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This paper describes an onsite comparison of two different digital impedance bridges when performing measurements on a quantum Hall resistance standard with the purpose of realizing the SI unit of capacitance, the farad. In the EMPIR Joint Research Project 18SIB07 GIQS, graphene impedance quantum standards, the Physikalisch-Technische Bundesanstalt (PTB), Germany, developed a Josephson impedance bridge, and the Istituto Nazionale di Ricerca Metrologica (INRIM) and the Politecnico di Torino (POLITO), Italy, developed an electronic digital impedance bridge. The former is based on Josephson waveform generators and the latter on an electronic waveform synthesizer. The INRIM–POLITO impedance bridge was moved to PTB and the two bridges were compared by measuring both temperature-controlled standards and a graphene AC quantized Hall resistance (QHR) standard. The uncertainties for the calibration of 10 nF capacitance standards at 1233 Hz are within 1 × 10−8 for the PTB’s bridge and around 1 × 10−7 for the INRIM–POLITO’s bridge. The comparison mutually validates the two bridges within the combined uncertainty. The result confirms that digital impedance bridges allow the realization of the SI farad from the QHR with uncertainties comparable with the best calibration capabilities of the BIPM and the major National Metrology Institutes.

Fully digital impedance bridges are emerging as measuring instruments for primary electrical impedance metrology and the realization of impedance units and scales. This article presents a comprehensive analysis of electronic fully digital impedance bridges for both generating (based on digital-to-analog converters) and digitizing (based on analog-to-digital converters) bridges. The sources of measurement error are analyzed in detail and expressed by explicit mathematical formulas ready to be applied to the specific bridge and measurement case of interest. The same can be employed also as a basis to optimize the design and the operating parameters of digital bridges and evaluate the measurement uncertainty. A practical application of the analysis to the digital bridges developed and measurements performed in the framework of an international research project is presented.

Measurements of fractional multiples of the ν = 2 plateau quantized Hall resistance (R(H) ≈ 12906 Ω) were enabled by the utilization of multiple current terminals on millimetre-scale graphene p-n junction devices fabricated with interfaces along both lateral directions. These quantum Hall resistance checkerboard devices have been demonstrated to match quantized resistance outputs numerically calculated with the LTspice circuit simulator. From the devices’ functionality, more complex embodiments of the quantum Hall resistance checkerboard were simulated to highlight the parameter space within which these devices could operate. Moreover, these measurements suggest that the scalability of p-n junction fabrication on millimetre or centimetre scales is feasible with regards to graphene device manufacturing by using the far more efficient process of standard ultraviolet lithography.

A mathematical approach is introduced for predicting quantized resistances in graphene p-n junction devices that utilize more than a single entry and exit point for electron flow. Depending on the configuration of an arbitrary number of terminals, electrical measurements yield nonconventional, fractional multiples of the typical quantized Hall resistance at the v = 2 plateau (R(H) ≈ 12906 Ω) and take the form: [Formula: see text]. This theoretical formulation is independent of material, and applications to other material systems that exhibit quantum Hall behaviors are to be expected. Furthermore, this formulation is supported with experimental data from graphene devices with multiple source and drain terminals.

We have demonstrated the millimeter-scale fabrication of monolayer epitaxial graphene p-n junction devices using simple ultraviolet photolithography, thereby significantly reducing device processing time compared to that of electron beam lithography typically used for obtaining sharp junctions. This work presents measurements yielding nonconventional, fractional multiples of the typical quantized Hall resistance at ν = 2 ( R H ≈ 12906 Ω) that take the form: a b R H . Here, a and b have been observed to take on values such 1, 2, 3, and 5 to form various coefficients of RH. Additionally, we provide a framework for exploring future device configurations using the LTspice circuit simulator as a guide to understand the abundance of available fractions one may be able to measure. These results support the potential for drastically simplifying device processing time and may be used for many other two-dimensional materials.

Quantum Hall array resistance standards (QHARSs) are integrated circuits composed of interconnected quantum Hall effect elements that allow the realization of virtually arbitrary resistance values. In recent years, techniques were presented to efficiently design QHARS networks. An open problem is that of the evaluation of the accuracy of a QHARS, which is affected by contact and wire resistances. In this work, we present a general and systematic procedure for the error modelling of QHARSs, which is based on modern circuit analysis techniques and Monte Carlo evaluation of the uncertainty. As a practical example, this method of analysis is applied to the characterization of a 1 MΩ QHARS developed by the National Metrology Institute of Japan. Software tools are provided to apply the procedure to other arrays.

