Perspectives of the generation and measurement of small electric currents

This paper provides a comprehensive overview of the historical background, current status, and future prospects related to the generation and measurement of small electric currents. It specifically caters to nonprofessional readers, with the aim of making the information comprehensible. A range of technologies are introduced, applicable in both basic research and industrial context. Quantum-mechanical approaches have been the focus of extensive efforts in this field, encompassing various types of single-electron pumps and combinations of two other quantum standards: the Josephson voltage standard and the quantized Hall resistance standard. These methods offer a reliable and precise means of generating and measuring small electric currents, minimizing uncertainties. However, operating complex cryogenic systems requires specialized expertise. Alternatively, conventional room-temperature systems are comparatively easier to handle. They employ low-noise amplifiers in conjunction with stable high-value resistors or capacitors charged with voltage ramps. This paper not only examines the characteristics of the both quantum and classical approaches from multiple perspectives but also outlines current and future applications for the generation and measurement of small electric currents.


Introduction: development of current generation and measurement technologies
Since the importance of small electric current measurement was recognized 100 years ago [1], several challenging studies have been conducted to evaluate its magnitude.In the 1930s, the possibilities of detecting small electric currents * Author to whom any correspondence should be addressed.
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as low as attoamperes (aA) or femtoamperes (fA) were addressed in the context of electrometers using thermionic vacuum tubes [2][3][4].To our surprise, the author of [4] also mentioned the possibility of highly sensitive current detection at the level of single electrons as 'individual electron pulses could be observed', although it was only two decades after Millikan's oil drop experiment which discovered the elementary charge [5,6].(Note that observation of individual electron pulses is markedly challenging even in the present day.)In the 20th century, especially prior to the development of nanofabrication technologies such as those used in semiconductor manufacturing, the early works on smallcurrent measurements were concerned with radiation currents, photo currents, and currents due to electron or ion beams.
For this purpose, vacuum tubes or analog electronic devices for precisely measuring small current in the range between 10 nA and 1 fA were intensively studied and developed [7][8][9][10][11].
Currently, small-current generation and measurement techniques are attracting increasing interest due to the industry demands from a practical point of view.More precise and quantitative roles of them in environmental surveys, ambient air pollution monitoring, cleanliness control of cleanrooms, radiation monitoring, and medical applications have similarly attracted increasing attention in recent years.
A century ago, it was possible to 'detect' electric currents of 10 −13 A (= 0.1 pA) and 'measure' 10 −9 A (= 1 nA).Today, we can generate and measure these ranges of small current traceable to the International System of Units (SI) as we now have state-of-the-art ultrastable lownoise amplifiers.Furthermore, transferring single electrons can be precisely controlled in nanostructure devices, which helps in realizing the recently redefined ampere in the revised SI [12]: The ampere, symbol A, is the SI unit of electric current.It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 × 10 −19 when expressed in the unit C, which is equal to A • s, where second is defined in terms of ∆ν Cs .
Here, ∆ν Cs is the unperturbed ground-state hyperfine transition frequency of the cesium-133 atom.The definition of 'measurement' today is not the same as that in the good old days as it currently denotes 'traceable measurement,' which is required in ISO and related regulations and legislation in most of the industries, e.g.automobile, electrical measurements, and medical devices.
Before the invention of a direct measurement technique with high-gain negative-feedback semiconductor amplifiers, a current divider was used to measure small currents by dividing the current into larger and smaller (e.g. the ratio of 100000 : 1) portions and by estimating the larger one, while a null detector is implemented in the other leg (the lower current circuit).The basic principle is still used mainly in the electric power industry to measure large (e.g.hundreds of A, mainly AC) currents, although it uses the opposite approach by measuring the divided small current to evaluate the large current in the main circuit.
In this paper, several selected techniques are presented; their characteristics and dedicated applications with regard to small-current generation and measurement are described.
