measuring the electron's charge and the fine-structure

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Volume 97, Number 2, March-April 1992

Journal of Research of the National Institute of Standards and Technology

[J. Res. Natl. Inst. Stand. Technol. 97, 299 (1992)]

Measuring the Electron's Charge and theFine-Structure Constant by Counting

Electrons on a Capacitor

Volume 97 March-April 1992Number 2

E. R. Williams andRuby N. Ghosh,

National Institute of Standardsand Technology,Gaithersburg, MD 20899

and

John M. Martinis

National Institute of Standardsand Technology,Boulder, CO 80303

The charge of the electron can be de-termined by simply placing a knownnumber of electrons on one electrodeof a capacitor and measuring thevoltage, V" across the capacitor. If V. ismeasured in terms of the Josephsonvolt and the capacitor is measured inSI units then the fine-structure constantis the quantity determined. Recent de-velopments involving single electrontunneling, SET, have shownhow tocount the electrons as well as how to

make an electrometer with sufficientsensitivity to measure the charge.

Key words: calculable capacitor; cou-lomb blockade; electron charge; elec-tron counting; fine-structure constant;single electron tunneling.

Accepted: February 3, 1992

1. Introduction

The recent development of single-electrondevices[1,2,3,4,5]has made possiblea newand veryprecise technique to measure the charge of an elec-tron. These devices are based on metaVinsulator/metal tunnel junctions whose current-voltage(I-V)characteristics are determined by individual elec-tron tunneling events. If such a device is used toplace n electrons on a capacitor of capacitance C,and the resulting voltage V, is measured, the elec-tron charge e can be determined in terms of thisvoltage and capacitance, ne = V,C,.In practice, V, ismost precisely determined in terms of the voltagegenerated by a Josephson junction and thus the ex-periment will not be a determination of e in SIunits, but a measurement of e in terms of theJosephson volt [6]. In this paper we showthat sucha measurement willsoonbe possibleand that an ac-curate result would make a useful contribution tothe field of fundamental physicalconstants.

1.1 Relating the Fine-Structure Constant and theElectric Charge

First, we address how this measurement relatesto the fine-structure constant, a. By relating themeasured voltage to a Josephson voltage and thecapacitance to SI capacitance as measured in a cal-culable capacitor experiment [7], we can calculatethe fine-structure constant from the followingsim-ple equations:

ne 'F. hV, = C, = JJj 2e

(1)

where Ii is the frequency producing the Josephsonsteps andj is an integer associated with the Joseph-son effect. (Note that in a Josephson array j is thesum of all the integer steps of each junction.) Solv-ing Eq. (1) for e2/h, a is given by

2 . Iia = ~ _lJLoc jC,2h - 4n (2)

299



where J.Lois the permeability of vacuum and C.must be measured in SI units. Specifically,the ca-pacitance must be related to the calculable capaci-tor experiment. In a calculable capacitorexperiment a change of capacitance of 0.5 pF canbe measured in terms of the meter with an accu-racy of 0.014 ppm [7] (1 ppm=1x 10-6; through-out, all uncertainties are one standard deviationestimates). This capacitance can then be relatedwith accuracies near the 0.01 ppm level to a 10 pFcapacitor that is stable and transportable.

