Semiconductor nanostructures self-assemble under a variety of growth conditions. The control of their shape, size and self-organization has been a major quest of modern nanoscience. In our work, we theoretically predict the formation and shape evolution of these nanostructures using models based on fundamental thermodynamic
principles and kinetic processes. We focus our attention on the widely studied heteroepitaxial Ge on Si system where a large variety of defect-free, single-crystalline nanostructures such as quantum dots, quantum wires, quantum dot molecules, etc. have been reported based on different experimental conditions. The surface chemical potential, which is the fundamental driving force for nanostructure evolution is explicitly calculated using atomistic or continuum models. After a brief introduction to the theoretical methods, we will showcase two recent results. We show how the ordering of quantum dots can be controlled by pre-patterned substrates. Then we will illustrate the spontaneous formation of a novel nanostructure called Quantum Dot Molecule which
can be directly attributed to kinetic effects.