In a recent letter, Chatterjee et al. (Nature Phys. 4, 108 (2008)) reported tunnelling data obtained below and above the critical temperature (Tc = 15 K) by STM in overdoped Bi2-yPbySr2CuO6+x (Bi2201) and some interpretations of the data1. The main experimental findings of their work are observations of native-impurity resonances which coexist spatially with the superconducting (SC) gap below Tc and survive unchanged on warming through Tc. No doubts, the data in the letter are an important piece of information for assembling the high-Tc SC jigsaw puzzle; however, there are inaccuracies in the discussion. In addition, I propose another interpretation of the data.
We predict a resonance impurity state generated by the substitution of one Cu atom with a nonmagnetic atom, such as Zn, in the pseudogap state of a high-T_c superconductor. The precise microscopic origin of the pseudogap is not important for this state to be formed, in particular this resonance will be present even in the absence of superconducting fluctuations in the normal state. In the presence of superconducting fluctuations, we predict the existence of a counterpart impurity peak on a symmetric bias. The nature of impurity resonance is similar to the previously studied resonance in the d-…
We analyze the anomalies of superconducting state (both s- and d-wave pairing) in the model of the pseudogap state induced by Heisenberg spin fluctuations of antiferromagnetic short-range order, and based on the scenario of strong scattering near the "hot spots" on the Fermi surface. We present microscopic derivation of Ginzburg-Landau expansion, taking into account all Feynman graphs of perturbation theory for electron interaction with fluctuations of short-range order and in the "ladder" approximation for electron scattering by normal (nonmagnetic) impurities. We determine the dependence of …
Magnetic impurities coupled to both fermionic and bosonic baths or to a fermionic bath with pseudogap density of states, described by the Fermi-Bose Kondo and pseudogap Kondo models, display non-trivial intermediate coupling fixed points associated with critical local-moment fluctuations and local non-Fermi liquid behavior. Based on renormalization group together with a renormalized perturbation expansion around the free-impurity limit, we calculate various impurity properties in the vicinity of those intermediate-coupling fixed points. In particular, we compute the conduction electron T matri…
We derive the polarizability of an electron system in (i) the superconducting phase, with d-wave symmetry, (ii) the pseudogap regime, within the precursor pairing scenario, and (iii) the d-density-wave (dDW) state, characterized by a d-wave hidden order parameter, but no pairing. Such a calculation is motivated by the recent proposals that imaging the effects of an isolated impurity may distinguish between precursor pairing and dDW order in the pseudogap regime of the high-Tc superconductors. In all three cases, the wave-vector dependence of the polarizability is characterized by an azymuthal …
Irrespective of the class they belong to, all the hole doped high-Tc cuprate superconductors show an anti-correlation between the superconducting transition temperature and the characteristic pseudogap energy in the underdoped region. The doping dependent pseudogap in the quasiparticle spectral density is believed to remove low-energy electronic states and thereby reduce the superconducting condensate. Impurities within the CuO2 plane, on the other hand, break Cooper pairs in the unitarity limit and diminish superfluid density. Both pseudogap in pure cuprates and impurity scattering in disorde…
The response to a local strong non-magnetic impurity in the pseudogap phase is examined in two distinctly different scenarios: phase-fluctuation (PF) of pairing field and d-density-wave (DDW) order. In the PF scenario, the resonance state is generally double-peaked near the Fermi level, and is abruptly broadened by vortex fluctuations slightly above the transition temperature. In the DDW scenario, the resonance is single-peaked and remains sharp up to gradual intrinsic thermal broadening, and the resonance energy is analytically determined to be at minus of the chemical potential.
Impurity substitution of Zn in La-214 and (Y,Ca)-123 high-T_c superconductors suppresses T_c but does not affect appreciably the onset of the pseudogap phase in the underdoped region nor optimal doping or the position of the inferred quantum critical point. Based on a 1/N expansion of the t-J model we explain these findings as well as the similar dependence on a magnetic field in terms of a quantum critical point scenario where a flux phase causes the pseudogap.
