Realization of the quantum ampere using the quantum anomalous Hall and
Josephson effects

Linsey K. Rodenbach1,2,3†, Ngoc Thanh Mai Tran3,4†, Jason M. Underwood3†,
Alireza R. Panna3, Molly P. Andersen2,5, Zachary S. Barcikowski3,4, Shamith U.
Payagala3, Peng Zhang6, Lixuan Tai6, Kang L. Wang6, Randolph E. Elmquist3,
Dean G. Jarrett3, David B. Newell3, Albert F. Rigosi3 and David Goldhaber-
Gordon1,2a

1Department of Physics, Stanford University, Stanford, California 94305, USA
2Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
3Physical Measurement Laboratory, National Institute of Standards and
Technology (NIST), Gaithersburg, Maryland 20899, USA
4Joint Quantum Institute, University of Maryland, College Park, Maryland
20742, USA
5Department of Materials Science and Engineering, Stanford University,
Stanford, California 94305, USA
6Department of Electrical and Computer Engineering, Department of Physics and
Astronomy, University of California, Los Angeles, California 90095, USA
†These authors contributed equally to this work.
aTo whom correspondence should be addressed; E-mail: goldhaber-
<EMAIL_ADDRESS>
By directly coupling a quantum anomalous Hall resistor to a programmable
Josephson voltage standard, we have implemented a quantum current sensor (QCS)
that operates within a single cryostat in zero magnetic field. Using this QCS
we determine values of current within the range 9.33 nA – 252 nA, providing a
realization of the ampere based on fundamental constants and quantum
phenomena. The relative Type A uncertainty is lowest, 2.30$\times$10-6 A/A, at
the highest current studied, 252 nA. The total root-sum-square combined
relative uncertainty ranges from 3.91 $\times$10-6 A/A at 252 nA to 41.2
$\times$10-6 A/A at 9.33 nA. No DC current standard is available in the
nanoampere range with relative uncertainty comparable to this, so we assess
our QCS accuracy by comparison to a traditional Ohm’s law measurement of the
same current source. We find closest agreement (1.46 $\pm$ 4.28)$\times$10-6
A/A for currents near 83.9 nA, for which the highest number of measurements
were made.

## I. Introduction

In 2019, the International System of Units (SI) was redefined to link the
seven base units to physical constants in nature [1, 2, 3]. Under the new SI,
units are defined solely through their relation to fixed-value fundamental
constants. This redefinition removed from unit definitions all volatile
artifacts (objects such as a 100 $\Omega$ resistor or 1 kg mass) and endorsed
seeking practical _realizations_ of base and derived units via quantum
phenomena [3, 4].

Within the field of metrology, ‘realization of a unit’ refers to a process by
which a value and associated uncertainty of a quantity are established in a
manner consistent with the definition of the unit [5]. A key requirement is
that the quantity being evaluated needs to be of the same kind as the unit
being realized. For example, an electrical current is a quantity of the same
kind as the unit ampere, so assigning a value and uncertainty to an electrical
current can realize the ampere so long as the process used to do so is
consistent with the ampere’s definition. This stands in contrast to providing
‘reference to a unit’, where the process used to establish a quantity’s value
is either not consistent with the definition of the unit, or uses one or more
transfer standards (artifacts which were previously calibrated against a
realization).

Prior to the 2019 redefinition, the ohm ($\Omega$) and the volt (V) were
referenced by precisely measuring the quantum Hall (QH) effect and Josephson
effect. However, these quantum effects could not realize their respective SI
derived units due to the stringent definitions for the SI base units the
ampere (A) and the kilogram (kg) (recall the unit relations: V =
kg$\cdot$m2$\cdot$s-3$\cdot$A-1 and $\Omega=$
kg$\cdot$m2$\cdot$s-3$\cdot$A-2). Regardless, precision measurements of the QH
effect and Josephson effect were used to maintain resistance [6, 7, 8, 9, 10]
and voltage [11, 12, 13] artifact standards [14]. These artifact standards
were in turn used to maintain electrical current [4] artifact standards, via
the unit relation A = V/$\Omega$ (Ohm’s law). This convoluted approach made
traceability of measurements burdensome.

Single-electron tunneling (SET) devices can function as excellent realizations
and standards in the sub-nanoampere range [15, 16, 17, 18, 19, 20, 21, 22],
but the calibration of higher currents has traditionally relied on applying
Ohm’s law to secondary voltage and resistive standards, traceably linked to
$K_{\mathrm{J}}$ and $R_{\mathrm{K}}$ respectively. The resultant long
calibration chains frustrate efforts to reduce uncertainty below a few parts
in $10^{6}$ [4, 23]. The new SI has opened a path for realizing the ampere in
higher current ranges. The ohm and the volt are now practically realized from
the von Klitzing constant $R_{\mathrm{K}}=h/e^{2}$ and the Josephson constant
$K_{\mathrm{J}}=2e/h$ through the QH effect and Josephson effect,
respectively. With $R_{\mathrm{K}}$ and $K_{\mathrm{J}}$ fixed and the
definition of the ampere no longer based on a singular pre-defined
experimental implementation of Ampere’s force law, a practical realization of
the ampere is now possible through application of Ohm’s law to the QH effect
and the Josephson effect [3, 24, 25, 26]. Recently, Brun-Picard et al. [27]
combined a quantum Hall resistance standard with a Josephson voltage standard
and a superconducting cryogenic amplifier to produce a current standard based
directly on primary quantum phenomena. However, due to differing magnetic
field requirements for the QH resistance standard and Josephson voltage
standard, the two quantum standards were operated in separate cryostats.
Despite this heavy infrastructural requirement, current realization with a
relative uncertainty on the order of 1 part in $10^{8}$ was demonstrated over
a range of 1 $\mu$A up to 10 mA. However, realization of the ampere for
currents between 1 nA and 1 $\mu$A has remained elusive.

Here we report the construction of a zero-field quantum current sensor (QCS)
based on direct integration of a quantum anomalous Hall (QAH) resistor and a
programmable Josephson voltage standard (PJVS) within a single cryostat. This
direct integration is made possible by the zero-field quantization of the Hall
resistivity ($\rho_{yx}=R_{yx}=R_{\mathrm{K}}$) characteristic of the QAH
state [28, 29, 30], which in recent years has been shown to be a viable and
attractive alternative to traditional QH resistance standards [31, 32, 33,
34]. The QCS is aimed at realizing the ampere and reducing the uncertainty of
current detection in a critical metrological parameter space that lies outside
ranges addressed with the methods outlined above. We confirm the accuracy of
our QCS for currents in the range of 9.33 nA – 252 nA via a comparison to a
more traditional Ohm’s-law measurement of the current. Using this first
iteration of the QCS, we find that our uncertainty of detection, for all
tested values of current, meets or exceeds that offered by the best
calibration and measurement capabilities reported by national metrology
institutes worldwide. In addition to reducing uncertainty of current detection
in a range heavily utilized by the nanofabrication industry [35], integration
into one cryostat opens the door to a combined realization of the ampere,
volt, and ohm in a single apparatus.

## II. Experimental Design and Implementation

Figure 1: Depiction of a current measurement using the QCS. Left: Simplified
schematic for the zero-field quantum current sensor. The current source $I$
and the nanovoltmeter are external to the cryostat and at room temperature. To
provide consistency between different types of measurements, all currents were
generated using the cryogenic current comparator’s digital current source (see
the last paragraph of Section II for details). Right: Optical micrograph of
Cr-doped (Bi,Sb)2Te3 QAH device.

A schematic depicting the conceptual design of the QCS is shown in Fig. 1
(left). The current to be determined, $I$, biases an ideal QAH resistor to
generate a transverse voltage $V_{\mathrm{QAH}}=IR_{\mathrm{K}}$.
Simultaneously, $N$ junctions of a PJVS are biased at a microwave frequency
$f$ to be on their first Shapiro steps, yielding a voltage across the array
$V_{\mathrm{J}}=NfK_{\mathrm{J}}^{-1}$ [11, 36, 37, 38]. $V_{\mathrm{J}}$ is
tied to the high-potential side of the QAH Hall bar so that the potential
difference $V_{\mathrm{null}}=V_{\mathrm{J}}-V_{\mathrm{QAH}}$ can be measured
by a nanovoltmeter at room temperature. For a given value of $N$, if $f$ is
tuned such that $V_{\mathrm{null}}=0$ the current is known to be

$I_{\mathrm{direct}}=\frac{Nf}{K_{\mathrm{J}}R_{\mathrm{K}}}.$ (1)

Here the subscript ‘direct’ is used to indicate that this measurement is a
‘direct realization’ of the ampere — i.e. the measurement of $I$ is done
through the integration of two quantum standards, without the use of any
transfer standards, in a manner consistent with the definition of the ampere.

Generalizing to the case of nonzero null voltage and imperfect quantization of
the QAH resistor, the current can still be calculated according to

$I_{\mathrm{direct}}=\frac{NfK_{\mathrm{J}}^{-1}-V_{\mathrm{null}}}{R_{\mathrm{QAH}}(I,f)},$
(2)

$R_{\mathrm{QAH}}(I,f)\simeq R_{\mathrm{K}}$ is the measured transverse-
resistance of the QAH device when it is biased with current $I$ and the PJVS
is biased with frequency $f$.

Radiative leakage of the microwave bias for the PJVS (mounted at the 4K stage)
may result in undesirable electromagnetic coupling to other parts of the
cryostat. In our realization of the QCS, such coupling resulted in electron
heating of the QAH device (connected to the lowest-temperature stage) and was
determined to be the main source of deviation of Hall resistance from
quantization (see Fig. E1 in the Extended data section). By limiting $f$ to
specific frequencies and measuring quantization of the QAH device for each
value of $f$ in situ (see Methods) the impact of radiative leakage on QCS
performance was made minimal. Details of the impact of leakage on the QAH
device can be found in the Supplemental Information.

In addition to functioning as a novel realization of the ampere as discussed
above, the QCS can also function as a current detector. To verify the QCS as
an accurate current detector, a second measurement of $I$ is made in a
standard Ohm’s law configuration. Here, $I$ still biases the QAH device as in
Fig. 1, but the microwave bias to the PJVS is disabled and the Josephson
junction array functions as a superconducting short on-chip
($V_{\mathrm{J}}=0$ V). A Keysight 3458A digital voltmeter, previously
calibrated against the PJVS, is used to measure the Hall voltage
$V_{\mathrm{DVM}}$. The current is then calculated according to

$I_{\mathrm{indirect}}=\frac{V_{\mathrm{DVM}}}{R_{\mathrm{K}}},$ (3)

and compared to $I_{\mathrm{direct}}$. Agreement of the two measurements
indicates that the QCS can accurately determine the magnitude of $I$ and
therefore functions as a novel current sensor, as well as a realization of the
ampere. A simplified schematic depicting the measurement of
$I_{\mathrm{indirect}}$ and the calibration of the DVM can be found in the
Supplemental Information.

A dry 3He/4He dilution refrigerator housed both the PJVS and QAH devices. The
PJVS was mounted on the 4 K stage and covered by a Cu radiation shield to
minimize stray microwave leakage to the QAH sample. The QAH device was mounted
to a sample stage at the end of a cold finger below the mixing chamber plate.
Base temperature at the sample stage was approximately 10 mK, as measured by a
ruthenium oxide resistance thermometer. Direct current (DC) wiring to the QAH
sample and to the PJVS consisted of copper twisted pair embedded within a
woven loom. Each 12-pair cryogenic woven loom cable was broken by mating pairs
of 25-pin micro-D connectors at the 60 K, 4 K, and mixing chamber temperature
stages to aid in thermalization.

