Role of Spatial Amplitude Fluctuations in Highly Disordereds-Wave Superconductors

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. [S0031-9007(98)07517-6] PACS numbers: 74.20.-z, 74.40.+ k, 74.80.-g The effect of strong disorder on superconductivity has been a subject of considerable interest, both theoretically [1,2] and experimentally [3,4], for a long time. A generally accepted physical picture of how the superconducting (SC) state is destroyed and the nature of the non-SC state has not yet emerged. Much of the theoretical work ("pairing of exact eigenstates" [1,5] or diagrammatics [2,6]) assumes that the pairing amplitude D͑r͒ is uniform in space (r independent) even for a highly disordered SC; see, however, [7,8]. Recent work on universal properties at the SC-insulator transition [9] has also ignored amplitude fluctuations, since phase fluctuations are presumably responsible for critical properties.In this paper we consider a simple model of a 2D s-wave superconductor at T 0 in a random potential, defined by Eq. (1) below, and analyze it in detail within a Bogoliubov-de Gennes (BdG) framework [10]. Our goal is to see how the local pairing amplitude D͑r͒ varies spatially in the presence of disorder, and the effect of this inhomogeneity on physically relevant correlation functions. Our results can be summarized as follows.(1) With increasing disorder, the distribution P͑D͒ of the local pairing amplitude D͑r͒ becomes very broad, eventually developing considerable weight near D ഠ 0.(2) The spectral gap in the one-particle density of states persists even at high disorder in spite of the growing number of sites with D͑r͒ ഠ 0. A detailed understanding of this surprising effect emerges from a study of the spatial variation of D͑r͒ and of the BdG eigenfunctions.(3) There is substantial reduction in the superfluid stiffness and off-diagonal correlations with increasing disorder; however, the amplitude fluctuations by themselves cannot destroy the superconductivity.(4) Phase fluctuations about the inhomogeneous BdG state are described by a quantum XY model whose parameters, compressibility and phase stiffness, are obtained from the BdG results. A simple analysis of this effective model within a self-consistent harmonic approximation leads to a transition to a non-SC state.We conclude with some comments on the implications of our results for experiments on disordered films.We model the 2D disordered s-wave SC by an attractive Hu...

Theory of the magnetoresistance of disordered superconducting films

“…This concept has already been used to interpret some experiments [11,13]. It was directly observed in other systems by STM measurements [15] and further corroborated by numerical simulations [16]. The second assumption is that as the magnetic field is increased, the concentration and size of these SCIs decrease.…”

Fulde-Ferrell-Larkin-Ovchinnikov state in disordereds-wave superconductors

“…This is consistent with previous studies. 37,40 On the other hand the order parameter ∆ has a single (positive) sign despite the fluctuations, as a result of which the Fourier transform of ∆ peaks sharply at k = 0; this is a hallmark of the BCS state. When h is turned on, there may be some sign changes in ∆; but as long as ∆ k=0 remains the dominant Fourier component, we can still identify the state as BCS, and a typical example is Fig.…”

“…In fact such spatial fluctuations have been observed previously in strongly disordered superconductors in the absence of Zeeman splitting. 36,37,38,39,40 Thus a more complete understanding of the FFLO state requires a more careful examination of the disorder induced order parameter fluctuations. A related conceptual issue is how to distinguish the FFLO state from other competing states (including the Bardeen-Cooper-Schrieffer, or BCS state) in the presence of impurities, where momentum is no longer a good quantum number.…”

Disorder and superconductivity: a new phase of bi-particle localized states

“…The case of stronger disorder and non-uniform order parameters can be treated within the Bogoliubov-de Gennes approach [7]. Such studies have been carried out by several groups [8,9] and the study for s-wave superconductors shows the persistence of a spectral gap with relatively strong disorder [9]. In the spirit of Anderson's theorem, it is expected that the superconducting phase penetrates the localized non-interacting phase until the disorder strength is sufficiently large to overcome the superconducting gap.…”

“…The eigenvectors and the quasiparticle excitation energies were obtained self-consistently, with good convergence. In addition, we obtained the BCS value for the order-parameter ∆ ≈ 1.36t in the limit of W/t = 0, at L = 12 and U/t = −4, as in [9]. In the limit U = 0, this method reproduces δρ i for finite disorder strengths W/t.…”