918 IEEb TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL 45, NO 6, DECEMBER 1996

AC Measurements of the Minimum Longitudinal Resistance of a QHE Sample from 10 Hz to 10 kHz

Franqois P. M. Piquemal, GCrard R

Abstract- AC measurements of the longitudinal resistance, R,,, of a quantum Hall effect (QHE) sample have been made in a frequency range from 10 Hz to 10 kHz. The results show no frequency effect on the minimum value of R,, corresponding to the quantum numbers z = 2 and 1 = 4, within the measurement resolution of 0.5 mR. Therefore, the influence of freduency on the value of the quantized Hall resistance, RIr , sliould not exceed a few parts in 10'. Some unwanted effects detected during the development of the resistance bridge have been pointed out.

I. INTRODUCTION

N up-to-date question concerns the occurrence of the A complete quantization of the Hall effect at a level of one part in 10' (or less) around 1 kHz. In the event of an affirmative reply, on the one hand, the direct determination of the von Klitzing constant RI( from the Thompson-Lampard calculable standard of capacitance could be simplified and, on the other hand, the maintenance of farad from the ac quantum Hall effect (QHE) could be envisaged.

Since 1983, there has been a controversy about the existence or the breakdown of the QHE in an ac regime (11 j, [2] and references cited therein). Around 1990, it was accepted that the physical phenomena remains quantized at low frequen- cies (lower than 1 MHz) for GaAs/AlGaAs heterostructures. However, the measurement uncertainties, of the order of one part in 10' or lo3, mentioned in the various publications up to 1990 were not small enough to allow a pertinent use of QHE for a metrological application around 1 kHz. More recently, experimental results of the value of the quantized Hall resistance (QHR), corresponding to the quantum number z = 2, measured at frequencies around 1600 Hz were very promising. They have shown an agreement with dc values within a few parts in IO7 [3j, 141. As a consequence of the past controversy, it seems more reasonable to begin the metrological study of the ac QHE by first checking on the longitudinal resistance, as it is usually done before performing any dc measurements of the QHR.

In this article, we present ac measurements of R,, on the z = 2 and z = 4 plateaus. The measurements have been carried out on one GaAs-based sample at frequencies up to 10 kHz. They show that the minimum value of R,, at 1.2 K is still very small, of the order of 0.5 mil (the limit of resolution), whatever the frequency is. Such a value should not introduce an error in the ac QHR exceeding a few parts in lo9.

Manuscript received July 1, 1994; revised June 28, 1996. The authors are with the Ldboratoire Central des Industries Electriques,

Publisher Item Identifier S 001 8-9456(96)08273-3. Fontenay aux Roses Cedex F92266, France.

Trapon, and GCrard P. J. Genevks

11. THE MEASUREMENT SYSTEM

A. Principle

Because the expected value of R,, for i = 2 and i = 4 plateaus is very low, it is necessary to define it as a four- terminal resistance. A Hall bar-shaped sample connected to a simple resistance bridge can be modeled as shown in Fig. 1 by taking into account the model of Ricketts and Kemeny [5]. The hypothesis of perfect contacts and nonnegligible residual value of R,,, which is the real part of the longitudinal impedance Z,,, is considered. The probes 1 and 4 of the sample are used as current terminals and the probes 2 and 3 as voltage terminals. The current across probes 1 and 4 is taken as the phase reference. The value of R,, is deduced from the in phase component V&(p) of the voltage between the probes 2 and 3, V23, by the relation

where V,, is the main voltage applied to the sample between probes 1 and 4, and i is an integer (the index of the Hall resistance plateau).

In order to cancel out the unwanted effects due to the high value of the series resistance of the potential probe ($) and due to the large capacitance between the inner and outer conductors of the cables (500 pF), the voltage Vz, and therefore V23 ( p ) have to be measured under conditions of zero current at the defining points of the leads. Because of the very low value of R,,, this condition is achieved with sufficient accuracy if one of the voltage probes is at ground potential ( V ~ G = 0), if the quadrature component of the voltage V& is nulled (V&(q) = 0), and by using a high input impedance voltmeter.

