The von Klitzing resistance standard

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Physica E 20 (2003) 14 – 23 www.elsevier.com/locate/physe

The von Klitzing resistance standard Hans Bachmair, Ernst O. G'obel∗ , G'unter Hein, J'urgen Melcher, Bernd Schumacher, J'urgen Schurr, Ludwig Schweitzer, Peter Warnecke Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

Abstract The present state of the art of reproducing the SI unit for the electrical resistance, the ohm, by the quantum Hall e3ect is summarised including recent developments like quantum Hall arrays and AC measurements of the quantum Hall e3ect and their application in metrology. We further brie6y discuss the role of the von Klitzing resistance standard for other SI units, in particular the kilogram. ? 2003 Elsevier B.V. All rights reserved. PACS: 06.20.−f; 73.43.−f; 73.40.−c; 72.15.Rn Keywords: Quantum Hall e3ect; von Klitzing constant; Resistance standard

1. Introduction The unit for the electrical resistance, the ohm, is a derived unit in the International System of Units (SI) implemented by the 11th General Conference on Weights and Measures (CGPM) in 1960. According to its deAnition, a conductor has a resistance of 1 B ohm if a voltage drop of 1 V occurs when a current of 1 A passes the conductor. A realisation of the ohm as given by this deAnition, however, is not possible with the required low uncertainty. However, the Australian scientists Thompson and Lampard discovered a theorem which allows the realisation in a di3erent way [1]. They showed that, in a Arst step, the realisation of a capacitance is possible via a cylindrical cross capacitor. With only a length being

∗

Corresponding author. Fax: +49-531-592-1005. E-mail address: [email protected] (E.O. G'obel).

precisely measured, the capacitance C can be calculated according to C = ‘ · (0 =) ln 2 where ‘ is the length of the cross capacitor and 0 is the permittivity of vacuum 0 = 1=(0 c2 ), with the magnetic Aeld constant 0 and the vacuum velocity of light c. In a second step, this capacitance C is linked at a certain angular frequency ! to an AC resistance R = 1=!C. In a third step, this AC resistance is linked to a DC resistance using a resistor with calculable AC/DC behaviour and Anally stepped down to 1 B, the value at which the as-maintained unit of resistance is kept at the National Metrology Institutes (NMIs). This realisation is pursued at various NMIs with a relative uncertainty of a few parts in 108 . The impact of the discovery of the quantum Hall e3ect (QHE) [2] on precision resistance metrology was immediately recognised by Klaus von Klitzing as it became evident to him that the resistance value of a quantum Hall plateau could be described by RH (i) = h=e2 (1=i), where i is an integer, h Planck’s constant and e the electron charge.

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H. Bachmair et al. / Physica E 20 (2003) 14 – 23

After the discovery of the e3ect and its Arst public presentation on the occasion of the Conference on Precision Electromagnetic Measurements in 1980, the Arst precise measurement of a quantised Hall resistance was performed with an uncertainty in the order of one part in 10−6 in a co-operation between Klaus von Klitzing being then at the University of W'urzburg and the PTB [3]. This uncertainty approximately agreed with the uncertainty associated with the Ane structure constant = (1=20 c) · e2 =h as known from the least-squares adjustment of fundamental constants at that time. Besides PTB, NMIs all over the world started with establishing the QHE in their laboratories and performing precision measurements against their as-maintained unit of resistance. Meanwhile, the QHE is used in more than 15 NMIs to reproduce, maintain and disseminate the SI unit of resistance, one of the most important electrical units.

