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A self-referenced single-electron quantized-current source

With the anticipated redefinition of the international system of units (SI) the base units will be linked to fundamental constants of nature [1]. As for the electrical base unit"Ampere", it will be linked to the elementary charge e, requiring a corresponding quantum standard [2, 3]. Many concepts for such a standard have been investigated [4-14] relying on controlling the time-dependent tunnelling of electrons. However, the stochastic nature of quantum mechanical tunnelling intrinsically evokes uncontrolled deviations from the nominally quantized current. Alternatively, the counting of electrons [15, 16] has been explored but is severely limited in current amplitude and uncertainty by the low detector bandwidth. The late M. Wulf proposed [17] that this fundamental problem of electrical quantum metrology could be overcome by combining serial single-electron pumps with charge detectors allowing the generation of a quantized current and the in-situ detection of its stochastic deviations. Here, we experimentally demonstrate such quantized-current generation with in-situ detection of tunnelling errors at low frequencies and a reduction of the total current uncertainty by more than one order of magnitude. After frequency scaling this should enable a validated primary standard for the redefined SI base unit Ampere.

The field of quantum metrology promises measurement techniques and standards providing highest precision and universality by making reference to fundamental constants of nature (1).
For electrical quantum metrology two macroscopic quantum effects -the Josephson effect (2) and the Quantum-Hall effect (3) -have become the basis of robust and versatile quantum standards for the units Volt and Ohm, respectively. The anticipated redefinition of the international system of units (SI) based on quantum metrology, however, requires a quantum standard for the SI base unit Ampere (4,5). As candidates for such a quantum current standard single-electron pumps and turnstiles (6)(7)(8)(9)(10)(11)(11)(12)(13)(14)(15)) (see as a review,e.g. (16)) have been explored. Among them dynamic semiconductor quantum dots (QDs) (13)(14)(15) have recently demonstrated very promising characteristics. However, the operation of all of these kinds of devices is not based on a robust macroscopic quantum effect, but on a quantum mechanical tunnelling process during the periodic capture of a charge e from a source reservoir (17)(18)(19). The stochastic nature of this process (20) inevitably leads to small random deviations from the ideally quantized current of I = ef , with f being the pumping rate, hindering the realization of a robust quantum current source with lowest uncertainty.
As an alternative approach towards quantum current metrology the detection and counting of electrons passing randomly through a nanostructure has been explored and demonstrated both for unipolar (21) and bipolar (22) currents. However, the sensitivity and bandwidth of present single-charge detectors places severe limits on the current amplitude and achievable uncertainty of this approach. The late M. Wulf has recently proposed (23) a way to overcome these two fundamental problems of quantum-current metrology by combining the two concepts of singleelectron pumping and single-electron detection in a single device: the so-called self-referenced single-electron current source. In this quantum circuit the nominally quantized single-electron current is generated in a serial arrangement of single-electron pumps and single-electron detectors (24). During self-referenced operation only the much less frequently occurring stochastic pumping errors are detected instead of detecting all charges passing the circuit. Doing so the deviations from the nominal current I = ef are reliably determined and accounted for allowing the realization of a low-uncertainty single-electron quantum current source with validated output current. However, such self-referenced current generation has not been experimentally demonstrated yet.
Here, we present a quantum circuit integrating QD-based single-electron pumps and singlecharge detectors to implement all basic features of a self-referenced single-electron current source. We demonstrate single-electron pumping and in-situ charge detection during singleelectron current generation at low frequencies. It is shown that single-electron error detection combined with statistical analysis yields corrected values of the quantized current with uncertainties that are reduced by more than one order of magnitude as compared to an individual single-electron pump. Quantum circuits of the type presented in this paper should allow scaling to higher currents and lower uncertainties for future applications as a robust quantized-current standard for the SI base unit ampere.
