Electron counting in a silicon single-electron pump

Recent development in the field of single-charge pumping has provided a basis for the emerging quantum standard of the ampere in the International System of Units (SI) [1]. This standard will be based on an agreed value for the elementary charge e and the frequency f, the product of which yields the ampere.

Single-charge pumps and turnstiles have been implemented in many different physical systems including normal-metal tunnel junction devices [2–4], superconducting devices [5–7], hybrid superconductor-normal-metal turnstiles [8, 9], semiconductor quantum dots [10–15], and single atom-sized impurities [16–19]. A satisfactory relative pumping accuracy at the 10⁻⁸ level has only been demonstrated in normal-metal devices in the picoampere range [3]. This current, however, falls significantly below 100 pA which is required for a practical realization of the quantum current standard [20]. The most accurate single-electron pumps that produce high enough current are thus far based on GaAs quantum dots [21]. Recently, an uncertainty 0.2 parts per million (ppm) levels has been reached at 87-pA current [22].

Silicon quantum dots [12, 14, 23–25] provide a promising alternative to the GaAs platform. Devices fully based on silicon have exhibited greatly suppressed $1/f$ noise and absence of large amplitude background charge jumps [26]. To date the most accurate silicon single-electron pumps produce a pumped current of 80 pA with uncertainty below 30 ppm [14].

The accuracy of the electron pump is essentially given by missed or excess electrons pumped per cycle. It is possible to arrange the electron pumps such that the pumping errors can be in situ observed with a nearby charge sensor, thus providing a self-referenced current source. Although several experiments [3, 18, 23, 27–35] provide observations on the pumping errors and the number of electrons transferred per cycle, a thorough comparison of the direct current provided by the electron pump and the results of the electron counting scheme is lacking. Only comparison between electron counting and the current flowing through a non-driven system has been reported [36].

In this paper, we demonstrate electron counting in a silicon electron pump utilizing the quantum dot architecture which has provided the most accurate results in silicon [14], thus providing a proof of concept for a self-referenced silicon charge pump. Furthermore, the average number of pumped electrons per cycle, n, extracted from our electron counting scheme agrees with that obtained from the pumped direct current. This result verifies the consistency between these two schemes.

Our device shown in figures 1(a), 1(b), and 2(a) is fabricated using metal-oxide-semiconductor (MOS) technology on a near-intrinsic silicon substrate with 8 nm thermally grown SiO₂ gate-oxide [14, 37]. The aluminum gates are defined with electron beam lithography in three layers isolated from each other by thermally grown Al_yO_x. The topmost layer of gates is used to accumulate a two-dimensional electron gas (2DEG) at the Si/SiO₂ interface and the two bottom layers are used to control the electrostatic confinement of the dot in the planar directions by locally depleting the 2DEG and forming tunnel barriers. A schematic potential landscape of the device is presented in figure 2(b).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We employ two different measurement schemes: the direct-current scheme and the electron counting scheme. In the direct-current scheme, we induced a 2DEG below the source lead (SL), drain lead (DL), and switch barrier (SB) gates (see figure 1(a)). The pump dot is induced with the plunger gate (PL) such that the left barrier (BL) and right barrier (BR) gates are used to define tunable tunnel barriers between the leads and the dot. The confining gates (C1 and C2) are set to negative voltage to tighten the dot potential as first demonstrated in [14]. Experiments in both schemes are carried out in a cryostat with a bath temperature of 180 mK.

The gates PL and BL are also connected to an arbitrary-waveform generator providing the voltage drive for the dot to pump the electrons from the source to the drain. As shown in figure 2(c) the waveforms of the pulses consist of three consecutive parts: (i) voltage ${s}_{1}(t)={A}_{\mathrm{PL}/\mathrm{BL}}[1-\mathrm{cos}(2\pi t/T)]/2$ for $0\leqslant t\lt T,$ (ii) voltage ${s}_{2}(t)=-{s}_{1}(2T-t)$ for $T\leqslant t\lt 2T,$ and (iii) zero voltage for $2T\leqslant t\leqslant 2T+{t}_{{\rm{w}}}.$ The period of the sinusoidal part is fixed at T = 50 ns and the pumping frequency $f=1/(2\;T+{t}_{{\rm{w}}})$ is adjusted by changing the wait time ${t}_{{\rm{w}}}\gg T.$ The temporal offset of the pulses in PL and BL is 13.6 ns and the voltage amplitudes at the sample are denoted by ${A}_{\mathrm{PL}}$ and ${A}_{\mathrm{BL}},$ respectively. The induced current is measured from the drain side using a room-temperature transimpedance amplifier. In the direct-current scheme, we have ${t}_{{\rm{w}}}=1.9$ μs that yields ef = 0.08 pA. The waveform has to be adjusted such that the integral of the positive and negative area vanishes. Otherwise we need to adjust the dc bias of the gates for each t_w to achieve the desired potential due to the loss of the dc component of the waveform in the capacitor of the bias tee.

