Shamith U. Payagala

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 EΩ, 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 ν=2 plateau (RH≈ 12 906.4 Ω) 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.

Advances in the development of graphene-based technology have enabled improvements in DC resistance metrology. Devices made from epitaxially grown graphene have replaced the GaAs-based counterparts, leading to an easier and more accessible realization of the ohm. By optimizing the scale of the growth, it has become possible to fabricate quantized Hall array resistance standards (QHARS) with nominal values between 1 kΩ and 1.29 MΩ. One of these QHARS device designs accommodates a value of about 1.01 MΩ, which made it an ideal candidate to pursue a proof-of-concept that graphene-based QHARS devices are suitable for forming wye-delta resistance networks. In this work, the 1.01 MΩ array output nearly 20.6 MΩ due to the wye-delta transformation, which itself is a special case of star-mesh transformations. These mathematical equivalence principles allow one to extend the QHR to the 100 MΩ and 10 GΩ resistance levels with fewer array elements than would be necessary for a single array with many more elements in series. The 1.01 MΩ device shows promise that the wye-delta transformation can shorten the calibration chain, and, more importantly, provide a chain with a more direct line to the quantum SI.

A precise high resistance comparison was performed between the traveling dual source bridge developed by the National Metrology Institute of Japan (NMIJ) and the adapted Wheatstone bridge of the National Institute of Standards and Technology (NIST) from 10 MΩ to 100 TΩ at NIST. The NMIJ traveling bridge was shipped to NIST Gaithersburg and was installed right next to the NIST bridge and these bridges alternately measured the resistance ratio of the high resistance standards, without moving the location of the resistors inside the temperature controlled air-bath. Having the bridges and resistance standards in the same location for the comparison decreased the transportation and temperature coefficient effects on the resistance standards, contributing to the excellent agreement of the measured values. The NMIJ traveling bridge used an 8.5-digit digital multimeter and a relay switch box to determine a resistance ratio by measuring the ratio of the voltages applied to the resistors. The comparison was started from 10 MΩ based on the same 1 MΩ standard resistor calibrated using a NIST two-terminal cryogenic current comparator bridge, and standard resistors from 10 MΩ to 100 TΩ were calibrated by repeating 10:1 scaling measurements with both systems. Excellent agreement was obtained within the uncertainty of all resistance ranges and the difference between both systems was less than 1 μΩ Ω⁻¹ up to 1 TΩ and the degrees of equivalence for 10 TΩ and 100 TΩ were less than 6 μΩ Ω⁻¹.

Since 2017, epitaxial graphene has been the base material for the US national standard for resistance. A future avenue of research within electrical metrology is to remove the need for strong magnetic fields, as is currently the case for devices exhibiting the quantum Hall effect. The quantum Hall effect is just one of many research endeavours that revolve around recent quantum physical phenomena like composite fermions, charge density waves, and topological properties [1-2]. New materials, like magnetically doped topological insulators (MTIs), offer access to the quantum anomalous Hall effect, which in its ideal form, could become a future resistance standard needing only a small permanent magnet to activate a quantized resistance value [3-5]. Furthermore, these devices could operate at zero-field for measurements, making the dissemination of the ohm more economical and portable. Here we present results on precision measurements of the h/e2 quantized plateau of Cr-Doped (BixSb1-x)2Te3 and give them context by comparing them to modern graphene-based resistance standards. Ultimately, MTI-based devices could be combined in a single system with magnetic-field-averse Josephson voltage standards to obtain an alternative quantum current standard.

We report the fabrication and measurement of top gated epitaxial graphene p-n junctions where exfoliated hexagonal boron nitride (h-BN) is used as the gate dielectric. The four-terminal longitudinal resistance across a single junction is well quantized at the von Klitzing constant R_K with a relative uncertainty of 10-7. After the exploration of numerous parameter spaces, we summarize the conditions upon which these devices could function as potential resistance standards. Furthermore, we offer designs of programmable electrical resistance standards over six orders of magnitude by using external gating.

We tested a digital impedance bridge in a hybrid structure for comparison of a capacitor with a resistor where the impedance ratio was measured in two separate parts. The modulus of the impedance ratio was matched arbitrarily close to the input-to-output ratio, in magnitude, of a two-stage inductive voltage divider by adjusting the operating frequency of the bridge; the residual deviation between the two together with the phase factor of the impedance ratio was measured using a custom detection system based on a four-channel 24-bit digitizer. The ratio of the inductive voltage divider was calibrated, in situ, using a conventional four-arm bridge with two known capacitors. Fluctuations of the source voltages were largely removed through postprocessing of the digitized data, and the measurement results were limited by the digitizer error. We have achieved an overall bridge resolution and stability of 0.02 μF/F in 2 hours for measuring a 100 pF capacitor relative to a 12906 Ω resistor at 1233 Hz. The relative combined standard uncertainty (k = 1) is 0.13 μF/F, dominated by the digitizer error.

A graphene quantized Hall resistance (QHR) device fabricated at the National Institute of Standards and Technology, Gaithersburg, MD, USA, was measured alongside a GaAs QHR device fabricated by the National Research Council of Canada, Ottawa, ON, Canada, by comparing them to a 1-kΩ standard resistor using a cryogenic current comparator. The two devices were mounted in a custom developed dual probe that was then assessed for its viability as a suitable apparatus for precision measurements. The charge carrier density of the graphene device exhibited controllable tunability when annealed after Cr(CO) ₃ functionalization. These initial measurement results suggest that making resistance comparisons is possible with a single- probe wired for two types of quantum standards-GaAs, the established material, and graphene - the newer material that may promote the development of more user-friendly equipment.