The quantum interference between two nonmagnetic impurities is studied numerically in both the d-wave superconducting (DSC) and the d-density wave (DDW) state. In all calculations we include the tunnelling through excited states from the CuO$_2$ planes to the BiO layer probed by the STM tip. Compared to the single impurity case, a systematic study of the modulations in the two-impurity local density of states can distinguish between the DSC or DDW states. This is important if the origin of the pseudogap phase is caused by preformed pairs or DDW order. Furthermore, in the DSC state the study of…
Suppression of T_c in bilayer bismuth and thallium cuprates, by substitution of Co impurities at Cu sites, are taken for examination. T_c suppression data on differently doped Bi2212 and Tl2212 are analysed within the unitary pair-breaking formalism due to Abrikosov and Gorkov, by fitting data points to a phenomenological relation valid for weak coupling d-wave superconductors. Values of the pseudogap magnitude at each doping are thereby estimated within a "fermi-level density of states suppression" picture. Pseudogap magnitude from our estimation is observed to have a correspondence with a re…
The pseudogap phenomena in the cuprate superconductors requires a theory beyond the mean field BCS level. A natural candidate is to include strong pairing fluctuations, and treat the two-particle and single particle Green's functions self-consistently. At the same time, impurities are present in even the cleanest samples of the cuprates. Some impurity effects can help reveal whether the pseudogap has a superconducting origin and thus test various theories. Here we extend the pairing fluctuation theory for a clean system [Phys. Rev. Lett. 81, 4708 (1998)] to the case with nonmagnetic impurities…
The influence of magnetic Ni and non-magnetic Zn impurities on the normal state pseudogap (PG) in the c-axis optical conductivity of NdBa$_{2}$\{Cu$% _{1-y}$(Ni,Zn)$_{y}\}_{3}$O$_{7-δ}$ crystals was studied by spectral ellipsometry. We find that these impurities which strongly suppress superconductivity have a profoundly different impact on the PG. Zn gives rise to a gradual and inhomogeneous PG suppression while Ni strongly enhances the PG. Our results challenge theories that relate the PG either to precursor superconductivity or to other phases with exotic order parameters, such as flux phas…
We report a new multi-nuclei based NMR method which allows us to image the staggered polarization induced by nonmagnetic Li impurities in underdoped O6.6 and slightly overdoped O7 YBa2Cu3O6+y above T_C. The spatial extension of the polarization xi_imp approximately follows a Curie law, increasing up to six lattice constants at T=80K at O6.6 in the pseudogap regime. Near optimal doping, the staggered magnetization has the same shape, with xi_imp reduced by a factor 2. xi_imp is argued to reveal the intrinsic magnetic correlation length of the pure system. It is found to display a smooth evoluti…
In order to explain how impurities affect the unconventional superconductivity, we study non-magnetic impurity effect on the transition temperature using on-site U Hubbard model within a fluctuation exchange (FLEX) approximation. We find that in appearance, the reduction of Tc roughly coincides with the well-known Abrikosov-Gor'kov formula. This coincidence results from the cancellation between two effects; one is the reduction of attractive force due to randomness, and another is the reduction of the damping rate of quasi-particle arising from electron interaction. As another problem, we also…
We consider a magnetic impurity coupled to both fermionic quasiparticles with a pseudogap density of states and bosonic spin fluctuations. Using renormalization group and large-N calculations we investigate the phase diagram of the resulting Fermi-Bose Kondo model. We show that the Kondo temperature is strongly reduced by low-energy spin fluctuations, and make connections to experiments in cuprate superconductors. Furthermore we derive an exact exponent for the critical behavior of the conduction electron T matrix, and propose our findings to be relevant for certain scenarios of local quantum …
Near a Mott transition, strong electron correlations may enhance Cooper pairing. This is demonstrated in the Dynamical Mean Field Theory solution of a twofold-orbital degenerate Hubbard model with inverted Hund's rules on-site exchange, which favors local spin-singlet configurations. Close to the Mott insulator, which is a local version of a valence bond insulator, a pseudogap non-Fermi-liquid metal, a superconductor, and a normal metal appear, in striking similarity with the physics of the cuprates. The strongly correlated superconducting state has a larger Drude weight than the corresponding…
We present a short review of our basic understanding of the physics of copper - oxide superconductors and formulate the list of "solved" and "unsolved" problems. The main problem remains theoretical description of the properties of the normal state, requiring clarification of the nature of the so called pseudogap state. We review simplified models of the pseudogap state, based on the scenario of strong electron scattering by (pseudogap) fluctuations of "dielectric" (AFM, CDW) short - range order and the concept of "hot" spots (patches) on the Fermi surface. Pseudogap fluctuations are described…