The device under investigation is a hybrid semiconducting-metallic nanostructure as shown in Fig. 1 (19). From the lower left to the upper right a semiconducting channel is formed by etching a GaAs/AlGaAs heterostructure with a high-mobility two-dimensional electron gas (2DEG) situated 90 nm underneath the surface. The resulting 2DEG channel consists of source and drain (green) contacts and intermediate nodes 1,2 (blue, red) each separated by one of the three single-electron pumps P1-P3. Each pump is defined by three metallic top gates (yellow) crossing the channel. The electrical potentials of the nodes and hence the nodes' charge states are monitored by two single-electron transistors (SET) (25,26) operated as electrometer detectors labelled D1 (blue) and D2 (red), respectively. The SETs are fabricated by two-angle shadow evaporation of aluminium (27). For enhanced capacitive coupling of the detectors to the nodes gates with floating potentials (FG, yellow) are deposited. The SET detectors are oper-ated at fixed bias voltage and the current, acting as detector signal, is measured by synchronized digitizer cards at a sampling rate of 12 kS/s after digital averaging. Doing so we oversample the detector response by about 20 times. The measurements presented are all performed in a dry dilution cryostat at nominal base temperature of about 25 mK.
The robust operation of these electron pumps in a serial arrangement has been demonstrated in Ref. (24) and has been adapted for this work as follows: By applying negative voltages of about −200 mV to the first two gates of each pump (entrance and exit gate, respectively) isolated quantum dots (QD) are formed between the gates. (The third gate of each pump is grounded and not used in our experiments.) For these applied voltages the QD ground state is well above the chemical potential of the semiconductor channel and the QD is empty. To induce pumping of a single electron, a single, cosine-shaped pulse with frequency f = 40 MHz and amplitude V = −75 mV at the output of the generator is superimposed onto the entrance gate of pump i. During the pulse the entrance barrier is first lowered to allow tunnelling onto the QD from the source and then raised back up. In this cycle n i electrons are captured from the pump's source side in the QD with probability P (i) n and subsequently emitted to the drain side. Note that whenever no pump pulse is applied the high gate barriers of the QD prevent electrons from tunnelling between the nodes.
A sketch of the pulse shape as well as the pulse sequence applied to the pumps during this work is shown in Fig. 2A. The pulses are delayed by τ = 20 ms each. The delay τ is about ten times larger than the response time of our detectors τ d ≈ 1.5 ms. Hence the resulting charge change following each pumping event can be reliably resolved by the detectors.
The pulses of the sequence can be separated in two groups. The first three pulses ((i)-(iii)) constitute what we refer to as a marker sequence. Using this pulse pattern, we pump n = 1 additional electron sequentially through the structure from source to node 1, to node 2, and to drain by subsequent application of a pump pulse to P1, P2, and P3. This allows for a calibration of D1 and D2 in terms of the signature of n = 1 (or more) additional electrons on node 1 and 2 during the measurement (see supplemental material). Additionally, we can extract the individual pump characteristics from these pulses for every pump i i.e. the probability P (i) n of pumping n i electrons per pumping cycle (see supplemental). The last two elements (pulses (iv)-(v)) of the pulse sequence contain two sets of pulses where all three pumps are triggered simultaneously. Here, we change the detection scheme from an absolute measure of number of transferred electrons to the detection of pumping errors as described below.
The expected detector signals D1 (blue) and D2 (red) during one sequence are sketched at the bottom of Fig. 2A. Here we assume that all pumps transfer exactly one electron per pulse.
