Quantum dot growth on a stripe-pattern

We study the development of the Asaro-Tiller-Grinfel'd instability on a thin strained film on top of a stripe-patterned substrate and the subsequent growth of self-organized quantum dots. We use a continuum model describing the evolution equation enforced by surface diffusion. We compute the elastic energy up to the first non-linear order and investigate the long time dynamics which describes the dot growth. We find different island locations depending on the substrate wavelength and thickness. As found in experiments, the instability long-time dynamics leads to islands located either on top or in the bottom of the pattern.