The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov–de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N × N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)2 coupled self-consistent BdG equations for a supercell of (N + s) × (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron–electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site.
Strong evidence is presented for the localization of low-energy quasiparticle states in disordered
d-wave superconductors. Within the framework of the Bogoliubov-de Gennes (BdG) theory applied to the extended Hubbard model with a finite concentration of nonmagnetic impurities, we carry out a fully self-consistent numerical diagonalization of the BdG equations on finite clusters containing up to 50\ifmmode\times\else\texttimes\fi{}50 sites. Localized states are identified by probing their sensitivity to the boundary conditions and by analyzing the finite-size dependence of inverse participation ratios.
We propose the ultra-fast numerical approach to large-scale inhomogeneous superconductors, which we call the Localized Krylov Bogoliubov-de Gennes method (LK-BdG). In the LK-BdG method, the computational complexity of the local Green's function, which is used to calculate the local density of states and the mean-fields, does not depend on the system size
N. The calculation cost of self-consistent calculations is
O(N), which enables us to open a new avenue for treating extremely large systems with millions of lattice sites. To show the power of the LK-BdG method, we demonstrate a self-consistent calculation on the 143806-site Penrose quasicrystal lattice with a vortex and a calculation on 1016064-site two-dimensional nearest-neighbor square-lattice tight-binding model with many vortices. We also demonstrate that it takes less than 30 seconds with one CPU core to calculate the local density of states with whole energy range in 100-millions-site tight-binding model.