In this paper, we investigated the Fano factor in two types of self-similar potential structures in a graphene monolayer. These structures are substrate-based potential and electrostatic-based potential. However, in order to determine the Fano factor in such structures, we solved the Dirac Hamiltonian by using the transfer matrix method. We found that the self-similar substrate-based potential structure manifests a self-similar behavior in the Fano factor and conductance. Therefore, we proposed scaling rules that represent a scale invariance between generations, main barrier heights, and total lengths of the structures. In particular, the maximum Fano factor value was reported for the self-similar electrostatic-based potentials. More analysis was given in terms of the generation, main barrier heights, and structure’s total lengths. These kinds of structures could be used to control the Fano factor.
Our study investigated the emergence of spatial Quasi-Bound States (QBSs) in graphene mono- layers induced by rectangular potential barriers. By solving the time-independent Dirac equation and using the transfer matrix formalism, we calculated the resonance energies and identify the QBSs based on probability density functions (PDF). We analyzed two types of structures: single and dou- ble barriers, and we find that the QBSs are located within the barrier region, at energies higher than the single barrier. Additionally, we observe QBSs in the double barrier and their position depends on the distance and width of the well between the two barriers. The width and height of the barrier significantly impact the QBSs while the well width influences the resonance energy levels of the QBSs in the double barrier. Interestingly, the QBSs can be manipulated in the graphene system, offering potential for optoelectronic devices. Finally, our results demonstrated that the spatial localization of these states is counter-intuitive and holds great promise for future research in optolectronic devices.
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