If this experiment to measure e in terms of theJosephson volt and the calculable capacitor can berealized with high accuracy, it will provide a newpath to a. This new approach is similar to the mea-surement of a via the quantum Hall effect [8]. Al-though both experiments require a connection tothe calculable capacitor, this new method is muchmore direct. The 0.5pF capacitance change used inmost calculable capacitor experiments is a goodmatch to the size of the capacitance that might beused to determine e; only one or two precision ra-tio transformer bridges need be involvedin the cal-ibration of C.. By contrast, in the quantum Hallcase a chain of calibrations might involvethese in-termediate standards: 0.5pF, 10pF, 100pF,1000pF, 100k n, 10 k n, 1 k n, and 6453.2n. Themost accurate value of a determined from the cal-culable capacitor and quantum Hall effect has anuncertainty of 0.024ppm [7,9]with 0.014ppm com-ing from the calculable capacitor and the rest fromthe chain of intermediate standards. An alternatedetermination of a using the proton gyromagneticratio in H20 and the quantum Hall effect but notthe calculable capacitor, has an uncertainty of0.037 ppm [9]. However, these two results differfrom each other by (0.10:t 0.043)ppm. A value of adetermined from the anomalous magnetic momentof the electron and quantum electrodynamictheory(QED) has an uncertainty of 0.007ppm [10]and itsvalue lies between the two non-QED values.Therefore, the unexplained 0.10 ppm difference inthe non-QED values limits the accuracy to whichQED theory is tested. Any measurement of a bythis new SET method at the 0.1ppm levelor betterwould be helpful in the field of fundamental con-stants [8].

1.2 SET Devices

K. K. Likharev and his colleagues [1] have longbeen proponents of the application of Coulombblockade effects arising from the discreet charge ofelectrons to realize a precise current source. Their

pioneering ideas and experiments paved the wayfor much of the recent progress. In the past fewyears, new devicesemployingmetal/insulator/metaltunnel junctions have been demonstrated whichcould allow a precise measurement of the electroncharge e and thus a. These include an electrometer[2],observation of single electron tunneling oscilla-tions [3], a "turnstile" current source [4], and a"pump" current source [5]. A brief sketch of sin-gle-electron tunneling is presented to describe theoperation of the electrometer and pump.

Consider a normal-metal tunnel junction biasedby a current source and having a capacitance C.The change in energy of the junction after anelectron tunnels through the barrier is ~E =eQ/C -e2/2C, where Q is the charge across thejunction and e2/2C is the Coulomb energy cost ofthe tunneling event. If the tunneling resistance ofthe junction is greater than the quantum unit ofresistance, Rt ~h/e2, and thermal fluctuations donot mask the charging energy, kT~e2I2C, then aCoulomb blockade appears in the junction I-Vcharacteristic where the tunneling probability isgreatly reduced for IVI< e/2C .At 50 mK, and using1 fF for the junction capacitance, one finds (kT)/(e2/2C)-1120, hence temperature effects are smallbut non-negligible.The single-electron devices ex-ploit this Coulomb blockade of the tunneling cur-rent.

The Fulton-Dolan (SET) electrometer [2] isshown schematicallyin Fig. 1a. Aside from tunnel-ing events, the electrode a between the two junc-tions and the gate capacitor Co is an "island"electricallyisolated from the circuit. The electrom-eter provides a very high impedance technique tomeasure the potential U. At a constant bias V, thedevice conductance as a function of gate voltage isperiodic with the period ~U =e/Co. The biasvoltagesV and U are chosen to maximizethe sensi-tivityof the electrometer so that the devicecurrentis linearlyproportional to small changes of the po-tential U. Such electrometers have demonstrated

I

~I

~a

TCo VU

Fig. 1a. This circuit shows an electrometer formed by two tun-nel junctions each having a capacitance C. The input is potentialU coupled to the isolated island a through Co.

300



charge sensitivity of 1.5x 10-4 e/Hzt/2 at a fre-quency between 2 and 200 Hz [11].

Convincing experimental evidence of the SEToscillations producing the controlled transfer ofelectrons with the relation I =ef was reported byDelsinget al. [3].Geerligs et al. [4]were the first todemonstrate flat plateaus and observe a currentwith accuracies of better than 1% using the turn-stile device. However, certain properties of thepump device suggest that it will be more precise incounting electrons and we therefore sketch itsoperation.

An electron pump [5] (see Fig. Ib) is a devicesimilar to the electrometer, in which a single elec-tron tunnels sequentially through the "islands" la-beled band c. For a given voltage V, a controllednumber of electrons can be transferred in eitherdirection by appropriately cyclingthe gate voltagesUt and U2at the frequency f. The pump also deliv-ers an electron current at the rate I =ef. The direc-tion of the current can be reversed simply byreversingthe phase of Ut and U2.A pump based onfour or five tunnel junctions operating at 1-10MHz has been theoretically predicated to be ableto deliver a known current with errors below the 1ppm level [12,13].

