Implementing Pseudofractal Designs in Graphene-Based Quantum Hall Arrays using Minkowski-Bouligand Algorithms

This work introduces a pseudofractal analysis for optimizing high-resistance graphene-based quantized Hall array resistance standards (QHARS). The development of resistance standard device designs through star-mesh transformations is detailed, aimed at minimizing element count. Building on a recent mathematical framework, the approach presented herein refines QHARS device concepts by considering designs incorporating pseudofractals (which may be expressed as star-mesh transformations). To understand how future QHARS pseudofractal designs enable varying sizes of neighborhoods of available quantized resistance, Minkowski-Bouligand algorithms are used to analyze fractal dimensions of the device design topologies. Three distinct partial recursion cases are explored in addition to the original full recursion design, and expressions for their total element counts are derived. These partial recursions, assessed through their fractal dimensions, offer enhanced flexibility in achieving specific resistance values within a desired neighborhood compared to full recursion methods, albeit with an increased number of required elements. The formalisms presented are material-independent, making them broadly applicable to other quantum Hall systems and artifact standards.