Self-similarity of quantum transport in graphene using electrostatic gate and substrate

A particular design for multibarrier structure in graphene, yielding a self-similar transport response, is proposed. The potential profile is based on rectangular wells and barriers, generated according independent nth order scaling laws for their heights and widths. The barriers are constructed by means of two distinct approaches (electrostatic or substrate). Dirac equation and transfer matrix approach are used to calculate transmission properties which, in turn, allow to evaluate the conductance via Landauer–Büttiker formalism. It is found that self-similarity with determined scaling rules between nth and (n+1)th generations of transport properties appears when the order of generating laws is equal or greater than n = 7. Our proposal would be the first in which the self-similarity property is transferred from geometry to the spectrum, and consequently, to the transport properties of a quantum heterostructure. Possible ways of practical realization for the proposed structures are commented.