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

Localization lengths of superconducting quasiparticles $\lambda_s$ are evaluated and compared with the corresponding normal state values $\lambda_n$ in one 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 $\lambda$ is strongly correlated with the density of states (DOS) of superconducting quasiparticles. $\lambda_s$ 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, $\lambda_s/ \lambda_n$ at the Fermi energy $E_F$ 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 $E_F$ are more strongly localized than other low energy quasiparticles.