 # The Resistance Unit Ohm

The unit of the electrical resistance, measured with direct current, is the ohm (abbreviated Ω), named after the German physicist and mathematician Georg Simon Ohm (1789-1854). According to ohm’s law, the resistance R is the ratio of the voltage U across a conductor and the current I flowing through it:

R = U / I.

Hence it follows: 1 Ω = 1 V/A. This definition of the SI ohm is indeed not viable for a realisation.

Due to the extraordinary reproducibility of the quantum Hall resistance, its perfect long-term stability and world-wide uniformity, the ohm can be realised as a certain fraction of the von-Klitzing constant. Already since 1990, on basis of a recommendation by the CIPM (Comité International des Poids et Mesures), resistance comparisons and calibrations world-wide had to be referred to a fixed numerical value of the von-Klitzing constant, RK-90 = 25812.807 Ω90. The introduction of this conventional reference value for the von Klitzing constant had considerable practical advantages in terms of maintainance and dissemination of the unit ohm. At the same time, however, this also meant that the conventional unit Ω90 was not compliant with the International System of Units (SI) valid at that time. An SI-realization of the ohm was, e.g., possible with a Thompson-Lampard capacitor (calculable capacitance; due to the complexity of the corresponding measurement setup, the achievable accuracies were inferior to the reproducibility of quantum Hall resistors.

On May 20, 2019, a revision of the SI came into force according to which an SI value for the von Klitzing constant RK = h/e2 can be derived using exactly defined values for the elementary charge e and the Planck constant h. This made it possible to realize the ohm via the use of quantum Hall resistors within the SI.

At PTB, the resistance unit is realised from the quantum Hall resistance. For this purpose, our working group operates a cryostat with a superconducting solenoid. To guarantee that the Hall resistance takes the precisely quantised value, some internationally accepted criteria have to be fulfilled [Delahaye, Jeckelmann, Metrologia 40, 217-223 (2003)].  Firstly, the longitudinal resistance should be zero because a vanishing longitudinal resistance is a measure for complete quantisation (otherwise a correction has to be applied). Furthermore, all contact resistances of the quantum Hall device have to be sufficiently small. Prior to every calibration, these criteria have to be verified experimentally. Moreover, the resistance values calibrated at PTB and at other national metrology institutes have to be compared from time to time, to guarantee a world-wide uniformity of the resistance unit ohm.

For the dissemination of the unit, it turned out to be practicable to calibrate a conventional 100Ω resistor with known drift behaviour about two times a year, using a cryogenic current comparator. With this 100Ω working resistor, the calibrations for PTB’s customers are carried out by Working Group 2.11. Only in the case of special calibrations requiring a relative uncertainty of 10-9 (or less), the resistor to be calibrated is directly measured against the quantum Hall resistance (i.e., without the intermediate step with the 100Ω resistor). An example is a precision measurement at graphene in the framework of a research project. Ensemble of 1Ω resistors from "The Leeds & Northrup Co." as used at former times for the preservation of the ohm. 