Electric or electrical current I can be described in three different ways: = ef SEP . ( Equation ( 1) describes the so-called Ohm's law, where the value of electric current is measured from the voltage V across a resistance R, through which the current I is flowing.Here, the value of the resistance can be precisely determined by the quantum Hall resistance standard (QHRS) based on the quantum Hall effect [13].The voltage is precisely measured using a digital multimeter (DMM), which is calibrated by comparing it with the Josephson voltage standard (JVS) based on the AC Josephson effect [14,15].Equation ( 2) describes a relation between the time derivative of the voltage and a current I fed into a capacitor C. Here, the value of capacitance C can be determined by measuring its impedance (AC resistance) by means of a quadrature bridge such that it is traceable to both the resistance (QHRS) and the frequency standards [16].Finally, equation (3) describes how a small quantum current can be determined by the operational frequency of the electron transfer f SEP (or the number of electrons flowing per unit of time) and elementary charge, e = 1.602 176 634 × 10 −19 C (see section 3).
The traceability routes of these three methods for measuring small electric currents are summarized in figure 1.The time in the time derivative of equation ( 2) and the value of f SEP in equation ( 3) are determined based on a time base or a reference frequency traceable to a frequency standard based on an atomic clock.Moreover, f JVS in the JVS and a reference frequency used in the Quadrature bridge are traceable to the frequency standard, as depicted in the figure.The von Klitzing constant h/e 2 and the Josephson constant 2e/h, relevant to the quantum Hall effect used as the resistance standard (QHRS) and the AC Josephson effect as the voltage standard (JVS), respectively, are defined by the elementary charge e and the Planck constant h = 6.626 070 15 × 10 −34 J Hz −1 , both of which had been redefined as exact values in the revised SI [12].The number of electrons flowing through a conductor per unit time can be potentially measured at the quantum accuracy of a single electron using a single-electron pump (SEP), although this technique is still under development for industrial purposes.
Figure 2 provides an overview of the small-current generation and measurement study discussed in this paper.Notably, the figure builds on the work presented by Chae et al [17, figure 3] and is supplemented with additional information regarding conventional room-temperature methods.Each technology described in sections 3-6 has its own advantages and disadvantages, serving specific purposes and targeting a desired range of uncertainties, including practical cost considerations.The inclusion of figure 2 aims to facilitate the readers' comprehension by aiding in the understanding of various methods and their interrelationships.
Section 3 reviews research progress in small electric or electrical current standards based on SEPs.Section 4 discusses small-current measurements based on precision resistors described in equation (1).Section 5 considers smallcurrent measurements based on a charging capacitor.In section 6, more advanced methods, in which QHRS and JVS are combined, are reviewed, and we summarize this perspective paper in section 7. Typical traceability routes of small electric current to the SI.i is an integer, which is usually equal to 2. Another type of capacitance standard based on a calculable cross capacitor, which is traceable to the length and frequency standards, is omitted [16].Small electric current generation/measurement versus their uncertainties of various methods [17].Colored data points depict conventional room-temperature methods, and black data points depict quantum-based methods.CONVENTIONAL ROOM-TEMPERATURE METHODS: Purple diagonal crosses ×: High-value resistor at AIST [18].Orange open circles ⃝: ULCA at PTB [19].Red open up-pointing triangle △ and green closed squares ■: charging capacitor with voltage ramp at PTB [20] and VSL, The Netherlands [21], respectively.QUANTUM METHODS: Circles with cross ⊕ : programmable JVS (PJVS) with 1 MΩ QHARS at KRISS-AIST [17].Open squares □: ULCA at PTB [22].Down-pointing open triangles ▽: current-to-voltage conversion with a 1 MΩ QHARS by KRISS-AIST [23].Square with vertical cross ⊞: current comparison between LNE and PTB [24].Open circles ⃝: programmable quantum current generator (PQCG) at LNE [25].Star ⋆, closed square ■, and up-pointing triangle △: SET currents at KRISS-AIST [26], National Physical Laboratory (NPL) [27] and PTB [28], respectively.Note that all measurement uncertainties in the graph are for k = 1.Refer to the following sections for abbreviations and technical terms used in this figure.Reproduced from [17].© IOP Publishing Ltd.All rights reserved.

Application of small electric current
Given several inventions described in this paper and the higher demand for precise evaluation of energy consumption, absolute radiation measurements of the environment, and for medical purposes and air pollution measurements, the generation and measurement of small electric current are becoming increasingly important.The accumulation of power during the standby mode of electrical equipment is relatively large as we are surrounded by various electrical appliances in our daily life.The latest legislation and social needs require us to monitor environmental radiation and air pollution that are to be traceable using calibrated ammeters or small electric current measurement techniques.Therefore, not only have small-current measurement techniques largely attracted engineers and researchers to perform precise electrical measurements but they have also shed some light on other fields, such as radiometry/dosimetry and airborne-particle monitoring.Existing and possible applications in the industry and related science and technology include:

