Re-evaluation of uncertainty for calibration of 100 M Ω and 1 G Ω resistors at NPL
RRe-evaluation of uncertainty for calibration of M Ω and G Ω resistors atNPL S. P. Giblin National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW,United Kingdom a) (Dated: 29 August 2018) The uncertainty for calibrating 100 MΩ and 1 GΩ resistors using the NPL high-resistance CCC bridge hasbeen re-evaluated, resulting in combined uncertainties of ∼ . µ Ω / Ω, considerably smaller than the CMCentries of 0 . µ Ω / Ω and 1 . µ Ω / Ω respectively. This re-evaluation supports measurement of single electronpump currents with combined uncertainties of a few parts in 10 .PACS numbers: 1234 I. INTRODUCTION
Traceable measurements of high resistance are re-quired for diverse applications including semiconductorwafer characterization, insulator testing and radiationdosimetry. Recently, research into semiconductor elec-tron pumps , motivated by the upcoming re-definition ofthe SI base unit ampere , has focused additional atten-tion onto high resistance and small current metrology.Semiconductor electron pumps are micron-scale, cryo-genic devices, which generate a current by transportingelectrons one at a time in response to a clock input withfrequency f , yielding a current ideally equal to ef , wheree is the elementary charge. Promising early experimentsusing conventional laboratory ammeters to measure thecurrent showed that these pumps could generate quan-tized current plateaus with f of order ∼ I P larger than 100 pA . Sincethen, metrological research has focused on measuring I P as accurately as possible to investigate possible error pro-cesses, and so far seven studies have shown the pump cur-rent to be accurately quantised within measurement un-certainties of 1 part-per-million or less . Five of thesestudies were performed at the National Physi-cal Laboratory (NPL), UK, using a measurement systemwhich compared I P to a reference current I R derived froma measured voltage across a calibrated 1 GΩ standardresistor . The focus of this paper is a precise evaluationof the uncertainty in calibrating this resistor.The high-resistance cryogenic current comparator(CCC) has been in routine use at NPL since 2001 forcalibration of decade-value standard resistors in the rangefrom 100 kΩ to 1 GΩ. The calibration and measure-ment capability (CMC) expanded relative uncertainties( k = 2) are 0 . × − , 0 . × − , 0 . × − , 0 . × − and 1 . × − for decade standard resistors of nominalvalue 100 kΩ, 1 MΩ, 10 MΩ, 100 MΩ and 1 GΩ respec-tively. For the first four of the NPL electron pump mea-surement campaigns , the type B uncertainty in the a) [email protected] e-mail address σ relative uncertainty contributionof 8 × − . This was the largest single contribution tothe uncertainty in I P in all these studies. The relative un-certainty in measuring the voltage across the resistor wasreduced to ∼ × − by frequently calibrating the volt-meter directly against a Josephson voltage standard .Averaging up to two days of data reduced the type Auncertainty in I P to a similar level, leaving the 1 GΩ un-certainty completely dominating the uncertainty budget.Thus, the present study was undertaken to re-evaluatethe type B uncertainty in the 1 GΩ calibration. The re-evaluated uncertainty was used in a recent measurementcampaign on a silicon electron pump, yielding a relativecombined uncertainty of 2 . × − for measuring a pumpcurrent of ∼
160 pA .An alternative approach to reducing the uncertaintycould have been to use a lower-value resistor to generate I R , for example 100 MΩ instead of 1 GΩ. This involvesa trade-off: lower type B uncertainty is attained at theexpense of larger type A uncertainty because Johnson-Nyquist current noise scales as 1 / √ R . For the NPL CMCcalibration uncertainties, the trade-off becomes favorablefor I P > ∼
150 pA . However, recent investigations haveshown that available 100 MΩ and 1 GΩ standard resistorsbased on thick-film elements may not be sufficiently sta-ble, on time-scales of hours to days, to routinely achieverelative uncertainties much less than ∼ × − inpractice . Much lower-value resistors are required toachieve sufficient stability, but as noted, these have un-acceptably high thermal current noise. One innovativeapproach is to combine a 1 MΩ current-to-voltage conver-sion resistor with a stable 1000 : 1 current-scaling resistornetwork. This package, known as the ultrastable low-noise current amplifier (ULCA) , combines the stabil-ity and low relative calibration uncertainty ( < × − )of the 1 MΩ resistor with the low thermal noise of thecurrent scaling network, which (in the standard ULCA)presents a 3 GΩ resistance to the input. The ULCA hasbeen used at the Physikalisch-Technische Bundesanstalt(PTB) , to measure electron pump currents with rela-tive combined uncertainties as low as 1 . × − .This paper is structured as follows: In section II, we a r X i v : . [ phy s i c s . i n s - d e t ] A ug FIG. 1.