B. Description o j the ac Bridge Fig. 2 shows the bridge circuit for ac measurements of R,,

The in-phase and quadrature components of the voltage V,G are both cancelled by adjusting, respectively, the inductive divider ID in conjunction with the resistor Re (Re E! *), and the inductive divider IDA in conjunction with the capacitor CA (10 pF). Another inductive divider IDB, associated with the capacitor Cg (10 pF) is used to cancel out the quadrature component V&(q) mainly due to the combination of the QHR of sample terminals with the capacitance C1 ( ~ 0 . 8 pF) between current and potential probes of the sample holder.

The measurement network is well isolated from the source by means of a transformer (Tr) with two shields connected

0018-9456/96$05.00 0 1996 IEEE

PIQUEMAL et al.: AC MEASUREMENTS OF MINIMUM LONGITUDINAL RESISTANCE 919

SAMPLE AND ITS HOLDER CONNECTION CABLES VOLTMETER (PSD) I A

-( -7

0 A

e

m

0-

I I I

Ii 1

Fig. 1. Modeled sample and schematic representation of connections and voltages. C1 is the small capacitance between current and potential probes of the sample holder (0.8 pF). C2 is the large capacitance of the potential cables (500 pF). L/2 is the inductance of each cable. CJ is the input capacitance of the voltmeter (25 pF).

as shown on Fig. 2. A phase-sensitive detector (PSD) is used for nulling the voltages &G (input A) and V23 ( 4 ) (differential input A-B) and for measuring V23(p) . The phase is locked in with the phase of the main current as reference (in practice with the voltage applied between the probes 1 and 4). Current equalizers are used for each mesh of the measurement circuit, especially in the reference signal transmission cable [6].

111. RESULTS

A. Main Results

The main ac measurements of R,, have been carried out at 1.2 K on a GaAs based sample provided by the Laboratoire d’ Electronique Philips (LEP). The measured values of carrier density and mobility are, respectively, 5.3 x lo1’ mp2 and 31 T-’ in good agreement with the mean values obtained on some samples diced from the same wafer (900514) [7].

At a magnetic flux density B of 10.52 T, corresponding to the center of the i = 2 plateau, R,, is found to be independent of frequency in the domain studied from 10 Hz to 10 kHz. For a main voltage V, of 200 mV rms, R,, is lower than 0.5 mi2 (the typical limit of resolution of the measurements) in good agreement with the values of R,, (less than 0.2 mR), obtained for a direct current of 40 PA. The range of magnetic flux density for which R,, is a minimum has been studied at

two frequencies, 10 Hz and 12311 Hz (1 233 Hz is the frequency used to relate the impedance of the QHR (i = 2) to that of a 10 nF capacitance). The range of magnetic flux density for which R,, 5 12.5 mR is 1.2 T at these two frequencies, as shown in Fig. 3. However, a significant variation of R,, (5.7 mQ) appears at 1233 Hz on the high-B edge. It is noteworthy that each measurement of R,, was carried out with a constant value of B because of some unbalances of the bridge occurred when the magnetic field was swept.

For the z = 4 plateau, with a main voltage V, of 100 mV rms, R,, is typically lower than 0.7 mR at frequencies from 400 Hz to 7 kHz. At 1233 Hz, R,, does not exceed 2 mR over a very narrow range of magnetic flux density of 0.15 T (centered on 5.26 T) similar to that observed in dc measurements.

B. Experimental Determination of Some Unwanted Impedance InJluences

During the development of the bridge, measurements of R,, were carried out on three other samples without compensating the quadrature component V23 ( 4 ) of the measured voltage V&. The results (presented at CPEM’94 during a poster session) show a frequency dependence of both V23(p) and V&(q) for all the samples. The results obtained for one of these samples are reported in Fig. 4. This dependence is in very good

920 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 45, NO. 6, DECEMBER 1996

Sync.(TTL), R e f . Siynul

t

ID IDA IDB

Fig. 2. Schematic diagram of the measurement bridge.

