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. Current through an atomic metallic contact STM fabricatedMCBJ technique AI  V d.c. current through the contact

3. The current through a metallic atomic contact  Non-linear generalization  Energy dependent transmission coefficient Same single-channel model L R t Left leadRight 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  Subtract:

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

8.  Using for the calculation of where G a and G r 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. Current noise in a metallic atomic contact Same single-channel model L R t Left leadRight lead  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  Poissonian limit (Schottky) binomial distribution charge of the carriers (electrons)

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

17. Anderson model out of equilibrium  Non-interacting case: U=0 00 LR

18.  Equilibrium case: L1 00 LR

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  As we have and

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

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

23.  Coulomb blockade and Kondo effects: -0.50.00.51.01.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LDOS  U/  10    U Equilibrium spectral density Coulomb blockade peaks Kondo resonance

24. Current through a finite mesoscopic region  As a preliminary problem let us first analyze Current through a finite 1D system 00 L R 00 00 00 t t t tLtL tRtR 12N

25.  Current (stationary) between L and 1: 00 L R 00 00 00 ttt tLtL tRtR 12N stationary current  In terms of diagonal Green functions in sites L and 1:

26.  Problem : same steps as in the single resonant level case: 00 L R tLtL tRtR 00 L R 00 00 00 ttt tLtL tRtR 12N

27.  Linear conductance:

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

30. Multi-channel generalization electron reservoirs EFEF E F +eV M mesoscopic region

31.  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 d-like N=5 transition metals Al atomic contact

32.  Same model than in the 1-channel case: tight-binding model including different orbitals i sites  orbitals

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

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

35. The PIN code of an atomic contact electron reservoirs EFEF E F +eV S S PIN code

36. 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