共Baletto, Ferrando, et
al., 2002兲. Doye and Wales 共1998b兲 modeled Ag by the
Sutton and Chen 共1990兲 potential and optimized Ag
clusters at N艋80, finding the same lowest-energy struc-
tures as Bonacic
´
-Koutecký, Cespiva, et al. 共1993兲 at N
=7,8,9, and icosahedral global minima at N =13 and 55.
Thirty of the global minima were icosahedral in charac-
ter, but several decahedron and fcc clusters were found,
the most stable being at N =38 共truncated octahedron兲
FIG. 14. ⌬ as a function of N for different cluster motifs in Ni,
Cu, Pd, Ag, Pt, and Au: 䊊, iscosahedra; 䊐, decahedra; 䉭,
trunacted octahedra. Data are taken from Baletto, Ferrando,
et al. 共2002兲, except for the data concerning Ni, which are origi-
nal. For decahedra and truncated octahedra only the most fa-
vorable clusters are shown in the different size ranges.
TABLE I. Potential parameters and cluster crossover sizes for several metals. Parameters p,q of the Rosato et al. 共1989兲 potential;
parameters
␴
1
=pq/2,
␴
2
,
␴
3
, and
␴
A
关see Eqs. 共16兲 and 共17兲兴; sizes where ⌬ is minimum 共N
⌬
Ih
for icosahedra and N
⌬
Dh
for
decahedra兲, and crossover sizes 共N
Ih→Dh
, N
Dh→fcc
, and N
Ih→fcc
兲. in the case of the different noble and quasi-noble metals. The
parameters of the potentials are taken from Baletto, Ferrando, et al. 共2002兲 except for those of Ni, which are found in Meunier
共2001兲 and Baletto, Mottet, and Ferrando 共2003兲. The results on the crossover sizes are taken from Baletto, Ferrando, et al. 共2002兲,
except those for Ni which are published here for the first time. The results for
␴
A
are taken from Soler et al. 共2000兲.
Metal pq
␴
1
␴
2
␴
3
␴
A
N
⌬
Ih
N
⌬
Dh
N
Ih→Dh
N
Dh→fcc
N
Ih→fcc
Cu 10.55 2.43 13.1 0.062 0.35 0.24 309 20000 1000 53000 1500
Ag 10.85 3.18 17.2 0.065 0.29 0.54 147 14000 ⬍300 20000 400
Au 10.53 4.30 22.6 0.082 0.16 1.86 147 1300 ⬍100 500 ⬍100
Ni 11.34 2.27 12.9 0.055 0.37 0.05 561 17000 1200 60000 2000
Pd 11.00 3.79 20.9 0.062 0.21 0.78 147 5300 ⬍100 6500 ⬍100
Pt 10.71 3.85 20.6 0.073 0.22 1.40 147 5300 ⬍100 6500 ⬍100
387
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and N=71,75 共decahedron兲. Mottet et al. 共1997兲 mod-
eled Ag by the Rosato et al. 共1989兲 semiempirical poten-
tial, reproducing essentially the same behavior as the
Sutton and Chen 共1990兲 potential at small sizes. They
showed that clusters of high stability can be obtained by
removing the central atom in perfect icosahedra. In fact,
a central vacancy allows neighboring atoms to better re-
lax, expanding their intrashell distance. The same effect
was also found in Cu and Au clusters. We note that, at
variance with Au, there is no indication in Ag in favor of
disordered structures at geometric magic numbers 共38,
55, 75兲. Finally Baletto, Ferrando, et al. 共2002兲 and Mot-
tet et al. 共2004兲 looked at the crossover sizes 共see Table I兲
by an extensive comparison of clusters of the different
motifs.
All the previous theoretical results agree in predicting
that Ag follows the general trend outlined in Sec. III.A,
with small icosahedra, intermediate-size decahedra, and
large truncated octahedra clusters. This is in clear con-
trast with the inert-gas aggregation experiments by
Reinhard, Hull, Ugarte, and Monot 共1997兲, in which
small clusters 共⬃2 nm of diameter兲 were mainly decahe-
dral 共or fcc兲 while large clusters 共above 5 nm兲 were
mostly icosahedral. This discrepancy can only be ex-
plained by taking into account kinetic trapping effects
leading to the growth of metastable structures, as dis-
cussed in Sec. V.B.3.
c. Copper clusters
The energetics of small Cu clusters has been exten-
sively treated in a recent review by Alonso 共2000兲; here
we give a brief summary of the results for small sizes and
then focus on larger clusters.
Several ab initio calculations
10
have been performed
for N 艋 10, again comparing a set of reasonably good
structures. There is evidence of planar structures up to
N=5 or N =6 maximum, as suggested by recent density-
functional calculations 共Jug, Zimmerman, Calaminici, et
al., 2002兲. For N =13, density-functional calculations
共Fujima and Yamaguchi, 1989兲 indicate that the icosahe-
dron is favored over the cuboctahedron.
For larger size, up to N⬃100, global optimization
studies by semiempirical potentials have been per-
formed. Doye and Wales 共1989b兲 used the Sutton and
Chen 共1990兲 potential, while Darby et al. 共2002兲 used the
Gupta 共1981兲 potential. In both cases, the best structure
at N =13 is the icosahedron, in agreement with density-
functional calculations, and there is no indication in fa-
vor of disordered structures at geometric magic num-
bers. Cu thus behaves like Ag, and very differently from
Au. Baletto, Ferrando, et al. 共2002兲 compared the ener-
gies of icosahedron, decahedron, and truncated octahe-
dron clusters at larger sizes, finding that the crossover
from icosahedral to decahedral structures is around 1000
atoms and that decahedron and truncated octahedron
structures are in close competition up to 30 000 atoms at
least 共see Table I兲. Contrary to the Ag case, this behav-
ior is in agreement with the inert-gas aggregation experi-
ments of Reinhard, Hall, Berthoud, et al. 共1997, 1998兲,
who were able to identify a prevalence of small icosahe-
dra, intermediate-size decahedra, and large fcc clusters,
with a wide size interval in which decahedra and fcc
clusters coexisted. The crossover size between icosahe-
dron and decahedron structures is in agreement with the
calculations.
d. Platinum clusters
The energetic stability of small Pt clusters has been
addressed by fewer studies than the other metals treated
here. Grönbeck and Andreoni 共2000兲 performed a
density-functional study in the size range 2艋 N 艋 5, find-
ing that the lowest-lying isomers are planar for both the
tetramer and the pentamer. The preference for planar
structures is in agreement with previous density-
functional calculations by Yang et al. 共1997兲, who also
found planar structures for N=6 by a dynamical quench-
ing procedure. In contrast, other ab initio calculations
共Dai and Balasubramanian, 1995兲 on the tetramer gave
preference to the tetrahedal structure, as in the density-
functional results of Fortunelli 共1999兲, who found that
the lowest-lying isomer is indeed the tetrahedron, with a
planar rhombic structure that was slightly higher in en-
ergy.
Sachdev et al. 共1992兲 modeled Pt at larger sizes, by an
embedded atom potential and made simulated anneal-
ing calculations, finding that both at 13 and at 55 atoms
low-symmetry isomers are the lowest in energy. On the
other hand, further global optimization results based on
semiempirical modeling 共Doye and Wales, 1998b; Bal-
etto, Ferrando, et al., 2002; Massen et al., 2002; Sebetci
and Güvenc, 2003兲 gave preference to the icosahedral
structure at N =13. Doye and Wales 共1998b兲 and Massen
et al. 共2002兲 found a low-symmetry structure at N=55,
while Baletto, Ferrando, et al. 共2002兲 found the icosahe-
dral structure also at this size, though with small cross-
over sizes among icosahedral, decahedral, and truncated
octahedral structural motifs 共see Table I兲. Ab initio re-
sults are even fewer. Yang et al. 共1997兲 found that at N
=13 several low-symmetry structures are lower in energy
than the icosahedron and the cuboctahedron. Watari
and Ohnishi 共1998兲 found that the cuboctahedron is
more stable than the icosahedron at N=13 by density-
functional calculations. Recently, Fortunelli and Aprà
共2003兲 extended their calculations to N=13, 38, and 55.
At N=13, they found that the Ino decahedron is lower
in energy than the icosahedron and the cuboctahedron,
but a structure of D
4h
symmetry group originating from
the symmetry breaking of the cuboctahedron is even
lower. At N =55 the order of the structures is inverted,
and the icosahedron becomes lower than the Ino deca-
hedron and the cuboctahedron. Fortunelli and Aprà
共2003兲 attributed the results at N=13 to the small size of
the molecule, while the behavior at N=55 is intermedi-
10
See Bauschlicher 共1989兲, Bauschlicher et al. 共1988, 1989,
1990兲, Fujima and Yamaguchi 共1989兲, Massobrio et al. 共1995兲,
Jug, Zimmerman, Calaminici, et al. 共2002兲, Jug, Zimmerman,
and Köster 共2002b兲.
388
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

