Inelastic effects of Josephson junctions

We have investigated the effects of the inelastic interaction of electrons with phonons in the barrier region of S-I-S and S-N-S Josephson junctions. We find that under suitable conditions this mechanism can cause substantial modifications of the temperature dependence of the critical current $j\sb{c}$ as the inevitable loss of coherence can be more than compensated by the enhancement of the tunneling probability resulting from the phonon absorption. The effect depends strongly on the ratio $q\sb{TF}a$ of the junction width a to the screening length in the barrier region. For a S-I-S junction, a monotonic decrease in the critical current with temperature is found for $q\sb{TF}a \gg 1$ whereas for $q\sb{TF}a \ll 1$, the appearance of a peak in $j\sb{c}(T)$ near $T\sb{c}$ is predicted. This new interesting effect is the consequence of the competition between the decrease of the superconducting gap function and the increase in the number of phonons with temperature. A wide range of parameter values has been explored and contact with relevant experimental results has been made. For an S-N-S junction, there is a large increase in the coherence length in the non-superconducting region leading to a substantial enhancement of the critical current over a wide range of temperature. It turns out that the entire temperature range can be divided broadly into two regimes. At low temperatures, the electron predominantly exchanges energy with just one phonon and it is this process that mainly determines the critical current. At higher temperatures the critical current is determined by processes in which the electrons exchange energy with many phonons during their under barrier motion