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We have developed an analytic model that describes in detail the establishment of self-ordered profiles during semiconductor epitaxy on corrugated surfaces. Lateral, self-ordered epitaxy derives from surface gradients in the chemical potential due to the nonplanarity of the profile (capillarity). The growth rate variation on the different facets composing the profile determines whether the profile sharpens up at the bottom of the grooves or at the apex of the corrugations. For alloy growth, additional entropy of mixing effects affect the profile shape and composition. The predictions of the model were applied to explain the self-limiting surface profiles obtained by organometallic chemical vapor deposition on [011]-oriented grooves. Finally, this model is used to design a variety of low-dimensional quantum-confined nanostructures.