Layer-by-layer nucleation mechanism for quantum dot formation in strained heteroepitaxy

We study the spontaneous formation of quantum dots in the form of three-dimensional (3D) islands on facetted surfaces in heteroepitaxy. Island development from fast kinetic Monte Carlo (KMC) simulations at low deposition rates is found to follow a layer-by-layer nucleation pathway characterized by energetics driven continuous lateral expansion interrupted by a sequence of independent two-dimensional (2D) upper-layer nucleation events. The process involves only unstable 2D upper-layer nuclei but no unstable 3D nucleus. We have calculated analytically the elastic strain energy of an island in the form of an axisymmetric stepped mound using a small-slope approximation. The total free energy of a system with a 3D island and an adatom bath is obtained. Our theory explains island formation via a free energy driven layer-by-layer nucleation mechanism. Upper-layer nucleation energy barrier, nucleation time, critical radius, and island step spacings are estimated. The relevance of entropic step-step repulsion is discussed. Our theory satisfactorily explains the 3D KMC simulations and may describe the initial evolution of islands in the form of stepped mounds observed in experiments.