An illustrative method includes forming a FinFET device above structure comprising a semiconductor substrate, a first epi semiconductor material and a second epi semiconductor material that includes forming an initial fin structure that comprises portions of the semiconductor substrate, the first epi material and the second epi material, recessing a layer of insulating material such that a portion, but not all, of the second epi material portion of the initial fin structure is exposed so as to define a final fin structure, forming a gate structure above and around the final fin structure, removing the first epi material of the initial fin structure and thereby define an under-fin cavity under the final fin structure and substantially filling the under-fin cavity with a stressed material.