2. DC-resistance standard based on the QHE Today the QHE allows the reproduction of the unit of resistance with an uncertainty that is about two orders of magnitude smaller than the uncertainty by which the SI unit of resistance is known. To make use of the inherent precision of macroscopic quantum e3ects in international comparisons and for calibrations—for the Josephson e3ect a similar situation exists—it seemed advisable to agree on Axed numerical values for the Josephson frequency-on-voltage quotient (called Josephson constant KJ ≡ 2e=h) and the quantised Hall resistance (called von Klitzing constant RK ≡ h=e2 ) for the purpose of establishing accurate and internationally uniform reference standards of voltage and resistance based on the Josephson and quantum Hall e3ects, respectively. The Comit/e International des Poids et Mesures (CIPM) at its 77th meeting in 1988—only 8 years after the discovery of the QHE—adopted particular values of KJ and RK , designated KJ-90 and RK-90 , for use by all laboratories beginning on 1 January 1990 [4]: KJ-90 = 483 597; 9 GHz=V RK-90 = 25 812; 807 B:

and

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These values were conArmed by the CGPM at its 19th meeting in 1991. These values have been valid since then and are supported by the latest least-squares adjustment of the fundamental constants. The beneAt drawn from the adoption of the von Klitzing constant RK-90 can be seen from Fig. 1 which shows the results of international comparisons of resistance in the time period from 1964 to 1985. In Fig. 1 the drift with time of the as-maintained units of resistance with respect to the SI unit derived from a determination of the unit of resistance at the CSIRO NML in Australia is given. This situation improved a lot with the Axation of the von Klitzing constant as can be seen from Fig. 2. With e3ect from 1 January 1990, the NMIs which had a QHE resistance standard at their disposal changed their as-maintained units by the values given in Fig. 2, and since that time those NMIs that used the von Klitzing constant RK-90 to reproduce their unit of resistance are free of any changes with time. Fig. 3 shows the results of the direct and indirect measurements of the von Klitzing constant which were used to derive the recommended value for RK . The results (1) – (7) represent direct determinations of RK based on the calculable cross capacitor at CSIRO NML (Australia (1)), NPL (United Kingdom (2)), BNM-LCIE (France (3), now BNM-LNE LAMA), ETL (Japan (4), now NMIJ), NIST (USA (5)), VNIIM (Russia (6)) and NIM (China (7)). The results (8) – (11) represent the indirect determinations which require the assumption RK = h=e2 =

−1 0:50 c being valid. Results (8) – (11) are based on the value of the inverse Ane-structure constant −1 obtained from the measurement of the electron magnetic moment anomaly and its theoretical expression based on QED calculations (8), from the ground-state hyperAne splitting of muonium (9), from the magnetic moment of the proton in units of the Bohr magneton (10) and from the gyromagnetic ratio of the proton in a weak magnetic Aeld (11). In addition, two determinations of the von Klitzing constant based on an international comparison of resistance ((12) and (13)) have been taken into account. For details see Ref. [5]. The dashed line together with the grey shadowed area in Fig. 3 represents the agreed value of the von Klitzing-constant RK-90 = 25812; 807 B and the one standard deviation uncertainty of 2 × 10−7 associated to it.

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H. Bachmair et al. / Physica E 20 (2003) 14 – 23

Fig. 1. Results of international comparisons on resistance in the time period from 1964 to 1985, BIPM Bureau International des Poids et Mesures, France; ASMW Amt f'ur Standardisierung MeRwesen und Warenpr'ufung of the former German Democratic Republic; ETL Electrotechnical Laboratory, Japan (now NMIJ National Metrology Institute of Japan); IEN Istituto Elettrotecnico Nazionale, Italy; IMM D. S I. Mendeleyev Institute for Metrology, Russia; LCIE Laboratoire Central des Industries Electriques, France (now LNE/LAMA Laboratoire National d’Essais/Laboratoire AndrTe Marie AmpTere); NBS National Bureau of Standards, USA (now NIST National Institute of Standards and Technology); NPL National Physical Laboratory, UK; NRC National Research Council, Canada; PTB Physikalisch-Technische Bundesanstalt, Germany.

