Main

With sufficient energy resolution, resonant inelastic X-ray scattering (RIXS) can be an ideal probe for revealing the CDW excitations in cuprates. By tuning the incident photon energy to the Cu L3-edge (Fig. 1a), the resonant absorption and emission processes can leave the system in excited final states, which couple to a variety of excitations arising from orbital, spin, charge, and lattice degrees of freedom15. Thus, information of these elementary excitations in energy and momentum space can be deduced from analysing the RIXS spectra as functions of the energy loss and the momentum transfer of the photons (Fig. 1a). This is highlighted by the pivotal role that RIXS has recently played in revealing orbital and magnetic excitations in cuprates16,17,18,19,20. In addition, RIXS provided the first X-ray scattering evidence for an incommensurate CDW in (Y, Nd)Ba2Cu3O6+δ (ref. 4), owing to energy resolution that separated the quasi-elastic CDW signal (bright spot in Fig. 1b, limited by the instrumental resolution 130 meV) from other intense higher-energy excitations. Notably this quasi-elastic signal is asymmetric with respect to zero energy loss (Fig. 1c), which indicates the possible existence of additional low-energy excitations near the CDW wavevector (QCDW).

Figure 1: RIXS process and a hint of lower-energy excitations near QCDW.
figure 1

a, Schematics of the coherent two-step RIXS process. The photon energy of the incident X-ray (i) is tuned to the Cu L3-edge. Upon absorption, a core electron in the Cu 2p orbital made a transition to the unoccupied 3d valence band near the Fermi energy EF, bringing the system to an intermediate state |m〉. Then, the emission process occurs that fills the 2p core hole by one of the valence electrons, emits a photon (f), and leaves the system in an excited final state |f〉, which couples to a variety of elementary excitations. The energy–momentum information of these elementary excitations can be deduced by tracking the peak and spectral weight in RIXS spectra as a function of the momentum transfer Q and the energy loss of the scattering photons. The momentum of the incident and scattered photons is represented by ki and kf, respectively. Since the electronic state in Bi2212 is quasi-two-directional (that is, almost independent along the c-axis), all data are plotted as a function of projected momentum transfer Q along the [100] direction (that is, along the Cu–O bond direction). b, RIXS intensity map of the underdoped Nd1.2Ba1.8Cu3O6+δ (UD-NBCO) compound around the CDW position, taken with a resolution of 130 meV. c, RIXS spectrum along the red dashed line shown in b. d, A high energy-resolution (40 meV) RIXS spectrum of UD-Bi2212 at Q = 0.31 r.l.u. An enlarged view of the black-dashed box near the elastic peak is shown in the lower panel.

In this work, we exploit the newly commissioned ultrahigh-resolution RIXS instrument at the European Synchrotron Radiation Facility to reveal these low-energy excitations near the CDW. We choose the double-layer cuprate Bi2.2Sr1.8Ca0.8Dy0.2Cu2O8+δ (Bi2212), whose electronic structure has been extensively studied by surface-sensitive spectroscopy, such as scanning tunnelling microscopy21 and angle-resolved photoemission22, and in which a short-range CDW order was recently reported7,8. With improved energy resolution up to 40 meV, we see additional features in the previous quasi-elastic region (Fig. 1d).

Figure 2a presents an energy–momentum RIXS intensity map of our high-resolution data. Two excitation branches are clearly observed with a momentum-dependent intensity distribution, also evident in the energy-loss spectra at representative momenta (Fig. 2b). The first branch centred at zero energy should contain the scattering signal due to an underlying CDW. Indeed, as plotted in Fig. 2c, the momentum distribution of the RIXS intensity averaged over a small energy window exhibits a symmetric peak at a finite momentum, with a peak-to-background ratio of 2, unambiguously confirming the existence of the CDW in Bi2212. The peak position is located at QCDW 0.3 reciprocal lattice units (r.l.u.), consistent with previous STM studies8 and with a full-width at half-maximum (FWHM) of approximately 0.085 r.l.u., corresponding to a short correlation length of approximately 15 Å. Interestingly, despite the short correlation length, the CDW strength, estimated by the integrated intensity of the CDW peak in the quasi-elastic region, is not weaker than that in YBa2Cu3O6+x (see Supplementary Information).

