An interconnect structure on a substrate is provided. The interconnect structure comprises electrically conductive interconnect elements on at least two interconnect levels on or above a substrate level. In the interconnect structure of the invention, at least one electrically conductive via connects a first interconnect element on one interconnect level or on the substrate level to a second interconnect element on a different interconnect level. The via extends in a via opening of a first dielectric layer and comprises an electrically conductive via material that contains electrically conductive cylindrical carbon nanostructures. At least one cover-layer segment reaches into a lateral extension of the via opening and defines a via aperture that is small enough to prevent a penetration of the carbon nanostructures through the via aperture. This structure enhances control of carbon nanostructure growth in a height direction during fabrication of the interconnect structure.