Accurate high speed single-electron quantum dot preparation

Using standard microfabrication techniques it is now possible to construct devices, which appear to reliably manipulate electrons one at a time. These devices have potential use as building blocks in quantum computing devices, or as a standard of electrical current derived only from a frequency and the fundamental charge. To date the error rate in semiconductor 'tuneable-barrier' pump devices, those which show most promise for high frequency operation, have not been tested in detail. We present high accuracy measurements of the current from an etched GaAs quantum dot pump, operated at zero source-drain bias voltage with a single AC-modulated gate driving the pump cycle. By comparison with a reference current derived from primary standards, we show that the electron transfer accuracy is better than 15 parts per million. High-resolution studies of the dependence of the pump current on the quantum dot tuning parameters also reveal possible deviations from a model used to describe the pumping cycle.

Devices that can reliably transfer electrons one at a time, electron pumps, have important applications in the fields of electrical metrology [1], and solid-state quantum computing [2,3].
In the former field, there is especially strong interest, motivated by re-defining the SI base unit Ampere in terms of the electron charge and a known frequency [4,5]. Pumps based on multiple metal-oxide tunnel junctions have demonstrated very high relative accuracies approaching 10 -8 [6], but the speed of transfer is limited to around 10 MHz by the intrinsic time-constant of the junctions. The resulting pumped current ≈1 pA is at least an order of magnitude too small for the pump to function as a useful current standard, although it was used to demonstrate a quantum capacitance standard by charging a capacitor with a known number of electrons [7]. More recently, an innovative device, the "hybrid turnstile" has been demonstrated, utilizing metal-oxide-superconductor tunnel junctions [8]. Unlike the multiplejunction pumps, the hybrid turnstile needs only one AC control signal, and consequently the current can be increased by operating many devices in parallel [9]. However, the hybrid turnstile needs to be operated at finite bias voltage ≈ 1 mV, and eliminating errors due to leakage currents is a challenging ongoing project [10].
Another class of electron pumps exploits the tunability of potential barriers in reduceddimensional semiconductor systems. Following the pioneering work of Kouwenhoven et al, on pumping electrons through a quantum dot at finite source-drain bias [11], it was found that electrons could be pumped through a dot at zero source-drain bias by applying a large AC modulation to just one of the gates [12,13], as illustrated schematically in fig. 1a for the simplest case of one electron pumped for each cycle. The experimental signature of pumping is a DC current I P ≈ I 0 ≡ n 0 ef, where f is the repetition frequency of the potential modulation, and n 0 is an integer. Pumping has been observed in etched GaAs 2-dimensional electron gas (2-DEG) quantum dots [12,14] and silicon nano-wire MOSFETs [13], for f up to the order of [15]. The high operation speed, zero source-drain bias, and possibility of parallel current scaling, makes these pumps promising candidates for a primary metrological current source [1], as well as a source of single electrons for semiconductor-based quantum logic gates [3].
A crucial unanswered question addressed in this work concerns the accuracy of the electron transfer in the semiconductor pump. Estimates of acceptable error rates for fault-tolerant quantum computing range from 1 in 10 2 to 10 6 qubit operations [16], while metrological application of electron pumps as quantum standards of current requires error rates less than 1 in 10 7 [4,5]. In contrast, normal laboratory measurements of the fractional error in the pumped current ∆I P =(I P -I 0 )/I 0 , have at best shown ⏐∆I P ⏐≤ 10 -2 . One study, using a calibrated ammeter to measure I P , set a lower limit to possible errors: ⏐∆I P ⏐≤ 10 -4 , but this level of accuracy was only observed over a very narrow range of gate voltages used to tune the pump operating point [14]. Theoretical treatments of the error rates in these pumps are much more difficult than for the case of low-frequency fixed-barrier pumps [17,18], partly due to the rapid nonadiabatic change in the tunnel coupling from the source reservoir to the dot [19]. Some features of the pump behaviour have been explained by considering the time dependence of the back-tunneling rates during the initial phase of the pump cycle (Fig. 1a, frame 2) [12,13,20,21], and predictions of ∆I P ≤ 10 -5 have been made using this approach [21]. Furthermore, recent experiments showed that application of a magnetic field of a few Tesla results in an improvement in plateau flatness in GaAs pumps [22,23]. This suggests that spin states within the dot [24] or edge states in the leads [25] may play a role in the transport. With this promising experimental and theoretical background, high-accuracy measurements of the pump current are clearly of great interest. In this work we compare the current from GaAs pumps with a reference current derived from primary electrical standards with relative accuracy approaching 10 -5 . This enables us to set much more stringent limits on error mechanisms than was possible from previously measured data.