Equations (1) and (2) imply that the QCS’s uncertainty directly depends on the
quality of quantization of the quantum anomalous Hall resistance to
$R_{\mathrm{K}}$ and Josephson voltage to $NfK_{\mathrm{J}}^{-1}$. Deviation
of $R_{yx}$ from $R_{\mathrm{K}}$ is known to vary with both current bias and
electron temperature [31, 32, 33, 39, 34]. To account for this we measure the
quantization of the QAH resistor for each value of $I$ and $f$ in situ. The
quantized voltage of the PJVS is also verified in situ. Details of the
independent characterizations of the PJVS and a QAH resistor can be found in
the Methods section, with extended details available in the Supplemental
Information.

For this work we used the 16-bit digital current source of a commercial
cryogenic current comparator [40, 41, 42] to provide the longitudinal current
bias to the QAH device. A cryogenic current comparator is a standard
instrument within the field of electrical metrology[43, 44] and is commonly
used for precision measurements of resistance [6, 44]. The cryogenic current
comparator’s current source was used for in situ characterization of the Hall
resistance in a bridge configuration (the most common use of a cryogenic
current comparator). The cryogenic current comparator’s current source was
also used to bias the QAH device during direct and indirect measurements. The
nominal output of the current source was set using the cryogenic current
comparator’s graphical user interface (GUI). Onward, the cryogenic current
comparator’s digital current source will be referred to simply as the ’current
source’

## III. Results and Discussion

The QCS has two main functions: (1) serving as realization of the ampere and
(2) acting as a novel form of current sensor that is capable of measuring an
unknown current. Below we highlight the subtle aspects that make these two
functionalities distinct despite their near-identical implementation, and we
assess the performance of the QCS in each mode.

As noted in the introduction, realization of a unit requires that a quantity
of the same kind as a unit, in this case an electrical current, be given a
value and associated uncertainty in a manner consistent with the unit’s
definition. Realization of the ampere has previously been achieved by applying
Ohm’s law to realizations of the SI derived units the volt and the ohm, based
on the Josephson effect and QH effect respectively. The QAH effect has emerged
as a viable replacement for the QH effect as a realization of the ohm [27],
and could play that role in realization of the ampere.

To establish a new realization of the ampere we first characterized the PJVS
and the QAH devices to confirm that they functioned as realizations of the
volt and the ohm, respectively, when placed in the QCS configuration (see
Methods). Next, we demonstrated that we could combine these components to
associate a value (and uncertainty) with an electrical current — as is
required to claim realization of the ampere. Using the current source’s GUI we
directed the digital current source to output a specific current $I$ which was
used to bias the QAH device. We chose in succession 11 different current
values between $I=9.33$ nA and $I=252$ nA. In each case, we biased $N$
junctions on the PJVS with the appropriate microwave bias frequency (either
$f_{0},f_{1}$ or $f_{2}$ depending on the value of $I$ — see Supplemental
Information) to be on their first Shapiro steps and hence to nearly match the
voltage expected across the QAH device. $V_{\mathrm{null}}$ was then measured
as shown in Fig. 1 .

To eliminate thermoelectric offsets in $V_{\mathrm{null}}$ the polarity of the
input current $I$ was reversed periodically during the measurement. At least
760 samples of $V_{\mathrm{null}}$ were averaged across at least 10 (up to 90)
of these polarity reversal cycles, with each cycle lasting an average of 60 s.
The final averaged value of $V_{\mathrm{null}}$ was used, along with the
corresponding value of $R_{\mathrm{QAH}}(I,f)$, to determine the magnitude of
$I_{\mathrm{direct}}$ according to equation (2). The relative combined
uncertainty ($k=1$, corresponding to a 68% confidence interval) for these
measurements is shown as a solid red trendline in Fig. 2(b). The combined
uncertainty was calculated as a root sum of squares of the Type A and Type B
contributions (see Section IV for a full discussion of the uncertainty
budget).

The measurements discussed above constitute a novel realization of the ampere
and, on their own, serve as a significant accomplishment in the field of
electrical metrology. Indeed, the relative uncertainty for current detection
using the QCS meets or exceeds that available using state-of-the-art
calibration and measurement capabilities from national metrology institutes
around the world (see Fig. 2(b) — a full discussion of this figure to follow)
in this current range. However, to be of _practical_ use for national
metrology institutes, this realization should perform as well as or better
than existing methods used for measuring an electrical current. It is
ultimately the _accuracy_ of the QCS that distinguishes functionalities (1)
and (2).

Figure 2: QCS operation and contextualization. (a) Relative agreement of
$I_{\mathrm{direct}}$ and $I_{\mathrm{indirect}}$:
$\left(I_{\mathrm{direct}}–I_{\mathrm{indirect}}\right)/I_{\mathrm{indirect}}$.
Agreement shown is the mean of multiple measurements. Shaded area shows the
combined Type A uncertainty ($k=1$) of the direct and indirect measurements.
(b) Worldwide comparison of relative uncertainty reported for precision
current generation and measurement within the range ($10^{-14}$ A to $10^{-5}$
A). The shaded region shows calibration and measurement capabilities (CMCs)
using a traditional Ohm’s law method [45] reported by the following national
metrology institutes: INRIM, KRISS, VSL, METAS, NIM, MIKES, LNE, PTB, and RISE
(the full names of these institutes can be found in the Supplemental
Information). Calibration and measurement capabilities from PTB and INRIM
using an ULCA (blue diamonds) [45]. SET results represent current realization
reported for metallic [15, 16] (green triangles), GaAs [17, 18, 19] (orange
triangles), and silicon [20, 21, 22] (purple triangles) devices. The QCS
agreement data (red crosses) are the same data shown in (a). The QCS
uncertainty data (red trendline) represent the total combined (Type A and Type
B) uncertainty ($k=1$) for measurements of $I_{\mathrm{direct}}$.

To assess the accuracy of the QCS, additional measurements of $I$ were made in
a traditional Ohm’s law configuration as described in Section II. The same 11
values of current (indicated by the current source’s GUI) were sourced and the
value of $I$ was calculated using equation (3). These measured values were
then compared to the corresponding measurement of $I_{\mathrm{direct}}$.

Figure 2(a) shows
$\left(I_{\mathrm{indirect}}-I_{\mathrm{direct}}\right)/I_{\mathrm{indirect}}$,
the relative agreement of the two current measurements, as red markers. The
combined Type A uncertainties ($k=1$) of the two measurements
($I_{\mathrm{indirect}}$ and $I_{\mathrm{direct}}$) is shown as a shaded
region centered around 0 $\mu$A$/$A. This combined uncertainty was calculated
as an root-sum-square total of the individual Type A uncertainties. Values of
$I$ within 400 $\mu$A$/$A of one another have been grouped into a single data
point. A table showing the exact GUI setting for $I$, and the corresponding,
individually-measured values of $I_{\mathrm{direct}}$ and
$I_{\mathrm{indirect}}$, can be found in the Supplemental Information.

If statistical uncertainty were the only factor contributing to the
differences between the measurements of $I_{\mathrm{direct}}$ and
$I_{\mathrm{indirect}}$, then their relative agreement should fall within the
combined Type A uncertainty as is the case for $I=9.33$ nA and $I=83.9$ nA.
Here it is important to note that the uncertainty budget is dominated by Type
A contributions to the extent that Fig. 2(a) would remain visually unchanged
if Type B (non-statistical) uncertainties shown in Section IV were included.
(The full uncertainty budget for $I_{\mathrm{direct}}$ can be found in Section
IV and the uncertainty budget for $I_{\mathrm{indirect}}$ can be found in the
Supplemental Information.) The fact that the relative agreement falls outside
the combined Type A uncertainty for $I=28.0$ nA, 51.9 nA, 132 nA, and 252 nA
could mean that the Type A uncertainty for one or both measurements was under-
evaluated or that a relatively large non-statistical source of error was
unaccounted for in the budget. Measurements made in close succession were
highly repeatable (see Section IV) and so the latter possibility is far more
likely.

We believe that the observed disagreement was caused by long-term drift in the
output of the current source. Multiple measurements of $I_{\mathrm{indirect}}$
were taken over the course of 8 days. During this time we observed that the
measured value of $I_{\mathrm{indirect}}$ differed by a relative amount of
28.0 $\mu$A$/$A for the same GUI setting of $I=251.7777$ nA (see Supplemental
Information). Importantly, this difference is greater than the average
disagreement between direct and indirect current measurements at any value of
$I$.

Due to experimental constraints, measurements of $I_{\mathrm{direct}}$ and
$I_{\mathrm{indirect}}$ were taken more than 1 week apart. Given the evidence
of drift discussed above, we conclude that despite the settings on the digital
current source being the same between measurements of $I_{\mathrm{indirect}}$
and $I_{\mathrm{direct}}$ for a given value of $I$, the magnitude of the
current source’s output likely drifted between those two measurements,
inflating the disagreement to be outside of their combined uncertainty.
Because the error due to drift is an artifact of the particular current source
being measured and not something fundamental to the QCS, it has not been
included in our error budget. This level of drift is not a reflection of the
quality of the current source nor is it an issue that would have been noticed
during normal operation of the cryogenic current comparator. This is because
long-term stability of a cryogenic current comparator’s current source is not
a necessary design criterion for that instrument given the compensation
network and the balancing procedure used in typical bridge comparisons. A
detailed discussion of this topic can be found in the Supplemental
Information.

Fig. 2(b) contextualizes our realization of the ampere by the QCS amongst the
best realizations and calibration and measurement capabilities reported by
national metrology institutes worldwide. State-of-the-art calibration and
measurement capabilities from the Physikalisch-Technische Bundesanstalt (PTB)
and the Istituto Nazionale di Ricerca Metrologica (INRIM) using an ultrastable
low-noise current amplifier (ULCA) [45] (blue diamond markers), as well as
calibration and measurement capabilities based on the traditional Ohm’s law
method (shaded region) [45] are shown across a broad current range of
currents. The list of institutes referenced by shaded region can be found in
the Supplemental Information. Several realizations of the ampere are also
shown. Metallic [15, 16] (green triangles), GaAs [17, 18, 19] (orange
triangles), and silicon [20, 21, 22] (purple triangles) SET devices are shown
in the fA to nA region. For direct comparison to the other data shown, the
relative uncertainty of our QCS measurements are depicted as a solid red
trendline. We also plot the relative agreement of $I_{\mathrm{indirect}}$ with
$I_{\mathrm{direct}}$ from Fig. 2(a) as red crosses. For a given value of
current, the larger of these two values (QCS agreement or QCS uncertainty) can
be viewed as a ‘worst-case scenario’ for performance of the QCS. Even when
assessed in this manner, our measurements meet or exceed the precision offered
by state-of-the-art calibration and measurement capabilities for all values of
current tested. Not shown in Fig. 2(b) are the results of the three-cryostat
integration of a Josephson voltage standard with QH resistance standard where
a relative uncertainty of $1\times 10^{-8}$ A/A was obtained for currents in
the mA range [27].

