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Disentangling the Electronic and Phononic Glue in a High-Tc Superconductor

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Science  30 Mar 2012:
Vol. 335, Issue 6076, pp. 1600-1603
DOI: 10.1126/science.1216765

Abstract

Unveiling the nature of the bosonic excitations that mediate the formation of Cooper pairs is a key issue for understanding unconventional superconductivity. A fundamental step toward this goal would be to identify the relative weight of the electronic and phononic contributions to the overall frequency (Ω)–dependent bosonic function, Π(Ω). We performed optical spectroscopy on Bi2Sr2Ca0.92Y0.08Cu2O8+δ crystals with simultaneous time and frequency resolution; this technique allowed us to disentangle the electronic and phononic contributions by their different temporal evolution. The spectral distribution of the electronic excitations and the strength of their interaction with fermionic quasiparticles fully account for the high critical temperature of the superconducting phase transition.

Lattice vibrations (1) and excitations of electronic origin, such as spin or electric polarizability fluctuations (2) and loop currents (3), are generally considered potential mediators of Cooper pairing in the copper oxide high-temperature superconductors (cuprates). The generic interaction of fermionic quasi-particles (QPs) with bosonic excitations is accounted for by the bosonic function Π(Ω) [usually indicated as α2F(Ω) for phonons and I2χ(Ω) for spin fluctuations], a dimensionless function that depends on the density of states of the excitations and the strength of their coupling to QPs. Because both the energy dispersion and lifetime of QPs are strongly affected by the interactions, signatures of QP-boson coupling have been observed in experiments that probe the electronic properties at equilibrium. The ubiquitous kinks in the QP dispersion at ~70 meV, measured by angle-resolved photoemission spectroscopy (ARPES) (4), have been interpreted in terms of coupling to either optical Cu-O lattice modes (5, 6) or spin excitations (7). Inelastic neutron and x-ray scattering experiments found evidence for both QP-phonon anomalies (8) and bosonic excitations attributed to spin fluctuations (7, 9) and loop currents (10). Dip features in tunneling experiments have been used to alternatively support the scenarios of dominant electron-phonon interactions (11) or antiferromagnetic spin fluctuations (12). The frequency-dependent dissipation of the Drude optical conductivity, σ(ω), measured by equilibrium optical spectroscopies, has been interpreted (1315) as the coupling of electrons to bosonic excitations, in which the separation of the phononic and electronic contributions is impeded by their partial coexistence on the same energy scale (<90 meV).

We have disentangled the electronic and phononic contributions to Π(Ω) through a nonequilibrium optical spectroscopy, in which femtosecond time resolution is combined with an energy resolution smaller than 10 meV over a wide photon energy range (0.5 to 2 eV). Our approach can be rationalized on the basis of the widely used assumption (16, 17) that, after the interaction between a superconductor and a short laser pulse (1.55 eV photon energy), the effective electronic temperature (Te) relaxes toward its equilibrium value through energy exchange with the different degrees of freedom that linearly contribute to Π(Ω). In a more formal description, the total bosonic function is given byΠ(Ω)=Πbe(Ω)+ΠSCP(Ω)+Πlat(Ω)(1)where Πbe refers to the bosonic excitations of electronic origin at the effective temperature Tbe, ΠSCP to the small fraction of strongly coupled phonons (SCPs) at TSCP (16), and Πlat to all other lattice vibrations at Tlat. Because the term that couples the rate equations (18) for Te and Tb (with b = be, SCP, lat) is Gb, Tb, Te)/Cb [where G is the functional described in (18) and Cb is the specific heat of the bosonic population], each subset of the bosonic excitations is characterized by different relaxation dynamics on the femtosecond time scale. The most convenient systems for such an experiment are the hole-doped cuprates close to the optimal dopant concentration needed to attain the maximum critical temperature Tc, in which the total Π(Ω) is maximum (15). Despite the magnitude of Π(Ω), vertex corrections beyond Eliashberg theory can be reliably neglected (9, 15) in this doping regime.

