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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  Superconducting model Hamiltonians   Assume an electronic system with Hamiltonian (in a site representation):     )( t  iii iiii             ccccnH       110   If due to some attractive interaction non included in H, the system becomes superconducting:         iiiii iii iiiiS   )(  )( t   ccccccccnH           110 t  0  0  0  0  0 t t t        = local pairing potential = gap parameter (homogeneous system)   iiii cccc  0  t  0  0  0  0  0 t t t           iiiii iii iiiiS   )(  )( t   ccccccccnH           110   Diagonalization of H S : Bogoliubov transformation:      iiiiiiiiii vuvu cc γ cc γ   A quasi-particle is a linear combination of electron and hole    2x2 space (Nambu space)    Matrix notation: spinor operator for a quasi particle of spin         iii cc ψ   iii  cc ψ   The usual causal propagator in this 2X2 space will be                )' t (  )t (  )' t (  )t (   )' t (  )t (  )' t (  )t (  i )' t  ,t (   ji ji  ji ji ij ccTccT ccTccT G   Which in an explicit 2x2 representation has the form       )' t (  )t ( i )' t  ,t (  iiij    ψψ TG
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