In this paper, we investigate a local density of state (LDOS) in a two-dimensional nano-structured superconductor. We solve the Bogoliubov-de Gennes equations self-consistently with the two-dimensional finite element method. In the nano-structured superconductor, the LDOS as a function of energy has many discrete peaks. Discretization of the LDOS comes from discretization of energy levels due to the quantum confinement effect in the nano-structured system. When temperature increases, a width of a peak in the LDOS is spread to a large energy range and neighbor peaks are overlapped. 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 length of a long side increases, a contribution of the quantum confinement effect from the long side is suppressed, while the contribution from a length of a short side remains large. Then, some peaks are left in the LDOS even when the length of the long side 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 that in the rectangular system.

在本文中，我们研究了二维纳米结构超导体中的局部状态密度（LDOS）。我们用二维有限元方法自洽地求解Bogoliubov-de Gennes方程。在纳米结构的超导体中，LDOS作为能量的函数具有许多离散的峰值。由于纳米结构系统中的量子限制效应，LDOS的离散化来自能级的离散化。当温度升高时，LDOS中峰的宽度扩展到较大的能量范围，并且相邻峰重叠。另一方面，对于固定温度，纳米尺度的矩形和正方形系统的LDOS行为是不同的。在纳米级矩形系统中，当仅长边的长度增加时，来自长边的量子限制效应的贡献被抑制，而来自短边的长度的贡献仍然很大。然后，即使长边的长度很大，LDOS中也会留下一些峰。这些峰形成周期性结构，由于量子限制效应，可以将其视为多间隙结构中的间隙。另一方面，平方系统中的能级趋向于平均分配。因此，在正方形系统中，存在于矩形系统中的LDOS中的峰很小。同样，在方形系统中，LDOS中的峰之间的周期小于矩形系统中。这些峰形成周期性结构，由于量子限制效应，可以将其视为多间隙结构中的间隙。另一方面，平方系统中的能级趋向于平均分配。因此，在正方形系统中，存在于矩形系统中的LDOS中的峰很小。同样，在方形系统中，LDOS中的峰之间的周期小于矩形系统中。这些峰形成周期性结构，由于量子限制效应，可以将其视为多间隙结构中的间隙。另一方面，平方系统中的能级趋向于平均分配。因此，在正方形系统中，存在于矩形系统中的LDOS中的峰很小。同样，在方形系统中，LDOS中的峰之间的周期小于矩形系统中。