ate between finite molecules and fully metallic systems.
Very recent density-functional calculations by Aprà et al.
共2004兲 show that rosette structures, originating from the
disordering of one or two vertices in icosahedra of 55
atoms, are considerably lower in energy than the icosa-
hedron itself. This result would support the preference
of Pt for low-symmetry structures at icosahedral magic
numbers.
e. Palladium clusters
The energetics of small Pd clusters have been investi-
gated by several groups employing different methods.
11
Besides their catalytic properties, these clusters are very
interesting because their lowest-lying isomers could
have nonzero spin 共Reddy et al., 1993; Watari and
Ohnishi, 1998兲, being thus magnetic. Zhang et al. 共2003兲
have recently performed an extensive study of the sta-
bility of Pd clusters within the density-functional ap-
proach at N艋 13. In this size range they compared sev-
eral 1D, 2D, and 3D isomers; moreover, they considered
a few selected structures at N=19 and 55. Their results
indicate that the lowest-lying isomers are 3D starting
with the tetramer 共Zacarias et al., 1999; Moseler et al.,
2001兲, and that icosahedral structures are favored over
both decahedral and cuboctahedral structures for N
=13 and 55. These results are in good agreement with
previous calculations by Kumar and Kawazoe 共2002兲.
On the other hand, Watari and Ohnishi 共1998兲 found
that the cuboctahedron is more stable than the icosahe-
dron at N =13 according to density-functional calcula-
tions. Moseler et al. 共2001兲 were able to compare their
results on vertical electron detachment energies for an-
ionic clusters with the experimental data 共Ervin et al.,
1988; Ho et al., 1991; Ganteför and Eberhardt, 1996兲,
obtaining a good agreement.
At larger sizes, ab initio results are few, being re-
stricted to the comparison of a small set of selected
structures at some special sizes. Kumar and Kawazoe
共2002兲 have compared icosahedron with cuboctahedron
and Ino decahedron clusters at N=55 and 147, finding
that slightly distorted icosahedral structures are the low-
est in energy in both cases. Nava et al. 共2003兲 also found
that a Jahn-Teller distorted icosahedron is lower in en-
ergy than a perfect icosahedron at N=55. Moreover,
Nava et al. 共2003兲 found that icosahedra and cuboctahe-
dra are very close in energy at both N=147 and N
=309, with the icosahedra prevailing at 147 and cuboc-
tahedra at 309. By tight-binding calculations, Barreteau
et al. 共Barreteau, Desjonquères, and Spanjaard, 2000;
Barrateau, Guirado-Lopez, et al., 2000兲 found that the
icosahedron is still lower in energy than the cuboctahe-
dron at N =309, but not at N=561. However, it must be
kept in mind that the cuboctahedron and the Ino deca-
hedron are not usually favorable fcc and decahedron
clusters. This observation is in agreement with previous
density-functional calculations by Jennison et al. 共1997兲,
who showed that for N =140, the truncated octahedron
is favored over the icosahedron structure obtained by
removing seven vertex atom from the Ih
147
, while in Ag
the opposite happens.
Global optimization studies up to N⯝100 by the Sut-
ton and Chen 共1990兲 potential give for Pd the same re-
sults as for Ag, with icosahedron structure at N =55, as
confirmed also in the optimization of the Gupta 共1981兲
potential by Massen et al. 共2002兲. At larger sizes, a com-
parison of the different structural motifs shows that Pd
behaves in a similar way to Pt 关with rather small cross-
over sizes 共see Table I兲, and fcc clusters already in close
competition with the other motifs at N ⬃100兴 when
modeled by the Rosato et al. 共1989兲 potential, while it
behaves very similarly to Ag when treated by means of
embedded atom potentials 共Baletto, Ferrando, et al.,
2002兲. Calculations by Jennison et al. 共1997兲 better sup-
port the results from the Rosato et al. 共1989兲 potential,
finding that fcc structures are more favored in Pd than in
Ag. However, this point would need further investiga-
tion, and it is not resolved by the analysis of the avail-
able experimental data. José-Yacamán, Marín-Almazo,
and Ascencio 共2001兲 observed by transmission electron
microscopy the thiol-passivated Pd nanoparticles in the
range of 1–5 nm diameter, seeing a variety of structures,
ranging from fcc 共simple and twinned兲, to icosahedral, to
decahedral, to amorphous structures. They were able to
observe rather large icosahedral clusters, explaining
their presence as being probably due to kinetic trapping
effects.
f. Nickel clusters
As in the case of Cu, small Ni clusters have been re-
cently reviewed by Alonso 共2000兲. Reuse and Khanna
共1995兲 and Reuse et al. 共1995兲 performed several density-
functional studies of small Ni clusters, concluding that
the lowest-lying isomers are nonplanar for N艌 4. At N
=7, two isomers, the pentagonal bipyramid and the
capped octahedron, are in close competition 共Nayak et
al., 1996兲. The former is favored by density-functional
共Nygren et al., 1992兲 and embedded-atom 共Boyakata et
al., 2001兲 calculations, while the latter is most likely to
be observed in the experiments 共Parks et al., 1994兲, al-
though the experimental data do not completely rule out
the pentagonal bipyramid 共Nayak et al., 1996兲. Nayak et
al. 共1996兲 performed ab initio calculations, finding that
these structures are almost degenerate in energy; then
they performed molecular-dynamics simulations by the
Finnis and Sinclair 共1984兲 semiempirical potential, find-
ing that the capped octahedron has a wider catchment
basin, becoming thus much more favorable at high tem-
peratures. This is clear evidence of an entropic effect
共see Sec. IV.A兲.AtN =13, Parks et al. 共1994兲 experimen-
tally identified the icosahedron as the most stable struc-
11
See Valerio and Toulhoat 共1996, 1997兲, Jennison et al.
共1997兲, Zacarias et al. 共1999兲, Barreteau, Desjonquères, and
Spanjaard 共2000兲, Barreteau, Guirado-López, et al. 共2000兲,
Efremenko and Sheintuck 共2000兲, Guirado-López et al. 共2000兲,
Efremenko 共2001兲, Krüger et al. 共2001兲, Moseler et al. 共2001兲,
Roques et al. 共2001兲, Kumar and Kawazoe 共2002兲, Nava et al.
共2003兲, Zhang et al. 共2003兲.
389
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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ture. Ab initio calculations at the same size 共Reuse and
Khanna, 1995; Reuse et al., 1995兲 have shown that the
icosahedron is favored over the cuboctahedron, but the
most stable isomer is obtained by distorting the icosahe-
dron slightly to obtain a cluster with D
3d
symmetry
group. The experimental observations 共Parks et al., 1995,
1997, 1998兲 indicate a truncated octahedron structure at
N=38, a capped decahedron structure at N =39, and
共possibly defected兲 icosahedral structures at N=55. The-
oretical results are in reasonable agreement with these
experimental findings. Different global optimization re-
sults within semiempirical schemes 共Doye and Wales,
1998b; Grygorian and Springborg, 2003兲 and tight-
binding calculations 共Lathiotakis et al., 1996兲 predict fcc
and icosahedral structures at N =38 and 55, respectively.
Moreover, capped decahedral structures just above size
38 are found 共Wetzel and DePristo, 1996; Andriotis and
Menon, 2004兲. At larger sizes, the comparison of the
different structural motifs by different semiempirical ap-
proaches 共Cleveland and Landman, 1991; Mottet et al.,
2004兲 shows that the crossovers icosahedral→decahedral
and decahedral→ fcc take place at large sizes as in the
case of Cu 共see Table I兲.
4. General structural properties of noble-metal and
quasi-noble-metal clusters
Here we sketch general trends among the six metals
of the previous section with respect to three features:
formation of planar clusters, magnitude of crossover
sizes among the motifs, and preference for low-
symmetry global minima at geometric magic sizes.
Concerning the preference for formation of planar
structures, there is a clear indication of an increasing
tendency going down from the 3d to the 5d series and
from left to right in the Periodic Table. Thus Au has the
strongest tendency, followed by Ag, Pd, and Pt, all three
being at the same level 共however, there is still some de-
bate about planar Pt clusters兲; Cu and Ni have no pref-
erence at all for planar structures. Several groups have
investigated the origin of this trend. Bravo-Pérez et al.
共1999b兲 correlated the preference for planar structures
in Au to the fact that nonadditive many-body interac-
tions are stronger than additive two-body ones in this
element. Bonacic
´
-Koutecký et al. 共2002兲 noticed that in
planar Au clusters, d electrons contribute more to the
bonding than in 3D structures; this would also qualita-
tively explain the weaker tendency to planar clusters for
Ag, because in Ag the bonding is more of s type. Finally,
Häkkinen et al. 共2002兲 investigated Au
7
−
,Ag
7
−
, and Cu
7
−
by density-functional calculations and demonstrated that
the propensity of Au clusters to favor planar structures
is correlated with the strong hybridization of the atomic
5d and 6s orbitals due to relativistic effects.
The tendency to form strained structures like icosahe-
dra and decahedra, leading to large crossover sizes, and
the tendency to present low-symmetry structures at geo-
metric magic numbers have been discussed, in the
framework of the same well-defined and quite reliable
energetic model, by Baletto, Ferrando, et al. 共2002兲 and
Soler et al. 共2000兲. What follows is a summary of these
works. The energetic model is the Gupta 共1981兲 or
Rosato et al. 共1989兲 potential, which predicts the correct
surface reconstructions 共Guillopé and Legrand, 1989兲
and reproduces quite accurately the diffusion barriers
共Montalenti and Ferrando, 1999a, 1999b; Ala-Nissila et
al., 2002兲 for these metals. In this framework, the poten-
tial energy E of a cluster of N atoms is E =兺
i=1
N
E
i
, where
the energy of atom i is given by
E
i
= A
兺
j=1
n
v
e
−p共r
ij
/r
0
−1兲
−
␰
冑
兺
j=1
n
v
e
−2q共r
ij
/r
0
−1兲
, 共11兲
where n
v
is the number of atoms within an appropriate
cutoff distance 共in the following we include only the first
neighbors兲, and A ,
␰
,p,q are parameters fitted to the
bulk properties of the element. The first term in Eq. 共11兲
is a repulsive Born-Mayer energy, while the second is
the attractive band energy. Following Tománek et al.
共1985兲, one can eliminate the parameters A and
␰
,in
order to have a function of 共p,q兲 and of the cohesive
energy per atom E
coh
. This is achieved by requiring that
in the bulk crystal at equilibrium E
i
=E
coh
and
⳵
E
i
/
⳵
r
=0 for r =r
0
. The result is
E
i
=
兩E
coh
兩
12共p − q兲
冋
q
兺
j=1
n
v
e
−p共r
ij
/r
0
−1兲
−
冑
12p
冑
兺
j=1
n
v
e
−2q共r
ij
/r
0
−1兲
册
. 共12兲
This explicitly shows that the properties of the potential
depend only on the parameters 共p,q兲, E
coh
and r
0
play-
ing the role of scale factors on energy and distance.
Since atomic distances r
ij
in noncrystalline structures are
not optimal, one can expect a metal that strongly in-
creases its energy for a change in r
ij
共i.e., that has a sticky
interatomic potential兲 to have small crossover sizes
共Doye et al., 1995兲. In a solid at equilibrium, all 12 first
neighbors are at r
ij
=r
0
. When one changes all r
ij
by a
common factor r
ij
→ 共1+␧兲r
ij
, developing the crystal en-
ergy per atom E
i
共␧兲 in Eq. 共12兲共with n
v
=12兲 to the sec-
ond order and dividing it by the equilibrium value 兩E
coh
兩,
one obtains 共Baletto, Ferrando, et al., 2002兲
␳
共␧兲 =
E
i
共␧兲 − E
i
共0兲
兩E
i
共0兲兩
=
1
2
␧
2
pq. 共13兲
Here
␴
1
=
␳
共␧兲/␧
2
=pq/2 is essentially the product of the
bulk modulus and the atomic volume divided by the co-
hesive energy per atom of the bulk crystal. Thus the
larger the pq, the smaller the crossover sizes from icosa-
hedra to decahedra N
Ih→Dh
and from decahedra to fcc
crystallites N
Dh→fcc
, as can be seen in Table I.
Let us now discuss the tendency to form low-
symmetry structures at geometric magic sizes 共referred
to as the tendency to amorphization in the following兲.
This tendency is due to a specific feature of the metallic
bonding, which derives from its many-body character
and relates the optimal distance of the bonding to the
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coordination of the atom 共bond-order/bond-length cor-
relation兲. This can be easily seen by taking a number of
neighbors 1艋n
v
艋12 in Eq. 共12兲, all at the same distance
r. Minimization with respect to r leads to the optimum
distance r
*
共n
v
兲,
r
*
共n
v
兲
r
0
=1+
1
2共p − q兲
ln
冉
n
v
12
冊
, 共14兲
which is more and more contracted for decreasing n
v
.It
also leads to the energy
E
*
共n
v
兲 =−兩E
coh
兩
冉
n
v
12
冊
共p−2q兲/2共p−q兲
. 共15兲
The atoms on the surface of the cluster try to contract in
order to minimize their elastic energy 共Soler et al., 2001兲.
In the case of highly coordinated structures like the
icosahedra 共which have intrashell expanded distances兲,
the optimization of the bond lengths is made at the ex-
pense of the number of bonds. Therefore one expects
amorphization to be easy in the elements with high elas-
tic energy 共i.e., large pq兲 and strong contraction of the
bonds with decreasing coordination, the latter leading to
a weak dependence of E
*
on n
v
. The contraction of the
bonds and the dependence of E
*
on the coordination
can be quantified by the following dimensionless param-
eters:
␴
2
=
n
v
r
0
dr
*
dn
v
=
1
2共p − q兲
,
␴
3
=
n
v
E
*
dE
*
dn
v
=
p −2q
2共p − q兲
. 共16兲
As is evident from Table I, all parameters agree in indi-
cating that Au has by far the strongest tendency to
amorphization, having the largest
␴
1
and
␴
2
and the
smallest
␴
3
, followed by Pt and Pd; Ag and especially Cu
and Ni have the weakest tendency.
On the same line of reasoning, Soler et al. 共2000兲 de-
fined a parameter based solely on experimental quanti-
ties to extend the estimate of the tendency to amor-
phization to other systems. This parameter is
␴
A
and is
defined as the elastic energy
␦
E
el
that is needed to form
an ordered surface with contracted atomic distances, di-
vided by the amorphization energy
␦
E
am
which is re-
quired to form a scattered distribution of bond lengths.
␦
E
el
is expressed by means of the 共Voigt-averaged兲 bulk
and shear moduli B and G. ⌬E
am
is approximated by the
enthalpy of melting ⌬H
melt
, which is an estimate of the
energetic cost to form amorphous structures resembling
those found in liquids. This leads to
␴
A
=
␦
E
el
⌬E
am
⯝
v共3B −5G兲
2
320B⌬H
melt
, 共17兲
where v is the atomic volume. Equation 共17兲 is valid for
small clusters having almost all atoms on their surface.
Elements with large
␴
A
are expected to have a stronger
tendency to amorphization. As can be seen in Table I,
the inspection of
␴
A
confirms the above trends, the only
difference being the indication that Pt should have a
stronger amorphization tendency than Pd, due essen-
tially to the comparatively weaker enthalpy of melting
of Pt 共the ratio ⌬H
melt
/兩E
coh
兩 is much smaller in Pt than
in Pd兲.
Finally, let us compare metals with Lennard-Jones
clusters, in order to sketch a qualitative picture of the
effects of many-body forces. If one applies the criterion
of Eq. 共13兲 to Lennard-Jones clusters, one finds a value
of
␴
1
, which is even larger than those for Au, Pt, and Pd.
This would suggest that Lennard-Jones systems have a
weaker tendency to form icosahedral clusters than these
metals, but this is not the case, since crossover sizes
icosahedron→ decahedron and icosahedron→ fcc are
much larger for Lennard-Jones clusters than for Au, Pt,
or Pd. This happens because the bond-order/bond-
length correlation of metallic elements tends to destabi-
lize icosahedral structures. In fact, metals would prefer
contracted interatomic distances for the low-
coordination surface atoms, and bulk distances for the
highly coordinated inner atoms. This is exactly the op-
posite of what happens in perfect icosahedral structures,
which have contracted internal distances and expanded
surface distances. Moreover, in some metals such as Pt,
bond directionality effects 共which are not included in the
potentials considered in Table I兲 can be important 共For-
tunelli and Velasco, 2004兲 and favor the appearance of
共111兲-like hexagonal facets on the cluster surface 共Aprà
et al., 2004兲, with a further destabilizing effect on icosa-
hedral clusters.
5. Silicon clusters
Silicon nanoclusters are of great practical interest be-
cause of their intense photoluminescence at room tem-
perature and the presence of quantum size effects.
There is an ongoing debate about the size at which the
most favorable clusters adopt the bulklike diamond
structure. Bachels and Schäfer 共2000兲 produced neutral
Si clusters in a laser vaporization source and measured
their binding energy in samples having average sizes N
¯
from 65 up to 890 atoms. They showed that the binding
energy per atom E
b
in this size range scales as N
1/3
, this
being characteristic of approximately spherical clusters
共Kaxiras and Jackson, 1993a兲. At smaller sizes, different
regimes in the behavior of E
b
with N were found, as can
be seen from Fig. 15. There, Bachels and Schäfer 共2000兲
plotted, besides their results at large sizes, E
b
for smaller
clusters 共N艋7兲 obtained from the results of other
groups 共Jarrold and Honea, 1991; Schmude et al., 1993,
1995; Fuke et al., 1994兲. Three regimes are clearly
shown. For N⬍10 the binding energy increases rapidly
with the cluster sizes, and compact elementary units are
built up. For 10⬍ N ⬍25, E
b
is practically independent
of N, as would happen for prolate structures 共Kaxiras
and Jackson, 1993a兲. This agrees with the previous ob-
servation of small prolate Si clusters by Jarrold and Con-
stant 共1991兲. Finally, for larger clusters the N
1/3
behavior
is recovered. Bachels and Schäfer 共2000兲 were also able
to produce metastable prolate structures at sizes up to
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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about 170 atoms. These structures were mostly present
in the part of the molecular beam with short dwell times.
How do these results compare to calculations? And is
the transition size to quasispherical structures the same
as the transition size to bulklike diamond structures? In
the following we address these questions.
Theoretical results on the energetics of small silicon
clusters have been obtained by quantum Monte Carlo,
density-functional, and tight-binding calculations. The
latter also permitted the application of global optimiza-
tion methods such as simulated annealing 共Yu et al.,
2002兲 and genetic algorithms 共Ho et al., 1998; Rata et al.,
2000; Wang, Wang, et al., 2001兲.
For very small clusters, Raghavachari and Logovinsky
共1985兲 found that the best clusters are planar only up to
N=4; several calculations 共Fournier et al., 1992; Li et al.,
1999; Zickfeld, et al., 1999; Yu et al., 2002兲 found a gen-
erally good agreement with the above results. An impor-
tant result at small sizes is that Si
10
is a very stable clus-
ter 共Raghavachari and McMichael-Rohlfing, 1988;
Grossman and Mitás, 1995a; Ramakrishna and Bahel,
1996; Li, Yin, et al., 2000兲 so that it can act as a subunit
of larger ones 共see Fig. 16兲. For example, quantum
Monte Carlo, density-functional 共Mitás et al., 2000兲, and
tight-binding calculations 共Yu et al., 2002兲 find that the
best structures at N =20 are simply formed by linking
two Si
10
subunits. Indeed, the most reliable explanation
for the observation of prolate clusters is that they are
built up by stacked subunits, whose structure is still un-
der debate. Kaxiras and Jackson 共1993a, 1993b兲 pro-
posed that the subunits are sixfold rings; Raghavachari
and McMichael-Rohlfing 共1988兲, Jarrold and Bower
共1992兲, and Ho et al. 共1998兲 were in favor of tricapped
trigonal prisms of nine atoms; Rata et al. 共2000兲 found
that Si
20
is made of six- and eight-atom subunits. They
also found that the dissociation energies and inverse mo-
bilities calculated from their global minima were in ex-
cellent agreement with the experiments 共Jarrold and
Honea, 1991; Hudgins et al., 1999兲 up to N=18, conclud-
ing that cluster formation is dominated by the energetics
up to these sizes.
In summary, experiments and calculations agree in
finding prolate structures of a few tens of atoms which
are built up by small subunits in the range 6–10 atoms.
This is also consistent with fragmentation experiments
of clusters of about 150 atoms by Ehbrecht and Huisken
共1999兲, yielding Si
6
+
–Si
11
+
products.
Clusters are quasispherical for about N ⬎ 25 共Hudgins
et al., 1999兲, even if they do not form crystalline struc-
tures at these sizes. Indeed, the experiments of Ehbrecht
and Huisken 共1999兲 could be interpreted as assuming
that compact shapes at N⬃150 are built up by small
subunits 关the prolate metastable structures observed by
Bachels and Schäfer 共2000兲 being possibly unfolded ver-
sions of these clusters兴, and this agrees with the simu-
lated annealing calculations by Yu et al. 共2002兲. But this
is not the only possibility, since Kaxiras and Jackson
共1993a兲,Hoet al. 共1998兲, and Mitás et al. 共2000兲 have
shown that quasispherical noncrystalline structures,
which are not built up by stacking smaller subunits, be-
come favorable at N艌20. These sizes are large enough
to allow the formation of cages containing at least one Si
atom inside 共Mitás et al., 2000兲. These results compare
rather well with the experiments of Bachels and Schäfer
共2000兲. However, the theoretical determination of the
most stable structures in medium-size Si clusters is a
very complex task: see, for example, the debate about
the structure of Si
36
in Sun et al. 共2003兲 and Bazterra et
al. 共2004兲.
Mélinon et al. 共1997, 1998兲 demonstrated that the
models based on quantum confinement in crystalline sili-
con clusters fail for films containing grains of less than
2 nm diameter 共say N ⯝200兲. They were not able to ob-
serve any crystallization by transmission electron mi-
croscopy. On the other hand, if the grains are of 3 nm
diameter, there is evidence of crystalline ordering 共Eh-
FIG. 15. Size dependence of the binding energies of neutral
silicon clusters: 䊐, binding energies of the two groups of clus-
ter isomers found by calorimetric measurements; 쎲, data for
neutral silicon clusters obtained from the collision-induced dis-
sociation experiments of Jarrold and Honea 共1991兲 on the cor-
responding silicon cluster cations, taking the experimentally
determined photoionization potentials into account as from
Fuke et al. 共1994兲; 䉱, the Knudsen mass spectrometric mea-
surements of Schmude et al. 共1993, 1995兲. The crossover from
elongated to the spherical neutral silicon structures can be es-
timated from the binding energies to be located around N
=25. The structure for an elongated Si
26
cluster is taken from
Grossman and Mitás 共1995b兲. Adapted from Bachels and
Schäfer, 2000.
FIG. 16. The best isomers of Si
20
according to Mitás et al.
共2000兲. Structure E is the lowest in energy according to quan-
tum Monte Carlo calculations. From Mitás et al., 2000.
392
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

brecht et al., 1997; Ehbrecht and Huisken, 1999; Ledoux
et al., 2000兲. This would indicate that the transition to
crystalline structures is around N=400 atoms, as Yu et al.
共2002兲 found by simulated annealing tight-binding calcu-
lations.
6. Clusters of fullerene molecules
The condensed-phase properties of C
60
molecules
have been the subject of considerable interest because
of the unusual character of molecular interactions. In
fact, above room temperature, C
60
molecules can be
considered as large spherical pseudoatoms which are
free to rotate; the effective interaction between these
pseudoatoms is extremely short-ranged relative to the
large equilibrium pair separation. This is at the origin of
peculiar properties. In fact, such sticky interactions 共see
Fig. 7兲 disfavor all strained configurations, like those oc-
curring in icosahedra or in the liquid phase. Indeed, the
existence of a liquid phase for bulk C
60
共Hagen et al.,
1993; Caccamo et al., 1997; Ferreira et al., 2000兲 is still
debated. Therefore clusters of C
60
molecules are on the
opposite side with respect to clusters of alkali metals,
and we expect noncrystalline structures to be the global
minima only for aggregates of a small number N of mol-
ecules.
This qualitative prediction has been confirmed by all
existing calculations. We can divide these studies into
two classes. The first uses an all-atom potential, with
each carbon atom interacting with atoms of other mol-
ecules by a Lennard-Jones potential. This approach in-
cludes deviations from spherical symmetry, but at
present allows global optimization only for small N. The
second class uses spherically averaged potentials, such as
the Girifalco 共1992兲 potential, in which Lennard-Jones
centers are continuously and uniformly distributed on
the surface of a sphere, or the potential developed by
Pacheco and Prates-Ramalho 共1997兲, which is derived by
fitting density-functional calculations. The Girifalco
共1992兲 potential is purely two-body, whereas the Pacheco
and Prates-Ramalho 共1997兲 potential includes two-body
and three-body terms 共the latter, however, being of
rather small importance兲.
All these approaches agree in predicting that icosahe-
dral structures are favored only at very small N. All-
atom potentials favor icosahedral structures up to N
=16 共Doye et al., 1997; Garcia-Rodeja et al., 1997; Rey et
al., 1997兲, while calculations with the Girifalco 共1992兲
potential, which is the stickiest, indicate that icosahedra
are the lowest in energy only up to N=13 共Rey et al.,
1994; Wales, 1994a; Doye and Wales, 1996b兲. Finally, the
Pacheco and Prates-Ramalho 共1997兲 potential favors
icosahedral structures up to N=15. Larger clusters are
either based on decahedra or on close-packed struc-
tures, as shown by Doye et al. 共2001兲, who performed an
exhaustive basin-hopping global optimization study up
to N=105 on the basis of both Pacheco and Prates-
Ramalho 共1997兲 and Girifalco 共1992兲 potentials. Close-
packed structures become more frequent as the size in-
creases. Both potentials indicate that the most stable
isomers are found at N =38 共truncated octahedron兲, N
=75 关共2,2,2兲 Marks decahedron兴 and 101 关共2,3,2兲 Marks
decahedron兴, and that N =55 is not a magic number. At
that size, the icosahedron is higher in energy than the
best decahedron structure by 0.3 and 2 eV according to
the Pacheco and Prates-Ramalho 共1997兲 and Girifalco
共1992兲 potentials, respectively.
These results completely disagree with the experimen-
tally observed structures. In fact, Martin et al. 共1993兲 and
Branz et al. 共2000, 2002兲 found that clusters grown at low
T and annealed at ⬃500 K 共to allow the less bound sur-
face molecules to evaporate兲 present a mass spectrum
with a clear sequence of icosahedral magic numbers up
to very large sizes, well above N =100 共see Fig. 17兲. The
peak at N=55 is very evident, and no peaks are found at
N=38 and 75. The mass spectra are qualitatively similar
for neutral, positively and negatively charged clusters
共Branz et al., 2002兲. Recently, Rey et al. 共2004兲 compared
the neutral and singly-ionized cluster structures in a
model including the Girifalco 共1992兲 potential plus a
point polarizable dipole electrostatic model. They found
that the structures of ionized clusters are very similar to
those of neutral clusters, thus confirming the observa-
tions. Branz et al. 共2002兲 found decahedra and close-
packed structures only after further annealing at higher
T. This result indicates the possible existence of kinetic
trapping effects in the formation of the clusters of C
60
molecules 共Baletto, Doye, and Ferrando, 2002兲,aswe
discuss in Sec. V.B.5. Trapping effects are overcome only
after a strong annealing.
The resulting high-T experimental structures, how-
ever, still do not coincide with those predicted by the
Girifalco 共1992兲 and Pacheco and Prates-Ramalho
共1997兲 potentials. In fact, there is a sequence of experi-
mental magic numbers that can be attributed to the
Leary tetrahedron at N =98 and to its fragments down to
N=48, while the calculations do not attribute special sta-
bility to these structures. This discrepancy is probably
due to the fact that even the Pacheco and Prates-
Ramalho 共1997兲 potential is too sticky; a slightly less
sticky Morse potential gives the Leary tetrahedron as a
magic structure 共Doye et al., 2001兲.
IV. THERMODYNAMICS OF FREE NANOCLUSTERS
In this section, we analyze the effects of raising the
temperature on cluster structures. These effects may in-
clude solid-solid structural transitions and the melting of
the cluster if the temperature is raised enough. Both
solid-solid transitions and melting are cases that show
the peculiar thermodynamic behavior of small finite sys-
tems such as clusters. Indeed, the thermodynamics of
finite systems is a fascinating and complex field, involv-
ing many subtleties. Here we have no intention of being
exhaustive and refer readers who need more complete
treatments to existing monographs, from the classical
book of Hill 共1964兲 to the excellent book by Wales
共2003兲. In the following we treat only those issues that
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have a direct bearing on temperature-dependent
changes in the cluster structures.
Generally speaking, two points must be kept in mind
about cluster thermodynamics. First, phase transitions in
small systems are gradual, not sharp 共Hill, 1964兲. A con-
sequence of this fact is that there are bands of tempera-
ture and pressure within which two or more cluster
structures may coexist. This coexistence is dynamic, like
that of coexisting chemical isomers.
12
Second, thermody-
namic properties 共like the melting point兲 can be strongly
size dependent, in analogy to what happens to the
global-minimum structures. General trends for thermo-
dynamic properties may be deduced on the basis of the
following expression for the Gibbs free energy G 共Hill,
1964兲. In a bulk system, G =Ng共p,T兲 where g共p,T兲 is the
specific Gibbs potential. For a small system, we have
also to consider contributions coming from surfaces,
edges, etc. Thus we write G as
G = Ng共p,T兲 + b共p,T兲N
2/3
+ c共p,T兲N
1/3
+ d共p,T兲,
共18兲
where the term in N
2/3
is a surface free energy, the term
in N
1/3
is due to edge contributions, and the last term
might be due, among other things, to the rotation of the
cluster. In the limit N → ⬁, one has G → Ng, the macro-
scopic relationship. But when the system is small all
terms are important. The finite size of the clusters also
implies that treatments in different thermodynamic en-
sembles 共microcanonical and canonical兲 may give differ-
ent behaviors for thermodynamic quantities such as the
heat capacity 共Bixon and Jortner, 1989; Jortner, 1992兲.
The thermodynamics of clusters have been studied by
a variety of theoretical and simulation tools. These in-
clude Monte Carlo and molecular-dynamics simulations
and analytical methods.
13
In Sec. IV.A we focus on the role of entropic contri-
butions to the free energy at increasing temperature,
which may cause solid-solid structural transitions when
the structure corresponding to the global energy mini-
mum ceases to be the most likely at high T, so that other
structures prevail. In Sec. IV.B we deal with the melting
transition. There, we first discuss what is meant by melt-
ing 共and premelting兲 in clusters, briefly reviewing experi-
mental and simulation methods. Then we focus on the
size dependence of the melting point, which may be very
complex, as in the case shown in Fig. 18. We treat phe-
nomenological theories for the 共average兲 monotonic de-
pendence of the melting point on size, and discuss the
origin of its nonmonotonic variations. Finally, in Sec.
IV.C we focus on some systems of particular interest.
A. Entropic effects and solid-solid transitions
Structural transitions can take place upon increasing
the temperature for a given size because of entropic ef-
12
See, for example, Honeycutt and Andersen 共1987兲, Berry et
al. 共1988兲, Labastie and Whetten 共1990兲, Bartell 共1992兲, Mat-
suoka et al. 共1992兲, Cleveland and Landman 共1994兲, Schmidt et
al. 共1998兲, Jellinek 共1999兲.
13
Early studies are those of Lee et al. 共1973兲, Briant and Bur-
ton 共1975兲,Imry共1980兲, Nautchil and Petsin 共1980兲, Natanson
et al. 共1983兲, Berry et al. 共1984兲, Quirke and Sheng 共1984兲,
Amar and Berry 共1986兲, Davis et al. 共1987兲, Luo et al. 共1987兲,
Beck and Berry 共1988兲, Reiss et al. 共1988兲, Bixon and Jortner
共1989兲.
FIG. 17. Mass spectra of C
60
clusters: panel 共a兲, clusters pro-
duced at low T do not reveal any magic number; panel 共b兲,
after mild annealing to evaporate the less bound molecules, a
clear series of icosahedral magic numbers emerges; panels 共c兲
and 共d兲, decahedra and close-packed magic numbers appear
only after a high-T annealing. Adapted from Branz et al., 2002.
394
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

