Effect of superconducting correlation on the localization of quasiparticles in low dimensions

Localization lengths of superconducting quasiparticles ${\lambda }_{\mathit{s}}$ are evaluated and compared with the corresponding normal-state values ${\lambda }_{\mathit{n}}$ in one-dimensional (1D) and two-dimensional lattices. The effect of superconducting correlation on the localization of quasiparticles is generally stronger in an off-site pairing state than in an on-site pairing state. The modification of superconducting correlation to λ is strongly correlated with the density of states (DOS) of superconducting quasiparticles. ${\lambda }_{\mathit{s}}$ drops within the energy gap but is largely enhanced around energies where DOS peaks appear. For a gapless pairing state in 1D or a d-wave pairing state in 2D, ${\lambda }_{\mathit{s}}$/${\lambda }_{\mathit{n}}$ at the Fermi energy ${\mathit{E}}_{\mathit{F}}$ is of order 1 and determined purely by the value of gap parameter not by the random potential. For the d-wave pairing state, the localization effect is largely weakened compared with the corresponding normal state and quasiparticles with energies close to ${\mathit{E}}_{\mathit{F}}$ are more strongly localized than other low-energy quasiparticles.