Disordered superconducting state in a nano-sized dirty superconductor: A microscopic simulation

Superconductivity in nano-size shows special properties. Previously, we investigated transition temperature of nano-sized clean and dirty superconductors [1,2]. We found that the transition temperature is increased with oscillation as size of the superconductor is decreased. This is quantum effect. Electrons are confined in small region and the energy levels become discrete. Decreasing the size, energy level distance becomes large and effective electron density within the range of the attractive interaction between electrons becomes large.

In a dirty superconductor, the transition temperature is strongly increased because of the localization of superconducting electrons and increase of local density of states [3]. In figure, we show the Tc-enhancement (a) and localized superconducting state (b). However, in such localized superconductivity, zero-resistivity may disappear because there remain normal regions. In order to obtain superconductivity in whole region, we should decrease the temperature.

Here, we investigate how localized superconductivity extend the whole region with decreasing the temperature. We solve the Bogoliubov-de Gennes equations with impurity potential using the finite element method. We have found that this localized superconductivity extend rapidly. This is because there are several localized states that have similar Tc and just below Tc these states appear. This behavior depends on the impurity potential.