60

A

A

W

I 10Hz A 1233Hz

A

0

-20 I I

9.5 10 10.5 ll

8 0 11.5

Fig. 3. Minimum of the longitudinal resistance R,, corresponding to the z = 2 plateau measured at 10 Hz and 1233 Hz for a current of 15 p A rms.

agreement with the calculated numerical values of V , 3 ( p ) and VZ3(q) deduced from the Kirchhoff law applied to a model of the measurement set-up, that include the descriptions of the QHE sample and of the associated bridge (Fig. 1). During the measurements, unwanted capacitances such as C1 and C2 have been modified, by changing the sample holder configuration or varying the length of the connecting cables.

The agreement between calculations and values obtained under these conditions confirm clearly why such a frequency behavior occurs when the quadrature component of V23 is not compensated:

1) The linear frequency variation of V23(q) is mainly due to the combination of the QHR of the sample potential terminal and capacitance C1. The predominant term in the calculated expression of the ratio V23 (4) /V, is the following

v23(q) v, %clrj 22 ( 2 )

where w is the pulsation of the main voltage V,. The intrinsic value if any, of the imaginary part of the longitudinal impedance 2,. (Fig. 1) is at most equal to the uncertainty of the measured value of the quadrature

PIQUEMAL et al.: AC MEASUREMENTS OF MINIMUM LONGITUDINAL RESISTANCE 92 1

I ’I / I

I I I

1000 1 . 1 0 ~ 1.10~ I

1 IO 100

Fig. 4. Frequency dependence of and ,,i,,’ observed on the z = 1, V, 2, and 4 plateaus for a sample provided by the University of Copenhagen [SI.

part of Z,, (depending on the uncertainty on Cl). This uncertainty is about 5 mb2 at 1 kHz.

2) The almost quadratiic frequency variation of V ~ 3 ( p ) is also explained by the combination of the QHR with C1 and depends on the value of the input capacitance C3 of the PSD (C, and C, form a capacitive voltage divider). This quadratic frequency variation does not depend on the value of the potential cable capacitances CZ, for a reasonable range of values (300 to 1000 pF). The ratio V23(p)/V,, has the following first order expression

where the influence of V23(q) is pointed out.

IV. DISCUSSION

Let us consider the linear relationship, often found by experiment [9], linking the QHR R H ( ~ , T ) and the minimum value of R,,, R,F:’I(T), measured at the center of the plateau, in the temperature range T = 0 to T = 4.2 K

RH(Z,T) = R ~ ( i , 0 ) [ 1 +aRr$’(T)] (4)

where RH(^, 0) is the von Klitzing constant RK divided by the index i of the plateau and a is a constant. For the central pair of Hall voltage contacts of the sample, the value of a corresponding to the z = 2 plateau amounts to -2 parts in 10’ and +5 parts in 10’ per mf2 depending on the magnetic field direction. These values, which are comparable with those observed on other LEP samples diced from the same wafer [7],,are estimated by carrying out measurements of R,(i) and R,”;” at T = 1.2 K and 7’ = 4.2 K with a direct current of 10 PA. If we suppose that the constant a does not depend on

frequency, the values of R:;” measured at the center of the i = 2 plateau should lead to a deviation in RH not exceeding a few parts in 10’. In the case of the observed deviation of R,, from its minimum value at the magnetic flux density of 11 T (Fig. 3), R,, M 5.7 mf2, the error introduced on RH should amount to about 2 parts in 10’. To confirm this, it will be necessary to carry out measurements both on R,, and RH with the same sample and in the same conditions of magnetic field, frequency, temperature and applied voltage. For the i = 4 plateau, the influence of R,, on RH should be also at a level of a few parts in 10’ if we use the values of a measured on another sample [7].