Ambient airborne-particle monitoring
Air pollution is a major issue affecting our quality of life, especially in overpopulated cities.An airborne-particle counter and spectrometer are widely utilized to monitor particulate pollution, and the devices are compliant with the national standard of the number concentration of airborne particles, e.g. an aerosol electrometer equipped with a Faraday cup.A current in the range of fA-pA from the Faraday cup is fed into an ammeter, which must be calibrated based on electrical standards.Another example where airborne-particle counters are employed is industrial cleanrooms.Since the invention of integrated circuit technology, cleanrooms have been introduced in semiconductor-manufacturing companies and institutes.Cleanroom classification is defined by the number of particles in a unit of air volume, which is the most important specification.Other parameters include temperature, humidity, pressure, number of air changes, and flow rate.This technology is now widespread to other fields that are sensitive to environmental contamination, including the rechargeable battery industry, nuclear power plants, and life sciences.It is now well-understood that the cleaner the rooms with controlled air purity and airflow, the higher the yield rate and reproducibility in the fabrication process.The traceable measurement devices used to measure air purity in cleanrooms include a particle counter.The higher the demand for precise small-current measurements in this area, the more distinct the underlying issues become.There are several ISO standards in this field, and the ISO 27 891:2015, Aerosol particle number concentration-Calibration of condensation particle counters, is one of the most important standard; it defines the range of particle concentration for which the calibration of airborne-particle counters using an aerosol electrometer is applicable.The lower limit of the concentration range is 10 3 cm −3 , which corresponds to approximately 3 fA for an aerosol electrometer operated at a flow rate of 1 L min −1 .Below this concentration, an aerosol electrometer cannot be applied as a concentration reference due to the lack of accuracy inherent to ammeters.
If the uncertainty of the current measurement could be lowered in the range below fA, air pollution could be monitored more precisely, contributing to better cleanliness of cleanrooms and better standardization and support of local legislation.

Radiation monitoring
Environmental and personal radiation monitoring for individuals treating radioactive materials or working with ionizing radiation are becoming increasingly important e.g. at nuclear power plants and surrounding areas.A particular example is the Fukushima nuclear power station accident caused by the 2011 Tohoku earthquake and tsunami.Consequently, the related legislation was amended and radiation monitoring in the area by the government, research groups, and private sectors are being continuously conducted; therefore, more accurate and reliable measurements are needed in Japan.Moreover, this trend has become more active in other countries with the recognition of the importance of monitoring radioactivity in the environment.The monitoring devices and systems are to be traceable to a national standard, which mostly consists of ionizing chambers and ammeters in the current range typically below 1 nA.The calibration of low dose-rate monitors, such as at monitoring posts, is often based on ISO/DIS 20 956:Radiological protection-Low dose-rate calibration of instruments for environmental and area monitoring.If a small current can be measured with a low uncertainty, e.g. 10 fA with 0.1%, a more precise and dense environmental radiation mapping could be obtained.

Medical applications
State-of-the-art x-ray medical imaging and x-ray and γray treatment instruments have become more sophisticated; the reduction of radiation dose to healthy body cells to optimize the performance and minimize the side effects of imaging and therapy is a key issue.Therefore, the traceability of radiation sources with smaller uncertainty (in the range of fA-µA) is important and further improvement of the technology to minimize the uncertainty is demanded.The evaluation and maintenance of cartridges of brachytherapy sources, such as 125 I for the treatment of prostate cancer, have gathered increasing attention in recent years.This evaluation, which is crucial for reliable treatment and management, was conducted with a welltype ionization chamber, the current from which is in the range of tens of fA, representing the main source of the uncertainty.
The applications extend beyond the aforementioned issues, encompassing the control of elementary particle beams in high-energy physics, manipulation of micro/nanofabrication systems such as electron-beam lithography and focused ionbeam lithography in semiconductor research and industry, and the material science of highly resistive materials, all of which require enhanced precision in small-current measurement and control.Subsequent sections provide a comprehensive overview of cutting-edge technologies that address these demands.

