Journal of Computer-Aided Materials Design, 7: 195–201, 2001.
KLUWER/ESCOM
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Three-dimensional Schwoebel–Ehrlich barrier
S. J. LIUa , E. G. WANGb, C. H. WOOa and HANCHEN HUANGa,∗
a Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong
b Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
Received and accepted 12 May 2001
Abstract. It is well known that the Schwoebel–Ehrlich barrier affects, and even dictates, surface microstructure evolution – such as the transition of growth modes from layer-by-layer to island growth. The conventional
Schwoebel–Ehrlich barrier refers to the case when an adatom diffuses down an island of one monolayer. During thin film deposition, an adatom often needs to diffuse down an island of multiple layers. For the latter,
we demonstrate and calculate the corresponding Schwoebel–Ehrlich barrier – which we call three-dimensional
Schwoebel–Ehrlich barrier. Our calculations show that the three-dimensional Schwoebel–Ehrlich barrier can be
large even if its conventional counterpart is small – as in aluminum. We further propose and demonstrate a possible
process of engineering surface faceting and film texture, by modifying the three-dimensional Schwoebel–Ehrlich
barrier.
Keywords: Faceting, Schwoebel–Ehrlich barrier, Surfactant, Texture competition.
Thin film microstructures, such as texture, are key factors that determine its performance
in various applications. For example, 111 texture is preferred when aluminum thin films
are deposited as interconnect metal lines [1]. Different textures could be preferred even for
the same thin film, TiN. When it is used as a barrier layer in semiconductor devices, 111
texture is preferred; while 100 texture is preferred in mechanical coating [2, 3]. The texture
formation is a result of many processing factors, one of the most important being adatom
diffusion. When an adatom has a small barrier, such as on flat aluminum {111} surfaces,
layer-by-layer growth prevails if the Schwoebel–Ehrlich barrier is negligible; as a natural
result, the 111 texture tends to dominate. Being a controlling factor in surface processing,
the Schwoebel–Ehrlich barrier has been a focus of research [4–22].
In their classical works [4, 5], Schwoebel and Ehrlich – together with their colleagues
– clearly demonstrated the existence of the extra barrier for an adatom to diffuse down a
step; the extra barrier was named after them. Since its discovery, this Schwoebel–Ehrlich
barrier has always been a major consideration in studying thin film deposition. For some FCC
metals, such as aluminum, this barrier is very small [23–25]. As a result, in our earlier studies
this Schwoebel–Ehrlich barrier was assumed to be zero [26–32]. A detailed examination of
surface structures, as shown in Figure 1, indicates that an adatom may need to diffuse across
two facets. In another word, an adatom may diffuse down a step of monolayer and that of
multiple layers. Even if the extra barrier for an adatom to diffuse down a step of monolayer
is zero, it is possible that large extra barrier exists for an adatom to diffuse down a step of
multiple layers. The latter case is the focus of this paper.
In comparison with the conventional Schwoebel–Ehrlich barrier – hereafter referred to
as two-dimensional (2D) Schwoebel–Ehrlich barrier, we propose a three-dimensional (3D)
Schwoebel–Ehrlich barrier for the diffusion of an adatom across two facets. This model is
∗ To whom correspondence should be addressed. E-mail: [email protected]
196 S. J. Liu et al.
Figure 1. (a) A typical equilibrium spherical crystal according to the Wulff construction. (b) Electron microscopy
of aluminum thin film surface after high temperature annealing, with various surface orientation labeled [35].
shown in Figure 2. The diffusion of an adatom down a step of multiple layers is essentially
the diffusion across two facets, when the number of layers is large; results for the intermediate
number of layers fall between the 2D and the 3D extremes and will be presented in another paper. Unlike its 2D counterpart, the 3D Schwoebel–Ehrlich barrier is defined as the total energy
barrier in diffusing across an edge between two facets; this definition is for the convenience
of description since the barriers of an adatom on the two facets are different.
Three-dimensional Schwoebel–Ehrlich barrier 197
Figure 2. 3D Schwoebel barrier model: (a) The cross section of edge between two vicinal surfaces; (b) Schematic
plot of potential energy of adatom diffusing across the edge from one surface to another; (c) Schematic plot of the
3D Schwoebel barrier.
