Fractal-like star-mesh transformations using graphene quantum Hall arrays

TL;DR

This paper introduces a mathematical framework for designing fractal-like graphene quantum Hall arrays using star-mesh transformations, enabling high resistance values with minimal device elements, suitable for practical fabrication.

Contribution

It extends star-mesh transformations to fractal device designs, reducing the number of elements needed for high resistance in graphene quantum Hall arrays.

Findings

Mathematically, fewer than 200 elements suffice for high resistance

Epitaxial graphene at nu=2 plateau used as a building block

Fractal-like designs achieve resistances around 1 EΩ

Abstract

A mathematical approach is adopted for optimizing the number of total device elements required for obtaining high effective quantized resistances in graphene-based quantum Hall array devices. This work explores an analytical extension to the use of star-mesh transformations such that fractal-like, or recursive, device designs can yield high enough resistances (like 1 E{\Omega}, arguably the highest resistance with meaningful applicability) while still being feasible to build with modern fabrication techniques. Epitaxial graphene elements are tested, whose quantized Hall resistance at the nu=2 plateau (R_H = 12906.4 {\Omega}) becomes the building block for larger effective, quantized resistances. It is demonstrated that, mathematically, one would not need more than 200 elements to achieve the highest pertinent resistances