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A unified realization of electrical quantities from the quantum International System of Units

Nature Electronics volume 8, pages 663–671 (2025)Cite this article

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Abstract

In the revised International System of Units (SI), the ohm and the volt are realized from the von Klitzing constant and the Josephson constant, and a practical realization of the ampere is possible by applying Ohm’s law directly to the quantum Hall and Josephson effects. As a result, it is possible to create an instrument capable of realizing all three primary electrical units, but the development of such a system remains challenging. Here we report a unified realization of the volt, ohm and ampere by integrating a quantum anomalous Hall resistor (QAHR) and a programmable Josephson voltage standard (PJVS) in a single cryostat. Our system has a quantum voltage output that ranges from 0.24 mV to 6.5 mV with combined relative uncertainties down to 3 μV V−1. The QAHR provides a realization of the ohm at zero magnetic field with uncertainties near 1 μΩ Ω−1. We use the QAHR to convert a longitudinal current to a quantized Hall voltage and then directly compare that against the PJVS to realize the ampere. We determine currents in the range of 9.33–252 nA, and our lowest uncertainty is 4.3 μA A−1 at 83.9 nA. For other current values, a systematic error that ranges from −10 μA A−1 to −30 μA A−1 is present due to the imperfect isolation of the PJVS microwave bias.

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Fig. 1: Illustration of a current measurement with the unified system.
Fig. 2: Validation of QAHR quantization.
Fig. 3: Direct realization of the ampere and contextualization.
Fig. 4: Time dependence of indirect realizations of the ampere and of QAHR longitudinal resistivity ρxx.

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Data availability

Data that support the findings of this study are included as Supplementary Information. Raw data files (for example, CCC output files) are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank T. Mai, F. Fei, G. J. Fitzpatrick and E. C. Benck for assistance with the NIST internal review process. We also thank I. T. Rosen and M. A. Kastner for enlightening discussions throughout this work. L.K.R., M.P.A. and D.G.-G. were supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary Research Program of the University Research Initiative (MURI) under grant no. FA9550-21-1-0429. At the initiation of the project, L.K.R., M.P.A. and D.G.-G. were supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract no. DE-AC02-76SF00515 and the Gordon and Betty Moore Foundation through grant no. GBMF9460. P.Z., L.T. and K.L.W. acknowledge support from the National Science Foundation (NSF) under the Accelerating Interdisciplinary Frontiers in Quantum Sciences and Technologies (grant no. 2125924, NRT-QISE) and Quantum Devices with Majorana Fermions in High-Quality Three-Dimensional Topological Insulator Heterostructure (grant no. 1936383, QII-TAQS) programmes. P.Z., L.T. and K.L.W. were also supported by the Army Research Office MURI under grant nos. W911NF16-1-0472 and W911NF-19-S-0008. Commercial equipment, instruments and materials are identified in this paper to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology or the US government, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose. Work presented herein was performed, for a subset of the authors, as part of their official duties for the US government. Funding is hence appropriated by the US Congress directly. Part of this work was performed at nano@stanford, supported by the National Science Foundation under grant no. ECCS-2026822.

Author information

Author notes

  1. These authors contributed equally: Linsey K. Rodenbach, Jason M. Underwood.

Authors and Affiliations

  1. Department of Physics, Stanford University, Stanford, CA, USA

    Linsey K. Rodenbach & David Goldhaber-Gordon

  2. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    Linsey K. Rodenbach, Molly P. Andersen & David Goldhaber-Gordon

  3. Physical Measurement Laboratory, National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA

    Jason M. Underwood, Ngoc Thanh Mai Tran, Alireza R. Panna, Zachary S. Barcikowski, Shamith U. Payagala, Dean G. Jarrett, Randolph E. Elmquist, David B. Newell & Albert F. Rigosi

  4. Joint Quantum Institute, University of Maryland, College Park, MD, USA

    Ngoc Thanh Mai Tran & Zachary S. Barcikowski

  5. Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA

    Molly P. Andersen

  6. Department of Electrical and Computer Engineering, University of California, Los Angeles, Los Angeles, CA, USA

    Peng Zhang, Lixuan Tai & Kang L. Wang

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Contributions

L.K.R., J.M.U., D.B.N., A.F.R. and D.G.-G. conceived and designed the experiments. L.K.R., J.M.U., N.T.M.T., A.R.P., M.P.A. and Z.S.B. performed the experiments. L.K.R., J.M.U., N.T.M.T., A.R.P., Z.S.B. and D.G.-G., analysed the data. P.Z., L.T. and K.L.W. contributed MTI thin film materials. L.K.R. and M.P.A. fabricated and qualified QAH devices. L.K.R., J.M.U., N.T.M.T., A.R.P., S.U.P., D.G.J, R.E.E., D.B.N., A.F.R. and D.G.-G. contributed specialized hardware and expertise to support metrology experiments. L.K.R. and J.M.U. wrote the paper. N.T.M.T., A.R.P., M.P.A., Z.S.B., A.F.R. and D.G.-G. reviewed and provided input on the paper.