Small-current generation based on the single-electron transfer effect
As mentioned in section 1, quantum electric current can be generated by the single-electron transfer effect, i.e. an SEP (or single-electron transistor (SET)), which is believed to inherently have an extremely small 'pump' error, provided that the pump is carefully designed and operated, as described by reformulated equation (3).In this section, the focus is given to DC current generation based on the SEPs, with the target range of nA or less.Figure 3 shows a schematic of one type of SEP.In the SEP device, a quantum dot is in a two-dimensional electron gas system formed in a GaAs/AlGaAs heterostructure substrate.By applying gate voltages V RF and V DC to metallic Schottky electrodes formed on top of the substrate as shown in the figure, an electron is isolated from the Fermi sea and loaded into the quantum dot by lowering the barrier potential (which is realized by raising the gate voltage V RF due to the negative charge of electrons).Only one electron is captured in the quantum dot due to the Coulomb blockade effect.The electron is released to the drain electrode by raising the barrier potential (i.e.lowering V RF ).When this process is repeated at a frequency of f SEP , the number of electrons carried per unit of time is equivalent to the value of f SEP , generating a quantum DC current of I = ef SEP .In addition to DC current generation, the generation of a quantized AC current (or an arbitrary quantized current) has been recently demonstrated by combining digital modulation and SEPs [29].
There are various types of SEPs [30], all of which are carefully designed under an optimized operational condition to precisely pump a fixed number (usually one) of electrons per one operational cycle, f −1 SEP , based on the Coulomb blockade effect.
• material of the substrate -Si, with which one can make use of Si-integration technology -GaAs/GaAlAs, which one can make use of twodimensional high mobility electron gas -Si pump, which has a highest controllability and works at the highest pump speed exceeding 1 GHz [34] -R-pump, which has an additional on-chip impedance (resistor R) exceeding R K = h/e 2 connected to a series of metallic SEPs and reduces co-tunneling events [35] -tunable barrier (Si pump [34], GaAs/GaAlAs pump [28,29,36]), which has an entrance barrier electrode to the quantum dot tunable to control the electron transfer (see figure 3).-SINIS pump (Hybrid SET, turnstile SEP) [32,33] -surface-acoustic-wave SEP, which has the potential of easy scalability by parallel driving (parallelization) of multiple SEPs; however, it exhibits a larger pump error [37] • f SEP , which is the most important factor to set the generated current • waveform of f SEP , with which one can reduce the error rate [34] • external magnetic field, which further confines the electrons, reducing the error • temperature, which is usually lower for a more effective reduction of the error The single-electron transfer effect is a quantum effect; however, the electron pumping mechanism itself is a stochastic process that is not quantum-mechanically protected.Hence, the precise control of electrons with an low error rate of 10 −7 A/A(= 0.1 ppm) or lower, when the current is high enough (>1nA), is a long-standing issue to meet the level of uncertainties of other electrical quantum standards based on the Josephson and quantum Hall effects.
The pump accuracy has been evaluated using various methods, all of which are based on precision measurement devices traceable to a JVS and an QHRS or a capacitance standard traceable to the length standard (unit: meter defined by the speed of light, c = 299 792 458 m s −1 ) [30,38].These experimental evaluation methods are referred to as the quantum metrology triangle, which is schematically shown in figure 4.
One approach to confirm accuracy is to use an electron counting capacitance standard (ECCS), which is a tailor-made cryogenic capacitor set in the vicinity of the SEP [39]. Figure 5 displays its schematic.The uncertainty of this method is limited by the evaluation of the frequency dependence of the capacitance of C ECCS , as this method requires the DC value of the ECCS.The ECCS has its traceability via a roomtemperature capacitance bridge (with long RF cables from the millikelvin stage to room temperature) to a capacitance standard or 'a calculable cross capacitor,' which is eventually traceable to a length standard realized by the speed of light, c.Because the capacitance bridge works in the frequency range of kHz, evaluating frequency dependence to manipulate its DC value is essential.The pumping error evaluation requires an 'electrometer' directly connected to the SET structure.The  electrometer consists of a quantum dot, which senses whether a single electron is actually pumped (transferred) in each pump event.
Other methods to evaluate the pump accuracy include: • a stable high-value standard resistor and a precision digital multimeter (DMM or voltmeter), routinely calibrated by QHRS and JVS, respectively • a stable low-noise amplifier, e.g.ULCA described in section 4, which is routinely calibrated by QHRS and cryogenic current comparator (CCC), • a direct (or almost direct) measurement using the combination of a JVS and an QHRS [40] The last method is represented in figure 4 and by the following equation, which connects three quantum effects via Ohm's law: where α is a trivial factor determined from the three parameters as α = i Nf JVS /f SEP (which is supposed to be exactly 1); the QHRS plateau number i (usually, = 2), the integration number of Josephson junctions N, and the frequency ratio f JVS /f SEP .Another issue is the upper limit of the pumping speed as high as GHz to serve in minimizing the pump error rate as low as 0.1 parts-per-million.Note that the GHz frequency range is equivalent to the generated current in the range of nA by equation ( 3) or (4).In the case of metallic SEPs (a series of metallic SEPs or so-called R-pump), this lower limit reaches up to 100 MHz (see references by Scherer and Camarota [38] and Kaneko et al [30]).Due to the practical upper limit of the generated current, it was considered challenging to meet conventional industrial needs in the range of mA or higher; however, in recent years, various applications have emerged in the industry of small-current generation and measurement in the range of aA-fA, as described in section 2.