Table 1 shows a comparison between the numerical values of the von Klitzing and Josephson constants, h=e2 and 2e=h, respectively as given by the least-squares adjustments in 1986 and 1998 and the international recommendation in 1990. The changes in the numerical values of the latest least-squares adjustment with respect to the values in 1986 and 1990 are small and lie within the uncertainties associated with these values. These small changes encouraged the / Comit/e Consultatif d’Electricit/ e et de Magn/etisme

(CCEM) to reduce the uncertainty associated with the von Klitzing constant by a factor of two without changing the numerical value [6,7]. For accurate measurements of quantised Hall resistances, mainly two techniques are in use today: the potentiometric method and the current comparator bridge technique [8]. With the potentiometric method the two resistances to be compared are connected in series and driven by the same current source. The voltage drop across the resistors is measured against a

H. Bachmair et al. / Physica E 20 (2003) 14 – 23

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Fig. 2. E3ect of the international adoption of the von Klitzing constant on the dissemination of the unit of resistance.

Fig. 3. Results of the direct and indirect measurements of the von Klitzing constant.

precisely adjustable voltage source. A signiAcant improvement can be obtained by using a Josephson array voltage standard (JAVS) to realise the adjustable

voltage source. With a newly designed double potentiometer based on two JAVS driven by the same microwave frequency [9], the uncertainty for the

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H. Bachmair et al. / Physica E 20 (2003) 14 – 23

Table 1 Comparison of the numerical values of the von Klitzing and Josephson constants h=e2 and 2e=h, respectively, as given by the least-squares adjustments in 1986 and 1998 and the international recommendation in 1990 LSA98 − LSA86 LSA98

LSA98 − IR90 LSA98

Constant

CODATA 1986 least-squares adjustment (LSA 86)

International recommendation 1990(IR90)

CODATA 1998 least-squares adjustment(LSA98)

2e=h

483 597; 67 GHz=V 3:0 × 10−7

483 597; 9 GHz=V (4:0 × 10−7 )

483 597; 898 GHz=V 3:9 × 10−8

+4:7 × 10−7

−4:1 × 10−9

h=e2

25 812; 805 6 B 4:5 × 10−8

25 812; 807 B (2:0 × 10−7 )

25 812; 807 572 B 3:7 × 10−9

+7:6 × 10−8

+2:2 × 10−8

Fig. 4. Comparative measurements between a Josephson double potentiometer and a cryogenic current comparator bridge (R1 =R2 = 12:9 kB=10 kB).

comparison of a 10 kB resistor against a quantised Hall resistance at the second plateau could be reduced to only 4 × 10−9 . In a current comparator bridge the two resistance standards to be compared are connected in two separate current loops which track one another. The ratio of the two currents is controlled by a current comparator which delineates it as a turns ratio and allows it to be measured with a very small uncertainty. The best ratio accuracy and the lowest random uncertainty can be attained with a cryogenic current comparator (CCC) which makes use of the Meissner e3ect and allows the ratio of two DC currents to be measured with an uncertainty in the order of about 1 × 10−12 . With a CCC bridge a 100-B resistor can be compared against a quantised Hall resistance at the second plateau with an uncertainty of 1:4 × 10−9 [10]. Only recently, comparative measurements between a Josephson double potentiometer and a CCC bridge as summarised in Fig. 4 [11] have conArmed the small uncertainties which can be obtained with these methods.

To check the world-wide consistency of the QHR measurements at the highest level of accuracy, the Bureau International des Poids et Mesures (BIPM) has started in 1993 to perform “on-site comparisons” of resistance ratio measurements using a transportable QHE standard and resistance bridge. After the “Multilateral Recognition Arrangement (MRA)” of the CIPM came into force on 14 October 1999, these bilateral comparisons were chosen to become so-called key comparisons which serve to (i) test the principal techniques in each Aeld, (ii) provide data for calculating the degree of equivalence of national standards and (iii) give mutual conAdence in the measurement capabilities of the participating NMIs. The results of key comparisons are made public by the BIPM in a key comparison data base which is accessible via the internet (www.bipm.org). The results of the QHE comparisons obtained so far are shown in Fig. 5. The agreement between each participating laboratory and the BIPM for the ratio RH (2)=100 B is in the order of one part in 109 which is well within the combined standard uncertainty of the comparisons. Nowadays, the QHE is used by all major NMIs to maintain and disseminate the unit of resistance and to perform calibrations of resistors at the highest level. The reproducibility of this quantum standard is two orders of magnitude better than the realisation of the ohm in the SI. By Axing conventional values for the von Klitzing constant RK and the Josephson constant KJ , the worldwide consistancy of the electrical measurements has improved considerably during the last decade. As mentioned above, the “nondecimal” value of RK-90 = 25812:807 B requires a number of steps to compare the value of a given plateau with the 1 B

