Pair-field susceptibility and superconducting tunneling: A macroscopic approach

Abstract

The dc $I\ensuremath{-}V$ characteristics of tunnel junctions between weak and strong superconductors characterized by transition temperatures ${T}_{c}$ and ${T}_{c}^{\ensuremath{'}}$, respectively ($T\ensuremath{\approx}{T}_{c}<<{T}_{c}^{\ensuremath{'}}$), exhibit an excess pair current. This current is proportional to the imaginary part of the generalized pair-field susceptibility $\ensuremath{\chi}(k,\ensuremath{\omega})$, which characterizes the dynamics of the weak superconductor, where $\ensuremath{\omega}$ 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 $I\ensuremath{-}V$ 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.