fects. When the minimum-free-energy structure is differ-
ent from the minimum-energy structure, we can define a
temperature T
ss
at which a solid-solid transformation oc-
curs. In other words, for T ⬎ T
ss
the minimum-energy
structure 共which is always the most likely for T → 0兲
ceases to be the most probable to the advantage of some
other structure, which prevails because of entropic ef-
fects. Structural changes from fcc to decahedral and
icosahedral structures as T increases have been pre-
dicted theoretically and seen in simulations of several
systems 关small Lennard-Jones 共Doye and Calvo, 2001兲,
Au 共Cleveland et al., 1998, 1999兲 and Cu clusters 共Bal-
etto et al., 2004兲兴. Moreover, solid-solid entropy-driven
structural transformations in Lennard-Jones clusters of
about 200 atoms and with different structures have been
observed in Monte Carlo simulations by Polak and Pa-
trykiejew 共2003兲. Recent experiments by Koga et al.
共2004兲 support the existence of entropy-driven solid-
solid transitions. Here we follow the theoretical treat-
ment of Doye and Calvo 共2001, 2002兲, dealing first with
the classical harmonic approximation, and then intro-
ducing anharmonic and quantum corrections.
1. Structural transitions in the harmonic approximation
Let H be the Hamiltonian of a cluster of size N:
H = T共兵p其兲 + U共兵r其兲, 共19兲
where T共兵p其兲=兺
i=1
N
p
i
2
/2m is the kinetic energy and
U共兵r其兲=U共r
1
,...,r
N
兲 is the potential energy. Assume
now that the cluster has a 共nonlinear兲 structure, corre-
sponding to a given minimum s of energy E
s
0
. If the cou-
pling of rotational and vibrational motions can be ne-
glected, Z
s
can be factored as
Z
s
= Z
s
tr
Z
s
rot
Z
s
vib
, 共20兲
where Z
s
tr
and Z
s
rot
are related to the center-of-mass
translation and to rigid rotational motions, respectively,
while Z
s
vib
is related to the vibrational motion around s.
Z
s
vib
depends on the internal coordinates
␰
s,i
where i
ranges from 1 to
␬
=3N−6. For small oscillations, the
vibrational motion can be treated within the harmonic
approximation. In this case, it is convenient to choose
the
␰
s,i
as the normal-mode coordinates, in order to write
U as
U = E
s
0
+
1
2
兺
i=1
␬
␻
s,i
2
␰
s,i
2
, 共21兲
where the
␻
s,i
are the normal-mode frequencies. The
transformation to normal modes allows an easy evalua-
tion of Z
s
vib
, which, in the classical case, is given by
Z
s
vib
⯝
e
−
␤
E
s
0
共2
␲
ប兲
␬
兿
i=i
␬
冕
−⬁
⬁
d
␰
s,i
d
␰
˙
s,i
e
−共
␤
/2兲共
␻
s,i
2
␰
s,i
2
+
␰
˙
s,i
2
兲
= e
−
␤
E
s
0
兿
i=i
␬
冉
k
B
T
ប
␻
s,i
冊
. 共22兲
The classical expressions for Z
s
tr
and Z
s
rot
are
Z
s
tr
= V
冉
Mk
B
T
2
␲
ប
2
冊
3/2
, Z
s
rot
=
冉
2
␲
k
B
TI
¯
s
ប
2
冊
3/2
, 共23兲
where V is the volume of the box in which the cluster is
contained, M is the cluster mass, and I
¯
s
is the average
inertial moment in s 关I
¯
s
=共I
s
xx
I
s
yy
I
s
zz
兲
1/3
, with I
s
xx
, I
s
yy
, and
I
s
xx
the principal moments of inertia兴.
To estimate the temperature dependence of the prob-
ability of finding the cluster in s, we have to compute the
total partition function Z. Before doing so, we note that
each minimum s has a number n
s
of equivalent permu-
tational isomers, of equivalent minima which are ob-
tained by exchanging the coordinates of atoms of the
same species. For homogeneous clusters, n
s
=2N!/h
s
,
where h
s
is the order of the symmetry group of mini-
mum s 共Wales, 2003兲. If, at a given T, there are M
min
minima with a non-negligible probability of being occu-
pied, the probability p
s
that the cluster is in the en-
semble of the permutational isomers of minimum s can
be evaluated by the superposition approximation 共Doye
and Wales, 1995兲. Within this approximation, Z is ob-
tained by summing up the contributions from all signifi-
cant minima. This gives
FIG. 18. Size-dependence of the measured melting tempera-
tures and latent heat of melting for sodium clusters. These
quantities are compared to the mass spectrum of the top panel,
which shows electronic magic numbers. The correlations be-
tween the electronic magic sizes and the peaks in the melting
temperature are rather weak, indicating that the melting be-
havior of small sodium clusters cannot be explained by simple
models. Adapted from Schmidt et al., 1998. Reprinted with
permission from Nature http://www.nature.com/
395
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

p
s
=
n
s
Z
s
Z
⯝
n
s
Z
s
兺
␴
=1
M
min
n
␴
Z
␴
⯝
n
s
I
¯
s
3/2
e
−
␤
E
s
0
⍀
s
−
␬
兺
␴
=1
M
min
n
␴
I
¯
␴
3/2
e
−
␤
E
␴
0
⍀
␴
−
␬
, 共24兲
where ⍀
s
=共⌸
i
␬
␻
i,s
兲
1/
␬
is the geometric average of the vi-
brational frequencies of s.
Let us now consider the simplest case of a potential-
energy surface having two minima, s and s
⬘
共with E
s
0
⬍E
s
⬘
0
兲, and calculate the ratio p
s
/p
s
⬘
between their occu-
pation probabilities as a function of T. Assuming that
I
¯
s
⯝I
¯
s
⬘
, one obtains
p
s
p
s
⬘
=
Z
s
n
s
Z
s
⬘
n
s
⬘
= e
−
␤
⌬E
0
冋
n
s
n
s
⬘
冉
⍀
s
⬘
⍀
s
冊
␬
册
, 共25兲
where ⌬E
0
=E
s
0
−E
s
⬘
0
. Since ⌬E
0
⬎0 we have two cases:
共i兲 if n
s
/⍀
s
␬
⬎n
s
⬘
/⍀
s
⬘
␬
, s is favored over s
⬘
for all tem-
peratures;
共ii兲 if n
s
/⍀
s
␬
⬍n
s
⬘
/⍀
s
⬘
␬
, for T⬎T
ss
, s
⬘
becomes more
favored than s and thus a solid-solid transition is
possible; see Fig. 19.
From the above equations, we estimate T
ss
as
T
ss
=
⌬E
0
k
B
关ln共n
s
⬘
/n
s
兲 +
␬
ln共⍀
s
/⍀
s
⬘
兲兴
. 共26兲
To find a solid-solid transition, not only must case 共ii兲
hold, but T
ss
must be below the melting range of the
cluster.
How do entropic contributions behave in typical
cases? Doye and Calvo 共2002兲 throughly analyzed
Lennard-Jones clusters within the harmonic superposi-
tion approximation. They grouped together the minima
pertaining to each motif 共icosahedron, decahedron, and
fcc兲, thus being able to estimate transition temperatures
between structural motifs and not simply between pairs
of minima. They found that the entropic contributions
shift upwards through icosahedral→ decahedral and
decahedral→ fcc crossover sizes with increasing tem-
perature, as can be seen in the 共coarse-grained兲 struc-
tural phase diagram of Fig. 20. This happens because
icosahedral structures have on average smaller vibra-
tional frequencies than decahedral ones, the latter hav-
ing softer vibrations than fcc clusters 共Doye and Calvo,
2002兲. Metals modeled by Sutton and Chen 共1990兲 po-
tentials behave in a qualitatively similar way 共Doye and
Calvo, 2001兲. A discussion of the observation of solid-
solid transitions in experiments and simulations on spe-
cific systems is given in Sec. IV.C. Here we only remark
that solid-solid transitions may be extremely slow in
typical situations, so that observing them may require a
substantial overheating of the initial structure 共Koga et
al., 2004兲, to surmount the energy barriers leading to
escape from the initial basin.
2. Anharmonic and quantum corrections
The harmonic superposition approximation is often
accurate for cluster thermodynamics up to temperatures
close to the melting range, as shown for Ag
38
and Cu
38
by Baletto et al. 共2004兲. However, there are cases in
which this approximation is not sufficient, and anhar-
monic effects must be included 共Doye and Wales, 1995兲.
Solid-solid transitions can be close to the melting tem-
perature, in which case anharmonic effects are not neg-
ligible. Moreover, the harmonic approximation may
strongly overestimate the probability of minima that
have low-barrier escape paths. An example of failure of
the harmonic approximation is found by Wang, Blastein-
Barojas, et al. 共2001兲 and Doye and Calvo 共2003兲.
One can introduce anharmonic effects via
T-dependent frequencies, i.e., ⍀
˜
s
共T兲=⍀
s
共1−
␤
s
0
/
␤
兲,
where
␤
0
is a measure of the anharmonicity. This then
gives for the transition temperature, neglecting the per-
mutational contribution,
␤
ss
anHA
=
␬
⌬E
0
冋
ln共⍀
s
/⍀
s
⬘
兲 +ln
冉
␤
−
␤
s
0
␤
−
␤
s
⬘
0
冊
册
, 共27兲
where the second term represents the first order ap-
proximation to the anharmonic correction to
␤
ss
anHA
FIG. 19. 共Color in online edition兲 Schematic representation of
the probabilities p
S
and p
S
⬘
of the local energy minima S and
S
⬘
, with S
⬘
having higher energy than S vs temperature. In the
temperature range close to the transition temperature T
SS
⬘
the
two structures coexist, being almost equally likely.
FIG. 20. Structural phase diagram in the N,T plane for
Lennard-Jones clusters in the harmonic superposition approxi-
mation. From Doye and Calvo, 2002.
396
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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=1/k
B
T
ss
anHA
. To apply the previous equation it is neces-
sary to estimate
␤
0
but the available methods do not
allow one to treat large sizes.
Quantum corrections can be inserted both in Z
s
rot
and
Z
s
vib
. Since quantum corrections to Z
s
rot
should become
important only at very low temperatures, we now con-
sider only the corrections to Z
s
vib
. These corrections are
very easily introduced within the harmonic approxima-
tion. The quantum partition function of a set of oscilla-
tors is given by
Z
s
vib
= e
−
␤
E
s
0
兿
i=1
␬
关2 sinh共
␤
ប
␻
i,s
/2兲兴
−1
. 共28兲
The classical limit is valid when it is possible to approxi-
mate the sinh共
␤
ប
␻
i,s
/2兲 with its argument; thus it is valid
for high temperatures. The introduction of quantum os-
cillators modifies the expression of the ratio between the
occupation probabilities of two minima s and s
⬘
in the
following way:
p
s
p
s
⬘
= e
−
␤
⌬E
0
冤
n
s
n
s
⬘
兿
i
␬
sinh共
␤
ប
␻
i,s
⬘
/2兲
兿
i
␬
sinh共
␤
ប
␻
i,s
/2兲
冥
. 共29兲
In particular, we note that here the temperature depen-
dence is also in the vibrational contribution and not only
in the Boltzmann factor. To understand when quantum
corrections are needed we may refer to the Debye tem-
perature ⌰
D
. If the temperature of the system is above
0.70⌰
D
the system can be treated as classical to a good
approximation, while at lower temperature we also need
to take quantum effects into account 共Ashcroft and Mer-
min, 1976兲. Calvo et al. 共2003兲 satisfactorily explained the
finite-temperature spectroscopic properties of CaAr
N
clusters including quantum and anharmonic corrections
in their calculations.
B. Melting of nanoclusters
The concepts of solid and liquid states, which are
commonly employed when discussing extended systems,
can be applied to clusters. In fact, at low temperature,
the particles of a cluster spend most of the time making
small-amplitude vibrations around the global minimum,
in analogy to what happens in bulk solids. If the tem-
perature is increased, other minima begin to be popu-
lated, and this is associated with the onset of some dif-
fusive motion. Finally, if the temperature becomes high
enough, the cluster explores the basins of a huge num-
ber of minima, with fast rearrangements, thus behaving
like a liquid droplet. Within this description, cluster
melting is seen as an isomerization transition, with the
number of probable isomers increasing drastically after
some threshold temperature. A nice example of this be-
havior for Ag
6
clusters is given by Garzón, Kaplan, et al.
共1998兲. However, there are several differences between
the solid-liquid transition of clusters and that of its bulk
counterpart:
共i兲 The melting point is reduced 共with a few known
exceptions兲 with a complex dependence on size.
共ii兲 The latent heat is smaller; this can be understood,
for example, by noting that disordering the sur-
face costs less than disordering inner atoms.
共iii兲 The transition does not take place sharply at one
definite temperature, but smoothly over a finite
temperature range. There, solid and liquid phases
may coexist dynamically in time 共Berry et al.,
1988; Lynden-Bell and Wales, 1994兲. When one
observes the cluster over a long time interval,
there will be subintervals in which the cluster ap-
pears to be solid and others in which the cluster is
fully liquid.
共iv兲 The heat capacity can become negative in micro-
canonical environments 共Schmidt et al., 2001兲.
This means that the microcanonical average ki-
netic energy may be a nonmonotonically increas-
ing function of the total energy in the range of the
transition 共Bixon and Jortner, 1989兲.
共v兲 The melting transition depends on cluster struc-
ture and chemical ordering 共this being, indeed, a
nonequilibrium effect兲.
Often the melting transition is preceded by premelting
phenomena. To quote Calvo and Spiegelman 共2000兲,
“premelting phenomena are characteristic of isomeriza-
tions taking place in a limited part of the configuration
space,” for example, isomerizations involving surface at-
oms only 共in this case the term surface melting is used兲.
Premelting phenomena are often singled out by addi-
tional peaks in the specific heat vs temperature.
1. Experimental methods
Following Haberland 共2002兲, experiments studying
cluster melting can be divided into two classes:
共i兲 study of the change of some physical property
across the melting point 共for example, changes in
photon or x-ray diffraction patterns兲
共ii兲 measurement of the caloric curve E =E共T兲, that is,
the cluster’s internal energy E as a function of T.
Takagi 共1954兲 made the first observation of the melt-
ing point depression by transmission electron micros-
copy. This is a standard technique for studying the size-
dependent melting point of small particles by
monitoring the changes in the diffraction pattern associ-
ated with the disordering of the structure. Electron and
x-ray diffraction and nanocalorimetry techniques have
been used to study melting in deposited clusters 共Lai et
al., 1996; Peters et al., 1998; Efremov et al., 2000兲.At
present these methods are, however, not applicable to
free, mass-selected clusters in vacuum for two reasons:
first, no method of temperature measurement is known
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Rev. Mod. Phys., Vol. 77, No. 1, January 2005

