Josephson tunnel effect and the order parameter of superconducting alloys containing paramagnetic impurities with local states within the gap

Abstract

Using the Green's function given by Shiba or Rusinov we have calculated some properties of a superconducting alloy containing paramagnetic impurities and having local states within the gap. These properties are (i) for a fixed impurity concentration $p$ and for different values of ${\epsilon }_0$ (the normalized position of a local state with a single impurity), the temperature dependence of the normalized order parameter $\delta \mathrm{}(p,t)\mathrm{}$ and the normalized maximum dc Josephson current $i(p,t)\mathrm{}$; (ii) for ${\epsilon }_0=0.6\mathrm{}$ and ${\epsilon }_0=1.0\mathrm{}$, and for different values of $p$, the temperature dependence of $\delta \mathrm{}(p,t)\mathrm{}$ and $i(p,t)\mathrm{}$; (iii) for ${\epsilon }_0=0.6\mathrm{}$ and ${\epsilon }_0=1.0\mathrm{}$, and for different fixed temperatures, the impurity concentration dependence of $\delta \mathrm{}(p,t)\mathrm{}$ and $i(p,t)\mathrm{}$. Our main results are (i) the impurity-concentration dependence of the normalized transition temperature $t_c$ of the alloy is the same both in the Shiba-Rusinov model (${\epsilon }_0=0.6\mathrm{}$) and the Abrikosov-Gorkov model (${\epsilon }_0=1.0\mathrm{}$); (ii) for a fixed value of the impurity concentration, and for temperatures low as compared to $t_c$, the differences between the values of $i(p,t)\mathrm{}$ [and also of $\delta \mathrm{}(p,t)\mathrm{}$] in the two models are significantly different; (iii) for a fixed temperature, the above noted difference becomes greater as the impurity concentration is increased.