Small-current amplifier with stable resistor network: ultra low current amplifier
This and subsequent sections describe the methods employed for measuring small currents using high-value resistors traceable to an QHRS.When a current I is applied to a resistor R, the current I = V/R is evaluated by measuring the voltage V across the resistor.Alternatively, the current I = V/R can also be generated by applying a voltage V to a resistor R. High-value resistances, such as GΩ and TΩ, are essential for measuring/generating small currents such as fA, pA, and nA.DMMs used for voltage measurements are calibrated at the measurement range 1 V; thus, DMMs perform optimally in this range (and around it).Hence, higher resistances are required to generate larger voltages in this range.Furthermore, the Johnson-Nyquist thermal noise imposes an unavoidable limitation on small-current measurements.For example, the current noise generated from a 1 TΩ resistor can be estimated as 0.13 fA Hz −0.5 at 300 K, and current signals below this value are buried in noise and difficult to be resolved.If the current to be measured is smaller than the thermal noise level, the measurements must be repeated to improve the signal-to-noise ratio, meaning that a larger resistor is required to attain the desired accuracy within a practical measurement time.
An ULCA, developed by PTB, is now a well-established instrument for small-current generation and measurement that has a well matched 1 : 1000 ohmic resistor network [19,[41][42][43].This instrument was primarily used to measure currents generated by SEPs and is now finding other fields of application, as described above (see section 2).
Figure 6 shows the simplified equivalent circuit of the ULCA, which is a current-to-voltage converter consisting of the input-stage current amplifier followed by the outputstage current-to-voltage converter (transimpedance amplifier).The input-stage current amplifier consists of a resistor network composed of connected identical surface-mount thinfilm resistors.
In the prototype ULCA [41], for example, 1500 elements of 2 MΩ resistors were serially connected to form R s = 3 GΩ.Almost the same number of the resistor elements are also connected in a combination of parallel and series to form R P = 3 MΩ, resulting in a resistance ratio of R S : R P = 1000 : 1.Consequently, the input stage can amplify the input current I in with a current gain of R S /R P = 1000, as shown in the figure.The output stage is a transimpedance amplifier (inverting amplifier) with a feedback resistor R T that converts current to voltage as V out = I in R T R S /R P .In the ULCA, R T = 1 MΩ is built-in; thus, the overall transfer gain of the ULCA is R T R S /R P = 1 GV/A.The factors that govern the small-current measurements with ULCA are the accuracy and temperature/long-term stability of the resistance ratio R S /R P as well as the accuracy and stability of R T .The resistance ratio can be calibrated using a CCC, and the absolute value of R T can be calibrated with high accuracy by directly comparing it with the QHRS via a CCC.
The stability in time of the ULCA is specified as an order of 10 −6 Ω/Ω per year and can measure/generate currents as small as 100 pA with an uncertainty of 10 −7 A/A.Recently, a low-noise version of ULCA has been developed, which was reported to achieve 40 fA-1.6 pA with relative type-A standard uncertainties below 1.2 × 10 −3 A/A, allowing the nonlinearity in the gain to be clearly resolved.The instability in the gain at the level of a few parts in 10 −3 A/A is resolved on timescales of approximately 1 day [44].

