Microscopic theory of Josephson mesoscopic constrictions

We present a microscopic theory for the dc Josephson effect in model mesoscopic constrictions. Our method is based on a non- equilibrium Green function formalism which allows for a self-consistent determination of the order parameter profile along the constriction. The various regimes defined by the different length scales (Fermi wavelength ${\lambda }_{\mathit{F}}$, coherence length ${\xi }_0$, and constriction length ${\mathit{L}}_{\mathit{C}}$) can be analyzed, including the case where all these lengths are comparable. For ${\lambda }_{\mathit{F}}$<̃(${\mathit{L}}_{\mathit{C}}$,${\xi }_0$) phase oscillations with spatial period ${\lambda }_{\mathit{F}}$/2 can be observed. For ${\mathit{L}}_{\mathit{C}}$>${\xi }_0$ solutions with a phase-slip center inside the constriction can be found, in agreement with previous phenomenological theories.