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arXiv:cond-mat/0201339 · Physical review. B, Condensed matter

Density of states in d -wave superconductors disordered by extended impurities

İnanç Adagideli, Daniel E. Sheehy, Paul M. Goldbart

Published Oct 31, 2002• 11 citations
arXivHTMLPDF

Abstract

The low-energy quasiparticle states of a disordered d-wave superconductor are investigated theoretically. A class of such states, formed via tunneling between the Andreev bound states that are localized around extended impurities (and result from scattering between pair-potential lobes that differ in sign), is identified. Its (divergent) contribution to the total density of states is determined by taking advantage of connections with certain one-dimensional random tight-binding models. The states under discussion should be distinguished from those associated with nodes in the pair potential.

DOI
10.1103/physrevb.66.140512

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The low-energy quasiparticle states of a disordered d-wave superconductor are investigated theoretically. A class of such states, formed via tunneling between the Andreev bound states that are localized around extended impurities (and result from scattering between pair-potential lobes that differ in sign), is identified. Its (divergent) contribution to the total density of states is determined by taking advantage of connections with certain one-dimensional random tight-binding models. The states under discussion should be distinguished from those associated with nodes in the pair potential.
The absence of localization of impurity-induced low-energy quasiparticle states in a 2D d-wave superconductor is shown for any amount of disorder in the limit of unitary scatterers. This result follows from the fact that a unitary impurity produces a marginally bound state at zero energy which decays as a power law along the nodes of the d-wave energy gap. Consequently, for finite density of impurities, the impurity-induced states are coupled by long-range overlaps yielding extended quasiparticle states below a characteristic energy scale ${\ensuremath{\omega}}_{c}$. Simple scaling arguments suggest that ${\ensuremath{\omega}}_{c}\ensuremath{\propto}{e}^{{\ensuremath{-}c/n}_{\mathrm{imp}}}$, with nimp{n}_{\mathrm{imp}}nimp​ the impurity density and c a positive constant.
We explore the effects of various kinds of random disorder on the quasiparticle density of states of two-dimensional ddd-wave superconductors using an exact real-space method, incorporating realistic details known about the cuprates. Random on-site energy and pointlike unitary impurity models are found to give rise to a vanishing DOS at the Fermi energy for narrow distributions and low concentrations, respectively, and lead to a finite, but suppressed, DOS at unrealistically large levels of disorder. Smooth disorder arising from impurities located away from the copper-oxide planes meanwhile gives rise to a finite DOS at realistic impurity concentrations. For the case of smooth disorder whose average potential is zero, a resonance is found at zero energy for the quasiparticle DOS at large impurity concentrations. We discuss the implications of these results on the computed low-temperature specific heat, the behavior of which we find is strongly affected by the amount of disorder present in the system. We also compute the localization length as a function of disorder strength for various types of disorder and find that intermediate- and high-energy states are quasiextended for low disorder, and that states near the Fermi energy are strongly localized and have a localization length that exhibits an unusual dependence on the amount of disorder. We comment on the origin of disorder in the cuprates and provide constraints on these based on known results from scanning tunneling spectroscopy and specific heat experiments.
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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.
We consider a disordered ddd-wave superconductor in two dimensions. Recently, we have shown in an exact calculation that for a lattice model with a Lorentzian distributed random chemical potential the quasiparticle density of states at the Fermi level is nonzero. As the exact result holds only for the special choice of the Lorentzian, we employ different methods to show that for a large class of distributions, including the Gaussian distribution, one can establish a nonzero lower bound for the Fermi-level density of states. The fact that the tails of the distributions are unimportant in deriving the lower bound shows that the exact result obtained before is generic.
Effects of nonmagnetic impurities with unitarity scattering in two-dimensional d -wave superconductors are investigated by the numerical diagonalization of finite-size systems of up to 35×35 sites. It is shown that a localized state with almost zero excitation energy is formed around an impurity site if it is isolated. However, its localized state is extended in real space along the direction of the node of the d -wave gap, so that the localized states of adjacent impurities easily interfere with each other forming a kind of impurity band, i.e., resonance states around the zero energy. It is also pointed out that there exists a strong association between the resonance density of states due to impurity scattering in the unitarity limit and the so-called zero-bias anomaly of tunneling conductance.