Multiorbital charge-density wave excitations and concomitant phonon anomalies in Bi2Sr2LaCuO6+δ

Significance Charge-density waves (CDWs) are a ubiquitous form of electron density modulation in cuprate superconductors. Unveiling the nature of quasistatic CDWs and their dynamical excitations is crucial for understanding their origin––similar to the study of antiferromagnetism in cuprates. However, dynamical CDW excitations remain largely unexplored due to the limited availability of suitable experimental probes. Here, using resonant inelastic X-ray scattering, we observe dynamical CDW excitations in Bi2Sr2LaCuO6+δ (Bi2201) superconductors through its interference with the lattice. The distinct anomalies of the bond-buckling and the bond-stretching phonons allow us to draw a clear picture of funnel-shaped dynamical CDW excitations in Bi2201. Our results of the interplay between CDWs and the phonon anomalies shed light on the nature of CDWs in cuprates.

In Fig. 1C, we highlight an excitation at ~60 meV whose intensity is enhanced for q// between QCDW and the zone boundary. At the O K (Fig. 1D) the excitation is much weaker in comparison to the CDW peak. To quantify the q// dependence of the inelastic component, we fitted the quasielastic peak and then subtracted it from the RIXS spectra (Method, and SI Appendix, Figs. S5-S6).
Fitting examples of the excitation spectra at the Cu L3-and O K-edges are shown in Fig. 2A and To elucidate the latter, we collected higher energy resolution RIXS spectra from the same sample and were able to clearly resolve the lower-energy peak (SI Appendix, 2G-2J. At the Cu L3 resonance (Fig. 2G), the inelastic peak is located around 65 meV at small q// then softens in the q// range of 0.2 ~ 0.4 r.l.u., with a broad 'dip' developing at about 50 meV near QCDW. The extracted dispersion matches well with those obtained using π polarized light and they are reminiscent of the phonon softening observed in Bi2212 (17) (SI Appendix, Fig. S8). Although the softening wavevector is close to QCDW in Bi2201 and Bi2212, a recent RIXS study of CDW correlations in LSCO shows that the phonon softening develops at Q > QCDW, possibly implying a delicate relationship between the momentum of the phonon softening and the CDW wavevectors (19). In Fig. 2H, the integrated intensity has a non-monotonic increase as a function of q// with a maximum around 0.35 r.l.u.. Noticeably, both the dispersion and the intensity profiles of the inelastic peak agree very well with that of the Cu-O bond-stretching phonon branch (the halfbreathing D1 mode along the (100) direction) in the under-doped Bi2212 (17). The intensity enhancement at wavevectors larger than QCDW resembles the Fano interference effect of the bond-stretching phonon in the under-doped Bi2212 (17). These observations suggest the existence of the dispersive CDW excitations that interact with the bond-stretching phonon in UD23. We highlight that the phonon intensities measured by RIXS and IXS are very different: the former is proportional to the strength of the electron-phonon coupling (EPC) in the reciprocal space while the latter measures the phonon self-energy (15)(16)(17)(20)(21).
Concerning the excitation at the O K-edge, the high-energy peak exhibits a downward dispersion to 0.2 r.l.u., before abruptly forming a dome-shaped enhancement peaked around QCDW of 0.25 r.l.u.. The central energy of the low-energy branch is comparable to that of a bond-buckling phonon in YBa2Cu3O7 (YBCO) identified by INS (22). In fact, a RIXS study on the undoped compound NdBa2Cu3O7 (NBCO) revealed the in-phase A1g mode of the bond-buckling phonon at ~ 30 meV in accord with our data (23). RIXS experiments on NBCO and the model calculations both confirm that the EPC of the in-phase bond-buckling phonon decreases from the Brillouin zone center towards the zone boundary while the EPC of the bond-stretching phonon shows the opposite trend (20,(23)(24). Comparing RIXS results between UD23 and NBCO, we found that the q-dependent EPC of the high-energy bond-stretching phonon agrees reasonably well. But an apparent discrepancy exists for the low-energy bond-buckling phonon: the dome-shaped intensity enhancement around QCDW in UD23 is in stark contrast to the monotonically decreasing EPC in parent NBCO (23). The anomalous softening of the buckling phonon in YBCO was suggested to be associated with a charge-density modulation, which is now understood to be omnipresent in under-doped cuprates (22). The giant intensity anomaly of the buckling mode in UD23 may, therefore, reflect an interplay with the dispersive CDW excitations.
To elucidate the anomalous intensity enhancement of the buckling phonon, we surveyed the other parts of the phase diagram by investigating a slightly over-doped Bi2Sr1.8La0.2CuO6+d superconducting compound (Tc = 30 K, OD30), anticipating that the CDW correlations and EPC may be rather different than its under-doped counterpart. In Figs. 3A-3B, Cu L3 and O K RIXS spectra of OD30 are plotted as a function of q// = (H, 0). No scattering peak is observed in the quasi-elastic region at either edge across the whole accessible q// range. To expand on this observation, we studied RIXS under various configurations including along the (-H, 0) direction, the (H, H) direction, and in off-resonance conditions (SI Appendix, Fig. S9). None of them reveal any CDW signatures, making this system distinct from under-doped and the extremely over-doped Bi2201, where the CDW was found (6,25). We illustrate these remarkable results in Figs. 3C-3D.
