Robust operation of a GaAs tunable barrier electron pump

We demonstrate the robust operation of a gallium arsenide tunable-barrier single-electron pump operating with 1 part-per-million accuracy at a temperature of $1.3$~K and a pumping frequency of $500$~MHz. The accuracy of current quantisation is investigated as a function of multiple control parameters, and robust plateaus are seen as a function of three control gate voltages and RF drive power. The electron capture is found to be in the decay-cascade, rather than the thermally-broadened regime. The observation of robust plateaus at an elevated temperature which does not require expensive refrigeration is an important step towards validating tunable-barrier pumps as practical current standards.


I. INTRODUCTION
The controlled transport of single electrons in mesoscopic devices has attracted much attention as a conceptually simple primary standard of electric current 1 . Very precise control of electrons has been achieved using chains of mescoscopic normal metal islands 2 , but limited to slow pumping rates < ∼ 10 MHz due to the fixed RC time-constant of the junctions between the islands. At the present time, the most practically useful combination of accuracy and high electron pumping rate has been achieved using electrostatically gated semiconductor quantum dots (QDs) operated as nonadiabatic tunable-barrier pumps 3 in the low-temperature decay cascade regime 4 . Using state-of-the-art current measurement techniques 5, 6 , there have been several reports of pumped current accurate at the part-per-million (ppm) level or better, at pump repetition rates in the range 0.5 GHz ≤ f ≤ 1 GHz, generating current 80 pA ≤ I P = ef ≤ 160 pA 5,[7][8][9][10][11] , where e is the elementary charge. These studies were performed on a variety of device architectures: etch-defined 5,8,10 and gate-defined 7 QDs in GaAs heterostructures, and silicon nano-wire MOSFETs 9 . While very promising for the metrological application of electron pumps, most of these studies were performed on carefully tuned devices. The required robustness of the current against changes in the pump control parameters has only recently begun to be investigated with high precision 11,12 , and only in one type of etch-defined pump.
In this study, we broaden the study of robustness, and investigate the gate-defined tunable barrier pump 7,13 . Most significantly for the application of pumps as practical current standards, we perform our measurements at ∼ 1.3 K, the temperature of pumped helium-4. This is in contrast to previous robustness studies 11,12,14 which were carried out at dilution refrigerator temperatures. Using a rigorous statistical approach to evaluate the plateau extension and flatness, we find robust plateaus in all the tuning parameters we investigated, flat to within the ∼ 2 × 10 −6 relative statistical uncertainty of each data point. Long measurements with the device in an optimally-tuned condition gave a current equal to ef within a relative total uncertainty of 8.6 × 10 −7 . We also show that despite the elevated temperature, the pump was operating in the decay-cascade regime and not the thermally-broadened regime predicted 15 and observed 16 at higher temperatures. Furthermore, the device was affected by a significant amount of charge noise. The robust performance of the pump under these non-ideal conditions is encouraging evidence that the semiconductor electron pump can fulfill a role as a practical current standard.
This paper is structured as follows: Section II describes the characterization and measurement technique. Section III presents the main experimental results in which we show that the pump current displays flat plateaus over a wide range of several tuning parameters. In section IV we analyze the statistical fluctuations of the current on the plateaus, and show that there is no indication of structure on the plateaus within the measurement uncertainty. Finally in section V we show that the pump is operating in the decay-cascade regime, and not in the thermal-equilibrium regime, even at the elevated temperature.

