Influences of the Minkowski-Bouligand Dimension on Graphene-Based Quantum Hall Array Designs

TL;DR

This paper investigates how the Minkowski-Bouligand fractal dimension influences the design of graphene-based quantum Hall array standards, proposing optimized recursive methods for resistance value tuning with fewer components.

Contribution

It introduces refined mathematical frameworks and partial recursion techniques based on fractal dimensions to improve QHARS device design and flexibility.

Findings

Partial recursions increase design flexibility.

Fractal dimension analysis guides device optimization.

Fewer devices needed for desired resistance values.

Abstract

This work elaborates on how one may develop high-resistance quantized Hall array resistance standards (QHARS) by using star-mesh transformations for element count minimization. Refinements are made on a recently developed mathematical framework optimizing QHARS device designs based on full, symmetric recursion by reconciling approximate device values with exact effective quantized resistances found by simulation and measurement. Furthermore, this work explores the concept of fractal dimension, clarifying the benefits of both full and partial recursions in QHARS devices. Three distinct partial recursion cases are visited for a near-1 Gigaohm QHARS device. These partial recursions, analyzed in the context of their fractal dimensions, offer increased flexibility in accessing desired resistance values within a specific neighborhood compared to full recursion methods, though at the cost of the number of required devices.