We study the behavior of the pseudogap in overdoped Bi$_{2}$Sr$_2$CaCu$_2$O$_{8+d}$ by electronic Raman scattering (ERS) and angle-resolved photoemission spectroscopy (ARPES) on the same single crystals. Using both techniques we find that, unlike the superconducting gap, the pseudogap related to the anti-bonding band vanishes above the critical doping p$_c$ = 0.22. Concomitantly, we show from ARPES measurements that the Fermi surface of the anti-bonding band is hole-like below pc and becomes electron-like above p$_c$. This reveals that the appearance of the pseudogap depends on the Fermi surface topology in Bi$_{2}$Sr$_2$CaCu$_2$O$_{8+d}$ , and more generally, puts strong constraint on theories of the pseudogap phase.
The heating-compensated interlayer tunneling spectroscopy is performed on stacks of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ intrinsic junctions. The high-accuracy spectra without the local heating, for varying temperatures and magnetic fields, reveal that the spectral weight forming the superconducting coherence peak is mainly contributed from the dip position of the hump structure of the pseudogap state. The observed U-shaped subgap structure is consistent with the weighted antinodal tunneling between Cu-O double layers in a junction as suggested in Ref. [17].
We report a fine tuned doping study of strongly overdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystals using electronic Raman scattering. Combined with theoretical calculations, we show that the doping, at which the normal state pseudogap closes, coincides with a Lifshitz quantum phase transition where the active hole-like Fermi surface becomes electron-like. This conclusion suggests that the microscopic cause of the pseudogap is sensitive to the Fermi surface topology. Furthermore, we find that the superconducting transition temperature is unaffected by this transition, demonstrating that th…
We report muon spin relaxation ($μ$SR) measurements of optimally-doped and overdoped Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+δ}$ (Bi2212) single crystals that reveal the presence of a weak temperature-dependent quasi-static internal magnetic field of electronic origin in the superconducting (SC) and pseudogap (PG) phases. In both samples the internal magnetic field persists up to 160~K, but muon diffusion prevents following the evolution of the field to higher temperatures. We consider the evidence from our measurments in support of PG order parameter candidates, namely, electronic loop currents and…
We report heating-compensated interlayer tunneling spectroscopy (ITS) performed on stacks of overdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ intrinsic junctions, where most of bias-induced heating in the ITS was eliminated. The onset temperature of the pseudogap (PG), revealed in the hump structure of the electronic excitation spectra, reached nearly room temperature for our overdoped intrinsic junctions, which represented the genuine PG onset. At a temperature below but close to $T_c$, both the superconducting coherence peak and the pseudogap hump coexisted, implying that the two gaps are of separate…
We report on magnetization experiments from which we obtain the field-induced kinetic energy density, $E_k$, in the superconducting phase of several Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystal samples. The kinetic energy magnitude changes according to the characteristic reduction of the single-electron density of states produced by the pseudogap in the underdoped limit. Moreover, a remarkable peak of $E_k$ occurring at the specific holes density $p\sim0.18$ is related to a van Hove singularity due to the pseudogap closure. We also extracted the superfluid density, $ρ_s$. We conclude that $E_k…
Photoexcited quasiparticle relaxation dynamics in overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ ($T_{c}$=65~K, hole doping $p$=0.22) single crystal is investigated as a function of temperature. We provide evidence of a $\sim$22~meV pseudogap ($T^{\ast}$$\approx$100~K) at this doping level. Our data support the scenario where both the superconducting gap and pseudogap coexist in the superconducting state. Our results also suggest an increased scattering rate between electrons and spin fluctuations as the sample enters the pseudogap phase.
