We have established a thermodynamic model to quantitatively address the nucleation of self-assembled quantum dots (QDs) on the patterned substrate and the shape transition from pyramid to dome of QDs in the heteroepitaxial system. In the nucleation case of QDs, the proposed theory not only elucidates the physical origins of the regular spatial arrangements of QDs on the patterned substrates but also predicts the position of the preferred nucleation. In the case of the shape transition of QDs, taking the Ge and ${\mathrm{Ge}}_x{\mathrm{Si}}_{1-x}$ strain islands on the Si (001) substrates as examples, we find that the free energy of a pyramid QD is lower than that of a dome QD at the early stage of growth, while it is completely opposite at the later stage. The predicted critical volume of shape transition of QDs is in good agreement with experiments, suggesting that the relationship between the surface energy and the relaxed energy of islands determines the shape selection of QDs.

• Received 11 February 2007

DOI:https://doi.org/10.1103/PhysRevB.75.245428