The electron counting scheme has the following differences from the scheme described above: We use a much lower ${V}_{\mathrm{SB}}$ to define a reservoir dot below the DL gate bounded by SB, C1, and BR gates. The charge state of the reservoir is monitored with a capacitively coupled single-electron transistor (SET). The SET is induced with the top gate (TG) and barrier gates (B1 and B2). The hold time of the charge state of the reservoir was measured at gate voltages similar to the one used for the counting experiments and showed stability of several hours. The current through the voltage biased SET is transimpedance amplified and channeled to a proportional-integral-derivative (PID) controller which keeps the operation point of the SET fixed by compensating ${V}_{\mathrm{TG}}.$ Electrons are pumped to the reservoir with an identical waveform as in the direct-current scheme but with relatively long wait time ${t}_{{\rm{w}}}=750$ ms. After a fixed number of subsequent pumping pulses, the reservoir is initialized by inducing a 2DEG below BL, PL, and BR so that the excess electrons flow from the reservoir back to the source.

Figure 3(a) shows a representative trace of the SET current signal as a function of time when electrons are pumped into the reservoir. At each pumping event, there is a clear peak in the signal which subsequently saturates back to the set point of the PID controller. The PID controller is employed to enhance the signal to noise ratio compared with the current threshold method used in other electron counting experiments [18, 33, 34]. The advantages of this method [3] are that the low-frequency noise is filtered out and the sensor works at its most sensitive operation point at all times. We characterize the electron transfers by evaluating the area between the SET current trace and the set point, as indicated in figure 3(a). The tunable parameters of the PID controller define the observed decay time.

Zoom In Zoom Out Reset image size

In figure 3(b), we show histograms of the SET signal area at plunger voltages corresponding to the maximum probability of achieving $n=0,1,2,$ or 3 electrons transferred per cycle. In order to evaluate the transfer probabilities P_n at different ${V}_{\mathrm{PL}},$ we fit the histograms with a function $f(x)={\displaystyle \sum }_{n}{A}_{n}g(x,{b}_{n},{\sigma }_{n}),$ where $g(x,{b}_{n},{\sigma }_{n})$ is a Gaussian distribution with mean b_n and standard deviation ${\sigma }_{n}.$ Since the mean and standard deviation of each distribution are essentially independent of the plunger gate voltage, we determine their values using the whole data set acquired for all different voltage ranges. For the mean values we obtain ${b}_{n}=n\times 1.15$ pC and for the standard deviations ${\sigma }_{n}=0.37$ pC for $n\ne 2$ and ${\sigma }_{2}=0.33$ pC. The probabilities P_n for n transferred electrons are extracted using the amplitudes A_n as fitting parameters for each ${V}_{\mathrm{PL}}$ and computing ${P}_{n}={A}_{n}/{\displaystyle \sum }_{j=0}^{4}{A}_{j}.$ In figure 3(c), a representative fit at ${V}_{\mathrm{PL}}=0.62\;{\rm{V}}$ is presented.

Figure 3(d) shows the probability of a single-electron transition as a function of the number of consecutively applied pumping pulses since the initialization of the reservoir. The error bars indicate 95% confidence interval obtained by taking into account two error sources independently: uncertainty related to the fit of the amplitude A_n and the one obtained from the Wilson score interval method. Each data point is derived from combined statistics of 2000 pulses and 200 reset events. The data show that we may inject up to 50 electrons into the reservoir without changing the probability more than 1%. This probability decreases with increasing number of pumped electrons into the reservoir due to its increasing electrochemical potential [28]. We estimate the capacitance between DL and the reservoir by assuming them to be parallel plate capacitors: ${C}_{\mathrm{RES}-\mathrm{DL}}=\epsilon A/d\approx 1.8\times {10}^{-15}$ F, where is the permittivity of SiO₂, A is the area of the reservoir dot, and d is the thickness of the SiO₂. Thus the charging energy of the reservoir is roughly ${E}_{{\rm{C}}}={e}^{2}/{C}_{\mathrm{RES}-\mathrm{DL}}\approx 87$ μeV leading to a potential difference of the reservoir due to 50 excess electrons in the island of ${{\rm{\Delta }}}_{\mathrm{RES}}\approx 4.3$ mV.

In order to extract the probabilities P_n for $n=0,1,2,3,$ and 4 as a function of plunger voltage, we apply a reset event followed by 22 consecutive pumping pulses. We repeat this procedure 12 times for each voltage value. Based on the data presented in figure 3(d) where ${P}_{1}\gt 99\%$ for up to 50 consecutive pumping pulses, we note that the choice of 22 pulses between each reset should not lead to observable underpumping for n = 1 within the uncertainty of the counting scheme. The average number of electrons pumped per cycle can be computed from the individual probabilities as ${n}_{\mathrm{count}}={\displaystyle \sum }_{n}{{nP}}_{n}.$ In figure 4(a) the probabilities P_n are shown as well as ${n}_{\mathrm{count}}.$ The error bars are computed the same way as in figure 3(d). The data indicate that, by adjusting the potential of the dot, it is possible to transfer with a single pulse up to three electrons with over 99% probability. However, single electron transfers are clearly more robust than multiple electron transfers since P₁ is insensitive to variations of ${V}_{\mathrm{PL}}$ in a significantly larger range than that of P₂ and P₃.

