Electron-phonon coupling revealed by charge density fluctuations in cuprate superconductors

TL;DR

This work investigates how electron–phonon coupling (EPC) evolves in the strongly correlated cuprate cuprate superconductor YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) and how it relates to dynamic charge fluctuations (CDF). Using Cu $L_3$-edge resonant inelastic x-ray scattering (RIXS), the authors track bond-stretching phonons and charge fluctuations across a wide range of dopings $p$, temperatures, and momenta, identifying a pronounced phonon softening and enhanced EPC at the CDF wave vector $\mathbf{q}_\mathrm{c}$ that peaks near $p \approx 0.19$, near optimal superconductivity. They distinguish dynamic CDF from quasi-static CDW, showing that the phonon renormalization tracks the CDF energy scale $\omega_0$, which decreases toward zero near optimal doping, correlating with the superconducting dome. The results support a cooperatively enhanced EPC–CDF mechanism that modulates lattice dynamics and suggests EPC in cuprates is an emergent, doping-dependent property shaped by the electronic environment, with implications for superconductivity and potentially a unifying framework across unconventional superconductors.

Abstract

Electron-phonon coupling (EPC) governs lattice dynamics, charge transport, and collective electronic phases in quantum materials. In several families of unconventional superconductors, including transition-metal dichalcogenides and kagome metals, growing evidence points to a cooperative role of EPC and dynamic charge-density fluctuations (CDF) in stabilizing superconductivity. However, how the EPC strength evolves across phase diagrams and relates to superconducting properties in strongly correlated systems remains an open question. Here we investigate the interplay between phonons and the CDF recently identified in cuprate superconductors. Using resonant inelastic x-ray scattering, we track the dispersion and intensity of bond-stretching phonons in YBa$_2$Cu$_3$O$_{7-δ}$ over wide ranges of doping, temperature, and momentum. We find that both the phonon softening at the CDF wave vector and the EPC strength, extracted from a pronounced phonon intensity anomaly, are maximized near $p = 0.19$, where superconducting properties are optimal and CDF intensity is strongest. These results identify dynamic charge-density fluctuations, rather than quasi-static charge density waves, as the dominant source of phonon renormalization in cuprates, and establish a direct correlation between EPC strength and the superconducting dome. More broadly, our measurements highlight EPC as a doping-dependent property of correlated materials, shaped by the electronic environment in which lattice vibrations are embedded.

Figures (4)

• Figure 1: Entwining of charge order and BS phonons measured by RIXS. (a)-(b) Intensity maps acquired at 20 K on YBCO, at doping levels where charge order is dominated respectively by CDW ($p=0.13$) and CDF ($p=0.19$). (c)-(d) Fits of representative RIXS spectra for YBCO with $p = 0.13$ at two momentum values (indicated by dashed lines in the intensity map): $q = 0.30$, where the softening is most pronounced, and $q = 0.44$, where it is significantly reduced. The change in energy of the BS phonons, represented by the green Gaussian, is highlighted by the light blue bar. The red, light violet, and blue Gaussians, as well as the region below the gray dashed line, correspond respectively to the elastic peak, the CDF contribution, the BS overtone, and the paramagnons. (e)-(f) Same as (c)-(d), but for YBCO $p=0.19$.
• Figure 2: Temperature evolution of charge order intensity and BS phonon energy. (a)-(b) The energy of the BS phonons, determined from the Gaussian fits as in Fig. \ref{['fig:figure1']}, plotted as a function of temperature and q, respectively for YBCO $p=0.13$ and $p=0.19$. Lines are guides to the eye. Previous INS measurements Pintschovius2005, performed at $T\approx10$ K, are overlaid in grey on our data for direct comparison. The grey shaded area indicates the 95% confidence interval extracted from the fitting procedure. (c)-(d) Energy-integrated intensities of the quasielastic peak, evaluated in the range [–0.1, 0.035] eV, respectively for YBCO $p=0.13$ and $p=0.19$. Only in the underdoped case the peak is dominated at low temperature (below 200 K) by CDW, while CDF persist at higher temperatures and at all temperatures in the fully oxygenated sample.
• Figure 3: Azimuthal evolution of charge order intensity and BS phonon energy. (a)-(b) Maps for YBCO $p=0.13$ and $p=0.19$, showing the BS phonon energy as a function of momentum $q$ and azimuthal angle $\phi$, within the wedge-shaped region of the Brillouin zone connecting the $\Gamma$–X and $\Gamma$–M directions, as sketched in the central inset. (c)-(d) Maps for YBCO $p=0.13$ and $p=0.19$, showing the integrated intensity of the quasielastic peak in the energy range [–100, 35] meV, across the same wedge-shaped region of the Brillouin zone as in (a)-(b).
• Figure 4: BS phonon softening driven by CDF: converging fingerprints of superconductivity and electron–phonon coupling. (a) Height of the quasielastic peak and (b) magnitude of the BS phonon energy softening $\Delta E$ as a function of temperature for YBCO $p = 0.13$ (diamonds) and $p = 0.19$ (circles). In panel (b), the temperature dependence of the CDF energy $\omega_0$ for $p = 0.19$ is also shown (red circles). The vertical dashed line indicates $T_\mathrm{c}$ at $p = 0.19$. (c) BS phonon energy as a function of momentum transfer $q$ and doping $p$, measured at $T = 20$ K. The 3D surface highlights a pronounced softening at $p = 0.19$, where both the CDF intensity and the superconducting strength reach their maximum. Experimental data points are overlaid on the surface. (d) Doping dependence of $\Delta E$ at $T=20$ K. The data closely follow the doping dependence of the CDF peak height, as reported in Ref. NatCommRArpaia (light blue region). (e) Momentum dependence of the BS phonon intensity for different doping levels at $T = 20$ K. The average intensity over a broad momentum range ($q > 0.3$ r.l.u.) defines the integrated quantity $I_{\mathrm{BS}}$. (f) Doping dependence of $I_{\mathrm{BS}}$ at $T = 20$ K, which mirrors the behavior of both the CDF peak height and the phonon softening $\Delta E$.