A defect-free, relaxed semiconductor covering layer (e.g., epitaxial SiGe) over a semiconductor substrate (e.g., Si) is provided having a strain relaxation degree above about 80% and a non-zero threading dislocation density of less than about 100/cm2. A lattice mismatch exists between the substrate and the covering layer. The covering layer also has a non-zero thickness that may be less than about 0.5 microns. The strain relaxation degree and threading dislocation are achieved by exposing defects at or near a surface of an initial semiconductor layer on the substrate (i.e., exposing defects via selective etch and filling-in any voids created), planarizing the filled-in surface, and creating the covering layer (e.g., growing epitaxy) on the planarized, filled-in surface, which is also planarized.