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

↑ 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.
↑ Nazarov, Y. V.; Blanter, Ya. M. (2009). Quantum transport: Introduction to Nanoscience. Cambridge University Press. pp. 29–41. ISBN 978-0521832465.