Impurity and hybridization effects on the symmetry classification and magnetic response function of a two-band superconductor with inter-band pairing order

The effects of hybridization and impurities (magnetic and nonmagnetic) potentials on the pairing symmetries and magnetic response of a two-band superconductor with equal-time s-wave inter-band pairing order parameter in the framework of Green's function technique are investigated theoretically. First, the effects of spin-independent and spin-dependent hybridization on the generation of even- or odd-frequency Cooper pairs which determines the symmetry classification and the response of superconductor are studied. Next, the impurity effect on creating different symmetry classes and the kernel response function of a two-band superconductor are discussed. By separating the contributions of even- and odd-frequency pairing to the Meissner kernel, it is shown that the competition between these two terms determines the total Meissner effect of the superconductor. For a two-band spin-singlet superconductor, nonmagnetic impurity scatterings do not change transition temperature according to Anderson's theorem, while both intra- and inter-band magnetic impurity scattering cause the superconducting transition temperature suppression with the rate following the Abrikosov-Gor'kov theory. For spin-triplet pairing, inter-band magnetic scattering has no impact on pair breaking, while intra-band magnetic scattering acts as a pair breaker and suppresses the transition temperature in the Born limit. In this case, the odd-frequency superconducting pairs can be induced in the simultaneous presence of both intra-and inter-band magnetic impurities. Thus by controlling the concentration of magnetic impurities, it is possible to engineer triplet-paring odd-frequency superconductors with the total diamagnetic Meissner response which stabilizes the superconducting state. This technique opens a road for designing stable odd-frequency superconductors.