Conformal mapping and shot noise in graphene

TL;DR

This paper analytically investigates ballistic transport in undoped graphene billiards using conformal mapping, revealing pseudodiffusive and tunneling regimes, with conductance and shot noise behaviors depending on boundary conditions and system size.

Contribution

It introduces an analytical approach using conformal mapping to study transport regimes in graphene billiards, confirming results with alternative methods for various boundary conditions.

Findings

Pseudodiffusive conductance matches classical conductivity σ₀=4e²/πh.

Tunneling regime shows power-law decay of conductance and Poissonian shot noise.

Conductance in crossover region relates to Fano factor as G≈(1-F)×se²/h.

Abstract

Ballistic transport through a collection of quantum billiards in undoped graphene is studied analytically within the conformal mapping technique. The billiards show pseudodiffusive behavior, with the conductance equal to that of a classical conductor characterized by the conductivity $\sigma_0=4e^2/\pi h$, and the Fano factor $F=1/3$. By shrinking at least one of the billiard openings, we observe a tunneling behavior, where the conductance shows a power-law decay with the system size, and the shot noise is Poissonian (F=1). In the crossover region between tunneling and pseudodiffusive regimes, the conductance $G\approx (1-F)\times se^2/h$. The degeneracy $s=8$ for the Corbino disk, which preserves the full symmetry of the Dirac equation, $s=4$ for billiards bounded with smooth edges which break the symplectic symmetry, and $s=2$ when abrupt edges lead to strong intervalley scattering. An alternative, analytical or numerical technique, is utilized for each of the billiards to confirm the applicability of the conformal mapping for various boundary conditions.