in this case, and the density of mass-selected clusters is
so small that it is extremely difficult to collect a diffrac-
tion signal. In any case, as the size decreases, the diffrac-
tion techniques become increasingly inaccurate due to
line broadening. Several experiments have tried to mea-
sure the melting behavior of free cluster, mainly by
methods belonging to class 共i兲. Even et al. 共1989兲 and
Buck and Ettischer 共1994兲 looked for some spectro-
scopic evidence. Electron diffraction from a 共not mass-
selected兲 supersonic expansion gives Debye-Scherrer
like diffraction rings, whose intensity is a measure of
cluster temperature. This method was pioneered by
Farges et al. 共1981兲 and later intensively studied by Hov-
ick and Bartell 共1997兲. Martin et al. 共1994兲 were the first
to publish a size dependence of the melting temperature
of free clusters. They showed that the structure on mass
spectra of large sodium clusters can depend sensitively
on the temperature. The disappearance of structure was
interpreted as being due to melting. Another method
has been proposed by Shvartsburg and Jarrold 共2000兲 to
measure the melting temperature T
m
for small tin clus-
ters, with the surprising result that they melt higher than
bulk tin. Here, cluster ions are injected into a helium-gas
atmosphere, and are pulled by an electric field through
the gas. The collisions with the gas produce an effective
friction force. Clusters having a small collision cross sec-
tion experience a smaller friction force and arrive first.
A new approach to the study of the melting processes in
gallium nanoparticles embedded in a matrix has been
recently developed by Parravicini et al. 共2003a, 2003b兲.
This method is based on capacitance measurements
through the derivative of the dielectric constant with re-
spect to T.
Calorimetry experiments have been performed on de-
posited clusters 共for example, Lai et al., 1996, considered
tin clusters on a SiN substrate兲 and free clusters. Free-
cluster calorimetry has been applied by Haberland’s
group 共Schmidt et al., 1997, 1998, 2001; Kusche et al.,
1999兲 to Na clusters and by Bachels et al. 共2000兲 to Sn
clusters. Haberland’s method consists of two steps, the
preparation of size-selected clusters of known tempera-
ture and the determination of their energy. A beam of
cluster ions is produced and thermalized in a heat bath
at temperature T
1
; a mass spectrometer is used to select
a single cluster size. Then the clusters are irradiated by a
laser beam and absorb several photons. The basic idea
of the experiment is that absorbtion of photons of en-
ergy
␦
U=h
␯
will raise the temperature beyond T
1
. The
photon energy quickly relaxes into vibrations and heats
the cluster to a temperature T
2
at which the clusters do
not emit atoms on the time scale of the experiment 共sev-
eral microseconds兲. Only the adsorption of more pho-
tons from the laser pulse raises the temperature above
T
evap
, the temperature needed for evaporation of atoms
from the cluster. The size distribution of the remaining
cluster ions is measured, and this is a very sensitive mea-
sure of the cluster internal energy. The increased tem-
perature of the cluster, T
2
, is then identified by increas-
ing the cluster source temperature until the thermally
heated clusters show the same photofragmentation be-
havior as the laser-heated ones. A second mass spec-
trometer measures the distribution of the fragment ions
produced. Different numbers of absorbed photons lead
to clearly separated groups of fragments in the mass
spectrum, with the distance between two groups corre-
sponding to exactly the energy of one photon. This al-
lows one to calibrate the mass scale in terms of energy. If
the temperature of the heat bath is varied, the internal
energy of the selected clusters changes and thus also the
number of evaporated atoms 共see Fig. 21兲.
In the method of Bachels et al. 共2000兲 Sn clusters are
produced by a laser ablation source using a pulsed
nozzle whose temperature is variable. Cluster tempera-
tures should deviate from nozzle temperatures only by
10–20 K. Neutral clusters are studied, so that there is no
mass selection; the width of the distribution of the clus-
ter sizes can be of the order of 60% of the average size.
Energy is measured by a sensitive pyroelectric foil in
which the clusters impinge, causing a temperature in-
crease which leads to a measurable voltage jump.
2. Computational methods
Here we first discuss the quantities that may be used
to single out the melting transition, and then we briefly
review the most common computational methods.
The most common method for studying the melting
transition is the calculation of the caloric curve, that is,
the total cluster energy E as a function of T, in a simu-
lation where the cluster is heated up from a low-T solid
configuration. E共T兲 may show a smooth jump in the
melting region 共see Fig. 22兲, corresponding to a peak in
the heat capacity c共T兲=
⳵
E/
⳵
T, as in Fig. 23. However,
the caloric curve is not always an efficient indicator, be-
cause the jump may be small and difficult to find, or
even absent. When properly calculated, peaks in the
heat capacity can indicate the melting temperature T
m
.
However, when multiple peaks are present, the assign-
ment of the melting point may become difficult. One
may identify T
m
with the temperature of the highest
peak, but sometimes this criterion fails 共see the discus-
sion in Frantz, 2001兲.
FIG. 21. Schematic representation of the experimental method
for measuring E共T兲 for free clusters. Adapted from Schmidt et
al., 1998. Reprinted with permission from Nature http://
www.nature.com/
398
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

In general, criteria for distinguishing solidlike and liq-
uidlike phases and sensitive indicators of the melting
transition are needed.
Solidlike and liquidlike phases can be distinguished if
an order parameter Q can be found such that the Lan-
dau free energy F
l
共Q兲 is bistable for a range of tempera-
tures. Then two distinct phases can be identified and can
be said to coexist 共Wales, 2003, and references therein兲.
The Landau free energy is defined as F
l
共Q兲
=k
B
T ln P共Q兲+const, where P共Q兲 is the probability of
observing a value Q of the order parameter for the tran-
sition in the simulation run. Thus, in an equivalent way,
one can find that the probability distribution of the or-
der parameter has two minima in the canonical en-
semble. As N → ⬁, the free-energy barrier between these
minima is expected to increase and the phase transition
tends to the first-order bulk phase transition.
The practical problem is now to find an adequate set
of order parameters. A possible order parameter is the
total potential energy E
p
or just the difference between
the configurational energy and the global-minimum en-
ergy 共Lynden-Bell and Wales, 1994; Shah et al., 2003兲.
Other parameters are related to the local ordering, such
as the orientational bond order parameters Q
L
共Stein-
hardt et al., 1983兲, or the signatures
14
of the common-
neighbor analysis 共Faken and Jónsson, 1994兲. The radial
distribution function is also routinely monitored. This is
useful also to single out surface disordering phenomena
共Qi et al., 2001; Huang and Balbuena, 2002兲.
Solidlike and liquidlike phases may also be distin-
guished by their different mobility. In bulk systems, the
Lindemann criterion is commonly used. The atomic vi-
brational amplitude 具⌬r
2
典
1/2
is defined as a disorder pa-
rameter ⌬
L
; experiments and simulations show that its
critical value is around 0.10–0.15 in units of the atomic
spacing 共Bilgram, 1987; Lowen, 1994兲. For irregular fi-
nite systems, however, the pure Lindemann criterion is
not appropriate; a better idea is to introduce the
distance-fluctuation measure ⌬
DF
, as proposed by Etters
and Kaelberer 共1977兲 and Berry et al. 共1988兲:
14
In the common-neighbor analysis a signature is assigned to
each pair of neighbors. This signature is a triplet of integers
共r ,s,t兲, where r is the number of common nearest neighbors of
two atoms of the pair, s is the number of nearest-neighbor
bonds among the r common neighbors, and t is the length of
the longest chain that can be formed with the s bonds. We have
found that the monitoring of the signatures 共r ,s ,t兲=共5,5,5兲,
共4,2,1兲, 共4,2,2兲 is sufficient to distinguish icosohedral, decahe-
dral, and fcc structures over a wide range of sizes 共Baletto,
Mottet, and Ferrando, 2001a; Baletto, 2003兲 during growth
simulations. The 共5,5,5兲 signature indicates pairs located along
a 共locally兲 fivefold symmetry axis. The 共4,2,1兲 signature is asso-
ciated with pairs with fcc neighborhood. The 共4,2,2兲 signature
is associated with pairs comprising an atom of a 共locally兲 five-
fold axis and an atom outside the axis. In perfect fcc clusters
one expects to find a large percentage of 共4,2,1兲 signatures and
no 共5,5,5兲 and 共4,2,2兲 signatures. Comparing icosahedral and
decahedral clusters in the same size range, one finds that the
percentage of 共5,5,5兲 signatures is much larger in icosahedra,
while the percentage of 共4,2,1兲 signatures is larger in decahe-
dra. Tables with the values of the percentages for these signa-
tures are found in the dissertation of Baletto 共2003; see also
Baletto, Mottet, and Ferrando, 2001a兲.
FIG. 23. Heat capacity and bond-length fluctuations in the 57-
atom Lennard-Jones cluster. Left panel, heat capacity curves.
Open circles and solid line represent the results of parallel
tempering simulations, while the dotted line and the thin solid
line are the results of Metropolis Monte Carlo runs of different
length. Right panel, ⌬
DF
calculated in two Monte Carlo simu-
lations of different length 共the longer one corresponds to the
open circles兲. The temperature at which ⌬
DF
starts its strong
increase corresponds well to the temperature of the heat ca-
pacity peak. The heat-capacity c and ⌬
DF
are given in reduced
units 共see Frantz, 2001兲. Adapted from Frantz, 2001.
FIG. 22. 共Color in online edition兲 Caloric curves from a
molecular-dynamics melting simulation: 쎲,Ag
38
; 䊏,Cu
38
. The
heating rate is of 0.1 K/ns. The clusters are modeled by the
Rosato et al. 共1989兲 potential. E
*
is the average total energy E
of the cluster after the subtraction of the global-minimum en-
ergy and of the kinetic and harmonic contributions: E
*
=E
−E
GM
−3共N −1兲k
B
T. The jump in the caloric curve is clear for
Ag
38
, and less evident for Cu
38
. Figure courtesy of Christine
Mottet.
399
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

⌬
DF
=
2
N共N −1兲
兺
i⬍j
冑
具⌬r
ij
2
典
具r
ij
典
, 共30兲
where r
ij
is the distance between atoms i and j, 具⌬r
ij
2
典
=具共r
ij
−具r
ij
典兲
2
典. The key difference between the Linde-
mann and distance-fluctuation criteria is that the latter is
based on the fluctuation of the distance between pairs of
atoms while the former is based on the fluctuation of
individual atoms relative to their average position. The
critical value of ⌬
DF
for a solid-liquid transition has been
suggested to be around 0.03–0.05 共Zhou et al., 2002兲.
Some caveats are necessary when using ⌬
DF
. As pointed
out by Frantz 共1995, 2001兲, ⌬
DF
depends on the length of
the simulation run. For very long simulations, a value
⌬
DF
⬃0.1 can be obtained even at low temperatures, at
which the cluster is simply making isomerizations among
permutational isomers in its solid form. However, com-
puting ⌬
DF
can be a very sensitive method for monitor-
ing qualitatively the melting transition from below
共Frantz, 2001兲.
In order to have quantitative information on the mo-
bility of the cluster particles, while avoiding the short-
comings of ⌬
DF
, one can consider quantities that do not
depend on the simulation length. Rytkönen, Valkealahti,
and Manninen 共1998兲 monitored the average rate of
change in the nearest neighbors of the cluster atoms.
Another possibility is to compute the number of distinct
basins visited per unit time.
The melting of nanoclusters has largely been studied
by standard simulation methods such as molecular dy-
namics and Monte Carlo 共see, for example, Frenkel and
Smit, 1996兲. Unlike the Monte Carlo method, which is
based on a fictitious dynamics, molecular dynamics
closely mimics the true dynamics of the system. This al-
lows the calculation of equilibrium time-dependent cor-
relation functions, like those related to the diffusion of
the cluster particles. Molecular dynamics also permits
the realistic simulation of melting and freezing pro-
cesses, which take place on finite time scales and thus
may include kinetic effects. On the other hand, Monte
Carlo can be faster in sampling the configuration space,
being thus more appropriate if one is interested in static
quantities only.
In fact, a major problem when calculating thermody-
namic properties by simulation is quasiergodicity, that is,
the incomplete sampling of the configurational space
that may occur in the phase-change region. Quasiergod-
icity can lead to overestimated transition temperatures
in molecular-dynamics caloric curves 共Calvo and Guet,
2000兲 if the heating rate is too fast.
Various methods have been employed to reduce the
systematic errors resulting from quasiergodicity, includ-
ing the histogram, jump-walking, smart-walking, and
parallel tempering methods.
15
Many of these methods
are based on the coupling of configurations obtained
from ergodic higher-T simulations to the quasiergodic
lower-T simulations. Monte Carlo J-walking methods,
for example, couple the usual small-scale Metropolis
moves made by a lower-T random walker with occa-
sional large-scale jumps that move the random walker
out of confined regions of configurational space. These
large-scale jumps are to configurations that are obtained
in higher-T simulations. Calvo et al. 共2000兲 and Neirotti
et al. 共2000兲 showed that parallel tempering is remark-
ably successful in overcoming quasiergodicity and in cal-
culating accurate heat-capacity curves. In parallel tem-
pering, several simulations are run in parallel at
different temperatures. Periodically, exchange moves are
attempted between configurations at nearby tempera-
tures, and accepted or rejected according to the Me-
tropolis rule.
3. Size dependence of the melting point
The size dependence of the cluster melting point for a
given material usually shows a monotonic decrease with
decreasing size, and has irregular variations on a fine
scale. Here we first deal with the justification for the
monotonic trend and then treat the fine-scale nonmono-
tonic variations.
The average dependence of a melting point with size
N has been derived by means of a few phenomenologi-
cal models. The classical calculation by Pawlow 共1909兲
has been further extended and modified by several
groups. A recent account of some of these developments
is given by Chushak and Bartell 共2001b兲. Here we derive
Pawlow’s formula following the approach of Buffat and
Borel 共1976兲. We consider a cluster of size N and of
spherical shape. At a given pressure p, its melting tem-
perature is T
m
共N兲, which has to be compared with the
bulk melting temperature T
m
共⬁兲. In analogy with bulk
melting, one identifies the solid-liquid transition by
equating the chemical potentials
␮
s
and
␮
l
of the solid
and of the liquid, so that T
m
共N兲 at a given pressure p
follows from the solution of this equation:
␮
s
共p,T兲 =
␮
l
共p,T兲. 共31兲
This equation means that the chemical potential of a
fully liquid and of a fully solid cluster are equal at melt-
ing. The chemical potential can be expanded around its
value at the triple point; we retain first-order terms only:
␮
共p,T兲 =
␮
共p
0
,T
0
兲 +
⳵
␮
⳵
T
共T − T
0
兲 +
⳵
␮
⳵
p
共p − p
0
兲. 共32兲
From the Gibbs-Duhem equation 共−Vdp+SdT+Nd
␮
=0兲 it follows that
⳵
␮
⳵
T
=−s,
⳵
␮
⳵
p
=
1
␳
, 共33兲
where s =S /N is the entropy per particle and
␳
=V/N is
the number density. From Eqs. 共31兲–共33兲, and taking into
account that
␮
s
共p
0
,T
0
兲=
␮
l
共p
0
,T
0
兲, one obtains
15
See, for example, Frantz et al. 共1990兲, Labastie and Whetten
共1990兲, Calvo et al. 共2000兲, Ghayal and Curotto 共2000兲, Neirotti
et al. 共2000兲, Frantz 共2001兲, Fthenakis et al. 共2003兲.
400
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

− s
l
共T − T
0
兲 +
1
␳
l
共p
l
− p
0
兲 =−s
s
共T − T
0
兲 +
1
␳
s
共p
s
− p
0
兲.
共34兲
Here, one must distinguish between the pressure p
s
of a
solid cluster and the pressure p
l
of a liquid cluster. In
fact, the pressure inside a small object of radius r is
larger than the external pressure because of the Laplace
contribution p
Lap
=2
␥
/r to the pressure, where
␥
is the
interface tension of the cluster and r is its radius. Since a
liquid and a solid cluster of N atoms can differ both in
interface tension 共
␥
lv
for the liquid-vapor interface,
␥
sv
for the solid-vapor interface兲 and in radius, they can
have different Laplace pressure terms,
p
l
= p
ext
+
2
␥
lv
r
l
, p
s
= p
ext
+
2
␥
sv
r
s
. 共35兲
For a cluster of r ⬃10 nm, typical interface tensions are
of the order of 10
3
erg/cm
2
, so that the Laplace pressure
is much larger than p
ext
under the usual conditions. Thus
p
ext
can be neglected in Eq. 共35兲. Moreover, for spherical
clusters,
r
s
r
l
=
冉
␳
l
␳
s
冊
1/3
. 共36兲
Substituting Eqs. 共35兲 and 共36兲 into Eq. 共34兲, neglecting
p
ext
, and taking into account that L =T
0
共s
l
−s
s
兲 is the la-
tent heat of melting per particle, one obtains
1−
T
m
共N兲
T
m
共⬁兲
=
2
␳
s
r
s
L
冋
␥
sv
−
␥
lv
冉
␳
s
␳
l
冊
2/3
册
. 共37兲
Since r
l
⬀N
1/3
, one finds that Eq. 共37兲 can be rear-
ranged to the form 共Reiss et al., 1988兲
T
m
共N兲 = T
m
共⬁兲
冋
1−
C
N
1/3
册
, 共38兲
where C is a constant. This formula gives a very simple
dependence for T
m
共N兲 which can thus be considerably
lower than T
m
共⬁兲共Buffat and Borel, 1976; Lewis et al.,
1997; Rytkönen, Valkealahti, and Manninen, 1998兲.An
expression for C easily follows from Eq. 共37兲, but the
quantities involved in that expression may not be easy to
evaluate. Moreover, several rather crude approxima-
tions are involved in Eq. 共37兲, so that usually C is con-
sidered as a fitting parameter.
Several attempts have been made to improve
Pawlow’s theory. Buffat and Borel 共1976兲 included
higher-order terms in Eq. 共32兲, and solved numerically
the resulting equation. Including higher-order terms is in
principle more accurate, but these terms increase the
number of unknown parameters to be evaluated. Reiss
and Wilson 共1948兲, Hanszen 共1960兲, and finally Sambles
共1971兲 refined Pawlow’s model by including the possibil-
ity of surface melting, that is, of having clusters 共of total
radius r兲 made of an inner core of radius r −
␦
and an
external liquid shell of thickness
␦
. The melting tem-
perature is found by imposing the equilibrium condition
on this solid-core/liquid-shell particle. A derivation for
the case of metallic particles was given by Kofman et al.
共1994兲 and Vanfleet and Mochel 共1995兲; here we briefly
sketch their approach. The extension of this approach to
molecular clusters 共by including van der Waals forces兲
can be found in Levi and Mazzarello 共2001兲.
Let us consider a particle like that in Fig. 24, contain-
ing N
l
particles in the liquid shell. Its free energy G is
given by
G = 共N − N
l
兲
␮
s
+ N
l
␮
l
+4
␲
r
2
冋
␥
sl
冉
r −
␦
r
冊
2
+
␥
lv
+ S
⬘
e
−
␦
/
␨
册
, 共39兲
where
S
⬘
=
␥
sv
−
冋
␥
lv
+
␥
sl
冉
r −
␦
r
冊
2
册
. 共40兲
Here
␨
is the characteristic length of the interaction
among atoms in liquid metals, and the term S
⬘
e
␦
/
␨
takes
into account the effective interaction between the solid-
liquid and liquid-vapor interfaces. This effective interac-
tion is repulsive and favors the formation of a liquid
shell between the solid core and the vapor. For T not too
far from T
m
共⬁兲 one may approximate N
l
共
␮
l
−
␮
s
兲
⯝V
l
␳
L关T
m
共⬁兲−T兴/T
m
共⬁兲, where V
l
=4
␲
关r
3
−共r −
␦
兲
3
兴/3 is
the volume of the liquid layer, and the density difference
between liquid and solid is neglected 共
␳
l
=
␳
s
=
␳
兲. Mini-
mizing G with respect to
␦
, one finds the following solu-
tion for T
m
共N兲:
1−
T
m
共N兲
T
m
共⬁兲
=
2
␥
sl
␳
L共r −
␦
兲
共1−e
−
␦
/
␨
兲 +
S
⬘
r
2
␳
L
␨
共r −
␦
兲
2
e
−
␦
/
␨
.
共41兲
Pawlow’s result of Eq. 共37兲 is recovered in the limit
␨
→ 0, which means a sharp interface, or equivalently, no
effective repulsive interaction between the solid-liquid
and the liquid-vapor interfaces. For molecular clusters,
this effective interaction is proportional to 1/
␦
2
instead
of e
−
␦
/
␨
共Levi and Mazzarello, 2001兲.
FIG. 24. A particle with a solid core of radius r −
␦
and a liquid
shell of thickness
␦
.
401
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

