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arXiv:cond-mat/0605424

Impurity clusters and localization of nodal quasiparticles in \emph{d}-wave superconductors

Yu. G. Pogorelov, M. C. Santos, В. М. Локтев

Published May 16, 2006• 0 citations
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Abstract

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 $\r(0)$ value is possible if the (finite) impurity perturbation $V$ is spin-dependent (magnetic). The found value has a power law dependence on the impurity concentration $c$: $\r(0) \propto \r_N c^{n}$, where $\r_N$ is the normal metal density of states and $n \geq 2$ 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 $V \to \infty$) 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.

DOI
10.48550/arxiv.cond-mat/0605424

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The long disputed issue of the limiting value of quasiparticle density of states (˚0)=(˚\e→0)\r(0) = \r (\e \to 0)(˚​0)=(˚​\e→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\e = 0\e=0 in a pure system with the gap parameter \D\D\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.
What are zero energy excitations in high-TcT_{c}Tc​ superconductors?
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.
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A comparative analysis has been done of the formerly established two self-consistent solutions for the density of quasiparticle states in doped d-wave superconductors, having strikingly different and disputed behavior near the Fermi energy. One of them (1) remains finite in this limit, while the other (2) tends to zero. To resolve this discrepancy, the known Ioffe-Regel criterion for band states, widely used for doped semiconductors, was applied to these solutions. It is shown that both them are valid in limited and different energy regions, where the corresponding quasiparticles are weakly damped. In particular, density of states of nodal quasiparticles near the Fermi level is provided by the (2) solution, while the (1) only applies far enough from this level.
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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 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.