Instrumentation & Measurement Magazine 24-3 - 8

world. The accuracy of these standards has been extensively
tested by direct comparison between similar systems which
typically agree within their experimental uncertainty, on the
order of 1 part in 1010 [11].

The Quantum Hall Effect
Fig. 4a illustrates the well-known Hall effect. If a magnetic field
is applied normally to the surface of a semiconductor carrying
a current I, the charge carriers experience a force perpendicular
to both the magnetic field and the current, causing the charge
carriers to be pushed toward one edge of the sample and producing a voltage difference across the width of the sample. The
ratio of the transverse voltage VH to the current is known as the
Hall resistance, RH  VH / I.
The classical Hall resistance depends linearly on the
magnetic field density B, a characteristic that is used in
common magnetic field sensors. However, the Hall effect
departs from its classical linear behavior under certain extreme conditions. If the charge carriers are confined to a

Fig. 4. (a) Depiction of the Hall effect. (b) Hall resistance versus B
characteristic of a two-dimensional sample at a temperature of 0.3 K (red
trace). Quantized Hall resistance plateaus are apparent at high values of B. The
largest plateau shown in the plot (centered around 10 T) corresponds to i=2.
The blue trace is the longitudinal resistance Rxx.
8

very thin layer (e.g., a two-dimensional electron gas), the
applied magnetic field is very large and the sample is cooled
down to a very low temperature, quantum effects impose
significant restrictions on charge transport through the sample and the Hall resistance becomes constant for a range of
operating conditions.
The quantum Hall effect illustrated in Fig. 4b was discovered and explained by Klaus von Klitzing in 1980 [12]. The plot
shows the transition from the classical (linear) Hall effect at
small magnetic fields to the quantized effect at high values of
B. When the sample is biased in one of the plateaus, the Hall resistance takes on the values

RQHR 

h
, (2)
i e 2

where the step number i is an integer number. This equation
is believed to be exact in the limit of zero temperature. At finite temperatures the quantization is not perfect but it is not
uncommon for practical systems to reach an uncertainty below 1 part in 109.
The longitudinal resistance, Rxx = Vxx / I, also shown in the
plot, drops to a very small value (ideally zero) in the regions
where the Hall resistance is quantized and it is a useful parameter to quantify the level of quantization.
A typical quantum Hall resistance (QHR) standard includes a superconducting magnet and a cryogenic system
to cool down the samples to temperatures below 1.5 K. The
sample is usually biased at the i = 2 plateau to produce a Hall
resistance h / 2e2 = 12 906.403 729 652 5 Ω. The quantum Hall
sample can then be treated as a precisely known 4-terminal resistor, with the Hall potential contacts taking the place of the
voltage (sense) terminals, and it can be used to measure standard resistors of decade values (e.g., 100 Ω, 1 k Ω, etc.) using a
regular resistance bridge.
Of course, the resistance bridge affects the accuracy of the
measurement, and in order to achieve the best uncertainty, it
is necessary to use a home-made bridge based on a Cryogenic
Current Comparator (CCC) [13]. The principle of the bridge is
illustrated in Fig. 5. The resistors to be compared are connected
in parallel (i.e., a nanovolt-meter is used to establish that the
resistors have the same voltage), and the CCC is used to determine the ratio of the currents. A CCC is essentially a current
transformer consisting of magnetically coupled windings surrounded by a superconducting shield which is used to achieve
a nearly perfect flux linkage through a detecting winding. The
ratio of the currents I1 / I2 can be made to be equal to N2 / N1
(the ratio of the turn-numbers) by adjusting one of the currents
until a magnetic flux null is sensed by the detection winding.
The magnetic flux is measured with a Superconducting Quantum Interference Device (SQUID), an extremely sensitive
magnetic flux sensor which allows the dc operation of the CCC
without the use of high permeability materials.
QHR standards are now widely used in a few dozen laboratories around the world. The precision and robustness of these
standards has been extensively demonstrated at extremely
low uncertainty levels by various tests, including many direct

IEEE Instrumentation & Measurement Magazine

May 2021

Instrumentation & Measurement Magazine 24-3

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 24-3