Table 1. The 3D Schwoebel–Ehrlich barriers across an edge from one facet to another
Direct hopping
Exchange
{111} → {111}
{111} → {100}
{100} → {100}
{100} → {111}
0.45 eV
0.22 eV
0.48 eV
0.30 eV
0.60 eV
0.25 eV
0.78 eV
0.68 eV
198 S. J. Liu et al.
Figure 3. Formation energy of an adatom along the diffusion coordinate from the {111} to the {100} facet.
Taking aluminum as the prototype, we have used the molecular statics method to examine
the 3D Schwoebel–Ehrlich barrier for an adatom to diffuse across two facets of {111} and
{100} types. Details of the molecular statics method are given in reference [25], and will be
summarized briefly here. Two mechanisms – direct hopping and diffusion by exchange – are
considered in our calculations. In the direct hopping mechanism, the adatom hops over the
facet edge. According to the exchange mechanism, the adatom replaces one atom at the facet
edge, and the latter becomes an adatom. A migrating atom is first chosen to be the adatom for
the direct hopping, and to be the atom at the facet edge for the exchange mechanism. Between
the initial and the final positions of the migrating atom, a straight line is drawn. The formation
energy of the migrating atom is calculated as a function of its coordinate along this line – the
reaction coordinate. At each point, the migrating atom is allowed to relax perpendicular to
the line but fixed along it. All other atoms in the substrate are free to relax in all directions to
minimize the formation energy. The formation energy of an adatom as a function of diffusion
coordinate, when it goes by exchange from a {111} (reaction coordinate = 0.0) to a {100} facet
(reaction coordinate = 0.5), is shown in Figure 3 as an example. The results are summarized in
Table 1. Even for the case of {111} → {111}, which has the smallest 3D Schwoebel–Ehrlich
barrier, the extra barrier is significantly large. Take the adatom diffusion barrier on the {111}
surface as 0.04 eV [23, 24]. This 3D Schwoebel–Ehrlich barrier means an extra barrier of
0.18 eV for an adatom to diffuse down a {111} surface when the island is of multiple layers in
thickness.
The 3D Schwoebel–Ehrlich barrier will in many aspects change the expected surface morphology. In this short Letter, we use the kinetic Monte Carlo method [27, 33] to demonstrate its
effects in connection with the use of surfactants. The surfactants would allow us to alter the 3D
Schwoebel–Ehrlich barrier [34]. The deposition onto a spherical seed, as studied in reference
Three-dimensional Schwoebel–Ehrlich barrier 199
Figure 4. (a) Snapshots after deposition of 104 atoms with the smaller barrier for diffusion from {100} to {111},
and (b) that for the smaller barrier for diffusion from {111} to {100}.
200 S. J. Liu et al.
[27], is repeated here with consideration of the 3D Schwoebel–Ehrlich barrier, as shown in
Figure 4. First, assume that the 3D Schwoebel–Ehrlich barrier for {100} → {111} is reduced
to only 0.1 eV – keeping all other barriers unchanged. The {100} facets overwhelmingly
dominate with this smaller 3D Schwoebel–Ehrlich barrier. In terms of texture competition,
this would lead to the dominance of the 100 texture – which is not thermodynamically
preferred. This dominance is easy to understand, since the smaller 3D Schwoebel–Ehrlich
barrier allows more adatoms to diffuse from {100} to {111} than the other way around. As one
would expect, the {111} facets overwhelmingly dominate when the 3D Schwoebel–Ehrlich
barrier for {111} → {100} is reduced to only 0.1 eV – keeping all other barriers unchanged.
In summary, we have proposed a 3D Schwoebel–Ehrlich barrier. Our calculations of the 3D
Schwoebel–Ehrlich barrier for aluminum show that it can be significantly larger than its 2D
counterpart. The effects of the 3D Schwoebel–Ehrlich barrier are demonstrated in connection
with the use of surfactants. Combining the concept of the 3D Schwoebel-Erlich barrier and the
effects of surfactants, we have proposed a new way of engineering textures and demonstrated
its feasibility.
Acknowledgement
The work described in this paper was substantially supported by a central research grant from
the Hong Kong PolyU (G-V943), partially by grants from the Research Grants Council of
the Hong Kong Special Administrative Region (PolyU 1/99C, PolyU 5146/99E and PolyU
5152/00E).
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