The doping-dependent evolution of the d-wave superconducting state is studied from the perspective of the angle-resolved photoemission spectra of a high-T_{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 doping-dependent screening.

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-T_{c} 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 T_{c} with underdoping and is open even at some temperatures above T_{c}[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[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 high-T_{c} cuprate superconductor, Bi_{2}Sr_{2}CaCu_{2}O_{8+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-T_{c} superconductivity.

Methods

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 T_{c} 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 hν = 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 relation between a bare electron and a renormalized quasiparticle is described in terms of self-energy Σ_{
k
}(t), which can be regarded as a factor of feedback on the wave function from past to present through the surrounding medium. Incorporating a feedback term into the Schrödinger equation, we obtain

where ψ_{
k
}(t) and${\omega}_{\mathbf{k}}^{0}$ denote a wave function and a bare-electron energy, respectively. It is obvious from Equation 1 that the self-energy is a linear response function. Therefore, its frequency representation, Σ_{
k
}(ω), obeys the Kramers-Kronig relation. As the solution of Equation 1, we obtain the form of dressed Green’s function,

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

In the superconducting state, the condensate of electron pairs allows the particle-like and hole-like excitations to turn into each other. Hence, the wave function of a hole-like excitation also comes back by way of the particle-like excitation. Such a feedback has a sign-reversed eigenenergy,$-{\omega}_{\mathbf{k}}^{0}$, and is expressed by${\mathrm{\Sigma}}_{\mathbf{k}}(t)=-i\mathrm{\Theta}(t){\mathrm{\Delta}}_{\mathbf{k}}^{2}{e}^{i{\omega}_{\mathbf{k}}^{0}t}{e}^{-{\gamma}_{\mathbf{k}}t}-i{\mathrm{\Gamma}}_{\mathbf{k}}\delta (t)$, where Θ(t), Δ_{
k
}, γ_{
k
} and Γ_{
k
} denote the step function, the particle-hole off-diagonal element, and the scattering rates of the intermediate and bare-particle states, respectively. The Fourier transform of Σ_{
k
}(t) gives the frequency representation of the self-energy of the BQPs,

Figure1 shows the ARPES spectra of BQPs for underdoped and overdoped Bi2212 samples with T_{c} = 66 and 80 K (UD66 and OD80, respectively)[8]. As shown in Figure1b,c, an energy distribution curve was extracted from the minimum gap locus for each off-node angle θ and symmetrized with respect to the Fermi energy ω = 0. These spectra were well fitted with a phenomenological function,

except for a featureless background. Equation 4 is deduced from Equation 3 and${\omega}_{\mathbf{k}}^{0}=0$, neglecting γ_{
k
} after Norman et al.[11]. Figure1b,c exemplifies that the superconducting gap energy Δ at each θ is definitely determined by sharp spectral peaks. In Figure1d,e, the obtained gap energies (small yellow circles) are plotted over the image of spectral intensity as a function of sin 2θ, so that the deviation from a d-wave gap is readily seen with reference to a straight line. While the superconducting gap of the overdoped sample almost follows the d-wave line, that of the underdoped sample is deeply curved against sin 2θ. Furthermore, Figure1d indicates that the deviation from the d-wave gap penetrates into the close vicinity of the node and that it is difficult to define the pure d-wave region near the node. Therefore, the next-order harmonic term, sin 6θ, has been introduced, so that the smooth experimental gap profile is properly parametrized[12–14]. The next-order high-harmonic function is also expressed as Δ(θ) = Δ_{N} sin 2θ + (Δ^{∗}-Δ_{N})(3 sin 2θ- sin 6θ)/4, where the antinodal and nodal gap energies are defined as Δ^{∗} = Δ(θ)|_{θ=45°} and${\mathrm{\Delta}}_{\mathrm{N}}={\left.\frac{1}{2}(d\mathrm{\Delta}/d\theta )\right|}_{\theta =0}$, respectively, so that Δ_{N}/Δ^{∗} = 1 is satisfied for a pure d-wave gap.

