During droplet epitaxy, the shape of final crystalline quantum structures is determined by the growth temperature and As flux density. Here we develop a theoretical model with which we elucidate that the diffusion of III-column element atoms away from the droplet and the trapping of As atoms critically govern the shape of final quantum structures, yielding a shape phase diagram in the space of growth temperature and As flux density. Theoretical investigation suggests that we not only control the basic shape of quantum structures but also design the heights of ring-shaped quantum structures through regulating growth temperature and As flux intensity. (C) 2013 Elsevier B.V. All rights reserved.