Computational Materials Science 23 (2002) 85–94
www.elsevier.com/locate/commatsci
Diffusion of clusters down ð1 1 1Þ aluminum islands
M. Bockstedte
a
a,*
, S.J. Liu b, Oleg Pankratov a, C.H. Woo b, Hanchen Huang
b
Lehrstuhl fur Theoretische Festk€orperphysik, Universit€at Erlangen-N€urnberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany
b
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong
Accepted 1 June 2001
Abstract
The key factor determining nucleation processes and faceting in homoepitaxial growth as well as texture competition
is the mobility of adatoms and small clusters across step edges and facets. Using a combination of molecular dynamics
and ab initio calculations, we investigate the mechanisms of small clusters (dimer and trimer) diffusion down the
aluminum (1 1 1) surface. In this paper we report results of molecular dynamics studies. Our study shows that the
clusters dissociate at the step-edge of compact islands. As a result, the clusters diffuse down the step by an exchange
mechanism with a small or medium Schwoebel barrier. The mechanism of this down-diffusion/dissociation is discussed
and the corresponding energetics are calculated using the molecular statics method. We find a large anisotropy between
the barriers at the two types of h1 1 0i oriented steps. Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Cluster diffusion; Schwoebel barrier; Island; Facet; Aluminum
1. Introduction
The microstructure evolution of thin films has
been a focus of research for years, driven by both
scientific interest and technological importance [1–
17]. For example, aluminum thin films, grown with
preferred texture and desirable density, are used in
the form of metal lines as interconnect in integrated circuits. It is well known that aluminum
films of h1 1 1i texture are resistant to electromigration, a process of atomic mass transport due to
electron wind [18,19]. During the growth of alu-
*
Corresponding author.
E-mail addresses: [email protected], and [email protected] (H. Huang) after summer 2002.
minum films, the texture competition is affected by
many factors; one of them being the faceting of
(1 1 1) surfaces [2].
The h1 1 1i texture dominates when large f1 1 1g
facets form at the early stage of deposition. The
direct result of the large facets is a large fraction of
substrate area covered by the h1 1 1i grains at the
nucleation stage. The large f1 1 1g facets form
when adatoms and clusters (e.g. dimers and trimers) diffuse fast and have low Schwoebel barriers
for diffusion hops over the facet edges. Both
ab initio calculations [13–15] and molecular dynamics simulations [2] have shown that: (1) an
adatom has a small migration barrier on the (1 1 1)
surface (<0.1 eV); and (2) the adatom has a small
Schwoebel barrier (<0.1 eV). If the small clusters,
in particular dimers and trimers, have similar
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 3 2 - 4
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M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
migration and Schwoebel barriers, large {1 1 1}
facets are expected. On the other hand, confinement of the small clusters on the (1 1 1) surfaces by
large Schwoebel barriers or by other mechanisms
may substantially suppress the faceting and therefore the dominance of the h1 1 1i texture.
Apart from the texture competition, the {1 1 1}
faceting also affects the density or uniformity of
aluminum thin films. This effect is of crucial importance in the filling of trenches and vias, as part
of the aluminum interconnections in integrated
circuits. Large {1 1 1} facets formed near shoulders
of the trenches and vias aggravate the geometrical
shadowing – an unavoidable effect in most physical vapor deposition processes. As a result, voids
form in the trenches and vias; the {1 1 1} faceting is
detrimental in this regard. With the presence of
voids, the effective cross-sectional area of the aluminum metal lines for electrical conduction is
smaller. Due to the resulting increased current
density – keeping the same current at the reduced
cross-sectional area, such metal lines will have a
shorter lifetime.
Despite the importance of the diffusion of small
clusters – on and down the {1 1 1} facets – the
details of their dynamics remained largely unknown, in particular for aluminum. Experimental
studies on Ag systems [20,21] indicate that small
clusters are mobile. The experiments clearly showed
initial and final configurations of the clusters at
various time intervals. However, the time spent in
the vicinity of the transition states is too short for
experimental observation of the migration process;
consequently, the mechanisms of diffusion remain
unclear. Our recent molecular dynamics studies [9]
indicate that small clusters, such as dimers and
trimers, rapidly diffuse on the aluminum (1 1 1)
surfaces via jumps of constituent atoms; the involved diffusion barriers are not much higher than
0.1 eV. We further showed that, if dimers are the
critical nuclei [2,5], the faceting is limited by dimers diffusion rather than by their dissociation.
