One method disclosed includes, among other things, covering the top surface and a portion of the sidewalls of an initial fin structure with etch stop material, forming a sacrificial gate structure around the initial fin structure, forming a sidewall spacer adjacent the sacrificial gate structure, removing the sacrificial gate structure, with the etch stop material in position, to thereby define a replacement gate cavity, performing at least one etching process through the replacement gate cavity to remove a portion of the semiconductor substrate material of the fin structure positioned under the replacement gate cavity that is not covered by the etch stop material so as to thereby define a final fin structure and a channel cavity positioned below the final fin structure and substantially filling the channel cavity with a stressed material.