Corresponding authors

Correspondence to Jason M. Underwood or David Goldhaber-Gordon.

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Extended data

Extended Data Fig. 1 Longitudinal resistivity as a function of top gate voltage.

Shown are measurements of QAHR longitudinal resistivity, ρxx, versus top gate voltage Vgate, performed with the CCC’s nanovoltmeter. First, Vgate was quickly swept toward negative voltages (blue circles, integration time τ = 10 s). Vgate was next brought back to zero and then swept toward positive voltages (orange squares, τ = 10 s). Each symbol represents an average of 30 observations per measurement. The time between measurements (symbols) was approximately 7 minutes. The curved arrows are intended as guides to the eye and show the sweep direction. Finally, Vgate was slowly swept over a narrow range (green crosses, τ = 60 s). For the narrow sweep, each symbol represents an average of 10 observations and the time between measurements was 12 minutes. The local minimum in ρxx seen near Vgate = 0 V indicates that the native Fermi level is well positioned near the center of the magnetic exchange gap. The mixing chamber temperature over the course of these sweeps (lasting about two hours total) only decreased by 0.2 mK. The decrease in ρxx near Vgate = 0 with each sweep suggests that either ρxx is strongly sensitive to temperature or that thermalization occurs over very long timescales.

Source data

Extended Data Fig. 2 Characterization of QAHR quantization in the presence of microwave leakage.

Deviation of the Hall resistance from quantization, δRyx/RK = (Ryx − RK)/RK, was measured as a function of bias current I and PJVS microwave bias. The impact of the microwave leakage varied with the microwave frequency f. For f = 0 (blue circles, microwave bias disabled), each symbol represents the average of at least three measurements, each based on at least 30 observations. For other frequencies only one measurement (again, based on at least 30 observations) was performed for each current. Error bars show the Type A (statistical) uncertainty. Note that the uncertainty for some points may be smaller than the symbol. For the f ≠ 0 measurements, the PJVS microwave excitation was active (input power P = 1 mW), but the PJVS output voltage was zero. The frequencies fn in the legend are as follows: f0 = 9.701 GHz (orange crosses), f1 = 18.00 GHz (yellow squares), and f2 = 14.00 GHz (violet diamonds).

Source data

Extended Data Fig. 3 Example raw measurement data of Vnull.

Shown are raw Vnull data taken for I = 83.9 nA. (a) 800 individual measurements of Vraw,null over several polarity reversal cycles are shown as red dots. Each raw measurement was acquired using a 60 s integration time. The triangular symbols show \({V}_{{\rm{null}}}^{+}\) (\({V}_{{\rm{null}}}^{-}\)), the average value of the raw null voltage readings for the positive (negative) portion of a polarity reversal cycle. (b) The 18 measurements of Vdiff calculated from the data in (a) via Eq. (4) are shown as cross-symbols. The solid green line is Vnull – the average of the 18 Vdiff values. The dashed line shows the corresponding null voltage expected from the indirect realization of the ampere, \({V}_{{\rm{null}},{\rm{expected}}}={I}_{\rm{indirect}}{R}_{K}-Nf{K}_{J}^{-1}\). (c) An expanded view of the two trendlines shown in (b).

Source data

Supplementary information

Supplementary Information

Supplementary discussion, Figs. 1–8 and Tables 1–3.

Supplementary Data 1

Source data for main article results.

Source data

Source Data Figs. 2–4

Tabular data in Excel format.

Source Data Extended Data Figs. 1 and 2

Tabular data in Excel format.

Source Data Extended Data Fig. 3

Tabular data in Excel format, sheets 1 and 2.

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Rodenbach, L.K., Underwood, J.M., Tran, N.T.M. et al. A unified realization of electrical quantities from the quantum International System of Units. Nat Electron 8, 663–671 (2025). https://doi.org/10.1038/s41928-025-01421-2

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