Doping dependence of low-energy quasiparticle excitations in superconducting Bi2212
© Ino et al.; licensee Springer. 2013
Received: 17 July 2013
Accepted: 20 November 2013
Published: 5 December 2013
The doping-dependent evolution of the d-wave superconducting state is studied from the perspective of the angle-resolved photoemission spectra of a high-Tc cuprate, Bi2Sr2CaCu2 O8+δ (Bi2212). The anisotropic evolution of the energy gap for Bogoliubov quasiparticles is parametrized by critical temperature and superfluid density. The renormalization of nodal quasiparticles is evaluated in terms of mass enhancement spectra. These quantities shed light on the strong coupling nature of electron pairing and the impact of forward elastic or inelastic scatterings. We suggest that the quasiparticle excitations in the superconducting cuprates are profoundly affected by doping-dependent screening.
KeywordsHigh-Tc cuprate Bi2212 ARPES Superconducting gap Effective mass Coupling strength 74.25.Jb 74.72.-h 79.60.i
Electronic excitations dressed by the interaction with the medium are called quasiparticles. They serve as a direct probe of the anisotropic order parameter of a superconducting phase and also as a clue to the electron-pairing glue responsible for the superconductivity. In fact, the major unresolved issues on the mechanism of high-Tc superconductivity depend on the low-energy quasiparticle excitations. The superconducting order parameter, which is typified by the particle-hole mixing and gives rise to Bogoliubov quasiparticles (BQPs), manifests itself as an energy gap in quasiparticle excitation spectra. In cuprate superconductors, however, the energy gap increases against the decrease in critical temperature Tc with underdoping and is open even at some temperatures above Tc[1–3]. In the direction where the d-wave order parameter disappears, renormalization features have been extracted quantitatively from the gapless continuous dispersion of nodal quasiparticles (NQPs), suggesting strong coupling with some collective modes. Nevertheless, the origins of these features remain controversial[4, 5].
In this paper, we address the doping dependence of BQP and NQP of a high-Tc cuprate superconductor, Bi2Sr2CaCu2O8+δ (Bi2212), on the basis of our recent angle-resolved photoemission (ARPES) data[6–8]. The use of low-energy synchrotron radiation brought about improvement in energy and momentum resolution and allowed us to optimize the excitation photon energy. After a brief description of BQP and NQP spectral functions, we survey the superconducting gap anisotropy on BQPs and the renormalization features in NQPs. In light of them, we discuss possible effects of doping-dependent electronic screening on the BQP, NQP, and high-Tc superconductivity.
High-quality single crystals of Bi2212 were prepared by a traveling-solvent floating-zone method, and hole concentration was regulated by a post-annealing procedure. In this paper, the samples are labeled by the Tc value in kelvin, together with the doping-level prefix, i.e. underdoped (UD), optimally doped (OP), or overdoped (OD). ARPES experiments were performed at HiSOR BL9A in Hiroshima Synchrotron Radiation Center. The ARPES data presented here were taken with excitation-photon energies of h ν = 8.5 and 8.1 eV for the BQP and NQP studies, respectively, and at a low temperature of T = 9 - 10 K in the superconducting state. Further details of the experiments have been described elsewhere[7–9].
The spectral function given by A k (ω) = - Im G k (ω)/π is directly observed by ARPES experiments. The extensive treatments of the ARPES data in terms of Green’s function are given elsewhere.
Superconducting gap anisotropy
As presented in Figure2e, the correlation between the nodal and antinodal gaps provides a perspective of crossover for our empirical formula (Equation 5). It is deduced from the conventional Bardeen-Cooper-Schrieffer (BCS) theory that 2Δ/kBTc = 4.3 in the weak coupling limit for the d-wave superconducting gap. However, the critical temperature Tc is often lower than that expected from the weak coupling constant and a given Δ as an effect of strong coupling. Thus, the gap-to- Tc ratio is widely regarded as an indicator for the coupling strength of electron pairing and adopted for the coordinate axes in Figure2e. As hole concentration decreases from overdoped to underdoped Bi2212, the experimental data point moves apart from the weak coupling point toward the strong coupling side, and a crossover occurs at 8.5, which is about twice the weak coupling constant. It appears that the evolution of ΔN is confined by two lines as ΔN ≤ 0.87Δ∗ and 2ΔN ≤ 8.5kBTc. As illustrated in the insets of Figure2e, the strong coupling allows the electrons to remain paired with incoherent excitations. As a result, the superconducting order parameter is reduced with respect to the pairing energy. Indeed, it has been shown that the reduction factor due to the incoherent pair excitations has a simple theoretical expression and that the nodal and antinodal spectra are peaked at the order parameter and at the pairing energy, respectively, taking into account a realistic lifetime effect[24, 25]. Therefore, the latter part of Equation 5 is consistent with the strong coupling scenario, and furthermore, the two distinct lines in Figure2e are naturally interpreted as the energies of the condensation and formation of the electron pairs.
