Unified Description of the Mid Gap States in Inhomogeneous Superconductors

Condition for the mid gap state to exist in inhomogeneous superconductor is investigated. On the basis of the Andreev equations, a general expression of the wavefunction of this bound state is derived. Then, the mid gap state is shown to exist only if signs of the pair potential is different, which an quasiparticle feels in the initial and final states in the scattering process described by the Andreev equations. It is also studied why some inhomogeneous systems have no mid gap state. Our results present a unified picture that the absence of the mid gap states results from their hybridization due to the spatial variation of the order parameter and the presence of the normal reflection.