II. CHARACTERISATION
The pump used in this study (see SEM image in Fig.  1a) was realised in a 2-dimensional electron gas (2-DEG) in a GaAs-AlGaAs heterostructure with metallic surface gates. The sample was fabricated using techniques described previously 7,13 , and measured at a temperature of ∼ 1.3 K. DC voltages V G1 − V G6 defined a quantum dot in the region between the gates, and a sinusoidal AC voltage at f = 500 MHz was added to gate 1 using a room-temperature bias-T, to pump electrons from the source to the drain. The AC source had an output power P RF , calibrated for a 50 Ω load, and the total attenuation of the 50 Ω co-axial line between the source and the device was ≈ 4 dB. A magnetic field B = 13.5 T was applied perpendicular to the plane of the sample [17][18][19] . The pump current I P was measured in two modes; normal-accuracy and high-accuracy. In normal-accuracy mode, used for rapid characterization, the current was amplified by a room-temperature transimpedance amplifier, with an uncertainty in the gain calibration of ∼ 2 × 10 −4 . For high-accuracy measure-ments, I P was compared with a reference current derived from applying a voltage across a calibrated 1 GΩ standard resistor. 5,7,9 . In this mode the amplifier measures the small difference between the pump and reference currents, and provided this difference is made less than 0.05% of I P , by tuning the reference current, the calibration uncertainty of the amplifier contributes less than 1 × 10 −7 to the total relative uncertainty. We are chiefly interested in the deviation of I P from its expected quantised value ef , so we define the dimensionless normalised deviation, ∆I P ≡ (I P − ef )/ef .Likewise, all uncertainties in ∆I P will be expressed as relative uncertainties in dimensionless units. The RF modulation of the entrance gate, and the reference current source are turned on and off synchronously with a cycle time of 40 seconds to eliminate instrumental offsets. The onoff cycle is repeated n cyc times. To reject linear drift in the offset current, our data analysis routine calculates ∆I P using the data from the 'off' part of the cycle and half of the data from the two adjacent 'on' parts, thus generating n cyc − 1 statistically independent values of ∆I P with standard deviation σ I 20 . These values are then averaged to yield a mean ∆I P with statistical uncertainty U ST = σ I / n cyc − 1 (all uncertainties reported in this paper are 1 sigma standard uncertainties). The relative systematic uncertainty in ∆I P is dominated by the calibration uncertainty of the standard resistor, U 1G = 8 × 10 −7 , with an additional small contribution due to the voltage measurement U V < ∼ 2 × 10 −7 so that the total uncertainty Fig. 1b shows the derivative dI P /V G2 as a function of V G1 and V G2 , obtained from a normalaccuracy measurement, following an iterative tuning procedure to find the optimum settings for the DC gate voltages: (V G1 , V G2 , V G3 , V G4 , V G5 , V G6 ) = (−0.96, −0.7, 0.39, −0.78, 0.53, −1) V, and P RF = 5.2 dBm.
During the tuning procedure, plots of I P (V G1 , V G2 ) similar to Fig. 1b were obtained first while systematically stepping V G3 and V G5 , with the aim of maximising the width of the 1ef plateau. At minimum, a 4 × 4 matrix of (V G3 , V G5 ) values were investigated. Having found the optimal values of V G3 and V G5 , the procedure was repeated stepping V G4 and V G6 . Note that the relatively large negative values of the voltages applied to the lower finger gates in Fig. 1a, combined with the positive voltage applied to the plunger gate V G3 , has the effect of shifting the QD position above the axis of symmetry defined by the trench gate V G5 . The approximate location of the QD is indicated by a dashed red circle in Fig. 1a 13 .
The data of Fig. 1b was taken as a series of V G2 scans at fixed V G1 , with V G1 incremented between scans. This plot, known as the 'pump map', shows clearly the regions of zero derivative, where the current is invariant in the two control voltages 21,22 . The mis-alignment of regions of maximum derivative in successive scans visible in this data also shows that the device operation is affected by a random telegraph signal (RTS) well known to affect this type of 2-DEG structure 23,24 and already observed in another sample 7 with a similar design to the one in this study. Despite the noise, a broad region can be identified on the one-electron plateau where the derivative is zero within the resolution of the data. In the next section, we use high-accuracy measurements to investigate the robustness of current quantization on the one-electron plateau.