Figure 2: CDW and phonons at 20 K.
figure 2

a, RIXS intensity map of the high energy-resolution data (ΔE 40 meV) as a function of energy loss and Q. The white circles represent the fitted phonon dispersion. b, Energy-loss spectra at selected momentum values ranging from Q = 0.235 r.l.u. to Q = 0.425 r.l.u. The fits (solid lines) are superimposed on the raw data (black circles). c, Averaged intensity in the quasi-elastic region, defined as the region between the two white dashed lines in a. The solid line is a Lorentzian fit to the data with a background consisting of a constant plus a Lorentzian to account for the tail of the specular reflection peak at Q = 0. d, Demonstration of the quality of the fit of a RIXS spectrum at a representative momentum. The fitting process is described in the Methods section. The phonon peak is highlighted in pink. e, Position and FWHM (de-convolved) of the measured phonon peak extracted from the fits. The red dashed lines on a and e indicate QCDW. Error bars in a and e (upper panel) are estimated by the uncertainty in determining the zero energy loss. The error bars for the FWHM in e are two standard deviations from the fit. Those in c are determined by the noise level of the spectra.

The second branch of excitations possesses an energy scale of approximately 60 meV, whose energy–momentum-dispersion can be reliably extracted (Fig. 2d and Supplementary Fig. 2). The extracted dispersion agrees well with that of the Cu–O bond-stretching phonon measured by non-resonant inelastic X-ray scattering23 (Supplementary Information). Owing to the high momentum-resolution of our data, Fig. 2e shows mode softening by approximately 25% at QCDW, with a corresponding broadening of the fitted peak width, indicating that the short-ranged ordered CDW unambiguously affects the lattice. This observation is reminiscent of the bond-stretching phonon anomaly reported in the striped-ordered cuprates24 and the acoustic phonon anomaly in YBCO25, where CDW correlation lengths are notably longer. Curiously, the phonon softening occurs gradually near QCDW, unlike the Kohn anomaly due to the reduced lattice symmetry from the CDW, which usually occurs more abruptly in a narrow momentum window. In addition, whereas the width of the phonon self-energy directly measured by inelastic neutron or non-resonant inelastic X-ray scattering is typically a few millielectronvolts23,24, the fitted width of peak shown in Fig. 2e progressively broadens near QCDW and is not resolution limited (40 meV), hinting at the existence of additional low-energy excitations.

Remarkably, the phonon intensity varies non-monotonically with momentum, having a maximum near QA 0.37 r.l.u = 0.07 r.l.u. + QCDW (Fig. 3a). It is important to note that the RIXS phonon cross-section directly reflects the momentum dependence of the electron–phonon coupling strength26,27,28 (Supplementary Information), in stark contrast with the phonon self-energy measured by other scattering techniques. As a consequence, RIXS is also directly sensitive to the interference between phonons and underlying charge excitations (that is, the Fano effect), which can manifest as an intensity anomaly. Thus, the intensity anomaly at QAQCDW (Figs 2a and 3a) provides evidence for the existence of dispersive CDW excitations that cause a strong Fano interference at the intersection QA with the bond-stretching phonon.

Figure 3: CDW excitations in Bi2212 and calculated RIXS intensity in a 1D model.
figure 3

a, RIXS phonon intensity extracted from the averaged intensity over the energy range between the two black-dashed lines in c. The black arrow highlights the intensity anomaly QA. Error bars are estimated by the noise level of the spectra. b, Momentum-distribution curves (MDCs) at fixed energy of the raw data (Fig. 2a) from E = 0 eV to E = 0.1 eV. Both the raw data (markers with dashed lines) and the smoothed curves (solid lines) are superimposed. The MDCs corresponding to the CDW and phonon energy regions are highlighted in blue. The red ticks serve as guides to the eye for the connection between CDW and the phonon intensity anomaly. QCDW represents the CDW wavevector. c, Elastic peak-subtracted intensity map. df, Calculations of RIXS phonon intensity for a 1D metallic system, with the same plotting format and notations used in ac. 2kF represents the perfect nesting wavevector of the 1D system. All the experimental (calculated) data are plotted as a function of QQCDW (Q − 2kF).

