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Journal of the Korean Physical Society

Self-consistent calculations of the effects of disorder in d-wave and s-wave superconductors

Long He, Yun Song

Published Jun 1, 2013• 4 citations
Publisher

Abstract

The kernel polynomial method, as an approach to deriving highly-accurate spectral properties of large sparse matrices, is extended to investigate the effects of disorder in two-dimensional inhomogeneous superconductors. Being able to solve the Bogoliubov-de Gennes equations for quite large square lattices self-consistently, we gain new microscopic insights and make intuitive observations of the localization of low-energy quasiparticles on the nanometer scale. We also compare the behaviors of the optical conductivities of superconductors with different pairing symmetries. We find that the Drude weight turns out to be finite only for the weakly-disordered d -wave superconductor while obvious suppression of the optical conductivity in the infrared range can be observed in the disordered s -wave superconductor.

DOI
10.3938/jkps.62.2223

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The kernel polynomial method, as an approach to deriving highly-accurate spectral properties of large sparse matrices, is extended to investigate the effects of disorder in two-dimensional inhomogeneous superconductors. Being able to solve the Bogoliubov-de Gennes equations for quite large square lattices self-consistently, we gain new microscopic insights and make intuitive observations of the localization of low-energy quasiparticles on the nanometer scale. We also compare the behaviors of the optical conductivities of superconductors with different pairing symmetries. We find that the Drude weight turns out to be finite only for the weakly-disordered d -wave superconductor while obvious suppression of the optical conductivity in the infrared range can be observed in the disordered s -wave superconductor.
We present the effect of disorder on the optical conductivity of two-dimensional inhomogeneous superconductors by applying the kernel polynomial method to solve the Bogoliubov-de Gennes equations. By means of the lattice size scaling of the generalized inverse participation ratio, we find that the localization length of the quasiparticle decreases significantly with the increase of the disorder strength. Meanwhile, the weak disorder can readily restrain the Drude weight, while the superconducting gap has the tendency to suppress the low-energy optical conductivity. We also employ the Lanczos exact diagonalization method to study the competition between the on-site repulsive interactions and disorder. It is shown that the screening effect of repulsive interactions significantly enhances the Drude weight in the normal phase.
We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d-wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d-wave to s-wave superconductor transition, and then an s-wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d-wave superconductors.
We report on a numerical study of disorder effects in 2D d-wave BCS superconductors. We compare exact numerical solutions of the Bogoliubov-de Gennes (BdG) equations for the density of states rho(E) with the standard T-matrix approximation. Local suppression of the order parameter near impurity sites, which occurs in self-consistent solutions of the BdG equations, leads to apparent power-law behavior rho(E) approximately |E|(alpha) with nonuniversal alpha over an energy scale comparable to the single-impurity resonance energy Omega(0). We show that the novel effects arise from static spatial correlations between the order parameter and the impurity distribution.
We study the superconductor-insulator transition (SIT) in ddd-wave superconducting ultrathin films. By means of the kernel polynomial method, the Bogoliubov-de Gennes equations are solved for square lattices with up to 360×360360\times 360360×360 unit cells self-consistently, making it possible to observe fully the nanoscale spatial fluctuations of the superconducting order parameters and discriminate accurately the localized quasiparticle states from the extended ones by the lattice-size scaling of the generalized inverse participation ratio. It is shown that Anderson localization can not entirely inhibit the occurrence of the local superconductivity in strongly-disordered ddd-wave superconductors. Separated by an insulating 'sea' completely, a few isolated superconducting 'islands' with significant enhancement of the local superconducting order parameters can survive across the SIT. The disorder-driven SIT, therefore, is a transition from a ddd-wave superconductor to a Bose insulator which consists of localized Cooper pairs. Unlike an sss-wave superconductor which presents a robust single-particle gap across the SIT, the optical conductivity of a ddd-wave superconductor reveals a gapless insulating phase, where the SIT can be detected by observing the disappearance of the Drude weight with the increasing disorder.
Strong evidence is presented for the localization of low-energy quasiparticle states in disordered ddd-wave superconductors. Within the framework of the Bogoliubov-de Gennes (BdG) theory applied to the extended Hubbard model with a finite concentration of nonmagnetic impurities, we carry out a fully self-consistent numerical diagonalization of the BdG equations on finite clusters containing up to 50\ifmmode\times\else\texttimes\fi{}50 sites. Localized states are identified by probing their sensitivity to the boundary conditions and by analyzing the finite-size dependence of inverse participation ratios.
The effect of nonmagnetic impurities on 2D sss-wave superconductors is studied beyond the weak disorder regime. Within the Bogoliubov--de Gennes (BdG) framework, the local pairing amplitude develops a broad distribution with significant weight near zero with increasing disorder. Surprisingly, the density of states continues to show a finite spectral gap. The persistence of the spectral gap at large disorder is shown to arise from the breakup of the system into superconducting ``islands.'' Superfluid density and off-diagonal correlations show a substantial reduction at high disorder. A simple analysis of phase fluctuations about the highly inhomogeneous BdG state is shown to lead to a transition to a nonsuperconducting state.