H. Bachmair et al. / Physica E 20 (2003) 14 – 23

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Fig. 5. Results of on-site comparisons of three di3erent resistance ratios using the BIPM transportable QHE system (LCIE LaboraS toire Central des Industries Electriques, France (now LNE/LAMA Laboratoire National d’Essais/Laboratoire AndrTe Marie AmpTere); OFMET OWce FSedSeral de MSetrologie, Switzerland (now METAS Swiss Federal OWce of Metrology and Accreditation); PTB Physikalisch-Technische Bundesanstalt, Germany; NPL National Physical Laboratory, UK; NIST National Institute of Standards and Technology, USA).

standard resistors. Also there are needs to get higher resistance values of the order of magnitude of about 1 MB with the precision delivered by the QHE at the plateau i = 2 or 4. In principal, any multiple or submultiple of RK-90 can be obtained by respective series and parallel connection of QHE samples. Indeed, it has been shown [12] that such QHE arrays consisting of more than 100 Hall bars show very well quantised resistance values when properly connected [13]. A QHE array with 10 Hall bars in parallel is shown in Fig. 6. Key feature of these QHE circuits is the multiple connection scheme to eliminate erroneous contributions due to contact resistances. A respective measurement of the Hall resistance R as a function of the applied magnetic 6ux density B is shown in Fig. 7. 3. AC-quantum Hall electrical resistance standard As mentioned above, the starting point for the realisation of the SI unit ohm is the calculable capacitor which traces back the capacitance measurement to a

Fig. 6. QHE array of 10 elements in parallel with dimensions 800 × 400 m2 for one sample and 6 × 6 mm2 for the whole array. Upper: design of the lithographic mask; yellow: insulator, red: crossing lines. Lower: corresponding microscopic image.

length measurement. These calculable capacitors are generally operated at a frequency of 1592 kHz corresponding to ! = 10; 000 s−1 . Apart from the fact that the calculable capacitor is a sophisticated instrument the chain ending up at the ohm is rather lengthy resulting in an uncertainty which is about one order of magnitude larger than the uncertainty for the capacitance value realised with the calculable capacitor. This chain

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H. Bachmair et al. / Physica E 20 (2003) 14 – 23 Array P585_01, 10 elements in parallel 1400

200 T = 2,2 K, ISD = 100 µA B>0 B<0

1200

RK-90 20

i=2

180 160 140 120

800

100 600

ρxx/Ω

ρxy/Ω

1000

80 60

400

40 200

20 0

0 0

1

2

3

4

5

6

7

8

T

9

B

Fig. 7. xx and xy of the array P585 01 with 10 elements in parallel. The mean deviation of xx from RK -90 at the plateau i = 2 is (+1:7 ± 0:6) × 10−8 .

