Interplay of disorder and magnetic field in the superconducting vortex state

We calculate the density of states of an inhomogeneous superconductor in a magnetic field where the positions of vortices are distributed completely at random. We consider both the cases of s-wave and d-wave pairing. For both pairing symmetries either the presence of disorder or increasing the density of vortices enhances the low-energy density of states. In the s-wave case the gap is filled and the density of states is a power law at low energies. In the d-wave case the density of states is finite at zero energy and it rises linearly at very low energies in the Dirac isotropic case (α_D=t/Δ₀=1, where t is the hopping integral and Δ₀ is the amplitude of the order parameter). For slightly higher energies the density of states crosses over to a quadratic behavior. As the Dirac anisotropy increases (as Δ₀ decreases with respect to the hopping term) the linear region decreases in width. Neglecting this small region the density of states interpolates between quadratic and back to linear as α_D increases. The low-energy states are strongly peaked near the vortex cores.