Magnetic Field Splitting of the Density of States of Thin Superconductors

Using Green's-function techniques, we present a theoretical calculation of the behavior of the density of states, free energy, and order parameter of very thin superconductors in a high magnetic field as a function of spin-orbit and spin-flip impurity scattering. In very thin superconducting films without spin scattering, the upper critical field is determined by the Pauli paramagnetism of the normal-metal electrons. Tunneling measurements by Meservey and Tedrow have shown a spin splitting by $2{\mu }_{B}H$ in energy space of the BCS peak in the density of states. Zero-temperature calculations of the separate up- and down-spin Green's function for a superconductor show that spin-orbit impurities destroy the magnetic field separation of the peaks in the density of states but do not destroy the energy gap. Spin-flip scattering is much more destructive and destroys both the separation of the peaks and the energy gap. We generalize the calculation to $T\ne 0$ and calculate and plot the critical field versus temperature and the magnetic field dependence of the free energy and order parameter for the various values of the spin-orbit and spin-flip parameter. We also use the theoretical calculations to obtain a fit to the low-temperature tunneling data of Meservey and Tedrow between thin Al and a normal metal and the spin-polarized tunneling between thin Al and ferromagnetic metals.