The total bosonic function Π(Ω) is directly determined by fitting an extended Drude model (18) and a sum of Lorentz oscillators accounting for the interband optical transitions in the visible region (19) to the equilibrium dielectric function of optimally doped Bi2Sr2Ca0.92Y0.08Cu2O8+δ (Y-Bi2212) high-quality crystals (20) (Tc = 96 K) measured at T = 300 K by conventional spectroscopic ellipsometry (21). If we assume a histogram-like form (18) (fig. S1) and impose an upper limit of 1 eV, the extracted Π(Ω) is characterized by (i) a low-energy part (up to 40 meV) compatible with the coupling to acoustic phonons (22) and Raman-active optical phonons involving c-axis motion of the Cu ions (23); (ii) a narrow, intense peak centered at ~60 meV, attributed to the anisotropic coupling to either out-of-plane buckling and in-plane breathing Cu-O optical modes (6) or bosonic excitations of electronic origin such as spin fluctuations (7); and (iii) a broad continuum extending up to 350 meV (1315), well above the characteristic phonon cutoff frequency (~90 meV).

The key point to extend this analysis to nonequilibrium experiments is that the electron self-energy, Σ(ω, t), used to calculate σ(ω, t) (18), can be factorized intoΣ(ω,T)=0Π(Ω)L(ω,Ω,T)dΩ(2)(24), where L(ω, Ω, T) is a material-independent kernel function accounting for the thermal activation of the bosonic excitations and of the QPs. The kernel function is given byL(ω,Ω,Te,b)=2πi[N(Ω,Tb)+12]+Ψ[12+i(ΩΩ)2πTe]Ψ[12i(Ω+Ω)2πTe] (3)where Ψ is the digamma function obtained by integrating the Fermi-Dirac functions, and N(Ω, Tb) is the Bose distribution at temperature Tb. The kernel function can be decomposed into different terms depending on the electronic (Te) and bosonic (Tb) temperatures. The independent variation of Te,b is expected to induce different modifications of the dielectric function. Figure 1 shows the expected relative variation of the reflectivity, expressed as δRR(ω,T)=R(ω,T+δT)R(ω,T)R(ω,T) (4)in the quasi-thermal (δTe = δTb > 0) and nonthermal (δTe > 0, δTb = 0) scenarios. Phenomenologically, in the first case the reflectivity variation is dominated by the increase of the QP-boson scattering, corresponding to a broadening of the Drude peak, whereas in the second case the decoupling between the QP and bosonic distributions can be rationalized in terms of a small increase of the plasma frequency without any change in the scattering rate. The difference between the two cases is more evident in the spectral region close to the dressed plasma frequency, Ωp ≈ 1 eV—that is, an energy scale much higher than the energy scale of the bosonic function.

Fig. 1

Nonequilibrium reflectivity. The relative reflectivity variation δR/R(ω, T) = [R(ω, T + δT) – R(ω, T)]/R, calculated for the incremental variations δTe and δTb in Eqs. 2 and 3 and using the Π(ω) value obtained from the fit to the equilibrium measurements, is reported. The change of sign of δR(ω, T)/R in the quasi-thermal scenario (gray line; δTe = δTb = 1 K) and the maximum δR(ω, T)/R in the nonthermal scenario (red line; δTe = 5 K, δTb = 0) occur at a frequency coinciding with the dressed plasma frequency, Ωp ~ 1 eV. The spectral regions probed by the optical parametric amplifier (OPA) and supercontinuum techniques (18) are indicated.

Time-resolved reflectivity measurements in the 0.5 to 2 eV photon-energy range (25) have been performed at T = 300 K on the same crystals (20) (OP96) used for the equilibrium optical spectroscopy. The δR/R(ω, t) two-dimensional matrix is reported in Fig. 2, A and B, along with the time traces at ~1.5 and 0.7 eV photon energies (white curves). After the pump excitation at t = 0, the temporal dynamics above and below Ωp are very similar, exhibiting a relaxation dynamics of about 200 fs, generally attributed to the thermalization of electrons with SCPs (16), and a slower decay on the picosecond time scale, related to the thermalization with all other lattice vibrations. Shown in Fig. 2C are the energy-resolved traces at fixed delay (t = 100 fs). Comparing the δR/R(ω) measured on OP96 to the relative variation of the reflectivity calculated in the nonthermal and quasi-thermal cases (Fig. 1), we come to the major point of our work: The electrons thermalize with part of the bosonic excitations, described by Π(Ω), on a time scale faster than the electron-phonon thermalization. The fast time scale (<<100 fs) of this thermalization implies a very large coupling and a relatively small specific heat. These overall observations strongly suggest that this process involves bosonic excitations of electronic origin.