Schematic diagrams of the bridge. The lower rightinset shows the general schematic circuit of the bridge, and themain figure shows the detailed connections specific to the casewhere R = 10 M Ω and R = 100 M Ω or G Ω . The screen isnot shown. Points H and L indicate the high and low potentialterminals of the voltage source, and points A, B, C and D arereferenced in the main text. The lead resistances r and r aredistributed between the junction L and the low terminals of R and R . The lower left inset illustrates the current scaling used inbuild-up measurements. Solid lines: routine traceability. Dashedline: SET G Ω measurement. The dashed line with gradient − shows 1 mW power dissipation give an overview of the CCC bridge, and describe itsmode of operation for calibrating 100 MΩ and 1 GΩ re-sistors. In section III we discuss the noise and type Auncertainty. In section IV we review independent tests ofthe bridge prior to this study. Section V forms the mainbody of the paper, in which we describe in detail our eval-uation of the main sources of bridge error: Leakage resis-tance, SQUID non-linearity, series resistance correctionand winding ratio error. In section VI we discuss furthererrors affecting the uncertainty in the unknown resistor.We aim to restrict our discussion to the CCC bridge it-self, but inevitably in this section we briefly discuss thestability of standard resistors in use at NPL. Finally insection VII we present the full uncertainty budget anddiscuss the possibilities for lower-noise measurements ofelectron pump currents. II. BRIDGE CIRCUIT AND OPERATION
A schematic diagram of the overall bridge circuit isshown in the lower right inset to figure 1. The design
FIG. 2.
Examples of raw SQUID and detector data for onemeasurement cycle. (a): M Ω : M Ω with V source = ± V.(b): M Ω : G Ω (the SET G Ω ) with V source = ± V. Verticalarrows indicate the difference voltages ∆ V S , ∆ V (cid:48) S and ∆ V D usedto calculate the resistance ratio using equation (4) of the maintext. Vertical dashed lines separate the two phases of the cyclewith N = 10000 (left side of the line) and N = 10001 (rightside of the line). Solid vertical bars provide a SQUID flux scale.(c): Example measurement of the SET G Ω : R = 1 G Ω(1 + δ R / ) . Each data point is calculated from one cycle of thetype shown in panel (b). (d): Allan deviation of the data shownin (c). The dashed line is a guide to the eye with slope / √ τ departs from more conventional CCC resistance bridges,which employ two isolated current sources to drive cur-rent in the two arms of the bridge . Our high-resistance CCC employs a single-source topology, whichplaces less stringent demands on the noise propertiesof the source . In our circuit, the source voltage V SOURCE drives current in the two resistors to be com-pared, R and R , and the CCC windings with N and N turns. Relays allow selection of N = 100 , , N is fixed at 10000 turns, with an additional turn addedperiodically by a relay, as will be detailed below. As inmore conventional dual-source CCC bridges , a Wag-ner circuit maintains the junction of R and N (pointA in figure 1) at screen potential to minimise errors dueto leakage resistance. A combining network composed ofresistors R C1 and R C2 reduces to a negligible level theeffect of current flowing in the high potential terminalsof R and R , but this plays no role for the high value,two-terminal resistors mostly considered in this paper. FIG. 3. (a): Amplitude spectrum of the voltage output of theSQUID control unit, referred to flux noise using the conversionfactor . V/ Φ . The high frequency data was acquired us-ing a -bit digitiser sampling at ksamples/s, and the lowfrequency data was obtained by block-averaging samplesfrom the digitiser. The intermediate frequency range was cov-ered using an integrating voltmeter with an aperture of µ s.High frequency data was obtained with the SQUID controllerfeedback loop set to high- and low-bandwidth modes. (b): filledcircles: Allan deviation calculated from the same time-domaindata set that yielded the low-frequency portion of the spectrumin plot (a). Open circles: Allan deviation calculated from a dataset obtained years prior to the data in this study. Open tri-angles: Allan deviation calculated from a data set obtained days after the data in panel (a). The single filled square showsthe data point from figure 2 (d) at τ = 100 s converted to fluxunits. The dashed lines in both plots show a white noise level of µ Φ / √ Hz . The routine build-up procedure (lower left inset to fig-ure 1) involves a series of 10:1 (or 100:1 for measuring1 GΩ) measurements with the lower-value resistor dis-sipating 1 mW of power. Measurements of 1 GΩ forthe single-electron current source used a lower value of V SOURCE = ±
33 V instead of ±
