A method of controlling threading dislocation densities in Ge on Si involving graded SiGe layers and chemical-mechanical polishing (CMP) is presented. This method has allowed us to grow a relaxed graded buffer to 100&percent; Ge without the increase in threading dislocation density normally observed in thick graded structures. This sample has been characterized by transmission electron microscopy, etch-pit density, atomic force microscopy, Nomarski optical microscopy, and triple-axis x-ray diffraction. Compared to other relaxed graded buffers in which CMP was not implemented, this sample exhibits improvements in threading dislocation density and surface roughness. We have also made process modifications in order to eliminate particles due to gas-phase nucleation and cracks due to thermal mismatch strain. We have achieved relaxed Ge on Si with a threading dislocation density of2.1×106 cm−2,and we expect that further process refinements will lead to lower threading dislocation densities on the order of bulk Ge substrates. ©1998 American Institute of Physics.