We report atomic scale characterization of the pseudogap state in a high-T$_c$ superconductor, Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$. The electronic states at low energies within the pseudogap exhibit spatial modulations having an energy-independent incommensurate periodicity. These patterns, which are oriented along the copper-oxygen bond directions, appear to be a consequence of an electronic ordering phenomenon--the observation of which correlates with the pseudogap in the density of electronic states. Our results provide a stringent test for various ordering scenarios in the cuprates, which have b…
We report the transverse relaxation rates 1/$T_2$'s of the $^{63}$Cu nuclear spin-echo envelope for double-layer high-$T_c$ cuprate superconductors HgBa$_{2}$CaCu$_{2}$O$_{6+d}$ from underdoped to overdoped. The relaxation rate 1/$T_{2L}$ of the exponential function (Lorentzian component) shows a peak at 220$-$240 K in the underdoped ($T_c$ = 103 K) and the optimally doped ($T_c$ = 127 K) samples but no peak in the overdoped ($T_c$ = 93 K) sample. The enhancement in 1/$T_{2L}$ suggests development of the zero frequency components of local field fluctuations. Ultraslow fluctuations are hidden i…
The tunneling spectra obtained within the ab-plane of Bi$_2$Sr$_2$CaCu$_2$O$_y$ (Bi2212) for temperatures below and above the critical temperature (T$_c$) are analyzed. We find that the tunneling conductance spectra for the underdoped compound in the superconducting state do not follow the conservation of states rule. There is a consistent loss of states for the underdoped BI2212 implying an underlying depression in the density of states (DOS) and hence the pseudogap near the Fermi energy (E$_F$). Such an underlying depression can also explain the peak-dip-hump structure observed in the spectr…
We present an extended zero-field muon spin relaxation (ZF-$μ$SR) study of overdoped Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+δ}$ (Bi2212) single crystals, intended to elucidate the origin of weak quasistatic magnetism previously detected by $μ$SR in the superconducting and normal states of optimally-doped and overdoped samples. New results on heavily-overdoped single crystals show a similar monotonically decreasing ZF-$μ$SR relaxation rate with increasing temperature that persists above the pseudogap (PG) temperature $T^*$ and does not evolve with hole doping ($p$). Additional measurements using an …
Tunneling spectra of near optimally doped, submicron Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ intrinsic Josephson junctions are presented, and examined in the region where the superconducting gap evolves into pseudogap. The spectra are analyzed using a self-energy model, proposed by Norman {\it et al.}, in which both quasiparticle scattering rate $Γ$ and pair decay rate $Γ_Δ$ are considered. The density of states derived from the model has the familiar Dynes' form with a simple replacement of $Γ$ by $γ_+$ = ($Γ$ + $Γ_Δ$)/2. The $γ_+$ parameter obtained from fitting the experimental spectra shows a roughl…
We investigate the electron-boson spectral density function, $I^2χ(ω,T)$, of CuO$_2$ plane in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ (Bi-2212) and underdoped YBa$_2$Cu$_3$O$_{6.50}$ (Y-123) using the Eliashberg formalism. We apply a new (in-plane) pseudogap model to extract the electron-boson spectral function. For extracting the spectral function we assume that the spectral density function consists of two components: a sharp mode and the broad Millis-Monien-Pines (MMP) mode. We observe that both the resulting spectral density function and the intensity of the pseudogap show strong temperat…
To elucidate the pseudogap phase diagram including the overdoped state of high transition temperature (high-$T_c$) cuprates, we must understand the origin of the positive out-of-plane magnetoconductivity (MC) observed in these compounds. For this purpose, the out-of-plane resistivity $ρ_c(T,H)$ of an overdoped Bi$_{1.6}$Pb$_{0.4}$Sr$_{2}$CaCu$_{1.96}$Fe$_{0.04}$O$_{8+δ}$ (Bi-2212) single crystal is measured under pulsed magnetic fields up to 60 T. We show that the superconductive density-of-states (DOS) depletion effect, in addition to the pseudogap effect, clearly appears below the supercondu…
In the high-temperature ($T_{c}$) cuprate superconductors, increasing evidence suggests that the pseudogap, existing below the pseudogap temperature $T$*, has a distinct broken electronic symmetry from that of superconductivity. Particularly, recent scattering experiments on the underdoped cuprates have suggested that a charge ordering competes with superconductivity. However, no direct link of this physics and the important low-energy excitations has been identified. Here we report an antagonistic singularity at $T_{c}$ in the spectral weight of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ as a compel…