III. HIGH-ACCURACY PLATEAU MEASUREMENTS
We made a total of 6 high-accuracy measurement scans as a function of the control parameters V G1 , V G2 , V G3 and P RF , denoted S1-S6, as well as normal-accuracy measurements over a wider range of each scanned parameter. We also made a further 4 measurements with the pump tuning parameters fixed to the optimal values and n cyc = 750, 1400, 900 and 983, denoted F1-F4. The six scans and four fixed-parameter measurements were made over a period of 14 days. In Fig. 2 we present data from four of the scans, with each set of high-accuracy data plotted (filled circles) on logarithmic (Figs. 2(a-d)) and linear (Figs. 2(e-h)) axes, with normal-accuracy data (open circles) also shown on the logarithmic plots. Each high-accuracy data point in the data of Fig. 2 is averaged from 70 on-off cycles. The error bars indicate the statistical uncertainty U ST ∼ 2×10 −6 , which for these relatively short averaging times is the largest component of the total uncertainty; U T ∼ U ST . The normal-accuracy data has sufficient accuracy and signal-to-noise ratio to resolve relative deviations of ∆I P from ef as small as 10 −4 , and the logarithmic plot is a useful way to visualize the data during the iterative gate tuning procedure. In each scan plotted in Fig. 2, the fixed parameters were set to the optimum values noted in section II. Two additional scans were performed, S1 and S2 (not shown in Fig. 2), with one fixed parameter slightly offset from the optimum: S1 was a V G2 scan, with V G1 = −0.975 V, and S2 was a V G1 scan with V G2 = −0.695 V.
The effect of RTS noise can be seen in the normalaccuracy data, particularly for scan S3, where individual RTS switching events are indicated by gray arrows in Fig. 2a. Nevertheless, for each scan, the high-accuracy data exhibits a plateau where ∆I P appears invariant in the control parameter within the uncertainty of the individual data points. Scans S3 and S4 can immediately be compared with similar data measured using an etchdefined pump 11 , and we note that the plateaus in our gate defined pump are approximately twice as wide in both entrance gate (V G1 ) and exit gate (V G2 ) as those in the etch-defined pump. This may reflect a higher charging energy of the gate-defined pump, but it could also be an artifact of different lever arms (gate voltage to QD energy conversion factors) resulting from the very different geometries of the two types of device. Comparing scans S4 and S5 (Fig. 2f,g) the effect of the different lever arms of V G2 and V G3 on the QD level is clear: both of these gates control the depth of the QD, so I P has a similar functional dependence on either gate, but because V G3 is coupled much more strongly to the QD than V G2 , the plateau occupies a smaller range of gate voltage.
To evaluate the plateau extension more quantitatively, two methods were used. Firstly (the 'exponential fit method'), we fitted the high-accuracy data I P (x) to a sum of two exponential functions 15 where α 1 , α 2 , x 1 , x 2 , δ I are fitting parameters. The parameter δ I is the best-fit offset of the plateau from ef . We include it because we do not assume a priori that the plateau is exactly quantised. For runs S1-S6, we found 0.23 × 10 −6 ≤ δ I ≤ 1.33 × 10 −6 . For runs S3 and S6, only the second exponential term was used for the fit because the data had no clear deviation from the plateau on the low-x axis side. We defined the plateau width as the range of the control parameter for which This choice of δ fit reflects the lower limit to the statistical uncertainty achievable for realistic measurement times of order 1 day. Other studies 8,11 used the same method to define the plateau, but without including the offset δ I , and with δ fit = 10 −8 . The fits are shown in the lower panels of Fig. 2 as solid lines 25 , and the resulting selections of data points (number of points = N exp ) are enclosed by a solid box. The standard deviation of the N exp data points in each scan is denoted σ(∆I P ), and the statistical uncertainty of ∆I P averaged over these points on the plateau is U ST,plat = σ(∆I P )/ N exp . The scatter of the data points inside the boxes appears to be consistent with their individual uncertainties, but we will address this point more quantitatively in section IV. Secondly, a purely empirical criterion was used, based on linear fits to sections of the high-accuracy data (the 'linear fit method'). This method does not make any assumptions about the functional form of the data. For each scan, we found the largest number N lin of consecutive data points for which |S| < U SLOPE , where S is slope of a linear fit to the N lin points, and U SLOPE is the uncertainty in the slope 26 . The resulting data ranges are enclosed by dashed boxes in the lower panels of Fig. 2, and the relevant parameters are shown in table 1. As with the exponential fit method, the statistical uncertainty of the averaged points is given by U ST,plat = σ(∆I P )/ √ N lin . The linear fit method allows us to assign a numerical value to the plateau flatness given by U SLOPE multiplied by the plateau width. The flatness is comparable to the uncertainty of the data points from which it is derived, because the scatter of the data points determines the uncertainty in the linear regression. The flatness therefore is roughly between 1×10 −6 and 2×10 −6 for all the scans irrespective of the plateau width in the scanned units. For example, scans S4 and S5 have plateau widths in gate voltage units differing by roughly a factor 3 due to the  different lever arms of V G2 and V G3 as noted above, but the flatness for both the plateaus is ∼ 2 × 10 −6 . To evaluate the flatness with 10 −7 uncertainty using the linear fit method would require long averaging times, but we note that this is the only unambiguous method of proving that a plateau is flat. The exponential fit method, on the other hand, allows the plateau extension to be estimated based on a much shorter measurement, under the strong assumption that the fitting function (in this case, an exponential) captures all of the physics relevant to the pump accuracy at the target level of uncertainty.