The doping dependences of the superconducting gap parameters are summarized in Figure2. One can see from Figure2a that as hole concentration decreases with underdoping, the nodal gap energy 2Δ_{N} closely follows the downward curve of 8.5k_{B}T_{c} in contrast to the monotonic increase in the antinodal gap energy 2Δ^{∗}. It seems reasonable that T_{c} primarily depends on Δ_{N} rather than Δ^{∗} for the underdoped Bi2212, because it follows from 2Δ^{∗} ≫ 4k_{B}T_{c} that the thermal quasiparticle excitations concentrate in the vicinity of the node and hardly occur around the antinode. The relevance of the nodal excitations has also been suggested by various experiments[15–19]. Then, the problem with T_{c} is that the nodal gap Δ_{N} is suppressed relative to the antinodal gap Δ^{∗}. This behavior can be associated with low superfluid density ρ_{s}[20]. Figure2b,c shows that the doping dependence of the nodal-to-antinodal gap ratio Δ_{N}/Δ^{∗} is quite similar to that of the square-root superfluid density$\sqrt{{\rho}_{\mathrm{s}}}$[8, 21, 22]. The normalized gap plot in Figure2d indicates that what occurs with underdoping is analogous to the nodal gap suppression observed with increasing temperature[17] in terms of the decrease in ρ_{s}. It is notable that the square-root dependence on ρ_{s} is a typical behavior of the order parameter as expected from the Ginzburg-Landau theory[23]. These findings can be written down in a simple relational formula,

where${\rho}_{\mathrm{s}}^{0}\simeq 31\mu {\mathrm{m}}^{-2}$, for a wide hole-concentration range of Bi2212.

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/k_{B}T_{c} = 4.3 in the weak coupling limit for the d-wave 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 gap-to- 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 the nodal direction where the order parameter disappears, one can investigate the fine renormalization features in dispersion. They reflect the intermediate-state energy in coupling between an electron and other excitations, and thus provide important clues to the pairing interaction. As for the electron-boson coupling, the intermediate state consists of a dressed electronic excitation and an additional bosonic excitation (Figure3a). Averaging the momentum dependence for simplicity, the energy distribution of the intermediate state is expressed by A(ω - Ω) Θ(ω - Ω)+A(ω + Ω) Θ(-ω - Ω) for a given boson energy Ω and for zero temperature, owing to the Pauli exclusion principle. Therefore, taking into account the effective energy distribution of the coupled boson, α^{2}F(Ω), the self-energy is written down as follows:

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 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) = [ω-ReΣ(ω)]/v_{0} and has a natural half width of Δk(ω) = - ImΣ(ω)/v_{0}.

We argue that the mass enhancement function defined as the energy derivative of the self-energy, λ(ω) ≡ -(d/dω)Σ(ω), is useful for the analysis of NQP[7, 26]. The real and imaginary parts of λ(ω) are directly obtained from the ARPES data as the inverse of group velocity, v_{g}(ω), and as the differential scattering rate, respectively.

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 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.

Figure4 shows the ARPES spectra along the nodal cut perpendicular to the Fermi surface for the superconducting Bi2212[7]. Although the splitting due to the CuO_{2} bilayer is minimum at the nodes, it has clearly been observed by using some specific low-energy photons[6–8]. A prominent kink in the NQP dispersion is observed at 65 meV for all the doping level, as has been reported since early years[4]. In addition to this, another small kink at 15 meV is discernible in the raw spectral image of the underdoped sample (UD66)[7, 27].

The fine renormalization features in the NQP dispersion were determined by fitting the momentum distribution curves with double Lorentzian. Figure5a,d shows the real and imaginary parts of λ(ω)/v_{0} experimentally obtained as the energy derivatives of the peak position and width, respectively. The KKT of Reλ(ω)/v_{0} in Figure5a is shown in Figure5b as Imλ(ω)/v_{0}, which is comparable with the data in Figure5d. A step-like mass enhancement in Figure5a and a peak-like coupling weight in Figure5b,d are consistently observed at 65 meV. This is a typical behavior of the mode coupling, as shown by the simulation in Figure3. It is also found that an additional feature around 15 meV is dramatically enhanced with underdoping. In order to deduce the partial coupling constant, we express the mass enhancement factor λ as the form of KKT,

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 low-energy (LE; 4 < |ω| < 40 meV), intermediate-energy (IE; 40 < |ω| < 130 meV), and high-energy (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 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 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 intermediate-energy 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 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[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 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[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 d-wave high-T_{c} 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.

Abbreviations

AB:

Antibonding band

ARPES:

Angle-resolved photoemission spectroscopy

BB:

Bonding band

BCS:

Bardeen-Cooper-Schrieffer

Bi2212:

Bi_{2}Sr_{2}CaCu_{2} O_{8+δ}

BQP:

Bogoliubov quasiparticle

HE:

High energy

IE:

Intermediate energy

KKT:

Kramers-Kronig 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. 07-A-2, 09-A-11 and 10-A-24).

Authors’ Affiliations

(1)

Graduate School of Science, Hiroshima University

(2)

Graduate School of Engineering, Osaka Prefecture University

(3)

Hiroshima Synchrotron Radiation Center, Hiroshima University

(4)

Department of Physics, University of Tokyo

(5)

National Institute of Advanced Industrial Science and Technology

(6)

Research Center for Neutron Science and Technology, CROSS

(7)

Laboratory for Atomic and Solid State Physics, Department of Physics, Cornell University

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