100 V for two reasons:firstly, to eliminate any possible voltage co-efficient inthe 10 MΩ reference resistor (since the 1 MΩ : 10 MΩand 10 MΩ : 1 GΩ measurements are made at the samevoltage), and secondly, to reduce the possible voltage co-efficient in the 1 GΩ, a point discussed further in sectionVI.B.For [ R : R ] = [10 kΩ : 100 kΩ] up to [1 MΩ : 10 MΩ],the squid output V S is used as the input to a digitalservo, which controls the gate voltage of a JFET in oneof the bridge arms. This controls the resistance of theJFET channel, and ensures the current balance condition I /I = N /N . The JFET resistance is balanced by afixed resistor R BAL in the other bridge arm. However,for the measurements we are mostly concerned with inthis paper, [ R : R ] = [10 MΩ : 100 MΩ] and [10 MΩ :1 GΩ], the servo is not used and the bridge functions asa passive current divider in which I /I = R B /R A with R A = R + r and R B = R + r . The series resistances r and r (main panel of figure 1) are due to wiring inthe room-temperature electronics, connecting cables andthe down-leads in the CCC cryostat.The SQUID output is proportional to the differenceof the ampere-turns product in the two windings: V S = k ( I N − I N ). The sensitivity co-efficient k is given by k = S SQUID /S CCC with S SQUID ≈ . / Φ and S CCC ≈ µ A turns / Φ . To eliminate k from the bridge equations,the measurement is performed in two phases, with N =10000 and N (cid:48) = 10001. For each phase, the voltagesource is reversed, yielding SQUID difference signals ∆ V S and ∆ V (cid:48) S . The detector voltage difference ∆ V D from thefirst phase is also extracted from the data. Examples ofraw data for 100 MΩ and 1 GΩ measurements are shownin figures 2 (a) and (b) respectively. The resistance ratioin the two arms of the bridge is then given by R B R A = N (cid:48) ∆ V S − N ∆ V (cid:48) S N (∆ V S − ∆ V (cid:48) S ) . (1)Since r (cid:28) R , r (cid:28) R , we can re-write equation (1)as R R = (cid:18) N (cid:48) ∆ V S − N ∆ V (cid:48) S N (∆ V S − ∆ V (cid:48) S ) (cid:19) (cid:18) r R − r R (cid:19) (2)It is necessary to measure r with an uncertainty lessthan 0 . × − to the relative uncertainty in R . Theseries resistances can be measured directly with a digitalmultimeter , and we found r ≈ . r ≈
15 Ω,but for our bridge it is more practical to use the bridgevoltage detector signal V D = I r − I r . Again, assuming r (cid:28) R , r (cid:28) R , the detector voltage difference ∆ V D when the source voltage is reversed is given by∆ V D = 2 V SOURCE (cid:18) r R − r R (cid:19) . (3)Substituting equation (3) into equation (2) yields thefinal equation for determining the resistance ratio, R R = (cid:18) N (cid:48) ∆ V S − N ∆ V (cid:48) S N (∆ V S − ∆ V (cid:48) S ) (cid:19) (cid:18) V D V SOURCE (cid:19) . (4)The main panel of figure 1 is a detailed schematic show-ing the resistor connections to the bridge for the case of[10 MΩ : 100 MΩ] and [10 MΩ : 1 GΩ] measurements.The 10 MΩ resistor R is a 2-terminal standard (usuallyFluke type 742A) in a temperature-controlled air bath,connected to the CCC with a 4-wire cable consisting oftwo individually screened twisted pairs. The 100 MΩor 1 GΩ resistor R usually has two coaxial terminals.The 1 GΩ resistor used for the single-electron measure-ments (of type Guildline 9336) has a measured temper-ature co-efficient of 5 . × − / ◦ C, and is in a custom-made temperature controlled enclosure with ± . ◦ Cstability. For the remainder of this paper we refer to thisartifact as the ‘SET 1 GΩ’.
III. NOISE AND TYPE A UNCERTAINTY
Figure 3 presents frequency-domain (panel(a)) and Al-lan deviation plots (panel (b)) of the SQUID flux noise,obtained from time traces of V S measured with the CCCwindings disconnected from the electronics (very simi-lar plots were obtained with the electronics connected).Several peaks are visible in the amplitude spectrum, themost prominent of which is a resonance at 80 Hz which isprobably of mechanical origin. The damped L-C wind-ing resonance is at ∼ . × − Φ / √ Hz (dashed line in both plots) is roughly20 times the bare SQUID noise. From the Allan devi-ation plot (filled circles), a transition from white noiseto random walk behavior occurs for times longer thanaround 10 seconds. However, we also show Allan devi-ation plots (open triangles and circles) obtained at dif-ferent times, which demonstrate that the low frequencySQUID noise behaviour exhibits considerable variationin time. A calibration of 1 GΩ with ±
