Quantum metrology and fundamental constants: international school at Les Houches Physics Centre, 1st to 12 October 2007.
> Highlights (291 Ko)
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
Hall standard connected on
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.
m* : effective electron mass
For metrological applications :
- electron density : ns = 3 - 5.1011 e/cm³,
- electron mobility µ = 20 - 100 T-1
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 .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
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:
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 . 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 , 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 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.
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