Fig. 2

Nonequilibrium optical spectroscopy. (A and B) Time- and frequency-resolved reflectivity measurements carried out on the OP96 sample at T = 300 K in the OPA (A) and supercontinuum (B) probe optical region (18). The white curves represent the time traces at photon energies of 1.55 eV and 0.65 eV, respectively. (C) Energy-resolved spectra for OP96, measured at fixed time t = 100 fs. The black lines are the maximum and minimum δR/R(ω, t = 100 fs) determined through the fitting procedure described in the text and in (18) and accounting for the experimental uncertainty in the laser fluence and spot dimensions. Most of the values of δR/R(t = 100 fs) measured on OP96 at each wavelength fall between the two lines. (D) Time-resolved trace for OP96 at fixed photon energy of 1.55 eV. The black line is the best fit to the data obtained through the fitting procedure described in (18).

The relative strengths of Πb(Ω) (where the index b refers to the three components in Eq. 1) determine (26) both the temporal evolution of the temperatures Tb, through the four-temperature model (4TM) (18), and the intensity of the reflectivity variation, through Eq. 2. As a consequence, the simultaneous fit of the calculated δR/R(ω, Te, Tbe, TSCP, Tlat) to the data reported in Fig. 2 in the time and frequency domain decisively narrows the phase space of the parameters of the model, as compared to single-color measurements. The fit procedure (18) allows us to unambiguously extract the different contributions to Π(Ω) and to estimate Cbe and CSCP.

Figure 3 summarizes the main results of this work. The pump pulse of 10 μJ/cm2 gently increases the electronic temperature by δTe ~ 2 K. The entire high-energy part and ~46% of the peak (red areas) instantaneously thermalize with electrons at a temperature TbeTe. The spectral distribution and the value of the specific heat of these excitations (Cbe < 0.1Ce) demonstrate their electronic origin. On a slower time scale (100 to 200 fs), the electrons thermalize with the SCPs that represent ~20% of the phonon density of states (CSCP = 0.2Clat) but are responsible for ~34% of the coupling (blue area) in the peak of the bosonic function at 40 to 75 meV, corresponding to ~17% of the total bosonic function. Prominent candidates as SCPs are the buckling and breathing Cu-O optical modes. The third and last measured time scale is related to the thermalization with all other lattice modes (80% of the total), which include all acoustic modes and the infrared- and Raman-active modes involving c-axis motion of the Cu ions and provide less than 20% of the coupling (green area) in the peak of Π(Ω).

Fig. 3

Disentangling the contributions to the total bosonic function. The electronic (Πbe, red areas), strongly coupled phonon (ΠSCP, blue area), and lattice (Πlat, green area) contributions to the total bosonic function are shown. The insets display the temporal evolution of the temperatures Tbe (red line), TSCP (blue line), and Tlat (green line) determined through the 4TM (18). Sketches of the possible microscopic mechanisms at the base of the different contributions to the total Π(Ω) are shown in the upper panels.