33 nA in the10000 turn winding presents a full-signal flux linkage of2 I N /S CCC = 110 Φ . Therefore, based on the mea-sured white flux noise of 7 . × − Φ / √ Hz , we expectto resolve 7 × − of full signal at the measurement cycletime of 10 s. Crucially, the measured noise at τ = 100 splotted in Fig. 3 (b) varies over almost two orders ofmagnitude and corresponds to relative uncertainties ina 1 GΩ measurement from 4 × − to 1 . × − . Re-sults of a measurement of the SET 1 GΩ are shown infigure 2 (c). This measurement consisted of 128 cycles,lasting a total 3 . τ = 100 s is 5 . × − , and this value, con-verted to flux, is plotted as the single square data pointin 3 (b), showing that it is consistent with the more pes-simistic SQUID noise data. It is immediately apparentthat a better control of the low frequency SQUID noisebehaviour, combined with an optimisation of the mea-surement cycle time, could lead to a much lower type Auncertainty for high resistance measurements.The Allan deviation plot of figure 2 (d) suggests thatthe resistor is stable on the time-scale of ∼ ∼ × − to be achieved withan overnight measurement. In fact, long measurementsof the SET 1 GΩ and also a sample of 100 MΩ stan-dard resistors typically show instability on time-scaleslonger than a few hours and consequently, the stan-dard error on the mean is not a meaningful measure ofthe type A uncertainty . For the recent high-precisionelectron pump measurements , relative type A uncer-tainties ∼ × − were evaluated on a case-by-case ba-sis by examining the Allan deviation of resistor calibra-tions performed immediately before and after the pumpmeasurements . The main focus of this paper is the un-certainty in the CCC measurement, and the limitationsto the standard resistor stability will be the subject offuture papers. However, we will touch on the subject of the 1 GΩ resistor stability again in section VI.C, as thiswas an important limiting factor in performing electronpump measurements with relative uncertainties much be-low 1 ppm. IV. INDEPENDENT TESTS
Before embarking on a detailed evaluation of the bridgeuncertainty, we briefly consider four cases where thebridge has been compared with other measurement meth-ods. Firstly, the most recent relevant intercomparison,conducted in 2005-2007, was Euromet EM-K2, of 10 MΩand 1 GΩ resistors . The degrees of equivalence andtheir associated k = 2 uncertainty are plotted in figure4 (a). Also plotted are the NPL CMC uncertainties. Itcan be seen that the comparison uncertainty was consid-erably larger than the CMC uncertainties for both theresistor values. This was mainly due to a large trans-port uncertainty component ∼ . Whilethe comparison did not find any gross errors in the CCCmeasurements, it did not test the CCC at its design un-certainty level.Secondly, the bridge was compared with a voltage ra-tio technique for measuring a 1 MΩ : 10 MΩ resis-tance ratio. A schematic diagram of the voltage ratiobridge is shown in figure 4 (b). Two voltage sourcesapplied 1 V and −
10 V to the two-terminal 1 MΩ and10 MΩ resistors respectively, and the voltages were mea-sured by two precision DVMs (HP3458A). A sensitiveammeter connected to the junction of the low currentresistor terminals measured the residual error current,and the value of the unknown resistor was given by R = R [ − ∆ V / ( R ∆ I + ∆ V )], where ∆ V ,∆ V and∆ I are the difference signals recorded by the two volt-meters and the ammeter when the signs of the voltageswere reversed. The link between the low current resistorterminals was made with a thick braid, 10 cm long, andthe ammeter was connected close to the 1 MΩ terminal,roughly dividing the length of the braid in the ratio 1 : 9.The voltmeters were calibrated directly against a Joseph-son array at least once every hour, and the maximum rel-ative type B uncertainty due to the voltage measurementwas ∼ − . In fact, the voltage ratio method offers acomparable uncertainty to the CCC, but is impracticalfor routine use due to the requirement for frequent cali-bration of two voltmeters. As can be seen in figure 4 (c),the voltage ratio bridge gave results in agreement withthe CCC, within the combined uncertainties of the twomeasurements.Thirdly, an ULCA calibrated at PTB with a relativeuncertainty of ∼ . . At NPL, the ULCA was used to mea-sure I R from the reference current generator employedin the electron pump measurements. Both 100 MΩ and1 GΩ resistors were used to generate the reference cur- FIG. 4. (a): degree of equivalence d P and associated uncertainty(large black error bars) for NPL participation in the euromet EM-K2 comparison of M Ω and G Ω resistors. The NPL CMCuncertainty is shown as the smaller red error bars. Both errorbars show expanded k = 2 uncertainties. (b): schematic circuitdiagram of a voltage ratio bridge used to measure a M Ω :10 M Ω resistance ratio. (c): Comparison of the voltage ratiobridge measurements with CCC measurements on two M Ω standards, where R = 10 M Ω(1 + δ R / ) . The left (right)pair of data points refer to the left (right) y-axis. Error barsfor the voltage ratio bridge measurement indicate the total σ uncertainty. The black error bars on the CCC data points indicatethe type A uncertainty, and the grey error bars indicate the totalCMC uncertainty. rent, and in both cases agreement of ∼ . , a growingnumber of ULCA transfers now indicate that the ULCAgain is stable at at least the part in 10 level under trans-portation, and the results of Ref. can be interpreted aschecking the NPL traceability to 100 MΩ and 1 GΩ. Ina fourth series of experiments, to be presented in an up-coming paper , an ULCA was calibrated at NPL andused to measure a 1 GΩ resistor. Comparison betweenthe ULCA measurement and a CCC measurement of thesame resistor produced agreement better than 1 part in10 .Collectively, these four tests rule out gross errors in theCCC and support the NPL CMC relative uncertainties(k=2) of 2 × − , 4 × − and 1 . × − for 10 MΩ,100 MΩ and 1 GΩ respectively. They provide a base-linefrom which we commence our detailed evaluation of theCCC uncertainty. V. CCC UNCERTAINTY EVALUATION
