Bound states in nonuniform superconductors

Abstract

This paper shows that the WKB quantization rule is suitable for the Andreev bound states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap functions Δ(x) in superconductors with the Bogoliubov quasiparticle energy E, the Fermi level EF, and the total momentum p at EF+E. The Andreev bound states in the well of slowly varying |Δ(x)| are studied, and the well may also be induced by the phase variation of Δ(x) for massless Dirac fermions. By applying the WKB method to the Bogoliubov–de Gennes equation, we obtain two main results: (i) For EF∼0, the bound states are determined by ∫LERE|pdx|=(12+n)πℏ, where n∈N0 and LE and RE are the boundary points between the classically allowed region and forbidden regions, and (ii) for EF≫E and |Δ(x)|, the bound states are given by ∫LERE|p±pF|dx=(12+n)πℏ with small py(z). Empirical quantization conditions are provided for broader parameter regions. In addition to applying the traditional WKB method, we also develop a generalized WKB method to tackle semimetals with parabolic dispersion relationships. The applications of our results are discussed, for example, Dirac π junctions or nonchiral Majorana wires, SNS junctions, the excitation threshold, and the tunneling rate in NSN junctions. In the π junction, the Majorana zero modes correspond to the zero-point energy in the WKB formalism. This observation may provide insights into the Majorana bound state in a vortex and the Majorana fermions in high-energy physics.

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