Mean-Field Equations of Superconductivity

Abstract

Cooper’s analysis clarified that an ideal gas of identical fermions becomes unstable in the presence of a mutual attractive potential, however small it may be. What is the new ground state, and how is it related to superconductivity? A breakthrough for this fundamental issue was achieved by Schrieffer, then a graduate student of Bardeen. Motivated by Cooper’s finding, he finally had the idea of applying Tomonaga’s intermediate coupling theory for mesons (Tomonaga, Prog Theor Phys 2:6, 1947) to describe the new ground state (Cooper LN, Feldman D (eds), BCS: 50 years. World Scientific, Hackensack, 2011). In this chapter, we construct this BCS wave function in the coordinate space in such a way that both the pair condensation and phase coherence are manifest. We then derive the Bogoliubov–Valatin operator that describes excitations based on the BCS wave function. These two ingredients are subsequently used to obtain the basic mean-field equations of superconductivity, called the Bogoliubov–de Gennes (BdG) equations, using the same methods as in Chap. 6 for the Hartree–Fock equations. Besides the Hartree–Fock potential, the BdG equations are characterized by a novel self-consistent potential we call the pair potential.