The invention provides methods which can be applied during the epitaxial growth of two or more layers of semiconductor materials so that the qualities of successive layer are successively improved. In preferred embodiments, surface defects present in one epitaxial layer are capped with a masking material. A following layer is then grown so it extends laterally above the caps according to the known phenomena of epitaxial lateral overgrowth. The methods of the invention can be repeated by capping surface defects in the following layer and then epitaxially growing a second following layer according to ELO. The invention also includes semiconductor structures fabricated by these methods.