The character of the electron-phonon interaction (EPI) in metals and superconductors with strongly correlated electrons has recently been the subject of intense research, especially with regard to its influence on high-temperature superconductivity in the copper oxides. Whereas density-functional theory (DFT) predicts only weak EPIs in these materials , various anomalies in the dispersion relations of both conduction electrons and lattice vibrations have been interpreted as evidence of an EPI strength far exceeding these predictions. The question whether strong correlations substantially modify the EPI in the cuprates with respect to standard DFT therefore remains open, as do more general questions about the role of the EPI in driving high-temperature superconductivity and/or competing instabilities such as "stripe" or charge density wave (CDW) order . Similar questions are being asked for other correlated metals including the recently discovered iron-based superconductors.
Fig. 1: (a) Diffuse scattering mapping of the Q=(0,K,L) plane at room temperature and (b) at T=90K. White arrows indicate the CDW superstructure peaks. (c) and (d) display cuts of these maps along (0,K,6.5) and (0, 2–K, 0.5)$, respectively. The red box corners correspond to the Γ, Y, T and Z points of the BZ centered at Q=(0,0,6).[less]
Fig. 1: (a) Diffuse scattering mapping of the Q=(0,K,L) plane at room temperature and (b) at T=90K. White arrows indicate the CDW superstructure peaks. (c) and (d) display cuts of these maps along (0,K,6.5) and (0, 2–K, 0.5)$, respectively. The red box corners correspond to the Γ, Y, T and Z points of the BZ centered at Q=(0,0,6).
We have used non-resonant inelastic x-ray scattering (IXS) with high energy resolution to carefully monitor the temperature dependence of low-energy lattice vibrations around the CDW ordering wavevector of underdoped YBa2Cu3O6.6 with Tc=61K . In such experiments, the phonon intensity depends on the total momentum transfer Q, rather than the reduced momentum q=Q–GHKL, where GHKL stands for the Brillouin zone (BZ) center, Γ, closest to Q. As high-resolution IXS measurements are performed with high-energy (≈20keV) photons, the Ewald sphere encompasses hundreds of BZs. Prior to the IXS experiments, we therefore assembled a comprehensive map of the diffuse scattering intensity to identify the BZs with the most intense x-ray signatures of CDW formation. In Fig. 1 (a), (b), we show maps of the (0,K,L) plane at room temperature and 80K, that reveal the emergence of extended features at low temperatures. The positions, qCDW=(0,0.31,0.5), and correlation lengths of these features are compatible with those of the CDW reflections reported previously on the same material. Their structure factor depends strongly on both K and L, mirroring the complex ionic displacement pattern associated with CDW formation. In the following, we will focus on transverse acoustic and optical phonons in the BZ adjacent to G006, where the CDW features are particularly intense.
Fig. 2: (a) Momentum dependence of the IXS spectra along the Z–T direction of reciprocal space at T=5K. (b) at T=Tc=61K. (c) at T=150K. (d) Temperature dependence of the inelastic part of the IXS spectra at q=(0, 0.25, 6.5). (e) Temperature dependence of the inelastic part of the IXS spectra at q=qCDW=(0, 0.31, 6.5). (f) Details of the fits of the IXS spectra at qCDW at 5, 61, and 295K. A logarithmic scale has been used for clarity. The dashed lines and tick marks indicate the individual phonon profiles resulting from the fits and their maxima, respectively.[less]
Fig. 2: (a) Momentum dependence of the IXS spectra along the Z–T direction of reciprocal space at T=5K. (b) at T=Tc=61K. (c) at T=150K. (d) Temperature dependence of the inelastic part of the IXS spectra at q=(0, 0.25, 6.5). (e) Temperature dependence of the inelastic part of the IXS spectra at q=qCDW=(0, 0.31, 6.5). (f) Details of the fits of the IXS spectra at qCDW at 5, 61, and 295K. A logarithmic scale has been used for clarity. The dashed lines and tick marks indicate the individual phonon profiles resulting from the fits and their maxima, respectively.
The temperature evolution of the phonon profiles in the vicinity of qCDW is displayed in Fig. 2. Whereas the phonon dispersions at T=150K are identical to those at room temperature within the experimental error, and phonon linewidths are limited by the instrumental energy resolution, marked anomalies are observed at lower temperatures. We first focus on the behavior in the superconducting state, where the dispersions of both phonons exhibit pronounced dips in the vicinity of q=qCDW (Figs. 2(a) and 3(a)). These anomalies are not visible in the results of the DFT calculation, and, as they are restricted to a very narrow q-range around qCDW, they were apparently not recognized in prior experimental work on the lattice dynamics of YBa2Cu3O6+x. Remarkably, the sharpness of the dispersion anomaly in YBa2Cu3O6.6 closely resembles those associated with "Kohn anomalies" in quasi-1D CDW compounds with strongly nested Fermi surfaces. In contrast, the phonon anomalies associated with quasi-2D CDWs are considerably broader, reflecting the less pronounced nesting of the associated Fermi surfaces. We now turn to the behavior at higher temperatures, which is even more surprising. Upon heating up to Tc, the phonon energy is weakly T-dependent (although a slight softening of the acoustic phonon at q=qCDW is noticeable, Fig. 2(c)), and the linewidth remains resolution limited over the entire BZ, presumably because the maximum of the superconducting gap exceeds the phonon energy. At Tc, however, the frequency of the TA phonon at qCDW abruptly jumps by about 15% to its normal-state value; the hardening of the optical mode even exceeds 20%. At the same time, the phonon linewidths become extremely large in a narrow range around q=qCDW, where the FWHM of the TA phonon at Tc amounts to 3.5meV, ≈40% of its energy (Fig. 2(b)). Upon further heating, the phonon frequency is approximately T-independent, while the mode gradually narrows. The linewidth becomes resolution limited around T≈150K, where we observed the onset of the CDW peak in the same sample using resonant x-ray scattering.
