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arXiv:cond-mat/0301630 · Physical review. B, Condensed matter

Quantum interference in nestedd-wave superconductors: A real-space perspective

W. A. Atkinson, P. J. Hirschfeld, Lingyin Zhu

Published Aug 4, 2003• 30 citations
arXivHTMLPDF

Abstract

We study the local density of states around potential scatterers in d-wave superconductors, and show that quantum interference between impurity states is not negligible for experimentally relevant impurity concentrations. The two-impurity model is used as a paradigm to understand these effects analytically and in interpreting numerical solutions of the Bogoliubov--de Gennes equations on fully disordered systems. We focus primarily on the globally particle-hole symmetric model which has been the subject of considerable controversy, and give evidence that a zero-energy delta function exists in the density of states. The anomalous spectral weight at zero energy is seen to arise from resonant impurity states belonging to a particular sublattice, exactly as in the two-impurity version of this model. We discuss the implications of these findings for realistic models of the cuprates. In particular, we show how apparently isolated impurity states can be observed in STM experiments, despite long-range interference, due to averaging over a finite energy range within the impurity band.

DOI
10.1103/physrevb.68.054501

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We study the local density of states around potential scatterers in d-wave superconductors, and show that quantum interference between impurity states is not negligible for experimentally relevant impurity concentrations. The two-impurity model is used as a paradigm to understand these effects analytically and in interpreting numerical solutions of the Bogoliubov--de Gennes equations on fully disordered systems. We focus primarily on the globally particle-hole symmetric model which has been the subject of considerable controversy, and give evidence that a zero-energy delta function exists in the density of states. The anomalous spectral weight at zero energy is seen to arise from resonant impurity states belonging to a particular sublattice, exactly as in the two-impurity version of this model. We discuss the implications of these findings for realistic models of the cuprates. In particular, we show how apparently isolated impurity states can be observed in STM experiments, despite long-range interference, due to averaging over a finite energy range within the impurity band.
We revisit the interpretation of quasiparticle scattering interference in cuprate high-Tc{T}_{c}Tc​ superconductors. This phenomenon has been very successful in reconstructing the dispersions of ddd-wave Bogoliubov excitations, but the successful identification and interpretation of quasiparticle interference (QPI) in scanning tunneling spectroscopy (STS) experiments rely on theoretical results obtained for the case of isolated impurities. We introduce a highly flexible technique to simulate STS measurements by computing the local density of states using real-space Green's functions defined on two-dimensional lattices with as many as 100 000 sites. We focus on the following question: to what extent can the experimental results be reproduced when various forms of distributed disorder are present? We consider randomly distributed pointlike impurities, smooth ``Coulombic'' disorder, and disorder arising from random on-site energies and superconducting gaps. We find an apparent paradox: the QPI peaks in the Fourier-transformed local density of states appear to be sharper and better defined in experiment than those seen in our simulations. We arrive at a no-go result for smooth-potential disorder since this does not reproduce the QPI peaks associated with large-momentum scattering. An ensemble of pointlike impurities gets closest to experiment, but this goes hand in hand with impurity cores that are not seen in experiment. We also study the effects of possible measurement artifacts (the ``fork mechanism''), which turn out to be of relatively minor consequence. It appears that a more microscopic model of the tunneling process needs to be incorporated in order to account for the sharpness of the experimentally obtained QPI peaks.
The quasiparticle resonant states around a single nonmagnetic impurity with unitary scattering in a d-wave superconductor is studied by solving the Bogoliubov--de Gennes equations based on a $t\ensuremath{-}J$ model. Both the spatial variation of the order parameter and the local density of states (LDOS) around the impurity have been investigated. We find that (i) a particle-hole symmetric system has a single symmetric zero-energy peak in the LDOS regardless of the size of the superconducting coherence length ${\ensuremath{\xi}}_{0};$ (ii) for the particle-hole asymmetric case, an asymmetric splitting of the zero-energy peak is intrinsic to a system with a small value of ${k}_{F}{\ensuremath{\xi}}_{0}.$
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.
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 discuss the dynamics of magnetic moments in d-wave superconductors. In particular, we focus on moments induced by doping nonmagnetic impurities into cuprates. The interaction of such moments with the Bogoliubov quasiparticles of the superconductors can be described by variants of the pseudogap Kondo model, characterized by a power-law density of states at the Fermi level. The numerical renormalization-group technique is employed to investigate this Kondo problem for realistic band structures and particle-hole asymmetries, both at zero and finite temperatures. In particular, we study the boundary quantum phase transition between the local-moment and asymmetric strong-coupling phases, and argue that this transition has been observed in recent nuclear magnetic resonance experiments. We determine the spectral properties of both phases and the location of the critical point as a function of Kondo coupling and doping, and discuss quantum-critical crossovers near this phase transition. In addition, changes in the local density of states around the impurity are calculated as functions of temperature, being relevant to scanning tunneling microscopy experiments.
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.