V. CONCLUSION

The results have shown that the residual value of the longitudinal resistance R,, corresponding to the center of the i = 2 and i = 4 plateaus is not affected by the frequency from dc to 10 kHz. Therefore the effect of R,, on the quantized Hall resistance RH should not exceed a few parts in lo9 around 1 kHz if the coefficient a does not depend on frequency.

ACKNOWLEDGMENT

The authors would like to thank J. P. AndrC, LEP, who pro- vided the sample, F. Delahaye, BIPM, for useful discussions and M. Bellon, B N M L C I E , for mathematical supports and software applications.

REFERENCES

[I] C. T. Van Degrift, M. E. Cage, and S. M. Girvin, “Resource letter QHE- 1: The integral and fractional quantum Hall effects,” Amer. J. Phys., vol. 58, pp. 109-123, 1990.

121 F. Piquemal, “Etalon quantique de risistance en courant alternatif,” Rep. BNM 1169, Jan. 1991.

[3] J. Melcher, P. Warnecke, and R. Hanke, “Comparison of precision AC and DC measurements with the quantized Hall resistance,” IEEE Trans. Instrum. Meas., vol. 42, pp. 292-294, 1993.

141 F. Delahaye, “Accurate AC-measurements of the quantized Hall re- sistance from I Hz to 1.6 kHz,” Metrologia, vol. 31, pp. 367-373, 1994.

151 B .W. Ricketts and P. C. Kemeny, “Quantum Hall effect devices as circuit elements,” J. Phys. D, vol. 21, pp. 483487, 1988.

161 B. P. Kibble and G. H. Rayner, Coaxial AC Bridges. Bristol, U.K.: Adam Hilger, 1984, pp. 166-167.

171 F. Piquemal, C. Cenevis, F. Delahaye, J. P. AndrC, J. N. Patillon, and P. Frijlink, “Report on a joint BIPM-EUROMET project for the fabrication of QHE samples by the L,EP,” IEEE Trans. Instrum. Meas., vol. 42, pp. 264-268, 1993.

181 A. Kristensen et al., “Fabrication and distribution of quantum Hall effect samples for metrology,” EUROMET Project 184, Final Rep., 1993.

191 F. Delahaye, “Technical guidelines for reliable measurements of the quantized Hall resistance,” Melrolugiu, vol. 26, pp. 63-68, 1989.

Fransois P. M. Piquemal was born in Bois D’arcy,

degree in solid state physics from the Universiti de Paris VII, Paris, France, in 1988.

Since 1988, he has been with the Laboratoire Pri- maire d’Electricitt MagnCtisme BNM/LCIE, Fonte- nay aux Roses, France, where he is involved mainly in the quantum ohm metrology.

France, in Octobcr 1960. Hc received the Docteur

922 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 45, NO. 6, DECEMBER 1996

GCrard R. Trapon was horn in Thiers, France, in 1945 He received the Engineer degree in 1976.

Since 1966, he has been with the Lahoratoire Central des Industries Electriques (LCIE), Fontenay aux Roses, France. For 25 years, he was in charge of the DC and LF Calibration Centre, LCIE In 1992, he joined the Lahoratoire Primaire d’ElectricitC MagnCtisme BNMLCIE where he works on the direct determination of the farad and the ohm

Gerard P. J. Genevks was born in AI&, France, in January 1954. He received the Doctorat de troisikme cycle degree in solid state physics from the UniversitC des Sciences et Techniques du Languedoc (USTL), Montpellier, France, in 1981.

From 1982 to 1990, he was involved in research on microwave and RF primary standards and calibration methods as a member of the Laboratoire Pri- maire d’ElectricitC-MagnCtisme, Laboratoire Central des Industries Electriques (LCIE), Fontenay aux Roses, France. He is now the head of the Fundamental and Applied Low Frequency and DC Electrical Metrology Group, LCIE. His main interests are quantum metrology and direct determination of electrical units.