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Quantum Hall effect

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Quantum metrology and fundamental constants: international school at Les Houches Physics Centre, 1st to 12 October 2007.

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The aim of this study is to improve representation of the French electrical resistance unit by developing both quantum standards and calibration chain instrumentation.


The quantum Hall effect (QHE) appears in a two-dimensional electron gas subjected to a perpendicular magnetic field [1-2]. In practice this gas is produced at the interface of an AlGaAs/GaAs heterostructure or along the drain-source channel of a silicon MOSFET (metal-oxide-semiconductor field effect transistor). The confinement potential well, whose thickness (~3-4 nm) is much less than the Fermi wavelength, forces the electrons to move in the plane of the interface.

two-dimensional electron gas with Hall bar geometry

two-dimensional electron gas
with Hall bar geometry

Hall standard connected
on ceramic support

Hall standard connected on
ceramic support



At low temperature (typically 1.5 K) and in a strong magnetic field, the Hall resistance defined by the quantity RH = VH/I takes the quantized values RH = RK/i, where i is an integer and theoretically RK = h/e². This effect is due both to the quantization of state density in Landau levels and to the existence of a conductivity gap linked to the formation by disordering of localized states. This localization is responsible for the cancellation of the longitudinal resistance Rxx=Vxx/I in the middle of the quantization plateau.

Quantization in Landau levels

m* : effective electron mass

Observation conditions

For metrological applications :
- electron density : ns = 3 - 5.1011 e/cm³,
- electron mobility µ = 20 - 100 T-1

Conservation of the ohm

Since 1990, France's national metrology laboratories have used the QHE for conservation of the ohm, following a recommendation by Le comité International des Poids et Mesures (CIPM). The QHE supplies a quantum resistance standard of value RK/i (i=2 ou 4), with a relative reproducibility of around 10-10. On the other hand, the determination of 10-7 is given with a relative uncertainty of RK in the Système International d'Unités (SI). In order to facilitate international comparisons and conservation of the ohm, a conventional true value RK-90 of RK was fixed by the CIPM on 1st January 1990. RK-90 equals exactly 25812,807 Ω [2].The bridge used for comparison of resistance standards at LNE is based on a cryogenic current comparator (CCC). It can calibrate standards with nominal values of 1 Ω, 100 Ω and 10 kΩW with a relative uncertainty of around 10-9 in terms of RK-90. Standard traceability is ensured first for France's primary calibration centre, which in turn ensures traceability for the secondary calibration centres. Resistance calibrations and all QHE studies are performed in a shielded, anti-vibration laboratory with temperature regulated +/- 0,3 °C . The Hall samples are cooled to 1,3 K and 0,3 K respectively with He4 and He3 refrigerators placed in cryostats fitted with 12/14T and 14/16T superconducting magnets. To perform the complete calibration chain, the Laboratory uses material resistance standards with temperature regulated to a few mK, either in oil baths or in climatic chambers.

 Diagram of cryogenic bridge for comparison of resistance

Diagram of cryogenic bridge for comparison of resistance

Bridge principle

The number of turns NP and NS of the primary and secondary coils of the CCC are chosen in a ratio NP/NS close to the ratio RP/RS. The resistive divider diverts a fraction ε of the current Is to an auxiliary coil with NA turns. With the counter-reaction of the SQUID on the current Is the ratio Is/Ip is adjusted to a few parts in 10-10 to the ratio NP/(NS + εNA). For a fraction ε giving zero voltage at the zero detector terminals, this gives:

Development of quantum resistance standards

International availability of quantum resistance standards is a key issue. BNM has always been highly active in the development of these standards. Working with the Philips electronics laboratory LEP/OMMIC, it has completed several fabrication projects in order to distribute standards to the other French national metrology laboratories [3]. At present BNM is leading the way in developing a new generation of quantum Hall array resistance standards (QHARS) which integrate multiple elementary Hall bars connected together on a single sample [4-7]. This new technique, based on the redundant connection of the Hall bars [8], makes it possible to develop quantum standards with nominal resistance values ranging from 100Ω and 1 MΩ.

Photograph and diagram of a nominal resistance standard

Photograph and diagram of a nominal resistance standard RK/200 (i=2) (QHARS129), with 100 Hall bars connected in parallel


Heterostructures are manufactured by MOCVD (LEP) or MBE (LPN)

- Ohmic contacts: annealed AuGeNi
- Connections: Pt/Au
- Insulating layers between connection levels: Si3N4 / SiO2

Quantum resistance standards are developed using epitaxy and microelectronic lithography techniques. Constructing QHARS is more difficult, as it is necessary both to obtain electron gases with homogeneous density on surfaces of around 1 cm² and to insert insulating layers between the connection levels.







References :
[1] K. von. Klitzing et al, Phys. Rev. Lett., 45, 494 (1980).
[2] F. Piquemal, Bull. Bur. Nat. Métrologie, 116, 5 (1995).
[3] F. Delahaye et al, IEEE Trans. Instrum. Meas., 44, 258 (1995).
[4] F. Piquemal et al, IEEE Trans. Instrum. Meas., 48, 296 (1999).
[5] W. Poirier et al, J. Appl. Phys., 92, 2844 (2002).
[6] A. Bounouh et al, IEEE Trans. Instrum. Meas., 52, 555, (2003).
[7] W. Poirier et al, Metrologia, 41, 285 (2004).
[8] F. Delahaye, , J. Appl. Phys., 73, 7914 (1993).


Wilfrid Poirier
Tel : 00 (33) 1 30 69 21 74

Félicien Schopfer
Tel : 00 (33) 1 30 69 21 69


  • F. Piquemal, "L'effet Hall quantique en métrologie", Bull. Bur. Nat. Métrologie, 116, 5 (1995).
  • F. Piquemal et al, "A first attempt to realise (multiple-QHE devices)-series array resistance standards", IEEE Trans. Instrum. Meas., 48, 296 (1999).
  • W. Poirier et al, "RK/100 and RK/200 quantum Hall array resistance standards", J. Appl. Phys., 92, 2844 (2002).
  • A. Bounouh et al, "Quantum resistance standards with double 2DEG", IEEE Trans. Instrum. Meas., 52, 555, (2003).
  • C. Chaubet et al, "Inter and Intra Landau Level scatterings as a mechanism for the onset of the voltage drop across the contact at high currents in the quantum Hall effect regime", Semicond. Sci. Technol. 18, 983, (2003).
  • Y. M. Meziani et al, "Heating process in the pre-breakdown regime of the quantum Hall effect: a size dependent effect", Physica B, 346-347, 446, (2004).
  • Y. M. Meziani et al, "Behavior of the contacts of quantum Hall effect devices at high currents", J. Appl. Phys., vol. 41, pp. 285-294, (2004).
  • W. Poirier et al, "A new generation of QHARS : discussion about the technical criteria for quantization ", Metrologia, 41, 285-294, (2004).

Current projects

  • Development of quantum resistance standards, in collaboration with OMMIC and the Laboratoire de Photonique et Nanostructures (LPN).
  • Study of quantization break phenomena in strong current, in scientific collaboration with Université Montpellier II, semiconductor physics group.