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A. V. Dvornichenko and O. M. Shchokotova
In order to provide a theoretical study with a change in the morphology of the growing surface during adsorption/desorption processes nowadays, several approaches
are used. These methods include atomistic models [4, 7–10] and citation therein,
reaction-diffusion models [11–13], approaches, considering formation of separated
islands [14], and devices with steps [15]. These models allow one to study evolution
of the surface morphology [16–18]. The continuous approaches allow one to study
evolution of the height of the growing surface. They are based on partial differential
equations for the height field. Among these models, one can issue: a model of Kardar,
Parisi and Zhang (KPZ) with nonconserved dynamics [19]; the Edwards–Wilkinson
(EW) model for conserved solid-on-solid growth [20]; the Mullins–Herring (MH)
model for conserved dynamics with significant surface diffusion [21, 22]; the model
for molecular-beam epitaxy, known as conserved Villain–Lai–Das Sarma model [10,
23]. At the same time, these continuous models do not take into account the formation
of step edges and, hence, could not be used to describe formation of pyramidal-like
surface structures with terraces.
One of the theoretical methods for the step flow, allowing one to describe formation of steps in adsorbate structures is the mesoscopic Burton–Cabrera–Franck (BCF)
model [24]. It allows one to represent island boundaries as steps, and is widely used
in the modeling of islands formation during epitaxial growth. This model is continuous in the horizontal direction (along a substrate) and discrete in the growth
(vertical) direction. Here the concentration of adatoms on any terrace is described
by a continuous function and the location of the step is given by a continuous curve.
According to this approach, the boundaries of islands are modeled by using diffusionlike equation for the concentration of adatoms. The boundary conditions are taken in
accordance with the microscopic details of atomic processes on islands edges. One
of the modifications of the BCF model is the phase-field approach. This method can
be considered as an extension of the known Ginzburg–Landau theory for phase transitions. According to this approach, one simultaneously considers coupled partially
differential equations for the adsorbate concentration and the phase field, which plays
a role of the order parameter. This method was used by Liu, Matiu and Karma, Plapp
(see [25, 26]) to perform theoretical study of adsorbate islands formation during
epitaxial growth.
The BCF model [24] considers nucleation processes but does not take into account
the interaction of adsorbed atoms. In real situations, adatoms are mobile interactive
spices, which can diffuse along a substrate and form adsorbate clusters due to their
interactions. Moreover, lattice mismatch between adatom and substrate and elastic
tensions are able to induce additional interactions of adatoms. These interactions
provide strong local bond, which is able to correct the rate for desorption and produce the lateral flux which is opposite to the standard diffusion flux. In theoretical
studies of the surface structures formation during adsorption/desorption processes,
the temperature of the growing surface is assumed to remain a constant. At the same
time, from a physical point of view one should take into account that adsorption
processes lead to local increase in the surface temperature; the desorption processes
result to local cooling of the surface. Moreover, local changes in the surface temperature can be realized during processes of adatoms clustering and formation of islands
A. V. Dvornichenko and O. M. Shchokotova
In order to provide a theoretical study with a change in the morphology of the growing surface during adsorption/desorption processes nowadays, several approaches
are used. These methods include atomistic models [4, 7–10] and citation therein,
reaction-diffusion models [11–13], approaches, considering formation of separated
islands [14], and devices with steps [15]. These models allow one to study evolution
of the surface morphology [16–18]. The continuous approaches allow one to study
evolution of the height of the growing surface. They are based on partial differential
equations for the height field. Among these models, one can issue: a model of Kardar,
Parisi and Zhang (KPZ) with nonconserved dynamics [19]; the Edwards–Wilkinson
(EW) model for conserved solid-on-solid growth [20]; the Mullins–Herring (MH)
model for conserved dynamics with significant surface diffusion [21, 22]; the model
for molecular-beam epitaxy, known as conserved Villain–Lai–Das Sarma model [10,
23]. At the same time, these continuous models do not take into account the formation
of step edges and, hence, could not be used to describe formation of pyramidal-like
surface structures with terraces.
One of the theoretical methods for the step flow, allowing one to describe formation of steps in adsorbate structures is the mesoscopic Burton–Cabrera–Franck (BCF)
model [24]. It allows one to represent island boundaries as steps, and is widely used
in the modeling of islands formation during epitaxial growth. This model is continuous in the horizontal direction (along a substrate) and discrete in the growth
(vertical) direction. Here the concentration of adatoms on any terrace is described
by a continuous function and the location of the step is given by a continuous curve.
According to this approach, the boundaries of islands are modeled by using diffusionlike equation for the concentration of adatoms. The boundary conditions are taken in
accordance with the microscopic details of atomic processes on islands edges. One
of the modifications of the BCF model is the phase-field approach. This method can
be considered as an extension of the known Ginzburg–Landau theory for phase transitions. According to this approach, one simultaneously considers coupled partially
differential equations for the adsorbate concentration and the phase field, which plays
a role of the order parameter. This method was used by Liu, Matiu and Karma, Plapp
(see [25, 26]) to perform theoretical study of adsorbate islands formation during
epitaxial growth.
The BCF model [24] considers nucleation processes but does not take into account
the interaction of adsorbed atoms. In real situations, adatoms are mobile interactive
spices, which can diffuse along a substrate and form adsorbate clusters due to their
interactions. Moreover, lattice mismatch between adatom and substrate and elastic
tensions are able to induce additional interactions of adatoms. These interactions
provide strong local bond, which is able to correct the rate for desorption and produce the lateral flux which is opposite to the standard diffusion flux. In theoretical
studies of the surface structures formation during adsorption/desorption processes,
the temperature of the growing surface is assumed to remain a constant. At the same
time, from a physical point of view one should take into account that adsorption
processes lead to local increase in the surface temperature; the desorption processes
result to local cooling of the surface. Moreover, local changes in the surface temperature can be realized during processes of adatoms clustering and formation of islands