We present here a comprehensive analysis of the error sources in electronic fully-digital impedance bridges, for both sourcing and digitizing bridges. This work can be used as a basis to optimize the design and the operating parameters of digital bridges, and in the evaluation of the uncertainty.

GIQS: Graphene Impedance Quantum Standard is a Joint Research Project of the European Metrology Programme for Innovation and Research (EMPIR). The project objective is to combine novel digital impedance measurement bridges with graphene-based ac quantum Hall resistance standards in a simplified cryogenic environment, to achieve simple, user-friendly quantum impedance standards suitable for primary realisation of impedance units in national metrology institutes, calibration centers, and the industry.

The quantum Hall effect is the basis for the realisation of the resistance and impedance units in the International System of units since 2019. This paper describes a cryogenic probe that allows to set graphene Hall devices in quantisation conditions in a helium bath (4.2 K) and magnetic fields up to 6 T, to perform precision measurements in the AC regime with impedance bridges. The probe has a full coaxial wiring, isolated from the probe structure, and holds the device in a TO-8 socket. First, characterization experiments are reported on a GaAs device, showing quantisation at 5.5 T. In the AC regime, multiple-series connections will be employed to minimize the residual error, quantified by electrical modelling of the probe.

This paper presents the implementation of an electronic fully-digital impedance bridge optimized for RC comparisons with equal impedance magnitudes, together with an evaluation of the uncertainty. This bridge has been designed with the goal of realizing the farad directly from the quantum Hall effect with a bridge uncertainty component at the 10−7 level. Thanks to its simple design, ease of operation and affordability, this bridge is suitable to be industrially manufactured. Together with the increasing availability of graphene quantum Hall resistance standards, this can provide an affordable quantum realization of the unit farad for metrology institutes and calibration centres. In this paper we present the uncertainty budget of an example measurement and the results of the validation of the bridge against a suitably modified version of the traceability chain of the Italian national standard of capacitance. The combined uncertainty of the bridge resulted from repeated measurements (overall measurement time of about 200 min) is 9.2 × 10−8, suitable for the primary realization of the unit of capacitance from a quantized Hall resistance standard. The crosstalk among the channels of the electrical generator is the most significant uncertainty component, possibly reducible with internal shielding and filtering of the electronic generator.

Measurements of fractional multiples of the ν = 2 plateau quantized Hall resistance ( R H ≈ 12 906 Ω) were enabled by the utilization of multiple current terminals on millimetre-scale graphene p–n junction (pnJ) devices fabricated with interfaces along both lateral directions. These quantum Hall resistance checkerboard devices have been demonstrated to match quantized resistance outputs numerically calculated with the LTspice circuit simulator. From the devices’ functionality, more complex embodiments of the quantum Hall resistance checkerboard were simulated to highlight the parameter space within which these devices could operate. Moreover, these measurements suggest that the scalability of pnJ fabrication on millimetre or centimetre scales is feasible with regards to graphene device manufacturing by using the far more efficient process of standard ultraviolet lithography.

The core objective of EMPIR project 17RPT04 VersICaL is to improve the European measurement infrastructure for electrical impedance, with particular emphasis on the capabilities of developing NMIs and calibration centres. The project will seek to exploit the results of existing research on digital impedance bridges (DIBs) by designing, constructing and validating simple, affordable versions suitable to realise the impedance scale in the range 1 nF to 10 μF and 1 mH to 10 H with relative uncertainties in the range 10-5 to 10-6. The first results of the research project, including the bridge designs and details of a polyphase digitally synthesized multichannel source capable of providing voltage outputs of precise ratio and phase are presented.

This work presents precision measurements of quantized Hall array resistance devices using superconducting, crossover-free, multiple interconnections as well as graphene split contacts. These new techniques successfully eliminate the accumulation of internal resistances and leakage currents that typically occur at interconnections and crossing leads between interconnected devices. As a result, a scalable quantized Hall resistance array is obtained with a nominal value that is as precise and stable as that from single-element quantized Hall resistance standards.