An intricate interplay between superconductivity, pseudogap and Mott transition, either bandwidth driven or doping driven, occurs in materials. Layered organic conductors and cuprates offer two prime examples. We provide a unified perspective of this interplay in the two-dimensional Hubbard model within cellular dynamical mean-field theory on a $2\times 2$ plaquette and using the continuous-time quantum Monte Carlo method as impurity solver. Both at half filling and at finite doping, the metallic normal state close to the Mott insulator is unstable to d-wave superconductivity. Superconductivit…
We use scanning tunneling microscopy (STM) to study magnetic Fe impurities intentionally doped into the high-temperature superconductor Bi$_{2}$Sr$_{2}$Ca$_{2}$CuO$_{8+δ}$. Our spectroscopic measurements reveal that Fe impurities introduce low-lying resonances in the density of states at Ω$_{1}$ $\approx$ 4meV and Ω$_{2}$ $\approx$ 15 meV allowing us to determine that, despite having a large magnetic moment, potential scattering of quasiparticles by Fe impurities dominates magnetic scattering. In addition, using high-resolution spatial characterizations of the local density of states near and …
We have calculated the optical conductivity and the Raman response for the d-density wave model, proposed as a possible explanation for the pseudogap seen in high Tc cuprates. The total optical spectral weight remains approximately constant on opening of the pseudogap for fixed temperature. This occurs because there is a transfer of weight from the Drude peak to interband transitions across the pseudogap. The interband peak in the optical conductivity is prominent but becomes progressively reduced with increasing temperature, with impurity scattering, which distributes it over a larger energy …
We report electronic Raman scattering measurements on optimally doped YBa2Cu3O7 where Zn or Ni impurities have been substituted by Cu. Using Raman selection rules, we have probed the superconducting gap in the nodal and antinodal regions. We show that under impurity substitutions, the energy of the antinodal peak detected in the superconducting state is not related to the critical temperature Tc and that signatures of superconductivity disappear in the nodal regions. Our experimental findings advocate in favor of gapless arcs around the nodes. The breakdown of the relationship between the anti…
Underdoped cuprates exhibit a normal-state pseudogap, and their spins and doped carriers tend to spatially separate into 1- or 2-D stripes. Some view these as central to superconductivity, others as peripheral and merely competing. Using La$_{2-x}$Sr$_x$Cu$_{1-y}$Zn$_y$O$_4$ we show that an oxygen isotope effect in $T_c$ and in the superfluid density can be used to distinguish between the roles of stripes and pseudogap and also to detect the presence of impurity scattering. We conclude that stripes and pseudogap are distinct, and both compete and coexist with superconductivity.
Superconductivity emerges from the cuprate antiferromagnetic Mott state with hole doping. The resulting electronic structure is not understood, although changes in the state of oxygen atoms appear paramount. Hole doping first destroys the Mott state yielding a weak insulator where electrons localize only at low temperatures without a full energy gap. At higher doping, the 'pseudogap', a weakly conducting state with an anisotropic energy gap and intra-unit-cell breaking of 90\degree-rotational (C4v) symmetry appears. However, a direct visualization of the emergence of these phenomena with incre…
A recent discovery of the pair density wave (PDW) order creates a stir in understanding an underlying physics in the pseudogap states of the cuprate high temperature superconductors. We have performed a Spectroscopic Imaging Scanning Tunneling Microscopy (SI-STM) measurements on Bi$_2$Sr$_2$CaCu$_2$O$_8$+$δ$ and identified that spatially uniform d-wave superconductivity (DSC), d-symmetry charge density wave (CDW), and the PDW play important roles in an intertwined fashion. In this review, we show how these different electronic degrees of freedom emerges in electronic structures of the cuprate …
High-temperature superconductivity emerges in the CuO$_2$ plane upon doping a Mott insulator. To ascertain the influence of Mott physics plus short-range correlations, we solve a three-band copper-oxide model in the charge-transfer regime using cellular dynamical mean-field theory with continuous-time quantum Monte Carlo as an impurity solver. We report the normal and superconducting phase diagram of this model as a function of doping, interaction strength and temperature. Upon hole doping of the charge-transfer insulator, the phase boundary between pseudogap and correlated metal consists of a…
