Results and Discussions

From Eq. 2, we have four contributions to the reduction of the free energy, which are surface energy, evaporation/condensation, elastic strain energy, and bulk free energy. Therefore, we would like to study each effect separately and we can combine them to simulate the quantum dot.

To achieve this, we set the coefficient in front of term of interest in Eq. 13 to be non zero while multiplying other term with zero. However, to suppress the effect of evaporation-condensation, we have to further set m l 2 /M < < 1 , which make the effect of surface diffusion more prominent.

1) Stress-free undulating surface under surface tension

In this simulation, we studied the effect of the surface energy. We set the initial perturbation, , no applied force, and .

Fig. 3. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 500 iterations

Fig. 3 shows that with only surface energy contribution, the lowest energy state is the flat film. This is expected because the flat film has the lower surface area.

2) Stressed undulating surface without surface tension

In this simulation, we study the effect of stress by setting the perturbation, . We added a constant force on the two ends and set , and .

Fig. 4. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 500 iterations

From Fig. 4, the surface moves down almost uniformly. There exist a small amplitude increase but it is small and we are not sure whether this is due to the numerical error.

The face that the surface moves down is reasonable. When including stress, the energy of atoms in the solid becomes higher relative to that in the environment. Therefore, evaporation becomes more significant.

3) Stressed undulating surface under surface tension

 In this simulation, we study the competition between stress and surface tension contributions. We used the same initial surface profile with both misfit and surface tension term non-zero. We set .

Fig. 5. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 500 iterations

Fig. 5. Shows that the surface moves downward as well as reduce the amplitude, which come from stress effect and surface tension effect, respectively.

4) Undulating surface with phase difference only

The remaining parameter is the bulk free energy, g . We use the same surface profile while setting the surface tension and stress contributions to zero. We experimented the value of g from negative to positive.

(a)

(b)

Fig. 6. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 300 iterations. (a) g<0 (b) g>0.

Fig. 6. shows that the sign of g determine the movement of the surface. When g<0, the surface has lower bulk energy than that of the vapor and therefore, it is energetically favorable for the vapor phase to condense to the solid phase as in Fig. 6(a). The situation is the opposite in Fig. 6(b).

Recalling Eq (2), the free energy variation can be written as:

(16)

Because the free energy variation is associated with unit volume of solid grown on the surface, define a driving force:

(17)

Then, if , the solid gains mass from the environment, causing upward movement. If , the solid loses mass to the environment, resulting in downward movement. As a result, we can balance choose and to switch between cases where solid evaporates, vapor condensates, or no phase change occurs [6].

5 ) Stressed undulating surface with diffusion, and phase difference, stress dominated movement

In this simulation, we increased the magnitude of stress contribution and adjust g to prevent phase change.

Fig. 6. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 500 iterations.

Fig. 6. shows the that with higher stress contribution, the amplitude of the film increases. The value of g was adjusted to prevent evaporation. This is the analogous to the formation of quantum dots.

Considering a dimensionless parameter which characterizing the relative significance of the elastic and surface energy:

(18)

If we set large enough, the stress effect will dominate the surface movement, the amplitude of surface will increase. In this simulation, s = 15.

6) Stressed undulating surface with diffusion and phase difference, diffusion dominated movement

If is small enough, the surface tension dominate the surface movement, eventually, the surface will become flat. In this simulation, .

Fig. 7. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 900 iterations.

7) Stressed rough surface with diffusion, and phase difference

In experiments, quantum dots form by vapor depositing film material on the substrate. Any seemingly flat substrate is rough when looking at the atomic scale. Therefore, we would like to use simulate the quantum dot from a rough substrate. We introduce random perturbation on the surface and use the same parameter as that of the previous simulation.

Fig. 8. The shape of the surface of the thin film. The red line indicate the original perturbation and the blue line indicate the shape after 500 iterations.

From Fig. 8, the amplitude of the surface increase, resulting in formation of 6 dots. The dots still look very rough since this is still very early in the evolution. Nevertheless, the simulation shows that the simulation of quantum dots is possible.

We summarized the important contributions in each simulation in the table below.

Table 1. Summary of significant contribution in each simulation

All parameters used in simulations can be view from this spreadsheet.

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[6] J. H. Prevost, T. J. Baker, J. Liang, and Z. Suo, Int. J. Solids Structures 38 , 5185 (2001).