Physics: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
- [math]\displaystyle{ G(\mu) = G_0 \sum_n T_n (\mu) \ , }[/math]
where [math]\displaystyle{ G }[/math] is the electrical conductance, [math]\displaystyle{ G_0 = e^2/(\pi\hbar) \approx 7.75\times 10^{-5} \Omega^{-1} }[/math] is the conductance quantum, [math]\displaystyle{ T_n }[/math] 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, [math]\displaystyle{ E=\mu }[/math].
A generalization of the Landauer formula for multiple probes is the Landauer–Büttiker formula,[3] proposed by Landauer and Markus Büttiker. If probe [math]\displaystyle{ j }[/math] has voltage [math]\displaystyle{ V_j }[/math] (that is, its chemical potential is [math]\displaystyle{ eV_j }[/math]), and [math]\displaystyle{ T_{i,j} }[/math] is the sum of transmission probabilities from probe [math]\displaystyle{ i }[/math] to probe [math]\displaystyle{ j }[/math] (note that [math]\displaystyle{ T_{i,j} }[/math] may or may not equal [math]\displaystyle{ T_{j,i} }[/math]), the net current leaving probe [math]\displaystyle{ i }[/math] is
- [math]\displaystyle{ I_i = \frac{e^2}{2 \pi \hbar} \sum_{j } \left( T_{j,i} V_j - T_{i,j} V_i \right) }[/math]
See also
References
- ↑ Landauer, R. (1957). "Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction". IBM Journal of Research and Development 1 (3): 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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