Modeling Multilayer Pyramidal-Like Adsorbate Structures …
437
Next, we perform numerical simulations by solving coupling equations (3) and
(11) on the quadratic grid according to the mechanism, presented above. In order to
discuss an influence of the introduced fluctuations onto growing surface morphology
let us analyze the dependencies of the correlation length ξ on the noise intensity ,
shown in Fig. 5.
From obtained results, it follows that small ε fluctuations cannot induce formation
of well-organized pyramidal patterns with terraces. In this case, the correlation length
does not depend significantly on the noise intensity (see empty squares in Fig. 5).
For the case of large values of interaction strength, the noise action has no essential
influence onto pattern formation processes at small intensity (see filled squares
in Fig. 5). At large values of the noise intensity, the introduced fluctuations lead to
formation of needle-like patterns. Here two nearest points are spaced far from each
other [49].
5 Oscillatory Dynamics in a Generalized Model
In this section, we will pay our attention onto dynamics of pyramidal-like adsorbate
islands formation during epitaxial growth by taking into local changes in the temperature of the growing surface, caused by adsorption/desorption processes. To that
end, we will solve the system (9) by setting χ = 10.
5.1 Stability Analysis of the Two-Component Model
By analyzing the evolution of the phase field φ, one can describe different regimes of
the system dynamics and the corresponding change in the surface morphology [25].
Indeed, if ∂ t φ = 0 then the height of the surface remains constant and, as a result,
surface does not evolve in time. In such a case adsorption and desorption processes
are characterized by the same probability.
If the speed of the surface growth is positive, i.e., ∂ t φ ≡ v φ > 0 then the height of
the surface grows in time. In such a case one can distinguish two different regimes
characterized by v φ = const and v φ = v φ (t). In the regime, when the surface grows
with the constant speed one gets the layer-by-layer growth, similar to the Frank-van
der Merwe mechanism of the surface growth. For the more complicated situation
of the time-dependent speed of the surface growth, v φ = v φ (t), dynamics of the
phase field φ can govern processes of adsorbate islands formation according to the
Volmer–Weber or Stranski–Krastanov mechanisms. In the case of negative values
of the speed ∂ t φ < 0 desorption and/or evaporation processes play a major role in
surface height dynamics, and as a result, the height of the surface decreases. In this
section, we will consider the case ∂ t φ > 0, only, assuming φ v φ t. The averaging
of the second equation of the system (9) gives the simple relation v φ λx 0 , where
x 0 ≡ ≡x(t → ∞) = const relates to the stationary value of the concentration field.
437
Next, we perform numerical simulations by solving coupling equations (3) and
(11) on the quadratic grid according to the mechanism, presented above. In order to
discuss an influence of the introduced fluctuations onto growing surface morphology
let us analyze the dependencies of the correlation length ξ on the noise intensity ,
shown in Fig. 5.
From obtained results, it follows that small ε fluctuations cannot induce formation
of well-organized pyramidal patterns with terraces. In this case, the correlation length
does not depend significantly on the noise intensity (see empty squares in Fig. 5).
For the case of large values of interaction strength, the noise action has no essential
influence onto pattern formation processes at small intensity (see filled squares
in Fig. 5). At large values of the noise intensity, the introduced fluctuations lead to
formation of needle-like patterns. Here two nearest points are spaced far from each
other [49].
5 Oscillatory Dynamics in a Generalized Model
In this section, we will pay our attention onto dynamics of pyramidal-like adsorbate
islands formation during epitaxial growth by taking into local changes in the temperature of the growing surface, caused by adsorption/desorption processes. To that
end, we will solve the system (9) by setting χ = 10.
5.1 Stability Analysis of the Two-Component Model
By analyzing the evolution of the phase field φ, one can describe different regimes of
the system dynamics and the corresponding change in the surface morphology [25].
Indeed, if ∂ t φ = 0 then the height of the surface remains constant and, as a result,
surface does not evolve in time. In such a case adsorption and desorption processes
are characterized by the same probability.
If the speed of the surface growth is positive, i.e., ∂ t φ ≡ v φ > 0 then the height of
the surface grows in time. In such a case one can distinguish two different regimes
characterized by v φ = const and v φ = v φ (t). In the regime, when the surface grows
with the constant speed one gets the layer-by-layer growth, similar to the Frank-van
der Merwe mechanism of the surface growth. For the more complicated situation
of the time-dependent speed of the surface growth, v φ = v φ (t), dynamics of the
phase field φ can govern processes of adsorbate islands formation according to the
Volmer–Weber or Stranski–Krastanov mechanisms. In the case of negative values
of the speed ∂ t φ < 0 desorption and/or evaporation processes play a major role in
surface height dynamics, and as a result, the height of the surface decreases. In this
section, we will consider the case ∂ t φ > 0, only, assuming φ v φ t. The averaging
of the second equation of the system (9) gives the simple relation v φ λx 0 , where
x 0 ≡ ≡x(t → ∞) = const relates to the stationary value of the concentration field.