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arXiv:1711.11540 · Physical review. B./Physical review. B

Quasiparticle density of states, localization, and distributed disorder in the cuprate superconductors

Miguel Antonio Sulangi, Jan Zaanen

Published Apr 16, 2018• 9 citations
arXivHTMLPDF

Abstract

We explore the effects of various kinds of random disorder on the quasiparticle density of states of two-dimensional $d$-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.

DOI
10.1103/physrevb.97.144512

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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.
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.
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The quantum interference effect on the quasiparticle density of states (DOS) is studied with the diagrammatic technique in two-dimensional d-wave superconductors with dilute nonmagnetic impurities both near the Born and near the unitary limits. We derive in details the expressions of the Goldstone modes (cooperon and diffuson) for quasiparticle diffusion. The DOS for generic Fermi surfaces is shown to be subject to a quantum interference correction of logarithmic suppression, but with various renormalization factors for the Born and unitary limits. Upon approaching the combined limit of unitarity and nested Fermi surface, the DOS correction is found to become a δδδ-function of the energy, which can be used to account for the resonant peak found by the numerical studies.
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
The long disputed issue of the limiting value of quasiparticle density of states $\r(0) = \r (\e \to 0)$ in a \emph{d-}wave superconductor with impurities (\emph{vs} its linear vanishing, $\r_0(\e) \propto |\e|/\D$, near the nodal point $\e = 0$ in a pure system with the gap parameter $\D$) is discussed. Using the technique of group expansions of Green functions in complexes of interacting impurities, it is shown that finite (˚0)\r(0)(˚​0) value is possible if the (finite) impurity perturbation VVV is spin-dependent (magnetic). The found value has a power law dependence on the impurity concentration ccc: $\r(0) \propto \r_N c^{n}$, where $\r_N$ is the normal metal density of states and n≥2n \geq 2n≥2 is the least number of impurities in a complex that can localize nodal quasiparticle. This result essentially differs from the known predictions of self-consistent approximation: $\r(0) \propto \r_N \sqrt {c/\r_N\D}$ (for the unitary limit V→∞V \to \inftyV→∞) or $\r(0) \propto (\D/cV^2)\exp(-\D/cV^2\r_N)$ (for the Born limit $|V|\r_N \ll 1$). We predict also existence of a narrow interval (mobility gap) around the Fermi energy, where all the states are localized on proper impurity clusters, leading to exponential suppression of low-temperature kinetics.
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.