I

17

C

V2

Ulm u2rd5JTime Time

Fig. lb. This circuit showsa three junction pump that pumpsone electron per cycleof the wave-formsshown.The electron isfirst pumped to island b by potential U1then to c by Uz.

The currents produced by these pumps are onlyabout 1O-t2A and thus precision measurements willbe a challenge. Direct measurement by sensing thevoltage across a high resistance is a problem be-cause it is difficult for metrologists to accuratelycalibrate the required high resistances. Capacitors,on the other hand, have already been accurately

calibrated in a range that will be useful in the fol-lowingexperiment.

2. Experimental Configurations [14]2.1 Measurements of the Electric Charge e

Figure 2a shows one possible configuration thatincorporates these new SET devices to measure theelectronic charge. A standard capacitor Cs is con-nected to a current source (pump) and also to theinput of a Fulton-Dolan electrometer via a couplingcapacitor C:, forming an isolated island a. Using theelectrometer as a null detector a voltage Vs is ap-plied across C. to maintain the island a at a fixedpotential (near ground), while pumping electronsonto or off of the island. The stray capacitance toground, Cg,limits the sensitivity of the electrometerto Vs.The voltage, Vs,on the capacitor is adjusted sothat the electrometer input is always at the samepotential (near ground). Thus the current from the

. - - - - - - - - - - - - - --,

--1

:--1Electrometer

/.

I!Ve:--1

FeedbackVsto keeppotential atisland aconstant.

I

r JIII

4 '- I

I

L__I'eI s

r JI

~ - - - - . - . . ..I

Vs

Fig. 2a. Proposed circuit for measuring a. The voltage V. re-quired to keep the potential of the island a constant when n elec-trons are pumped onto the island can be used to measure theelectron charge in terms of C. and V.. The electrometer monitorsthe potential at a.

301

C CI

c m

-L L.l.J -L L.l.J

T TVI V2 V

2

Volume 97,Number 2,March-April1992

Journal of Research of the National Instituteof Standards and Technology

pump serves as the charging current of the capaci-tor as the voltage is ramped up. A specific exampleis illustrated in Fig. 2b. With the pump operating at6.2 MHz for 10 s, the voltage across a 1 pF capaci-tor standard will charge to 10 V at a linear rate inorder to keep the electrometer input constant. Westop the pump and measure the 10 V signal as wellas any deviation of the electrometer from null.Reversing the pump for 20 s then brings thevoltage to -10 V.

--1 205 r-

Fig.2b. The measurement sequence for measuring a,showingthe potential V. as a function of time.

In order to use a capacitor as a standard it musthave a well defined capacitance; it thus is necessaryto understand how precision capacitors are mea-sured and under what circumstances they are welldefined. Figure 3 shows the essential features ofone side of a standard bridge that is used to com-pare capacitors C1 and C2.The dashed lines repre-sent the grounded shield that prevents thepotentials VI and V2from affecting the detector ex-cept through their respective capacitances. Thisshielding is critical to making precise capacitancecomparisons. The bridge is balanced by adjustingthe potentials VtlV2=CJC1. CgI and Cg2representthe stray capacitance to ground. The bridge bal-ance is not affected by CgI because there is novoltage across CgI when the bridge is balanced.However, CgI does affect the detector sensitivity soit should be kept small. The capacitance Cg2doesnot affect the balance as long as the impedancefrom the source to C1 is small compared to theimpedance of Cg2. Present day precision 10 pFcapacitance standards with Cgl =200 pF are com-pared at the 0.01 ppm level at a frequency Cd=104

S-I (1592 Hz) using a room temperature detectorwith a noise figure of 20 el Hz1l2.A SET electrome-ter should have similar noise performance at thesame frequency and capacitance. Further improve-ments in the sensitivity of the SET electrometerare expected when measuring smaller capacitances.

