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arXiv:cond-mat/9806060 · Physical Review Letters

Role of Spatial Amplitude Fluctuations in Highly Disordered s -Wave Superconductors

Amit Ghosal, Mohit Randeria, Nandini Trivedi

Published Nov 2, 1998• 284 citations
arXivHTMLPDF

Abstract

The effect of nonmagnetic impurities on 2D $s$-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.

DOI
10.1103/physrevlett.81.3940

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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.
We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ≈0. At high disorder, the system is found to form superconducting "islands" separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.
We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential as a function of disorder strength. We first use the Bogoliubov--de Gennes (BdG) approach to show that, with increasing disorder, the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors that assume a uniform order parameter. In the high-disorder regime, we find that the system breaks up into superconducting islands, with large pairing amplitude, separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states, even in the weak-coupling regime, for all values of disorder, and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by quantum phase fluctuations about the inhomogeneous superconducting state.
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.
The effect of random on-site disorder on s-wave (BCS) superconductors described by a two dimensional negative-U Hubbard model is studied using Bogoliubov-de Gennes (BdG) method. The mean field pairing amplitudes become spatially inhomogeneous at large values of disorder where the system breaks up into superconducting islands with large pairing amplitudes, separated by insulating strips. The amplitude fluctuations are correctly accounted for via BdG calculations, however it misses phase fluctuations which are inherent to low dimensions. The phase fluctuations affect superconducting properties strongly, and the effect is more pronounced in the limit of large disorder. We provide a close estimate of the actual transition temperature, Tcby incorporating phase fluctuations about the inhomogeneous BdG state. This is obtained by relating the jump in renormalized Ds(obtained from a self consistent Harmonic approximation on a phase-only Hamiltonian) at the Kosterlitz-Thouless (KT) transition and the KT transition temperature, TKTobtained from Ds(T) = (2/π)TKTas temperature tends to TKTfrom below. This yields opening of a large region sandwiched between [Formula: see text] (obtained by the vanishing of [Formula: see text]) and TKTwhere there is no phase coherence between the pairs, however amplitude correlations continue to exist, reminiscent of a pseudogap phase in high-Tcsuperconductors which is marked by short ranged preformed pairs without any definite phase relation between them. Further, the appearance of the superconducting islands from a homogeneous phase indicates the evolution of a system consisting of large and overlapping pairs to one that contains short and tightly bound pairs - a scenario termed as BCS-BEC crossover. We have investigated in details the crossover phenomenon as a function of disorder and confirmed it's existence at small values of electron concentration, however, absent at larger densities.
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.
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.