In contrast to the well-defined CDW peak in the UD23 sample, the integrated intensity of the quasi-elastic peak in OD30 has a simple background-like profile demonstrating the complete obliteration of CDW correlations.
Correspondingly, the phonon excitations manifest differently in the absence of CDW correlations.
First, the bond-stretching phonon softening is largely suppressed, regardless of whether it is probed on the Cu (Fig. 3E) or O sublattices (Fig. 3F). Second, the EPC anomaly of the bond-stretching mode, i.e., the broad intensity enhancement at q// > QCDW, diminishes in both sublattices resulting in a simple upward increase (Figs. 3G-3H). Similarly, the intensity of the bond-buckling phonon is significantly altered with no appreciable change in its dispersion (Fig. 3I). The entire domeshaped enhancement vanishes in OD30 and a monotonically decreasing intensity profile is formed as a function of q// (Fig. 3J), opposite to the trend of the bond-stretching phonon.
It becomes clear now that the momentum-dependent EPC of each phonon branch in the absence of CDW correlations in OD30 coincides with that of the parent NBCO where the bond-stretching and the bond-buckling phonon intensities scale with sin 2 (pH) and cos 2 (pH) functions, respectively (20,(23)(24). We found that these functional forms describe the observed momentum-dependent phonon intensities quite well (Figs. 3G, and 3J). The good consistency between OD30 and NBCO provides compelling evidence that the phonon intensity anomaly, in UD23, is not owing to straightforward increase in the EPC, but rather a complex reflection of the Fano interference effect induced by the dispersive CDW excitations. In Fig. 4, we illustrate a comprehensive picture of the quasi-static CDWs, the dispersive CDW excitations, and concomitant electron-phonon anomalies in the momentum-energy space. Emanating from the quasi-static CDWs, the CDW excitations quickly disperse and intersect firstly with the bond-buckling phonon at low energy. The narrow CDW excitations in the momentum space result in a dome-shaped phonon intensity enhancement closely confined around QCDW. After reaching higher energy and greater q// (> QCDW), the dispersive CDW excitations significantly broaden in the momentum space and intersect with the bond-stretching phonon inducing a diffused intensity anomaly. The intensity enhancement at q// > QCDW side is due to the momentum-dependent EPC of the bond stretching phonon. It is worth mentioning that excitations of a conventional CDW can be gapped by the periodically distorted crystal lattice or by impurities (26). Similarly, the dispersive CDW excitations in Bi2201 may exhibit a gap falling below the current detection limit.
The strong intensity anomalies of the buckling and stretching phonons in UD23 allow us to deduce the characteristic velocity of the dispersive CDW excitations. To do so, we first fitted the momentum-dependent phonon intensities and retrieved the wavevector (QA) of the maximal intensity anomaly (Fig. 3G, 3J, and SI Appendix, Fig. S10). By connecting QA = 0.34 r.l.u. with QCDW = 0.259 r.l.u. at the Cu L3 edge, we extracted the velocity of the CDW excitations, VCDWstretching ~ 0.45 ± 0.05 eV Å, close to the bond-stretching phonon (~ 60 meV). Near the bondbuckling phonon (~ 30 meV), we joined the intensity anomalies at QA = 0.238 r.l.u. and 0.267 r.l.u.
with QCDW = 0.25 r.l.u. from the O K edge, and obtained an averaged velocity of the CDW excitations, VCDW-buckling ~ 1.3 ± 0.3 eV Å. Remarkably, VCDW-buckling is about four times larger of VCDW-stretching, demonstrating unambiguously that the CDW excitation disperses steeply after arising from the quasi-static CDW, then gradually flattens at higher energy. Overall, the trend extracted by the bond-stretching and the bond-buckling phonon describes funnel-shaped dispersive CDW excitations as highlighted in Fig.4. We notice that the VCDW-stretching in Bi2201 is comparable to that of the under-doped Bi2212 (~ 0.6 ± 0.2 eV Å) at the energy ~ 60 meV (17). Interestingly, at the energy ~30 meV, the deduced CDW excitations velocity in Bi2201 is close to that of the electron band dispersion, ~1.7 ± 0.2 eV Å, retrieved from angular-resolved photoemission data in under-doped Bi2212 (27). The similarity of the velocities may indicate comparable self-energies between the ordinary and the periodically modulated charge carriers. However, this connection should not be simply viewed as an evidence of the weak-coupling (i.e., Fermi surface nesting) picture to describe the emergence of CDWs correlations. Rather, it is crucial for theoretical models to take into account the velocity values for the description of the CDWs in cuprates.