Figure 3: Comparison of the expected and measured relative change in
$I_{\mathrm{direct}}$ when one or more system parameters were adjusted.
Measurements were made for currents near (a) 132 nA and (b) 83.9 nA. The
adjusted parameters were: $I$, the bias current being supplied by the current
source; $f$, the PJVS drive frequency; or a combination of the two (indicated
by $f+I$). In all cases the Type A uncertainties ($k=1$) are smaller than data
points

Additional measurements of $I_{\mathrm{direct}}$ were made to check the
accuracy and overall performance of the QCS in response to small variations of
the input parameters $I$ and $f$, an example of which is shown in Fig. 3(a).
For these measurements, a nominal current of $I=132.3984$ nA was set using the
current source’s GUI and a standard measurement of $I_{\mathrm{direct}}$ was
performed. Immediately following this measurement, the bias current was
decreased by the smallest allowed step on the GUI so that the new nominal
current value was $I=132.3893$ nA — a relative change of roughly $69$
$\mu$A$/$A. The measurement of $I_{\mathrm{direct}}$ was then repeated. The
measurement relative difference between these two measurements of
$I_{\mathrm{direct}}$ is shown as a red circular marker in Fig. 3(a), the
expected relative change based on the difference between the two nominal
values of $I$ indicated by the GUI setting is shown as a blue cross.

A similar procedure was repeated using $f$ as the varied parameter. Again, an
initial measurement of $I_{\mathrm{direct}}$ was made with the nominal bias
current $I=$ 132.3984 nA. Immediately after this measurement the bias
frequency $f$ was changed and $\Delta V_{\mathrm{null}}=\Delta
V_{\mathrm{J}}-V_{\mathrm{QAH}}=N(\Delta
f)K_{\mathrm{J}}^{-1}-IR_{\mathrm{QAH}}$ was calculated such that the
resulting relative change in $I_{\mathrm{direct}}$ was approximately 130
$\mu$A$/$A ($\Delta V$ and $\Delta f$ are the changes in the null voltage and
frequency respectively). The expected change is shown as a blue cross in Fig.
3(a). This change in $f$ was small enough that the value of
$R_{\mathrm{QAH}}(I,f)$ was unaffected. Measurement of $I_{\mathrm{direct}}$
was then repeated. The measured change in $V_{\mathrm{null}}$ is shown as a
red, circular marker Fig. 3(a).

When either $I$ or $f$ was adjusted, the measured change in
$I_{\mathrm{direct}}$ ($V_{\mathrm{null}}$) was equal, within uncertainty, to
the expected change. A similar set of measurements were repeated when the
nominal current set by the GUI was $I\sim 83.9$ nA. In addition to varying $I$
or $f$ alone, these tests also included combined variations of both
parameters, $f+I$. Again, the measured and expected changes were equal for all
cases.

The measurements described above have implications when assessing
functionalities (1) and (2) of the QCS. In the case of (1) these measurements
definitively show that the PJVS and QAH device are directly integrated with
one another during operation of the QCS. If this was not the case, and the
devices were independent from one another, varying $f$ while keeping the
magnitude of the bias current $I$ constant would not result in a measurable
change that is consistent with the predicted value. Regarding functionality
(2), it is important to note that the these measurements were done using one
value of current where the relative agreement of $I_{\mathrm{direct}}$ and
$I_{\mathrm{indirect}}$ fell within their combined uncertainty ($I=83.9$ nA)
and one current where the relative agreement was outside the combined
uncertainty ($I=132$ nA). Despite this, measured changes in $I$ matched the
expected change in either case. Though the changes in $I$ shown here were
larger than the disagreement between $I_{\mathrm{direct}}$ and
$I_{\mathrm{indirect}}$, these measurements still demonstrate that the QCS is
able to accurately detect small variations in current even for those values of
$I$ where comparisons of $I_{\mathrm{indirect}}$ with $I_{\mathrm{direct}}$
suggest the possibility of diminished accuracy (again, we believe this
disagreement is a result of uncontrollable drift of the current source over
time).

## IV. Uncertainty Budget

The overall uncertainty budget for the QCS data shown in Fig. 2(a) is given in
Table 1. The dominant contribution is that due to statistical fluctuations of
the nanovoltmeter readings of $V_{\mathrm{null}}$. This Type A contribution
$u_{\mathrm{null}}$, is defined as the standard deviation of means of multiple
polarity reversals within a single experimental run. For all direct
measurements, $u_{\mathrm{null}}\leq 20$ nV, which is close to the noise floor
for our measurement system. Thus, since we are noise-limited, the relative
uncertainty $u_{\mathrm{null}}/V_{\mathrm{J}}$ decreases as the magnitude of
the bias current increases. This is illustrated in Table 1: the lowest
(highest) relative Type A uncertainties correspond to the highest (lowest)
value of the bias current $I$.

Type B contributions include: offsets to measurements of $V_{\mathrm{null}}$
due to the finite input impedance, gain and linearity of the nanovoltmeter
(referred to as NVM in the table), uncertainty of the tone used for PJVS
microwave bias frequency $f$, and the combined uncertainty associated with
measurement of $\delta R_{yx}$. The total root-sum-square (RSS) combined
uncertainty ($k=1$) is given in the final row of the table. A detailed
description of each uncertainty source can be found in the Supplemental
Information.

Table 1: Uncertainty budget for measurements of $I_{\mathrm{direct}}$. Columns
one and two give the source and the type of the uncertainty, respectively.
Column three gives the magnitude of the relative contribution in units of
$\mu$A/A.

Source | Type | Contribution $\left(\frac{\mu\textrm{A}}{\textrm{A}}\right)$
---|---|---
Null detector reading11footnotemark: 1 | A | 2.3 – 40.0
$R_{yx}$ quantization | A+B | 1.0
NVM input impedance | B | 3.0
NVM gain & linearity | B | 0.001
PJVS microwave bias | B | 0.001
Total11footnotemark: 1 | RSS | 3.91 – 41.2

1For currents ranging from $I=252$ nA to $I=9.33$ nA.

Whereas the contribution of the null voltage fluctuations range from 2.4
$\mu$A$/$A to 40 $\mu$A$/$A, multiple measurements at the same nominal current
(i.e., same GUI setting) are highly repeatable, particularly at the highest
currents. For different evaluations of $I_{\mathrm{direct}}$ the relative
difference of the means ranged from 1.0 $\mu$A$/$A at $I=251$ nA to $11.0$
$\mu$A$/$A at $I=9.33$ nA. To gain a more complete understanding of the
factors contributing to our noise floor, we independently measured the Allan
deviation $\sigma(\tau)_{\mathrm{NVM}}$ for different experimental
configurations of the nanovoltmeter in a separate laboratory. (A detailed
discussion of the Allan deviation calculations can be found in the
Supplemental Information.) In one configuration, we shorted the nanovoltmeter
terminals together using a ‘low thermal EMF’ plug provided by the manufacturer
that connects directly to the nanovoltmeter’s input connector. In another, we
shorted the copper spade terminals together at the end of the low-thermal
cable that is furnished with the nanovoltmeter.

Figure 4: Allan deviations for measurements of $V_{\mathrm{null}}$. Depicted
are the overlapping, detrended, Allan deviations (see Supplemental Information
for details) calculated for: the QCS measurements $I=53.7889$ nA (red
asterisks) and $I=83.9473$ nA (blue x-markers) shown in Fig. 2(b),
measurements of $V_{\mathrm{null}}$ taken with the nanovoltmeter input shorted
with a low-thermal plug (gray upward-triangles), and with a short at the end
of leads of 1 m length (yellow downward-triangles). The values of $I$
indicated in the legend correspond to the setting in the current source’s GUI.
Shown for reference is a $\tau^{-1/2}$ trendline characteristic of noise with
a white spectrum, Measurements with a short were performed in a temperature-
and humidity-controlled laboratory in which temperature fluctuations were less
than $\pm$0.5 K. Error bars represent the $k=1$ Type A uncertainty of the
Allan deviation under the assumption of a white noise process.

The Allan deviations for both measurements as a function of $\tau$ are shown
on a $\log$-$\log$ scale in Fig. 4. The ‘low-thermal short’ data, can be
viewed as a baseline contribution of the nanovoltmeter to the overall
stability and noise of the measurement setup. For $\tau\lesssim 10^{2}$ s, the
data clearly scale as $\tau^{-1/2}$ indicating a white-noise-dominated
spectrum [46]. For longer times, the influence of the finite sampling time
(evidenced by the rapid growth in the uncertainty of $\sigma_{\mathrm{NVM}}$)
and of noise with different spectra can be seen. Comparing this measurement to
that done with the shorted, low-thermal cable (‘shorted leads’) is
particularly enlightening. Here the initial magnitude of
$\sigma_{\mathrm{NVM}}$ is smaller and the spectrum again appears white noise
dominated. However, this functional behavior quickly transitions to a spectrum
more similar to that of $1/f$ noise where $\sigma(\tau)$ is approximately
constant, before eventually ($\tau\sim 2\times 10^{2}$ s) transitioning back
to a seemingly white-noise-dominated spectrum.

Comparing the ‘shorted leads’ Allan deviation trend to that of QCS data at
$I=53.8$ nA and $I=83.9$ nA suggests that the cabling needed to connect the
nanovoltmeter to the devices in the fridge introduces a sizable amount of
noise to the QCS measurements. The magnitude and functional form of the QCS
Allan deviation is comparable to that when the leads were shorted with the 1 m
cable. The variation in magnitude depicted around $\tau=300$ s between the two
QCS measurements is similar to that observed between multiple measurements of
the shorted leads. Thus, even though a longer length of cable (internal and
external to fridge) was required to obtain the QCS data, one should not
necessarily expect the QCS Allan deviation to always exceed that of the
‘shorted leads’ case because of the variability between different Allan
deviation measurements. The variability is likely due to changes in the lab
environment — for example, microphonic and/or electromagnetic pickup from
adjacent experiments or occupants. These results suggest that uncertainty
reductions may be possible in future QCS measurements by meticulous design of
experimental wiring.

## V. Conclusions

We have successfully integrated a PJVS and QAH resistance standard within a
single cryostat to realize the quantum ampere for currents within the range
9.33 nA to 252 nA. For the range of currents tested, the uncertainty of the
measurements made by this first iteration of the QCS meet or exceed what is
currently offered worldwide by national metrology institutes today.
Furthermore, the QCS stands as one of the only _realizations_ of the ampere
within the range of $10^{-9}$ A to $10^{-6}$ A.

By conducting a comprehensive analysis of the uncertainty budget and noise
spectra associated with the QCS measurements, we were able to establish
definitive routes for improving future iterations of the QCS design. Improving
the electromagnetic shielding and filtering for both the QAH device and the
PJVS would allow the bias frequency $f$ to be continuously adjusted without
needing to measure $R_{\mathrm{QAH}}$. Specialized wiring inside and outside
the fridge would reduce the effects of triboelectric and thermoelectric noise
sources. A wider Hall bar structure should increase the operating range of the
QCS to currents beyond 250 nA. Finally, assuming a predominantly white noise
spectrum, significantly increased measurement times should greatly reduce Type
A uncertainty.

Having demonstrated that the QCS is capable of measuring an electrical current
with roughly the same precision and accuracy as a traditional Ohm’s law
measurement, future implementations need not be limited to a single current
source. However, follow-up measurements of the cryogenic current comparator’s
digital current source used here may be done to provide additional
confirmation of long-term drift.