Following a somewhat different line of reasoning,
Chushak and Bartell 共2001b兲 derived a similar expres-
sion, which is close to those found by Hanszen 共1960兲
and Sambles 共1971兲:
1−
T
m
共N兲
T
m
共⬁兲
=
2
␳
l
L
再
␥
sl
r −
␦
+
␥
lv
r
冋
1−
冉
␳
s
␳
l
冊
2/3
册
冎
. 共42兲
If
␥
sl
is evaluated as
␥
sv
−
␥
lv
共that is, if the liquid shell
perfectly wets the solid core兲, and
␦
=0, Eq. 共42兲 reduces
to Eq. 共37兲.
Both Eqs. 共41兲 and 共42兲 are more accurate than
Pawlow’s result down to small sizes 共see Fig. 25兲. Peters
et al. 共1998兲 found that the solid-core/liquid-shell model
gives a better fit of the experimental melting tempera-
tures of supported lead clusters. Chushak and Bartell
共2001b兲 and Wang, Zhang, et al. 共2003兲 melted a series of
selected fcc clusters of gold and copper, respectively, by
molecular-dynamics simulations within an embedded-
atom energetic model. They found that Eq. 共38兲 is accu-
rate down to N⬃10
3
and 5⫻10
2
for Au and Cu, respec-
tively. At smaller sizes, Eq. 共42兲 is in better agreement
with the simulation data in the case of Au. Other simu-
lation results are given by Ercolessi et al. 共1991兲, Lewis et
al. 共1997兲, and Qi et al. 共2001兲. Lai et al. 共1996兲 obtained
an excellent fit to their experimental data on the melting
of tin clusters 共see Fig. 26兲 by the formula of Hanszen
共1960兲; the same data were successfully refitted by means
of Eq. 共41兲共Bachels et al., 2000兲.
When the size dependence of the melting point is ex-
amined on a fine scale, irregular variations are found,
especially at small sizes, where the addition or the re-
moval of a single atom can have dramatic effects. There
is a good agreement among several simulation results in
predicting that clusters of special stability, such as icosa-
hedral clusters at magic sizes 共55, 147, etc.兲, melt at
higher temperatures than predicted by Eq. 共38兲关see, for
example, the cases of Lennard-Jones, Na, and noble-
metal clusters 共Rytkönen, Valkealahti, and Manninen,
1988; Valkealahti and Manninen, 1993; Frantz, 2001兲兴.
However, the situation can be more complicated.
Schmidt et al. 共1998兲 measured the melting point of Na
clusters up to 200 atoms. They found an irregular behav-
ior of T
m
, with few well-defined peaks, as shown in Fig.
18, and L showed the same kind of behavior. The peaks
in T
m
共N兲 and L were not well correlated with those in
the mass spectrum, corresponding to the closing of elec-
tronic shells. These results showed that the most abun-
dant clusters and the highest-melting-point clusters do
not always coincide. In a recent experiment, Schmidt et
al. 共2003兲 were able to measure the energy and the en-
tropy change in the melting of Na clusters. They found
that the peaks in T
m
共N兲 are driven by the energy differ-
ence between the liquid and the solid phases. The en-
tropy difference is closely correlated to the energy dif-
ference and simply causes a damping of the energetic
effects. This would show that the main indicator for a
high T
m
is the energetic separation of the global mini-
mum from higher isomers. However, it is not yet clear
whether entropic differences are always correlated to
energetic differences, or whether this is a trait of Na
clusters.
The irregular variations of the melting point in
Lennard-Jones clusters were studied by Frantz 共2001兲,
who thoroughly analyzed the range 26艋 N 艋60 by the
parallel tempering technique. Frantz 共2001兲 found that
the peaks in T
m
共N兲 were generally well correlated with
those of other stability indicators, such as the energetic
separation of the second isomer from the global mini-
mum E
SM
共N兲−E
GM
共N兲, ⌬
1
共N兲, and ⌬
2
共N兲关see Eq. 共6兲兴.
An exception was N =58, a magic size for the stability
indicators, but not corresponding to a peak of T
m
共N兲.
Finally, even though the melting point in nanoclusters
is usually depressed, some evidence of exceptions to this
rule is beginning to accumulate. The first experimental
evidence of an exception is due to Shvartsburg and Jar-
rold 共2000兲, who studied the melting of Sn clusters. Small
Sn clusters have a rather elongated structure, which
should change to nearly spherical upon melting. No sig-
nature of this change was observed by Shvartsburg and
Jarrold 共2000兲, so they concluded that Sn cluster ions
containing 10–30 atoms have a melting point at least
50 K above the bulk one. This behavior has been con-
FIG. 25. Melting point of gold clusters from molecular-
dynamics simulations 共black dots兲, and comparison with differ-
ent theoretical results: solid line, Pawlow’s theory; ⫻, second-
order corrections from Buffat and Borel 共1976兲; heavy dashed
curves, the liquid-shell model by Sambles 共1971兲,Eq.共42兲; thin
dashed curves, second-order corrections to Sambles’s formula.
Reprinted with permission from Chushak and Bartell 共2001b兲.
Copyright 2001 American Chemical Society.
FIG. 26. Comparison of theoretical and experimental melting
points of supported tin clusters: 쎲, experiment; solid line, the
fitting by means of Eq. 共42兲. From Lai et al., 1996.
402
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

firmed by Joshi et al. 共2002兲, who performed ab initio
molecular-dynamics simulations of the melting of Sn
10
.
They found that binding in such a small cluster is indeed
covalent, and that the specific heat of Sn
10
shows a
shoulder around 500 K due to a permutational rear-
rangement of atoms that preserves the trigonal prism
core of the ground state. Only at much higher tempera-
tures, T ⯝1500 K, does this core distort and break up,
yielding a peak in the specific heat around 2300 K. Joshi
et al. 共2003兲 also simulated the melting of Sn
20
, finding
that it has a lower melting point than Sn
10
, but one that
is still much higher than that of bulk Sn. Chuang et al.
共2004兲 have found by ab initio Langevin molecular dy-
namics that Sn
6
,Sn
7
, and Sn
13
also melt at a higher T
than bulk Sn.
Small Sn clusters seem not to be the only ones to melt
at higher temperature than the bulk crystal. There is
recent experimental and theoretical evidence for gallium
clusters 共Breaux et al., 2003; Chacko et al., 2004兲, whose
high melting point is attributed to the fact that, in con-
trast to bulk Ga, binding in small clusters is fully cova-
lent. Finally, in their simulations, Akola and Manninen
共2001兲 observed a bulklike behavior of Al
13
−
above the
Al bulk melting temperature.
As a final remark, we note that thermodynamic effects
on structural stability and the melting of nanoclusters
suggest a more general definition of the stability, includ-
ing temperature effects 共Baletto, Rapallo, et al., 2004兲.
C. Studies of selected systems
1. Lennard-Jones clusters
There are several computational studies of melting in
Lennard-Jones clusters, especially at magic icosahedral
sizes. Early molecular-dynamics studies on the melting
of Ar clusters by Briant and Burton 共1975兲 indicated a
relatively sharp first-order-like transition at T below the
bulk melting temperature. This result was later con-
firmed. At N=55 and N =147 all simulations indicate
relatively sharp jumps in the caloric curves at T⯝35 K
and T ⯝42 K, and the radial distributions confirm that
whole clusters melted during the transitions 共Etters and
Kaelberer, 1975; Honeycutt and Andersen, 1987; Mat-
suoka et al., 1992兲.AtN =13 the cluster fluctuates back
and forth between the two phases and a consistent de-
termination of T
m
is very difficult. The breadth of the
solid-liquid transition is also revealed for Ar
13
adsorbed
on a surface 共Blaisten-Barojas et al., 1987兲, where the
cluster changes from an icosahedral, solidlike structure
at low T to a set of liquidlike structures at high T. Clus-
ters containing 309 or more atoms are observed to des-
orb atoms at temperatures where the core is still solid
共Rytkönen et al., 1997兲. Thus it is very complicated to
define their melting point, if the heating rate is slow. Fast
heating rates can, however, cause the superheating of
the cluster. Rytkönen, Valkealahti, and Manninen 共1998兲
found a rather good agreement with Eq. 共38兲, except
that T
m
at small sizes and for icosahedral structures is
higher than predicted analytically. To investigate how
the melting takes place, Rytkönen et al. calculated the
radial distribution function, finding that, at nonmagic
icosahedral sizes, there is a clear tendency to surface
melting while the cores preserve an icosahedral mor-
phology.
Frantz 共1995, 2001兲 made a detailed systematic study
of small Lennard-Jones clusters by Monte Carlo simula-
tions. He found that the smallest clusters presented an
irregular dependence of their thermodynamic properties
on size. For N ⬎ 25 some trends were pronounced. Icosa-
hedral packing was dominant 共with the exception of the
truncated octahedron at N =38兲 and the heat capacity
peak parameters formed two overlapping sequences as a
function of N, depending on whether the overlayer had
Mackay or anti-Mackay stacking. In clusters with an
anti-Mackay overlayer, the heat-capacity peak shifted
towards higher T and became smaller at increasing N,
while clusters with Mackay rearrangements had small,
low-T peaks that generally shifted to higher T and grew
in size as N increased. There was a sequence of magic
numbers, N =36,39,43,46,49,55, at which the heat-
capacity peak was stronger. As already discussed in Sec.
IV.B.3, this sequence correlates well with that extracted
from the binding energy differences.
2. Sodium clusters
Experiments by Martin et al. 共1994兲 and Schmidt et al.
共1998兲 have revealed a complex dependence of T
m
共N兲
for Na clusters. Its irregular small-size behavior has al-
ready been discussed in Secs. IV.B.3. Liu et al. 共2002兲
performed a systematic simulation study of the melting
of different morphologies over a wide size range. They
found that the melting points for all sizes and structural
types were in a narrow T range 共200–300 K兲 and all
clusters presented a liquid-gas transition around 1000 K.
Another interesting point about the melting of Na
clusters, which is of great general interest, is the pres-
ence of premelting effects. A detailed study of this topic
at 8艋N艋147 atoms has been performed by Calvo and
Spiegelman 共2000兲 by a Monte Carlo thermodynamical
analysis close to the solid-liquid transition, within a
semiempirical modeling by Gupta 共1981兲 potentials. Up
to 147 atoms, the thermodynamics appears to be directly
related to the lowest-energy structures, and melting by
steps is favored by the presence of surface defects. For
N⬍75 the T
m
presents a strong nonmonotonic behavior
with N, which is typical of geometric size effects. For
larger sizes the transition becomes more and more simi-
lar to the bulk case. Calvo and Spiegelman 共2000兲 find
evidence of premelting phenomena at small sizes, with
the caloric curves presenting multimodal behavior with
increasing T. Simple isomerization between a few struc-
tures, surface melting, or competition between several
funnels on the energy landscape may be the causes of
this premelting. Premelting seems to be the rule for
small Na clusters, the only exceptions being the very
stable magic icosahedral structures. The secondary
peaks in the heat capacity become less pronounced at
N⬎100, indicating that premelting becomes less impor-
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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tant. Calculations by Gupta potentials and ab initio
schemes 共Rytkönen, Häkkinen, and Manninen 1998,
1999; Reyes-Nava et al., 2002兲 agree reasonably well
with the experiments, but several aspects of the melting
behavior of Na clusters are still to be understood. Calvo
and Spiegelman 共2004兲 have recently reexamined the
melting behavior of Gupta Na clusters in the size range
around 130 atoms, finding that the potential is somewhat
inadequate because it predicts strong premelting effects
that are not observed in the experiments. These pre-
melting effects are related to the surface melting of the
outer icosahedral shell.
3. Noble-metal and transition-metal clusters
In the case of metallic clusters, the literature is exten-
sive, especially for Au clusters.
16
To begin, we remark
that, although the melting point is depressed, its reduc-
tion is smaller than that of Lennard-Jones clusters, and
it is strongly material dependent 共Jellinek et al., 1986;
Garzón and Jellinek, 1991; Jellinek and Garzón, 1991兲.
Usually melting is accompanied by a peak in the specific
heat, and by a substantial rearrangement of the cluster.
But there are exceptions. For example, the simulations
by Westergren et al. 共2003兲 revealed that Pd
34
melts with-
out an accompanying peak in the heat capacity, and the
atoms become mobile without any significant change in
geometric structure. García-Rodeja et al. 共1994兲 and Lee
et al. 共2001兲 studied several transition and noble metals,
at very small sizes, around 13 atoms, modeled by Gupta
potentials. Their simulation results were in agreement
with those obtained by Güvenc and Jellinek 共1992兲. The
salient result is that the behavior of Ni, Cu, Pd, Ag, and
Pt is similar, the main difference being that Pd
13
presents
a more complex anharmonic behavior than Ag 共Wester-
gren and Nordholm, 2003兲. The 13-atom clusters un-
dergo a structural transition from a rigid, solidlike icosa-
hedral structure to a nonrigid, liquidlike one via an
intermediate temperature range in which both forms co-
exist. The caloric curves do not present any sharp tran-
sition, and T
m
is found by monitoring ⌬
DF
关Eq. 共30兲兴 and
looking for a maximum in the specific heat. The instabil-
ity of some specific atom can play a key role in melting,
as in the case of the capped atoms at N=14. The atom
added to the 13-atom icosahedron can diffuse rapidly at
T below the melting region, and this causes a small peak
in the specific heat 共García-Rodeja et al., 1994; Lee et al.,
2001兲 so that T
m
is smaller than for the 13-atom cluster.
Similar behavior is found at N=20 for the capped
double icosahedron. In contrast, when the double icosa-
hedron is not the global minimum, the heat capacity has
an unexpected behavior. For example, Pd
19
shows a dis-
tinct abrupt change at T ⬃400 K and a rounded-off
broad peak at higher T 共Lee et al., 2001兲.
Li, Ji, et al. 共2000兲 made a thorough analysis at N
=55 for Au, Ag, and Cu by means of molecular-
dynamics simulations. Ag
55
and Cu
55
showed abrupt
changes during the meltinglike transitions, while the
transition of Au
55
seemed to proceed for a relatively
broader interval, with ⌬
DF
increasing gently from 300 up
to 600 K and with a small, ladderlike energy jump in the
caloric curve. The degrees of reduction for T
m
were dif-
ferent among the three metals. Compared to Ag and Cu,
Au exhibited the largest size-induced drop of T
m
. This
confirms the trends of Table I, since Cu also gives the
most stable icosahedron cluster from the thermody-
namic point of view.
At N =38 a complex melting behavior for Cu was
found by Baletto et al. 共2004兲, using molecular-dynamics
simulations. A solid-solid transition took place from the
truncated octahedron at the global minimum. Then the
cluster rearranged itself into defected decahedron struc-
tures before melting, in analogy with the behavior of the
Lennard-Jones cluster of the same size 共Doye et al.,
1998兲. On the other hand, Ag
38
, which has the same
global-minimum structure as Cu
38
, does not exhibit any
solid-solid transition to defected decahedral structures.
At this size Au presents a coexistence phase region
among different isomers already at T⬃250 K 共Garzón et
al., 1999兲.
Molecular-dynamics simulations of the melting transi-
tions for crystalline nickel clusters above N⬃750 show
that T
m
共N兲 closely follows Eq. 共38兲共Qi et al., 2001兲,
while, for N ⬍ 500, icosahedral structures present higher
T
m
and large latent heat. Similar results are found also
for Cu 共Wang, Zhang, et al., 2003兲. For smaller N, the
icosahedral packing is more stable and presents higher
melting points than those derived from Eq. 共38兲
共Valkealahti and Manninen, 1993兲.
Gold clusters have been studied intensively in recent
years 共see Cleveland et al., 1998, 1999; Lee et al., 2001; Li
et al., 2002兲. All these studies agree that a solid-solid
structural transformation from the low-T optimal struc-
tures to icosahedral structures takes place below the
melting temperature. Detailed analysis of the atomic
trajectories and of the structural evolution indicates that
this solid-to-solid transition is essentially without diffu-
sive motion, occurring quickly and involving many
共small兲 cooperative displacements of the atoms. The
structural transformation is driven by the vibrational
and configurational entropy at elevated T 共Luo et al.,
1987; Ajayan and Marks, 1988兲. A thorough molecular-
dynamics simulation study was made by Liu et al. 共2001兲.
Different morphologies were compared up to N
⬃25 000. At intermediate sizes, for all type of mor-
phologies, Liu et al. 共2001兲 found that the melting pro-
cess occurs in three stages: a relatively long time of sur-
face disordering and reordering, a relatively short time
of surface melting, and finally a rapid overall melting.
Concerning the differences among the structural motifs,
starlike decahedra are the hardest to melt, while icosa-
hedron clusters are the easiest, with regular decahedra
being in between. Small cuboctahedra, up to N =309 at
16
See Buffat and Borel 共1976兲, Garzón and Jellinek 共1993兲,
Cleveland et al. 共1997, 1998, 1999兲, Li, Ji, et al. 共2000; Li, Lee, et
al., 2002兲, Lee et al. 共2001兲, and Koga et al. 共2004兲.
404
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

