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Arbeitsgruppe 2.62 Quanten-Hall-Effekt, Widerstand

The Quantum Hall Resistance

The tasks of the working group 2.62 are notably based on the so-called quantum Hall resistance which is characterised by quantised values of the Hall resistance RH:

RH = RK/i    with  RK = h/e2  and   i = 1, 2, 3, 4, …

i.e., the quantum Hall resistance is an integer fraction i of the von-Klitzing constant RK. The von-Klitzing constant depends only on the Planck constant h and the charge of the electron e. Klaus von Klitzing discovered this effect in 1980 und was honoured in 1985 with the Nobel Prize in Physics.


Whereas conventional resistance artefacts depend on the material and the geometrical dimensions, alter with time, temperature and atmospheric pressure, and are sensitive to mechanical shock, the quantum Hall resistance depends only on fundamental constants and is unchangeable and reproducible with an extraordinarily precision of at least 10-10. Due to its unique properties, the quantum Hall resistance has become the primary standard of resistance metrology worldwide. For practical reasons, the quantum Hall resistance usually is used at i = 2.

As defined by the CGPM (Conférence Général des Poids et Mesures) and valid from 20 May 2019, the revised International System of units (SI) is based on exact numerical values of seven fundamental constants according to the best measurements available at that time, amongst others the Planck constant h and the elementary charge e (see BIPM article). Consequently, all calibrations of resistance, capacitance and inductance refer to the exactly defined numerical value of the von Klitzing constant RK = h/e2 = 25812,8074593… Ω. The relative difference to the previously used value of RK-90 amounts to only 17,8·10-9 and is smaller than the uncertainty of the former SI resistance unit. Thus, the revised SI is an important progress for the electrical units ohm and farad (and, of course, for many other electrical and non-electrical units) and allows world-wide uniform calibrations at the highest level.
 
To the physics of the Quantum Hall resistance

In the case of the normal Hall Effect (named after its discoverer Edwin Hall), a thin metal film is located in a magnetic field perpendicular to the metal film. When an electric current is applied to the metal film, the electrons are deflected due to the Lorentzian force and create an electrical voltage perpendicular to the current, the so-called Hall voltage. From the Hall voltage UH and the longitudinal voltage Uxx, the Hall resistance RH and the longitudinal resistance Rxx are calculated according to:

RH = UH/  and  Rxx = Uxx/I  

with I as the Hall current.

 

Scheme of a GaAs heterostructure with bonded lithographic contacts on a device holder

Scheme of a GaAs heterostructure with bonded lithographic contacts on a device holder. The Hall and longitudinal voltage resulting from the Hall current are indicated.  

Whereas the Hall voltage increases linearly with the magnetic field in the case of the normal Hall effect, it can happen under special conditions that steps in the Hall voltage, so-called Hall plateaus, occur.

 

Hall and longitudinal voltage of a GaAs heterostructure as a function of the magnetic field

Hall and longitudinal voltage of a GaAs heterostructure as a function of the magnetic field B at a temperature of T = 0.3 K.

One essential requirement for the occurrence of this phenomenon is a restriction of the electron motion to a plane. Such a two-dimensional electron system can be formed, for example, at a semiconductor-isolator boundary of a GaAs heterostructure. In a strong magnetic field, the electrons move on circular orbits, so-called cyclotron orbits, each with the same discrete energy Ei. Every energy level corresponds to a plateau i of the Hall resistance. To avoid that this macroscopic electronic state gets destroyed by thermally stimulated scattering processes, a sufficiently low temperature is required. Under these conditions, the above mentioned equation for the Hall resistance applies:

RH = RK/i    with  RK = h/e2  and   i = 1, 2, 3, 4, …

Since the quantum Hall effect occurs only at low temperatures and in strong magnetic fields, a corresponding cryogenic system and a superconducting solenoid are required. GaAs heterostructures became the established material because at a temperature below 1.5 K and in a magnetic field of typically 10 T it yields a precise and robust quantum Hall effect. Based on the novel material graphene discovered in 2004, it could become possible in the future to realise a precise quantum Hall resistance at more moderate temperatures (≥ 4 K) and more moderate magnetic fields (≤ 4 T), which would simplify the measuring set-up.

 

GaAs heterostructure on a carrier

GaAs heterostructure on a carrier for ac measurements.

 

 

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