Small-current amplifier with a metal-glaze thick-film high-value standard resistor
In principle, using a stable and ohmic standard resistor and a precision voltmeter, each of which is traceable to an QHRS and a JVS, respectively, with small enough measurement uncertainties, can generate or measure any electric current of any range.However, a small-current range requires a set of high-value standard resistors, which are prone to instability over time and under environmental disturbances and tend to behave nonohmically in the low voltage range, as might be expected.Thus, if the nonlinearity in the range led by the Simplified diagram of the small-current measurement method based on a high-value standard resistor [18].nonohmic behavior can be calibrated, the succinct measurement circuitry can be exploited, which facilitates real operation and provides robustness.Note that the non-linearity behavior does not change over time, in principle, thus regular recalibration should not be necessary.
In contrast to the ohmic large-scale resistor network of ULCA, the National Metrology Institute of Japan, NMIJ/AIST, made use of a single stable high-value standard resistor (or a matching set of a couple of these resistors to compensate for their temperature coefficients) and has developed a small-current amplifier that can measure currents down to the 0.5 fA level [18].A schematic of this current measurement system is shown in figure 7.This current measurement system is a kind of feedback-type current comparator, where the current to be measured is compared with a reference current generated using a calibrated precision high-value resistor (feedback resistor), as described above.This technique has the advantage that the SI traceability of the measured current can be ensured by the voltage and resistance alone, without requiring calibration of the transimpedance gain.However, the high-value resistors made of a metal glaze thick film exhibit nonlinearity (nonohmic behavior in low voltage ranges).This nonlinearity comes from the electron transport mechanism (known as hopping conduction) [45,46] in the metal-glaze resistors, which is a notable uncertainty component in this system.When the transimpedance gain is set to 1 × 10 12 V A −1 and the feedback resistor of 1 TΩ is used for measuring a current of 1 fA, the corresponding output voltage is 1 mV.This means that the output voltage of 1 mV, which is a considerably low applied (bias) voltage compared to the normal voltage range (>1 V) in the calibration of such high-value resistors, is applied across the feedback resistor.Therefore, the nonlinearity of the feedback resistors must be evaluated carefully in the small-current measurements with this system.Okazaki et al [18] clearly evaluated the nonlinearity of the feedback resistor made of a metal glaze thick film in the bias voltage range of 1 mV-100 V using another type of transimpedance amplifier (ULCA).The feedback resistor of 3 GΩ in ULCA is implemented as an array of thin-film resistors network, which mitigates the nonlinearity.Okazaki et al demonstrated that uncertainties in the current measurement, including the nonlinearity component, were between 2.2 × 10 −2 A/A at 0.72 fA and 1.4 × 10 −3 A/A at 333 fA, respectively.

Small-current generation based on the charging of a capacitor with a voltage ramp
A stable capacitor is often used to generate an accurate reference current in the range between pA and fA to calibrate picoammeters or electrometers.In this capacitor-based current source, a linear voltage ramp at a known constant rate dV/dt is applied to a differentiating capacitor of known capacitance C, after which a DC current I is generated in accordance with the following re-formulated equation ( 2).This equation clearly shows that the current generated in this method is traceable to SI units through the units of capacitance, voltage, and time.To generate a current in the pA and fA ranges, capacitors ranging between 1 pF and 1000 pF are usually used with a practical voltage ramp of approximately 0.1 mV s −1 1 V s −1 .A schematic of this method is depicted in figure 8. Some voltage ramp generators have been proposed by several research groups [20,21,[47][48][49][50][51], in which the voltage ramp can be adjusted between 10 mV s −1 and 100 mV s −1 in the voltage range of −10 V to +10 V.The voltage ramp can be reduced below 10 mV s −1 (down to 0.1 mV s −1 ) using a voltage divider [20,49].
The stability of the generated current crucially depends on the linearity of the voltage ramp generator and the stability of the differential capacitor.The nonlinearity of the voltage ramp generator leads to a corresponding change in the generated DC current.Some voltage ramp generators are based on the electronic integrator circuit, and the nonlinearity in the voltage ramp is mainly due to the nonideal properties of integrating capacitors, such as dielectric absorption and leakage.The dielectric absorption, which is also referred to as soakage and sometimes as dielectric hysteresis, leads to asymmetric deformations in the voltage ramp decreasing with time.In turn, the leakage, which is often modeled as insulation resistors, causes a slower current change.
Methods to improve the nonlinearity in the voltage ramp have been reported.Willenberg et al [20] reported an analog feedback circuit, which compensates for the parasitic currents caused by the nonideal properties mentioned above.In contrast, van den Brom et al [21] compensated for the nonlinearity using a digital-to-analog converter (DAC), which injects software-controlled correction currents into the integrator circuit.To overcome the nonlinearity caused by the analog integrator circuit, Willenberg and Norbert Tauscher [49] also proposed a digital voltage ramp generator without any analog integrators, which is based on two DACs controlled by one computer; one (primary DAC) that synthesizes the voltage ramp and the other that compensates for the nonlinearity of the primary DAC.This digitalization of the voltage ramp generator allows to adjust voltage ramp easily and rapidly using software commands.However, the DAC error is inherent in the digital voltage ramp generators.To minimize the DAC error, Bergsten et al [50] adopted a ∆-Σ modulation technique [52] in the voltage source and an optimized low-pass filter.
As mentioned above, differential capacitors are also a key component for the capacitor-based DC current source.Lowloss air or sealed-gas dielectric capacitors are preferably used owing to their long-term stability of the ppm (or µF F −1 ) level, low sensitivity to temperature, and low leakage current.However, it must be noted that a possible frequency dependence in the differential capacitor is one of the largest contributions to the uncertainty in capacitor-based current generation.The calibration of capacitors is usually performed at 1.0 kHz, whereas this current source operates at a frequency considerably lower than 1 Hz.Giblin and Lorusso [53] indicated that some capacitors had an unexpectedly large frequency dependence, up to several hundred ppm, ranging between approximately 10 mHz and 1 kHz.The other notable contribution to the uncertainty in this current source is the humidity dependence in the capacitance of the unsealed air-dielectric capacitors.Some studies have shown that the capacitance of the unsealed air-dielectric capacitors varied in a few ppm levels, depending on the relative humidity [54][55][56].On the contrary, sealedgas dielectric capacitors are considered to have no humidity dependence.
A capacitor-based current source consisting of a voltage ramp generator and a capacitor can generate small DC currents, typically ranging from 10 fA to 100 pA with small uncertainty.For example, Willenberg et al [20] reported that the relative uncertainties in the DC current generation were between 2.2 × 10 −2 A/A at 100 aA and 9.4 × 10 −5 A/A at 10 pA, respectively.