least, transform into icosahedra before melting, while
large ones do not. Contrary to the other motifs, T
m
of
icosahedra saturates at N⬎ 12 000. Liu et al. 共2001兲 also
compared different structures at identical sizes, finding
that transition temperatures could differ by as much as
75 K. This nonequilibrium effect indicates the differ-
ences in kinetic stability of the different structures
against heating.
Very recently Koga et al. 共2004兲 were able to heat up
metastable gold icosahedra produced in an inert-gas ag-
gregation source 共see Sec. V.B.3兲. Upon heating to T
=1273 K, which is only 64 K below the bulk melting
temperature of Au, they were able to produce a clearly
dominant fraction of decahedron structures in the range
6–12 nm diameter. A considerable fraction of fcc clus-
ters was obtained only heating above the bulk melting
temperature, and only for diameters above 10 nm. Since
all calculated crossover sizes decahedron→ fcc at 0 K
are much smaller 共Cleveland et al., 1997; Baletto, Fer-
rando, et al., 2002兲, these experimental results give sup-
port to the hypothesis of an upward displacement of the
crossover size with increasing temperature, in agreement
with the predictions of the harmonic theory 共Doye and
Calvo, 2001兲, as can be seen in Fig. 20.
4. Silicon clusters
Wang, Wang, et al. 共2001兲 performed a systematic
simulation study of structural transitions and thermody-
namic properties of small Si clusters within a tight-
binding molecular-dynamics scheme. They confirmed
previous results on the thermodynamics of very small
clusters found by means of empirical many-body poten-
tials 共Stillinger and Weber, 1985; Blaisten-Barojas and
Levesque, 1986; Tchofo-Dinda et al., 1995兲, and also con-
sidered larger sizes. Their canonical Monte Carlo simu-
lations revealed that the melting point of Si clusters
changes dramatically when the global-minimum struc-
tures change from prolate and cagelike 共for N=17 in this
study兲 to an atom-centered and nearly spherical mor-
phology. The nearly spherical clusters present a much
broadened melting region, extending from 750 to
1300 K. This is because, in nearly-spherical structures,
the cluster core is more tightly bonded than the surface
atoms, and a much higher temperature is needed to dis-
order it. In the range 650⬍T⬍1050 K, the nearly
spherical clusters keep their structure, although there is
a noticeable surface diffusion. After that, melting can
take place via two pathways. The first possibility is that,
as T increases, the clusters develop to the prolate mor-
phology up to 1200 K, then break into subunits, which
finally become less and less stable until the clusters dis-
order completely. The other possibility is that the clus-
ters develop from quasispherical to prolate to molten
oblate structures. Atoms in prolate cagelike structures
are more stable than the surface atoms of the nearly
spherical structures. The melting of cagelike structures
often accompanies the overall deformation and frag-
mentation of the cage framework. Thus a much higher
temperature is needed to break the cage, but the whole
cluster melts at temperatures comparable with other
nearly spherical structures.
V. KINETIC EFFECTS IN THE FORMATION OF
NANOCLUSTERS
From the results reported in the previous sections,
one can see that there are discrepancies between the
results of energetic and/or thermodynamic calculations
and the real outcomes of the experiments, viz., the struc-
tures that are actually observed in the production of free
solid clusters. In some cases, the disagreement is of such
a qualitative nature that it is unlikely to be due to the
failure of the energetic or of the thermodynamic model-
ing. To cite just the most striking cases, we mention the
observation of small decahedra and large icosahedra in
the inert-gas aggregation experiments on the production
of Ag clusters 共Reinhard, Hall, Ugarte, and Monot,
1997兲, the production of large icosahedral clusters of C
60
molecules 共Martin et al., 1993; Branz et al., 2000, 2002兲,
and the detection of octahedral Al clusters presenting
only 共111兲 facets 共Martin et al., 1992兲. Moreover, there is
still quantitative disagreement between the theoretical
estimates 共based on the total energy optimization兲 and
the experimental data about the crossover sizes for Ar
clusters 共Ikeshoji et al., 2001兲. These results indicate that
kinetic effects must be taken into account to explain the
actual free-cluster formation in the experiments. As dis-
cussed in Sec. II, the time scale of nanocluster formation
in typical sources ranges from a fraction of a millisecond
to a few milliseconds. On this time scale, clusters may
not be able to reach the minimum free-energy structure,
thus remaining trapped in some metastable configura-
tion that can have a very long lifetime, especially when
the clusters are further cooled down after their solidifi-
cation.
In studying the formation process of solid clusters in
contact with a thermal bath, such as the inert-gas atmo-
sphere in inert-gas aggregation sources, we can think of
two alternative models. In both models, clusters grow
mainly by the addition of single atoms 共de Heer, 1993兲.
In the first model 共Sec. V.A兲, which is suited for high
growth temperatures, the cluster solidifies after its
growth is completed, that is, at a further cooling stage
taking place outside the growth chamber. In this case,
the final cluster structure does not depend on the kinet-
ics of the growth process, because the cluster remains a
liquid droplet while growing, but rather depends on the
kinetics of the cooling after growth, which may take
place on time scales like those discussed in Sec. V.A,
namely, in the range of 1–10
−2
K/ns. This model will be
referred to as the liquid-state growth model. It is simu-
lated by freezing a liquid droplet until it solidifies, usu-
ally at constant N and decreasing T. In fact, there are
systems 共such as the noble gases兲 in which evaporation
of atoms is non-negligible at temperatures close to the
melting point, because melting and boiling temperatures
are close to each other. Evaporating atoms can thus play
an important role even in the cooling of solid clusters. In
contrast, metals present huge differences between melt-
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

ing and boiling points, so that evaporative cooling is
negligible for solid metallic clusters.
In the second model 共Sec. V.B兲, which is suited for
relatively low growth temperatures, clusters solidify
while they are still growing.
17
This model will be re-
ferred to as the solid-state growth model. In this model,
the final outcome is determined by the kinetics of the
growth process itself. Solid-state growth is simulated by
adding single atoms to a small initial seed at constant
temperature 共Baletto, Mottet, and Ferrando, 2000a,
2000b兲.
Finally 共see Sec. V.C兲, there are cases in which growth
proceeds not only by the addition of single atoms, but
also by the collision and coalescence of already formed
clusters.
A. Freezing of liquid nanodroplets
When simulating cluster freezing, the relevant param-
eter is the rate r
c
at which the temperature T is rescaled,
in order to mimic a thermal contact with a cold atmo-
sphere. For example, the cluster can be cooled down by
small steps
␦
T at each time interval
␦
t so that r
c
=
␦
T/
␦
t. In an inert-gas atmosphere, one can estimate
that a cluster of radius R and area A
eff
=4
␲
R
2
will collide
with gas atoms with a frequency
␾
exp
G
given by
␾
exp
G
⬃
P
G
A
eff
冑
2
␲
m
G
k
B
T
G
, 共43兲
where P
G
and T
G
are the pressure and the temperature,
respectively, of the inert gas of mass m
G
. In the har-
monic approximation, the energy loss is given by
␦
T
␦
t
⬃
␾
exp
G
␦
E
3Nk
B
, 共44兲
where
␦
E is the energy transfer at each collision, and
since A
eff
⬀N
2
and N⬀R
3
we obtain
␦
T/
␦
t⬀1/R. Follow-
ing Westergren et al. 共1998兲, we can estimate that the loss
for each collision 共for example with a helium atom兲
␦
E is
1–10 meV. Using typical parameters for the gas, P
G
⬃1–100 mbar, T
G
⬃300 K 共Reinhard, Hall, Ugarte, and
Monot, 1997; Koga and Sugawara, 2003兲 and considering
R⬃1 nm, we have
␦
T/
␦
t in the range 0.01–1 K/ns.
1. Lennard-Jones clusters
Ikeshoji et al. 共2001兲 studied the freezing of Lennard-
Jones clusters by molecular-dynamics simulations. Con-
trary to what happens for metals, the melting and boil-
ing points of these clusters are rather close in
temperature, so that the evaporation of atoms during
the solidification process is non-negligible. Because of
that, Ikeshoji et al. 共2001兲 considered two ways of cool-
ing clusters down. The thermostat method decreased
temperature in a canonical simulation, while the evapo-
ration method let the cluster evaporate atoms in a simu-
lation at constant energy. They considered 380 clusters
in the range 160⬍N ⬍ 2200, finding that there was a
transition at increasing size from icosahedra to a mixture
of structures 共decahedron, fcc, hcp, and icosahedron兲.
The transition did not depend on the cooling method,
and took place for N⯝450. This is lower than the tran-
sition size 共750 atoms兲 observed in experiments on argon
clusters 共Farges et al., 1986兲, but it is closer than any
other estimate based on total energy optimization, thus
indicating the possible presence of kinetic trapping ef-
fects. Moreover, calculated and experimental diffraction
patterns were in good agreement. Besides pure clusters,
Ikeshoji et al. 共2001兲 considered binary Lennard-Jones
systems, finding that the formation of large icosahedra
was favored by the size mismatch. This finding may fur-
nish a qualitative explanation of the experimental obser-
vation of large icosahedron clusters in binary 共metallic兲
systems 共Saha et al., 1997, 1999兲.
2. Silver clusters
Baletto, Mottet, and Ferrando 共2002兲 studied the
freezing of Ag liquid nanodroplets by molecular-
dynamics simulations with realistic cooling rates r
c
in the
range 0.1–5 K/ns. They made a systematic study of Ag
freezing at 130艋N艋 310, and in addition considered the
freezing at the icosahedral magic numbers 147, 309, 561,
and 923. The latter simulations were made to investigate
the possibility of obtaining large metastable Ag icosahe-
dra by freezing liquid droplets, in order to ascertain
whether the liquid-state growth model could explain the
experimental observation of metastable icosahedra by
Reinhard, Hall, Ugarte, and Monot 共1997兲.
The results for 130艋N艋310 are summarized in Fig.
27. On a coarse-grained description, the energetic calcu-
lations by Baletto, Ferrando, et al. 共2002兲 showed that
icosahedral clusters are the best up to N ⯝170; then the
best structures are decahedral, except for the icosahe-
dral magic number N =309, while fcc clusters become
competitive with the decahedra for N ⬎ 600. The freez-
ing simulations of Baletto, Mottet, and Ferrando 共2002兲
17
Clusters may solidify during growth at constant tempera-
ture after reaching a critical size.
FIG. 27. 共Color in online edition兲 Percentages of the different
structures obtained in molecular-dynamics simulations of Ag
cluster freezing at r
c
=1 K/ns for 130⬍N⬍ 310: 쎲, icosahedral
structures; 䊏, decahedra; 䉱, fcc clusters. Adapted from Bal-
etto, Mottet, and Ferrando, 2002.
406
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

gave the following results. At r
c
=1 K/ns, particles so-
lidified as icosahedra in the range 135⬍ N ⬍ 165. Around
N=165 there was a transition to decahedron clusters,
which were the most frequent for 170⬍N⬍ 245. About
20% of the runs gave fcc structures in this range, and a
few icosahedron clusters were formed. For 245⬍N
⬍310, the dominant structures at the end of the freezing
were icosahedra, but fcc clusters were also common, es-
pecially in the range N ⬃250–260 and N ⬃280–300, in
which they made up more than 50%. At slower cooling
rates, r
c
=0.12 K/ns, the percentage of fcc and icosahe-
dron particles in the range 170⬍N⬍ 245 dropped prac-
tically to zero, indicating that decahedron clusters are
likely to be the most favored from the thermodynamic
point of view in this size range. On the other hand, at
faster cooling rates, r
c
=5 K/ns, the percentage of fcc
clusters in the range 245⬍ N ⬍310 decreased in favor of
decahedron clusters, while the dominant proportion of
icosahedra remained practically constant.
From the freezing results at the icosahedral magic
numbers 147, 309, 561, and 923, it was possible to extract
the following tendency 共see Table II and Fig. 28兲. The
percentage of icosahedral structures at the end of the
freezing process decreased with cluster size for all cool-
ing rates, passing from 100% at N =147 to less than 20%
at N =923. There was no evidence of changes in the re-
sults when r
c
ranged within 1–20 K/ns, probably indi-
cating that these rates are sufficient to mimic a freezing
process taking place close to equilibrium.
In summary, the simulations showed that it is not pos-
sible to avoid the formation of a large percentage of
small icosahedra 共2–3 nm diameter兲 if freezing takes
place after the growth is completed. Moreover, the prob-
ability of forming large icosahedra by freezing liquid Ag
droplets with realistic cooling rates is small. Both find-
ings are in contrast with the experimental results of
Reinhard, Hall, Ugarte, and Monot 共1997兲, who did not
find strong evidence of small icosahedra, while they ob-
served a dominant percentage of icosahedra at large
sizes. Therefore, one can rule out the liquid-state growth
model as an explanation for the outcome of this experi-
ment.
3. Gold clusters
Chushak and Bartell 共2001a, 2001b兲 studied the solidi-
fication of Au nanoclusters by molecular-dynamics simu-
lations. However, their method is rather different from
that applied to Ag clusters by Baletto, Mottet, and Fer-
rando 共2002兲. Chushak and Bartell 共2001a, 2001b兲
started from high-T liquid clusters of rather large sizes
共459, 1157, and 3943 atoms兲. These clusters were then
cooled down very quickly, at a rate r
c
=5⫻10
4
K/ns, to a
temperature of about 700 K, at which their most stable
form is solid. At this temperature, production runs of
1 ns were performed. After that, clusters were further
quenched down to 300 K, again with a fast rate
共300 K/ns兲. As a result of this cooling procedure
Chushak and Bartell found that icosahedron clusters
were preferentially produced, even though they should
not be the lowest-energy structures at these sizes 共see
Baletto, Ferrando, et al., 2002兲. In this case, icosahedron
clusters were produced with high probability during
freezing because of the width of the icosahedron funnel
共Doye, 2004兲 with respect to the funnels leading to ei-
ther close-packed or decahedral structures. This is a ki-
netic trapping effect, which is stronger at rapid cooling
rates, and agrees with the results on Ag freezing by Bal-
etto, Mottet, and Ferrando 共2002兲.
Nam et al. 共2002兲 have studied the mechanism by
which metastable gold icosahedron clusters are frozen
out of nanodroplets using molecular-dynamics simula-
tions within the embedded-atom model. Their simula-
TABLE II. Numbers of icosahedron 共Ih兲, decahedron 共Dh兲,
and fcc structures at magic icosahedral sizes. The results are
obtained from five molecular-dynamics simulations for each
cooling rate r
c
and for each size. Adapted from Baletto, Mot-
tet, and Ferrando, 2002.
r
c
共K/ns兲 NN
Ih
N
Dh
N
fcc
1 147 5
309 4 1
561 2 2 1
923 1 4
5 147 5
309 3 2
561 2 3
923 1 4
20 147 5
309 2 1 2
561 1 3 1
923 1 3 1
FIG. 28. 共Color in online edition兲 Molecular-dynamics simula-
tion of silver cluster freezing. On the left, we show the initial
configurations of Ag liquid nanodroplets at magic icosahedral
numbers 共147, 309, 561, and 923兲, while on the right, there are
typical final structures obtained by cooling at 1 K/ns: from top
to bottom, 147-atom icosahedron; 309-atom icosahedron; a
decahedron at 561 关strongly asymmetric and with an island on
hcp stacking above 共Baletto and Ferrando, 2001兲兴; a fcc poly-
hedron at 923.
407
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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tions started from a liquid droplet at 1500 K, which was
cooled down at a rapid rate of 100 K/ns. Again, the ma-
jority of clusters solidified in the icosahedral symmetry,
with the solidification starting at the cluster surface, in
such a way that a well-ordered, close-packed surface
with fivefold symmetry points was formed while the
cluster was still amorphous inside. The formation of
such a surface triggered the solidification process to-
wards the formation of icosahedra: Nam et al. 共2002兲
noticed that when ordering began at the surface, the fi-
nal result was an icosahedron cluster. Conversely, if the
surface did not order first, the final cluster was either
decahedral or fcc.
4. Entropic effects and kinetic trapping in Pt
55
As a paradigm showing the peculiar interplay between
entropic and kinetic trapping effects, we report on a
molecular-dynamics study of the melting of Pt
55
共Bal-
etto, 2003兲, modeled by the Rosato et al. 共1989兲 poten-
tial. Within this model, Pt
55
presents the competing
structures of Fig. 29: the icosahedron, the single rosette
共Sr兲, and the double rosette 共Dr兲, which are, respectively,
the minimum-energy structure, the high-T minimum-
free-energy structure, and a structure frequently found
in growth simulations 共Baletto, 2003兲. The single and
double rosettes are obtained by modifying the external
shell of the Ih
55
. In the single rosette, a single vertex
atom is displaced to form a hexagonal ring 共the rosette兲
together with its nearest-neighbor atoms on the surface.
The two rosettes of the double rosette are at nearby
vertices, giving a close resemblance to the global mini-
mum of Au
55
found by Garzón, Michaelian, et al. 共1998兲
and shown in Fig. 13. In the melting simulations 共see Fig.
30兲, the single rosette becomes the most favored struc-
ture for 600⬍T⬍ 700 K. In this range, the cluster still
oscillates among icosahedral, single-rosette, and double-
rosette structures. Finally, it melts above 750 K. This is
due to entropic effects. In fact, calculations within the
harmonic approximation 共see Sec. IV.A.1兲 show that the
probability of finding the single rosette becomes consid-
erable above 600 K, while the probability of the double
rosette always remains much smaller. Constant-T simu-
lations on long time scales 共several
␮
s兲 of the evolution
of Pt
55
above 600 K confirm this scenario, showing also
that the harmonic approximation is qualitatively right in
predicting that the single rosette is more probable than
the double, but it is quantitatively poor because it un-
derestimates the probability of both with respect to the
probability of the icosahedron 共see Fig. 31兲. Another
entropy-driven effect concerns decahedral structures.
Even though some decahedral structures are lower in
energy than the single and double rosette, we find that
the entropic contribution favors single rosettes and
FIG. 29. 共Color in online edition兲 Platinum cluster structures
at N =55. From left to right icosahedron 共Ih兲, single rosette
共Sr兲, and double rosette 共Dr兲. The Ih is the lowest in energy,
the Sr and the Dr are, respectively, 0.48 and 0.42 eV higher in
energy.
FIG. 30. Melting and freezing molecular-dynamics simulations
of Pt
55
clusters. In the upper panel we show the melting caloric
curves for for three structures: dashed line, icosahedral 共Ih兲;
solid line, single rosette 共Sr兲; dash-dotted line, double rosette
共Dr兲. For 600⬍ T ⬍700 K, all melting curves exhibit several
oscillations among the three structures. The lower panel shows
two freezing curves. The dashed line ends with an icosahedral
structure, while the solid line ends with a single-rosette struc-
ture. This is an example of the interplay of entropic and kinetic
effects. Due to entropic effects, the Sr structure is the most
likely in the temperature range in which the cluster begins to
solidify. This structure may then be preserved down to low
temperatures by a trapping effect.
FIG. 31. Probabilities of the different structures of Pt
55
vs T in
the harmonic superposition approximation: solid line, the
probability of the single rosette; dashed line, probability of the
icosahedron; dash-dotted line, probability of the double ro-
sette; dotted line 共which coincides with the T axis on this
scale兲, probability of the decahedron. The last always have a
very small probability, even if they are energetically favored
with respect to the single rosette, which is the best structure
from the entropic point of view and becomes the most likely at
high T.
408
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