We have found a quantitative connection between the evolution of the inhomogeneous nanoscale electronic gaps (INSEG) state detected in Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ by scanning tunneling microscopy/spectroscopy (STM/S) and the two universal, the upper and the lower, pseudogaps in high-temperature cuprate superconductors (HTCS). When the doping and temperature dependent INSEG map were analyzed by using our proposed hole-scale, we find that the two pseudogaps are connected to two specific coverages of the CuO$_2$ plane by INSEG: the 50\% and 100\% coverages of the CuO$_2$ planes by INSEG corresp…
To investigate the relationship between the pseudogap and superconductivity, we measured both the in-plane ($ρ_{ab}$) and out-of-plane ($ρ_c$) resistivity for oxygen-controlled Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ single crystals subject to magnetic fields (parallel to the $c$ axis) of up to 17.5 T. The onset temperature for the superconductive fluctuation, $T_{scf}$, is determined by the large positive in-plane magnetoresistance (MR) and negative out-of-plane MR observed near $T_c$, whereas the pseudogap opening temperature $T^*$ is determined by the semiconductive upturn of the zero-field $ρ_…
Although phase diagrams can be leveraged to investigate high transition temperature (high-$T_c$) superconductivity, the issue has not been discussed thoroughly. In this study, we elucidate the phase diagram of the overdoped side of high-$T_c$ cuprates via systematic anisotropic transport measurements for Pb-doped Bi-2212 single crystals. We demonstrate that the characteristic temperatures of the "weak" pseudogap opening and electronic coherence cross each other at a critical doping level, while those of the "strong" pseudogap merges into that of superconducting fluctuations above the critical …
We consider the problem of c-axis transport in double-layered cuprates, in particular with reference to Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ compounds. We exploit the effect of the two barriers on the thermal and tunnel transport. The resulting model is able to describe accurately the normal state c-axis resistivity in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$, from the underdoped side up to the strongly overdoped. We extend the model, without introducing additional parameters, in order to allow for the decrease of the barrier when an external voltage bias is applied. The extended model is found to d…
We report tunneling spectra of near optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ intrinsic Josephson junctions with area of 0.09 $μ$m$^2$, which avoid some fundamental difficulties in the previous tunneling experiments and allow a stable temperature-dependent measurement. A d-wave Eliashberg analysis shows that the spectrum at 4.2 K can be well fitted by considering electron couplings to a bosonic magnetic resonance mode and a broad high-energy continuum. Above $T_c$, the spectra show a clear pseudogap that persists up to 230 K, and a crossover can be seen indicating two different pseudogap p…
We find that peaks in the autocorrelation of angle resolved photoemission spectroscopy data of Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ in the superconducting state show dispersive behavior for binding energies smaller than the maximum superconducting energy gap. For higher energies, though, a striking anomalous dispersion is observed that is a consequence of the interaction of the electrons with collective excitations. In contrast, in the pseudogap phase, we only observe dispersionless behavior for the autocorrelation peaks. The implications of our findings in regards to Fourier transformed scanning tun…
The dynamics of excitations with different symmetry is investigated in the superconducting (SC) and normal state of the high-temperature superconductor Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ (Bi2212) using optical pump-probe (Pp) experiments with different light polarizations at different doping levels. The observation of distinct selection rules for SC excitations, present in A$_{\rm 1g}$ and B$_{\rm 1g}$ symmetries, and for the PG excitations, present in A$_{\rm 1g}$ and B$_{\rm 2g}$ symmetries, by the probe and absence of any dependence on the pump beam polarization leads to the unequivocal co…
We express the superconducting gap, $Δ(T)$, in terms of thermodynamic functions in both $s$- and d-wave symmetries. Applying to Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ and Y$_{0.8}$Ca$_{0.2}$Ba$_2$Cu$_3$O$_{7-δ}$ we find that for all dopings $Δ(T)$ persists, as a partial gap, high above $T_c$ due to strong superconducting fluctuations. Therefore in general two gaps are present above $T_c$, the superconducting gap and the pseudogap, effectively reconciling two highly polarized views concerning pseudogap physics.
