A method for manufacturing a semiconductor heterostructure by first manufacturing a donor wafer having a first substrate with a first in-plane lattice parameter, a spatially graded buffer layer having a second in-plane lattice parameter, and a strained smoothing layer of a semiconductor material having a third in-plane lattice parameter which has a value between that of the first and second lattice parameters. A top layer is formed on the ungraded layer a top layer of a semiconductor material having a top surface, optionally with a superficial layer present on the top surface and having a thickness that is equal to or smaller than 10 nanometers. Next, a handle wafer of a second substrate having an insulator layer thereon is bonded with the donor wafer in such way that (a) the insulator layer of the handle wafer is bonded directly onto the top surface of the top layer of the donor wafer, or (b) the insulator layer of the handle wafer is bonded onto the superficial layer.