# 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

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

where ${\displaystyle G}$ is the electrical conductance, ${\displaystyle G_{0}=e^{2}/(\pi \hbar )\approx 7.75\times 10^{-5}\Omega ^{-1}}$ is the conductance quantum, ${\displaystyle T_{n}}$ 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, ${\displaystyle E=\mu }$.

## 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.