Quantum mechanical current-to-voltage conversion with quantum Hall resistance array

Accurate measurement of the electric current requires a stable and calculable resistor for an ideal current to voltage conversion. However, the temporal resistance drift of a physical resistor is unavoidable, unlike the quantum Hall resistance directly linked to the Planck constant h and the elementary charge e. Lack of an invariant high resistance leads to a challenge in making small current measurements below 1 muA with an uncertainty better than one part in 106. In this work, we demonstrate a current to voltage conversion in the range from a few nano amps to one microamp with an invariant quantized Hall array resistance. The converted voltage is directly compared with the Josephson voltage reference in the framework of Ohm's law. Markedly distinct from the classical conversion, which relies on an artifact resistance reference, this current-to-voltage conversion does not demand timely resistance calibrations. It improves the precision of current measurement down to 8 10 -8 at 1 muA.


Introduction
The precise measurement of small-currents is of prime importance for fundamental research and practical applications. The International System of Units (SI) has been redefined with advances in quantum metrology for the past half-century and the new SI has come into effect since 20 May 2019 [1]. The revision of the SI is based on the philosophy that the SI units are inherently linked to invariant constants of nature via the laws of physics, not to variable physical artifacts. In the revised SI, the ampere is a flow of 1 (1.602 176 634 × 10 −19 ) ⁄ individual elementary charges per second. Existing single-electron current sources following this definition can hardly increase their current value accurately above 1 nA owing to the absence of a robust correlated quantum mechanical state of matter and the uncontrollable highorder processes [2]. This results in a poor precision of the electrical current in any accessible range far from the uncertainty of a part in 10 9 by orders of magnitude. In contrast, the voltage and the resistance can be realized stably and reproducibly below a part in 10 9 , stemming from the macroscopic quantum mechanical states of matters, i.e., the weakly coupled superconducting states leading to the Josephson effect [3] and the topologically protected quantum Hall state [4], respectively. For an accurate current, the well-established voltage and resistance quantum standards can be utilized with Ohm's law.
Needs for accurate small-current measurements are ubiquitous. Environmental monitoring of particulate matter and aerosol nanoparticles requires small-current measurements at the femto-ampere level [5]. The current level of radiation dosimetry applications for medical diagnoses and therapies is smaller than 10 pA [6]. Semiconductor industries also require small-current measurements, including leakage current characterization in nanoscale devices from femto-amperes to micro-amperes, an electron-beam intensity of lithography acquired by a Faraday-cup electrometer down to the pico-ampere level, and ionic conduction measurements in organic materials at the pico-ampere level.
In practice, however, achieving electrical current precision smaller than a part in 10 6 is challenging in most ranges. Conventional methods of current evaluation are categorized as either a capacitor-charging method for current below 1 nA or a voltage-to-resistor method for higher current [7]. In the capacitor-charging method, an accurate current is generated for the current measurement by charging a stable capacitor with a linearly ramped voltage with time.
A dominant uncertainty is attributed to a deviation of the capacitance value at the direct current limit from its value determined at the alternative current with a frequency of 1 kHz as well as a drift in time. The uncertainty is consequently larger than a part in 10 5 . For the measurement and generation of a current larger than 1 nA, artifact resistance and voltage references traceable to the quantum Hall and Josephson effects, respectively, are employed, relying on Ohm's law.
Uncertainty regarding the current value also arises from the intrinsic temporal drift and the environmental change of artifacts as well as the long calibration chains. Hence, the overall uncertainty typically becomes larger than a part in 10 6 .
Recently at the National Metrology Institute of Japan has recently been investigated systematically [13].
It turned out that the 1 MΩ QHRA is almost invariant for at least approximately 10 months within the measurement uncertainty of a few parts in 10 8 [13]. Furthermore, the realization of 10 MΩ QHRA and long-term stability are under investigation [14]. This high-value QHRA can bridge the single-electron current source and the Josephson voltage reference through Ohm's law. A consistency test of three electrical quantities realized by different physical effects is known as the quantum metrology triangle experiment [15][16][17].
Here we demonstrate a current-to-voltage conversion from a few nano amps to one that the QHRA can be exploited for an ideal current-to-voltage conversion for an unknown small-current measurement below 1 μA without frequent resistance calibrations. An Allan deviation study shows that the precision of the current measurement can also be improved beyond that of the classical conversion based on a physical resistor by at least two orders of magnitude.

Stability of quantum Hall resistance array
The QHRA was designed by the continued fractional expansion method and realized by integrating 88 quantum Hall devices with the following nominal value, almost 100-fold larger than a single quantum Hall resistance, at filling factor 2 ( The temporal drift is smaller than 30 × 10 −9 month ⁄ as shown in Fig. 1d and the temperature coefficient is smaller than 50 × 10 −9 ℃ ⁄ [19].

