1. Superconducting transport  Superconducting model Hamiltonians:  Nambu formalism  Current through a N/S junction  Supercurrent in an atomic contact  Finite bias current and shot noise:  The MAR mechanism

2. Superconducting model Hamiltonians  Assume an electronic system with Hamiltonian (in a site representation):  If due to some attractive interaction non included in H, the system becomes superconducting: t 00 00 00 00 00 ttt   = local pairing potential = gap parameter (homogeneous system)

3. t 00 00 00 00 00 ttt   Diagonalization of H S : Bogoliubov transformation:  A quasi-particle is a linear combination of electron and hole 2x2 space (Nambu space)

4.  Matrix notation: spinor operator for a quasi particle of spin  The usual causal propagator in this 2X2 space will be  Which in an explicit 2x2 representation has the form

5.  From a practical point of view of the quantum mechanical calculation:  Doubling up of the Hilbert space: t 00 00 00 00 00 ttt  Formally like a normal system with two orbitals per site

6.  Problem: surface Green functions in the superconducting state t h0h0 h0h0 h0h0 h0h0 h0h0 ttt Simple model: semi-infinite tight-binding chain t 00 00 00 00 tt surface site e-h symmetry

7.  Adding an extra identical site,, and solving the Dyson equation Normal case Superconducting case  In a superconductor the energies of interest are Wide band approximation Normal state Superconducting state BCS density of states

8.  A word on notation: Nambu space + Keldish space SuperconductivityNon-equilibrium Keldish Nambu

9. N/S superconducting contact Single-channel model perturbation L R t Left leadRight lead Superconductor

10.  Superconducting right lead (uncoupled): R Nambu space

11.  Normal metal left lead L hole distribution  Important point

12. I V  N/S quasi-particle tunnel: tunnel limit Differential conductance standard BCS picture

13. d Tunnel regime Contact regime Conductance saturation

14. Normal metal Superconductor Andreev Reflection ProbabilityTransmitted charge

15. perturbation LR t Left leadRight lead Superconductor Normal metal

16. Current due to Andreev reflections (eV 

17. Differential conductance saturation value

18. Josephson current in a S/S contact Zero bias case L R t Left leadRight lead Superconductor Superconducting phase difference BCS superconductors

19. SQUID configuration t ransmission L Nambu space Uncoupled superconductors

20. perturbation LR t Left leadRight lead Superconductor

21.  The zero bias case, V=0, is specially simple, because the system is in equilibrium  Even in the perturbed system:

22. Tunnel limit Ambegaokar-Baratoff Nambu space

23. Andreev states

24. Supercurrent Two level system

25. Josephson supercurrent Josephson (1962) Kulik-Omelyanchuk (1977)

26. S/S atomic contact with finite bias Multiple Andreev reflections (MAR) Sub-gap structure : qualitative explanation  e a) 1 quasi-particle eV >  e h b) eV >  e e h c) 3 quasi-particles eV >2  2 quasi-particles I V  a  b  c n quasi-particles eV >2  n

27. Conduction in a superconducting junction 22 22 I eV 22 E F,L E F,L - E F,R = eV > 2  22 E F,R I

28. Experimental IV curves in superconducting contacts  Al 1 atom contact

29. Superconductor Andreev reflection in a superconducting junction eV>  I eV 22  ProbabilityTransmitted charge

30. Superconductor Multiple Andreev reflection eV > 2  /3 I eV 22  2  /3 ProbabilityTransmitted charge

31. Theoretical model Gauge choice time dependent perturbation LR t Left leadRight lead Superconductor

32. dc component of the current I 0 (V) Calculation of the current  Non-linear and non-stationary current Experiments

33. Theoretical IV curves   

34.    Al “one-atom” contact Sub-gap structure (SGS) in:

35. Fitting of the curves I 0 (V)

36. I 0 (V) characteristics Atomic Al contacts Atomic Pb contacts

37. Mechanical break junction Superconducting IV in last contact before breaking Theoretical curves Determination of conduction channels of an atomic contact Scheer et al, PRL 78, 3535 (97) (Saclay)

38. The PIN code of an atomic contact PIN code

39.  Correlation between number of channels and number of valence atomic orbitals 3s 3p Al eV7 ~ Al 3 Pb 3 Nb 5 Au 1 (Saclay) (Leiden) (Madrid) MCBJ STM Proximity effect Determination of conduction channels of an atomic contact

40. Shot noise in superconducting atomic contacts Poissonian limit Charge of the carriers  What is the transmitted charge in a Andreev reflection?  e eV >  e h eV >  e e h eV >2  ??

41. Huge increase of S/2eI for V 0 Theoretical curves

42.  Effective charge carried by a multiple Andreev reflection:

43. Shot noise measurements in atomic contacts Cron, Goffman, Esteve and Urbina, Phys.Rev.Lett. 86, 4104, (2001). superconducting Al contact effective charge

44. SC F SS Superconducting transport through a magnetic region Superconducting transport through a correlated quantum dot