Therefore, whether dimers or trimers are effective
nuclei for three-dimensional growth – a mechanism that limits the faceting – is dictated by their
diffusion down {1 1 1 } facets or islands.
In this paper we investigate these aspects using
compact islands on a (1 1 1) aluminum surface.
The formation and time evolution of such islands
during the initial stages of homoepitaxial growth
have been a focus of investigation recently (cf.
[1,12–17] and references therein). It is a model
system to study the relevant microscopic processes
involved in the kinetics of growth. The main attention, however, has been directed to the diffusion
and kinetics of adatoms on the flat surface and at
the step edges of islands. Here we address the
diffusion mechanism of small clusters (dimers and
trimers) down step edges. In particular we focus
on the details of the mechanism and its energetics.
To answer these questions, we use a combination
of (classical) molecular dynamics and ab initio
methods to investigate the diffusion of the dimer
and the trimer. The molecular dynamics and molecular statics methods are used to elucidate the
mechanisms of diffusing down a (1 1 1) island and
to estimate the corresponding energetics. As a
supplementary tool, ab initio calculations using a
plane-wave pseudopotential method [22] provide
confirmation/correction of the estimated energetics. As the first part of this investigation, we present
the results from molecular dynamics simulations
and molecular statics calculations. The ab initio
calculations will be presented in another paper. In
Section 2, we will describe the simulation method,
and present the simulation results in Section 3.
The conclusions are summarized and discussed in
Section 4.
2. Simulation method
As in our previous studies [2], the interatomic
interactions of aluminum are described by the
EAM potential of Ercolessi and Adams [23–25].
This interatomic potential provides a fairly reliable
description of bulk properties, and more importantly the surface energetics. The range of the
potential extends up to third neighbors in our
). The simulation
calculations (the cutoff is 6.0 A
cell consists of three regions, as shown in Fig. 1.
Periodic boundary conditions are applied along
the two horizontal directions, that is h1 1 0i and
h1 1 2i. Within three layers from the bottom of the
simulation cell, atoms are fixed to their perfect
lattice positions to mimic a large bulk underneath
M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
87
Fig. 1. Schematic setup of the simulation cell with a flat surface.
the top free surface. Atoms in the three layers right
above the fixed region are subject to a Langevin
force, to maintain a constant temperature in this
region; further they serve as a thermal bath for
atoms above.
Starting from the simulation cell with a flat
(1 1 1) surface, a compact hexagonal island is
formed by an extra monolayer of atoms limited by
h1 1 0i=f1 0 0g- and h1 1 0i=f1 1 1g-faceted steps
right above the flat surface. A dimer or trimer is
then introduced on the island to study its diffusion
down the island. The island is chosen to be large
enough so that the diffusion of a dimer or trimer
down one faceted step is not dominated by interaction with other steps; barrier calculation will be
carried out at the middle of each step.
In the MD simulations, both h1 1 0i=f1 1 1g- and
h1 1 0i=f1 0 0g-faceted steps are included so that
diffusion down both of them can be sampled in a
simulation; all the facets are along h1 1 0i to minimize the potential energy. For simplicity, these two
faceted steps will be referred to as {1 1 1}- and
{1 0 0}-faceted steps from now on. We choose a step
length of 17.0 times of the interaction cutoff length.
For our studies of diffusion mechanisms we expect
that size effects will be unimportant with this choice.
Once the diffusion mechanisms are identified,
triangular shaped islands including either {1 1 1}or {1 0 0}-faceted steps are used for the investigation of the migration barriers within static
calculations; thereby the simulation cell is effectively reduced maintaining the same separation of
islands. To validate the use of the smaller com-
putational cell, we also calculate formation energies of a trimer using computational cells of
different sizes, and make sure that the variation is
less than 0.01 eV. The formation energy of a defect
is defined with respective to a reservoir of kinks;
that is, the energy of each reference atom is )3.36
eV in aluminum. The simulation cells with all the
three types of islands are shown in Fig. 2.
Both dynamics and statics simulations of the
aforementioned systems are carried out at constant volume. To account for the thermal expansion at finite temperatures, the dimension of
computational cell is fixed according to values of
the lattice constant at corresponding temperatures.