Zoom In Zoom Out Reset image size

Figure 4(a) also shows the average number of electrons transferred per cycle measured with the direct-current scheme ${n}_{\mathrm{dc}}=I/{ef}.$ Interestingly, these data are in good agreement with the counting method. Note that the curves for P_n are shifted by $-7.00$ mV in ${V}_{\mathrm{PL}}$ justified by the capacitive coupling between the SB gate and the pump dot and the fact that we need to use a different gate voltage in the direct-current scheme (${V}_{\mathrm{SB}}=0.39$ V) compared with the electron counting scheme (${V}_{\mathrm{SB}}=0.20$ V). We verified that the magnitude of the applied shift is in agreement with the observed shift of the current plateaux in the direct-current scheme (see figure 4(b)). The electron channel under the switch barrier turns off completely around ${V}_{\mathrm{SB}}=0.30\;{\rm{V}}$ which prevented us from measuring the shift in this scheme at lower voltages. We neglect the shift of the plateaux due to different ${V}_{\mathrm{TG}}$ used in the two schemes since it is much smaller than the shift due to ${V}_{\mathrm{SB}}.$

In the electron counting scheme, the rising edge to the first plateau shifts in ${V}_{\mathrm{PL}}$ as a function of number of excess electrons in the reservoir. Since we average over 22 pumped electrons this shift broadens the rise to the first plateau in figure 4(a). This effect is not clearly visible for the other, notably broad, steps.

In the inset of figure 4(a), the quantized electron pumping at the n = 1 plateau is compared in detail between the two measurement schemes. The positive and negative errorbars of ${n}_{\mathrm{dc}}$ each indicate two standard deviations of the shown data at the n = 1 plateau. The two data sets well agree within the experimental uncertainty. Averaging this data yields our best estimates for the average number of pumped electrons at the first plateau ${n}_{\mathrm{dc}}=1.000\pm 0.006$ and ${n}_{\mathrm{count}}=0.998\pm 0.004$ where we employ the 95% uncertainty level.

Finally, the probability P₁ as a function of ${V}_{{\rm{C}}2}$ and ${V}_{\mathrm{PL}}$ is presented in figure 4(c). We observe that the robustness of the single-electron transfer with respect to ${V}_{\mathrm{PL}}$ increases with decreasing ${V}_{{\rm{C}}2}.$ This phenomenon is due to an increase in the charging energy caused by a tightening of the electrostatic confinement of the pump dot. Here, we show this effect in the electron counting scheme as a consistency check of similar behaviour previously observed in the direct-current scheme [14, 38].

In this work, we compare the direct current generated with a quantum dot pump with electron counting scheme at a relative uncertainty below a per cent. The main limiting factor of our experimental approach is the relatively low sensitivity of the charge detector. Typically, in order to confidently assess single-electron counting statistics, one has to trade between the size of the storage reservoir and the sensitivity of the sensor. Our device is designed to have a fairly large reservoir to minimize the back-action on the pumping mechanism.

The disadvantage of this choice is the reduced performance of the readout. We estimate that the sensitivity of our detector is about 90 me/$\sqrt{\mathrm{Hz}}.$ This indicates that it is possible to sense a single electron in about 8 ms of averaging time. However, we have chosen to integrate up to 750 ms between pumping pulses to reduce the uncertainty in the readout. In this context, the employed PID controller reduces the slow drifts in the SET current. Nevertheless, the limited reservoir-to-sensor capacitive coupling of about 0.005e is mainly responsible for the non-ideal readout fidelity. The observed distributions of the signals for different numbers of pumped electrons are separated only by 3.2σ. Hence, those counting events that fall further than 1.6σ from the centre of the distribution should be considered as misattributions in the most conservative scenario.

In the near future, we will integrate a metallic SET sensor next to the silicon reservoir. In this way, we estimate that the capacitive coupling and, hence, the sensitivity will be improved up to an order of magnitude. This will allow us to enhance the readout fidelity and reduce the counting uncertainty down to ppm levels, while keeping the back action on the pump insignificantly small.

Ultimately, a precise electron pump verified by error counting would, not only provide a supreme candidate for the realization of the quantum ampere [1], but could also be harnessed in the quantum metrological triangle experiment [39, 40] to test the fundamental constants of nature.

We thank F Hudson, C H Yang, Y Sun, G C Tettamanzi, I Iisakka, A Manninen, and A Kemppinen for useful discussions. We acknowledge financial support from the Australian Research Council (Grant No. DP120104710), the Academy of Finland (Grant No. 251748, 135794, 272806, 276528), Jenny and Antti Wihuri Foundation, The Finnish Cultural Foundation, and the Australian National Fabrication Facility. We acknowledge the provision of facilities and technical support by Aalto University at Micronova Nanofabrication Centre. AR acknowledges support from the University of New South Wales Early Career Research Grant scheme.