Upper critical field in nanostructured Nb: Competing effects of the reduction in density of states and the mean free path

We show that the upper critical field in nanometer-sized Nb particles is governed by the changes in the effective Ginzburg-Landau coherence length occurring due to two competing factors: (i) the decrease in the grain size and consequent increase of disorder, and (ii) the effective decrease in the density of states at the Fermi level due to the formation of a Kubo gap. As a result, the upper critical field (HC2) and irreversibility fields (Hirr) in nanostructured Nb show non-monotonic grain size dependences. Between 60nm to 20nm, HC2 is found to increase by 2.5times while there is no appreciable decrease in the superconducting transition temperature (TC) from its bulk value of 9.4K. This can be ascribed to a decrease in the coherence length due to a reduction in the mean free path with decreasing size. Below 20 nm, however, HC2 decreases with decreasing size. In this size range (<20 nm), there also occurs a decrease in the TC as well as the superconducting energy gap. The decrease in HC2 in this regime can be ascribed to the decrease in the density of states at the Fermi level due to a quantization in the electronic energy levels.