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arXiv:cond-mat/0108171 · Physical review. B, Condensed matter

Kondo effect of impurity moments in d -wave superconductors: Quantum phase transition and spectral properties

Matthias Vojta C, R. Bulla

Published Dec 4, 2001• 76 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.65.014511

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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.
We consider the screening of an impurity moment in a d-wave superconductor under the influence of a Zeeman magnetic field. Using the numerical renormalization group technique, we investigate the resulting pseudogap Kondo problem, in particular the field-induced crossover behavior in the vicinity of the zero-field boundary quantum phase transition. The impurity spectral function and the resulting changes in the local host density of states are calculated, giving specific predictions for high-field scanning tunneling microscopy measurements on impurity-doped cuprates.
We study a pseudogap Anderson-Holstein model of a magnetic impurity level that hybridizes with a conduction band whose density of states vanishes in power-law fashion at the Fermi energy, and couples, via its charge, to a nondispersive bosonic mode (e.g., an optical phonon). The model, which we treat using poor-man's scaling and the numerical renormalization group, exhibits quantum phase transitions of different types depending on the strength of the impurity-boson coupling. For weak impurity-boson coupling, the suppression of the density of states near the Fermi energy leads to quantum phase transitions between strong-coupling (Kondo) and local-moment phases. For sufficiently strong impurity-boson coupling, however, the bare repulsion between a pair of electrons in the impurity level becomes an effective attraction, leading to quantum phase transitions between strong-coupling (charge Kondo) and local-charge phases. Even though the Hamiltonian exhibits different symmetries in the spin and charge sectors, the thermodynamic properties near the two types of quantum phase transition are closely related under spin-charge interchange. Moreover, the critical responses to a local magnetic field (for small impurity-boson coupling) and to an electric potential (for large impurity-boson coupling) are characterized by the same exponents, whose values place these quantum-critical points in the universality class of the pseudogap Anderson model. One specific case of the pseudogap Anderson-Holstein model may be realized in a double-quantum-dot device, where the quantum phase transitions manifest themselves in the finite-temperature linear electrical conductance.
The variational approach of Gunnarsson and Sch\"onhammer to the Anderson impurity model is generalized to study d-wave superconductors in the presence of dilute spin-12\frac{1}{2}21​ impurities. We show that the local moment is screened when the hybridization exceeds a nonzero critical value at which the ground state changes from a spin doublet to a spin singlet. The electron spectral functions are calculated in both phases. We find that while a Kondo resonance develops above the Fermi level in the singlet phase, the spectral function exhibits a low-energy spectral peak below the Fermi level in the spin-doublet phase. The origin of such a ``virtual Kondo resonance'' is the existence of low-lying collective excitations in the spin-singlet sector. We discuss our results in connection with recent spectroscopic experiments in Zn-doped high-Tc{T}_{c}Tc​ superconductors.
We discuss a multichannel SU(N) Kondo model which displays nontrivial zero-temperature phase transitions due to a conduction electron density of states vanishing with a power law at the Fermi level. In a particular large- N limit, the system is described by coupled integral equations corresponding to a dynamic saddle point. We exactly determine the universal low-energy behavior of spectral densities at the scale-invariant fixed points, obtain anomalous exponents, and compute scaling functions describing the crossover near the quantum-critical points. We argue that our findings are relevant to recent experiments on impurity-doped d-wave superconductors.
We investigate the impurity resonant state induced by nonmagnetic impurities in ddd-wave cuprate superconductors in two different impurity models: (i) in a pure potential scattering model within the TTT-matrix approach and (ii) in a Bose--Fermi Kondo model within the large-N\mathcal{N}N formalism. We modify the superconducting host to resemble the nanoscale electronic inhomogeneities seen in the scanning tunneling microscopy (STM) spectra (i.e., the small-and large-gap domains) and study how this influences the impurity resonance. In the pure potential scattering model, the resonant state is found to be a rather robust feature appearing in all host domains. On the other hand, in the Kondo-like model the impurity resonance can appear in the small-gap regions of the host and be completely suppressed in the large-gap regions; this is in agreement with the STM experimental data for Zn-doped ${\text{Bi}}_{2}{\text{Sr}}_{2}{\text{CaCu}}_{2}{\text{O}}_{8+\ensuremath{\delta}}$. Our results imply that the Kondo spin dynamics of the impurity moment is the origin of the impurity resonant state in ddd-wave superconductors, rather than the pure potential scattering.
We study the competition between Kondo physics and dissipation within an Anderson model of a magnetic impurity level that hybridizes with a metallic host and is also coupled, via the impurity charge, to the displacement of a bosonic bath having a spectral density proportional to ${\ensuremath{\omega}}^{s}$. As the impurity-bath coupling increases from zero, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional spin-sector Kondo effect to a charge Kondo effect. At particle-hole symmetry, and for sub-Ohmic bath exponents 0<s<10<s<10<s<1, further increase in the impurity-bath coupling results in a continuous zero-temperature transition to a broken-symmetry phase in which the ground-state impurity occupancy ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}$ acquires an expectation value ${⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}⟩}_{0}\ensuremath{\ne}1$. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and $\ensuremath{\omega}∕T$ scaling that are expected at an interacting critical point. The numerical values of the critical exponents suggest that the transition lies in the same universality class as that of the sub-Ohmic spin-boson model. For the Ohmic case s=1s=1s=1, the transition is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.