1. A method of making an electronic and/or optoelectronic device structure by layer transfer, comprising: providing a crystalline donor having a first free surface and a plurality of intrinsic cleavage planes spaced from one another and each substantially parallel to the first surface, wherein the first free surface and a particular one of the plurality of intrinsic cleavage planes defining a first donee layer; fabricating a device layer containing at least one device upon the first donee layer so that the at least one device is monolithic with the first donee layer, said at least one device being either one of an electronic type and an optoelectronic type; securing a first handle to either the device layer or the first free surface of the first donee layer; and cleaving the first donee layer from the crystalline donor along the particular one of the plurality of intrinsic cleavage planes so as to liberate the first donee layer, in combination with the first handle, from the crystalline donor.