These values were derived by using the Parrinello–
Rahman algorithm [26], and they are given in [9].
A constant temperature is kept during dynamics
simulation through the application of the Langevin force in the thermostat, as shown in Fig. 1.
In the dynamic simulations, coordinates of atoms in a dimer or trimer are tracked as a function
of time. Analysis of the atomic trajectories reveals
the mechanisms of diffusions. Following the identified mechanisms, molecular statics calculations
are carried out to estimate the corresponding energy barriers. In these calculations, along a path of
dissociation and/or diffusion, we draw a straight
line between the initial and final configurations. At
each point in between, the coordinate of one atom
parallel to the line is fixed (this coordinate is our
reaction coordinate) and the other two coordinates
perpendicular to the line are free to relax. Other
atoms in the cluster and substrate are free to relax
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M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
Fig. 2. Top view of the simulation cell with a cluster (dimer): (a) both kinds of steps h1 1 0i=f1 0 0g – the top step, and h1 1 0i=f1 1 1g –
the bottom step are present; (b) only h1 1 0i=f1 1 1g-faceted steps are present, and (c) only h1 1 0i=f1 0 0g-faceted steps are present. The
dark circles represent substrate atoms, the open circles represent atoms in the island, and the grey circles represent atoms in a cluster.
in all directions to minimize the total potential
energy.
A continuous path of dissociation and/or diffusion is constructed based on the relaxed configurations; in cases where a discontinuous path
appeared, another path is drawn between the two
discontinuous points. In this way the energy barrier is obtained as the maximum potential energy
along the path. If the migration mechanism possesses more than one barrier, further reaction
coordinates are constructed from straight lines
connecting adjacent minima on the path. All the
M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
energy states are given in terms of the formation
energy of the relevant cluster. For dimer (n ¼ 2) or
trimer (n ¼ 3), the formation energy Ef is defined
as
Ef ¼ EðN þ nÞ EðN Þ n ð3:36 eVÞ;
where EðN þ nÞ is the total potential energy of
the simulation cell containing a cluster and EðN Þ
is the potential energy of the simulation cell
without the cluster.
3. Simulation results
We start with the dynamic simulation results to
demonstrate the mechanisms of cluster diffusing
down the island. They are supplemented by the
static calculations of energy barriers. The trajectories of atoms in a dimer from a simulation of 350
ps at 600 K are shown in Fig. 3(a); their time dependence is shown in Fig. 3(b). The dimer was
initially placed at the center of an island, and finally diffuses down a step. The simulation is repeated at 300 K for a much longer time. In these
two simulations, we have observed two representative mechanisms of dimer diffusion on the island.
One is a diffusive concerted sliding and the other is
a non-diffusive (confined) atom-by-atom rotation.
The energy barrier of atom-by-atom rotation is
very small [12]. As indicated by the dark regions
in Fig. 3(a) and the plateaus in Fig. 3(b), relative
rotations are much more frequent than diffusive
jumps. In both simulations, we have never observed the dissociation of a dimer on the island
before it reached step edge. The edge evaporation,
corner break and surface melting are also absent
even in the high temperature (600 K) simulation.
These dynamic simulations indicate that: (1) one
atom of the dimer diffuses down the {1 1 1}-faceted
step through the exchange mechanism (the mechanism is also valid for adatoms) while the dimer
bond remains intact, and (2) the dimer dissociates
after this atom has been incorporated into the step
edge.
According to the identified mechanism of dimer
diffusing down a step, we carried out molecular
statics calculations of the corresponding energy
89
barrier. In Fig. 4, the calculated potential energy
of the dimer is plotted versus the reaction coordinate along the path. The initial configuration
(coordinate ¼ 0.0) corresponds to the dimer at the
edge of the island, and the final configuration
(coordinate ¼ 0.5) corresponds to the dimer with
one atom incorporated into the step and the other
on the island. The potential energy at the initial
configuration corresponds to the formation energy
of the dimer in this configuration. The corresponding relaxed configurations are shown below
the coordinate axis. At first one of the dimer atoms
approaches the step, displaces a step-atom from its
FCC-site into the adjacent HCP-site. In order to
fully replace the step-atom this dimer-atom has to
push the step-atom further away. The latter moves
to the nearest FCC-site at the step thereby forming
a kink and optimizing its coordination. The migration mechanism possesses two transition states,
corresponding to bridge-sites between the adjacent
FCC- and HCP-sites. The intermediate minimum
corresponds to the configuration when the stepatom passes through the HCP-site. The whole
process is described by a separate reaction coordinates for each transition state. The two barriers
of this diffusion jump are 0.10 and 0.08 eV, respectively. It is worth noting that according to our
simulation the dimer prefers to dissociate at the
step, with one of the atom diffusing down the step
at once, instead of both atoms diffusing down simultaneously.