Small-current measurement with an integrating electrometer
A low-loss gas dielectric capacitor (regardless of sealing) is also used for measuring small currents, typically below 10 nA; this instrument is referred to as an integrating electrometer.In this type of electrometer, the capacitor is used as a feedback element across the high-gain preamplifier of the electrometer.The current to be measured then generates a voltage ramp at the output of the preamplifier, which is essentially the reverse process of the charging of a capacitor with a voltage ramp, as stated above.The integrating method is free from Johnson-Nyquist noise contribution, which is an advantage compared to feedback ammeters using highvalue resistors as a feedback element.An integrating electrometer is, however, time-consuming.This method is shown in figure 9.
One of the notable contributions to the uncertainty with the integrating electrometers is the stray capacitance in parallel with the capacitor.This capacitance can be evaluated by measuring a stable input current with different values of the external capacitor.Fletcher et al [48] evaluated the stray capacitance in their current measurement system as approximately 0.25 pF.The capacitance of the feedback capacitor must be chosen such that the stray capacitance would not become a significant contribution to the overall uncertainty.They used a commercially available electrometer in combination with the air-dielectric capacitors ranging from 1 pF to 1 nF to demonstrate a traceable small current measurement using the integrating method.They reported relative uncertainties in their current measurement system between 2.5 × 10 −2 A/A at 100 fA and 5 × 10 −4 A/A at 1 nA.
Finally, another current measurement technique using a capacitor, which is known as a vibrating-reed electrometer [8,10], has been employed.This electrometer converts input DC current into an AC signal using a capacitance oscillating with a single frequency and synchronously amplifies the single-frequency signal.The advantages of the vibratingreed electrometers are, for example, exceptional sensitivity (or extremely high input impedance of approximately 10 15 Ω), very low leakage current, and excellent zero (offset) stability.Rietveld and van den Brom [57,58] developed homebuild vibrating-reed electrometers and reported uncertainty in the measurement of currents of below 10 pA at 30 ppm (= 30 µA A −1 ).
Notably, all these capacitor-based systems are prone to be affected by external mechanical vibration and impacts; thus, they need special care during transportation and on-site measurement.

