Modeling Multilayer Pyramidal-Like Adsorbate Structures …
429
In the framework of self-consistent approximation, an interaction potential for adsorbate can be represented in the following way: U (x(r)) = −
u(r − r
)x(r
)dr
,
where u(r ) is a symmetric binary attractive interaction potential for two adatoms
separated by a distance r . By assuming that an interaction length of adatoms r 0 is
small comparing to a diffusion length L d =
√
Dτ x0 (r 0 ∼ 1nm, L d ∼ 1μm for metals), i.e., r 0 , one can write U (x(r )) −x(r ), where =
u(r )dr stands for
the interaction strength and depends on concrete material properties.
It is known that the existence of the steps in the direct numerical simulations of
the sharp interface problem are not simple tasks [38]. Previously it was shown that
the phase-field model is convenient to simplify the analysis of the processes of the
surface growth and perform numerical simulations [25, 26].
The main idea of the phase-field approach is in the introduction of an effective
order parameter φ(r, t) which indicates phase at a particular position (see, for example, [26, 39–41]). In case under consideration, the phase field φ corresponds to the
surface height in units of the lattice constant a of the adsorbed atoms and defines the
number of layers. Following [25], local minima of the order parameter correspond to
terraces of the pyramidal-like structures, whereas rapid spatial variation of φ relates
to position of the step. According to [25, 26], the evolution equation for the phase
field φ is of the form
τ φ ∂ t φ = −
δ H
δφ
, H =
dr[
2
(∇φ)
2
/2 + h(φ, x)],
(3)
with the free energy density
h(φ, x) =
1
2π
cos(2π [φ − φ s ]) − λx
φ +
1
2π
sin(2π [φ − φ s ])
.
(4)
Here is the width of the step, λ stands for the dimensionless coupling constant,
φ s /2 is the initial height of the substrate. Here the term 1 + cos(π [φ − φ s ]) sets that
minima of the free energy H are possible at φ − φ s = 2n + 1, independently of the
concentration of adatoms [26]. Therefore, the phase-field model for the adsorbate
islands growth during epitaxy becomes the form:
∂ x
∂t
= F −
x
τ x0
e
−εx
+ D∇ · {[1 − εx(1 − x)] ∇x}
τ φ
∂φ
∂t
= −
δ H
δφ
=
2
φ + sin(π [φ − φ s ]) + λx(1 + cos(π [φ − φ s ])),
(5)
where we have introduced the dimensionless interaction strength ε = /T ; τ φ is the
relaxation time of the phase field φ.
In order to take into account local variations in the surface temperature during
adsorption/desorption processes, we will generalize the derived phase-field model.
According to the linear Onsager theory, one can represent the total lateral flux of
adsorbate in the following form J tot = −M∇(μ/T ), defined through the kinetic
429
In the framework of self-consistent approximation, an interaction potential for adsorbate can be represented in the following way: U (x(r)) = −
u(r − r
)x(r
)dr
,
where u(r ) is a symmetric binary attractive interaction potential for two adatoms
separated by a distance r . By assuming that an interaction length of adatoms r 0 is
small comparing to a diffusion length L d =
√
Dτ x0 (r 0 ∼ 1nm, L d ∼ 1μm for metals), i.e., r 0 , one can write U (x(r )) −x(r ), where =
u(r )dr stands for
the interaction strength and depends on concrete material properties.
It is known that the existence of the steps in the direct numerical simulations of
the sharp interface problem are not simple tasks [38]. Previously it was shown that
the phase-field model is convenient to simplify the analysis of the processes of the
surface growth and perform numerical simulations [25, 26].
The main idea of the phase-field approach is in the introduction of an effective
order parameter φ(r, t) which indicates phase at a particular position (see, for example, [26, 39–41]). In case under consideration, the phase field φ corresponds to the
surface height in units of the lattice constant a of the adsorbed atoms and defines the
number of layers. Following [25], local minima of the order parameter correspond to
terraces of the pyramidal-like structures, whereas rapid spatial variation of φ relates
to position of the step. According to [25, 26], the evolution equation for the phase
field φ is of the form
τ φ ∂ t φ = −
δ H
δφ
, H =
dr[
2
(∇φ)
2
/2 + h(φ, x)],
(3)
with the free energy density
h(φ, x) =
1
2π
cos(2π [φ − φ s ]) − λx
φ +
1
2π
sin(2π [φ − φ s ])
.
(4)
Here is the width of the step, λ stands for the dimensionless coupling constant,
φ s /2 is the initial height of the substrate. Here the term 1 + cos(π [φ − φ s ]) sets that
minima of the free energy H are possible at φ − φ s = 2n + 1, independently of the
concentration of adatoms [26]. Therefore, the phase-field model for the adsorbate
islands growth during epitaxy becomes the form:
∂ x
∂t
= F −
x
τ x0
e
−εx
+ D∇ · {[1 − εx(1 − x)] ∇x}
τ φ
∂φ
∂t
= −
δ H
δφ
=
2
φ + sin(π [φ − φ s ]) + λx(1 + cos(π [φ − φ s ])),
(5)
where we have introduced the dimensionless interaction strength ε = /T ; τ φ is the
relaxation time of the phase field φ.
In order to take into account local variations in the surface temperature during
adsorption/desorption processes, we will generalize the derived phase-field model.
According to the linear Onsager theory, one can represent the total lateral flux of
adsorbate in the following form J tot = −M∇(μ/T ), defined through the kinetic