Now we turn to the main aim of this paper: an evalua-tion of the type B uncertainty in the measurement of theratio R /R with the CCC operated as a passive currentdivider without a current servo. This is due to four con-tributions: leakage resistance, SQUID non-linearity, the correction due to series resistances r and r and finallythe winding ratio error which is evaluated using 1 : 1ratio error tests (RATs). A. leakage resistance
Referring to figure 1, the bridge is sensitive to leak-age at points A and B, in between the resistors and theCCC windings. As already noted, point A is maintainedat screen potential using a Wagner network, which min-imises the effect of a leakage resistance between this pointand screen. However, the lead resistances between pointsA and B, and the low terminals of R and R respectively,result in a small voltage between the low resistor termi-nals and screen which will drive a current in the leakageresistance distributed between the resistor terminals andleads. To asses the magnitude of leakage effects, the ’low’side of the bridge circuit was isolated by disconnectingit from the remainder of the bridge at points L, C andD. The Wagner amplifier was also disconnected at pointA, and the detector was disconnected inside the CCCelectronics enclosure (the detector itself has a differentialinput resistance > N winding selection relays, low sideresistor connecting cables and the resistors themselves. Atotal of 8 measurements were made: leakage from point Ato screen with N = 100 , , N = 10000 , N = 100 , , ±
10 V and measuring the difference cur-rent using a source-measure unit (Keithley 6430), whichin all cases was less than 20 pA, setting a lower limitto the leakage resistances R L of 1 TΩ. Leakage currentscause errors ∼ r/R L , where r is the lead resistance .Since winding and lead resistances are roughly 10 Ω, weconclude that the effect of leakages on the relative un-certainty in measuring R /R is at the 10 − level, com-pletely negligible compared to other uncertainty compo-nents. B. Linearity of SQUID readout
The CCC employs a commercial SQUID and associ-ated feedback electronics (Quantum Design model 550).The SQUID difference readings ∆ V S and ∆ V (cid:48) S are useddirectly to calculate the resistance ratio, so the linear-ity of the SQUID must be considered as a contributionto the uncertainty. Referring to figure 1, possible er-rors due to SQUID non-linearity were evaluated by re-moving R and measuring the SQUID output V S whilesweeping V SOURCE through a small range (such that N I /S CCC < ). V SOURCE was measured using an8.5-digit voltmeter with better than 10 − linearity. A FIG. 5.
Residual of a linear fit to the SQUID output voltage V S as a function of the current in the -turn winding. linear fit was performed to V S as a function of I . Anexample data set is shown in figure 5, which shows thefit residuals as a function of I . There is no structurevisible in the residuals indicative of non-linearity withinthe standard deviation ( ∼ . . − / . × − ∆ V S .The relative uncertainty in the resistance ratio is then2 × − ∆ V S /kN I . For a typical case of a 100 MΩ cal-ibration at 100 V (figure 2 (a)), ∆ V S ∼ . < × − . Similarly, for the1 GΩ raw data shown in figure 2 (b), ∆ V S ≈ × − . However, forthe specific case of the SET 1 GΩ this is clearly an over-estimate for the fortuitous reason that ∆ V S ≈ − ∆ V (cid:48) S ,a consequence of the resistor deviating from its nominalvalue by ≈ × − . The two parts of the calibration,with N = 10000 and N = 10001, therefore sample thesame part of the SQUID V − Φ characteristic, and thelinearity error becomes less than 10 − . C. series resistance correction
The CCC source voltage V SOURCE and the detectorvoltage V D were both calibrated with a relative un-certainty of 10 − using calibrated voltage sources andDMMs. For calibrations of the SET 1 GΩ, ∆ V D / V S =5 × − with a relative uncertainty of 0 . × − to the uncertainty in R /R usingequation (4). As an additional check, the value of r ob-tained from ∆ V D data and equation (3) (after measuring r with modest precision using a hand-held multimeter)was compared with the value measured directly by con-necting a 4-wire DMM to appropriate circuit test points.The agreement was better than 0 .
05 Ω.