Finally, the development of novel realizations of the SI base and derived
units — especially realizations that utilize quantum phenomena — was an
initiative set by the General Conference on Weights and Measures during the
2019 redefinition of the SI. In keeping with this initiative, it is our hope
that the design and methodology presented here can be adapted and enhanced to
further improve and diversify the field of electrical metrology.

## Methods

PJVS device and characterization: The PJVS device used in this work was based
on the same triple-stacked, Nb/NbSix/Nb junction process used in NIST’s
Standard Reference Instruments [47, 48]. The cryopackaged chip contains 59,140
junctions and is capable of an output voltage of approximately 2 V at a
microwave bias frequency of 18 GHz. The junctions are arranged into two
symmetric arrays and share a common microwave bias, which is distributed using
on-chip microwave splitters. This allows the chip to be operated as a dual-
channel (1 V + 1 V) source and is the configuration used in this work. Each
array is further divided into ternary-weighted subarrays.

It is well established [49] that PJVSs can suffer diminished critical currents
and/or operating margins when subjected to magnetic fields. In some instances,
the effect may be reversed once the field is removed. But certain scenarios
result in an irreversible loss of quantization. For the latter, quantization
can be restored by briefly increasing the temperature of the PJVS to above the
critical temperature of Nb. To account for this possibility, a heater was
attached to the fridge’s 4 K stage and operated in an open-loop manner.

To assess the performance of the PJVS in the environment of the fridge, we
swept the field of the fridge’s solenoid magnet up to 1 T (within solenoid)
and measured the stray field outside the vacuum envelope, but near the PJVS
(the radial separation between PJVS and probe was approximately 10 cm). For a
1 T solenoid field, we measured a stray field magnitude of approximately 1 mT
near the cryopackage, a value which is consistent with the known field map of
the bare solenoid (with integrated bucking coil) and the passive shielding
added by the fridge’s manufacturer (no additional magnetic shielding was
installed around the PJVS cryopackage). We also monitored the critical current
and Shapiro step width of one of the 8 400-junction subarrays for a bias
frequency of 9.7 GHz. At the maximum 1 T field, we found no observable change
in the PJVS critical current or Shapiro step width.

To allow comparison to historical performance at liquid helium temperatures,
we heated the fridge’s 4 K stage to approximately 4.2 K. The critical current
and Shapiro step widths were comparable to those measured in liquid helium:
$I_{c}=6.4$ mA, $\Delta I_{SS}=2$ mA for the fridge at 9.7 GHz and
approximately 4 K, compared with $I_{c}=6.2$ mA, $\Delta I_{SS}=2.5$ mA at 18
GHz in liquid helium. The difference in step width could be due to the
response of the junction array vs frequency or the fact that the power
amplifier used in this work has a lower maximum output power. During the field
sweep, we did observe a slight decrease in critical current each time the
field was stepped. Since we also observed increases in the temperature of the
4 K stage at each field increment, we attribute the reduction in critical
current to changes in temperature, rather than a direct result of the field
itself.

QAH film composition and device fabrication: The QAH insulator sample studied
in this was work was a high quality, 6-quintuple-layer (QL) Cr-doped
(Bi,Sb)2Te3 thin film grown by molecular beam epitaxy on a semi-insulating
GaAs (111)B substrate. The first and sixth QLs featured increased Cr
concentration, (Cr0.24Bi0.26Sb0.62)2Te3, compared to the inner four QLs,
(Cr0.12Bi0.26Sb0.62)2Te3. A lock-in characterization of the QAH state of this
film is given in the Supplemental Information.

The 300 $\mu$m wide Hall bar used in this study (Fig. 1) was fabricated using
a combination of photolithography, metal evaporation, wet and dry etching and
atomic layer deposition. Prior to all photolithography steps, the chip was
cleaned in acetone and isopropyl alcohol, coated in SPR 3612 photoresist (spun
at spun at 5.5 krpm for 45 s) and baked at 80 °C for 2 minutes. After
exposure, the resist was developed in a phosphate salt developer followed by
two water rinses (extended fabrication details can be found in the
Supplemental Information).

Each pair of Hall contacts was separated by a lateral, center-to-center
distance of 300 $\mu$m. The device featured a top gate (40 nm alumina
dielectric covered by a Ti/Au electrode) for tuning the chemical potential in
the film. However, measurements of the longitudinal resistance as a function
of gate voltage indicated that the native Fermi level was well positioned near
the center of the exchange gap (see Supplemental Information). Therefore, all
measurements were taken with 0 V applied to the top gate.

QAH characterization: Prior to characterization, the QAH device was magnetized
by applying a 1 T out-of-plane magnetic field and then reducing the field to 0
T.

To characterize the QAH device, quantization of the Hall resistance relative
to $R_{\mathrm{K}}$, $\delta R_{yx}=(R_{yx}-R_{\mathrm{K}})/R_{\mathrm{K}}$,
was measured as a function of bias current $I$ and PJVS microwave bias
frequency $f$, using a cryogenic current comparator. All measurements took
place with the cryostat operating at base temperature base temperature with
$T\approx 10$ mK at the sample stage. Extensive details of the methodology
used to measure $\delta R_{yx}$ and an estimation of the uncertainty budget
for the same cryogenic current comparator and measurement set-up, can be found
in previous publications [31, 33] as well as in the Supplemental Information.

Measurements of $\delta R_{yx}$ as a function of $I$ were first done with the
microwave excitation to the PJVS turned off ($f=0$), as a baseline measure of
sample quality. The measurements were then repeated with the microwave
excitation to the PJVS turned on but with the PJVS inactive (i.e. $f\neq 0$
but $V_{\mathrm{J}}=0$ V). This was done to simulate the environment that the
QAH device would experience while being used in the QCS configuration. By
comparing these two sets of measurements, the effects of microwave leakage to
the QAH sample could be qualified. Indeed, it was found that certain
frequencies caused large deviations from baseline behavior, whereas others had
only moderate effects (full discussion on this topic can be found in the
Supplemental Information). As a result, unless otherwise stated, all
measurements taken with the microwave excitation turned on were done so at one
of three fixed frequencies (0 dBm): $f_{\mathrm{0}}=9.701102806$ GHz,
$f_{1}=18.651341$ GHz, or $f_{2}=15.3020182$ GHz. These frequencies were shown
to have only modest impact on the quantization of the QAH device. A table
showing which of the three frequencies were used during each QCS measurement
of $I$ is given in the Supplement Information. The measured values of $\delta
R_{yx}(I,f)$ are shown Fig. E1 of the Extended Data section.

## Extended Data

Figure E1: Independent in situ characterization of the QAH device. Deviation
of the Hall resistance from quantization $\delta R_{yx}$, was measured as a
function of bias current $I$ with (open markers) and without (blue, filled
circles) a microwave excitation being supplied to the PJVS. Heating of the QAH
device due to microwave radiation leakage is seen as an increase in $|\delta
R_{yx}|$ when the microwave excitation is on compared to when it is off, for a
given value of $I$. Heating of the QAH device was heavily dependant on the
value of the bias frequency $f$. Heating effects were minimal for frequencies
$f_{\mathrm{0}}=9.701102806$ GHz, $f_{1}=18.651341$ GHz, or $f_{2}=15.3020182$
GHz. Measurements of $\delta R_{yx}$ are used to determine
$R_{\mathrm{QAH}}(I,f)$ seen in equation 2 of via
$R_{\mathrm{QAH}}(I,f)=R_{\mathrm{K}}+\delta R_{yx}(I,f)$. See the
Supplemental Information for further discussion.

Supplementary information See supplementary material for full details
regarding: independent characterizations of the digital voltmeter, the QAH
device (film growth, extended fabrication details, and longitudinal
resistivity measurements as a function of top gate voltage) and QCS
measurement protocols. Also included are names of the national metrology
institutes referenced for calibration and measurement capability data, the
indirect current measurement schematic, evidence of long term drift of the
cryogenic current comparator’s digital current source, a discussion on the
impacts of the heating due to microwave leakage, and details on changes to the
internal wiring of the dilution refrgerator to facilitate in situ
characterization and QCS measurements during a single cooldown, tables
detailing how measurements of $I_{\mathrm{direct}}$,$I_{\mathrm{direct}}$
correspond to the GUI setting of $I$, evidence of radiative heating of the QAH
device via microwave leakage, and Allan deviation calculations.

Acknowledgments The authors thank T. Mai, F. Fei, G. J. Fitzpatrick, and E. C.
Benck for assistance with the NIST internal review process. The authors also
thank Ilan T. Rosen and Marc 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 number
FA9550-21-1-0429. At the initiation of the project L.K.R., M.P.A. and D.G.-G.
were supported by the U.S. 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 the support from the National
Science Foundation (NSF) (DMR-1411085 and DMR-1810163) and the Army Research
Office MURI under grant numbers W911NF16-1-0472 and W911NF-19-S-0008.
Commercial equipment, instruments, and materials are identified in this paper
in order 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 United States 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 United States government. Funding is hence appropriated by the United
States Congress directly. Part of this work was performed at nano@stanford,
supported by the National Science Foundation under award ECCS-2026822.

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Supplemental information for: Realization of the quantum ampere using the
quantum anomalous Hall and Josephson effects

Linsey K. Rodenbach1,2,3†, Ngoc Thanh Mai Tran3,4†, Jason M. Underwood3†,
Alireza R. Panna3, Molly P. Andersen2,5, Zachary S. Barcikowski3,4, Shamith U.
Payagala3, Peng Zhang6, Lixuan Tai6, Kang L. Wang6, Randolph E. Elmquist3,
Dean G. Jarrett3, David B. Newell3, Albert F. Rigosi3 and David Goldhaber-
Gordon1,2a

1Department of Physics, Stanford University, Stanford, California 94305, USA
2Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
3Physical Measurement Laboratory, National Institute of Standards and
Technology (NIST), Gaithersburg, Maryland 20899, USA
4Joint Quantum Institute, University of Maryland, College Park, Maryland
20742, USA
5Department of Materials Science and Engineering, Stanford University,
Stanford, California 94305, USA
6Department of Electrical and Computer Engineering, Department of Physics and
Astronomy, University of California, Los Angeles, Los Angeles, California
90095, USA
†These authors contributed equally to this work.
aTo whom correspondence should be addressed; E-mail: goldhaber-
<EMAIL_ADDRESS>
###### Contents

  1. S1 Indirect Current Measurement Schematic
  2. S2 Independent Characterizations
    1. S2.1 Digital voltmeter characterization
    2. S2.2 Extended discussion: QAH characterization
  3. S3 QAH Device and Measurements
    1. S3.1 Extended details: QAH film growth and Hall bar fabrication
    2. S3.2 Lock-in characterization of Cr-BST thin-film
    3. S3.3 Measuring $\rho_{xx}$ using the CCC’s nanovoltmeter
    4. S3.4 Determining the optimal gate voltage of the QAH device
  4. S4 QCS Measurements
    1. S4.1 The CCC’s digital current source
    2. S4.2 Corresponding values of $I$, $N$, $f$ and $V_{\mathrm{J}}$
    3. S4.3 Determination of $V_{\mathrm{{null}}}$ and $I_{\mathrm{direct}}$
    4. S4.4 Comparison of $I_{\mathrm{direct}}$, $I_{\mathrm{indirect}}$, and $I_{\mathrm{GUI}}$
  5. S5 Heating of the QAH device via microwave leakage
  6. S6 National Metrology Institutes Referenced for Calibration and Measurement Capability Data
  7. S7 Long Term Drift of the CCC’s Digital Current Source
  8. S8 Uncertainty Budget and Noise Analysis
    1. S8.1 Uncertainty budget for measurements of $I_{\mathrm{direct}}$
    2. S8.2 Uncertainty budget for measurements of ${I}_{\mathrm{indirect}}$
    3. S8.3 Allan deviation calculation
  9. S9 The Interposer

## S1 Indirect Current Measurement Schematic

Fig. S1 depicts the of the Ohm’s law based current measurement used to
determine $I_{\mathrm{indirect}}$. The current $I$ was generated using the
cryogenic current comparator’s (CCC) digital current source as discussed in
the main text.