Our pumps were fabricated by wet-chemical etching of sub-micron width wires in a GaAs 2-DEG system, followed by deposition and patterning of Ti/Au surface gates [22]. An SEM image of a pump is shown in Fig. 1c, together with the circuitry for biasing the gates and measuring the current. The key feature of our measurement setup is the reference current I R , with opposite polarity to I P , which we generated by applying a linear voltage ramp to a lowloss capacitor [26]: I R =CdV CAL /dt. I R was traceable to primary maintained standards of capacitance, voltage and time, and had a relative systematic uncertainty of 15 parts per million (ppm). The magnitude of I R was adjusted to be within 0.2% of I P ; consequently the ammeter current I=I P -I R was small and variations in the ammeter gain (for example due to ambient temperature changes) had a negligible affect on the result. To remove offsets in the measurement circuitry and reference current source, both the pump and reference currents were switched on and off with a cycle period of 60 s, and the pump current calculated from the difference signal. The raw ammeter readings from one pump cycle are shown in Fig. 1d.
A sine wave at frequency f = 340 MHz applied to one gate implemented the pumping cycle, illustrated schematically in fig. 1 a and b. The pumps were mounted in a dilution refrigerator, and all data was taken at a mixing-chamber temperature of ≈30 mK and a perpendicular magnetic field of 5 T. We investigated the pump behaviour as a function of four adjustable control parameters: the DC voltages applied to the two gates, V GS and V GD , the RF generator power P RF , and the source-drain bias voltage V B . In all data apart from Fig. 3c, V B = 0.
Following each cool-down, the parameters were tuned iteratively to yield maximally flat quantised plateaus before making the measurements presented in this work.  [15,22,23]. We fitted the I P (V GD ) data to a back-tunneling model [21], (solid lines in the plot), and obtained minimum values of the error from the fits of ∆I P,MIN = -4×10 -6 and -1×10 -5 for samples A and B respectively. The minimum error is obtained at the value of V GD for which dI P /dV GD is minimum. Next we used our high-resolution measurement technique to zoom in on the plateau region. Fig 2b shows  can be resolved as a finite gradient dI P /dV B . This leakage does not constitute a significant source of error when the pump is operated close to zero V B . Otherwise, each scan shows a plateau region flat to within the typical ≈10 ppm error in the slope of a linear fit. Fig. 3e shows average values of ∆I P for all 6 high-resolution data sets presented in this paper, with shading to indicate the systematic uncertainty at 68% confidence (dark grey) and 95% confidence (light grey) intervals. The data points in this figure are the weighted averages of four consecutive points from each data set, chosen from the centre of each plateau (indicated by vertical lines in fig3 a-d). The difference between the mean currents for samples A and B is ∆I P (A) -∆I P (B) = (-5±2.5)×10 -6 , indicating possible sample-dependent errors at the ppm level which will be investigated more closely in future work. Note that the comparison of the currents from two samples is limited only by the random uncertainty. A weighted average of all data points in the figure yields an overall estimate of the pump error, ∆I P =(-14.8±15)×10 -6 .
Because we operate the pump in a regime where all the electrons loaded into the dot are ejected [28], we can also interpret our result as probing the reliability of loading the dot with just one electron in repeated operations: n=0.9999852±0.000015. This data is convincing evidence that the electron transport in tunable-barrier pumps is robustly quantised at the 10 -5 level or better, over a useful range of parameter space.
The accuracy in our experiment is close to the limit of what can be achieved with conventional room-temperature instrumentation. To further improve the systematic uncertainty, a cryogenic current comparator (CCC) could be used to compare the pump and reference currents [29]. This would enable a direct test of the error rates of a few ppm predicted by the model in ref. 21, and place experimental limits on other types of error such as thermally activated tunnelling, which have been predicted to be negligible [20]. Furthermore, if independent confirmation of the pump transport accuracy could be obtained, for example by a shuttle-type experiment with an on-chip charge detector to detect individual transport errors [6], the same experimental setup would constitute realisation of the metrological triangle [5] which is one of the long-standing goals of fundamental metrology.
The authors would like to thank Bernd Kaestner and Chris Ford for stimulating discussions.