For all the scans, N exp < N lin , which is to be expected since we chose δ fit U ST ; the exponential fit method estimates the plateau extension to be smaller than the lin-ear fit method, because the latter is only constrained by U ST ∼ 2 × 10 −6 . For scan S3, scatter of some of the data points strongly constrained the range of points which satisfied the linear fit criterion. As can be seen from table I, a similar scan, S2, exhibited a plateau in V G1 more than twice as wide in gate voltage. The question of whether the scatter in run S3 is excessively large is addressed in section IV. Regarding the P RF scan S6, there are some indications in Fig. 2h that an exponential function does not adequately describe the increase of the current for P RF > 5.7 dBm, and we speculate that rectification 27 or heating may play a role in the breakdown of quantised pumping at large gate drive amplitudes.
The current averaged over the plateaus, with the ST,plat + U 2 V for scans S1-S6, and U 2 ST + U 2 V for fixedpoints runs F1-F4. The uncertainty in the resistor calibration, U 1G = 8 × 10 −7 , is shown as a shaded grey box. For scans S1-S6, the plotted value is an average over a range of a control parameter, with the range selected using the linear fit method (open circles) and exponential fit method (filled triangles). (b): Statistical properties of the data in plot (a). Data points corresponding to the same run are aligned vertically in (a) and (b). Open circles show the mean U ST for the data points on-plateau, as selected by the exponential fit method for runs S1-S6. For runs F1-F4, they show the mean U ST for the data set analyzed in blocks of 70 on-off cycles. Filled triangles show the standard deviation of the data points on-plateau, and for runs F1-F4 the standard deviation of the data analyzed in blocks of 70 cycles. Horizontal bars show the 68% coverage upper and lower bounds for Nexp measurements of ∆I P to have a given standard deviation, assuming that ∆I P is normally distributed with standard deviation U ST . (c): Allan deviation of pumped current as a function of the number of on-off cycles, calculated from 3 of the runs at fixed operating point. The Allan deviation for run F1 (not shown) exhibited similar behavior. The gray dashed line shows the expected 1/ √ t dependence for frequency-independent Johnson-Nyquist noise in the reference resistor.
plateaus defined using both the exponential (closed triangles) and linear (open circles) fit methods, is plotted in Fig. 3a for runs S1-S6. Error bars show the un-correlated uncertainty U 2 ST,plat + U 2 V . The current measured in runs F1-F4 with the pump at fixed operating point is also plotted on the same graph (closed circles), with error bars indicating U 2 ST + U 2 V . The un-correlated uncertainty does not include U 1G , which is shown as a grey box centred on ∆I P = 0. The resistor was calibrated before and after the measurement campaign and its value was assumed constant during the campaign based on its long-term drift rate of ∼ 0.01(µΩ/Ω)/day 5 . In contrast, the voltage measurement was calibrated before and after each run. The un-correlated uncertainty thus allows the different measurements of ∆I P to be compared with each other without the additional uncertainty associated with linking to the SI unit system. For example, the two fixed-point runs with the lowest uncertainty, F3 and F4, are consistent within their combined uncertainty of 7.8 × 10 −7 . If the plateau is defined using the exponential fit method, the average current is consistent with ef within the uncertainties, and furthermore there are no major inconsistencies between the data points when only the un-correlated uncertainty is considered. Averaging all the data from the four fixed-point runs (a total of 4033 cycles lasting 47 hours) reduced U ST such that U T ∼ U 1G and yielded a best estimate of the pump current: ∆I P = 0.28 ± 0.86 × 10 −6 . This is marginally more accurate than the previous best electron pump measurement using the current measurement system at NPL 9 , although it falls short of the record low uncertainty of 1.6 × 10 −7 recently reported 11 using a measurement system based on a new type of ultra-stable current preamplifier known as an 'ULCA' 6 . Future efforts will aim to reduce U 1G to around 2 × 10 −7 as well as implementing an ULCA-based measurement system at NPL. It is interesting to note that the accumulated precision measurements and associated theoretical fit lines 5,7-9,11 , suggest that a tunable-barrier electron pump operated at an optimal working point is accurate at the 1 × 10 −7 level. With this premise, we could hypothetically consider the pump as a primary current standard, and the data of runs F1-F4 as constituting a calibration of the reference resistor with total uncertainty ∼ 3 × 10 −7 , almost a factor 3 lower than the U 1G presently achievable at NPL. However, we believe such a step would be premature, and that the robustness of these pumps requires further extensive investigation before a consensus can be reached on the required set of conditions for operation at a given accuracy level.