Small-current generation based on the Josephson and quantum hall effects
A programmable quantum current generator (PQCG) has been developed by the Laboratoire National de Metrologie et d'Essais (LNE) [24,25], which has been successful for small-and medium-range (up to mA) current generation based on two quantum-based standards-PJVS and QHRS-and a CCC.The PJVS device has a set of binary segmented junctions, each of which can be independently biased, such that the device, in total, can generate arbitrary voltage waveforms in an approximated stepwise manner.
This system consists of a PJVS that biases the QHRS via multiple connections (to render the wire resistances negligible).One of the low-voltage sides of the wires is connected to one of the CCC windings, e.g.129 turns.The CCC amplifies the current flowing between the PJVS and the QHRS, e.g.few tens of µA, with another small-number winding of e.g. 4 turns, such that the resulting current e.g.I PQCG = 30 µA × 129  4 ≈ 1 mA is generated.A schematic of the current generation system is shown in figure 10.
This PQCG approach has a clear advantage as it is able to cover most industrial needs for a wide electric current range with quantum-mechanically protected accuracy.However, it requires one 4 K cryostat (or a liquid helium dewar) for the PJVS, one 3 He cryostat for the QHRS (in the case of GaAs/AlGaAs-based QHRS), and one 4 K cryostat (or a liquid helium dewar) for the CCC.This complicated setup might be difficult to realize in most of the metrology laboratories offering calibration services, which require robust and continuous operation.Therefore, the development of a cryogen-free all-in-one-integrated system will be a key solution in electrical calibration.
Another approach was realized by the Korea Research Institute of Standards and Science (KRISS) and the NMIJ/AIST using a quantum Hall 'array' resistance standard (QHARS) [23,59,60] of 1 MΩ and a 2 V PJVS [17].With this setup, they succeeded in the generation of 1 µA with 0.12 µA A −1 relative uncertainty, as evaluated by ULCA.This relatively simple setup consists of serially connected PJVS

Perspectives
This paper provides nonprofessional researchers and engineers who are not familiar with the small-current generation and measurement with a historical overview and a perspective on the latest technologies.There are two important branches of the field: basic research and real measurement application.The former includes techniques with quantum effects, i.e. the purely single-electron transfer effect only or a combination of the Josephson and quantum Hall effects.These measures to date still require multiple cryogenic setups connecting each other, which could be difficult to implement in small laboratories.However, researchers can reach the lowest level of uncertainty in small-current generation and measurement using this method.The techniques have clear and direct SI traceability to the fundamental physical constants via quantum standards, the quantized Hall resistance standard (h/e 2 ), the JVS (h/(2e)), and the frequency standard (∆ν Cs ).These could be utilized as primary standards in the future in national metrology institutes or used as a reference current or detector under cryogenic conditions in basic science (e.g.nanophysics and material science).In the future, it is hoped that every ammeter will be a compact turn-key system with quantummechanical accuracy.This will enable us to improve various environmental-and health-related measurements with the best possible measurement technology to meet demands in social problems, however, it requires innovative ideas.
The latter includes more conventional room-temperature setups: a small-current generation method based on the charging of a capacitor with a voltage ramp, a method with an ultralow noise current amplifier that consists of a stable ohmic resistor network, a method with a (conventional) high-value standard resistor, an integrating electrometer, and a vibratingreed electrometer.These techniques also have clear SI traceability to resistance or capacitance standards and are relatively easy to install for calibration services and internal references in laboratories.
The science of measurement or metrology, as described in sections 1 and 2, continues to attract more researchers and engineers in various fields, which are expanding.The authors would like researchers and engineers, who are interested in any aspects covered in this paper, to share ideas for applications or even innovative tricks to further improve small electric current generation and measurement techniques.

Figure 1 .
Figure 1.Typical traceability routes of small electric current to the SI.i is an integer, which is usually equal to 2. Another type of capacitance standard based on a calculable cross capacitor, which is traceable to the length and frequency standards, is omitted[16].

Figure 3 .
Figure 3. Schematic of the tunable barrier (quantum dot) GaAs/GaAlGs SEP.A single electron is transferred from the source to the drain electrode sequentially by tuning the barrier electrode V RF .

Figure 4 .
Figure 4.A Schematic representation of the quantum metrology triangle (QMT) experiment.The consistency of measurement systems based on the three quantum effects is verifiable via Ohm's law.There are several variances in the QMT representation and experiments.

Figure 5 .
Figure 5. Simplified diagram of a SEP with an ECCS.

Figure 7 .
Figure 7. Simplified diagram of the small-current measurement method based on a high-value standard resistor[18].

Figure 8 .
Figure 8. Circuit diagram of the small-current generation method based on the charging of a capacitor with a voltage ramp.

Figure 10 .
Figure 10.Schematic principle of the programmable quantum current generator.Reproduced from [25].© IOP Publishing Ltd.All rights reserved.