Theory

4. Normalization of the weak statement: Please refer to this pdf file if interested.

2. Physical process

Stranski-Krastanov method is a typical growth mode of the chemical vapor deposition. It involves both surface diffusion process and interface migration process.  The S-K method usually uses MBE (Molecular Beam Epitaxy) to deposit atoms on the substrate layer by layer. The depositing speed, i.e. the condensation speed, can be entirely controlled by MBE. In most cases the atoms are deposited from a impinging flux on the substrate, about 0.1-1 layer of atoms per second. Once the atoms are deposited on the substrate instantly, then the surface diffusion plays the dominating role. So it is reasonable to assume the evaporation/ condensation processes are always in equilibrium, which means the surface diffusion dominates the process, while the evaporation/condensation process should be relatively restrained.

1. Linear stability analysis

We have learned in class that a flat surface will become unstable if subject to an uniaxial stress parallel to the surface.
Competition between the reduction of elastic energy and increase of the surface energy due to undulation determines a critical wavelength. It is given by the linear stability analysis, namely,

Therefore, the instability of a flat surface under stress field elucidate the origin of the quantum dot formation.

3. Models

We have learned the framework for solving problems of interface migration and surface diffusion in class. The weak statement
 
essentially puts the governing equations into an integration form. Then the finite element method is ready to be applied the weak statement to solve the problem.  
A previous group has studied the quantum dot formation in 2D case, and the basic ideas and formulations are similar here. For 3D case in our project, the numerical implementation part is quite different with 2D case because we have to use a higher-order element. Details about the FEM will be provided in next part.