could be considerably shortened if the QHE could be measured with AC currents in the kHz regime, with the same precision as with DC currents. In addition, this would, vice versa, provide a simple way to derive the capacitance in terms of RK-90 . From a theoretical point of view one has to consider that the quantisation of the Hall resistance is intimately connected with the localisation of electronic eigenstates in the tails of the disorder-broadened Landau bands. Consequentially, there is no dissipative component of the DC conductivity at zero temperature. This situation changes completely if a time-dependent current is driven through the sample. Now, even localised states contribute to the frequency dependent longitudinal conductivity [14,15], and a deviation of the Hall conductivity from the quantised value e2 =h has to be expected [16]. A quadratic frequency dependence of the deviation was proposed near Alling factor  = 2 from an approximate calculation [17]. The important question is, how large the deviation will be at a frequency of about 1 kHz, which is usually applied in metrological experiments. Of course, in addition to the deviation caused by intrinsic processes, external e3ects like imperfect sample properties, contacts and leads with the accompanying capacitive and inductive couplings will also in6uence the quantisation in actual measurements [18,19]. Recent numerical calculations [20] for noninteracting electrons within a two-dimensional lattice model in the presence of disorder and a strong perpendicular magnetic Aeld have shown that both the

Fig. 8. AC Hall conductivity versus Alling factor  for uncorrelated ( ) and spatially correlated (=a = 1) disorder potentials ( ).

◦

real and imaginary parts of the longitudinal and Hall conductivity have to be considered when comparing with resistances obtained from experiment. In Fig. 8 the AC Hall conductivity of two systems, one with uncorrelated and the other with spatially correlated disorder potentials, is plotted versus Alling factor . For  = 1 deviations from the quantised value are discernible. The deviation of the Hall conductivity from its quantised value increases with frequency !, but the e3ective relation depends on the spatial correlations of the disorder potential considered. The conductivities have been evaluated within linear response theory by means of a recursive Green function technique [21–24]. For uncorrelated disorder potentials the relative deviation of the Hall conductivity follows xy (!)=xy (0) − 1 ∼ !0:5 for high frequencies, while it grows linearly with frequency in the presence of Gaussian correlated disorder potentials with correlation length  of the order of the magnetic length as shown in Fig. 9. As a result, the extrapolated deviation from the quantised value is about 5×10−6 at ∼ 1 kHz in the uncorrelated case whereas it drops well below 10−8 for spatially correlated disorder potentials. First experimental investigations with the required high resolution of the order of 1 part in 108 on Hall bars with so-called multiple series connections were not really promising. A typical result is shown in Fig. 10, where the i = 2 “plateau” is shown for di3erent frequencies between 1 and 6 kHz. The “plateaux” had actually transformed to broad minima, often with reproducible, sample dependent structures and,

H. Bachmair et al. / Physica E 20 (2003) 14 – 23

Fig. 9. Frequency dependence of the Hall conductivity. Shown is the relative deviation Xxy (!)=xy (0) close to a Alling factor  ∼ 0:9 for uncorrelated (=a = 0:0) and for spatially correlated (=a = 1:0) disorder potentials.

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Fig. 11. Hall bar with multiple connected 2 DEG. The 2 DEG is shown in light grey, the contacts are dark grey. A “window” is inserted into the original Hall bar which itself is arranged in the “window” of an outer Hall bar. The window contacts are separately supplied by adjustable voltages. Edge currents are indicated by thick grey lines.

0.1

(R-R0)/R0 / 10-6

0.0 -0.1 -0.2

0 -3

-0.3

-6

-0.4 -0.5

8.5

8.4

8.6

9.0

8.8

9.5

9.0

9.2

9.4

B/T

Fig. 10. The AC von Klitzing plateau i = 2 as a function of frequency, measured at a bath temperature of 0:32 K and a current of 10 A. R0 indicates the von Klitzing constant RK -90 , divided by 2.

moreover, the resistance values at the minima and the curvature exhibited a dependence on frequency and current. Following a suggestion of Klaus von Klitzing [25] we have then fabricated special structures with multiple connected two-dimensional electron

Fig. 12. The AC von Klitzing plateau i =2 of a multiple connected 2DEG. The plateaux have been measured at a bath temperature of 0:32 K, a current of 10 A and di3erent frequencies (from inside to outside 6, 5, 3 and 1 kHz). R0 indicates the von Klitzing constant RK -90 , divided by 2. The plateaux are 6at within a statistical uncertainty of 1:5 × 10−8 but the total uncertainty of the AC von Klitzing resistance is larger as indicated by the error bar.