IV. STATISTICAL EVALUATION OF PLATEAU CURRENT
The data points on the plateaus in Fig. 2 show some scatter, and we now evaluate whether this scatter is con-sistent with statistical scatter about a stationary mean or whether it is a sign of structure on the plateau, or possibly drift in the pump current or the measurement system. We note that recent developments in the metrology of small currents 6,28 have focused attention on the stability of high-value thick-film standard resistors, principally those of 100 MΩ value. The 1 GΩ standard resistor used in the reference current source is also a thick-film design, and may suffer from short-term instability at the sub-ppm level. However, the uncertainties in the data of Figs. 2 and 3 are too large for this to have a significant effect on the scatter of the data points. We focus on the more conservative (narrower) plateaus defined using the exponential fit method. For each scan, the mean of the N exp values of the statistical uncertainty U ST , denoted U ST , is plotted as the open points in Fig 3b. We also plot as solid points the standard deviation σ(∆I P ) of the N exp values of ∆I P . If the current on the plateau was drifting on the time-scale of the scan, or if the plateau was not flat, we expect σ(∆I P ) > U ST . To assign a statistical significance to the ratio σ(∆I P )/ U ST , we used a numerical simulation to assign a 68% confidence interval to the distribution of σ(∆I P ) expected for N exp normally-distributed measurements with standard deviation U ST 29 . This is plotted as upper and lower horizontal bars in Fig. 3b. The fixed-parameter runs F1-F4 were evaluated in the same way as the scans, by dividing the data into blocks of 70 cycles and analyzing each block separately. Over the whole data set, there is no statistically significant deviation of the ratio σ(∆I P )/ U ST from 1. One particular run, S3, appeared to have anomalously large scatter, visible in Fig. 2(e) and already discussed in section III. This scatter is apparent in Fig. 3(b), in the relatively large ratio of σ(∆I P )/ U ST . However, σ(∆I P ) is still just within the 68% confidence interval, clarifying that the data at different V G1 values cannot be distinguished from data drawn from the same distribution. Overall, we conclude from this analysis that the scatter of the data points on the plateaus is consistent with statistical fluctuations about a stationary mean. This conclusion is supported by the Allan deviation of the current measured from runs F1-F4, all of which exhibited similar behavior. The Allan deviation plots for runs F2-F4 are shown in Fig. 3c. They show no significant deviation from the expected √ t behavior for frequency-independent noise out to the longest averaging times probed by the Allan deviation analysis 30 , roughly one quarter of the total measurement time, or ∼ 3 hours. For comparison, the dashed line shows the expected Allan deviation of frequency independent Johnson-Nyquist noise in the 1 GΩ resistor, (4.2 fA/ √ Hz)/( √ 2τ ), where τ = 40 s is the time for one on-off cycle. The Allan deviation of the pump current is increased above this theoretical level due to three inefficiencies in the duty cycle which reduce the effective averaging time: The onoff cycle means the pump current is only measured for half the time, auto zero in the readout voltmeters halves the measurement time again, and rejection of data points at the start of each half-cycle, to eliminate transient effects, further reduces the duty cycle. The latter two of these effects need to be optimized in future experiments to yield a lower overall statistical uncertainty 11 .

