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arXiv:cond-mat/9909363 · Physical review. B, Condensed matter

Quasiparticle resonant states induced by a unitary impurity in ad-wave superconductor

Jian‐Xin Zhu, T. K. Lee, C. S. Ting, Chia-Ren Hu

Published Apr 1, 2000• 35 citations
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Abstract

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 ${k}_{F}{\ensuremath{\xi}}_{0}.$

DOI
10.1103/physrevb.61.8667

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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−Jt\ensuremath{-}Jt\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 \ensuremathξ0;{\ensuremath{\xi}}_{0};\ensuremathξ0​; (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 kF\ensuremathξ0.{k}_{F}{\ensuremath{\xi}}_{0}.kF​\ensuremathξ0​.
The superconducting state around a well isolated nonmagnetic impurity in the high-Tc superconductors is studied using the two-dimensional t-J model. The spatial dependence of the order parameter and the local density of states are obtained from the numerical diagonalization of the Bogoliubov-de Gennes equation derived using the Gutzwiller approximation. We find a zero-energy peak in the local density of states around the impurity. Different from the previous results on a vortex or surfaces in the t-J model, the splitting of zero-energy peak is not found. The zero-energy states corresponding to the zero-energy peak is approximately localized around the impurity.
Within numerically exact solutions of the Bogoliubov-de Gennes equations, we demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal state particle-hole symmetry. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-state suppression below an energy scale which appears to be correlated with the corresponding single impurity resonance.
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.
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.
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.
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.