Lin et al. recently reported little signature of dispersive CDW excitations in LSCO compounds
despite the phonon-softening across a wide doping range (19). The bond-stretching phonon intensities, with and without the presence of CDW correlation, show similar momentum dependence, in contrast to our Bi2201 RIXS data (Figs. 3G-3H, 3J). As the bond-buckling phonon intensity anomaly is much more confined around QCDW differentiating drastically to the bare EPC, we corroborate the highly sensitive O K RIXS in the detection of the coupling of CDW excitations with phonons and its complementary with Cu L RIXS.
The co-existence of CDW at the Cu and O sublattices in Bi2201 demonstrates that the modulated charge density carries both Cu-3d and O-2p orbital character. REXS studies on LBCO and YBCO showed that the CDW order has s'-or d-wave symmetry which depict a bond-centered charge order naturally explaining the projection onto Cu and O sublattices (14,28). Given the correlation length of the CDW in Bi2201 is more than an order of magnitude shorter than in LBCO (29), we suggest that the multi-orbital nature is universal to the CDWs in all hole-doped cuprates. From a theoretical perspective it is unclear whether the CDWs can be captured properly using a singleband model given the significant oxygen character of the density modulation and the phonons involved. Our work, therefore, highlights the need to use multiorbital Hubbard models (specifically 3-bands Hubbard models) to describe the cuprates in terms of electron dynamics and orders (13,(30)(31)(32).
Our findings on the rich interplay between the multi-orbital CDW and the electron-phonon anomalies in Bi2201 substantiate the existence of funnel-shaped CDW excitations dispersing in the energy-momentum space. This is a major step forward in characterising the dispersive CDW excitations comparing to the previous study on Bi2212, which was solely based on the interference effect from a single phonon branch (17). The dispersive CDW excitations are in line with the shortrange dynamical charge density fluctuations found in a large portion of the phase diagram in YBCO, postulating its ubiquity in cuprates (12). Further experiments aimed at uncovering dynamical CDW excitations in different cuprates may help elucidate the relevance of CDWs for the anomalous normal state and the unconventional superconducting properties. Concerning the underlying mechanism of CDWs, the most recent CDW studies on YBCO and LSCO add mounting evidence that the Fermi-surface nesting alone is unlikely to be the primary driving force (12,19). Moreover, the fact that the CDW dynamics interfere with the phonons points towards an important role of the EPC in the formation of the CDWs (33)(34)(35)(36). As suggested by Refs. (20-21, 23, 37-38), the phonon intensity measured by RIXS is scaled to M 2 , where M is the EPC matrix element. A simple comparison of phonon intensities between UD23 and OD30 (Figs. 3G-3H, and 3J) would imply that M at momenta far away from the phonon anomalies is comparable between two Bi2201 compounds. However, determining M at QCDW faces difficulties as the RIXS intensity is not anymore a simple proportion to M 2 but rather reflects a complex interplay with CDW excitations. It is currently challenging to extract reliably the EPC at CDW wavevector and a much more sophisticated theoretical modelling is required in future.

Conclusion
We combined high resolution RIXS at the O K-and Cu L3-edges to study the quasi-static CDWs, the collective CDW excitations, and the electron-phonon coupling in Bi2201. The quasi-static CDWs are present at both Cu and O sublattices and carry comparable periodicity and correlation length implying their multi-orbital nature to be universal in hole-doped cuprates. Both the bondstretching and the bond-buckling phonons exhibit strong anomalies concomitant to CDWs. The confined intensity anomaly of the bond-buckling phonon, together with the diffused intensity enhancement of the bond-stretching phonon, uncovered unambiguously funnel-shaped dispersive CDW excitations. The significant interference effects suggest that CDWs are intimately connected to the electron-phonon coupling, which needs to be considered as a crucial element for the underlying mechanism leading to CDWs in cuprates.