V. PUMPING REGIME AND NOISE BROADENING
The relatively high temperature of these measurements compared to previous high-precision studies motivated us to consider the mechanism of charge capture by the pump. At low temperatures, this occurs by a cascade of one-way tunneling events whereby electrons tunnel back to the source electrode as the QD is progressively isolated from the source 4,31 . The experimental signature of the decay cascade is a characteristic double-exponential shape to the pump current as a function of the QD depthtuning parameter. This tuning parameter can be the 'exit gate' voltage in simple two-gate pumps 5,8,14 , or a global top gate voltage 9,31 , and in this work its role can be fulfilled by either V G2 or V G3 . At higher temperatures, experimental 16 and theoretical 15,32 work has indicated a cross-over to a thermal regime, in which back-tunneling is accompanied by forward tunneling into the QD from the source. This results in a symmetric shape to the current as a function of QD depth tuning parameter, reflecting the Fermi distribution of electrons in the leads. The cross-over to the thermal regime has been predicted to occur for 10 × k B T > ∼ ∆ ptb 15 . Here, ∆ ptb is defined as the change in energy of the QD level when the entrance barrier transmission changes by a factor of Euler's number ∼ 2.718... and it thus quantifies the devicespecific cross coupling between the modulated entrance barrier, and the QD energy level 32 . We crudely estimate ∆ ptb ∼ 1 meV= 8.9 × k B T for our device, based on the slope of representative conductance pinch-off data and typical lever arm factors between a gate voltage and QD energy level. From this estimate we expect the device to be between the two regimes, and we next examine experimental data to clarify the capture mechanism.
In Fig. 4a, we plot the normalized pump current as a function of V G3 , which functions as a QD depth-tuning gate. A RTS is visible in the transition between the plateaus, where the pump current is a sensitive probe of changes in the electrostatic potential. On the plateau, the current is insensitive to the state of the RTS. For the data of Fig. 4a, in the transition region between I P = 0 and I P = ef , the charge state causing the RTS noise appears to be in one state for the majority of the data points (filled points), and the points affected by a switch to the other state (open points) were excluded from fitting. The data is fitted to the decay cascade model 4 : over the full range (solid line), with reduced χ 2 = 2.6 × 10 −5 , yielding the fit parameter δ 2 ≡ (∆ 2 − ∆ 1 ) = 15.2, and a thermal equilibrium (Fermi function) model 16,32 : in the range 0.32 ≤ V G3 ≤ 0.38 (dotted line) with reduced χ 2 = 3.5 × 10 −4 . Close inspection of the fit lines shows that the decay-cascade model gives a better fit, and the thermal equilibrium model fails to reproduce the asymmetric plateau shape, with a sharp riser from I P = 0 and a more gradual approach to I P = ef . The reduced χ 2 for the decay-cascade fit is more than a factor 10 smaller than the thermal equilibrium fit, suggesting that the pump is operating in the decay cascade regime.
A similar conclusion was drawn by fitting equations (2) and (3) to a I P (V G2 ) scan 33 which was obtained with a faster sweep rate to the data of Fig 4(a). This data was not so much affected by RTS switching events, at the expense of a much smaller number of data points (∼ 15) in the region between I P = 0 and I P = ef . For this data, equation (2) yielded δ 2 = 15.6 with χ 2 = 2.6 × 10 −4 and equation (3) yielded a fit with χ 2 = 6.0 × 10 −4 . The high RF power levels used in this experiment, corresponding to on-chip peak-to-peak gate voltages of order 1 V, raise the possibility that the electron temperature in the leads is elevated from the refrigerator bath temperature, for example by RF currents from the entrance gate flowing to ground through stray capacitances and parts of the leads. We did not estimate the electron temperature in the leads, but some insight can be gained by cooling the device to 300 mK. If RF-induced heating was the dominant mechanism determining the electron temperature at a bath temperature of 1.3 K, we would not expect further reduction of the bath temperature to have any effect on the device characteristics. In fact, we observe a considerable sharpening of the plateau when the device is cooled to 300 mK; fits of I P (V G2 ) to equation (2) yield δ 2 ∼ 20, compared to δ 2 ∼ 15 at 1.3 K 34 . We can conclude that RF-induced heating is not a dominant mechanism determining the device characteristics at a bath temperature of 1.3 K, although it may play a role at 300 mK.