Subtle but non-negligible spectral features further support the existence of dispersive CDW excitations. Examining the momentum-distribution curves (Fig. 3b), the peak at zero energy disperses and broadens with increasing energy loss, smoothly connecting to the phonon at QA. To visualize this connection in an alternative way, we remove the elastic contribution from the energy-loss spectra (Fig. 3c) to reveal a funnel-shaped RIXS intensity emanating from QCDW, suggesting that these excitations are associated with the CDW. This funnel-shaped intensity essentially reflects the anomalous broadening of the peak width near QCDW shown in Fig. 2e. These CDW excitations disperse to high energy and intersect with the phonon at QA, causing the phonon intensity anomaly due to the strong Fano interference effect.

To demonstrate the Fano resonance in the RIXS phonon cross-section, we calculate the RIXS intensity map for a one-dimensional (1D) metallic system with coupling between electrons and a bond-stretching phonon mode (Fig. 3f). Perfect Fermi surface nesting at 2kF (that is, twice the Fermi momentum) creates a particle–hole continuum, whose presence softens the phonon dispersion (that is, not a Kohn-like phonon anomaly) and creates an intensity anomaly at the intersection between the continuum and the phonon (Fig. 3d–f). This result qualitatively agrees with our observations in Bi2212. We also note that the strength of the Fano interference is dictated by the electron–phonon coupling strength at the intersecting momentum. Since the coupling strength of the bond-stretching phonon is zero at Q = 0 and increases monotonically with momentum, the Fano interference is weaker at Q < QCDW than at Q > QCDW. Finally, by connecting QCDW and QA, we deduce a characteristic velocity of these CDW excitations: VCDW 0.6 ± 0.2 eV Å. We find that VCDW is neither equivalent to the Fermi velocity seen in ARPES nor the quasi-particle-interference dispersion seen in STM21 (Supplementary Information), indicating that the weak-coupling fermiology may be irrelevant for describing the observed dispersion.

Temperature-dependent measurements were conducted on another Bi2212 crystal with a similar Tc (40 K), with results shown in Fig. 4. We first confirm that the low-temperature measurement at 20 K reproduces all the observations shown in Fig. 2 (Supplementary Fig. 4). Upon increasing the temperature to 240 K (near the pseudogap temperature22), we find that the CDW signal in the quasi-elastic region persists (Fig. 4a) with reduced peak intensity (Fig. 4b). We also observed that whereas the phonon softening still remains (Supplementary Fig. 4), the intensity anomaly at QA diminishes and broadens, as shown by the momentum distribution of the phonon intensity in Fig. 4c. This indicates that the CDW excitations still notably affect the charge dynamics, but become poorly defined in energy–momentum space as the CDW weakens. Interestingly, QCDW appears to change from incommensurate 0.3 r.l.u. at 20 K to nearly commensurate 0.26 r.l.u. at 240 K (Fig. 4b, d), which corresponds to a real-space periodicity of approximately four lattice constants (4ao), similar to the stripe-ordered cuprates1,3. This indication of temperature-dependent CDW commensuration requires future investigation to further confirm these findings.