D. winding ratio error
The basic configuration of the ratio error test (RAT)is well known: the same current is passed through twowindings with equal numbers of turns, N = N = N but opposite sense so that zero flux will be measuredby a perfect comparator. The current is periodically re-versed, giving a difference in the excitation current ∆ I .The squid signal is recorded and the SQUID differencevoltage ∆ V S is extracted. The error in the turns ratio isgiven by δ N ≡ ∆ V S kN ∆ I . RATs have historically been usedmainly for two purposes: firstly to check for gross errorsin a new comparator such as cracks in the shield, or in-correct numbers of turns, and secondly to evaluate smallerrors due to flux penetration into the comparator as-sociated with the finite number of CCC shield overlaps.Recent work further identified a ’low-flux’ regime inwhich a finite SQUID difference signal could be obtainedin a RAT due to non-linear rectification of noise in theSQUID itself, effectively setting a limit of ∼ − Φ tothe resolution of a CCC.A full binary build-up was performed on our CCC priorto its commissioning, and no errors were found in Ratioerror tests (RATs) at the 10 − level. However, the con-figuration of the electronics, connecting cables and CCCdown-leads employed for routine use only allows accessto a sub-set of the windings, and a repeat of the fullbuild-up was not performed in this study. Instead, wefocused on sensitive RATs with N = 1, 2, and 10000turns. We performed sensitive RATs using two differ-ent current sources: the bridge’s internal voltage sourcein series with a 10 MΩ standard resistor for generatingcurrents up to 10 µ A, and a commercial current source(Yokogawa GS200) with a single-pole 30 Hz RC filter onthe output for larger currents. RATs were performedwith the current sources connected with both polarities,where ‘polarity’ refers to the sense of the physical con-nection between the current source and the windings.Altogether, the RATs spanned a range of 10 mAturns ≤ N ∆ I ≤ N ∆ I /S CCC = 1 . µ Φ , close to the onset of non-linearrectification effects .Focusing first on the high-turns RATs, we performeda total of 8 RATs, with parameters given in table I, anda current reversal every 16 s. The individual ∆ V S mea-surements for one test are plotted in figure 6(a), with theAllan deviation of these data in figure 6(b). The meanof these measurements is ∆ V S = − . µ V, with a stan-dard error on the mean of 1 . µ V. This corresponds toa finite ratio error of δ N = − . × − , which is qual-itatively consistent with the number of shield overlaps(3) in the CCC construction. Using the same test setup,consistent results were obtained in 3 RATs spread over 3years (RATs 1,2 and 8). Additional 10000-turn RATs (3-7) were performed using the commercial current sourcewith currents up to 100 µ A, and these showed that thereis no marked dependence of δ N on ∆ I or the currentsource polarity. We conclude that the high-turns RATs TABLE I. Parameters for 10 -turn RATsTest Current source Current ( µ A) Polarity1 Internal ±
10 F2 Internal ±
10 F3 GS200 ±
50 F4 GS200 ±
50 R5 GS200 ±
10 R6 GS200 ±
100 R7 GS200 ±
100 F8 Internal ±
10 F are measuring the geometric error due to the finite CCCshield overlap, with a relatively small contribution fromnon-linear rectification errors.In contrast, the 1- and 2-turn RATs (figure 7 and tableII) show a much larger ratio error, as large as δ N ∼ − .Moreover, δ N was seen to fluctuate on time-scales longerthan a few hours, as clearly evidenced by the data setof figure 7(a). The beginning and end of this data set isanalysed to yield RAT measurements 1 and 2 (figure 7(c)and table 2) which are clearly different within their sta-tistical uncertainties. Considering all the low-turn RATs,while δ N does not correlate with δ I , there is a clear influ-ence of current polarity: RATs 4, 5 and 11, which em-ployed reverse polarity, yielded much smaller δ N than theother RATs. We are led to conclude that non-linear rec-tification effects have a bigger effect for these low-turnRATs than the high-turn RATs This is not an unex-pected discovery, because the low-turn RATs employ alower overall ampere-turn linkage in the comparator. Forthe smallest current used (tests 5 and 6), δ I = 10 mA,and an error δ N = 10 − corresponds to a SQUID flux dif-ference of 1 . µ Φ . This is close to the threshold whereerrors in RATs have been associated with rectificationeffects . A further possibility is that the 1- and 2-turnRATs are probing the geometrical error, but the singleturns are much more sensitive to the geometry of theflux penetration than the large-turn windings. If the sin-gle turns were mechanically unstable, fluctuating resultscould be obtained in RATs.Because we were not able to perform a full build-up,we cannot quantitatively assess the error associated withthe turns ratio N /N that enters equation (4). However,we can set an upper limit to possible turns-ratio errors byusing the low-turns RATs (figure 7 (c)) as representinga worst-case. From this, we assign a relative uncertaintycontribution of 1 × − to the ratio R /R . VI. ADDITIONAL CONSIDERATIONS
Up until now we have evaluated contributions to theuncertainty in the measured resistance ratio R /R dueto the CCC instrument. It remains to consider the un-certainty in the known 10 MΩ resistor R . For mea- FIG. 6. (a): Squid difference voltage for a -turn RAT with ± µ A. Each data point is calculated from seconds of rawdata; seconds with each current polarity. (b): Allan deviationof the data plotted in (a). The dashed line is a guide to the eyewith gradient / √ τ . (c): Ratio error δ N for RATs. Error barsare the statistical uncertainty, evaluated as the standard error onthe mean. The right panel shows RATs - on an expanded timeaxis. TABLE II. Parameters for 1- and 2-turn RATsTest Turns Current (mA) Polarity1 2 ±
10 F2 2 ±
10 F3 1 ±
10 F4 1 ±
10 R5 1 ± ± ±
10 F8 1 ±
10 F9 1 ±
50 F10 1 ±
100 F11 1 ±
100 R surements of the SET 1 GΩ, the voltage coefficient andtime dependence were also significant contributions tothe combined uncertainty, and these are also discussedbelow.
