Three-dimensional theory of quantum dot growth is constructed from energetics of coherent nanometer-size islands in one-component strained layer epitaxy using nonlinear elasticity. The theory is amenable to statistical mechanical treatment and thus shows a great promise for predicting island size distribution. This attribute is crucial for ensuring timely technological applications of quantum dots. which rely on a uniform size distribution.

Construction of the theory is motivated by our 2D model that incorporates surface and elastic strain energies. The 2D model predicts nucleationless islands of cellular morphology for strained lavers grown near critical misfit strain. Qualitative predictions from the 2D model agree with measured size distributions and island slope histograms of SiGe islands. The 2D model cannot predict the thickness of wetting laver that forms in Stranski-Krastanow growth mode.

A new methodology for representing time variable is proposed to analyze the evolution of island size distributions in multilayer deposition. The measured size distributions of SiGe islands exhibit a slow-decay at small sizes and an asymmetric bell-shaped distribution peaked at a large size. A size distribution model based on random percolation and Smoluchowski [coalescence) ripening gives a good agreement with the measured distributions. SiGe islands of low Ge fractions grow critically, while those with large Ge fractions develop size bimodality. Defect-mediated, first-order transitions occur for the latter and are confirmed by their slope histograms.

The 3D theory simultaneously solves for the wetting layer thickness and equilibrium shapes of quantum dots. Thickness-dependent equilibrium morphologies are caused by a nonlocal surface force equilibrium. Order parameter for coherent islanding transition is shear strains along the growth direction. Coupling between adatoms and the order parameter is crucial for producing bulk shear deformation. First-order transition is predicted and adatom diffusion barrier determines the magnitude of discontinuity of the transition. The concept of transition thickness—the flat laver coverage needed by tetragonal strain to balance surface force from surface and interface tension—is introduced to explain the misfit sign dependence of strained layer stability. Under typical conditions, coherent layers in compression are unstable above the transition thickness. but become stable in tension.