C2rI

IL____ .........- --

- - -- I V2~

IIIII II..J

I..J

- --

- --

Fig. 3. Circuit showing half of a bridge used to compare capaci-tors CI and Cz. The potentials VI and Vzare usually supplied bya precision transformer. The effect of stray capacitances C,I andC,z are discussed in the text.

The key to this experiment to measure e using acapacitor is to replace the detector in Fig. 3 with aFulton-Dolan electrometer as shown in Fig. 2a. Byplacing the electrometer at 20 mK near the sourceof charge, we anticipate fewer problems from leak-age and less stray capacitance to ground. In addi-tion, we expect to gain from the impressivesensitivityof this device, reduce the leakage dra-matically, and still make a well defined capacitorthat can be calibrated in situ.

302

>'"c:

>'":: >'"0) 0 a. 0):; "0 :J :;

"E :J "EenI gj I

en1113 I 1113I I :; I 0) I :; 10)

1:iE I:iE

r I u u 1

rI II I L____"

I V1

y-: I Ily-'-___..JCg1 C1 Cg2



Because of the requirement to shield thestandard capacitor, it is easier initiallyto have thiscapacitor on a different chip from the electrometerand pump. As multilayer technology becomesincorporated into the junction fabrication processthis capacitor could be placed on the same chip.Havingit on a separate chip leads to somepracticallimits in the optimum choice for Cs.The sensitivityof the electrometer is approximately given byCJ(C.+Cg), where Cgrepresents all other capaci-tances to ground. This two chip arrangement willincrease Cg,but we expect to keep it near 1 pF.ChoosingC. to be smallboth increases V. and max-imizes the electrometer sensitivity.A 1 pF capaci-tance for C. constitutes a practical compromise.

Calibration of the capacitance requires makingaconnection to room temperature. However, theFulton-Dolan electrometer could still be used bymeans of the calibration input switch shown inFig. 2. Note that care must be taken to keep thecapacitance from the island to ground low.

2.2 Dual Pump Bridge

Another interesting experiment (see Fig. 4) is tohave two current sources (drawn as two, four-junc-tion "pumps" in Fig. 4) feeding into an island awith a total capacitance of about 10 fF. The chargestate of island a is then measured with an electrom-eter. This geometry will not measure e but will al-low the very precise comparison of two currentsources. Any difference in current from the twosources will show up as an accumulation of the is-land charge. The island capacitance is chosen to belarge enough to greatly reduce the interaction ofthe two pumps but small enough to still have singlecharge resolution. Because the error is detected asan integrated charge on the island, only a shorttime is required to obtain extremelyhigh precision.It would also be interesting to compare the errorrate of a "pump" and a "turnstile."

3. Future Prospects

A new precision technique to measure the fine-structure constant via Eqs. (1) and (2) has beendescribed. At present SET experiments have beenused to determine currents only at the 0.1 to 1.0%level, hence it is riskyto predict the ultimate accu-racy of the technique. Based on theoretical predic-tions [12,13] we expect that the number ofelectrons could be measured with 1 ppm accuracyin a four junction pump. A five junction pump

-----------VI

Pump 1

Electrometer1- - - - - - - -

1 ~ I

r=I

I111II

I Pum p 2I11I

V2 I- - - - - - - - - _I

--------

Fig. 4. A bridge circuit used to compare two pump circuits. Theelectrometer detects any accumulation of charge on island a.

should be more accurate still. At present the elec-trical standards needed to support this measure-ment do not appear to limit its accuracy. Forexample, capacitance standards of 10 pF are com-pared between national laboratories near the 0.02ppm level and 10 V Zener voltage references arealso compared with 0.02 ppm accuracy. In the firstexperiments these accuracies will likely not bereached, but much physics will be learned aboutSET devicesas we make another accurate determi-nation of a.

