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Universal quantum transport and impurity band super metallicity in self-similar graphene carpets

Fractals, a fascinating mathematical concept made popular in the eighties, remained for decades a beautiful scientific curiosity mainly. With the tremendous advances in nanofabrication techniques, such as nanolithography, it has become possible to design self-similar materials with fine structures down to nanometer scale. Here, we investigate the effects of self similarity on quantum electronic transport in graphene Sierpinski carpets. We find that a gap opens up in the electron spectrum in the middle of which lies a flat band of zeros energy modes. Although these states have a zero velocity, a supermetallic phase is found at the neutrality point. For Fermi energy located in the valence/conduction band and in the presence of a small inelastic scattering the system stays metallic and the transport is found strongly anisotropic.