Self-similar Charge Transport in Gapped Graphene

TL;DR

This paper investigates a self-similar potential in graphene, revealing a unique self-similar transmission pattern for charge carriers described by the Dirac equation, unlike in traditional semiconductors.

Contribution

It introduces a novel self-similar potential based on the Cantor set for graphene and analyzes its unique self-similar charge transport properties.

Findings

Self-similar transmission patterns in graphene with the new potential.

Scaling properties depend on barrier height, system length, and generation number.

Analytic expressions for the scaling behavior are derived.

Abstract

A new type of self-similar potential is used to study a multibarrier system made of graphene. Such potential is based on the traditional middle third Cantor set rule combined with a scaling of the barriers height. The resulting transmission coefficient for charge carriers, obtained using the quantum relativistic Dirac equation, shows a surprising self-similar structure. The same potential does not lead to a self-similar transmission when applied to the typical semiconductors described by the non-relativistic Schr\"odinger equation. The proposed system is one of the few examples in which a self-similar structure produces the same pattern in a physical property. The resulting scaling properties are investigated as a function of three parameters: the height of the main barrier, the total length of the system and the generation number of the potential. These scaling properties are first identified individually and then combined to find general analytic scaling expressions.