Acknowledgment

We thank M. H. Devoret, D. Esteve, C. Urbina,H. Pothier, and P. Lafarge of Service de Physiquedu Solide et de Resonance Magnetique, Centred'Etudes Nucleaires des Saclay, for many informa-tive discussions about SET devices. We thank J. Q.Shields for crucial discussions about the techniquesused to measure standard capacitors. Discussionswith B. N. Taylor, Hans Jensen, and R. L. Kautzwere also very useful.

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4. References

[1] D. V. Averin and K. K. Likharev, in Mesoscopic Phenom-ena in Solids, B. AJ'tshuler, P. A. Lee, and R. A. Webb,eds., Elsevier, Amsterdam (1991) Chapter 6.

[2] T. A Fulton and G. J. Dolan, Observation of Single-Electron Charging Effects in Small Tunnel Junctions,Phys. Rev. LeU. 59, 109-112 (1987).

[3] P. Delsing, K. K. Likharev, L. S. Kuzmin, and T. Claeson,Time-Correlated Single-Electron Tunneling in One-Dimensional Arrays of Ultrasmall Tunnel Junctions, Phys.Rev. LeU. 63, 1861-1864 (1989) and Phys. Rev. B 42,7439-7449 (1990).

[4] L. J. Geerligs,V. F. Anderegg,P. A. M. Holweg,J. E.Mooji,H. Pothier,D. Esteve,C. Urbina,and M. H.Devoret, Frequency-Locked Turnstile Device for SingleElectrons, Phys.Rev. Leu. 64, 2691-2694 (1990).

[5] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H.Devoret, Single-Electron Pump Based on ChargingEffects, Europhys. Leu. 17,249-254 (1992).

[6] Resolution 6 of the 18th Conference Generale des Poids

et Mesures. See B. N. Taylor, ed., The International Sys-tem of Units (SI), Natl. Inst. Stand. Technol. Spec. Pub.330 (1991) p. 45.

[7] J. Q. Shields, R. F. Dziuba, H. P. Layer, New Realizationof the Ohm and Farad Using the NBS Calculable Capaci-tor, IEEE Trans. Instrum. Meas. 38, 249-251 (1989).

[8] B. N. Taylor and E. R. Cohen, How accurate are theJosephson and Quantum Hall Effects and QED?, Phys.Leu. A 153, 308 (1991).

[9] M. E. Cage, R. F. Dziuba, R. E. Elmquist, B. F. Field, G.R. Jones, Jr., P. T. Olsen, W. D. Phillips, J. Q. Shields, R.L. Steiner, B. N. Taylor, and E. R. Williams, NBSDetermination of the Fine-Structure Constant and the

Quantized Hall Resistance and Josephson Frequency-to-Voltage Quotient in SI Units, IEEE Trans. Instrum. Meas.38, 284-289 (1989).

[10] T. Kinoshita and W. B. Lindquist, Eighth-order MagneticMoment of the Electron. V. Diagrams Containing No Vac-uum-Polarization Loop, Phys. Rev. D 42, 636-655 (1990).

[11] L. J. Geerligs, V. F. Anderegg, and J. E. Mooij, TunnelingTime and Offset Charging in Small Tunnel Junctions,Physica 8165/166, 973-974 (1990).

[12] H. Pothier, 1991, Blocage de Coulomb et Transfertd'Electons Un Par Un, PhD Thesis, de l'Universite Paris6.

[13] Hans Jansen and J. Martinis, private communications.[14] The configurations shown here were first presented by the

authors as a poster paper at the NATO Advanced StudyInstitute on Single Charge Tunneling, Les Houches,France, March 1991. J. Martinis will present a paper onthis subject at the Conference on Precision Electromag-netic Measurements, June 1992, in Paris France.

About the authors: Edwin R Williams is a physicistin the Electricity Division at NIST. Ruby N. Ghosh isan American Society of Engineering Education Post-doctoral Fellow in the Electricity Division at NIST.John M. Martinis is a physicist in the ElectromagneticTechnology Division at NIST. The National Institute

of Standards and Technology is an agency of the

Technology Administration, U.S. Department of Com-merce.

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measuring the electron's charge and the fine-structure

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