Theory of Dirty Superconductors

A B.C.S. type of theory (see BARDEEX, COOPER and SCHREIFFER, Phys. Rev. 108, 1175 (1957)) is sketched for very dirty superconductors, where elastic scattering from physical and chemical impurities is large compared with the energy gap. This theory is based on pairing each one-electron state with its exact time reverse, a generalization of the k up, -k down pairing of the B.C.S. theory which is independent of such scattering. Such a theory has many qualitative and a few quantitative points of agreement with experiment, in particular with specific-heat data, energygap measurements, and transition-temperature versus impurity cumes. Other types of pairing which have been suggested are not compatible with the existence of dirty superconductors. ONE of the most striking experimental facts about superconductivity is that it is often insensitive to enormous amounts of physical and chemical impurities. For one example, several substances in essentially an amorphous state have been shown to be superconductors, such as bismuth and beryllium films laid down at liquid-helium temperatures.(l) As another example, there arc disordered alloy systems with 20-50 per cent of chemical scattering centers, but with transition tempcratures comparable with those of pure elements.(*) These quantities of crystal imperfections are large enough to scatter the electrons at an extremely rapid rate. In fact, if we were to take the mean free time before scattering for the electron as a measure of the electrons’ uncertainty in energy, that uncertainty in energy is large compared not only with the energy gap ~0, but with the Debye energy Rwo. Plane-wave states for the electrons definitely have this very large degree of energy uncertainty. On the other hand, the experiments of SERIN et uZ.(3) have shown that, starting with a pure single crystal of a superconducting material, there is usually a rather sharp initial drop in the supcrconducting transition temperature as the first small percentage of chemical imperfection is added. They show that this initial drop is proportional to the extra rcsistivity caused by these imperfections, and therefore proportional to the amount of scattering. If the impurities which are introduced are magnetic ions rather than ordinary chemical impurities, MATTHIAS et al@) have shown that this initial sharp drop continues, and superconductivity is very soon destroyed. On the other hand, for ordinary impurities the sharp drop stops rather soon and is replaced by a more gradual behavior, which seems to be determined primarily by the fact that the impurity adds or subtracts electrons from the band, changes the density of states, and in various ways gradually varies the parameters of the free electrons. Thus WC: may divide superconductivity into two regions: (1) the region of relatively pure superconductors where scattering has a rather sharp effect on superconducting transition temperatures; and (2) the region of very imperfect superconductors, where additional scattering has very little effect. It is the purpose of the present paper to give a theory of this region of the “dirty” superconductor. The fundamental assumption WC will make is that in this region the problem of the electron wave functions is best solved by first diagonalizing the scattering interaction between the electrons and the impurities, and then calculating the phonon interactions between electrons. Finally, one calculates from this the superconducting properties. That is, we find a new set of one-electron wave functions for the electrons, and then solve the problem of the interactions of the electrons in terms of these, rather than in terms of ordinary