Calculation of vertical correlation probability in Ge/Si(001) shallow island quantum dot multilayer systems

We investigate the behavior of the island vertical pairing probability in multilayer systems of Ge island quantum dots (QDs) in Si(001). By combining a simple kinetic rate model with our previously reported atomistic simulation results on the nature of the stress field from buried shallow Ge islands having {105}-oriented sidewalls, we derive an analytical expression for correlation probability as a function of the parameters characterizing the multi-QD systems. The approach is based upon continuum mechanochemical potential model, which allows one to introduce necessary elements of the kinetics of island formation in a simple way. We compare the model predictions with available experimental data and find that the model provides a satisfactory description of the coupling probability. The correlation probability behavior as a function of capping layer thickness, Ge island size, interisland distance, and Ge adatom diffusion length is investigated within the framework of the developed model.