Fig. 3: (a) Dispersion of the two low-energy phonons in the ZT direction at T=295, 150, 45, and 5K. (b) Momentum dependence of the intensity of the central peak at T=150, 61 and 5K.[less]
Fig. 3: (a) Dispersion of the two low-energy phonons in the ZT direction at T=295, 150, 45, and 5K. (b) Momentum dependence of the intensity of the central peak at T=150, 61 and 5K.
On a qualitative level, the observed superconductivity-induced phonon anomaly is in line with the generic behavior of low-energy phonons in superconductors, which are expected to broaden and harden when the energy gap collapses and low-energy electron-phonon decay channels open up upon heating above Tc. The magnitude of the superconductivity-induced phonon anomalies reported here are by far the largest reported in cuprates, undoubtedly as a consequence of the close competition between superconducting and CDW ground states in this material. In order to further elucidate the nature of this competition, we now take a detailed look at the IXS data above Tc. The elastic line, centered at zero energy, is T-independent and smoothly Q-dependent over most of the BZ (Fig. 3(b)), and can thus be attributed to incoherent scattering from defects. For q=qCDW, however, we observe an additional contribution to the elastic intensity. The T-dependent intensity and q-width of this contribution are in excellent agreement with those inferred from the quasi-elastic scattering previously determined by resonant x-ray scattering experiments. Detailed analysis of the elastic peak and of the Stokes and anti-Stokes part of spectra allows us to put an upper bound of ≈100μeV on the intrinsic energy width of the elastic component of the CDW signal. This implies that CDW domains with characteristic fluctuation energies below ≈100μeV and typical dimensions of 1–10nm (inferred from the momentum widths of the CDW peaks) are present in a wide temperature range both above and below Tc.
The momentum and temperature dependence of the elastic intensity in the IXS spectra of YBa2Cu3O6.6 is reminiscent of the behavior of other materials undergoing structural phase transitions, including insulating SrTiO3, superconducting Nb3Sn and metallic ZrTe3. In these compounds, an elastic "central peak" appears in the fluctuation regime above the critical temperature, where it is understood as defect-induced nucleation of finite size domains of the low-temperature phase. In YBa2Cu3O6+x, both local lattice distortions generated by oxygen defects in the CuO chains and extended defects such as dislocations may act as pinning centers for CDW nanodomains. The extremely large phonon linewidths in the normal state can then be attributed to inhomogeneous broadening.
Since doping-induced lattice defects are present in all superconducting cuprates, the gradual onset of a spatially inhomogeneous CDW domain state with decreasing temperature may be a generic feature of the "pseudogap" regime in the cuprate phase diagram, although the temperature and doping dependence of the corresponding volume fractions may depend on the specific realization of lattice disorder in different materials. Whereas CDW nanodomains will surely contribute to the anomalous normal-state properties observed in this regime, their gradual nucleation explains the absence of thermodynamic singularities associated with CDW order, at least in the absence of a magnetic field. The persistence of this domain state over a much wider temperature range than corresponding phenomena in classical materials probably reflects the strong competition between CDW correlations and superconductivity. In the presence of superconducting long-range order, the inhomogeneity is strongly reduced. Conversely, thermodynamic singularities and NMR signals due to CDW long-range order have been reported in external magnetic fields strong enough to weaken or obliterate superconductivity. The fragility of the CDW nanodomain state in zero field and its competition with superconductivity explain the isotope effect on the superconducting penetration depth observed in the YBa2Cu3O6+x system.
We end our discussion with some remarks about the implications of our results for the mechanism of high-temperature superconductivity. As it involves low energy phonons, the large superconductivity induced renormalization revealed by our IXS study likely accounts for a large part of the total electron-phonon coupling. It appears strong enough to be a major contributor to at least some of the "kinks" observed in the dispersions of fermionic quasiparticles in the cuprates. Since it is very sharply concentrated in momentum space, however, its momentum-averaged strength seems insufficient to be a significant driving force for Cooper pair formation. Rather, the EPI favors a CDW instability that strongly competes with superconductivity and reduces the superconducting Tc at moderate doping levels. We have confirmed these conclusions by repeating some of the IXS experiments on a fully oxygenated YBa2Cu3O7 crystal with Tc=90K, where we found neither phonon anomalies characteristic of incipient CDW formation nor a central peak heralding CDW nanodomains.