Nanoscale superconductors: Dual Friedel oscillations of single quasiparticles and corresponding influence on determination of the gap energy

By numerically solving the Bogoliubov-de Gennes equations, Friedel oscillations of single quasiparticles in nanoscale superconducting square disks are systematically studied. We find that Anderson's approximation, which assumes the electron- and hole-like wave functions of a quasiparticle are proportional to a certain singleelectron wave function, breaks down under strong quantum confinement, i.e. the electron- and hole-like components of single quasiparticles may show Friedel oscillations with different amplitude and normalized profile, especially when the corresponding single-electron energy moves away from the Fermi level. Such oscillation behavior is referred as dual Friedel oscillations of single quasiparticles, and is expected to be observed the local density of states spectrum. The resulting density of states (DOS) becomes asymmetric and shows significant oscillations with respect to the applied bias.