In reality the formation of quantum dots takes place on a substrate by the evolution of the surface of  epitaxial film or droplet which is exposed to misfit strains at the interface between the substrate and the film. There are two distinct processes for the formation of  QD, one involves  nucleation and growth phenomenon, and the other takes place without nucleation on the surfaces, which are disturbed by the   scholastically formed roughness and ripples .  We have done extensive computer experiments using the second mode with or without ripples by selecting either droplet shape or the rectangular shape film, respectively.  GOOD LUCK

Thank you for the answer....... I am working on single atom transistor (SAT) and i want to model this device. I am facing problem in modeling (mathematically) it .....so please tell me which path to follow for modeling SAT... I am trying to model using MATLAB....

Dear Amit, first one needs an irreversible thermodynamic theory of surface evolution induced by surface drift  surface drift fusion driven by  the capillary forces and  the gradient of the stored strain energy created by the lattice misfit at the interface between the film (or droplet) and the substrate.  That is not enough still one can't get the formation of quantum dots. What one needs a special type of  surface free energy as such that it should be depend on the thickness of the  film when the film thickness becomes smaller than a threshold value.  Actually it should increase monotonically when film thickness decreases.  

At the appendix of the second paper you will find the rigorous variational  derivation of the problem, which sets the well-posed boundary value problem to be solved. Actually, the original  derivation was done using the discrete finite micro-elements formulation of irreversible thermodynamics of surfaces and interfaces published in two separate articles in 2006.  BEST REGARDS

You have to  read the following three papers very thoroughly to get the full picture. 

There is no other simulator available what so ever. Every thing including the numerical methods are home made, which uses the direct boundary element method to handle solutions of the elastic and electrostatic problems using the respective Greens Functions very efficiently.  

Thank you very much for providing the papers ..... and its helpful.....