Horizontal dashed lines indicate the two detector states d (1,2) , d (1,2) +1 of node (1, 2) reflecting a change in the number of electrons on the corresponding node by one electron. When triggering pump P1 (pulse (i)), one electron is transferred from the source lead to node 1, leading to a step in D1 while D2 remains unaffected. By the following pulse (ii) one electron is transferred across P2 from node 1 to node 2, resulting in a step in both detectors with opposite sign. D1 returns to the initial value d 1 and D2 increases to d 2 + 1. Lastly in the marker sequence, pump P3 is triggered (iii) removing one electron from node 2 to drain with D2 returning to its initial state. In the following synchronized pumping of P1-P3 (pulses (iv)-(v)), D1 and D2 both remain in their initial state as one electron is pumped on to and off each node leaving the node charge unchanged. The same behaviour is also observed in the experimental data of error free pumping operation shown in Fig. 2B. Here the measured signals of D1 (blue) and D2 (red) are shown during five consecutive pumping sequences (1-5) as described above. The coloured vertical lines indicate the different pulses (i) -(iv) of the sequence as marked in Fig As a consequence D1 and D2 remain in the same state before and after pulse (i). However, during pulse (ii) P2 transfers an electron from node 1 to node 2 thereby shifting the baseline of detector D1 by −1e. In the same manner pumping errors of P2, P3 in the marker sequence can be detected reliably and analyzed statistically due to their characteristic signature (not shown).
Also during synchronous operation (pulses (iv),(v)) pumping errors can be detected and We also include in our analysis that for the given device and operation parameters specific events can occur which cannot be identified with high reliability. Fig. 3B shows an event leading to the signature (d 1 + 1, d 2 ).
with N the number of pumps in series and P 1 the average probability of all pumps to transfer exactly one electron per cycle. In the case of N = 3 as used here this formula simplifies to To evaluate the present technical limits of this technique we assume a circuit with five serial pumps and an increased detector bandwidth of 1/τ d ≈ 50 kHz which can be obtained using RF-SET technology (30). Assuming further a pumping frequency of f = 1 GHz and a pump error probability of |1 − P 1 | ≈ 1 · 10 −6 as experimentally demonstrated (15)   The semiconductor part of the device between source and drain (green) consists of three pumps P1 -P3 and two charge nodes 1,2 (blue, red) in between. Each pump is defined by three metallic top gates (yellow) defining a QD in the semiconductor channel. Between P2 and P3, another pump structure (grey) is visible, which is not used in the experiment and electrically grounded. In the lower part, the single-electron transistor (SET) detectors are shown (D1 and D2, coloured blue and red, respectively). These are capacitively coupled via metallic floating gates (FG, yellow) to the two charge nodes allowing to detect the node charge on the single-electron level.

Detector calibration
The SETs used here are operated at fixed bias voltage, the current varying as a function of external potentials is measured. This periodic SET response for a fixed voltage bias is shown in Figs. S.1A and S.1C for detector D1 and D2, respectively. Instead of creating an electrostatic potential on the semiconducting nodes by adding electrons, we continuously change the voltage on the SETs' gates here. To cope with the periodicity, we divide the SET response in two parts for positive and negative slope, respectively, and analyze the response during the marker sequence separately for both parts. At the extrema (above and below the dashed horizontal lines), the SETs are insensitive to variations in the external potentials, finally limiting the observation time. Due to the unidirectionality of the pump's charge transfer and the exact timing between digitizers and pump-pulse generators, we are able to identify the jumps observed in the detector traces as surplus or missing electrons. This knowledge also enables us to determine whether we measure at the rising or the falling edge of the corresponding detector.
As an example, during the marker sequence we expect to add electrons to node 1 by the first pulse (via P1) and to remove electrons by the following pulse to node 2 via pump P2. If the detector current shifts upwards after the first pulse (electrons added to node 1) and back down after the second pulse, we know for certain that the SET is operated on the falling edge.
Since the pumps' working points are chosen close to n i = 1, we expect to transfer mainly one electron per cycle. By creating pairs of SET current before the pulse and the corresponding response after the pulse, we obtain a frequency distribution of the response as a function of SET working point which should show maxima of the response at ±1 electron.  for 10 minutes, which corresponds to about 6000 sequence periods, but the detectors are not always sensitive as explained above), we estimate an upper bound of the unobserved transfer rates by the inverse number of total marker events for the specific pump and working point.