Modeling Multilayer Pyramidal-Like Adsorbate Structures …
431
term in (8), that also relates to local heating (∂ x/∂t > 0) or cooling (∂ x/∂t < 0) the
surface during adsorption/desorption processes with intensity β.
For further consideration, let us move to dimensionless quantities and measure
time and space in units τ x0 and diffusion length L d . The phase-field model describing
processes of adsorbate islands formation during epitaxial growth and local variations
in the surface temperature becomes the form
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ t x =F − xe
−εx/θ
− ∇ · J tot −
1
2
∂ t φ + ξ(t),
∂ t φ =
2
φ − ∂ φ h(x, φ),
ν∂ t θ =1 − θ + χχθ + α F x + β∂ t x.
(9)
Here we have introduced dimensionless quantities F ≡ Fτ x0 , ε ≡ /T 0 , θ ≡ T /T 0 ,
χ ≡ χτ T , α ≡ ατ T /T 0 , β ≡ βτ T /T 0 and put τ φ /τ x0 = 1 The last term in the evolution equation for the adsorbate concentration represents additive Gaussian zero-mean
and delta-correlated noise.
3 Pyramidal Islands Growth at Constant Temperature
In this section, we will discuss a possibility to control dynamics of pattern formation
at epitaxial growth assuming the constant temperature of the growing surface. In order
to provide a detailed study of the processes of the surface growth at epitaxy, we will
exploit numerical simulations. To that end, we will solve numerically the dynamical
system (5) on two-dimensional quadratic lattice with the linear size L = 256, mesh
size l = 1.0, time step δt = 0.005 and periodical boundary conditions. Here and
thereafter we put
2
= 2.0, λ = 10.0.
It is known that at epitaxial growth processing the morphology of the growing
layer will repeat the morphology of the substrate. Moreover additional monolayers of
adsorbate will appear as additional terraces during epitaxial processes. For the actual
phase-field model at φ s = 0 no surface structures can be realized during exposing. An
edge defect (one edge dislocation) will promote formation of terraces of the pyramid
in parallel directions to the edge defect. A dot defect will induce formation of “kidlike pyramid,” Pyramidal-like holes will realize on a vacancy on the substrate. The
screw dislocation located on the substrate will stimulate the formation of pyramidallike structure [26].
In order to provide a general description, we will consider the initial substrate with
a lot of surface defects, by setting initial conditions of the order parameter field in a
form of Gaussian distribution with small dispersion, i.e., φ(r, t = 0) = =φ s = 0,
dispersion (δφ(r, t = 0))
2
= 0.1. For the adsorbate concentration, we put x(r, t =
0) = 0. Snapshots of the system evolution at different values of the deposition rate
F and the interaction strength ε are shown in Fig. 1.
431
term in (8), that also relates to local heating (∂ x/∂t > 0) or cooling (∂ x/∂t < 0) the
surface during adsorption/desorption processes with intensity β.
For further consideration, let us move to dimensionless quantities and measure
time and space in units τ x0 and diffusion length L d . The phase-field model describing
processes of adsorbate islands formation during epitaxial growth and local variations
in the surface temperature becomes the form
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ t x =F − xe
−εx/θ
− ∇ · J tot −
1
2
∂ t φ + ξ(t),
∂ t φ =
2
φ − ∂ φ h(x, φ),
ν∂ t θ =1 − θ + χχθ + α F x + β∂ t x.
(9)
Here we have introduced dimensionless quantities F ≡ Fτ x0 , ε ≡ /T 0 , θ ≡ T /T 0 ,
χ ≡ χτ T , α ≡ ατ T /T 0 , β ≡ βτ T /T 0 and put τ φ /τ x0 = 1 The last term in the evolution equation for the adsorbate concentration represents additive Gaussian zero-mean
and delta-correlated noise.
3 Pyramidal Islands Growth at Constant Temperature
In this section, we will discuss a possibility to control dynamics of pattern formation
at epitaxial growth assuming the constant temperature of the growing surface. In order
to provide a detailed study of the processes of the surface growth at epitaxy, we will
exploit numerical simulations. To that end, we will solve numerically the dynamical
system (5) on two-dimensional quadratic lattice with the linear size L = 256, mesh
size l = 1.0, time step δt = 0.005 and periodical boundary conditions. Here and
thereafter we put
2
= 2.0, λ = 10.0.
It is known that at epitaxial growth processing the morphology of the growing
layer will repeat the morphology of the substrate. Moreover additional monolayers of
adsorbate will appear as additional terraces during epitaxial processes. For the actual
phase-field model at φ s = 0 no surface structures can be realized during exposing. An
edge defect (one edge dislocation) will promote formation of terraces of the pyramid
in parallel directions to the edge defect. A dot defect will induce formation of “kidlike pyramid,” Pyramidal-like holes will realize on a vacancy on the substrate. The
screw dislocation located on the substrate will stimulate the formation of pyramidallike structure [26].
In order to provide a general description, we will consider the initial substrate with
a lot of surface defects, by setting initial conditions of the order parameter field in a
form of Gaussian distribution with small dispersion, i.e., φ(r, t = 0) = =φ s = 0,
dispersion (δφ(r, t = 0))
2
= 0.1. For the adsorbate concentration, we put x(r, t =
0) = 0. Snapshots of the system evolution at different values of the deposition rate
F and the interaction strength ε are shown in Fig. 1.