In this paper, we show that quantum Hall resistance measurements using two terminals may be as precise as four-terminal measurements when applying superconducting split contacts. The described sample designs eliminate resistance contributions of terminals and contacts such that the size and complexity of next-generation quantized Hall resistance devices can be significantly improved.

A mathematical approach is introduced for predicting quantized resistances in graphene p-n junction devices that utilize more than a single entry and exit point for electron flow. Depending on the configuration of an arbitrary number of terminals, electrical measurements yield nonconventional, fractional multiples of the typical quantized Hall resistance at the ν = 2 plateau ( R H ≈ 12906 Ω) and take the form: a b R H . This theoretical formulation is independent of material, and applications to other material systems that exhibit quantum Hall behaviors are to be expected. Furthermore, this formulation is supported with experimental data from graphene devices with multiple source and drain terminals.

The use of multiple current terminals on millimeter-scale graphene p-n junction devices fabricated with Corbino geometries, or quantum Hall resistance dartboards, has enabled the measurement of several fractional multiples of the quantized Hall resistance at the ν = 2 plateau (RH ≈ 12 906 Ω). Experimentally obtained values agreed with the corresponding numerical simulations performed with the LTspice circuit simulator. More complicated designs of the quantum Hall resistance dartboard were simulated to establish the potential parameter space within which these Corbino-type devices could output resistance. Most importantly, these measurements support simpler processes of ultraviolet lithography as a more efficient means of scaling up graphene-based device sizes while maintaining sufficiently narrow junctions.

We describe a resistance ratio bridge suitable for calibrating a four-terminal resistor of value close to the quantum Hall resistance. The bridge includes three quantum Hall elements, wired with multiple connections. The unique properties of the quantum Hall effect provide a Kelvin-like combining network and the rejection of wiring stray resistances. The bridge has a simple schematic and does not include any adjustable element: its reading is the deviation from equilibrium, measured as a small voltage ratio. Circuit network analysis predicts that a basic implementation of the bridge can reach a calibration accuracy in the 10−8 range.

It has already been reported that 1/f noise in graphene can be dominated by fluctuations of charge carrier mobility. We show here that the increasing damage induced by oxygen plasma on graphene samples result in two trends: at low doses, the magnitude of the 1/f noise increases with the dose; and at high doses, it decreases with the dose. This behaviour is interpreted in the framework of 1/f noise generated by carrier mobility fluctuations where the concentration of mobility fluctuation centers and the mean free path of the carriers are competing factors.

GRACE - Developing electrical characterisation methods for future graphene electronics is a 2016 Normative Joint Research Project of the European Metrology Programme for Innovation and Research. The project focuses on the measurement of the electrical properties of graphene. Its objectives are to develop validated measurement methods and protocols, including fast-throughput examples. The work is performed in collaboration with international standardisation committees, with an aim to initiate and develop dedicated documentary standards.

The utilization of multiple current terminals on millimeter-scale graphene p–n junction devices has enabled the measurement of many atypical, fractional multiples of the quantized Hall resistance at the ν = 2 plateau (RH ≈ 12 906 Ω). These fractions take the form abRH and can be determined both analytically and by simulations. These experiments validate the use of either the LTspice circuit simulator or the analytical framework recently presented in similar work. Furthermore, the production of several devices with large-scale junctions substantiates the approach of using simple ultraviolet lithography to obtain junctions of sufficient sharpness.

The unique properties of the quantum Hall effect allow one to revisit traditional measurement circuits with a new flavour. In this paper we present the first realization of a quantum Hall Kelvin bridge for the calibration of standard resistors directly against the quantum Hall resistance. The bridge design is particularly simple and requires a minimal number of instruments. The implementation here proposed is based on the bridge-on-a-chip, an integrated circuit composed of three graphene quantum Hall elements and superconducting wiring. The accuracy achieved in the calibration of a 12 906Ω standard resistor is of a few parts in 10(8), at present mainly limited by the prototype device and the interferences in the current implementation, with the potential to achieve few parts in 10(9), which is the level of the systematic uncertainty of the quantum Hall Kelvin bridge itself.