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Application to transport phenomena

Application to transport phenomena. Current through an atomic metallic contact Shot noise in an atomic contact Current through a resonant level Current through a finite 1D region Multi-channel generalization: Concept of conduction eigenchannel . ». m. I. A. V.

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Application to transport phenomena

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  1. Application to transport phenomena • Current through an atomic metallic contact • Shot noise in an atomic contact • Current through a resonant level • Current through a finite 1D region • Multi-channel generalization: • Concept of conduction eigenchannel

  2. » m I A V Current through an atomic metallic contact STM fabricated MCBJ technique d.c. current through the contact

  3. L R t The current through a metallic atomic contact • Non-linear generalization • Energy dependent transmission coefficient Same single-channel model Left lead Right lead perturbation

  4. We use, though, the full energy dependent Green functions of the uncoupled electrodes: previous calculation • Then

  5. For a more general calculation it is useful to express the current in terms of the electrodes diagonal Green functions • It is also convenient to use the specific Dyson equation for (in terms of )

  6. Problem: derivation of expression: • Start from • Use for • Use for • Subtract:

  7. With this expression the tunnel limit is immediately reproduced: lowest order tunnel expression (low transmission)

  8. Using for the calculation of where Ga and Gr are calculated from • Problem

  9. First notice that higher order process in t are included in the denominator Tunnel limit • It is possible to identify the energy dependent transmission Landauer-like

  10. L R Left lead Right lead t Current noise in a metallic atomic contact Same single-channel model • We define the spectral density of the current fluctuations: where

  11. The noise at zero frequency will be given by: • Remembering that the current operator has the form in this model: • The current-current correlation averages contains terms of the form: • However in a non-interacting system they can be factorized (Wick’s theorem) in the form • As the averages of the form are related to

  12. A simple algebra leads to: • Wide-band approximation (symmetrical contact): Keldish space • Direct “unsophisticated” attack: Dyson equation in Keldish space

  13. Problem: solve Dyson equation for the Green functions • Problem: substituting in expression of noise

  14. Identifying the transmission coefficient:

  15. Shot noise limit: Fano reduction factor binomial distribution • Poissonian limit (Schottky) charge of the carriers (electrons)

  16. M M R L Resonant tunneling through a discrete level resonant level Quantum Dot

  17. L R e0 Anderson model out of equilibrium • Non-interacting case: U=0

  18. L R e0 • Equilibrium case: L1

  19. stationary current • As in the contact case: useful expression in terms of diagonal functions: • And now we use the specific Dyson equation for

  20. Problem: substitution in expression of current: • Linear conductance and • As we have

  21. For a symmetrical junction: • Resonant condition: Irrespective of

  22. R L • A more interesting case: e-e interaction in the level resonant level Quantum Dot • Coulomb blockade and Kondo effects

  23. 3.5 3.0 e0 e0 + U G = U / 10 2.5 2.0 LDOS 1.5 1.0 0.5 0.0 -0.5 0.0 0.5 1.0 1.5 w • Coulomb blockade and Kondo effects: Kondo resonance Equilibrium spectral density Coulomb blockade peaks

  24. L R tL tR t t t e0 e0 e0 e0 1 2 N Current through a finite mesoscopic region • As a preliminary problem let us first analyze Current through a finite 1D system

  25. R L tL tR t t t e0 e0 e0 e0 1 2 N • Current (stationary) between L and 1: stationary current • In terms of diagonal Green functions in sites L and 1:

  26. R L tL tR t t t e0 e0 e0 e0 1 2 N L R tL tR e0 • Problem: same steps as in the single resonant level case:

  27. Linear conductance:

  28. Self-consistent determination of electrostatic potential profile Oscillations with wave-length

  29. electron reservoirs M EF+eV EF mesoscopic region left lead right lead Multi-channel generalization

  30. Even a one-atom contact has several channels if the detailed atomic orbital structure is included s-like N=1 simple metals alkali metals sp-like N=3 III-IV group Al atomic contact d-like N=5 transition metals

  31. left lead right lead • Same model than in the 1-channel case: tight-binding model including different orbitals sites i a orbitals

  32. In practice, the effect of a finite central region can be taken into account in a matrix notation : 1D chain finite region

  33. Linear regime Hermitian matrix diagonalization: eigenvalues & eigenvectors conduction channels

  34. S S PIN code The PIN code of an atomic contact electron reservoirs EF+eV EF

  35. Microscopic origin of conduction channels s-like N=1 simple metals alkali metals sp-like N=3 III-IV group d-like N=5 transition metals

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