Figure S1: A simplified schematic for the indirect quantum current measurement
scheme is shown. The Hall voltage $V_{\mathrm{QAH}}$ is measured with a
commercial digital voltmeter (DVM). The current $I$ is determined using Ohm’s
law via Eq. (3) of the main text.

## S2 Independent Characterizations

### S2.1 Digital voltmeter characterization

Figure S2: Depiction of DVM calibration with the PJVS embedded within the
dilution refrigerator.

Measurements of $I_{\mathrm{indirect}}$ require both a quantized $R_{yx}$ and
a sufficiently characterized voltmeter to enable accurate measurements of the
transverse voltage $V_{yx}$. We used the same programmable Josephson voltage
standard (PJVS) array located at the dilution refrigerator’s (DR) 4 K stage to
characterize the digital voltmeter (DVM), as shown in Fig. S2. This was
facilitated by a separate set of voltage leads that run from the cryopackage
directly to room-temperature connections (i.e., $R_{yx}$ is not in the
measurement loop). We set the PJVS voltage to the nominal values used in the
direct measurements and compared the calculated (expected) voltage with the
DVM’s measurement. The purpose here was to check that the error was within an
acceptable range—less than the measurement’s Type A uncertainty and less than
the overall uncertainty of the direct measurements alone. This condition was
satisfied for all voltages considered.

Additional sources of error include the input resistance of the DVM and the
interaction of the DVM’s pumpout current with resistances in the measurement
circuit (e.g., $R_{yx}$, RC filtering, or wiring). We characterized both of
these phenomena in separate measurements in a different lab. For the DVM’s 100
mV range, we found the input resistance was greater than 1 T$\Omega\gg
R_{\mathrm{K}}$. Pumpout current associated with the DVM’s autozero cycle can
be mitigated by disabling autozero. However, we found that this increased the
Type A uncertainty of the measurements. Even though our indirect measurements
incorporated current reversals, these would not correct for the DVM’s internal
drift. Instead, we added a delay between the autozero and the start of the
DVM’s integration cycle to ensure that residual pumpout currents are
negligible.

### S2.2 Extended discussion: QAH characterization

From the Methods section of the main text: prior to characterization, the
quantum anomalous Hall (QAH) device was magnetized by applying a 1 T out-of-
plane magnetic field and then reducing the field to 0 T.

To characterize the QAH device, quantization of the Hall resistance relative
to $R_{\mathrm{K}}$, $\delta
R_{yx}(I,f)=(R_{yx}(I,f)-R_{\mathrm{K}})/R_{\mathrm{K}}$, was measured using
commercial CCC resistance bridge – designed by Physikalisch-Technische
Bundesanstalt (PTB) and manufactured by Magnicon [S1–S3]. A general
description of the methodology used to measure $\delta R_{yx}(I,f)$ is given
below but extensive details can be found in previous publications by Fox, et
al. [S4] and Rodenbach, et al. [S5], both of whom used the CCC from this work
to characterize the Hall resistance of similar QAH samples.

A simplified schematic depicting how the CCC was used in the measurement of
$\delta R_{\mathrm{yx}}(I,f)$ is shown in Fig. S3(a). The main components of
the CCC are: two low-noise digital current sources, $I_{1}$ and $I_{2}$, used
to source currents across two resistors that are to be compared, $R_{1}$ and
$R_{2}$; a nanovoltmeter (NVM) used to measure the difference in the potential
drops across the the two resistors; two sets of superconducting windings,
$N_{1}$ and $N_{2}$, which are housed in a superconducting toroidal screen
(not pictured); and a superconducting quantum interference device (SQUID) to
which the windings are inductively coupled.

The CCC works by comparing an unknown resistance to one that is precisely
known. In this case, $R_{1}=R_{yx}$ is the unknown resistance which is biased
by current source $I_{1}$. The known resistor, biased by current source
$I_{2}$, was chosen to be NIST’s 100 $\Omega$ calibrated standard resistor
($R_{2}=R_{100}$) with a known drift of 32.7 n$\Omega$/$\Omega$/year. The
windings are chosen such that their ratio is as close as possible to the ratio
of the resistors. In this work, windings $N_{\mathrm{1}}=2065$ and
$N_{\mathrm{2}}=8$ were chosen. As the two currents pass through their
respective windings, any imbalance results in the presence of a net current
$|N_{1}I_{1}-N_{2}I_{2}|$ on the surface of the superconducting screen due to
the Meissner effect. The surface current produces a magnetic flux that can be
measured by the SQUID. This measurement is fed through a compensation network
to produce a feedback signal that keeps the net flux constant and thereby
holds the ratio of the two currents fixed. With the proper choice of $N_{1}$
and $N_{2}$, the voltage drops across the resistors are approximately the
same. The small difference in the two potential drops is measured with the NVM
via $V_{\mathrm{NVM}}=I_{1}R_{1}-I_{2}R_{2}$. A precise measurement of the
ratio of the two resistors is then obtained and the value of the unknown
resistance can be determined.

To limit the the effects of thermal instability, $R_{100}$ was placed in a
temperature-controlled oil bath. To eliminate thermoelectric offsets caused by
linear thermal drift, the currents, $I_{1}$ and $I_{2}$, reversed polarity
every half-cycle (30 s). A total of 30 samples of $V_{\mathrm{NVM}}$ were
taken every half-cycle, the first 14 of which were ignored to allow for
stabilization after current polarity reversal. Each measurements of $\delta
R_{yx}$ consisted of at least 30 cycles.

Figure S3: Measurements of the Hall resistance using the CCC. (a) Shown is a
simplified schematics of the CCC as it was used to measure $\delta R_{yx}$.
(b) (Fig. E1 from the Extended Data section) Independent in situ
characterization of the QAH device. Deviation of the Hall resistance from
quantization $\delta R_{yx}$, was measured as a function of bias current $I$
with (red, open circles) and without (blue, filled circles an microwave
excitation being supplied to the PJVS. Heating of the QAH device due to
microwave radiation leakage is seen as an increase in $|\delta R_{yx}|$ when
the microwave excitation is on compared to when it is off, for a given value
of $I$. Heating of the QAH device was heavily dependant on the value of the
bias frequency $f$. Heating effects were minimal for
$f=f_{0}$,$f_{1}$,$f_{2}$. Measurements of $\delta R_{yx}$ are used to
determine $R_{\mathrm{QAH}}(I,f)$ seen in Eq. (2) of the main text via
$R_{\mathrm{QAH}}(I,f)=R_{\mathrm{K}}+\delta R_{yx}(I,f)$

The PJVS’s microwave bias frequency had no impact on the electrical set up
shown in Fig. S3(a) or on the measurement procedure described above. The only
effect of this bias was increasing the electron temperature of the QAH device
(Section S5). This effect is undesirable and future implementations of the QCS
will feature improved radiative shielding to mitigate microwave leakage.

Measurements of $\delta R_{yx}$ as a function of $I$ were first done with the
microwave excitation to the PJVS turned off ($f=0$), as a baseline measure of
sample quality. The measurements were then repeated with the microwave
excitation to the PJVS turned on but with the PJVS inactive (i.e. $f\neq 0$
but $V_{\mathrm{J}}=0$ V). This was done to simulate the environment that the
QAH device would experience while being used in the QCS configuration. By
comparing these two sets of measurements, the effects of microwave leakage to
the QAH sample could be qualified. Indeed, it was found that certain
frequencies caused large deviations from baseline behavior, while others had
only moderate effects (full discussion on this topic can be found in Section
S5). Furthermore, in some instances, the equilibration time of $\delta R_{yx}$
was found to be on the order of several hours, even for small ($\sim 1\%$)
changes in $f$. As a result, unless otherwise stated, all measurements taken
with the microwave excitation turned on were done so at one of three fixed
frequencies (0 dBm): $f_{\mathrm{0}}=9.701102806$ GHz, $f_{1}=18.651341$ GHz,
or $f_{2}=15.3020182$ GHz, all three of which were shown have only modest
impact on the quantization of the QAH device.For clarity, Table 3 in Section
S4.2 shows which of the three frequencies were were used during each QCS
measurement of $I$.

Figure S3(b) shows the measurements of $\delta R_{yx}(I,f)$ taken as a
function of bias current, $I$. When the microwave excitation was off (blue
filled circles) $|\delta R_{yx}|\leq(0.90\pm 0.68)$ $\mu\Omega/\Omega$ for
currents ranging from $I=9.33$ nA to $I=252$ nA. In comparison, when the
microwave excitation to the PJVS turned on with $f\neq 0$ (open markers),
heating of the QAH device due to radiation leakage was observed as an increase
in $|\delta R_{yx}|$ beyond the baseline value for a given $I$. Regardless,
for $f=f_{0}$,$f_{1}$, or $f_{2}$ the deviations for all $I$ were small enough
that they did not impact the overall function of the QCS (i.e. they are not
the leading source of error). These deviations were taken into account during
operation of the QCS though use of the measured value of
$R_{\mathrm{QAH}}(I,f)=R_{\mathrm{K}}+\delta R_{yx}(I,f)$ as opposed to a
fixed value of $R_{\mathrm{K}}$.

In summary, measurements of $\delta R_{yx}(I,f)$ were made for all values of
$I$ measured using the QCS. The frequency $f$ was one of the three found to
have minimal heating effects ($f_{\mathrm{0}}=9.701102806$ GHz,
$f_{1}=18.651341$ GHz, or $f_{2}=15.3020182$ GHz). The particular choice of
$f$ for a given $I$ was such that $V_{\mathrm{J}}$ was as close as possible to
$V_{\mathrm{QAH}}=IR_{\mathrm{K}}$ (see Section S4.2). Figure S3(b) shows the
measured values of $\delta R_{yx}(I,f)$ used in this work. These values were
used to compute the overall Hall resistance of the QAH device used in Eq. (2)
of the main text via:

$R_{\mathrm{QAH}}(I,f)=R_{\mathrm{K}}+\delta R_{yx}(I,f).$ (S1)

## S3 QAH Device and Measurements

### S3.1 Extended details: QAH film growth and Hall bar fabrication

The Cr-doped (Bi,Sb)2Te3 (Cr-BST) material used in this work was grown on epi-
ready semi-insulating GaAs (111)B substrate in an ultra-high vacuum, Perkin-
Elmer molecular beam epitaxy (MBE) system. Before growth, the substrate was
loaded into the MBE chamber and pre-annealed at a temperature of 630 °C in a
Te-rich environment to remove the oxide on the surface. During growth, high-
purity Bi, Sb, Cr and Te were evaporated from standard Knudsen cells. The
substrate was kept at 200 °C. The reflection high-energy electron diffraction
(RHEED) in situ was used to monitor the quality and thickness of the
materials.

