Relativistic particles in super-periodic potentials: exploring graphene and fractal systems

TL;DR

This paper investigates relativistic particles in super-periodic potentials using transfer matrix methods, analyzing their tunneling, transmission, and conductance properties, especially in graphene and fractal systems.

Contribution

It introduces a detailed analysis of relativistic particle behavior in super-periodic and fractal potentials, including analytical and numerical results for graphene and Cantor systems.

Findings

Relativistic particles exhibit Klein tunneling with higher reflection than non-relativistic particles.

Transmission probabilities show resonances depending on barrier number and super-periodicity.

Fractal systems display sharp transmission peaks and saturation effects at higher stages.

Abstract

In this article, we employ the transfer matrix method to investigate relativistic particles in super-periodic potentials (SPPs) of arbitrary order $n \in I^{+}$. We calculate the reflection and transmission probabilities for spinless Klein particles encountering rectangular potential barriers with super-periodic repetition. It is found that spinless relativistic particles exhibit Klein tunneling and a significantly higher degree of reflection compared to their non-relativistic counterparts. Additionally, we analytically explore the behavior of experimentally realizable massless Dirac electrons as they encounter rectangular potential barriers with a super-periodic pattern in a monolayer of graphene. In this system, the transmission probability, conductance, and Fano factor are evaluated as functions of the number of barriers, the order of super-periodicity, and the angle of incidence. Our findings reveal that the transmission probability shows a series of resonances that depend on the number of barriers and the order of super-periodicity. We extend our analysis to specific cases within the Unified Cantor Potentials (UCPs)-$\gamma$ system ($\gamma$ is a scaling parameter greater than $1$), focusing on the General Cantor fractal system and the General Smith-Volterra-Cantor (GSVC) system. For the General Cantor fractal system, we calculate the tunneling probability, which reveals sharp transmission peaks and progressively thinner unit cell potentials as $G$ increases. In the GSVC system, we analyze the potential segment length and tunneling probability, observing nearly unity tunneling coefficients when $\gamma \approx 1$, as well as saturation behavior in transmission coefficients at higher stages $G$.