### The Landauer formula—named after Rolf Landauer, who first suggested its prototype in 1957—is a formula relating the electrical resistance of a quantum conductor to the scattering properties of the conductor. 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 that an electron has when propagating with an energy equal to the chemical potential, $$E=\mu$$ .

A generalization of the Landauer formula for multiple probes is the Landauer–Büttiker formula, proposed by Landauer and Markus Büttiker [de]. If probe j has voltage $$V_{j}}$$ (that is, its chemical potential is $$eV_{j}})$$ , and $$T_{i,j}}$$ is the sum of transmission probabilities from probe i to probe j (note that $$T_{i,j}}$$ may or may not equal $$T_{j,i}})$$, the net current leaving probe i i is

$$I_{i}={\frac {e^{2}}{2\pi \hbar }}\sum _{j}\left(T_{j,i}V_{j}-T_{i,j}V_{i}\right)}$$

Ballistic conduction

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.
Bestwick, Andrew J. (2015). Quantum Edge Transport in Topological Insulators (Thesis). Stanford University.

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