The Hall bar device (shown on the right in Fig. 1 of the main text) was
fabricated using a combination of photolithography, metal evaporation, wet and
dry etching and atomic layer deposition. Prior to all photolithography steps,
the chip was cleaned in acetone and isopropyl alcohol, coated in SPR 3612
photoresist (spun at spun at 5.5 krpm for 45 s) and baked at 80 °C for 2 min.
After exposure, the resist was developed in a phosphate salt developer
followed by two water rinses.

First, the device mesa was patterned using direct write photolithography and
defined using Ar ion milling. Sidewalls from the mesa definition process were
removed using a 30 s acetone sonication and solvent rinse. Metallic contact
regions were defined using photolithography. The 5 nm / 90 nm Ti/Au contacts
were deposited using electron-beam (e-beam) evaporation following a 10 s in
situ Ar pre-etch to remove photoresist residue from the contact region. After
metal lift-off in acetone, a 1 nm Al seed layer was evaporated globally at a
rate of 0.2 Å/s. Low temperature atomic layer deposition was then used to grow
the 40 nm alumina gate dielectric over the entire surface of the thin film.
The gate electrode was patterned using photolithography and the gate metal (5
nm / 80 nm Ti/Au) was deposited using e-beam evaporation. After metal liftoff,
a 150 s masked wet etch was used to remove the alumina covering the contact
pads. The mask was defined using photolithography and a tetramethylammonium
hydroxide (TMAH)-based developer served as the etchant.

### S3.2 Lock-in characterization of Cr-BST thin-film

In an ideal QAH sample, zero longitudinal resistivity, $\rho_{xx}=0$, occurs
simultaneously with quantized Hall resistivity, $\rho_{yx}=h/e^{2}$ (note that
in 2D $R_{yx}=\rho_{yx}$). Measurements of $\rho_{xx}$ as a function of gate
voltage and temperature are often used to determine when the electrostatic top
gate has tuned the Fermi level to the center of the magnetic exchange gap,
where the QAH state is most robust.

Empirically, we have observed that position of the native Fermi level in our
Cr-BST films shifts over time, with optimal gate voltages moving towards more
positive values. This observation holds true for the sample studied here where
the optimal gate voltage appears to have shifted to be near $V_{g}=0$ V
(described in Section S3.3 below) compared to the optimal gate voltage of
$V_{g}=-1.35$ V observed in different sample made from the same film two years
earlier.

Two years prior to the measurements presented in the main text, a 100 $\mu$m
wide Hall bar was fabricated from the same 6-QL Cr-BST thin-film described in
Section S3.1. This device was was characterized using standard lock-in
amplifier techniques. A source current of $I=5$ nA RMS biased the Hall bar.
Measurements of the longitudinal voltage $V_{xx}$ and Hall (transverse)
voltage $V_{yx}$ were made as a function of top gate voltage $V_{g}$ and
temperature $T$. From these measurements, he longitudinal resistivity
$\rho_{xx}$ was calculated as:

$\rho_{xx}=\frac{w}{l}\frac{V_{xx}}{I}$ (S2)

where $w$ is the transverse width of the Hall bar device and $l$ is the
center-to-center distance between the voltage probes used to measure $V_{xx}$.
The Hall resistivity $\rho_{yx}$ was also calculated:

$\rho_{yx}=\frac{V_{yx}}{I}$ (S3)

along with the longitudinal conductivity:

$\sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^{2}+\rho_{yx}^{2}}$ (S4)

and the Hall conductivity:

$\sigma_{xy}=\frac{\rho_{yx}}{\rho_{xx}^{2}+\rho_{yx}^{2}}$ (S5)

.

These measurements are shown in Fig. S4(a)–(d). At base temperature we observe
a large plateau spanning $V_{g}\approx-6$ V to $V_{g}\approx 3$ V where
$\rho_{xx}=\sigma_{xx}\approx 0$, $\rho_{yx}=h/e^{2}$, and
$\sigma_{xy}=e^{2}/h$ indicating that the Fermi level is positioned within the
magnetic exchange gap. Beyond these gate voltages, increased dissipation and
deviations from quantization become apparent as a the Femi level approaches
the dissipative surface states. Within the plateau, dissipation appears
thermally activated with $\sigma_{xx}\propto e^{-T_{0}/T}$ above $T\approx
160$ mK as seen in Fig. S4(d). The fitted thermal activation temperature scale
$T_{0}$ peaks at $V_{g}=-1.35$ V with $T_{0}=1041$ mK as shown in Fig. S4(e).

Figure S4: Lock-in characterization of a Hall bar device fabricated from the
same film that used to make the QAH device used in the QCS. Fabrication and
measurement of this device took place two years prior to the measurements
outlined in the main text. The longitudinal resistivity (a) and longitudinal
conductivity (c), as well as Hall resistivity (b) and Hall conductivity (d)
are plotted as a function of top gate bias voltage $V_{g}$ for different
values of the mixing chamber temperature $T$. Base temperature was
approximately $30$ mK. (e) Fitted thermal activation temperature scale $T_{0}$
as a function of $V_{g}$ shows that by modulating the potential of the top
gate we are able to tune the Fermi level through the center of the magnetic
exchange gap. (f) The longitudinal conductivity $\sigma_{xx}$ measured at
$V_{g}=-1.35$ V is plotted as a function of inverse temperature on a
$\log$-$\log$ scale. A fit to thermally activated conduction
$\sigma_{xx}\propto e^{-T_{0}/T}$ for $T>160$ mK is shown as a solid line.
Empirical evidence suggests that the film use in the main text should have a
magnetic exchange gap comparable to magnitude that shown in (e) but shifted
such that $T_{0}^{\mathrm{max}}$ is near $V_{g}=0$ V.

### S3.3 Measuring $\rho_{xx}$ using the CCC’s nanovoltmeter

After the QAH device was cooled to base temperature, the device was magnetized
by applying a 1 T field out of plane and then reducing the field to 0 T. The
CCC’s NVM was used measure $\rho_{xx}$ as was done in a previous work [S5]. A
schematic depicting the longitudinal measurement configuration is shown in
Fig. S5. It is important to note that Fig. S5 is only depicting the QAH device
within the context of longitudinal resistivity measurements and does not
include the PJVS or its associate electronics. The PJVS was present during
these longitudinal measurements, and connected to the QAH device via the
interposer (see Section S9) but was inactive (i.e. the microwave excitation
was turned off and $V_{\mathrm{J}}=0$ V).

Figure S5: Using the CCC’s nanovoltmeter to measure longitudinal resistivity
of the QAH device. A simplified schematic of the CCC as it was used for
longitudinal voltage measurements is shown. The QAH device is biased with
current $I$ supplied by digital current source $I_{1}$. $R_{2}$ is shorted,
indicated by the red wire, so the nanovoltmeter can be accessed directly.
Current source $I_{2}$ is turned off as is the CCC’s compensation network.
$N_{1}=1$ and $N_{2}=0$. This schematic does not show the PJVS or its
connection to the QAH device as discussed in the main text.

To measure $\rho_{xx}$, using the CCC’s NVM, the digital current source
$I_{1}$ biased the QAH device. The second resistor network was shorted so the
CCC’s NVM could be accessed independently and used to measure $V_{xx}$. The
CCC’s compensation network and digital current source $I_{2}$ were both
inactive. Measurements of $V_{xx}$ were then used to compute the longitudinal
resistivity via Eq. S2.

### S3.4 Determining the optimal gate voltage of the QAH device

Measurements of $\rho_{xx}$ as a function of top gate voltage $V_{g}$ were
made with a fixed 100 nA bias current and are shown in Fig. S6. Measurements
of $\rho_{xx}$ made just after the device had been magnetized when the
temperature at the sample stage was 9 mK, show a local minimum near $V_{g}=0$
V(red dots). Fig. S6 also shows measurements of $\rho_{xx}$ made when the
sample stage temperature was 8.8 mK (blue squares). This second set of
measurements took place after the system was given several hours to
equilibrate since being heated by ramping the superconducting magnet during
the initial magnetization process.

Figure S6: Longitudinal resistivity as a function of top gate voltage measured
using the CCC’s NVM. Measurements of $\rho_{xx}$ were taken directly after
magnetizing the device (red dots) and after the system was given several hours
to equilibrate (blue squares). The sensitivity of $\rho_{xx}$ to temperature
can seen from the decrease in the magnitude of $\rho_{xx}$ around $V_{g}=0$ V
between the two sets of data. The local minimum in $\rho_{xx}$ seen near
$V_{g}=0$ V indicates that the native Fermi level is well positioned near the
center of the magnetic exchange gap.

Given that the minimum value of $\rho_{xx}$ was observed for top gate voltages
very near 0 V, we concluded that the native Fermi level was well positioned
near the center of the magnetic exchange gap. Therefore, all measurements were
taken with zero volts applied to the top gate.

## S4 QCS Measurements

### S4.1 The CCC’s digital current source

As stated in the main text, the CCC’s digital current source acted as the bias
source for the QAH device during QCS measurements and was also the current
source to be measured by the QCS.

Importantly, the current source operates using a 16-bit digital-to-analog
converter (DAC)[40]. Therefore $I$ could not be changed continuously but had
to be incremented in pre-defined steps using a graphical user interface (GUI).

In the manuscript the source currents $I$ are referenced to 3 significant
figures for simplicity. Additionally, and any currents that were within 500
$\mu$A$/$A of one another were grouped into a signal data point (using the
arithmetic mean) for clarity. Table 2 below shows the 11 values of current
measured in this work. The first column, $I_{\mathrm{GUI}}$, gives the value
of sourced current as it was shown on the CCC’s GUI. The second column,
$\Delta I_{GUI}$, gives the step size to the next highest value on the GUI
that was measured (if applicable for the manuscript). The third column gives
the value $I$ that was used to reference the corresponding value of
$I_{\mathrm{GUI}}$ in the manuscript.

Generally it was found that the CCC’s GUI indicated a slightly larger value of
nominal current than what was measured by either $I_{\mathrm{direct}}$ or
$I_{\mathrm{indirect}}$. The largest disagreement between $I_{\mathrm{GUI}}$
and $I_{\mathrm{direct}}$ or $I_{\mathrm{indirect}}$ was roughly -300
$\mu$A$/$A for $I=83.9$ nA. As described in Section S2.2, in normal operation
of the CCC it is ratio of $I_{1}$ and $I_{2}$ that matters, not the absolute
value of either bias current. Accordingly these small deviations (on the order
of tens or hundreds of $\mu$A$/$A) do not impact the overall function of the
CCC. It is therefore unsurprising that these deviations have not been noticed
during prior operation of the instrument.