These results have important implications for the identification of the pairing mechanism in cuprates. The electron-boson coupling λb = 2∫Πb(Ω)/Ω dΩ is calculated for each subset b of the bosonic excitations, considering the experimental uncertainties. In the strong-coupling regime (λb < 1.5), the critical temperature can be estimated (18) through an extended version of McMillan’s equation (27), containing the logarithmic-averaged frequency (Ω˜) and the portion of Πb(Ω) that contributes to the d-wave pairing (28). The maximal critical temperature attainable is calculated assuming that each Πb(Ω) entirely contributes to the d-wave pairing. The coupling with SCPs (λSCP = 0.4 ± 0.2) is in complete agreement with the values measured on similar materials via different techniques, such as time-resolved photoemission spectroscopy (16), time-resolved electron diffraction (17), and single-color high-resolution time-resolved reflectivity (29). Although this value is rather close to the threshold of the strong-coupling regime (30, 31), the small value of Ω˜ gives Tc = 2 to 30 K, which is far from being able to account for the high-temperature superconductivity of the system. The coupling of electrons with all other lattice vibrations is even smaller in strength (λlat = 0.2 ± 0.2) and provides an upper bound of the critical temperature of Tc ≈ 12 K. Finally, the large coupling constant (λbe = 1.1 ± 0.2) and the larger Ω˜ value of the electronic excitations give Tc = 105 to 135 K, and this alone accounts for the high critical temperature. We note that, whereas ΠSCP(Ω) and Πlat(Ω) are expected to be temperature-independent, Πbe(Ω) increases as new magnetic excitations emerge when approaching Tc, particularly in the pseudogap phase (10). Therefore, the use of Πbe(Ω) determined at T = 300 K to estimate the QP-boson coupling and Tc underestimates the electronic contribution to the pairing, further supporting our conclusion of a dominant electronic mechanism in the superconductivity of cuprates. All the λb values, maximum attainable critical temperatures, and important parameters for each subset of the total bosonic function are reported in table S1.

The measured values of λbe, λSCP, and λlat and the spectral distribution of the bosonic excitations strongly indicate that the antiferromagnetic spin fluctuations (7, 9) and the loop currents (3) are the most probable mediators for the formation of Cooper pairs. An isotope effect in the dispersion of nodal QPs has been observed by ARPES measurements on optimally doped Bi2212 (32). Analysis of these data (33) indicates that the nodal isotope effect can be explained by assuming that the QP-phonon coupling represents about 10% of the total contribution of other bosonic excitations. From Fig. 3, we estimate that the contribution of SCP is less than 20% of the total bosonic function. Hence, our results fully explain the observed isotope effect, providing a consistent interpretation of the most important experimental results about the QP-boson coupling in cuprates.

Our conclusions are rather independent of the assumption of the histogram-like form of Π(Ω) and are robust against modifications of the details of the equilibrium dielectric function. In fact, the outcome of this work strongly supports the factorization of the self-energy at T = 300 K into a temperature-dependent kernel function and the glue function Π(Ω) (see Eq. 2), even under nonequilibrium conditions. Although we do not exclude a priori that the upper limit used in the determination of the bosonic function can hide possible contributions to Π(Ω) even above 1 eV and that the electron-phonon coupling may cooperate in driving the superconducting phase transition, we demonstrate that bosonic excitations of electronic origin are the most important factor in the formation of the superconducting state at high temperatures in the cuprates. Our findings pave the way for the investigation of electron-boson coupling in a variety of complex materials, such as transition-metal oxides and iron-based superconductors.

Supporting Online Material

www.sciencemag.org/cgi/content/full/335/6076/1600/DC1

Materials and Methods

Fig. S1

Table S1

References (34, 35)

References and Notes

  1. See supporting material on Science Online.
  2. Acknowledgments: We thank M. Capone and A. Chubukov for discussions and suggestions. Partially supported by the Università Cattolica del Sacro Cuore (grant D2.2 2010) (C.G., F.B., and G.F.), by the Italian Ministry of University and Research (grants FIRBRBAP045JF2 and FIRB-RBAP06AWK3) (F.C., G.C., and F.P.) and by the Swiss National Science Foundation (grant 200020-130052) and its Materials with Novel Electronic Properties (MaNEP) program (D.v.d.M.). The crystal growth work was performed in M.G.’s prior laboratory at Stanford University, supported by the U.S. Department of Energy under contract DE-AC03-76SF00515. Work at the University of British Columbia was supported by the Killam, Sloan Foundation, and Canada Research Chair Programs, the Steacie Fellowship Program of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI), NSERC, the Canadian Institute for Advanced Research (CIFAR) Quantum Materials Program, and the British Columbia Synchrotron Institute (BCSI).
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