Figure 4: Temperature dependence of the CDW and the phonon anomaly.
figure 4

a, RIXS intensity map taken at T = 240 K. The white dashed lines define the quasi-elastic region (centred at zero energy loss) and the phonon energy region (centred at 60 meV) used for calculating the averaged intensity profile shown in b and c. b, Averaged intensity of the quasi-elastic region (indicated in a) at 240 K (red) and 20 K (blue). The dashed lines indicate the position of the CDW. The RIXS intensity map taken at T = 20 K is presented in Supplementary Fig. 4. c, Averaged intensity of the phonon energy regions, as defined in a, at 240 K and 20 K. The black arrow highlights the intensity anomaly at 20 K. Error bars in b and c are estimated by the noise level of the spectra. d, Raw energy-loss spectra (markers) and the corresponding fits (solid lines) of RIXS data taken at T = 240 K (top) and T = 20 K (bottom). The elastic peak fits are highlighted with filled areas. The RIXS spectra where the elastic peak intensity is maximal (that is, the CDW position) are indicated by red and blue boxes for T = 240 K and T = 20 K, respectively.

Our results have several important implications. They provide the first direct evidence for the existence of dispersive CDW excitations in the charge–charge correlation function with a characteristic velocity VCDW. Furthermore, these excitations persist to high energy, at least up to 60 meV, causing the phonon intensity anomaly, indicating that the CDW influences the charge and lattice degrees of freedom, despite its short correlation length. Whether the observed CDW excitations are collective due to a symmetry breaking29 or rather a reflection of the particle–hole continuum remains an open question for future investigation. Additionally, if confirmed, the surprising change of QCDW from incommensurate at low temperature towards commensurate at high temperature is reminiscent of the temperature dependence observed in stripe-phased nickelates, where the charge order is strongly coupled to a spin order, which originates from strong real-space interactions30. These results may be consistent with the proposal that the incommensurate QCDW forms from a discommensuration of a universal period 4ao modulations31. Finally, the CDW signatures survive up to at least 240 K T, which taken together with the similarities to striped nickelates indicates a close relationship between the CDW, spins, and the pseudogap.

Methods

Sample growth, preparation and characterization.

The high-quality UD-Bi2212 (Bi2.2Sr1.8Ca0.8Dy0.2Cu2O8+δ) single crystals (Supplementary Fig. 1a, sample 1 and 2) were grown by floating-zone (FZ) methods and annealed in a nitrogen atmosphere. The samples were characterized and roughly aligned using Laue diffraction prior to RIXS measurements (Supplementary Fig. 1a). The samples were cleaved in air immediately before loading into the high-vacuum measurement chamber. The transition temperatures TC are 45 K and 40 K for samples 1 and 2, respectively (Supplementary Fig. 1b), corresponding to doping concentration p 8–9%. The X-ray absorption curves for samples 1 and 2 are shown in Supplementary Fig. 1c, showing no apparent contamination. Data shown in Figs 2 and 3 were taken on sample 1. Data in Fig. 4 were taken on sample 2.

Low-resolution RIXS measurements on UD-NBCO.

The low energy-resolution RIXS data on the UD-NBCO (Nd1.2Ba1.8Cu3O6+δ) sample were taken at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI, Switzerland) using the SAXES spectrometer32,33. The RIXS spectra were measured at T = 60 K, just below Tc = 65 K to maximize the CDW peak intensity, and using linear vertical (σ) polarized incident photons. The total energy resolution was 130 meV, the scattering angle was 130°. The RIXS map is obtained by combining spectra measured every 2° in the rotation of the incidence angle on the sample surface. The NBCO films, 100 nm in thickness, were deposited by high-oxygen-pressure diode sputtering on SrTiO3 (100) single crystals. The lattice parameters measured by X-ray diffraction are a = b = 3.84 Å and c = 11.7 Å.

Ultrahigh-resolution RIXS measurements on UD-Bi2212.