We also rule out the possibility that the data of Fig.  4a is broadened by noise leading to erroneous conclusions from the fits. We calculated numerically the effect of Gaussian fluctuations in V G3 , with standard deviation V N , on the ideal decay-cascade behavior described by equation (2). Fig. 4b shows eq. (2) with δ 2 = 20 (solid line), and after broadening with V N = 12 mV (dotted line). The broadened characteristic is more symmetric and resembles a thermal distribution. Fitting the noisebroadened characteristic to the decay-cascade formula results in a decreasing δ 2 parameter as V N is increased (Fig.  4c, solid symbols), but also a progressive reduction in the quality of the fit, reflected in an increase in χ 2 (Fig. 4d, solid symbols). Fitting to the thermal function, eq. (3), the reverse is true: the thermal fit becomes a more accurate description of the simulated data for larger noise amplitudes (Fig. 4d, open symbols). Comparing the actual χ 2 values obtained from fitting the data of Fig. 4(a) (horizontal dashed lines in Fig. 4(d)) with those calculated from noise broadening, we conclude that the experimentally measured data is not consistent with more than a few mV of noise broadening, and the pump is indeed operating in the decay cascade regime at our experimental temperature.

VI. CONCLUSIONS
In conclusion, pumping in a GaAs tunable-barrier electron pump is robust against changes in the gate control parameters, and the RF drive amplitude, at the partper-million level at a temperature of 1.3 K. The presence of two-level flucutators did not affect the accuracy of the pump current. Compared to previous studies, this relaxes the experimental conditions required to observe quantised pumping at the part-per-million accuracy level, which is a promising step towards adoption of quantised charge pumps as current standards.

VII. SUPPLEMENTARY INFORMATION
The purpose of this supplementary information is to provide more detail on the analysis process for the highaccuracy measurements ( Figure S1), and the calculation of statistical quantities used in the main text ( Figure  S2). We also show a comparison between exit gate characteristics at two different temperatures ( Figure S3), illustrating the sharpening of the plateau when the pump is cooled from 1.3 K to 300 mK. Figure S1 illustrates the process of analysing raw data. As explained in the main text, the high-accuracy measurement system compares the unknown pump current I P with a reference current generated by applying a voltage across a calibrated 1 GΩ resistor. The raw data are readings from two instruments: an ammeter which measures the difference between the pump and reference currents ( Fig. S1(a)) and a voltmeter which measures the voltage across the 1 GΩ resistor (Fig. S1(b)). The currents are switched on and off synchronously with a cycle time of 40 seconds to remove instrumental offsets. Note that due to careful tuning of the reference current source, the (on-off) ammeter difference signal ∼ 10 fA is only just visible in the raw data. In this study, each half-cycle consisted of 50 readings from each instrument, triggered synchronously, with the instruments set to integrate for 10 power line cycles (at a nominal power line frequency of 50 Hz) and including an auto zero measurement with each reading. Thus, each instrument reading takes 0.4 s, each half-cycle takes 20 s and the 70 cycles shown in the figure take 2800 s.
