Electrical Resistance
Electrical resistance is a measure of the opposition to the flow of electric current through a material. It is a fundamental property of most electrical components and conductive materials, arising from the collisions between charge carriers (like electrons) and the atoms of the conductive lattice.
The SI derived unit of electrical resistance is the ohm ($\Omega$), named after the German physicist Georg Simon Ohm. He published the foundational relationship between current, voltage, and resistance in 1827.
The Ohm ($\Omega$) and Ohm's Law
The ohm is defined as an electrical resistance between two points of a conductor when a constant potential difference of one volt, applied to these points, produces in the conductor a current of one ampere. This is formalized by Ohm's Law.
Where $R$ is resistance in ohms, $V$ is voltage in volts, and $I$ is current in amperes. An ohm can also be expressed in terms of SI base units.
Resistance is dependent not only on the intrinsic resistivity of a material but also its geometry. A longer wire will have a higher resistance, while a thicker wire will have a lower resistance, akin to water flowing through pipes of different sizes. Additionally, the resistance of most materials changes significantly with temperature.
Quantum Realization: The Quantum Hall Effect
Historically, the ohm was maintained using artifact wire-wound resistors. Much like chemical voltage cells or standard masses, these physical artifacts drifted over time due to metallurgical changes and environmental factors, requiring constant international comparisons.
The solution to an intrinsically stable resistance standard was discovered in 1980 by Klaus von Klitzing, who observed the Quantum Hall Effect (QHE). The classical Hall effect occurs when a magnetic field is applied perpendicular to a current flowing in a conductor, creating a transverse voltage (the Hall voltage).
At extremely low temperatures (near absolute zero) and in the presence of very strong magnetic fields, when conduction electrons are confined to a two-dimensional plane (such as in a MOSFET or specialized semiconductor heterostructure), the Hall resistance ($R_H$) becomes strictly quantized.
Instead of increasing linearly with the magnetic field, the Hall resistance forms discrete plateaus. The resistance value at these plateaus is completely independent of the material properties or geometry of the semiconductor and depends solely on fundamental constants.
Where $i$ is an integer representing the plateau number, $h$ is the Planck constant, $e$ is the elementary charge, and $R_K$ is the von Klitzing constant.
Since the 2019 redefinition of SI base units, where $h$ and $e$ were assigned exact values, the von Klitzing constant is an exact, invariant value of $25812.807... \text{ }\Omega$. Today, National Metrology Institutes utilize cryogenic Quantum Hall systems as primary standards to realize the ohm with extraordinary precision, completely free from long-term artifact drift.