Transport currents and persistent currents in a mesoscopic loop connected with two reservoirs with two superconducting mirrors

We have studied the effect of the Andreev reflection on transport currents and persistent currents in a normal-metallic loop connected to two normal metals with two superconducting mirrors. It consists of two parts, one contributed by electrons and one by holes. The transport currents are periodic with a period equal to half a flux quantum, , in addition to being periodic with a period equal to a single flux quantum, , in the threading magnetic flux when the strength of the barriers at the normal-metal-superconductor interfaces is finite. This result is consistent with earlier experiments. The persistent currents have a period equal to a flux quantum, , in any case. The absolute values of the hole parts have sharp peaks at the gap edge and decrease to zero as the barrier strength increases to infinity, but those of the electronic parts reach a minimum and a constant respectively for these two cases.