Table 2:  Currents measured by the QCS. The first column shows $I_{\mathrm{GUI}}$, the nominal value of the current sourced by the CCC as indicated by the CCC’s GUI. The second column shows $\Delta I_{\mathrm{GUI}}$, the change in $I_{\mathrm{GUI}}$ between adjacent rows (if applicable). The third column shows how each value of $I_{\mathrm{GUI}}$ was referenced in the main text. For ease of reading, $I$ was rounded to three significant figures and currents within 500 $\mu$A$/$A of one another (i.e. $\Delta I_{\mathrm{GUI}}<500$ $\mu$A$/$A) were grouped together as indicated in column three.  $I_{\mathrm{GUI}}$ (nA) |  $\Delta I_{\mathrm{GUI}}$ ($\mu$A$/$A) |  $I$ (nA)
---|---|---
9.329500 | – | 9.33
27.97940 | 325.2 | 28.0
27.98850 | – | 28.0
53.78890 | – | 53.8
83.92895 | 109.0 | 83.9
83.93810 | 109.6 | 83.9
83.94730 | – | 83.9
132.3893 | 68.7 | 132
132.3984 | – | 132
251.8235 | 36.1 | 252
251.8326 | – | 252

### S4.2 Corresponding values of $I$, $N$, $f$ and $V_{\mathrm{J}}$

Three frequencies were shown to have limited impact on quantization
$R_{\mathrm{QAH}}$ (see Section S5). These frequencies were
$f_{\mathrm{0}}=9.701102806$ GHz, $f_{1}=18.651341$ GHz, and
$f_{2}=15.3020182$ GHz. All data were acquired at one of these three
frequencies, with the exception the variational measurements of Fig. 3.

The PJVS output voltage is given by:

$V_{\mathrm{J}}=\frac{Nf}{K_{\mathrm{J}}}$ (S6)

where $N$ is the number of active Josephson junctions (JJs) biased at
frequency $f$ to be on their first Shapiro step, and $K_{\mathrm{J}}=2e/h$ is
the Josephson constant. As discussed in the Methods section of the main text,
individual JJs are grouped into subarrays on the PJVS and so $N$ can take
corresponding values of 12, 36, 108, and 324. The constraints on $f$ and $N$
limited the possible values of $V_{\mathrm{J}}$ accessible during QCS
measurements. We emphasize that constraints on $V_{\mathrm{J}}$ do not limit
the accuracy or precision of QCS because the offset of $V_{\mathrm{J}}$ from
$V_{\mathrm{QAH}}=IR_{\mathrm{QAH}}$ is measured directly as described in the
main text. However, choosing $V_{\mathrm{J}}$ to be as close as possible to
$V_{\mathrm{QAH}}$ allows one to make use of the benefits associate with
making a null measurement as opposed to a full-scale measurement of the
voltage. The values of $f$, $N$, $V_{\mathrm{J}}$ used during QCS measurements
of current $I$ are given in Table 3.

Table 3:  PJVS settings for each QAH bias current $I$ (first column). Second column: $f$. Third column: $N$. Fourth column: resulting PJVS output voltage calculated using Eq. S6. $I$ (nA) | $f$ | N |  $V_{\mathrm{J}}$ (mV)
---|---|---|---
9.33 | $f_{0}$ | 12 | 0.2407232
28.0 | $f_{0}$ | 36 | 0.7221697
53.8 | $f_{1}$ | 36 | 1.388444
83.9 | $f_{0}$ | 108 | 2.166442
132 | $f_{2}$ | 108 | 3.416889
252 | $f_{0}$ | 324 | 6.499527

### S4.3 Determination of ${V_{\mathrm{\textbf{null}}}}$ and
${I_{\mathrm{\textbf{direct}}}}$

Measurements of $I$ using the QCS ultimately amount to measurement of
$I_{\mathrm{direct}}$ via Eq. (2) of the main text. Measurement of
$I_{\mathrm{direct}}$ requires accurate measurement of $V_{\mathrm{null}}$. To
eliminate thermoelectric offsets in $V_{\mathrm{null}}$ the polarity of the
input current $I$ was reversed periodically for a total of $n$ cycles. For all
measurements $10\leq n\leq 90$. Fig. S7(a) shows an example measurement of
$V_{\mathrm{null}}$ plotted as a function of of the number of samples taken
(red dots) taken when the bias current $I=83.9$ nA. For this particular
measurement $n=10$ polarity cycles were preformed.

Figure S7: Example measurement of $V_{\mathrm{null}}$ taken for $I=83.9$ nA.
(a) 800 Individual measurements of $V_{\mathrm{null}}$ over several polarity
reversal cycles are shown as red dots. The triangular markers show
$V^{+}_{\mathrm{null}}$ ($V^{-}_{\mathrm{null}}$), the average value of
$V_{\mathrm{null}}$ for the positive (negative) portion of a polarity reversal
cycle. (b) The 18 measurements of $V_{\mathrm{diff}}$ calculated from the data
in (a) via Eq. (S7) are shown as cross-markers. The solid green line shows the
overall average value of $V_{\mathrm{null}}$ calculated from the average of
the measurements of $V_{\mathrm{diff}}$. The dashed line shows the expected
value of $V_{\mathrm{null}}$ calculated from Eq. (S8). (c) An expanded view of
the trendline data shown in (b).

For the measurement shown in Fig. S7, current polarity reversal occurred every
60 seconds. After each reversal the system was given 10 seconds to
equilibrate. $V_{\mathrm{null}}$ was then sampled 40 times per polarity at a
rate of one sample per second.

The overall value of $V_{\mathrm{null}}$ used in Eq. (2) of the main text was
determined using the following procedure. First, the average value of
$V_{\mathrm{null}}$ for each polarity of the cycles was calculated. Let
$V^{+}_{\mathrm{null}}$ ($V^{-}_{\mathrm{null}}$) be the average value of
$V_{\mathrm{null}}$ for the positive (negative) portion of a polarity reversal
cycle (shown a triangular markers in Fig. S7(a)). The difference between
$V^{+}_{\mathrm{null}}$ and $V^{-}_{\mathrm{null}}$ for consecutive
consecutive polarities was then calculated as:

$V_{\mathrm{diff}}=\frac{1}{2}\left(V_{\mathrm{null}}^{+}-V_{\mathrm{null}}^{-}\right)$
(S7)

yielding a total of $2n-1$ values for $V_{\mathrm{diff}}$ each with an
independent standard deviation $\sigma_{\mathrm{diff}}$. These 2n-1 values of
$V_{\mathrm{diff}}$ are again averaged using an inverse variance weighting to
produce a final averaged value of $V_{\mathrm{null}}$. It is this final
averaged value that is used to calculate $I_{\mathrm{direct}}$ via Eq. (2) of
the main text.

Fig. S7(b) shows the 19 values of $V_{\mathrm{diff}}$ obtained using the
$V_{\mathrm{null}}$ data from Fig. S7(a). The final averaged value of these
data points is shown as a solid green line
(‘$V_{\mathrm{null}}^{\mathrm{avg}}$ measured’).

Also shown in Fig. S7(b) is the expected average value of $V_{\mathrm{null}}$
(black dashed lined). This is the final averaged value of $V_{\mathrm{null}}$
that one expects to measured based on the measurements of $I$ made using the
indirect measurement scheme shown in Fig. 2(a) of the main text. Concretely,
‘$V_{\mathrm{null}}^{\mathrm{avg}}$ expected’ was determined by:

$V_{\mathrm{null}}^{\mathrm{avg}}\mathrm{(expected)}=I_{\mathrm{indirect}}R_{\mathrm{K}}-NfK_{\mathrm{J}}^{-1}.$
(S8)

### S4.4 Comparison of $I_{\mathrm{\textbf{direct}}}$,
$I_{\mathrm{\textbf{indirect}}}$, and $I_{\mathrm{\textbf{GUI}}}$

As discussed in the main text, one current source was used for all
measurements presented in this work. Table 4 below shows $I_{\mathrm{GUI}}$,
the nominal sourced current as indicated by the CCC’s GUI. In the main text
currents within one setting of another were grouped into an average value for
ease of display. The individual values are given in Table 4 below.

Table 4:  Comparison of $I_{\mathrm{direct}}$, $I_{\mathrm{indirect}}$, and $I_{\mathrm{GUI}}$. First column: the value $I$ used to reference $I_{\mathrm{GUI}}$ in the main text (discussed in Section S4.1). Second column: $I_{\mathrm{GUI}}$. Third and fourth column: measured values of $I_{\mathrm{direct}}$ and $I_{\mathrm{indirect}}$ respectively. $I$ (nA) |  $I_{\mathrm{GUI}}$ (nA) |  $I_{\mathrm{direct}_{\textrm{ }}}^{\mathrm{avg}}$ (nA) |  $I_{\mathrm{indirect}}^{\mathrm{avg}^{\textrm{ }^{\textrm{ }}}}$ (nA)
---|---|---|---
9.3 | 9.329500 | 9.328131 | 9.328337
27.8 | 27.97940 | 27.97921 | 27.97988
27.8 | 27.98850 | 27.98659 | 27.98659
53.8 | 53.78890 | 53.78686 | 53.78732
83.9 | 83.92895 | 83.91331 | 83.91356
83.9 | 83.93810 | 83.92265 | 83.92263
83.9 | 83.94730 | 83.92946 | 83.92964
132.4 | 132.3893 | 132.3733 | 132.3750
132.4 | 132.3984 | 132.3832 | 132.3843
251.8 | 251.8235 | 251.7836 | 251.7877
251.8 | 251.8326 | 251.7938 | 251.7972

## S5 Heating of the QAH device via microwave leakage

It is well documented that both the longitudinal and Hall resistivities of QAH
devices are strongly dependent on temperature. This can be seen in Section
S3.2 where $\rho_{xx}$ at $V_{g}=-1.35$ V increased by two orders of magnitude
between $T=30$ mK and $T=1.7$ K while $\rho_{yx}$ decreased by roughly 20%.
The sensitivity of QAH properties to changes in either lattice or electron
temperature must therefore be considered whenever these materials are used.

As noted in the main text, heating of the QAH device was observed due to
leakage of the PJVS’s microwave bias. The measurements shown below were taken
before those presented in the main text. Initially, the system was cooled
without any shielding placed around the PJVS or the QAH device. It immediately
became apparent that the QAH device was experiencing significant heating when
the microwave bias to the PJVS was on. This can be seen in Fig. S8 by
comparing measurements of $\rho_{xx}$ taken with the microwave bias turned off
($f=0$, blue triangles) and those taken with the microwave bias on when there
was no radiative shielding present (red squares). At the lowest currents the
longitudinal resistivity measured with the microwave on was approximately
double that measured when it was off. At the highest currents the discrepancy
rose to be roughly a factor of 20. The same measurements were performed after
warming up the system and adding copper shielding around the PJVS, revealing a
substantial reduction in dissipation compared to measurements without the
shielding. We believe that the strong dependence on $f$ is related to
excitation of the cavity modes of the dilution refrigerator.

Figure S8: Effects of microwave leakage on the electron temperature of the QAH
device. (a) $\rho_{xx}$ measured as a function of bias current $I$ when the
microwave excitation to the PJVS was turned off (blue triangles), and when the
microwave bias was on. Measurements taken when the microwave bias was turned
on were done before (red squares) and after (purple circles) additional
radiative shielding was added. (b) The same data as in (a) but shown on a
semi-log scale. Error bars represent Type A ($k=1$) uncertainty and are
smaller than the markers in most cases. As in the main text,
$f_{\mathrm{0}}=9.701102806$ GHz (0 dBm). (c) Cooling of the QAH device after
a small variation in $f$. Thermalization of the device took more than 4 hours.
(d) Longitudinal resistivity (left axis) and sample stage thermometer reading
(right axis) as a function of time after turning on the PJVS RF bias
frequency. Here $f=16$ GHz, 0 dBm.