Similar to simulations of the dimer, molecular
dynamics simulations at 300 and 600 K indicate
that a trimer diffuses very fast on the island. Its
diffusion down the {1 1 1}-faceted step is more
complex, and consists of two events. At the first
event (referred to as process 1), one of the three
atoms moves down the step by an exchange mechanism (similar to that of the dimer), forming a
kink-site. The trimer thereby dissociates and the
remaining dimer starts to diffuse on the island. At
the second event the dimer itself diffuses down the
step as described before or returns to the kink-site
and by another exchange process a double kink is
formed (referred to as process 2); both processes
are observed in the dynamic simulations.
For the first event – one of the three atoms
diffusing down the facet – the potential energy of
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M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
Fig. 3. Atomic trajectories of a dimer at 600 K: (a) a top view, and (b) time dependence of positions of the two atoms. The open circles
represent atoms in the island, and the thin solid line and the thick dotted line correspond to the two atoms, respectively.
the trimer is shown in Fig. 5(a) (the potential energy of the initial configuration is related to the
formation energy of the trimer). Again two transition states occur, when the step atom passes
through the bridge-sites as described before. The
two energy barriers are 0.10 and 0.12 eV, respectively. For the second process – the remaining dimer diffusing down the facet – either of the two
alternatives may occur. The energy barriers for the
first alternative have already been described above.
The potential energy curve for the other alternative – process 2 – is shown in Fig. 5(b). In this case
the relevant step-atom is not free to move through
the adjacent HCP-site, which is partially blocked
by the kink at the neighboring FCC-site. At the
same time a complex motion of the dimer-atoms is
observed. This explains the pronounced shoulder
in the potential energy curve and the higher single
M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
91
Fig. 4. Diffusion of a dimer down the {1 1 1}-faceted step: potential energy along the path, with energy barriers and configurations
indicated. The energy at the initial configuration corresponds to the formation energy of the dimer at this position.
energy barrier is 0.26 eV. The insets in Fig. 5
indicate the intermediate atomic configurations
along the diffusion paths.
All these barriers are much smaller than the
0.47 eV for kink incorporation, 0.62 eV for corner
breaking, and 0.80 eV for edge evaporation [15]. It
is interesting to note that the energy gain of 0.63
eV in process 2 is much larger than the 0.04 eV in
process 1. As a result, the reverse process of process 2 has an activation energy of about 1.0 eV. In
contrast to the much lower barriers of other processes, such an event is highly unlikely. On the
other hand, the inverse process of process 1 has a
much lower barrier of about 0.15 eV and indeed,
this process is observed in our dynamic simulations.
When part of the clusters (one of the dimer
atoms or two of the trimer atoms) diffuse down the
step, the rest is just the adatom diffusion at the step
with one kink (dimer) or double kinks (trimer).
There are two possibilities of the adatom diffusion:
(1) the adatom diffuses down the kink-step following the first part of the clusters and (2) the
adatom migrates away from the kink-step. The
calculated energy barriers of the first possible diffusion are 0.11 and 0.16 eV for dimer and trimer,
respectively. Indeed, in our molecular dynamics
simulations, we observed these two kinds of diffusion of the adatom. Because the adatom has a
very small migration barrier on the (1 1 1) surface
(<0.1 eV) and small Schwoebel barrier (<0.1 eV)
at the steps, it is likely to stay on the island,
migrate away from the kink-step, and finally diffuse down the steps by exchanging mechanism.
This mechanism has been discussed in detail in
[13,14].
For completeness we have also calculated the
migration barriers of dimers and trimers diffusing
down {1 0 0}-faceted steps, following similar diffusion mechanisms as at {1 1 1}-faceted steps. The
results are summarized in Table 1. Consistent with
the dynamics simulations, the energy barriers
clearly show that diffusion down {1 0 0}-faceted
steps is much more difficult than {1 1 1}-faceted
steps; the minimum barrier is 0.40 eV for {1 0 0}faceted steps in contrast to 0.10 eV for {1 1 1}faceted steps.
