Summary

We have employed the finite element method to study the contribution from the surface tension, the strain energy, and the bulk free energy as well as simulated quantum dot formation. With the surface tension dominating, the system will try to reduce the energy by reducing the surface area, which results in a flat surface. The role of the strain energy is to promote an increase in the amplitude of modulation. At the same time, the stress affects an evaporation rate, causing the surface to move downward. In order to simulate the quantum dot formation, the downward movement is counterbalanced by the bulk energy term which promotes the formation the solid phase from the vapor phase. We show that it is possible to simulate the quantum dot behavior although longer simulation time is needed.

Future Work

•  In our simulation we only study each parameter without making a connection to physical parameters. It is very useful to employ non-dimensionalization scheme in order to better simulate experimental system.

•  We could improve our time evolution scale by using an implicit method. One efficient method is Runge-Kutta which is 4 th -orderd accurate.

•  Our program doesn't have flexibility in eliminating note or adding node to properly represent the surface. This adaptation scheme is useful when simulating for a long time.

•  If there is electrical current going across the film, the currency-induced electromigration will change the shape of surface. Simulating surface change due to electromigration by using the combination of Abaqus and Matlab will be an interesting topic. This will help us understand the interaction effects among surface diffusion, elastic stressing and electrical current stressing.

•  We could also look at the case where the surface energy is dependent on surface orientation. This results in facets in which are present in many quantum dot systems.