Abstract
We present a microscopic theory for the DC Josephson tunnelling currents between a superconducting STM tip and the surface of a superconductor. Within this theory, the spatial variation of the order parameter in the presence of a stationary current is calculated in a self-consistent way at the interface region. The Hamiltonian modelling this system is written in a tight-binding representation and the superconducting state is treated in a mean-field (BCS-like) approximation. We use a non-equilibrium Green functions technique for the calculation of the phase-parameter profile and currents. The results are analyzed as a function of the tip-sample distance, tunnelling current intensity, total phase gradient through the interface and coherence length of the superconductors. The effects of self-consistency are found to be crucial in the close-contact regime when the tip-sample junction has a bottle-neck geometry.
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