Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation?

Abstract

The solutions of the Bogoliubov–de Gennes (BdG) equation are usually interpreted as the excitations from the superconducting ground state. This viewpoint is not easily applied to a strongly coupled heterojunction since the ground state changes across the interface and it is not clear how the ground state should be connected across the heterointerface. In this paper, we present a different viewpoint that does not suffer from this conceptual drawback. We show that the BdG equation can be viewed as a ‘one-particle’ wave equation whose eigenstates (including the negative energy states) can be filled up systematically to describe the superconducting state, in much the same way that we fill the eigenstates of the Schrödinger equation to describe normal conductors. The only difference is that we need to start from a special vacuum | V〉, consisting of a full band of down-spin electrons, instead of the usual vacuum devoid of all particles. Any quantity of interest, A (such as the charge density or the current density), can be interpreted as the sum of a ‘vacuum contribution’AV ACdue to the vacuum | V〉 and a one-particle contribution ABdGdue to the filled eigenstates of the BdG equation. This picture is easily applied even to strongly coupled heterojunctions since the vacuum | V〉 is the same on both sides of a heterointerface. As such, we believe it puts the scattering theory of transport for superconductors on a firmer conceptual basis.