The most intriguing observation of cuprate experiments is most likely the metal-insulator-crossover (MIC), seen in the underdome region of the temperature-doping phase diagram of copper-oxides under a strong magnetic field, when the superconductivity is suppressed. This MIC, which results in such phenomena as heat conductivity downturn, anomalous Lorentz ratio, nonlinear entropy, insulating ground state, nematicity- and stripe-phases and Fermi pockets, reveals the nonconventional dielectric property of the pseudogap-normal phase. Since conventional superconductivity appears from a conducting n…
We present muon spin relaxation (muSR) measurements on a large YBa2Cu3O6.6 single crystal in which two kinds of unusual magnetic order have been detected in the pseudogap region by neutron scattering. A comparison is made to measurements on smaller, higher quality YBa2Cu3Oy single crystals. One type of magnetic order is observed in all samples, but does not evolve significantly with hole doping. A second type of unusual magnetic order is observed only in the YBa2Cu3O6.6 single crystal. This magnetism has an ordered magnetic moment that is quantitatively consistent with the neutron experiments,…
Layered doped Mott insulators, such as the cuprates, show unusual temperature dependence of the resistivity. Intriguingly, the resistivity perpendicular to the CuO$_2$ planes, $ρ_c(T)$, shows both metallic ($dρ_c/dT > 0$) and semi-conducting ($dρ_c/dT<0$) behavior. We shed light on this puzzle by calculating $ρ_c$ for the two-dimensional Hubbard model within plaquette cellular dynamical mean-field theory and strong-coupling continuous-time quantum Monte Carlo as the impurity solver. The temperature, $T$, and doping, $δ$, dependencies of $ρ_c$ are controlled by the first-order transition betwee…
Recently Chakravarty {\em et al.} proposed an ordered $d$-density wave (DDW) state as an explanation of the pseudogap phase in underdoped high-temperature cuprates. We study the competition between the DDW and superconducting ordering based on an effective mean-field Hamiltonian. We are mainly concerned with the effect of the DDW ordering on the electronic state around a single nonmagnetic impurity. We find that a single subgap resonance peak appears in the local density of state around the impurity. In the unitary limit, the position of this resonance peak is always located at $E_r=-μ$ with r…
We study the evolution with temperature of quasiparticle bound states around non-magnetic impurities in d-wave superconductors. The associated local density of states has a fourfold symmetry which has recently been observed in Zn-doped Bi2212 using scanning tunneling microscopy (STM). From the corresponding Bogoliubov-de Gennes equation we find that with increasing temperature the magnitude of the bound state energy increases and the amplitude of the fourfold contribution in the spinor wave functions decreases. In the pseudogap regime above T_c the fourfold angular dependence of the local tunn…
Impurity doping like Zn atoms in cuprates were systematically studied to provide important information on the pseudogap phase because this process substantially reduces $T_c$ without effect $T^*$. Despite many important results and advances, the normal phase of these superconductors is still subject of a great debate. We show that the observed Zn-doped data can be reproduced by constructing a nanoscale granular superconductor whose resistivity transition is achieved by Josephson coupling, what provides also a simple interpretation to the pseudogap phase.
Within the phase fluctuation picture for the pseudogap state of a high-$T_{c}$ superconductor, we incorporate the phase fluctuations generated by the classical XY model with the Bogoliubov-de Gennes formalism utilizing a field-theoretical method. This picture delineates the inhomogeneous characteristics of local order parameters observed in high-$T_{c}$ superconductors above $T_{c}$. We also compute the local density of states near a non-magnetic impurity with a strong scattering potential. The resonance peak smoothly evolves as temperature increases through $T_{c}$ without showing any sudden …
Extensive research into high temperature superconducting cuprates is now focused upon identifying the relationship between the classic 'pseudogap' phenomenon$^{1,2}$ and the more recently investigated density wave state$^{3-13}$. This state always exhibits wave vector $Q$ parallel to the planar Cu-O-Cu bonds$^{4-13}$ along with a predominantly $d$-symmetry form factor$^{14-17}$ (dFF-DW). Finding its microscopic mechanism has now become a key objective$^{18-30}$ of this field. To accomplish this, one must identify the momentum-space ($k$-space) states contributing to the dFF-DW spectral weight,…
Scanning tunneling spectroscopy (STS) is a useful probe for studying the cuprates in the superconducting and pseudogap states. Here we present a theoretical study of the Z-map, defined as the ratio of the local density of states at positive and negative bias energies, which frequently is used to analyze STS data. We show how the evolution of the quasiparticle interference peaks in the Fourier transform Z-map can be understood by considering different types of impurity scatterers, as well as particle-hole asymmetry in the underlying bandstructure. We also explore the effects of density wave ord…