Apart from the diffusion down a step, the dynamic simulations also indicate that the clusters
have preference to reside near the steps. Based on
the calculated formation energies at various locations on the island, we estimate that a dimer or
trimer has about 0.1 eV lower formation energy
near the steps than on the flat island.
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M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
Fig. 5. Diffusion of a trimer down the {1 1 1}-faceted step: potential energy along the path (a) when the first atom diffuses (process 1),
and (b) when the second atoms diffuses (process 2); the corresponding energy barriers and atom configurations are indicated. The
energy of the initial configuration corresponds to the formation energy of the trimer and dimer with kink in this configuration.
Table 1
Energetics of a cluster diffusing down a (1 1 1) island
Diffusion barrier (eV)
Energy gain (eV)
Facet
h1 1 0i=f1 0 0g
h1 1 0i=f1 1 1g
h1 1 0i=f1 0 0g
h1 1 0i=f1 1 1g
Dimer
Trimer (process 1)
Trimer (process 2)
Last atom in the dimer
Last atom in the trimer
0.39
0.38
0.13
0.15
0.16
0.10/0.08
0.10/0.12
0.26
0.11
0.16
0.28
0.09
0.65
1.08
0.94
0.27
0.04
0.63
0.95
0.92
For a trimer, one atom diffuses down in process 1, and the second atom follows in process 2. When double barriers exist for one
diffusion process, both are listed.
M. Bockstedte et al. / Computational Materials Science 23 (2002) 85–94
4. Conclusions and discussions
Using the molecular dynamics and molecular
statics methods, we have studied the processes of
dimer and trimer diffusing down a (1 1 1) island in
aluminum. Our study shows:
(1) Diffusion down the h1 1 0i=f1 1 1g-faceted
steps is much easier than the h1 1 0i=f1 0 0g-faceted steps.
(2) A dimer dissociates during the process of
diffusing down the h1 1 0i=f1 1 1g-faceted step.
One of the two atoms diffuses down the step
by an exchange mechanism. Two transition states
are encountered along the path with barriers of
0.10 and 0.08 eV. The other atom remains on the
island as a single adatom; if it diffuses down the
step following the first, the barrier is 0.11 eV.
(3) A trimer also dissociates in the process of
diffusing down the h1 1 0i=f1 1 1g-faceted step
in two separate diffusion events. At first, one
of the three atoms diffuses down the facet with
double barriers of 0.10 and 0.12 eV. The other
atoms remain on the island as a dimer; if the
second atom diffuses down the facet following
the first one, the barrier is 0.26 eV. The third
atom remains on the island as a single adatom;
if it diffuses down the step following the first
two, the barrier is 0.16 eV.
(4) Diffusion down the other faceted step,
h1 1 0i=f1 0 0g, is more difficult. In particular,
the diffusion barrier of the first atom is about
0.2 eV higher than down the h1 1 0i=f1 1 1g step.
For practical purposes, this process can be ignored in surface evolution of aluminum below
half of the melting temperature.
As in any classical molecular dynamics simulations, the reliability of the results depends on the
interatomic potential. The EAM potential of aluminum [23] used in this paper provides an accurate
description of surface formation energies, adatom
formation energy and migration energy on the
(1 1 1) surface [9,23]. Therefore, we expect that (a)
the identified mechanisms are reliable, and (b) the
calculated energetics on the (1 1 1) surface are
reliable at least in relative magnitude. Further,
diffusion processes strongly depend on the under-
93
lying crystal structure, further ensuring the reliability of the identified mechanisms and the
calculated energetics. More accurate determination of the absolute magnitude of the energetics is
being carried out using ab initio methods.
Acknowledgements
The work described in this paper was substantially supported by a central research grant from
the Hong Kong PolyU (G-V943), partially by
a grant from Deutsche Forschungsgemeinschaft
under SFB 292 ‘‘Multicomponent Layered Systems’’, and partially by grants from the Research
Grants Council of the Hong Kong Special Administrative Region (PolyU 5146/99E and Group
Research project: Computer Aided Materials Engineering). The joint financial support from the
Hong Kong RGC and the German DAAD is also
gratefully acknowledged.
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