The role of fractal aperiodic order in the transmittance, conductance and electronic structure of graphene-based systems

• We study the transmittance, conductance and electronic structure of aperiodic Cantor system in graphene.
• The conductance curves exhibit symmetrical and self-similar features as a function of the generation number.
• The conductance presents an oscillatory behaviour that can be described directly by means of the bound states.
• We obtain clusters of three subbands that are degenerated and occluded-degenerated.
• The geometrical characteristics of the aperiodic Cantor structures are manifested in the physical properties.

In this work we investigate the peculiar tunneling characteristics of Dirac electrons through quasi-regular or aperiodic single-layer graphene structures. In particular, we have implemented the aperiodic order to the barriers and potential wells arranged according to the Cantor sequence. We deal with two types of quasi-regular systems: the first one Electrostatic Cantor Graphene Structures (ECGSs), structures formed with electrostatic potentials, and the second one Substrate Cantor Graphene Structures (SCGSs), obtained by alternating substrates that can open and non-open, such as SiC and SiO2, an energy bandgap on graphene. We have used the transfer matrix method to compute the transmission probability, linear-regime conductance and spectrum of bound states of both systems, which are calculated for different combinations of the fundamental parameters: starting width, sequence generation number, electron incident energy and electron incident angle. The transmission spectra of ECGSs and SCGSs show self-similar patterns, which are more remarkable when the generation number increases. The conductance of ECGSs becomes smoother as the generation increases. Furthermore, in the case of SCGSs the conductance exhibits self-similar features as a function of the generation number. In general, the conductance of both systems presents an oscillatory behaviour that can be described directly by means of the bound states.