Quasi-particle spectrum of nano-scale superconducting plate under a magnetic field

Using the finite element method, we have solved the Bogoliubov-de Gennes equation for superconducting nano-scale plate under a magnetic field, numerically. The local density of states for these systems shows two types of the bound states inside of the superconducting energy gap. We have investigated their wave functions and identified them as the vortex bound states and the surface bound states. Under the zero external field, the surface bound states do not appear inside of the energy gap, and their energies decrease with increasing the external field.