Quantum transport and unified scaling law in graphene with polyadic Cantor electrostatic barriers

Abstract:We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$--$n$--$n$, $n$--$p$--$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

Mesoscale and Nanoscale Physics (cond-mat.mes-hall); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2608.28398 [cond-mat.mes-hall]

(or arXiv:2608.28398v1 [cond-mat.mes-hall] for this version)

https://doi.org/10.48550/arXiv.2608.28398 arXiv-issued DOI via DataCite (pending registration)

From: Bhabani Prasad Mandal Prof. [view email] [v1] Fri, 28 Aug 2026 14:51:41 UTC (15,218 KB)