Landauer's original 1957 formula [2],
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expresses the conductance of a one-dimensional system as the ratio of transmission and reflection probabilities. As explained by Imry [3], this formula gives infinity for unit transmission because it excludes the finite contact conductance contained in [30]
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Extension to higher dimensions [31] is achieved by replacing the transmission probability t by the eigenvalue tn of the transmission matrix product, and summing over n.
Generalizations of the Landauer formula have been found for a variety of other transport properties, besides the conductance [32]. At zero temperature, these expressions are of the form:
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- conductance G: A0=2e2/h, a(t)=t.
- shot-noise power P: A0=4e3V/h, a(t)=t(1-t).
- conductance
of a normal-metal - superconductor junction: A0=4e2/h, a(t)=t2(2-t)-2.
- supercurrent I through a Josephson junction with phase difference
:
,
.
The expressions for the thermo-electric coefficients involve an integration over energies around the Fermi energy
, weighted by the derivative f'=df/dE of the Fermi-Dirac distribution function at temperature T:
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If the energy-dependence of the transmission eigenvalues is small on the scale of the thermal energy kT, one has approximately K=-L0TG (the Wiedemann-Franz law) and
- electrical conductance G: p=0, A0=2e2/h.
- thermo-electric coefficient L: p=1, A0=2e/hT.
- thermal conductance K: p=2, A0=-2/hT.
, with
the Lorentz number.
Each Landauer formula predicts a specific quantum-size effect in a ballistic constriction, for which tn equals 0 or 1. The effect is a step-function dependence on the width of the constriction in the case of G,
,
, K, and an oscillatory dependence in the case of P, L.