An atomistic model for diffusion limited etching: Brownian and
deterministic effects in nanosurfaces formation
A.P. Reverberi**, A.G. Bruzzone*, L. Maga**, M. Massei***
**DICheP - University of Genoa - via Opera Pia 15, 16145 Genova (ITALY)
*DIPEM - University of Genoa - via Opera Pia 15, 16145 Genova (ITALY)
***Liophant Simulation - via Molinero 1, 17100 Savona (ITALY)
Abstract
In this paper, a Montecarlo simulation of a
diffusion-limited surface deconstruction is
presented in three dimensions. The process
takes into account the presence of an external
field by means of a tuning parameter that
matches the role of the Brownian and
deterministic components of the etchant
particle motion. The time trend of the surface
hull variance is analysed for several solid
substrate dimensions and for two different
values of the tuning parameter governing the
etchant particle trajectory.
Finally, the results are discussed and
compared with the analytic ones related to
surface growth dynamics.
1. Introduction
In the last few years, the research devoted to
the study of surface morphology and
characterization has attracted an increasing
interest. We refer in particular to growth and
etching processes, with a marked prevalence
of the first topic to the second one. An
excellent reference text concerning interface
growth dynamics is the book of Stanley and
Barabàsi [1], an we refer the reader to the
references quoted there. Many atomistic
models have been proposed [2] in order to
investigate the role of simple geometrical
rules on the time evolution of a growing
substrate, and the relevant Montecarlo
simulations led to a wide scenario of scaling
laws. By the way, a classical approach
generally adopted in interface characterization
relies upon the study of the standard deviation
of surface sites according to the well known
Family-Vicsek criterion, that is:
) ( ) , ( z L
t
f L t L w a = (1)
with b u u f ® ) ( for u®0 and
. ) ( const u f ® for u®¥.
The exponents a and b, defined as static
(spatial) and temporal (dynamic) scaling
parameters, have been used to define several
universality classes of non-equilibrium
phenomena.
A different, though complementary, approach
is based on a continuum modelling where the
height h(x,t) of a general surface site of spatial
coordinate x at time t is subject to the
following stochastic differential equation [3]:
( ) ) , ( ,.... ) ( , ) ( , , , 2 2 2 4 2 t x h h h h h g
t
h h + Ñ Ñ Ñ Ñ Ñ Ñ =
¶
¶
(2)
where h(x,t) is a space and time uncorrelated
gaussian noise according to:
) ( ) ( 2 ) 2 ( ) 1 ( 2 1 2 1 t t x x D - - >= < d d h h (3)
The form of the operator g containing the
spatial derivative of the height h(x,t) allow to
establish a link between the formalism
described by eq. (1) and the aforementioned
continuum approach. In particular, the
presence of the h 2 Ñ term accounts for
surface relaxation and annealing which have
been firstly investigated in random deposition
with restructuring (RDR) [4]. Besides, the
fourth-order dependence on surface height
proved to be a useful approach to model
diffusion in kink sites according to the Wolf-
Villain scheme [5]. Other intriguing
phenomenologies were further investigated
after the pioneering work of Kardar-Parisi-
Zhang leading to the following anisotropic
growth equation [6]:
) , ( ) ( 2 2 t x h b h a
t
h h + Ñ + Ñ =
¶
¶
(4)
.....