One illustrative device disclosed herein includes a plurality of fins separated by a trench formed in a semiconducting substrate, a first layer of insulating material positioned in the trench, the first layer of insulating material having an upper surface that is below an upper surface of the substrate, an isolation layer positioned within the trench above the first layer of insulating material, the isolation layer having an upper surface that is below the upper surface of the substrate, a second layer of insulating material positioned within the trench above the isolation layer, the second layer of insulating material having an upper surface that is below the upper surface of the substrate, and a gate structure positioned above the second layer of insulating material.