Josephson effect in superconductive bridges: microscopic theory

A theory is developed for the stationary Josephson effect in superconductive constrictions (microbridges).The theory is based on the microscopic Eilenberger equations and is valid for arbitrary temperatures 0<T<Tc. The current-phase relationships I(Q) and critical current Ic(T) obtained differ from the corresponding expressions for a tunnel junction. The difference is greatest at low temperatures and for limitingly pure bridges. In the pure limit l≫a (where l is the free path length; a is the orifice radius) the current is expressed by the formula I(φ)=πΔ0(T)eRNsin(φ/2) th Δn(T)cos(φ/2)2T. The critical current of a pure microbridge at t = 0 is twice the critical current of a tunnel junction with the same normal resistance RN, and the phase-current relationship at the points Q = ±π is discontinuous. The properties of microbridges in the pure (l≫a) and dirty (l≪a) limits are compared.