Generalized Bogolubov-de Gennes theory for nanoscale superconductors

Abstract

The generalized Bogolubov-de Gennes (BdG) theory, including explicitly the Zeeman energy of electrons, is developed for nanoscale superconductors. To this end the system of four BdG equations is derived, corresponding to four coherent functions (instead of two in conventional BdG theory), two for electron-like excitations and two for hole-like excitations. These equations are transformed into matrix equations by using the basis set of particle-in-the-box problem and solved self-consistently with the equation for the order parameter and the chemical potential. The proposed microscopic approach is suitable for the study of unconventional vortex states and the appearance of FFLO phase in thin nanoscale superconductors.