Theory of tunneling anomalies in superconductors above the paramagnetic limit

We Study the tunneling density of states (DOS) in superconducting systems driven by a Zeeman splitting E-Z into the paramagnetic phase. We show that, even though the BCS gap disappears, superconducting fluctuations cause a strong DOS singularity in the vicinity of energies -E* for electrons polarized along the magnetic field and E* for the opposite polarization. The position of this singularity E* = 1/2(E-Z + root E-Z(2) - Delta(2)) (where Delta is BCS gap at E-Z=0) is universal. We found analytically the shape of the DOS for different dimensionalities of the system. For ultrasmall grains the singularity has the shape of a hard gap, while in higher dimensions it appears as a significant though finite dip. Spin-orbit scattering, and an orbital magnetic field suppress the singularity. Our results are qualitatively consistent with recent experiments in superconducting films.