One illustrative method disclosed includes, among other things, forming a first dielectric layer and forming first and second conductive structures comprising cobalt embedded in the first dielectric layer. A second dielectric layer is formed above and contacting the first dielectric layer. The first and second dielectric layers comprise different materials, and a portion of the second dielectric layer comprises carbon or nitrogen. A first cap layer is formed above the first and second conductive structures and the second dielectric layer.