Electron-electron interactions of the multi-Cooper-pairs in the 1D limit and their role in the formation of global phase coherence in quasi-one-dimensional superconducting nanowire arrays

Nanostructuring of superconducting materials to form dense arrays of thin parallel nanowires with significantly large transverse Josephson coupling has proven to be an effective way to increase the upper critical field of superconducting elements by as much as two orders of magnitude as compared to the corresponding bulk materials and, in addition, may cause considerable enhancements in their critical temperatures. Such materials have been realized in the linear pores of mesoporous substrates or exist intrinsically in the form of various quasi-1D crystalline materials. The transverse coupling between the superconducting nanowires is determined by the size-dependent coherence length ξ0. In order to obtain ξ0 over the Langer-Ambegaokar-McCumber-Halperin (LAMH) theory, extensive experimental fitting parameters have been required over the last 40 years. We propose a novel Monte Carlo algorithm for determining ξ0 of the multi-Cooper pair system in the 1D limit. The concepts of uncertainty principle, Pauli-limit, spin flip mechanism, electrostatic interaction, thermal perturbation and co-rotating of electrons are considered in the model. We use Pb nanowires as an example to monitor the size effect of ξ0 as a result of the modified electron-electron interaction without the need for experimental fitting parameters. We investigate how the coherence length determines the transverse coupling of nanowires in dense arrays. This determines whether or not a global phase-coherent state with zero resistance can be formed in such arrays. Our Monte Carlo results are in very good agreement with experimental data from various types of superconducting nanowire arrays.