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

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 (13–15) 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 (T_e) 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 Π_be refers to the bosonic excitations of electronic origin at the effective temperature T_be, Π_SCP to the small fraction of strongly coupled phonons (SCPs) at T_SCP (16), and Π_lat to all other lattice vibrations at T_lat. Because the term that couples the rate equations (18) for T_e and T_b (with b = be, SCP, lat) is G(Π_b, T_b, T_e)/C_b [where G is the functional described in (18) and C_b 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 T_c, 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 Bi₂Sr₂Ca_0.92Y_0.08Cu₂O_8+δ (Y-Bi2212) high-quality crystals (20) (T_c = 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 (13–15), 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)=∫0∞Π(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(Ω−Ω′)2πTe]−Ψ[12−i(Ω+Ω′)2πTe] (3)where Ψ is the digamma function obtained by integrating the Fermi-Dirac functions, and N(Ω, T_b) is the Bose distribution at temperature T_b. The kernel function can be decomposed into different terms depending on the electronic (T_e) and bosonic (T_b) temperatures. The independent variation of T_e,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 (δT_e = δT_b > 0) and nonthermal (δT_e > 0, δT_b = 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, Ω_p ≈ 1 eV—that is, an energy scale much higher than the energy scale of the bosonic function.

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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 Ω_p 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.

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The relative strengths of Π_b(Ω) (where the index b refers to the three components in Eq. 1) determine (26) both the temporal evolution of the temperatures T_b, 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, T_e, T_be, T_SCP, T_lat) 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 C_be and C_SCP.

Figure 3 summarizes the main results of this work. The pump pulse of 10 μJ/cm² gently increases the electronic temperature by δT_e ~ 2 K. The entire high-energy part and ~46% of the peak (red areas) instantaneously thermalize with electrons at a temperature T_be ≈ T_e. The spectral distribution and the value of the specific heat of these excitations (C_be < 0.1C_e) 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 (C_SCP = 0.2C_lat) 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).

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These results have important implications for the identification of the pairing mechanism in cuprates. The electron-boson coupling λ_b = 2∫Π_b(Ω)/Ω dΩ is calculated for each subset b of the bosonic excitations, considering the experimental uncertainties. In the strong-coupling regime (λ_b < 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 Π_b(Ω) that contributes to the d-wave pairing (28). The maximal critical temperature attainable is calculated assuming that each Π_b(Ω) entirely contributes to the d-wave pairing. The coupling with SCPs (λ_SCP = 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 T_c = 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 (λ_lat = 0.2 ± 0.2) and provides an upper bound of the critical temperature of T_c ≈ 12 K. Finally, the large coupling constant (λ_be = 1.1 ± 0.2) and the larger Ω˜ value of the electronic excitations give T_c = 105 to 135 K, and this alone accounts for the high critical temperature. We note that, whereas Π_SCP(Ω) and Π_lat(Ω) are expected to be temperature-independent, Π_be(Ω) increases as new magnetic excitations emerge when approaching T_c, particularly in the pseudogap phase (10). Therefore, the use of Π_be(Ω) determined at T = 300 K to estimate the QP-boson coupling and T_c underestimates the electronic contribution to the pairing, further supporting our conclusion of a dominant electronic mechanism in the superconductivity of cuprates. All the λ_b 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 λ_be, λ_SCP, and λ_lat 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.