Renormalization features in dispersion
In ARPES spectra, the real and imaginary parts of self-energy manifest themselves as the shift and width of spectral peak, respectively. Specifically, provided that the momentum dependence of Σ k (ω) along the cut is negligible, and introducing bare electron velocity v0 by, it follows from Equation 2 that the momentum distribution curve for a given quasiparticle energy ω is peaked at k(ω) = [ω-ReΣ(ω)]/v0 and has a natural half width of Δk(ω) = - ImΣ(ω)/v0.
We note that -Imλ(ω) represents the energy distribution of the impact of coupling with other excitations and can be taken as a kind of coupling spectrum. However, it should be emphasized that -Imλ(ω) is expressed as a function of quasiparticle energy ω, whereas the widely used Eliashberg coupling function α2F(Ω) is expressed as a function of boson energy Ω. For example, a simulation of λ(ω) using Equations 7 to 9 is presented in Figure3b,c, where a single coupling mode is given at Ω = 40 meV. One can see that the peak of α2F(-ω) is reproduced by -Imλ(ω), provided that A(ω) is gapless and approximated by a constant. As an energy gap of Δ opens in A(ω), the peak in -Imλ(ω) is shifted from Ω into Ω + Δ. Nevertheless, irrespective of A(ω), the causality of Σ(ω) is inherited by λ(ω), so that Reλ(ω) and Imλ(ω) are mutually convertible through the Kramers-Kronig transform (KKT). The directness and causality of λ(ω) enable us to decompose the quasiparticle effective mass without tackling the integral inversion problem in Equation 7.
Dividing the energy range of the integral in Equation 10, one can quantify the contribution from a particular energy part. We refer to the KKT integrals of Im λ(ω)/v0 for the low-energy (LE; 4 < |ω| < 40 meV), intermediate-energy (IE; 40 < |ω| < 130 meV), and high-energy (HE; 130 < |ω| < 250 meV) parts as λLE/v0 (red circles), λIE/v0 (blue triangles), and λHE/v0 (green diamonds), respectively. Those obtained from the data in Figure5b,d are plotted in Figure5c,e, respectively. Also shown in Figure5c are the inverse group velocities at ω = 0 meV (black circles) and at ω = -40 meV (black triangles). Figure5c and Figure5e consistently indicate that as hole concentration decreases, the contribution of the low-energy part rapidly increases and becomes dominant over the other parts.
Possible origins of the low-energy kink are considered from the energy of 15 meV and the evolution with underdoping. The quasiparticles that can be involved in the intermediate states are limited within the energy range of |ω| ≤ 15 meV, and the irrelevance of the antinodal states is deduced from the simulation in Figure3c. Therefore, the low-energy kink is due to the near-nodal scatterings with small momentum transfer. The candidates for bosonic forward scatterers are the low-frequency phonons, such as the acoustic phonons and the c-axis optical phonons involving heavy cations[7, 28–31]. On the other hand, it has also been argued that the elastic forward scattering by off-plane impurities may give rise to the low-energy kink for the d-wave superconductors[7, 32]. In usual metal, both the potentials of the low-frequency phonons and the static impurities are strongly screened by the rapid response of electronic excitations. Therefore, the enhancement of the low-energy kink suggests the breakdown of electronic screening at low hole concentrations[7, 28].
The dispersion kink at 65 meV has been ascribed to an intermediate state consisting of an antinodal quasiparticle and the B1g buckling phonon of Ω ∼ 35 meV. However, the mass enhancement spectra in Figure5a,b,d are suggestive of the presence of multiple components in the intermediate-energy range.
We found that both the superconducting gap anisotropy and the renormalized dispersion show the striking evolution with underdoping. These behaviors are considered to be dependent on the extent of the screening. In association with the forward elastic or inelastic scatterings, the screening breakdown would enhance the low-energy kink. From the aspect of the impact of off-plane impurities, the inadequacy of static screening would inevitably lead to the nanoscale inhomogeneities, as observed by scanning tunneling microscopy experiments. The forward scatterings by the remaining potential would generate additional incoherent pair excitations, as expected from the nodal gap suppression at low temperatures[8, 25]. From the aspect of the electron-phonon coupling, the inhomogeneous depletion of the electrons for screening may considerably increase the coupling strength, providing an account for the unexpectedly strong dispersion kink and a candidate for the strong pairing interaction. The former and latter aspects have negative and positive effects, respectively, on the superconductivity. Thus, we speculate that the doping dependence of Tc is eventually determined by the balance between these effects.
Summarizing, the evolution of a d-wave high-Tc superconducting state with hole concentration has been depicted on the basis of the high-resolution ARPES spectra of the quasiparticles and discussed in relation to the screening by electronic excitations. The divergence between the nodal and antinodal gaps can be interpreted as an effect of the incoherent pair excitations inherent in the strong coupling superconductivity. The low-energy kink, which rapidly increases with underdoping, should be caused by the forward elastic or inelastic scatterings, although it remains as an open question which scattering is more dominant. The quantitative simulation of the doping-dependent effect will be helpful for resolving this problem.
Angle-resolved photoemission spectroscopy
We thank Z.-X. Shen and A. Fujimori for useful discussions and K. Ichiki, Y. Nakashima, and T. Fujita for their help with the experimental study. The ARPES experiments were performed under the approval of HRSC (Proposal No. 07-A-2, 09-A-11 and 10-A-24).
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