Methods of making FinFET device comprising a piezoelectric liner for generating a surface charge

One illustrative method disclosed herein includes, among other things, forming a conformal piezoelectric material liner layer on at least the opposing lateral sidewalls of a fin, forming a recessed layer of insulating material on opposite sides of the fin and on the conformal piezoelectric material liner layer, removing portions of the conformal piezoelectric material liner layer positioned above the recessed layer of insulating material to thereby expose a portion of the fin above the recessed upper surface, and forming a gate structure above the recessed layer of insulating material and around a portion of the fin positioned above the recessed upper surface.