# 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 that an electron has when propagating with an energy equal to the chemical potential, ${\displaystyle E=\mu }$.

A generalization of the Landauer formula for multiple probes is the Landauer-Büttiker formula,[3] proposed by Landauer and Markus Büttiker [de]. If probe ${\displaystyle j}$ has voltage ${\displaystyle V_{j}}$ (that is, its chemical potential is ${\displaystyle eV_{j}}$), and ${\displaystyle T_{i,j}}$ is the sum of transmission probabilities from probe ${\displaystyle i}$ to probe ${\displaystyle j}$ (note that ${\displaystyle T_{i,j}}$ may or may not equal ${\displaystyle T_{j,i}}$), the net current leaving probe ${\displaystyle i}$ is

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