Dependences of local density of state on temperature, size, and shape in two-dimensional nano-structured superconductors
In this paper, we investigate a local density of state (LDOS) in two-dimensional nano-structured superconductors. We solve the Bogoliubov-de Gennes equations self-consistently with the two-dimensional finite element method. In nano-structured superconductors, the LDOS as a function of the energy has many discrete peaks. A discretization of the LDOS comes from a discretization of energy levels due to the quantum confinement effect in nano-structured systems. When the temperature increases, a width of a peak in the LDOS is spread to a large energy range and neighbor peaks are superposed due to the thermal effect. On the other hand, for the fixed temperature, the behavior of the LDOS is different between nano-scaled rectangular and square systems. In the nano-scaled rectangular system, when only a lateral length increases, a contribution of the quantum confinement effect from the lateral side is suppressed, while the contribution from a longitudinal side remains large. Then, some peaks are left in the LDOS even when the lateral length is very large. These peaks form a periodic structure and can be regarded as gaps in a multi gap structure due to the quantum confinement effect. On the other hand, energy levels in the square system tend to arrange equally. Then, in the square system, peaks in the LDOS which exist in the rectangular system are small. Also, the period between peaks in the LDOS in the square system is smaller than the period in the rectangular system.