Doping dependence of lowenergy quasiparticle excitations in superconducting Bi2212
 Akihiro Ino^{1}Email author,
 Hiroaki Anzai^{2},
 Masashi Arita^{3},
 Hirofumi Namatame^{3},
 Masaki Taniguchi^{1, 3},
 Motoyuki Ishikado^{4, 5, 6},
 Kazuhiro Fujita^{4, 7},
 Shigeyuki Ishida^{4, 5} and
 Shinichi Uchida^{4}
DOI: 10.1186/1556276X8515
© Ino et al.; licensee Springer. 2013
Received: 17 July 2013
Accepted: 20 November 2013
Published: 5 December 2013
Abstract
The dopingdependent evolution of the dwave superconducting state is studied from the perspective of the angleresolved photoemission spectra of a highT_{c} cuprate, Bi_{2}Sr_{2}CaCu_{2} O_{8+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 dopingdependent screening.
Keywords
HighT_{c} cuprate Bi2212 ARPES Superconducting gap Effective mass Coupling strength 74.25.Jb 74.72.h 79.60.iBackground
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 electronpairing glue responsible for the superconductivity. In fact, the major unresolved issues on the mechanism of highT_{c} superconductivity depend on the lowenergy quasiparticle excitations. The superconducting order parameter, which is typified by the particlehole 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 T_{c} with underdoping and is open even at some temperatures above T_{c}[1–3]. In the direction where the dwave 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[4]. Nevertheless, the origins of these features remain controversial[4, 5].
In this paper, we address the doping dependence of BQP and NQP of a highT_{c} cuprate superconductor, Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212), on the basis of our recent angleresolved photoemission (ARPES) data[6–8]. The use of lowenergy 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 dopingdependent electronic screening on the BQP, NQP, and highT_{c} superconductivity.
Methods
Highquality single crystals of Bi2212 were prepared by a travelingsolvent floatingzone method, and hole concentration was regulated by a postannealing procedure. In this paper, the samples are labeled by the T_{c} value in kelvin, together with the dopinglevel 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 excitationphoton 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[10].
Results
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 BardeenCooperSchrieffer (BCS) theory that 2Δ/k_{B}T_{c} = 4.3 in the weak coupling limit for the dwave superconducting gap[23]. However, the critical temperature T_{c} is often lower than that expected from the weak coupling constant and a given Δ as an effect of strong coupling. Thus, the gapto T_{c} 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 Δ_{N} is confined by two lines as Δ_{N} ≤ 0.87Δ^{∗} and 2Δ_{N} ≤ 8.5k_{B}T_{c}. 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$\sqrt{{\rho}_{\mathrm{s}}/{\rho}_{\mathrm{s}}^{\text{BCS}}}$ 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 selfenergy 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 v_{0} by${\omega}_{k}^{0}={v}_{0}k$, it follows from Equation 2 that the momentum distribution curve for a given quasiparticle energy ω is peaked at k(ω) = [ωReΣ(ω)]/v_{0} and has a natural half width of Δk(ω) =  ImΣ(ω)/v_{0}.
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 α^{2}F(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 α^{2}F(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 KramersKronig 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)/v_{0} for the lowenergy (LE; 4 < ω < 40 meV), intermediateenergy (IE; 40 < ω < 130 meV), and highenergy (HE; 130 < ω < 250 meV) parts as λ^{LE}/v_{0} (red circles), λ^{IE}/v_{0} (blue triangles), and λ^{HE}/v_{0} (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 lowenergy part rapidly increases and becomes dominant over the other parts.
Possible origins of the lowenergy 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 lowenergy kink is due to the nearnodal scatterings with small momentum transfer. The candidates for bosonic forward scatterers are the lowfrequency phonons, such as the acoustic phonons and the caxis optical phonons involving heavy cations[7, 28–31]. On the other hand, it has also been argued that the elastic forward scattering by offplane impurities may give rise to the lowenergy kink for the dwave superconductors[7, 32]. In usual metal, both the potentials of the lowfrequency phonons and the static impurities are strongly screened by the rapid response of electronic excitations. Therefore, the enhancement of the lowenergy 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 B_{1g} buckling phonon of Ω ∼ 35 meV[33]. However, the mass enhancement spectra in Figure5a,b,d are suggestive of the presence of multiple components in the intermediateenergy range.
Discussion
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 lowenergy kink. From the aspect of the impact of offplane impurities, the inadequacy of static screening would inevitably lead to the nanoscale inhomogeneities, as observed by scanning tunneling microscopy experiments[34]. 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 electronphonon coupling, the inhomogeneous depletion of the electrons for screening may considerably increase the coupling strength, providing an account for the unexpectedly strong dispersion kink[35] and a candidate for the strong pairing interaction[8]. The former and latter aspects have negative and positive effects, respectively, on the superconductivity. Thus, we speculate that the doping dependence of T_{c} is eventually determined by the balance between these effects.
Conclusions
Summarizing, the evolution of a dwave highT_{c} superconducting state with hole concentration has been depicted on the basis of the highresolution 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 lowenergy 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 dopingdependent effect will be helpful for resolving this problem.
Abbreviations
 AB:

Antibonding band
 ARPES:

Angleresolved photoemission spectroscopy
 BB:

Bonding band
 BCS:

BardeenCooperSchrieffer
 Bi2212:

Bi_{2}Sr_{2}CaCu_{2} O_{8+δ}
 BQP:

Bogoliubov quasiparticle
 HE:

High energy
 IE:

Intermediate energy
 KKT:

KramersKronig transform
 LE:

Low energy
 NQP:

Nodal quasiparticle
 OD:

Overdoped
 OP:

Optimally doped
 UD:

Underdoped.
Declarations
Acknowledgements
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. 07A2, 09A11 and 10A24).
Authors’ Affiliations
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