Electronic properties of Cantor random box distribution of impurities in graphene

• We study graphene with on-site local impurities with a Cantor set random distribution.
• T-matrix approximation for the Green function is applied for different values of the iteration parameter of the Cantor set.
• The electronic density of states is obtained for different values of the iteration parameter of the Cantor set.
• We study the spectrum rearrangement and the Cantor set signatures in the electronic density of states.
• We compute the electronic density of states in the infinite limit for the iteration parameter.

The aim of this work is to study the electronic properties of graphene under random impurities which are distributed in the energy line following the Cantor set box distribution. This implies that for each iteration k, the possible energy values of the random impurities lie in the line segment of the Cantor set in the interval (−α/2,α/2). By applying the full T-matrix approximation, the electronic density of states is obtained for each iteration k and the limit k → ∞ limit is taken. A metal-insulator transition is obtained for critical values of α, where a resonance peak in the DOS at the Fermi level is split in two bands that shift towards the band edges when the width α increases. In turn, the electronic density of states for k ≥ 2 only enhance the van Hove singularities, resonant and antiresonant states for k = 2. In the other side, the Cantor set signatures are shown through a spectrum rearrangement for different values of α, where resonant states split in two narrow peaks for k = ∞. These results are important to study the transport properties in graphene with doped-based fractal superlattices, magnetic or electric barriers or multilayers with triadic patterns.