The RIXS measurements on the UD-Bi2212 samples 1 and 2 were performed using the ERIXS spectrometer at the ID32 beamline of the European Synchrotron Radiation Facility (ESRF, France). The RIXS spectra were taken with the photon energy of the incident X-rays tuned to the maximum of the absorption curve near the Cu L3-edge (Supplementary Fig. 1c). The scattering geometry is sketched in Supplementary Fig. 1d. The data were collected at T = 20 K for sample 1 and T = 20 K/240 K for sample 2 with a linear vertical polarization (σ-polarization) of the incident beam. The energy resolution was approximately ΔE 40 meV (sample 1) and ΔE 45 meV (sample 2). The scattering angle of the endstation was set at 2θ = 149.5°. Since the electronic state in Bi2212 is quasi-two-directional (that is, almost independent along the c-axis), the data shown in this report are plotted as a function of projected momentum transfer Q (that is, projection of Q = kfki on the CuO2 plane). Different in-plane momentum transfers, Q (projection of the scattering vector Q along [100]), were obtained by rotating the samples around the vertical b-axis. Note that the scattering vector Q is denoted using the pseudotetragonal unit cell with a = b = 3.82 Å and c = 30.84 Å, where the c-axis is normal to the sample surface. In our convention, positive Q corresponds to grazing-emission geometry and negative Q corresponds to grazing-incidence geometry. Supplementary Tables 1 and 2 summarize the L values (that is, projection of the scattering vector Q along [001]) for each momentum in the data of sample 1 and 2, respectively.

Data analysis and fitting.

The ultrahigh-resolution data (UD-Bi2212) were normalized to I0 and corrected for self-absorption effects using the formalism described in the Supplementary Methods of ref. 34. The zero energy positions were determined by comparing the spectrum recorded on a small amount of silver paint (at each Q) near the sample surface and fine adjusted by the fitted elastic peak position. The fitting model involves a Gaussian (elastic peak), a Lorentzian (phonon peak) and a background that fits an anti-symmetrized Lorentzian to the paramagnon at high energy loss. The model is convolved with the energy resolution (Gaussian convolution) and fitted to the data. The results of these fits are presented in Supplementary Fig. 2 for sample 1 (T = 20 K) and Fig. 4d and Supplementary Fig. 3 for sample 2 (T = 20 K and T = 240 K). Please note the plotted peaks are also already convolved with the energy resolution for ease of perceiving the fit quality.

From this analysis, we extracted the dispersion (peak position) and FWHM (resolution de-convolved) of the phonon peak at T = 20 K in sample 1 (Fig. 2e) and at T = 20 K and T = 240 K in sample 2 (Supplementary Fig. 4c, d). The dispersions and FWHM found at T = 20 K for both sample 1 and sample 2 are similar, with the presence of a softening and a broadening around the CDW wavevector (QCDW 0.3 r.l.u.). The reproducibility of the low-temperature data in two independent measurements and samples confirms our observations of the CDW and the phonon intensity anomaly due to Fano interference (Supplementary Fig. 4a). In addition, we note that the phonon intensity extracted from the fitting procedure agrees with the averaged intensity around the phonon energy (Supplementary Fig. 5b).

Theory.

The calculations in Fig. 3d–f were carried out using a recent framework for phonon contributions to RIXS given in ref. 28. Details are given in that reference. The theory developed in ref. 28 considered lowest-order diagrams for RIXS in the weak electron–phonon limit, giving the first contribution to RIXS coming from the emission of one phonon. The strength of the signal is directly tied to the momentum-dependent electron–phonon coupling. In the limit of small coupling, the intensity smoothly varies in momentum space, with a position that closely follows the bare phonon dispersion. However, if the phonon dispersion crosses the electron–hole continuum, a Fano effect occurs, leading to a strong interference between the dispersions of the two excitations, as shown in Fig. 3d–f.

Since we have used a simple 1D metallic model having a Peierls instability at 2kF, the particle–hole excitations disperse in energy and momentum in a cone from zero energy, with a dispersion set by the Fermi velocity. When the cone of charge excitations crosses the phonon line, an interference occurs that exploits the momentum-dependent electron–phonon coupling g(q). For the case of the bond-stretching phonon, g(q) determined from the nature of deformation coupling between Cu and O in-plane displacements is largest for large momentum transfers. Therefore, the intensity is stronger on the higher-momentum side of the intersection of the phonon line with the charge excitations. This effect reproduces the experimental observations well.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Additional Information

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