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arXiv:cond-mat/0612364 · Journal of Statistical Mechanics Theory and Experiment

The role of the impurity-potential range in disordered d-wave superconductors

Luca Dell’Anna, Michele Fabrizio, Claudio Castellani

Published Feb 19, 2007• 2 citations
arXivHTMLPDF

Abstract

We analyse how the range of disorder affects the localization properties of quasiparticles in a two-dimensional d-wave superconductor within the standard non-linear σ-model approach to disordered systems. We show that for purely long range disorder, which only induces intra-node scattering processes, the approach is free from the ambiguities which often beset the disordered Dirac-fermion theories, and gives rise to a Wess–Zumino–Novikov–Witten action leading to vanishing density of states and finite conductivities. We also study the crossover induced by inter-node scattering due to a short range component of the disorder, thus providing a coherent non-linear σ-model description in agreement with all the various findings of different approaches.

DOI
10.1088/1742-5468/2007/02/p02014

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We analyse how the range of disorder affects the localization properties of quasiparticles in a two-dimensional d-wave superconductor within the standard non-linear σ-model approach to disordered systems. We show that for purely long range disorder, which only induces intra-node scattering processes, the approach is free from the ambiguities which often beset the disordered Dirac-fermion theories, and gives rise to a Wess–Zumino–Novikov–Witten action leading to vanishing density of states and finite conductivities. We also study the crossover induced by inter-node scattering due to a short range component of the disorder, thus providing a coherent non-linear σ-model description in agreement with all the various findings of different approaches.
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.
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.
A lattice model for disordered d-wave superconductors with spin-rotational and time-reversal symmetries is reconsidered. Near the band center, the lattice model can be described by Dirac fermions with several species. Each fermion yields a Wess-Zumino-Witten (WZW) term for an effective action of the Goldstone mode, which is known to play a crucial role in critical phenomena in two dimensions. The WZW terms cancel out each other because of the fourfold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining a WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.
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.
Disorder in electronic systems suppresses diffusion and can lead to localization. The origin of this behavior is enhanced backscattering due to quantum interference. In addition, disorder leads to a broadening of the density of states (DoS). However, the localization properties of a disordered system are usually not discernible from a study of the DoS. It is necessary to calculate two-particle properties or consider the influence of boundary conditions on energy eigenvalues as in the study of Thouless numbers to obtain information about localization. Another feature of disordered systems is the independence of their localization properties from details of the disorder in the system. For example, the range of the disorder affects non-universal quantities like the mean-free path but does not influence whether or not the system shows localization at all. Recently, new symmetry classes came into focus that do not conform to the standard expectations about disordered systems outlined above. As we will show in this contribution, details of disorder do matter for these systems and localization properties leave their mark on the DoS. In particular, we look at a system that presents an idealization of a disordered d-wave superconductor. In our study we neglect effects of self-consistency and concentrate on the d-wave symmetry and influence of different kinds of disorder. We consider the lattice quasi-particle Hamiltonian
In this paper we present a method to compute the exact density of states induced by NNN non magnetic impurities in a system of two dimensional Dirac fermions in the unitarity limit. We review the case of the πππ-flux phase of the Heisenberg model and also treat the disordered d-wave superconductor. In both case we find additional states in the gap with δρ(ω)≃ni/∣ω(ln⁡2∣ω/Δ0∣+(π/2)2)∣δρ(ω) \simeq n_i/ |ω(\ln^2 |ω/Δ_0| + (π/2)^2)|δρ(ω)≃ni​/∣ω(ln2∣ω/Δ0​∣+(π/2)2)∣.