Theory of the long-range proximity effect

The theory of the proximity effect in superconducting heterostructures is revised and extended to junctions where the thickness of the normal layers is much larger than the superconductor's coherence length. The origin of this effect is found in a phenomenon called Andreev reflection, wherefore an electron (hole) is reflected as a hole (electron) by the superconductor, at its interface with the normal metal. It is shown that because of the phase differences between the incident and Andreev reflected particles for each energy, integrated over the energy distribution, the induced pairing amplitude becomes zero over a distance of the order of the superconducting coherence length. If, however, the incident and Andreev reflected particles are brought back in phase by some process for all energies, then the induced pairing amplitude acquires a large value at the distance where the constructive interference occurs. Such distance is not limited to the superconductor's coherence length, but to the length over which the phase memory of the particles is preserved. For clean metals at low temperatures this mean free path is an order of magnitude larger that the superconducting coherence length, or more. Hence, the proximity effect can acquire a very long range, not previously recognized by other theories. We develop here a new formalism that encompasses both the conventional view of the proximity effect, and also its long range extension. The model introduced is based on the solutions of the Bogoliubov-De Gennes equations for a one dimensional proximity junction. It is shown that the model explains unusual spikes found in the differential resistance of point contacts in high temperature superconductors. It is argued that this theory provides insight in the nature of phase coherent transport in mesoscopic structures.