Using a new variant of photoelectron spectroscopy, we measure the homogeneous near-nodal pairing ($Δ$) and pair-breaking self-energy ($Γ_S$) processes for a wide range of doping levels of Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$. For all samples we find that the pairing extends above the superconducting transition T$_c$ to a scale T$_{Pair}$ that is distinct from the antinodal pseudogap scale T$^*$ and near but slightly above T$_c$. We find that $Δ$ and T$_{Pair}$ are related with a strong coupling ratio 2$Δ$ /k$_B$T$_{Pair}\approx6$ across the entire doping phase diagram, i.e. independent of the effects…
A Fermi arc is a disconnected segment of a Fermi surface observed in the pseudogap phase of cuprate superconductors. This simple description belies the fundamental inconsistency in the physics of Fermi arcs, specifically that such segments violate the topological integrity of the band. Efforts to resolve this contradiction of experiment and theory have focused on connecting the ends of the Fermi arc back on itself to form a pocket, with limited and controversial success. Here we show the Fermi arc, while composed of real spectral weight, lacks the quasiparticles to be a true Fermi surface. To …
The real part of the in-plane optical self-energy data in underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ (Bi-2212) and ortho II YBa$_{2}$Cu$_{3}$O$_{6.5}$ contains new and important information on the pseudogap. Using a theoretical model approach we find that the density of state lost below the pseudogap $Δ_{pg}$ is accompanied with a pileup just above this energy. The pileup along with a sharp mode in the bosonic spectral function leads to an unusually rapid increase in the optical scattering rate and a characteristically sloped peak in the real part of the optical self-energy. These features…
Interlayer tunneling resistivity is used to probe the low-energy density-of-states (DOS) depletion due to the pseudogap in the normal state of Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$. Measurements up to 60 T reveal that a field that restores DOS to its ungapped state shows strikingly different temperature and doping dependencies from the characteristic fields of the superconducting state. The pseudogap closing field and the pseudogap temperature $T^{\star}$ evaluated independently are related through a simple Zeeman energy scaling. These findings indicate a predominant role of spins over the orbital eff…
A pseudogap (PG) was introduced by Mott to describe a state of matter which has a minimum in the density of states at the Fermi level, deep enough for states to become localized. It can arise either from Coulomb repulsion between electrons, or due to an incipient charge or spin order, or a combination of the two. These states are rapidly fluctuating in time with random phase, so they are hard to observe experimentally. Here we present the first coherent quench measurements of the dynamical transition to the pseudogap state in the prototype high temperature superconductor Bi$_{2}$Sr$_{2}$CaCu$_…
We reanalyze the data on the in-plane far infrared conductivity of underdoped orthoII YBa$_{2}$Cu$_{3}$O$_{6.50}$ (YBCO$_{6.50}$) in terms of a model in which a pseudogap opens on part of the Fermi surface with the remaining ungaped piece proportional to the temperature. The motivation for our model comes from recent angle-resolved photoemission spectroscopy data in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ (Bi-2212), which have revealed the existence of arcs on the Fermi surface. We find the optical data to be consistent with arc formation. In addition we find some evidence that the electronic…
Both electronic Raman scattering (ERS) and angle-resolved photoemission spectra (ARPES) revealed two energy scales for the gap in different momentum spaces in the cuprates. However, the interpretations were different, and the gap values were also different in two experiments. In order to clarify the origin of these discrepancies, we directly compared ERS and ARPES by calculating ERS from the experimental data of ARPES through the Kubo formula. The calculated ERS spectra were in good agreement with the experimental results except for the B$_{1g}$ peak energies. The doping-dependent B$_{2g}$ pea…
We propose that, if the pseudogap state in the cuprate superconductors can be described in terms of the phase-incoherent preformed pairs, there should exist Andreev reflection from these pairs even above the superconducting transition temperature, $T_c$. After giving qualitative arguments for this effect, we present more quantitative calculations based on the Bogoliubov--de Gennes equation. Experimental observations of the effects of Andreev reflections above $T_c$---such as an enhanced tunneling conductance below the gap along the copper oxide plane---could provide unambiguous evidence for th…
Since the discovery of the pseudogap and Fermi arc states in underdoped cuprates, the understanding of such non-Fermi-liquid states and the associated violation of Luttinger's theorem have been the central theme in correlated electron systems. However, still lacking is a well-accepted theoretical framework to unambiguously explain these metallic states that are clearly beyond Landau's Fermi liquid and Luttinger's theorem of a Fermi surface and electron filling. Here, we design a lattice model of orthogonal metals with fermion and Ising matter fields coupled to topological order and, by solving…
The pairing and superfluid phenomena in a two-component ultracold atomic Fermi gas is an analogue of Cooper pairing and superconductivity in an electron system, in particular, the high $T_c$ superconductors. Owing to the various tunable parameters that have been made accessible experimentally in recent years, atomic Fermi gases can be explored as a prototype or quantum simulator of superconductors. It is hoped that, utilizing such an analogy, the study of atomic Fermi gases may shed light to the mysteries of high $T_c$ superconductivity. One obstacle to the ultimate understanding of high $T_c$…
The study of interaction between different degrees of freedom in solids is of fundamental importance to understand the functionalities of materials. One striking example of such interaction is the intertwined coupling or competition between superconductivity (SC), charge density wave (CDW), pseudogap state (PG), and other exotic phases in cuprate superconductors. Recent emergence of nonlinear Terahertz (THz) third harmonic generation (THG) spectroscopy provides a powerful tool for exploring the collective (Higgs) modes of superconductivity order parameters, and its interaction with intertwined…
Thermodynamic phase transition, pairing symmetry and Fermi surface topology in Ruddlesden-Popper nickelate films
University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（中国科学院微观结构国家重点实验室，中国科学技术大学）
