Charge imbalance and Josephson effects in superconductor-normal metal mesoscopic structures

We consider a $SBS$ Josephson junction the superconducting electrodes $S$ of which are in contact with normal metal reservoirs ($B$ means a barrier). For temperatures near $T_c$ we calculate an effective critical current $% I_{c}^{\ast}$andtheresistanceofthesystematthecurrents$I<$ $% I_{c}^{\ast}$and$I>>I_{c}^{\ast}$. It is found that the charge imbalance, which arises due to injection of quasiparticles from the $N$ reservoirs into the $S$ wire, affects essentially the characteristics of the structure. The effective critical current $I_c^{\ast }$ is always larger than the critical current $I_c$ in the absence of the normal reservoirs and increases with decreasing the ratio of the length of the $S$ wire $2L$ to the charge imbalance relaxation length $l_Q$. It is shown that a series of peaks arises on the $I-V$ characteristics due to excitation of the Carlson-Goldman collective modes. We find the position of Shapiro steps which deviates from that given by the Josephson relation.