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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
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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 00 00 00 00 00 ttt = local pairing potential = gap parameter (homogeneous system)
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t 00 00 00 00 00 ttt Diagonalization of H S : Bogoliubov transformation: A quasi-particle is a linear combination of electron and hole 2x2 space (Nambu space)
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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
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From a practical point of view of the quantum mechanical calculation: Doubling up of the Hilbert space: t 00 00 00 00 00 ttt Formally like a normal system with two orbitals per site
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Problem: surface Green functions in the superconducting state t h0h0 h0h0 h0h0 h0h0 h0h0 ttt Simple model: semi-infinite tight-binding chain t 00 00 00 00 tt surface site e-h symmetry
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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
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A word on notation: Nambu space + Keldish space SuperconductivityNon-equilibrium Keldish Nambu
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N/S superconducting contact Single-channel model perturbation L R t Left leadRight lead Superconductor
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Superconducting right lead (uncoupled): R Nambu space
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Normal metal left lead L hole distribution Important point
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I V N/S quasi-particle tunnel: tunnel limit Differential conductance standard BCS picture
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d Tunnel regime Contact regime Conductance saturation
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Normal metal Superconductor Andreev Reflection ProbabilityTransmitted charge
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perturbation LR t Left leadRight lead Superconductor Normal metal
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Current due to Andreev reflections (eV
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Differential conductance saturation value
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Josephson current in a S/S contact Zero bias case L R t Left leadRight lead Superconductor Superconducting phase difference BCS superconductors
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SQUID configuration t ransmission L Nambu space Uncoupled superconductors
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perturbation LR t Left leadRight lead Superconductor
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The zero bias case, V=0, is specially simple, because the system is in equilibrium Even in the perturbed system:
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Tunnel limit Ambegaokar-Baratoff Nambu space
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Andreev states
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Supercurrent Two level system
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Josephson supercurrent Josephson (1962) Kulik-Omelyanchuk (1977)
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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
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Conduction in a superconducting junction 22 22 I eV 22 E F,L E F,L - E F,R = eV > 2 22 E F,R I
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Experimental IV curves in superconducting contacts Al 1 atom contact
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Superconductor Andreev reflection in a superconducting junction eV> I eV 22 ProbabilityTransmitted charge
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Superconductor Multiple Andreev reflection eV > 2 /3 I eV 22 2 /3 ProbabilityTransmitted charge
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Theoretical model Gauge choice time dependent perturbation LR t Left leadRight lead Superconductor
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dc component of the current I 0 (V) Calculation of the current Non-linear and non-stationary current Experiments
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Theoretical IV curves
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Al “one-atom” contact Sub-gap structure (SGS) in:
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Fitting of the curves I 0 (V)
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I 0 (V) characteristics Atomic Al contacts Atomic Pb contacts
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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)
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The PIN code of an atomic contact PIN code
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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
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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 ??
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Huge increase of S/2eI for V 0 Theoretical curves
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Effective charge carried by a multiple Andreev reflection:
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Shot noise measurements in atomic contacts Cron, Goffman, Esteve and Urbina, Phys.Rev.Lett. 86, 4104, (2001). superconducting Al contact effective charge
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SC F SS Superconducting transport through a magnetic region Superconducting transport through a correlated quantum dot
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