Materials and Methods
Sample growth and characterization. High-quality single crystals of UD23 and OD30 Bi2Sr2-xLaxCuO6+δ cuprates with x = 0.6 and 0.2, respectively, were grown by the traveling-solvent floating-zone method. The as-grown samples were annealed at 650℃ in oxygen atmosphere for two days to improve sample homogeneity. The samples were pre-characterized and aligned using     Fig.S1A and Fig. S1B. The superconducting transition temperature, Tc, were determined from the magnetization measurements shown in Fig. S1C and S1D. Please note that 10 Oe is already a low field limit therefore the use of the magnetic field values (10 and 1 Oe) does not make considerable difference for determining Tc between two samples. A set of super-structural (SS) points due to the structural distortion of BiO planes are clearly resolved and used as the reference to orient samples.

The self-absorption correction for RIXS spectra
The self-absorption can distort the spectral line profile severely. To eliminate this effect, we follow the procedure described in Ref. S1. Considering that the elastic peak and the low-energy phonon do not involve the spin-flip process meaning the X-ray polarization is preserved, i.e., # !→# = # #→! = 0, the intensity correction can be described as: Here, * ' (* ( ) and -.⃗ ' (-.⃗ ( ) characterize the absorption coefficient and the direction of the incoming (outgoing) X-rays, respectively. 1 .⃗ is the surface normal direction, 3 ) and 3 $ are the Cu or O atom scattering factor along a or c-axes which can be extracted from the XAS with 3 ) ≫ 3 $ due to the quasi-two-dimensional nature of the sample. It is clear that the self-absorption correction for A polarized RIXS spectra is only related to the scattering geometry and scaled by the in-plane scattering factor 3 ) . Whereas for the p polarized spectra, both 3 ) and 3 $ contribute to the correction.
The self-absorption correction was applied to all RIXS data presented in the paper.

CDW excitations in various configurations
We present the observation of the CDW excitation in various experimental configurations alongside the results shown in the main text. In Fig. S3 we show the RIXS map excited using the

Repeatability of the presence of the CDW peak in UD23
To confirm the observation of CDW at both the Cu L3-and O K-edges in UD23, we repeated the measurements on a second sample of UD23 which show again CDW scattering peak at both edges.
Using the fitting analysis described in the Methods of the main text, we show the fitting results in the table below and the integrated intensity of the quasi-elastic peak (± 30 meV) in Fig. S4. The error bars shown in the table are the convolution of the instrumental momentum resolution and the standard deviation of the fitting errors.

Fittings of the bond-stretching and the bond-buckling phonons
We show detailed fitting results of the bond-stretching and the bond-buckling phonons in q// dependent spectra in UD23 at the Cu L3 and the O K in Fig. S5, and Fig. S6, respectively. For the Cu L3-edge data, a Gaussian function with the instrumental resolution is used for fitting the elastic peak, a Lorentzian function is applied to the bond-stretching phonon, the tail of paramagnon is fitted by a damped harmonic oscillator multiplied by the Bose factor, and a linear function is used for fitting the general background. For the O K-edge data, two Gaussian functions are used to fit the residual spectra after subtracting the fitted elastic peak to track the bond-stretching and the bond-buckling phonon modes.

Ultra-high energy resolution O K-edge RIXS of UD23
The fitting analysis for the O K RIXS spectra of UD23 shows that the phonon excitations contain a two-peak structure though the lower-energy phonon is less visible comparing to the higherenergy phonon peak in the raw data. To confirm the observation, we re-measured the same UD23 sample using a higher energy resolution (FWHM = 18 meV) set-up than the normal energy resolution (FWHM = 26 meV) set-up used for the main measurements. Fig. S7 shows the comparison of the raw RIXS spectra at q// of (0.23, 0). A low energy phonon excitation below 50 meV is clearly resolved in the higher energy resolution RIXS spectrum.

Extracted dispersion of the bond-stretching phonon in UD23 at Cu L3
Using the same fitting analysis described sample, no CDW is observed as shown in Fig. 3. To further explore the existence of the CDW, we performed the Cu L3 RIXS measurements at the resonance (931.6 eV) along the (-H, 0) (Fig. S9A), at an energy off-resonance (932 eV) along the (H, 0) direction (Fig. S9B), and at the resonance (931.6 eV) along the (H, H) direction (Fig. S9C). For O K RIXS, we collected data at an energy off-resonance of the hole peak (528.1 eV) (Fig. S9E). Figures S9D and Fig. S9F summarize the integrated intensity of the quasi-elastic peak within the energy window (± 30 meV) defined by white dashed lines at the Cu L3 and O K edges, respectively. None of them shows any signature of CDW scattering peak. Strong CDW peak from UD23 is shown in both Fig. S9D and Fig. S9F for comparison.

Fittings of the momentum-dependent intensities of the bond-stretching and the bondbuckling phonons in UD23
To extract the velocity of dispersive CDW excitations in UD23, we fitted the momentumdependent intensities of the bond-stretching (bond-buckling) phonon mode obtained from the Cu