Bound states in nonuniform superconductors

This paper shows that the WKB quantization rule is suitable for Andreev bound states in nonuniform superconductors. We consider nonhomogeneous superconductivity gap functions $\mathrm{\ensuremath{\Delta}}(x)$ superconductors with Bogoliubov quasiparticle energy $E$, Fermi level ${E}_{F}$, and total momentum $\mathbf{p}$ at ${E}_{F}+E$. The well of slowly varying $|\mathrm{\ensuremath{\Delta}}(x)|$ are studied, may also be induced by phase variation massless Dirac fermions. By applying method to Bogoliubov--de Gennes equation, we obtain two main results: (i) For ${E}_{F}\ensuremath{\sim}0$, determined ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\phantom{\rule{0.16em}{0ex}}d\mathbf{x}|=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$, where $n\ensuremath{\in}{\mathbb{N}}_{0}$ ${L}_{E}$ ${R}_{E}$ boundary points between classically allowed region forbidden regions, (ii) ${E}_{F}\ensuremath{\gg}E$ $|\mathrm{\ensuremath{\Delta}}(x)|$, given ${\ensuremath{\int}}_{{L}_{E}}^{{R}_{E}}|\mathbf{p}\ifmmode\pm\else\textpm\fi{}{\mathbf{p}}_{F}|dx=(\frac{1}{2}+n)\ensuremath{\pi}\ensuremath{\hbar}$ small ${p}_{y(z)}$. Empirical conditions provided broader parameter regions. In addition traditional method, develop a generalized tackle semimetals parabolic dispersion relationships. applications our results discussed, example, $\ensuremath{\pi}$ junctions or nonchiral Majorana wires, SNS junctions, excitation threshold, tunneling rate NSN junctions. junction, zero modes correspond zero-point formalism. observation provide insights into state vortex fermions high-energy physics.