One illustrative method disclosed herein includes forming a silicon/germanium fin in a layer of insulating material, wherein the fin has a first germanium concentration, recessing an upper surface of the layer of insulating material so as to expose a portion of the fin, performing an oxidation process so as to oxidize at least a portion of the fin and form a region in the exposed portion of the fin that has a second germanium concentration that is greater than the first germanium concentration, removing the oxide materials from the fin that was formed during the oxidation process and forming a gate structure that is positioned around at least the region having the second germanium concentration.