# Fundamentals

The â€želectric triangleâ€œ is of fundamental interest in electricity: it illustrates the relation between the units Volt, Ohm and Ampere as stated by Ohmâ€™s law *I*Â = *UÂ /Â R*. Quantum standards are used in metrology to realise these units with the smallest possible measuring uncertainty: The Josephson effect for voltage, the quantum Hall effect for the realisation of resistance and the single electron tunneling for current.

## Josephson Effect

The Josephson effect was predicted in theory by Brian D. Josephson in 1962 and shortly after proved by S. Shapiroâ€™s experiments. The Josephson effect appears between two superconductors separated by a very thin insulating or normal conductive barrier. Superconductivity is the phenomenon that in some materials when cooled below a very low temperature the DC resistance is practically immeasurable and, simultaneously, the magnetic fields are eliminated from the interior of the superconductor. Irradiation with microwaves of appropriate frequency changes the voltage-current characteristic of a Josephson junction in such a way that voltage steps (plateaus) are created depending entirely on fundamental constants and the frequency of the microwaves. Since the frequency is exactly measurable, it is possible to achieve an extremely high repeatability of the DC voltage. The quantity of the voltage *U _{J}* is described in the simple correlation:

where *n* denotes the number of steps, *h* the Planckâ€™s constant, *e* the elementary charge, *f* the frequency of the microwaves and *K _{J}* the Josephson constant.

On 1^{st} January 1990 an international agreement stipulated the value of the Josephson constant with *K _{J}*Â = 483Â 597.9Â GHz/V.

Production methods similar to those used in semiconductor technology permitted to generate output voltages up to 10Â V by serial connection of thousands of contacts. The Josephson DC standard of the BEV is operated with a microwave frequency in the range of 70 GHz to 75 GHz creating a voltage of about 150 ÂµV per step. This allows to generate DC voltages in steps of 150 ÂµV. Precision adjustment of the desired voltage is effected by modification of the microwave frequency. The microwave frequency is measured through connection to the time and frequency standard of BEV (â€žatomic clockâ€œ).

## Quantum Hall Effect

The realisation of the electric unit of resistance Ohm by the quantum Hall effect is cutting-edge technology in many metrology institutes. Due to the large amount of apparatus involved this effect has not been used at BEV up to now. The Quantum Hall effect is observed in a two-dimensional electron gas of high mobility and in the presence of a strong magnetic field at temperatures close to absolute zero. Two-dimensional electron systems for metrological applications can be realised in MOS-field effect transistors or in GaAs/AlGaAs-heterostructures. When the Hall voltage as a function of magnetic flux density is measured in such a structure the plateaus in the Hall voltage can be observed at certain values of the magnetic field. These plateaus of constant Hall voltage correspond to a Hall resistance value *R _{H}* of

where *h* is Planckâ€™s constant, *e* the elementary charge and *i* is an integer. The term *h/e*^{2} is denoted as the von Klitzing constant *R** _{K}* and the value

*R*

_{K}*Â*=Â 25812.807Â Â Â Â was internationally assigned to it on 1.1.1990. The nondecadic value of the quantised Hall resistance demands special methods to calibrate standard resistors with decadic values. For comparisons of the highest accuracy a cryogenic current comparator is used which functions like a supersensitive measuring bridge.

The Josephson and quantum Hall effect are expected to play a decisive role in the upcoming new definition of the mass unit kilogram by means of the watt balance project. With the electric new definition of the mass unit the long-standing goal of metrologists to relate all base units to natural constants would be achieved.

## Single Electron Tunneling

The definition of the unit of electric current Ampere in the International System of Units (SI) serves primarily for determining the magnetic field constant *Âµ _{0} *and is not suitable for the realisation of the Ampere.

Since the beginning of the eighties efforts have been made to completely represent Ohmâ€™s Law by quantum standards and to use quantum standards also for the realisation of electric current. By the application of special semiconductor components it is possible to constrain a defined number of electrons to move through this structure in a defined time. The electric current results as ratio of transferred charge carriers per time unit. In the future the parallel connection of a large number of these components should enable the realisation of a technically applicable current standard based on quantum effects.