Josephson current in low-dimensional proximity systems and the field effect

The system $S_{\alpha }$-$M_{\beta }$-$S_{\gamma }$ ($S_{\alpha }$ and $S_{\gamma }$ are superconductors, $M_{\beta }$ is a semiconductor, a semimetal, or a normal metal) is studied. Particular attention is paid to the case when $M_{\beta }$ contains a two-dimensional electron gas (e.g., an inversion layer). A one-dimensional (1D) case is also considered. The Josephson current is evaluated and the main factors determining the field effect are studied. A special diagrammatic method allowing one to calculate the thermodynamic Green’s function and, consequently, the Josephson current, has been developed. The current depends strongly on the electron concentration which leads to a noticeable field effect. The dependence of $j_m$ on other factors, such as temperature, mobility, effect mass, etc. is also studied. The field effect appears to be stronger for low-dimensional systems. An analysis of the experimental data obtained recently for the Nb-InAs-Nb system is carried out.