Growth morphology for a ballistic deposition model for multiple species
Hassan F. El-Nashar
Department of Physics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
Wei Wang
Physics Department and National Laboratory of Solid State Microstructure, Institute of Solid State Physics, Nanjing University,
Nanjing 210093, China
Hilda A. Cerdeira
The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy
Received 20 August 1997; revised manuscript received 10 December 1997
The kinetics and morphology of surface growth are studied for a ballistic deposition model with two kinds
of particles A and C in 1 +1 and 2 +1 dimensions. A morphological structural transition is found as the
probability of being a particle C increases. This transition is well defined by the different behavior of the
surface width when it is plotted versus time and probability. The calculated exponents and for different
values of probability show the same behavior. We attribute this transition to the formation of wide vacancies
during the growth while the interface advances. S1063-651X9802008-X
PACS numbers: 68.10.Jy, 05.40.+j, 05.70.Ln, 68.35.Ct
I. INTRODUCTION
The growth of surfaces and interfaces remains a challeng-
ing problem in physics. It attracts much interest due to its
technological importance as well as its relevance in under-
standing nonequilibrium statistical mechanics at the funda-
mental level 1–3. The study of the kinetics of crystal
growth helps us understand this phenomenon since it de-
scribes how the surface evolves with time, while the study of
the morphology provides a clear interpretation of the growth.
Most of the studies contain rough surfaces and stochastically
growing interfaces in the context of the ballistic deposition
BD as well as other models and continuum growth equa-
tions 4.
A different feature of this phenomenon is the existence of
dynamic scaling 5, i.e., if we start at t =0 from a flat sub-
strate of length L , we have
W L , t =L
f t / L
z
, 1
where W( L , t ) is the surface width
W
2
L , t =
1
L
d -1
r
h r , t - h t
2
. 2
Here h ( r , t ) is the height of the surface at position r and time
t , h ( t ) is the average height at time t , and d ' =d -1 is the
substrate dimension. The dynamical scaling behavior is char-
acterized by the roughness exponent and the dynamical
exponent with z = / . The scaling function f ( x ) behaves
as f ( x ) =x
for x 1 and f ( x ) =const for x 1. The scaling
behavior has been studied in various systems and models and
has been argued to be universal 1–4. One of the successful
theoretical approaches describing the BD model is that of
Kardar, Parisi, and Zhang 6, which is based on Edwards
and Wilkinson’s theory 7. The Kardar-Parisi-Zhang equa-
tion is a nonlinear Langevin equation
h
t
=
2
h +
2
h
2
+ x , t 3
for the local growth of the profile h ( r , t ) of a moving inter-
face about a d ' -dimensional flat substrate.
The BD model represents an example of well studied
growth models. Here particles rain down vertically onto a
d ' -dimensional substrate and aggregate upon first contact
8. Such a model gives rise to a rather interesting structure:
The surface is a self-affine fractal, although the bulk is com-
pact 3. Most previous studies have dealt with the surface
growth of one kind of particle 1–4. Generally, in the
growth of real materials one should take into consideration
that different kinds of particles are deposited. Thus, in the
growing system, there may exist different interactions for
different particles, which in turn yield a different kinetics of
growth associated with a change in the morphological struc-
ture of the aggregate. Pelligrini and Jullien PJ9,10 de-
scribed a surface growth according to a model with two kind
of particles, sticky and sliding, where both are active. This
model interpolates between a diffusive model that incorpo-
rates surface diffusion and the usual ballistic deposition
model. They used a parameter c to control the process of
diffusion on the surface. When c =0 their model is similar to
that of Family 11, i.e., a model with surface reconstruction,
while when c =1 it is equivalent to the plain ballistic model.
In our work we describe the kinetic growth in 1 +1 and
2 +1 dimensions as well as the morphological structure in
order to interpret the results that have been obtained from the
kinetics for two kinds of particles that are active and inac-
tive. We hold the growth rules of the ballistic model for
different condition and in all cases 12–14. So we do not
introduce any kind of surface diffusion for the inactive par-
ticles or any reconstructuring processes on the surface. In 2
+1 dimensions, which simulates the real surface growth, we
extend the interaction between particles from nearest-
PHYSICAL REVIEW E OCTOBER 1998 VOLUME 58, NUMBER 4
PRE 58 1063-651X/98/584/44617/$15.00 4461 © 1998 The American Physical Society