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 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,[3] proposed by Landauer and Markus Büttiker [de]. If probe 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 i to probe j (note that \( {\displaystyle T_{i,j}} \) may or may not equal \( {\displaystyle T_{j,i}}) \), the net current leaving probe i i is

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