Theory of Tunneling Anomaly in Superconductor above Paramagnetic Limit

We study the tunneling density of states (DoS) in the superconducting systems driven by 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 $E^*=\case{1}{2}(E_Z + \sqrt{E_Z^2- \Delta^2})$ (where $\Delta$ is BCS gap at $E_Z=0$) is universal. We found analytically the shape of the DoS for different dimensionality of the system. For ultra-small grains the singularity has the shape of the hard gap, while in higher dimensions it appears as a significant though finite dip. The spin-orbit scattering, and the orbital magnetic field suppress the singularity. Our results are qualitatively consistent with recent experiments in superconducting films.