A. Reference resistor
For the recently reported electron pumpmeasurements , the 1 σ uncertainty in the 10 MΩreference resistor was taken simply to be the CMCrelative uncertainty, 1 × − . FIG. 7. (a): Squid difference voltage for a 2-turn RAT with ± mA. Each data point is calculated from seconds of rawdata; seconds with each current polarity. (b): Allan deviationof the data plotted in (a). The dashed line is a guide to theeye with gradient / √ τ . (c): Ratio error δ N for RATs. Tests and respectively were evaluated from the first and last cycles of the data shown in panel (a). The statistical uncertaintyevaluated as the standard error on the mean is typically the samesize as the data points. B. Voltage coefficient
For the electron pump measurements, possible voltageco-efficients in the 1 GΩ resistor must be considered. Theresistor is calibrated with an excitation voltage of ±
33 V,but measurements of electron pump current I P ≈
160 pArequire a much smaller voltage, ≈
160 mV across theresistor. In evaluating the uncertainty due to possiblevoltage coefficients we take a conservative approach, andattempt to set an upper limit to possible voltage effectsby measurement of the resistor over a range of voltages.Earlier CCC measurements at 100 V and 10 V ruled outrelative voltage co-efficients at the level of ∼ × − .In figure 8 we present a recent set of 1 GΩ measure-ments with different excitation voltages. No systematicvoltage-dependent effect can be discerned in this data,although the poor signal-to-noise ratio at low voltagescombined with time-dependent drifts in the resistance make it difficult to eliminate the possibility of voltage-dependence at the 10 − level on the basis of this data.The ULCA provided another method for low-voltagemeasurements of 1 GΩ resistors, because the resistor canbe compared directly with the ULCA trans-resistancegain which is also nominally 1 GΩ. ULCA measure-ments with ± . ±
33 V agreed within relative combined uncertain-ties of ∼ − in three sets of measurements made ondifferent days . The uncertainty in the comparison waslimited by the need to perform the measurements withina short time-scale to avoid resistor drift effects. Basedon this data set, we assign a relative uncertainty due topossible voltage co-efficients of 10 − / √ . × − . A future measurement strategy for the SET 1 GΩ couldbe to measure it using the ULCA at low voltage, and as-sume that there is no further voltage co-efficient betweenthe ULCA measurement voltage ( ∼ ∼ .
16 V). Alternatively, the ULCA could beused to measure the electron pump current directly , apossibility further discussed in section VII.
C. Resistor drift
As already mentioned, standard resistors of 100 MΩand 1 GΩ based on thick-film resistive elements exhib-ited instability on time-scales of hours to days. This in-stability was clearly visible as a flattening (characteristicof 1 /f noise) of the Allan deviation of the measured re-sistance on time-scales longer than 1 hour, and detaileddata will be presented elsewhere . Here, in figure 9, wepresent a plot of the measured values of the SET 1 GΩover a time-base of several months, covering the electronpump measurement campaign reported in . During themost intensive periods of pump measurements, the re-sistor was calibrated every 1 − R after + R before ) / | R after − R before | / √
3. Here, R before and R after are thevalues of the resistor before and after the pump mea-surement respectively. The two key measurement runsreported in were made in the time period highlightedby the blue shading in figure 9, and the relative uncer-tainty components due to drift resistor drift were 1 × − for both runs. Nevertheless, we did not feel that thisfairly small number captured our uncertainty in know-ing the resistor value in between calibrations, because asalready noted, in some calibration runs lasting severalhours, the Allan deviation exhibited a transition to 1 /f noise. For this reason, we assigned a type A uncertaintyin the resistor value based on the Allan deviation, andnot the standard error on the mean, resulting in relativeuncertainty contributions of 1 . × − and 7 × − forthe two key runs reported in . At present, the stabil-ity of high-value resistors is still an area of investigation,extrapolation of linear drift lines is not supported by theavailable data (such as figure 9) and each case needs tobe treated individually. VII. DISCUSSION
In table III we present a summary of the relative typeB uncertainty contributions (1 σ ) evaluated in this paper,ordered by decreasing size, for the case of a 1 GΩ resistorused at low voltage to generate a reference current for asingle-electron experiment. We have taken as our start-ing point the 1 × − CMC uncertainty for the 10 MΩ
FIG. 8.
