A method of fabricating a semiconductor structure. According to one aspect of the invention, on a first semiconductor substrate, a first compositionally graded Si_1-xGe_x buffer is deposited where the Ge composition x is increasing from about zero to a value less than about 20%. Then a first etch-stop Si_1-yGe_y layer is deposited where the Ge composition y is larger than about 20% so that the layer is an effective etch-stop. A second etch-stop layer of strained Si is then grown. The deposited layer is bonded to a second substrate. After that the first substrate is removed to release said first etch-stop Si_1-yGe_y layer. The remaining structure is then removed in another step to release the second etch-stop layer. According to another aspect of the invention, a semiconductor structure is provided. The structure has a layer in which semiconductor devices are to be formed. The semiconductor structure includes a substrate, an insulating layer, a relaxed SiGe layer where the Ge composition is larger than approximately 15%, and a device layer selected from a group consisting of, but not limited to, strained-Si, relaxed Si_1-yGe_y layer, strained S_1-zGe_z layer, Ge, GaAs, III-V materials, and II-VI materials, where Ge compositions y and z are values between 0 and 1.