Landauer formula

The Landauer formula—named after Rolf Landauer, who first suggested its prototype in 1957^[1]—is a formula relating the electrical resistance of a quantum conductor to the scattering properties of the conductor.^[2] In the simplest case where the system only has two terminals, and the scattering matrix of the conductor does not depend on energy, the formula reads

$G(\mu )=G_{0}\sum _{n}T_{n}(\mu )\ ,$

where $G$ is the electrical conductance, $G_{0}=e^{2}/(\pi \hbar )\approx 7.75\times 10^{{-5}}\Omega ^{{-1}}$ is the conductance quantum, $T_{n}$ are the transmission eigenvalues of the channels, and the sum runs over all transport channels in the conductor. This formula is very simple and physically sensible: The conductance of a nanoscale conductor is given by the sum of all the transmission possibilities an electron has when propagating with an energy equal to the chemical potential, $E=\mu$.

References

1. Landauer, R. (1957). "Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction". IBM Journal of Research and Development. 1: 223–231. doi:10.1147/rd.13.0223.
2. Nazarov, Y. V.; Blanter, Ya. M. (2009). Quantum transport: Introduction to Nanoscience. Cambridge University Press. pp. 29–41. ISBN 978-0521832465.