Proximity and Josephson effects in superconductor - two dimensional electron gas planar junctions

The DC Josephson effect is theoretically studied in a planar junction in which a two dimensional electron gas (2DEG) infinite in lateral directions is in contact with two superconducting electrodes placed on top of the 2DEG. An energy gap in the excitation spectrum is created in the 2DEG due to the proximity effect. It is shown that under certain conditions, the region of the 2DEG underneath the superconductors is analogous to a superconducting region with an order parameter $\varepsilon_g\exp(i\phi)$\/, where $\varepsilon_g~(\varepsilon_g<\Delta)$\/ depends on the interface transmittance and the Fermi velocity mismatch between the superconductors and the 2DEG.