One method includes forming a barrier layer in a trench/opening in an insulating material, forming a first region of a copper material above the barrier layer, forming a metal layer in the trench/opening on the first region of copper material, forming a second region of copper material on the metal layer, performing at least one CMP process to remove any materials positioned above a planarized upper surface of the layer of insulating material outside of the trench/opening so as to thereby define a structure comprised of the metal layer positioned between the first and second regions of copper material, forming a dielectric cap layer above the layer of insulating material and above the structure, and performing a metal diffusion anneal process to form a metal cap layer adjacent at least the upper surface of a conductive copper structure.