In some cases, after changing $f$ several hours were needed for the system to
come to equilibrium. This can be seen in Fig. S8(c) where, after changing $f$
by less than 1% from an initial frequency of $f_{0}$, more than four hours
were needed for the QAH device to equilibrate. Waiting for such long time
scales was not feasible and therefore the frequency was fixed to one of the
three for which heating was found to have minimal effects on
$R_{\mathrm{QAH}}$ (see Section S2.2).

Radiative leakage was also seen to affect the temperature reading of the
ruthenium oxide resistance thermometer near the sample stage where the QAH
device was located. Fig. S8(d) shows simultaneous measurements of $\rho_{xx}$
and $T$ (where $T$ is the ruthenium oxide resistance thermometer’s temperature
reading) as a function of time after the PJVS bias frequency was turned on
($f=16$ GHz, 0 dBm).

## S6 National Metrology Institutes Referenced for Calibration and Measurement
Capability Data

The blue shaded region in Figure 2(b) of the main text shows the typical
calibration precision ranges offered by the following national netrology
institutes: Istituto Nazionale di Ricerca Metrologica (INRIM), Korea Research
Institute of Standards and Science (KRISS), Van Swinden Laboratory (VSL), the
Swiss Federal Institute of Metrology (METAS), the National Institute of
Metrology, China (NIMC), VTT Technical Research Centre of Finland’s Centre for
Metrology (VTT), Laboratoire national de métrologie et d’essais (LNE),
Physikalisch-Technische Bundesanstalt (PTB), and Research Institutes of Sweden
(RISE).

## S7 Long Term Drift of the CCC’s Digital Current Source

Fig. S9 shows evidence of drift in the output current of the CCC’s digital
current source. The y-axis shows the deviation of the measured value of
$I_{\mathrm{indirect}}$ from the nominal value of the output current
$I_{\mathrm{GUI}}$ in units of $\mu$A$/$A. The x-axis shows the time (in days)
each measurement was taken. The error bars represent the Type A statistical
uncertainty of the measurements ($k=1$).

Two measurements of $I_{\mathrm{indirect}}$ were taken on day zero. Both of
these measurements indicated that the the output current of the CCC was
slightly lower in magnitude than the GUI indicated value but, within
uncertainty, these measurements were equal to one another. Eight days later,
the exact same measurement was repeated. Again the measurement indicated that
the output current of the CCC was slightly lower than the GUI indicated value.
However, the amount of deviation measured on day eight was not consistent with
what was measured on day zero. Instead, the measurement on day eight indicated
that the output current was slightly larger in magnitude than what is was on
day zero. This drift could explain the discrepancy between some of the
measurements of $I_{\mathrm{direct}}$ and $I_{\mathrm{indirect}}$ as discussed
in the main text.

Figure S9: Relative deviation of the measured value of $I_{\mathrm{indirect}}$
from the nominal output current $I_{\mathrm{GUI}}$ taken eight days apart. For
all three measurements the CCC’s digital current source was set to output
251.7777 nA. Measurements taken on the same day agree within uncertainty. When
the same measurement was repeated eight days later, the output current
appeared to have drifted to a slightly higher value so that the relative
difference between $I_{\mathrm{indirect}}$ and $I_{\mathrm{GUI}}$ was smaller
than on day zero.

## S8 Uncertainty Budget and Noise Analysis

### S8.1 Uncertainty budget for measurements of
$\mathbb{I}_{\mathrm{\mathbb{direct}}}$

This section provides additional details regarding the contributions to the
total uncertainty budget for measurements of $I_{\mathrm{direct}}$.

Type A contribution:

  * •

Null detector reading, $V_{\mathrm{null}}$: calculated as the standard
deviation of means of multiple polarity reversals within a single experimental
run. Lower values of current are naturally subject to larger amounts of
relative dispersion given the lower signal to noise ratio and thus this
contribution is largest for the lowest values of current.

Type B contributions:

  * •

NVM input impedance: When a digital voltmeter (DVM) is used to measure
$V_{yx}$, or the null between $V_{yx}$ and $V_{\mathrm{J}}$, a voltage divider
is formed between source impedances—such as the QAH device itself and
resistive filters in the wiring—and the input impedance of the DVM. The same
is true of any leakage resistance in the wiring. The Keysight 3458A DVM used
for the indirect measurements—in which only $V_{yx}$ is measured—was
experimentally determined to have an input impedance approaching 1 T$\Omega$.
Thus, its contribution to the uncertainty of the indirect current measurements
is less than 0.1 $\mu$A/A. In a null measurement, the uncertainty contribution
of the divider to the voltage estimate is diminished by the ratio of the null
voltage to $V_{\mathrm{J}}$ (or $V_{yx}$). The Keithley 2182A NVM used for
direct measurements of current was specified by its manufacturer to have an
input impedance greater than 10 G$\Omega$. Assuming a worst-case null ratio of
$10^{-3}$, the contribution of the NVM input impedance to the uncertainty
would be less than 0.005 $\mu$A/A.

  * •

NVM gain and linearity: Similar to above, for direct/null measurements, the
contribution of the nanovoltmeter’s gain and nonlinearity to the overall
uncertainty is diminished by the ratio of the null voltage to
$V_{\mathrm{J}}$. Even so, the Keithley 2182A used in these measurements was
calibrated using NIST’s conventional digital multimeter calibration service.
For the 10 mV range, the gain error was 24 $\mu$V/V and the nonlinearity was
less than 4 $\mu$V/V (note that minimum calibrated voltage was 1 mV). Assuming
the same worst-case null ratio of $10^{-3}$, the contribution of the
uncorrected NVM gain error and nonlinearity to the uncertainty would be less
than 0.03 $\mu$A/A.

  * •

PJVS microwave bias error: The laboratory where the DR was installed did not
have access to a GPS-disciplined frequency standard. Instead, we used a
commercial waveform generator (Keysight 33622A) with an onboard oven-
controlled quartz crystal oscillator (OCXO) as our 10 MHz frequency reference.
The RF bias for the PJVS was phase locked to this standard for all
measurements. We checked the frequency of the OCXO reference against a GPS-
disciplined standard in a separate laboratory prior to and after our
measurement campaign. Its frequency offset at both times was $-10\pm 5$
nHz/Hz. We also checked for phase locked loop deficiencies within the RF
generator by measurements against a high frequency counter. No issues were
found. We did not correct for the frequency offset of our reference in the
uncertainty budget because its contribution amounts to only 0.01 $\mu$A/A

  * •

$\delta R_{yx}$ combined uncertainty: $\delta R_{yx}$ is measured using a CCC
as a ratio measurement against a standard resistor. The following are the
sources of uncertainty for this measurement.

    * –

Type A uncertainty related to the dispersion of the bridge voltage difference
(BVD) measurements and its repeatability.

    * –

Type B uncertainty related to the SQUIDs rectification of noise causing an
offset in the BVD measurement. This can be measured by measuring the BVD with
the current sources off.

    * –

Compensation network drift: Type B uncertainty due to the drift of the
compensation network. The compensation network is a series-parallel
combination of resistors which act to bring the BVD as close to zero as
possible. Typically this network is calibrated once a year.

    * –

Standard resistor: Combined uncertainty of the standard resistor used to
characterize the QAH device. We used a calibrated 100 $\Omega$ standard
resistor as the reference for these measurements with an estimated uncertainty
of 3 n$\Omega/\Omega$ (k = 1)

    * –

Leakage: We estimated electric isolation of the lines and connections to be on
the order of 100 $\mathrm{G\SIUnitSymbolOhm}$. This results in a relative
error of a few $10^{-1}$ $\mu\Omega/\Omega$ and mostly limits the precision of
the resistance measurements.

    * –

CCC winding ratio error: The windings of the CCC used in this work were set by
connecting a series of windings with different turns in an arithmetic
combination. For ratios other than 1:1, the errors must be determined by a
step-up calibration procedure based on $1:1$ intercomparisons of all windings
[S6,S7] This error is small and on the order of a few parts in $10^{-10}$ but
must be taken into account.

### S8.2 Uncertainty budget for measurements of
$\mathbb{I}_{\mathrm{\mathbb{indirect}}}$

The uncertainty budget for measurements of $I_{\mathrm{indirect}}$ consisted
of the Type A uncertainty of the DVM readings ranged from 0.60 $\mu$A$/$A at
252 nA to 31.0 $\mu$A$/$A at 9.33 nA. The budget also includes the relative
Type B uncertainty of the DVM due to its finite input impedance, gain and non-
linearity. The total Type B relative uncertainty of $I_{\mathrm{indirect}}$
was estimated to be 1.01 $\mu$A$/$A.

### S8.3 Allan deviation calculation

For time-series or time-domain measurements (like the measurements of
$V_{\mathrm{null}}$ sampled at a fixed interval $\tau_{0}$), the magnitude of
the Allan deviation (referred to as $\sigma_{\mathrm{NVM}}$ in the main text),
is inversely indicative of the stability of the measurement (i.e. a smaller
magnitude indicates greater measurement stability). When plotted on a
$\log$-$\log$ scale, the functional dependence of $\sigma_{\mathrm{NVM}}$ on
$\tau$ is commonly used to determine the type of noise that dominates a
measurement over a given time scale [S8]. For example, the Allan deviation for
white noise limited data will scale as $\tau^{-1/2}$ whereas data dominated by
$1/f$ noise will have an Allan deviation that appears constant as a function
of $\tau$.

NIST special publication 1065 by Riley et al. [S9] provides extensive details
on stability analysis methods for time and frequency domain measurements. The
stability and noise analysis protocols used in this work largely followed the
guidance of that publication. Specifically, the time series data for
$V_{\mathrm{null}}$ was detrended by subtracting a linear function in time.
This removes correlations caused by linear thermal drift [S9].

Next, the overlapping Allan variance was calculated using equations (11) from
reference [S9]:

$\sigma_{\mathrm{NVM}}^{2}(\tau)=\frac{1}{2\left(N-2m\right)\tau^{2}}\sum\limits_{i=1}^{N-2m}\left(V_{\mathrm{null}_{i+2m}}-2V_{\mathrm{null}_{i+m}}+V_{\mathrm{null}_{i}}\right)^{2}$
(S9)

Here $V_{\mathrm{null}_{i}}$ is the $i^{\textrm{th}}$ sample of
$V_{\mathrm{null}}$ for $i=1,2,3,\ldots,N$, $m$ is the averaging factor, and
$\tau=m\tau_{0}$ is the general sampling interval. Use of the overlapping
Allan variance is increase the confidence in stability estimates by utilizing
all possible combinations of the data compared to the non-overlapping form
[S9].

## S9 The Interposer

To facilitate independent, in situ measurement of the QAH device, a custom
interposer was placed between the PJVS and the QAH device at the 4 K stage
micro-D connector. The interposer functioned as a straight-thru fnr cable,
which was shunted with a 1 nF ceramic capacitor to reduce differential noise
originating from the PJVS during direct mode measurements. When QAH
quantization or indirect current measurements were performed, the capacitor
was shunted by the superconducting state of the PJVS.

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[S6] K. Grohmann et al., IEEE Trans. Instrum. Meas., 1979, 28, 205–211.
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