Graphene-based quantum Hall effect metrology
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and prospects for quantum Hall effect metrology using graphene Metrological context Metrology aims to establish definitions and representations of units, with the constraint that they be as universal and as convenient as possible so that they can be easily used both by scientists and in industry. In terms of universality, the discovery of several quantum phenomena in solids has revolutionized electrical metrology in the past few decades. Based on the Josephson effect and the quantum Hall effect1 (QHE), the volt and the ohm have been represented since 1990 in terms of only Planck’s constant h and the electron charge e, which is preferable to the use of material artifacts (namely, Weston standard cells for the volt and resistors for the ohm). The resistance standard is based on the initially surprising observation that the transverse resistance, RH ≡ Rxy, of a two-dimensional electron gas (2DEG) placed in a perpendicular magnetic field B (strictly speaking magnetic induction) (see Figure 1a) is quantized at rational fractions of the von Klitzing constant, RK ≡ h/e2. At the same time, the longitudinal resistance, Rxx, drops to zero, reflecting the absence of dissipation in the 2DEG (Figure 1b).
This integer quantum Hall effect relies on the charge carriers in the system occupying a series of discrete energy levels known as Landau levels (LLs), which correspond to the quantization of the cyclotron motion of charge carriers in the magnetic field. The value at which RH is quantized depends on the number of filled LLs, described by the filling factor ν = nsh/eB, where ns is the carrier density. The universal character of the quantum Hall resistance standard is supported by very strong theoretical arguments.2 Nonetheless, it is critical for dependable metrology that this universality be experimentally validated. The quantum resistance standard is based on a Hall bar usually fabricated from a 2DEG formed in a GaAs/AlGaAs semiconductor structure (Figure 1a). When operated at low temperature (T ≤ 1.5 K) and high magnetic field (B ≈ 10 T), with measurement currents of a few tens of microamperes on the RH plateau at RK/2 (ν = 2), such a Hall bar allows the calibration of a wire resistor in terms of a conventional value of RK (i.e., 25,812.807 Ω) with a typical accuracy of 10–9. This performance is possible owing to specific instrumentation based on a resistance comparison bridge equipped with superconducting transformers (cryogenic current comparator, CCC) and
Félicien Schopfer, Laboratoire National de Métrologie et d’Essais, France; [email protected] Wilfrid Poirier, Laboratoire National de Métrologie et d’Essais, France; [email protected] DOI: 10.1557/mrs.2012.199
© 2012 Materials Research Society
MRS BULLETIN • VOLUME 37 • DECEMBER 2012 • www.mrs.org/bulletin
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GRAPHENE-BASED QUANTUM HALL EFFECT METROLOGY
Figure 1. (a) Drawing of a typical Hall bar, fabricated from a two-dimensional electron gas (formed in a GaAs/AlGaAs heterostructure, for example). (b) At a temperature of T = 1.3 K, when the perpendicular magn
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