The effect of Magnetic Impurities in Superconductors.

The problem of magnetic impurities in superconductors has been of great interest for a long time. It involves the competition of two distinct physical phenomena. The superconducting state, characterized by a transition temperature $T_{c0}$, is formed by the coherent pairing of electrons with time-reversal symmetry. On the other hand, the magnetic moment on the impurity, characterized by a Kondo temperature $T_{K}$, couples antiferromagnetically to the conduction electrons and locally breaks the time-reversal symmetry. Thus even a small amount of magnetic impurities can strongly reduce the transition temperature or, below the transition, break the pairs and form states within the superconducting gap.

Although many theoretical attempts have been made study this problem, most of them are based upon perturbative approaches specialized for either $T_{K}\ll T_{c0}$ or $T_{K}\gg T_{c0}$. To bridge the gap between these two regimes, I employed QMC to simulate the impurity and Eliashberg-Migdal formalism to treat the superconducting host. This gives me an essentially exact treatment of the problem over the entire region of interest and provides the only treatment of the problem which is consistent with experiment. Among other things, I find that both the initial suppression of the transition temperature and the superconducting transition temperature are universal functions of $T_K/ T_{c0}$ (with a prefactor depending upon the electron-phonon coupling strength). Below the transition, the suppression of the superconducting gap as well as the location of the gap states are again universal. However, the ground state of the impurity changes from a singlet, when $T_{K}< T_{c0}$, to a doublet when $T_{K}> T_{c0}$. Much of the physics of the system can be simply understood from this the crossover.