Experimental setup
The experimental setup is illustrated in Fig. 2a. Experiments were performed in a temperature-stabilized laboratory with a temperature change of less than 0.1 ℃ around 23 ℃.
To perform a current-to-voltage conversion experiment with the 1 MΩ QHRA, we need an ideal and invariant current source. Owing to the absence of an ideal current source, we have

Precision measurements of current-to-voltage conversion
Data sets were acquired for approximately 9 hours to determine the average difference between two current values. We note that the limited holding time of the base temperature near 0.3 K is approximately 9 hours for the single-shot 3 He refrigerator employed in this work.   [23]. Square, pentagon, and diamond symbols depict the measurement uncertainties for the ULCA (PTB) [11], the measurement of single electron pump with a timely calibrated resistor (NPL) [9], and the programmable quantum mechanical current generator (LNE) [8], respectively. The x-axis is the nominal current value driven in the circuit.
The red hexagon and green square symbols represent the relative difference and uncertainty, respectively. The relative difference is smaller than the uncertainty. This indicates that the current values measured by the 1-MΩ-resistance reference and 1 MΩ QHRA coincide with each other within the uncertainty. This reflects the fact that the 1 MΩ QHRA can be utilized for an ideal current-to-voltage conversion. The achieved uncertainty at the nominal current of 1 μA is smaller than one part in 10 7 . This value is smaller than that of the CMC based on an artifact resistance reference by at least one order of magnitude. The uncertainty increases logarithmically with decreasing current. Detailed uncertainty analysis is presented in the next section.

Uncertainty budget
The relative current difference ( ) is expressed as follows;  Table 1 summarizes important uncertainty contributions for nominal current values of 1 µA and 10 nA. The uncertainty for the 1 MΩ resistance calibrated by a CCC resistance bridge is described in detail in a previous report [13]. The corresponding relative uncertainty is smaller than 20×10 -9 . The limited resolution (10 nV) of the digital voltmeter (DVM) (Keysight 3458A) for the voltage measurement may also introduce uncertainty. The corresponding relative uncertainties for 1 µA and 10 nA become 10×10 -9 and 1×10 -6 , respectively. The finite input impedance of the used DVM introduces an error in the voltage measurement. We evaluated the input resistance [24]. It is approximately 10 TΩ. For the direct voltage measurement by the DVM, the relative uncertainty becomes 0.1×10 -6 for the measurement of the voltage drop across the 1-MΩ resistor. However, since the input voltage is typically reduced down to tens of µV by the balancing PJVS, the relative uncertainty is reduced accordingly by orders of magnitude. For instance, the relative uncertainties of the input impedance for 1 µA and 10 nA become 1×10 -11 and 1×10 -9 , respectively. Finally, the statistical type-A uncertainties of the voltage ratios become 76×10 -9 and 4.0×10 -6 , respectively. The measurement uncertainty is calculated by taking into account the dominant uncertainty contributions as shown in Table 1.

Discussions
The precision of the current-to-voltage conversion in this experiment may not be the fundamental limit. Two improvements for measurement can be conceived. For measurement of a voltage from high impedance, a charge injection [22,23] from the input circuitry of a digital voltmeter needs to be taken into account. The temporal behaviour of the injection current of the employed digital voltmeter is plotted in Supplementary material. To suppress an excess voltage in high resistance induced by the injection current at the pico-ampere level, a differential nanovoltmeter with an input chopper [25] can be employed for future experiments.
Another experimental factor for the limited precision is the data acquisition time stemming from the restricted operational time of the single-shot 3 He cryostat used. Therefore, the statistical uncertainty can be reduced according to the Allan deviation plot in Fig.3b if a longer measurement is allowed with a 3 He cryostat that is operational in a continuous flow mode.
Quantized Hall resistance in array does not require timely calibrations for the currentto-voltage conversion according to the stability investigation even though the measurement uncertainty is comparable with that of timely calibrated ULCA [11], which is a state-of-the-art

Conclusions
In   Long-term stability of the quantized Hall resistance in the thermally cycled array. The y-axis is the relative deviation of the quantized Hall resistance at 9.5 T corresponding to the filling factor 2 from the designed nominal value. The first three data points are taken from a previous work [13]. (d) Temporal drift of a stable 1-MΩ-resistance reference in an ULCA at 23 ℃ employed in the experiment. The y-axis represents the relative resistance difference from 1 MΩ. The shaded region indicates the experimental period of the subsequent measurements.     The CMC values of PTB registered in the BIPM are plotted by black circles for comparison. Square, pentagon, and diamond symbols depict the measurement uncertainties for the ULCA (PTB) [11], the measurement of single electron pump with a timely calibrated resistor (NPL) [9], and the programmable quantum mechanical current generator (LNE) [8], respectively. Resolution of DVM 10×10 -9 1×10 -6 Input impedance of DVM 1×10 -11 1×10 -9 Type-A 76×10 -9 4×10 -6 Measurement uncertainty (k=1) 7.8×10 -8 4.1×10 -6