Normal-inverse-Wishart distribution: Difference between revisions
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The '''Landauer formula'''—named after [[Rolf Landauer]], who first suggested its prototype in 1957<ref>{{cite journal|last1=Landauer|first1=R. |title=Spatial Variation of Currents and Fields Due to Localized Scatterers in Metallic Conduction|year=1957|journal=IBM Journal of Research and Development|volume=1|pages=223–231|doi=10.1147/rd.13.0223}}</ref>—is a formula relating the [[Electrical resistance and conductance|electrical resistance]] of a quantum conductor to the [[S-matrix|scattering properties]] of the conductor.<ref>{{cite book|last1=Nazarov|first1=Y. V.|last2=Blanter|first2=Ya. M. |title=Quantum transport: Introduction to Nanoscience|year=2009|publisher=Cambridge University Press|isbn=978-0521832465|pages=29–41}}</ref> | |||
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> | |||
G(\mu) = G_0 \sum_n T_n (\mu) \ , | |||
</math> | |||
where <math>G</math> is the electrical conductance, <math>G_0 = e^2/(\pi\hbar) \approx 7.75\times 10^{-5} \Omega^{-1}</math> is the [[conductance quantum]], <math>T_n</math> are the [[Transmission coefficient|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 [[Electrical resistance and conductance|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, <math> E=\mu </math>. | |||
<!-- The eigenvalues <math> T_n </math> should not be confused with the transmision probabilities, i.e. the probability that an electron in given incoming state n end up on the other side. However, the sume of both of these are the same quantity. --> | |||
==References== | |||
{{reflist}} | |||
[[Category:Mesoscopic physics]] | |||
[[Category:Quantum mechanics]] | |||
[[Category:Nanoelectronics]] | |||
Latest revision as of 05:05, 10 December 2013
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
where is the electrical conductance, is the conductance quantum, 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, .
References
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