Values of the SET G Ω obtained from CCC measure-ments at different voltages. R = 1 G Ω(1 + δ R / ) . Thevoltage is indicated next to each data point, and the error barsshow the type A uncertainty. reference resistor, and this is by far the largest contribu-tor to the relative combined uncertainty of 1 . × − .The second-largest term comes from empirical evaluationof the voltage co-efficient of the resistor. As noted in sec-tion VI.B, our ability to evaluate this term is limited bythe long averaging times required to measure the resistorat low voltage, combined with instability in the resis-tor itself. The remaining terms contribute an insignifi-cant amount to the combined uncertainty. In a practicalelectron pump measurement, we must add to the com-bined uncertainty in table III some significant additionalterms related to the voltmeter calibration and drift be-tween calibrations ( ∼ × − ), the type A uncertaintyin the resistor calibration ( ∼ × − ) and the type Auncertainty in the measurement of I P itself ( ∼ × − for a typical experimental run lasting 10-15 hours). Thislast, largest, term arises from the Johnson-Nyquist cur-rent noise = (cid:112) k B T /R ≈ / √ Hz in the 1 GΩ resis-tor, and we note that it could be reduced by a factor of √ I P directly.The data in table III are for the specific case relevantto the SET 1 GΩ. For routine calibrations of 100 MΩand 1 GΩ, the voltage co-efficient term can be neglectedbecause calibration values are quoted at the calibrationvoltage, usually 100 V. The SQUID linearity term maybe as large as a few parts in 10 for resistors with a largedeviation from nominal, but to a good approximationthe 1 σ relative standard uncertainty is ≈ × − , domi-nated by the uncertainty in the 10 MΩ reference resistor.This re-evaluated uncertainty is smaller than the CMCuncertainties by a factor 2 and 8 for 100 MΩ and 1 GΩresistors respectively. The reason for the large CMC un-certainties is that loop-closure tests played a major role inthe original evaluation of the bridge performance underroutine calibration conditions. In these tests, for exam-ple, a 1 GΩ resistor would be measured against a 10 MΩreference resistor in a single 100 : 1 steps, and two 10 : 1steps using an intermediate 100 MΩ standard. Thesetests can be re-evaluated in the light of recent findingsof the short-term instability of high-value standard re- sistors. Discrepancies in the original 1 GΩ loop-closuretests were most likely caused by drifts in the resistors onthe time-scales of a few hours required to perform themeasurements. For this reason, the CMC uncertaintiesare an accurate representation of the realistically achiev-able uncertainty in transferring a resistance value to acustomer’s laboratory, although they are a pessimisticexpression of the CCC performance.The question naturally arises as to whether the CMCuncertainty in the 10 MΩ reference resistor is over-estimated, and whether an extension of the investiga-tions of this paper to 10 MΩ and lower resistance val-ues would yield a profitable reduction in uncertainty forSET measurements. The problem is that reduction oftype B uncertainty terms leaves the stability of the 1 GΩresistor (figure 9) increasingly exposed as the most prob-lematic factor in the SET experiment. The requirementfor almost daily calibration of the resistor interrupts themeasurements and makes it difficult to perform long con-tinuous measurements, for example, to evaluate the flat-ness of an electron pump plateau. Unless more stablehigh-value resistors can be found, using the ULCA tomeasure I P directly appears to be the best route. Thedrift in the trans-resistance gain of the ULCA is smaller,and more predictable , than the drift of a 1 GΩ stan-dard resistor. Furthermore, variants of the ULCA havebeen tested with larger resistances in the input current-scaling network , offering even lower Johnson-Nyquistcurrent noise than the standard ULCA. It is conceivablethat a measurement of the present generation of electronpumps, with I P ∼
160 pA, could be made with a relativeuncertainty of 5 × − . Reduction of the uncertaintysignificantly below this level requires more frequent cali-bration of the ULCA, implying shorter averaging times,and higher pumps currents, such as may be offered bypumps based on single traps in silicon . VIII. CONCLUSIONS
We conclude that 100 MΩ and 1 GΩ resistors can bemeasured using the NPL high-resistance CCC with typeB uncertainties not significantly larger than 1 × − .This capability has allowed the investigation of prototypesingle-electron current standards with combined uncer-tainties dominated by the type A component, about 2parts in 10 for 160 pA averaged over 10 hours. ACKNOWLEDGMENTS
This research was supported by the UK department forBusiness, Energy and Industrial Strategy, the Joint Re-search Project ’Quantum Ampere’ (JRP SIB07) withinthe European Metrology Research Programme (EMRP)and the EMPIR Joint Research Project ’e-SI-Amp’(15SIB08). The The EMRP is jointly funded by theEMRP participating countries within EURAMET and0
FIG. 9.
Values of the SET G Ω obtained from CCC mea-surements during a measurement campaign on a silicon electronpump , where R = 1 G Ω(1 + δ R / ) . The shaded area in-dicates the time during which the highest-accuracy pump mea-surements were performed. TABLE III. Relative type B uncertainty contributions formeasuring 1 GΩComponent Section reference Uncertainty10 MΩ reference I (CMC) 1 × − Voltage coefficient VI.B 5 . × − Turns ratio V.D 1 × − Series resistance V.C 7 × − Squid linearity V.B 1 × − leakage resistance V.A 1 × − Total 1 . × − the European Union. The European Metrology Pro-gramme for Innovation and Research (EMPIR) is co-financed by the Participating States and from the Eu-ropean Union’s Horizon 2020 research and innovationprogramme. The author would like to thank JonathanWilliams, Nick Fletcher, Dietmar Drung and MartinGoetz for stimulating discussions. B. Kaestner and V. 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