double rosettes over decahedral structures even at low
temperatures.
The freezing simulations in Fig. 30 show an interplay
of kinetic and entropic effects. The cluster starts to so-
lidify at T⯝700 K, a temperature range in which the
single rosette is the most probable structure. Then the
cluster is likely to remain kinetically trapped in this
structure down to low temperatures.
5. Copper, nickel, and lead clusters
Valkealahti and Manninen 共1997兲 performed a system-
atic molecular-dynamics study of the freezing of Cu
nanodroplets within the effective-medium model. They
varied r
c
in the range 1000–10 K/ns, thus also consider-
ing rates that are rather close to the experimental ones,
and analyzed a wide range of sizes up to about 4000
atoms. In their simulations, the clusters solidified as
icosahedra at small icosahedral magic numbers 共up to
N=147兲; small clusters with other sizes and large clusters
could solidify as twinned fcc structures. No evidence of
unfaulted truncated octahedron or decahedron clusters
was found. The fact that icosahedron clusters are domi-
nant at small sizes but not at larger ones is in agreement
with the experimental findings of Reinhard, Hall, Ber-
thoud, et al. 共1997兲, who found, however, some evidence
in favor of intermediate-size decahedron clusters.
Qi et al. 共2001兲 have studied the freezing of several Ni
clusters at sizes from 336 to 8007 atoms by molecular-
dynamics simulations within the embedded-atom poten-
tial. In their simulations, they performed heating and
cooling cycles at very fast rates 共4000 K/ns兲: starting
from a solid cluster they increased T to melt the struc-
ture, and then cooled it down again. As a result, they
found icosahedron clusters after freezing for N⬍500,
and fcc clusters for larger sizes. All these studies confirm
the tendency to form icosahedral clusters only after
freezing small liquid droplets, in agreement with the pre-
vious cases, whereas at larger sizes other structures are
dominant.
The first study of the solidification of Pb clusters was
made by Lim et al. 共1994兲; they considered a single clus-
ter of about 8000 atoms, obtaining an icosahedron after
fast quenching, in disagreement with their own energetic
calculations 共Lim et al., 1992兲, which predicted that the
cuboctahedron would be lower in energy than the icosa-
hedron. This topic has been recently analyzed in a more
systematic way by Hendy and Hall 共2002兲, who
quenched a large variety of liquid Pb droplets in the size
range between 600 and 6000 atoms. In their simulations,
the liquid droplet was quenched suddenly below the
cluster melting point, and then equilibrated for 10 ns.
Up to 4000 atoms, they found mostly icosahedra with a
reconstructed surface, showing that they are lower in
energy than any other known structure in the same size
range. The freezing of clusters of about 6000 atoms pro-
duced faulted fcc structures. The production of icosahe-
dron clusters agrees with the experimental observation
of icosahedron particles of 3–6 nm diameter by Hyslop
et al. 共2001兲 in inert-gas aggregation experiments.
B. Solid-state growth
Even though the liquid-state growth model can ex-
plain some kinetic effects in the growth of nanoclusters
共see, for instance, the cases of lead and argon clusters
treated in Sec. V.A兲, there are still several experimental
results that cannot be explained by it: the growth of
large icosahedra in clusters of Ag and of C
60
molecules,
and the growth of octahedra in Al clusters. For these
systems, it turns out that the solid-state growth model is
much more appropriate.
In the following, we deal first with a very general
mechanism for the solid-state growth of metastable
icosahedra 共Sec. V.B.1兲; then we treat 共Sec. V.B.2兲 the
molecular-dynamics simulation results for noble-metal
and quasi-noble-metal clusters at small sizes 共Baletto,
Mottet, and Ferrando, 2001b兲. These simulations were
run with very slow fluxes, in order to test the solid-state
growth model against the energetics and thermodynam-
ics results. In Sec. V.B.3, results on the growth of Ag and
Au clusters of intermediate and large size are reported.
Finally Secs. V.B.4 and V.B.5 are devoted to the growth
of aluminum and C
60
clusters, respectively.
1. Universal mechanism of the solid-state growth of
large icosahedra
Before focusing on specific systems, we treat a very
general mechanism which we believe to be responsible
for the observation of large metastable icosahedra in
several systems 共see Ag, Au, and C
60
below兲. This
mechanism consists of two steps. The first is the solid-
solid transformation of a decahedron into an icosahe-
dron, the second the shell-by-shell growth of the icosa-
hedron to reach large sizes. This is a very peculiar
example of structural transformation, because it has a
clear kinetic origin, occurs only through solid states, and
does not depend on the interparticle potential, but is
essentially due to geometric reasons. First of all, we note
that this transformation is possible because a decahe-
dron is a fragment of a larger icosahedron, so that by
proper addition of atoms, a decahedron can grow to-
wards an icosahedron. However, this is only a possibility,
and a further ingredient is necessary to render this trans-
formation kinetically favored. This key ingredient is the
fact that on the 共111兲-like facets of a decahedron are
found a larger number of stable adsorption sites of hcp
stacking than of fcc stacking 共Baletto and Ferrando,
2001兲.
Islands of fcc stacking preserve the decahedral rear-
rangement in columns, around the fivefold symmetry
axis, while hcp islands break the decahedral symmetry
共see Fig. 32兲 and can lead to the transformation towards
a larger icosahedron. A decahedron plus a hcp island is a
part of a larger icosahedron. The displacement of a hcp
island to the fcc stacking costs a considerable amount of
energy, which increases with the stickiness of the poten-
tial. While the growth of islands of fcc stacking leads
simply to a larger decahedron 关this is the umbrella
growth model 共Martin et al., 1991b兲兴, the nucleation of
hcp islands can be the starting point for transforming a
409
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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decahedron into a larger icosahedron. The island can
be either one or two layers thick, leading to trans-
formations to icosahedra of different sizes, through dif-
ferent rearrangement pathways 共Baletto, Mottet, and
Ferrando, 2001a兲. The size at which a solid-state
decahedron→icosahedron transformation can take place
depends strongly on temperature. A larger starting
decahedron requires a higher temperature.
While a growing decahedron can transform into an
icosahedron, there is no natural growth sequence from
icosahedron to decahedron structures, because an icosa-
hedron is not a fragment of a larger decahedron. To
transform a growing icosahedron into a decahedron, a
complete rearrangement of the cluster is necessary. This
rearrangement becomes likely only at high tempera-
tures, close to the melting point, as proved by the ex-
periments of Koga et al. 共2004兲. Therefore we can well
understand that it is rather common to find wide tem-
perature ranges in which kinetic trapping into icosahe-
dral structures dominates the growth sequence. The
solid-state decahedron→ icosahedron transformation
thus allows us to explain several experimental results:
from small decahedra, larger icosahedra are grown,
which can then grow further in a shell-by-shell mode
preserving their symmetry.
The above considerations lead to a general observa-
tion about the possibility of building up noncrystalline
共especially icosahedral兲 structures in nanoclusters. In
fact, even though a sticky potential disfavors these struc-
tures, at the same time it enhances kinetic trapping ef-
fects, because diffusion barriers are high and rearrange-
ments involving many particles become very difficult
共Wales, 1994b; Baletto, Doye, et al., 2003兲. These kinetic
effects very likely lead to the growth of icosahedra 共see
Fig. 33兲.
2. Growth of small noble-metal and transition-metal
clusters
Here we analyze simulations of the growth of small
Cu, Ag, Au, Ni, Pd, and Pt clusters modeled by the Ro-
sato et al. 共1989兲 potential, in order to single out quali-
tative differences in the behavior of these elements. To
this purpose, we show in Fig. 34 the quantity ⌬ 关Eq. 共5兲兴
as a function of the cluster size for growth simulations at
different temperatures, depending on the metals, with
fixed
␶
dep
=98 ns. In this case, ⌬ is defined by putting into
Eq. 共5兲 the temperature-dependent total energy E of the
cluster, averaged at each size over several snapshots, in-
stead of the zero-temperature binding energy E
b
. Stable
structures are singled out by minima in ⌬. Finally we
focus on the growth of Ag and Cu clusters at N =38, to
show that in this case kinetic trapping effects are over-
whelming. In fact, the global-minimum structure, which
is a truncated octahedron according to this energetic
model, is grown with a non-negligible probability only in
a narrow temperature range for a given flux.
Growth sequences for Cu and Ni clusters are clearly
dominated by icosahedral or polyicosahedral structures,
FIG. 33. The solid-state decahedron→ icosahedron transfor-
mation obtained in growth simulation of clusters of C
60
mol-
ecules. The same kind of transformation is also found in simu-
lations of Ag cluster growth. Ag and C
60
present completely
different interactions, soft and sticky, respectively. Adapted
from Baletto, 2003.
FIG. 32. A 146-atom decahedron plus an hcp island above a
cap. The black dots indicate fcc adsorption sites, the gray dots
the hcp sites. The fcc sites on the edges between facets are not
stable for single adatoms. Compared to adsorption on icosahe-
dral clusters, the fcc and hcp sites correspond to Mackay and
anti-Mackay sites, respectively.
FIG. 34. Growth simulations of small metallic clusters. ⌬ as a
function of cluster size at different temperatures: low tempera-
tures 共200 K for Au and Ag, 300 K for Ni and Pt, 400 K for Cu
and Pd兲 on the left and high temperatures 共400 K for Au and
Ag, 600 K for Ni, 700 K for Pd, Pt, and Cu兲. Minima in ⌬
single out the most stable structures.
410
F. Baletto and R. Ferrando: Structural properties of nanoclusters
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as can be seen in Fig. 34. The magic sizes are 13, 19, 25,
40, 43, 46, 49, and 55, with 13, 19, and 55 being of special
stability. The sequences are well reproducible over a
wide range of growth temperatures. The main difference
between Ni and Cu is that in Ni it is easier to obtain
truncated octahedra of 38 atoms, even at relatively low
temperatures 共400 K兲. In this case, growth continues
with the cluster trapped in non icosahedral structures.
Growth sequences for Ag show the same kind of struc-
tures, but they are reproducible over a narrower tem-
perature range. In contrast, the growth sequences of Au
reveal completely different magic sizes. Icosahedral
structures are completely unfavored. The minima at N
=13 and 19 are absent, replaced by peaks at N=16 and
22. At N=30 we find a decahedral minimum, which sur-
vives up to T ⬃300 K. At higher temperatures the clus-
ters oscillate among different structures over a time
scale shorter than
␶
dep
so that they can be considered
either melted or quasimelted. There is no evidence of
entropic effects favoring the transition to icosahedral
structures. Finally, Pt and Pd show an intermediate be-
havior between Cu, Ni, and Ag on the one side, and Au
on the other. For Pd, there is a minimum at N=13, but
there is no evidence of any other magic size up to the
Ih
55
, which is likely to be grown. In Pt, minima are found
at N=13 共as in Cu兲 and 22 共as in Au兲. At high T,itis
possible to grow truncated octahedra of 38 atoms. Very
interesting behavior takes place around for Pt
55
,asan-
ticipated in Sec. V.A.4, with the probable growth of ro-
sette structures, either because of kinetic trapping in in-
complete icosahedral structures formed at lower sizes,
or because of the onset of entropic effects at high tem-
peratures.
A complex interplay of thermodynamic and kinetic
effects takes place in the growth simulations of Ag
38
and
Cu
38
共Baletto, Rapallo, et al., 2004兲. At this size, the glo-
bal minimum is a truncated octahedron, which is in com-
petition with a series of defected decahedron structures
共mostly based on decahedra of 23 atoms covered by is-
lands兲, and in the case of Ag, with a low-symmetry struc-
ture that is neither a defected decahedron nor a close-
packed cluster. In the case of copper, these decahedron
structures become favored over the truncated octahedra
at temperatures around 350 K, because of entropic ef-
fects. Moreover, for sizes just below N =38, the global
minima are defected decahedra in both Ag and Cu. For
these reasons, the truncated octahedron is not likely to
be grown in the simulations. Performing several simula-
tions at T=200, 250, 300, 350, and 400 K, and observing
the structures grown at N=38, one obtains the results
reported in Figs. 35 and 36. At 200–250 K, kinetic trap-
ping into decahedron structures is the fate of essentially
all simulations, so that the magic structure never coin-
cides with the global minimum, which is also the free-
energy minimum in this temperature range. This kinetic
trapping is due to the fact that the global minima in the
range 30–37 atoms are decahedral 共with the single ex-
ception of Cu
37
兲. In the interval 300–350 K, kinetic trap-
ping becomes less effective, even if it is still somewhat
present. In this range, the truncated structures are still
FIG. 35. 共Color in online edition兲 Ag
38
growth simulation re-
sults. In the top row, snapshots from a simulation at T
=200 K are shown at different sizes. The structures of the
snapshots pertain to the global minima up to N =33; then
growth is kinetically trapped into defected decahedron struc-
tures, which are neither energetically nor entropically favored.
The graph shows the structural frequencies f
TO
, f
Dh
, f
LS
of
different structures at N=38 and at different growth tempera-
tures: open stars, truncated octahedra; solid stars, decahedra;
triangles, low-symmetry structures. The lines are only guides to
the eyes. At low temperatures, strong kinetic trapping into
decahedral structures occurs. This is followed by an interme-
diate temperature regime where the three structural motifs are
essentially equally likely. At high temperatures, entropic ef-
fects favor decahedral and low-sym-metry structures at the ex-
pense of the truncated octahedron. Adapted from Baletto et
al., 2004.
FIG. 36. 共Color in online edition兲 Cu
38
growth simulation re-
sults. Symbols as in Fig. 35. In the top row, snapshots from a
Cu simulation at T =400 K are reported. Again, the snapshots
reproduce the global minima up to N=33. On the contrary, the
growth structure at N=38 is not the global minimum but a Dh,
which is entropically favored at this temperature. Around
200 K, there is a strong kinetic trapping into Dh structures
formed at smaller sizes. At intermediate temperatures the TO
structure is preferentially grown; at high T, Dh structures again
prevail due to entropic effects. Adapted from Baletto et al.,
2004
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
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quite likely at equilibrium, and they are indeed ob-
served, together with decahedra and low-symmetry
structures 共the latter in Ag only兲. Above 350 K, again
decahedral structures prevail in Cu
38
, as at low tempera-
ture. However, this is not due to kinetic trapping, but to
an entropic effect, because at these temperatures deca-
hedral structures are the most likely at equilibrium.
3. Growth of intermediate- and large-size Ag and Au
clusters
The experiments of production of Ag clusters in inert-
gas aggregation sources 共Reinhard, Hall, Ugarte, and
Monot, 1997兲 show that the abundances of small clusters
共2 nm diameter兲 are not dominated by icosahedral struc-
tures, while icosahedra are the most abundant structures
for much larger sizes. Here we discuss how these results
can be explained in the framework of a solid-state
growth model 共Baletto, Mottet, and Ferrando, 2000b,
2001a兲.
Let us consider first the sizes around 150 atoms, that
is, diameters of about 2 nm. In this range, the molecular-
dynamics growth simulations show that a reentrant mor-
phology transition takes place. This means that, at a
given deposition flux, there is an intermediate tempera-
ture window 共400–500 K for fluxes of experimental in-
terest兲 in which decahedral structures are preferentially
grown, while at low and high temperatures icosahedra
are obtained.
In fact, at intermediate temperatures, Ag clusters are
able to optimize their shape up to N ⬃100 关the best
structure is the 共2,3,2兲 decahedron at 101 atoms兴 and
then remain trapped in decahedral structures. This leads
to the formation of a metastable decahedron cluster
around N =150 关the 共3,2,2兲 decahedron cluster at 146 at-
oms is the best in this range兴 instead of the lower-energy
icosahedra related to the N=147 icosahedron. At higher
temperatures 共T⬎550 K兲, the clusters are liquid up to
N⯝130, and the final outcomes at N⯝150 are icosahe-
dra. Finally, at low temperatures 共T⬍450 K兲, the cluster
has sufficient kinetic energy to optimize its shape only
up to 75 atoms. Then it remains trapped in this structure
and evolves towards the 共6,1,1兲 decahedron at N =100.
When this decahedron is almost completed, the addition
of further atoms causes the nucleation of hcp islands on
the facets and starts the formation of larger icosahedra.
This is an example of a solid-state decahedron
→ icosahedron transformation 共Baletto and Ferrando,
2001兲. Typical simulation results are summarized by the
snapshots shown in Fig. 37.
The metastable decahedra at N ⯝150 can have very
long lifetimes. Baletto, Mottet, and Ferrando 共2000b兲
calculated the lifetime
␶
Dh
of the 共3,2,2兲 decahedron in
the range 550艋T艋650 K, finding an activated behavior
of the kind
␶
Dh
=
␶
Dh
0
exp关⌬E/共k
B
T兲兴. After estimating
the prefactor
␶
Dh
0
and the activation barrier ⌬E, they
were able to extrapolate a lifetime of several millisec-
onds at 450 K. This should indicate that this structures is
likely to survive on the experimental time scale.
The reentrant morphology transition is due to kinetic
trapping and not to entropic effects. In fact, on the basis
of the considerations reported in Sec. IV.A, from Eq.
共25兲 it follows that, for N =147, p
Dh
/p
Ih
monotonically
increases with T.
Let us now consider a growth sequence leading to the
formation of small decahedra and of large metastable
icosahedra, in agreement with the results of the experi-
ments by Reinhard, Hall, Ugarte, and Monot 共1997兲.
This sequence is obtained for 400⬍T⬍ 500 K and is es-
sentially a solid-state decahedron→ icosahedron trans-
formation followed by a shell-by-shell trapping in icosa-
hedral structures, as reported in Fig. 38. In fact,
continuing the deposition on a metastable decahedron
FIG. 37. 共Color in online edition兲 Snapshots of silver cluster
growth at
␶
dep
=7 ns from three simulations at three different
temperatures: left column, 400 K, middle column, 500 K, right
column, 600 K. In each column, the snapshots are taken at
sizes of 55, 105, and 147 atoms from top to bottom. At 75
decahedra are preferentially grown up to 500 K. Icosahedral
structures of 147 atoms are obtained in the simulations at 400
and 600 K, and a decahedron at 500 K. At 55 atoms, we obtain
an icosahedron at 400 and 500 K 共the structure at 600 K is
rapidly fluctuating兲.
FIG. 38. Growth sequences of intermediate-size Ag clusters at
different T: from left to right, 400, 450, and 600 K. In the top
row, N ⯝150, in the middle row N ⯝200, and in the bottom row
N⯝300.
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F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

