A nano-sized clean superconductor shows enhanced its transition temperature Tc [1]. This is because confinement of wave functions and discreteness of energy levels of electrons. For a bulk dirty superconductor where non-magnetic disorders exist, Tc is not affected by these disorders This is the Anderson’s theorem. However, we showed nano-sized dirty superconductors sometime show higher T_c, when the Cooper pairs are localized [2,3]. In such case, superconducting electrons form islands in the superconductor, at Tc. So, superconductivity does not appear in the entire superconductor. For applications, superconductivity should extend over the superconductor.

In this study, therefore, we investigate how superconducting islands in the nano-sized dirty superconductor at Tc extend and connected when temperature is decreased form the enhanced Tc. We solve the Bogoliubov-de Gennes equations numerically with the finite element method. We consider the impurity potential which is shown in Figure 1 (a). And we investigate temperature dependence of the order parameter structure below Tc. In Figure 1(b), we show an example of the order parameter structure.

Figure1. Distributions of random potential (a) and the order parameter at T/Tc=0.9 (b).

References
[1] M. Umeda and M. Kato, Physics C, 560 45 (2019).
[2] M. Umeda and M. Kato, J. Appl. Phys. 126 143905 (2019).
[3] M. Umeda and M. Kato, J. Phys. Comm. 3 105002 (2019).

Keywords: Nano-sized superconductor, Tc enhancement, Bogolibov-de Gennes equations, Finite element method