The dc $I-V$ characteristics of tunnel junctions between weak and strong superconductors characterized by transition temperatures $T_c$ and $T_c^{\prime }$, respectively ($T\approx T_c<<T_c^{\prime }$), exhibit an excess pair current. This current is proportional to the imaginary part of the generalized pair-field susceptibility $\chi (k,\omega )$, which characterizes the dynamics of the weak superconductor, where $\omega$ and $k$ are proportional, respectively, to the voltage across and the magnetic field in the junction. We show here that this result follows in a simple way from the interaction that gives rise to the usual Josephson current below $T_c$, and can in fact be viewed as a second-order Josephson effect. Above $T_c$ the order parameter on the strong side acts as an effective conjugate field to induce an order parameter on the weak side, yielding a proportional Josephson current between the two electrodes. We also interpret this result to yield a charge-imbalance relaxation time for the effective gapless super-conductor on the weak side. Below $T_c$ the induced part of the order parameter interacts with the usual Josephson current to produce an excess current in a manner analogous to the way an external oscillator coupled to a junction results in a constant-voltage step in the $I-V$ characteristic. In both temperature regimes the magnitude of this excess current is related to an appropriate time-dependent Ginzburg-Landau equation. Finally, the effect of thermal voltage noise on the excess current is considered. Throughout, a heuristic approach is used to bring out new aspects of the problem and to make the physical content of the theory accessible.

• Received 15 October 1981

DOI:https://doi.org/10.1103/PhysRevB.25.6701