of ⬃150 atoms, an island of hcp stacking is nucleated.
This island can be either one or two layers thick, leading
to the formation of icosahedra with N=309 or 561, re-
spectively 共Baletto, Mottet, and Ferrando 2001a兲. For
T⬎600 K, icosahedral structures are never obtained for
N⬎180. The deposition of a few tens of atoms above
N=147 causes a sharp transition towards decahedra of
192 atoms. This grows passing through different decahe-
dral structures, towards one with N =318 by the nucle-
ation of islands of fcc stacking. For T ⬎ 650 K, fcc struc-
tures are preferentially grown. This happens because
around N=201 and N =314, decahedral and fcc clusters
are in close competition from the energetic point of
view. Because of that, when temperature is high enough
and the energy differences become less important, the
growing cluster can pass through different faulted fcc
and asymmetric decahedral structures, growing finally as
a fcc structure. At low temperatures, around 400 K,
icosahedra are preferentially grown around N =150, and
these icosahedra continue growing in a shell-by-shell
mode.
The growth of Au clusters has also been investigated,
and there is evidence for kinetic trapping effects of the
same kind as those found in Ag. In a recent experiment,
Koga and Sugawara 共2003兲 generated and analyzed a
huge population of Au clusters with diameters in the
range 3–18 nm. The clusters were produced in an inert-
gas aggregation source with carrier helium gas, then de-
posited on an amorphous carbon substrate and analyzed
by high-resolution electron microscopy. Some of these
clusters are shown in Figs. 2 and 39. Analysis of the
cluster population revealed a striking feature: for all
sizes, most of the clusters were icosahedral 共see Fig. 40兲,
with a proportion of about 90% at small sizes, slowly
decreasing to 60–70 % at large sizes. The remaining
clusters were mostly of decahedral symmetry, while a
very few fcc clusters were observed. This result is in dis-
agreement both with the energetic calculations 共Cleve-
land et al., 1997; Baletto, Ferrando, et al., 2002兲 and with
the experimental observation of Patil et al. 共1993兲 indi-
cating very low crossover sizes from the icosahedron to
the other motifs. Patil et al. 共1993兲 observed Au clusters
after melting and very slow refreezing; they were thus
very likely observing equilibrated clusters. Koga and
Sugawara 共2003兲 attributed the observation of large
icosahedra to a kinetic trapping effect 共Baletto, Mottet,
and Ferrando, 2000a, 2001a兲, caused by the shell-by-
shell growth of small icosahedral clusters. At the mo-
ment, simulations on Au cluster growth are not available
to rule out the possibility of a liquid-state growth pro-
cess, but the formation of such large icosahedra by the
freezing of liquid droplets seems unlikely in the light of
the results reported in Sec. V.A.2 and by Baletto, Mot-
tet, and Ferrando 共2002兲.
4. Aluminum clusters
The formation of Al clusters was studied experimen-
tally by different groups 共Lermé et al., 1992; Martin et
al., 1992兲. The mass spectra presented regular oscilla-
tions, whose maxima were equally spaced by a quantity
␦
N
1/3
when the cluster abundance was plotted as a func-
tion of N
1/3
共see Fig. 41兲. The numerical value of the
spacing turned out to be
␦
N
1/3
⯝0.22. Valkealahti et al.
共1995兲 nicely explained these features as being of geo-
metric origin within the octahedral growth model. In this
model, each maximum corresponds to the addition of a
single close-packed layer on one of the eight facets of
the octahedron. Let us consider an octahedron with n
1
close-packed layers in the direction of one of its eight
facets. This octahedron has N
oct
共n
1
兲 atoms. When we
FIG. 39. High-resolution electron microscopy image of a large
icosahedral Au cluster grown in an inert-gas aggregation
source and then deposited on an inert substrate. Adapted from
Koga and Sugawara, 2003.
FIG. 40. Experimental population distributions of icosahedral,
decahedral, and fcc Au clusters grown in an inert-gas aggrega-
tion experiment. Adapted from Koga and Sugawara, 2003.
FIG. 41. Experimental mass spectrum of aluminum clusters.
Adapted from Valkealahti et al., 1995.
413
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

add a further layer above one facet, the number of at-
oms becomes N
oct
共n
1
+1兲. Valkealahti et al. 共1995兲 dem-
onstrated that
␦
N
1/3
= 关N
oct
共n
1
+1兲 − N
oct
共n
1
兲兴
1/3
=
1
6
冉
3
2
冊
2/3
共45兲
in very good agreement with the experiments.
In a subsequent molecular-dynamics study,
Valkealahti and Manninen 共1998兲 analyzed the possibil-
ity of growing octahedral clusters starting from trun-
cated octahedra. As a first step towards understanding
the growth mechanism, they calculated diffusion barriers
for adatoms and dimers, finding very interesting mecha-
nisms such as chain diffusion, which allows direct mass
transport between two nonadjacent 共111兲 facets through
an intermediate 共100兲 facet. Then, in their growth simu-
lations, they took a truncated octahedron of 586 atoms
as a starting seed, and deposited atoms one by one at
constant temperature 共T =400 K兲 with a rather fast
deposition rate 共
␶
dep
=0.5 ns兲 up to a total of 201 depos-
ited atoms. The results demonstrated the possibility of
transforming a truncated octahedron into an octahedron
by the following mechanism. When depositing on the
truncated octahedron, atoms are very likely to fall on
共111兲 facets, where they can diffuse fast. When an ada-
tom reaches the border with a 共100兲 facet, it can ex-
change easily with an edge atom, which is then trapped
on the 共100兲 facet. The reverse process 关exchange from a
共100兲 to a 共111兲 facet兴 is very difficult because the ad-
sorption energy is much more favorable on 共100兲 than on
共111兲 facets. The accumulation of atoms on 共100兲 facets
leads naturally to the formation of an octahedron expos-
ing only 共111兲 facets. This mechanism was confirmed
later by molecular-dynamics growth simulations of Ag
and Au clusters 共Baletto, Mottet, and Ferrando, 2000a兲
that were performed starting from a truncated octahe-
dron of N=201 but with a much slower deposition rate.
The growth simulations of truncated octahedron clusters
have revealed the formation of stacking faults
共Valkealahti and Manninen, 1998; Baletto, Mottet, and
Ferrando, 2000a兲. Manninen and Manninen 共2002兲 and
Manninen et al. 共2003兲 showed that clusters with stacking
faults are also obtained in global optimization on an fcc
lattice with all possible 共111兲 stacking faults allowed, in
the case of different model potentials.
5. Clusters of C
60
molecules
In the case of C
60
clusters, growth takes place in the
solid state. Baletto, Doye, and Ferrando 共2002兲 simu-
lated their growth with
␶
dep
between 100 and 200 ns and
T⬍600 K, modeling the interactions by the Girifalco
共1992兲 potential. Some results are shown in Fig. 42,
where snapshots from typical growth simulations at dif-
ferent T and
␶
dep
=100 ns are reported at some signifi-
cant sizes. All the sequences T ⬍ 525 K develop along
the same line and lead to the formation of a well-
ordered icosahedron of 55 atoms. At N=13 the structure
is always icosahedral, in agreement with the global opti-
mization results. Similarly, around N=25 the structure
usually resembles the global minimum, although this is
not necessarily the case in between these sizes. Only at
very low T 共⬍300 K兲 does the structure remain trapped
in the icosahedral shape after 13 molecules. The 25-
molecule structure is decahedral but with an island on
the bottom 共111兲 faces of the cluster. This structure plays
a key role, because growth is dominated by kinetic trap-
ping effects beyond this size, since a complete solid-state
decahedron→ icosahedron transformation takes place.
Furthermore, the structure is a fragment of the icosahe-
dron with N=55, and continued growth around the bot-
tom apex of the decahedron provides a pathway to this
structure, with its apex ending up at the center of the
resulting icosahedron 共Baletto et al., 2001a兲.
This growth pattern bears no resemblance to the se-
quence of global minima, which develops through either
close-packed or decahedral structures. In particular, the
共C
60
兲
38
global minimum is a truncated octahedron, while
the growth simulations always give structures with five-
fold symmetries. The same happens at N =45, and finally
a Mackay icosahedron results at N=55. If the growth is
continued for N⬎ 55, a well-ordered anti-Mackay over-
layer 共Martin, 1996兲 develops on the surface of the icosa-
hedron, in agreement with the sequence of magic num-
bers observed experimentally for 55⬍N⬍100 共Branz et
al., 2000, 2002兲. Decahedral and fcc structures are only
obtained in a few cases at rather high growth tempera-
tures 共525–550 K兲, as found by Baletto, Doye, and Fer-
rando 共2002兲. If one uses the Pacheco and Prates-
Ramalho 共1997兲 potential, the growth of nonicosahedral
structures becomes more likely, but icosahedra are still
obtained in most cases.
Let us discuss how these simulation results compare
to the experiments by Branz et al. 共2000, 2002兲 discussed
in Sec. III.E.6. The experimental data concerning the
formation of icosahedral clusters are easily explained by
the outcome of the simulations by Baletto, Doye, and
FIG. 42. Growth sequences of C
60
clusters for
␶
dep
=100 ns and
different temperatures. From top to bottom snapshots at N
=13, 25, 38, 45, and 55 particles are shown. Adapted from
Baletto, Doye, and Ferrando, 2002.
414
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

Ferrando 共2002兲: further annealing of the growth struc-
tures indicates that it is indeed quite easy to eliminate
the less bound molecules from the surfaces of the clus-
ters, thus leaving only icosahedral magic clusters.
On the other hand, the experimental results obtained
after annealing at higher T 共close to 600 K兲 are more
difficult to explain: close-packed or decahedral clusters
were mostly observed in the experiments 共Branz et al.,
2000, 2002兲. In principle, these clusters could be formed
in two ways:
共i兲 the high-T annealing allows some of the icosahe-
dral clusters obtained at lower temperatures to re-
arrange in such a way that they are able to reach
the structure with minimum free energy;
共ii兲 the high-T annealing causes the evaporation of al-
most all icosahedral clusters, and only the few 共al-
ready existing兲 nonicosahedral clusters, which are
more stable, survive and are observed.
Moelcular-dynamics simulations of the annealing of
icosahedral clusters of different sizes at high tempera-
tures 共650–750 K兲 have never produced a rearrange-
ment of the cluster to either decahedral or close-packed
structures 共Baletto, Doye, and Ferrando, 2002兲. Instead,
dissolution by desorption of molecules 共with some local
rearrangement within the structure兲 occurs. These re-
sults are consistent with mechanism 共ii兲 but we cannot
draw a firm conclusion. Some structural transformations
from icosahedral to decahedral structures have been ob-
served in extremely long 共0.1 ms兲 simulations by Branz
共2001兲 and Branz et al. 共2002兲 at T ⬃700 K, where, how-
ever, the number of atoms was forced to remain constant
in a closed simulation box, and this forbade cluster dis-
solution, which is, at this temperature, dominant. Unfor-
tunately, at the moment it is not possible to simulate
dissolution at 600 K, because the time scales involved
are too long.
C. Coalescence of nanoclusters
The coalescence of supported clusters is of great im-
portance in the field of surface nanostructuring. This
topic has been the subject of an excellent review paper
by Jensen 共1999兲. Here we intend to focus on the coales-
cence of free clusters, a mechanism that can be impor-
tant for free-cluster formation, especially in the late
stages of the growth process, when already formed clus-
ters can collide and join together. Some experimental
evidence in favor of this mechanism may be inferred
from the results of Patil et al. 共1993兲, who found the
formation of polycrystalline Au clusters in an inert-gas
aggregation source and attributed them to the encounter
and coalescence of different smaller units. However, we
remark that polycrystalline clusters may also result as
the outcome of the freezing of single liquid droplets
共Valkealahti and Manninen, 1997兲.
Lewis et al. 共1997兲 studied the coalescence of free Au
clusters by molecular-dynamics simulations. They
coupled the clusters to a thermostat, to keep the tem-
perature constant during the coalescence process, and
therefore their results are probably better applied to
supported clusters, which can exchange energy with the
substrate at a rapid rate. They considered the coales-
cence between two solid clusters, a liquid and a solid
cluster, and two liquid clusters. The coalescence of two
liquid clusters takes place rapidly. A single spherical
cluster is formed by the deformation of the two clusters
in such a way as to optimize the contact surface, without
interdiffusion. Later on, interdiffusion takes place, but
the spherical shape is reached over much shorter time
scales by a collective rearrangement phenomenon. The
coalescence of a solid and a liquid cluster proceeds in
two stages. At first the contact surface is maximized rap-
idly, on the same time scale as the coalescence of liquid
clusters. At this stage the cluster is far from being
spherical but has a faceted ovoidal shape. After that the
spherical shape is reached by a slow process driven
mainly by surface diffusion. The rapid changes seen at
short times are due to elastic and plastic deformations.
At long times the presence of facets slows down the
diffusion 共Baletto and Ferrando, 2001兲, so that coales-
cence times are much longer 共Mazzone, 2000兲 than pre-
dicted by the macroscopic theory of sintering 共Nichols
FIG. 43. Molecular-dynamics simulation of the coalescence of
two 565-atom lead icosahedra initially at 300 K. The sequence
of snapshots at 3.75-ps intervals shows the early growth of the
neck after the initial contact. Adapted from Hendy et al., 2003.
FIG. 44. Evolution of the temperature and of the aspect ratio
during the molecular-dynamics simulation of the coalescence
of two 565-atom lead icosahedra at the initial temperature of
300 K. At the point of first contact between clusters 共approxi-
mately 30 ps after the beginning of the simulation兲 the tem-
perature rises sharply due to the release of surface energy. The
inset shows the final cluster structure, which is almost spheri-
cal. The spherical shape is quickly reached, since the aspect
ratio is very close to one after 1 ns. Adapted from Hendy et al.,
2003.
415
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

and Mullins, 1965; Jensen, 1999兲 via surface diffusion.
Finally, the coalescence of two solid clusters 共simulated
also by Zhu, 1996 for Cu clusters兲 is a complex phenom-
enon, which takes place on a slow time scale and may
involve either the formation of a single domain cluster
or of complicated structures presenting grains. This de-
pends on size and structure of the initial clusters, at vari-
ance with the previous cases. Moreover, the nucleation
of a critical island on the surface can govern the rear-
rangement kinetics 共Combe et al., 2000兲.
Recently Hendy et al. 共2003兲 simulated the coales-
cence of free lead clusters in the microcanonical en-
semble, without coupling the clusters to a thermostat.
This seems to be the most appropriate method when
dealing with free clusters produced in inert-gas aggrega-
tion sources. In fact, when two clusters come into con-
tact, many new bonds form, causing a considerable re-
lease of surface energy. This can cause a noticeable
temperature increase 共more than 100 K for clusters of
N⯝500兲. Since cooling rates are less than 1 K/ns 关see
Eq. 共44兲兴, and the initial stage of coalescence develops in
a few ns 共see Figs. 43 and 44兲, the inert gas is not likely
to affect the process strongly by taking energy away. The
temperature increase makes the coalescence process
much faster, by enhancing surface diffusion. Hendy et al.
共2003兲 considered the coalescence of two solid surface-
reconstructed icosahedral clusters 共Hendy and Hall,
2001兲 of 565 atoms. They found that, depending on the
initial temperature and the sizes of the two clusters, the
final aggregate could be either solid or liquid. When the
final temperature T
f
of the resulting cluster of 1130 at-
oms was below T
m
of the Ih
565
, the coalescence took
place through solid states. On the other hand, when T
f
was above T
m
of the resulting cluster of 1130 atoms, the
final cluster was liquid. Finally, when T
f
was in between
the two melting temperatures, the aggregate melted at
first and solidified again at a later stage.
VI. CONCLUSIONS AND PERSPECTIVES
In this review we have tried to present an overview of
the physical properties of nanoclusters, with the aim of
showing the interplay of energetic, thermodynamic, and
kinetic factors in building up the structures that are pro-
duced in the experiments or observed in the simulations.
We hope that we have demonstrated that the physics of
nanoclusters can be understood only by taking into ac-
count all three factors and relating them to the features
of the interparticle potential. Among these features, the
potential range is always a guideline for understanding
the qualitative features of structural properties and
transformations. For example, we have seen that soft
interparticle interactions admit noncrystalline clusters as
minimum-energy isomers. In contrast, for sticky inter-
particle potentials, crystalline structures are energeti-
cally favored, but at the same time kinetic trapping phe-
nomena are enhanced. The latter often cause the growth
of noncrystalline structures in the actual experiments. In
metallic systems, the bond-order/bond-length correla-
tion plays a crucial role in destabilizing some cluster
structures, such as the icosahedra, and favoring the for-
mation of low-symmetry structures. Finally, bond direc-
tionality is crucial in semiconductor clusters and in some
metals, too.
Several examples have shown that the interplay of en-
ergetics, thermodynamics, and kinetics is crucial, and
that a satisfactory explanation of the experimental out-
comes is very often impossible on the basis of energetic
considerations alone. This can have deep consequences
for the very important issue of controlling the shapes of
the produced nanoclusters, which is of great technologi-
cal importance. Given that the interplay of the three
factors is crucial, each one is extremely interesting by
itself and poses stimulating theoretical challenges.
The development of reliable methods for modeling
the energetics of nanoclusters is a rapidly developing
field, both from the point of view of ab initio calcula-
tions, well suited for precise calculations on small sys-
tems, and from the point of view of semiempirical mod-
eling, which is needed to treat larger systems and to
simulate long time scales. In cluster science, these ap-
proaches are complementary and are both necessary.
Since the best structure of a cluster of given size and
composition is generally unknown, the semiempirical
modeling, which has a much lower computational cost, is
the starting point for selecting good candidates for sub-
sequent ab initio local structural optimization. We have
seen that there are already several examples in the lit-
erature showing that this approach is extremely fruitful,
and now the challenge is to extend it to large sizes and
complex systems 共for example, nanoalloys兲. The search
for good structural candidates is performed by global
optimization methods. The literature on these methods
and their applications to clusters has exploded in the last
few years; several different procedures have been pro-
posed, with important progress towards the understand-
ing of the requirements for building up efficient global
optimization algorithms. The main point here is that the
most efficient algorithms work after transforming the
original PES to a multidimensional staircase. In the field
of global optimization, there is still the need to develop
algorithms for complicated systems and large sizes. The
latter task is, however, limited by the intrinsic NP-hard
nature of the global optimization problem itself 共see
footnote 3 and Wille and Vennik, 1985兲. Thus there is no
hope in optimizing very large clusters, although good
putative global minima are found by the present algo-
rithms by exploring only a very small fraction of the
huge number of local minima. The reason why these
algorithms work lies in the features of the PES, which
may present funnels in which optimization is fast, and in
the favorable transformation of the thermodynamics of
the system when working on the staircase PES.
The thermodynamics of nanoclusters is a very active
research field, with several problems under debate. For
example, the relation between elementary interactions
and the type of phase changes in clusters is still to be
understood to a large extent for realistic model poten-
tials. Moreover, the effect of chemical composition on
melting and on structural transitions has been the sub-
416
F. Baletto and R. Ferrando: Structural properties of nanoclusters
Rev. Mod. Phys., Vol. 77, No. 1, January 2005

ject of very few studies. Finally, the approach of the bulk
limit is not yet well understood. Each of these problems
poses complex theoretical and computational chal-
lenges.
The study of the growth kinetics of nanoclusters is a
field that is just starting to develop. Here, a systematic
study of small systems by the available methods is still to
be performed. Moreover, new methods should be devel-
oped, first of all to extend the size of the simulated sys-
tems and the time scale of the simulations, and then to
treat the formation of clusters in complex environments,
such as formation in liquid solutions or on surfaces, in-
teraction with passivating agents and adsorbed mol-
ecules, and so on.
In conclusion, we hope that our review article has
given at least some idea of the present development of
the fascinating and lively subject of cluster science.
ACKNOWLEDGMENTS
The authors are grateful to Jonathan Doye, Alessan-
dro Fortunelli, Ignacio Garzón, Hellmut Haberland,
Roy Johnston, Andrea Levi, Christine Mottet, Arnaldo
Rapallo, and Giulia Rossi for their help and suggestions.
The authors acknowledge financial support from the
Italian MIUR under the project 2001021133. R.F. ac-
knowledges financial support from CNR for the project
SSA-TMN in the framework of ESF EUROCORES-
SONS.
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