Green’s function in proximity-contact superconducting-normal double layers

The Green’s function in a proximity-contact superconducting-normal (S-N) finite double layer with a spatially varying pair function and with finite reflection coefficient R at the interface is discussed in the clean limit. We first obtain a solution of the Gor’kov equation in a form including a quasiclassical evolution operator that can fully describe the spatial variation of the quasiclassical Green’s function. Then we perform analytically the averages of the Green’s function over rapidly oscillating phase factors due to the quantum interference effects in the finite double layers. The averaged results of the Matsubara Green’s function and the density of states are written in terms of elliptic integrals. The effect of finite R is illustrated on the tunneling density of states. The applicability of the present theory to S-N superlattice is mentioned. We show that the conventional normalization condition of the quasiclassical Green’s function does not hold in the double-layer system with finite R.