gas (2 DEG) which allowed for spatial tuning of the potential of the 2 DEG via capacitive coupling [18]. The respective new Hall bar design is shown in Fig. 11. By appropriate biasing and again applying the multiple series connection [13] both, the frequency dependence of the resistance value at the minima and the curvature of the plateaux could be removed, as can be seen in the result depicted in Fig. 12. Similar

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H. Bachmair et al. / Physica E 20 (2003) 14 – 23

results were achieved at the BIPM [26] using back gates. However, at present still some discrepant results have been found and a complete and commonly accepted understanding of the AC-QHE needs further investigations. Nevertheless, within an international cooperation [27] meanwhile a complete system for the calibration of standard capacitors in terms of RK-90 has been developed, which, e.g. allows calibration of 10 pF capacitors within an uncertainty of 1 part in 107 . 4. Further applications of the QHE for the representation of SI units Taking ohm’s law to be valid, the QHE together with the Josephson e3ect for representation of the unit volt can be used for representation of the unit ampere for the electrical current (note that again this is not a realisation of the ampere according to the SI deAnition). Since the reproducibility of the voltage representation by the Josephson e3ect is of the order of a few parts in 1010 , the ampere can be represented and reproduced with an uncertainty of few parts in 109 . In contrast to these afore discussed electrical units where considerable progress had occurred in recent years based on developments in modern quantum physics, the kilogram, the SI unit of mass, is still unchanged since its original introduction in 1889. However, there are indications, that the Pt/Ir kilogram prototype maintained at the BIPM in STevres close to Paris may not be stable. Drifts of the mass of the national kg prototypes against the international prototype of the order of 0:5 × 10−9 kg=a have been observed, see, e.g. Ref. [28]. Consequently, methods to monitor the stability of the kilogram with the Anal goal to provide a new deAnition of the kilogram are being developed at several NMIs. Two of them, the so-called Watt balance and, respectively, the ion accumulation experiment require the precise measurement of an electrical current, which will be done by using a Josephson voltage standard and a von Klitzing resistance standard. In the Watt balance experiment, originally proposed and realised by Brian Kibble [29], mechanical power and electrical power are being compared. This experiment, initially, provides a value for the Planck constant h but, in a second step, because we trust that the

Planck constant is really constant, it could be applied to monitor the stability of the kilogram, provided the achieved uncertainty for the value of h is smaller than 1 part in 108 . In the ion accumulation experiment [30] ions, preferably gold or bismuth, will be separated in a mass separator and collected in a Faraday cup until a weighable amount has been collected in the cup. By measuring and integrating over time the ion current and the determination of the accumulated mass of ions by a suitable balance, the kilogram can be traced back to the atomic mass u. A proof of principle of this experiment has been reported recently [31] and provided a determination of the atomic mass of gold in units of the kilogram with an uncertainty of 10−2 . However, to become relevant for monitoring the stability of the kilogram the uncertainty must be reduced by at least six orders of magnitude. In either case the Watt balance as well as in the ion accumulation experiment, the von Klitzing resistance standard is indispensable to reach the required small uncertainties. 5. Summary The discovery of the QHE and its application for the reproduction of the derived SI unit for the electrical resistance by the von Klitzing resistance standard has provided, together with the Josephson e3ect, the base for a worldwide harmonised reproduction of the important electrical units with a reproducibility being considerably higher than their realisation in terms of the SI deAnition. The importance of the QHE for the realisation of the SI units, however, meanwhile goes far beyond electrical units, as brie6y discussed for the kilogram, the SI unit of mass. Acknowledgements We gratefully acknowledge the continuous interest of Klaus von Klitzing in our work devoted to metrological applications of the QHE and are indebted to him for numerous suggestions and advice. References [1] A.M. Thompson, D.G. Lampard, Nature 177 (1956) 888.

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