Discrete atomistic model to simulate etching of a crystalline solid
Bernardo A. Mello*
Physics Department, Catholic University of Brası ´lia, 72030-170, Brası ´lia-DF, Brazil
Alaor S. Chaves and Fernando A. Oliveira
Institute of Physics and International Centre of Condensed Matter Physics, University of Brası ´lia, Caixa Postal 04513,
70919-970, Brası ´lia-DF, Brazil
Received 15 November 2000; published 28 March 2001
A discrete atomistic solid-on-solid model is proposed to describe dissolution of a crystalline solid by a
liquid. The model is based on the simple assumption that the probability per unit time of a unit cell being
removed is proportional to its exposed area. Numerical simulations in one dimension demonstrate that the
model has very good scaling properties. After removal of only about 10
2
monolayers, independently of the
substrate size, the etched surface shows almost time-independent short-range correlations and the receding
surface presents the Family-Vicsek scaling behavior. The scaling parameters =0.4910.002 and =0.330
0.001 indicate that the system belongs to the Kardar-Parisi-Zhang universality class. The imposition of
periodic boundary conditions on the simulations reduces the effective system size by a factor of 0.68 without
changing the exponents and . Surprisingly, the periodic condition changes drastically the statistics of the
surface height fluctuations and the short-range correlations. Without periodic conditions, that statistics is, up to
3 standard deviations, an asymmetric Le ´vy distribution with =1.820.01, and outside this region the
statistics is Gaussian. With periodic conditions, that statistics is Gaussian, except for large negative fluctua-
tions.
DOI: 10.1103/PhysRevE.63.041113 PACS numbers: 05.40.Fb, 05.45.Df, 68.35.Ct, 68.08.-p
I. INTRODUCTION
In the last 15 years there has been intensive research on
far from equilibrium moving surfaces 1–3. Those surfaces
can be related to expanding fire, fluid flowing in a porous
media, growing bacteria colonies, growth of colloidal aggre-
gates, epitaxial growth of crystalline solids on a flat sub-
strate, or etching of a crystalline solid by a liquid, a gas, a
plasma, or a bean of atomic ions sputtering, and others. It is
well established that many of those moving surfaces have
self-affine morphologies with simple space and time scaling
properties. The morphology of the surface is more easily
described in the case of epitaxial growth on an initially flat
horizontal substrate of dimensionality d and linear dimension
L. The vertical position of the grown film surface at the
horizontal coordinate r, elapsed a time t after the growth
start, is h ( r, t ). The roughness of the surface is measured by
the rms deviation of the surface height, w ( L , t ) = h ( r, t )
-h
¯
( t )
2
1/2
, where h
¯
( t ) is the average film thickness. For a
large class or growing systems, the roughness kinetics satisfy
the Family-Vicsek scaling ansatz 4
w L , t =L
f t / L
/
, 1
where the function f ( u ) has the limiting behavior f ( u )
u
for u 1 and f ( u ) =constant for u 1. Therefore, for
t t
=C
t
L
/
, w =C
w
t
, and for t t
, w
s
=C
w
s
L
,
where the subscript s in w stands for saturated.
Theoretically, the growth kinetics has been widely de-
scribed by continuum Langevin equations of the type
h
t
= “
2
h + “ h
2
+••• +
n
“
2 n
h + “
2
h “ h
2
+••• +
kj
“
2 k
h “ h
2 j
+F + r, t , 2
where n, k, and j are positive integers, F is the average par-
ticle flux on the substrate, and ( r, t ) is an uncorrelated
noise. This family of equations leads to the scaling expressed
by Eq. 1, and the exponents and depend on the coef-
ficients , ,
n
, , and
kj
, which have a nonzero value;
the F and terms are of course always present. Thus, each
equation of the family defines one universality class of mov-
ing surfaces, each family presenting well-defined values for
and . The most important universality classes are defined
by the Edwards-Wilkinson EW equation 5, which con-
tains only the “
2
h surface tension term in Eq. 2, and the
Kardar-Parisi-Zhang KPZ equation 6 which also contains
the nonlinear term ( “ h )
2
.
A very large number of discrete atomistic models for nu-
merical simulation of the moving surfaces has also been pro-
posed, and many of them demonstrated to present the
Family-Vicsek scaling. Some of these are very simple mod-
els intended not to describe in detail any specific system, but
to demonstrate on a microscopic basis the generation of a
self-affine moving surface. The most widely investigated
atomistic models are the ballistic deposition BD7 and
extensions of the random deposition model, which allow par-
ticle diffusion, as for example the Wolf-Villain WV model
8for reviews, see 1,2. Those models present some char-
acteristics that limit their usefulness in the investigation of
surface morphology. The BD model generates films with va-
cancies and overhangs that create difficulties for the analysis
of the film; not only the film surface but also its body are
fractal systems, and it is also difficult to define univocally the
*Electronic addresses: bernardo@iccmp.br, fao@iccmp.br,
alaor@iccmp.br
PHYSICAL REVIEW E, VOLUME 63, 041113
1063-651X/2001/634/0411138/$20.00 ©2001 The American Physical Society 63 041113-1