The inset to Fig. 1(a) shows a small portion of the ammeter data from the main panel; the first one and a half on-off cycles. The points are color-coded to illustrate the data analysis procedure. The first 16 data points following each current switch (grey) are rejected to eliminate transient effects. The remaining 34 data points from the off half-cycle (green) are averaged to yield I OFF . The 34 points from each on half-cycle are divided into two equal portions, and the two blocks of 17 points adjacent to the off cycle are averaged to yield I ON . The ammeter difference signal extracted from the illustrated data is ∆I(1) = I ON − I OFF . The first block from the first on half-cycle (first section of black points) is discarded altogether, and the second block from the second on halfcycle (second section of black points) is analysed with the second off half cycle, and the first block from the third on half-cycle to yield ∆I (2), and so on up to ∆I(n cyc − 1). The voltmeter data is analyzed in a similar way to yield ∆V (j), with 1 ≤ j ≤ (n − 1), and the pump current is given by I P (j) = ∆V (j)/R + ∆I(j) 5 . Breaking the data set up in this manner makes the measurement of I P insensitive to linear drift in the offset of the measured signals, at the expense of discarding data from one on-off cycle. The data analysis thus yields n cyc − 1 values of ∆I P from a raw data set of n cyc cycles (Fig. S1(c)). The mean and standard deviation σ I of the data of Fig. S1(c) yield one data point in Fig. S1 The mean of these 6 data points is indicated by a horizontal dashed line, and the standard deviation by a vertical double arrow. The red data point to the right shows the mean with the error bar indicating statistical uncertainty U ST,plat = σ(∆I P )/ Nexp. This is the same data point plotted as a solid triangle labeled 'S4' in Fig. 3(a) of the main text, although in the main text the error bar is marginally larger because it includes U V . with ∆I P = 3.79 × 10 −6 ± U ST = (3.79 ± 2.33) × 10 −6 . Here, the statistical uncertainty U ST is given by the standard error on the mean σ I / n cyc − 1. For this data point the relative total uncertainty U T = U 2 Referring to the scatter of the data points in Fig.  S1(d), we define two more statistical terms, with reference to Fig. S1(e). Here, we have re-plotted the N exp = 6 data points in Fig. S1(d) which are determined to be on the ef plateau by using the exponential fit method (equation (1) of the main text). The mean of the data points is indicated by a horizontal dashed line. Each of the data points has a statistical uncertainty, and we calculate the mean of these statistical uncertainties, denoted U ST = 2.18 × 10 −6 . We also calculate the standard deviation of the 6 data points, denoted σ(∆I P ), from which we derive the standard error of the mean = σ(∆I P )/ N exp = U ST,plat . To recap, the symbol U ST denotes the statistical uncertainty for a measurement of ∆I P at fixed pump operating point, while U ST,plat denotes the statistical uncertainty of an average of several measurements of ∆I P at different operating points along a plateau.
If the plateau is truly flat, the scatter of the data points on the plateau given by σ(∆I P ) should be on average the same as the uncertainty U ST of a single data point. In other words, data points measured at different points along the plateau are sampling the same stationary mean value with a standard deviation given by the same underlying noise process. The dominant source of noise in our experiment comes from the measurement system: Johnson noise in the 1 GΩ reference resistor, with additional small contributions from the current pre-amplifier and cryogenic wiring. This is reflected in the almost constant (within ∼ 10%) values of U ST for n cyc = 70 visible in Fig.  3(b) (open circles) of the main text. We can therefore state that on a true plateau, data points with n cyc = 70 should be drawn from a normal (Gaussian) parent distribution with standard deviation U ST . Since we measure a limited number N exp of data points on the plateau, we can compare the standard deviation σ(∆I P ) of these points with the expected distribution of the standard deviation, if we randomly selected N exp points from the parent distribution. To do this, we numerically generated a parent distribution with a large number of data points, and randomly selected N exp points from it. The random selection was repeated 10 6 times, and in Fig. S2 (a), we plot the histogram of the σ(∆I P ) values for N exp = 6 and U ST = 2.18 × 10 −6 (the parameters corresponding to run S4, Fig. 2(f) of the main text). The value of σ(∆I P ) measured for run S4 is shown as a dashed vertical white line super-imposed on the histogram. In the inset to Fig.  S2(a) we plot a histogram of (N − 1)[σ(∆I P )] 2 / U ST 2 (black bars) for the same random data set as the main panel, and for comparison, the χ 2 distribution with 5 degrees of freedom (solid red line). This illustrates a standard text-book result, namely that the variance of N randomly selected points is distributed according to the χ 2 distribution with N − 1 degrees of freedom. Fig.  S2(b) shows the cumulative sum of the histogram in (a), with vertical dashed lines showing the 1σ (68 % coverage) upper and lower limits to σ(I P ). These are plotted as horizontal bars for run S4 in Fig. 3(b) of the main text. The process illustrated in Fig. S2 was repeated with the parameters [ U ST , N exp ]for the remaining 5 scans to derive the horizontal bars in Fig. 3(b) of the main text. . The gate voltages are tuned to the optimal working point used for the data of the main text, apart from V G1 = −0.94 V. Each data set has been fitted to the decay cascade model (equation (2) of the main text) and a Fermi function (equation (3) of the main text), with the independent variable V G3 in the equations replaced by V G2 . χ 2 values for the fits are indicated on the plot. As for the χ 2 values reported in the main text, χ 2 is the sum of the square of the fit residuals divided by the number degrees of freedom.