Quantum transport in self-similar graphene carpets

Fractals, a fascinating mathematical concept made popular in the 1980s, has remained for decades mainly a beautiful scientific curiosity. 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 zero-energy modes. Despite the vanishing velocity of these states, a supermetallic phase is revealed at the neutrality point with a conductivity that coincides within a few percent with sigma(0) = 4e(2)/pi h. For Fermi energy located in the valence or conduction bands and in the presence of a small inelastic scattering rate, the system stays metallic and the transport appears to be strongly anisotropic.