Introduction

High-Tc superconductivity (SC) is closely proximal to a charge order (CO) electronic state in the cuprate phase diagram1, which exhibits an incommensurate wavevector QCO that varies with hole concentration (Fig. 1a and ref. 2). While there is growing evidence suggesting that CO competes with SC since its amplitude is weakened below Tc3,4,5,6, it might actually be the first emerged phase during the doping of a Mott insulator1, acting as a precursor to SC. Conforming to this scenario, scanning tunneling microscopy (STM) studies on Bi2Sr2−xLaxCuO6+δ (La-Bi2201)7 and Ca2−xNaxCuO2Cl2 (Na-CCOC)8 reveal that a surface-form CO has already become manifest even in the antiferromagnetic regime, with only a small amount of holes (p = 0.07~0.08) introduced into the parent compounds. However, whether the CO exists in insulating cuprates has not been confirmed by bulk-sensitive probes such as resonant inelastic X-ray scattering (RIXS). Thus, it remains unclear whether the CO has a bulk nature in the non-superconducting regime or exists only in a surface form, and whether it extends to even lower doping levels. Furthermore, it is important to determine whether the CO wavevector QCO is commensurate, as reported in refs. 7,8, to understand its relation with the underlying lattice. Answering these questions is critical in delineating a clear story about the relationship between unconventional SC and CO in cuprates.

Fig. 1: Demonstrations of the research with combined RIXS and STM techniques.
figure 1

a QCO of Bi2212 as a function of doping. The blue shaded line is a guide to the eye. Inset: the schematic crystal structure of Dy-Bi2212. b Scheduled measurements of this work, displaying the three paths (black lines and arrows) in the schematic phase diagram. The orange and purple circles denote the specific temperature and doping levels at which the RIXS and STM data were respectively collected. The blue-filled (Bi2212) and open (Bi2201) stars, together with the quadratic fitting, are suggested by ref. 2. The T* line (light gray) is adapted from ref. 13. AF: antiferromagnetic, SM: strange metal, SC: superconducting. c Representative RIXS spectra of the p = 0.04 sample at 17 K. The data are plotted as functions of energy loss distinguishing different excitations (indicated), as well as of the projected in-plane momentum transfer Q// in the (H, 0) direction. Inset: schematics of the RIXS process, displaying the intermediate (∣m >, resonant absorption from Cu 2p to Cu 3d) and final (∣f >, inelastic emission) states. d Representative tunneling spectra dI/dV (r, V) (roughly proportional to the local density-of-states) of the p = 0.04 sample at 5 K, which has extended the lower doping limit of STM's capability in measuring insulating cuprates at liquid-helium temperature. Inset: schematics of STM and the topographic image that clearly resolves the superstructure and Bi atoms of Bi2212.

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One more piece of the puzzle that remains elusive is the origin of CO. This CO is typically observed at relatively high doping levels and is intertwined with pseudogap (PG) and SC through a truncated Fermi surface. For example, angle-resolved photoemission spectroscopy (ARPES) experiments have found discontinuous segments known as Fermi arcs9, and momentum-space nesting mechanism has been suggested as a possible explanation for the origin4,10. On a different front, quantum oscillations11 and magnetoresistance12 measurements suggest that the CO is closely connected with a reconstructed Fermi pocket. However, all these previous pictures fail to explain the above-mentioned charge-ordered insulating phase, where the Fermi surface is absent13,14. To address these issues, we have designed a comprehensive set of temperature paths (Fig. 1b) to explore the property, universality, and extension of the CO, and how the SC emerges in the vicinity of its onset doping level.

We choose lightly doped Bi2Sr2Ca1−xDyxCu2O8+δ (Dy-Bi2212, x = 0.4) to study utilizing the state-of-art RIXS and STM techniques. Dy-Bi2212 has proved to be a feasible system for our purposes that can achieve hole concentration as low as the Mott insulating limit15. Importantly, its crystal structure (Fig. 1a inset) allows an atomically flat cleavage that is ideal for both STM and RIXS. A recent RIXS study on underdoped Dy-Bi2212 (p â‰ˆ 0.09) has identified dispersive excitations associated with the CO, which is proposed to induce a peak of electron-phonon coupling (EPC) strength at the in-plane bond-stretching (BS) phonon branch16. This EPC peak grows once the system enters the SC dome6, providing intriguing connections between SC, CO, and EPC. However, the implications of these findings remain contentious among different cuprate compounds when the inelastic spectral weight is solely attributed to the BS phonon mode16,17,18. Moreover, the extension of the EPC “anomaly" to lower doping levels, especially to the insulating phase diagram, remains largely unexplored. In this work, we investigate this lightly doped region with p = 0.04 and p = 0.06 (Methods and Supplementary Fig. S1) across the edge of the superconducting dome (p = 0.05), where the Fermi surface is found missing in both samples from ARPES studies13.

Results

RIXS and STM measurements

Figure 1 c shows a series of high-resolution RIXS spectra (ΔE = 36 meV, ΔQ = 0.006 r.l.u., see Methods)19 taken in the insulating sample (p = 0.04) at 17 K. The incident X-ray was tuned to the Cu-L3 edge, corresponding to resonant absorption and emission between Cu 2p and Cu 3d orbitals (Fig. 1c inset and Supplementary Fig. 1), which provides energy- and momentum-resolved information of elementary excitations, such as the paramagnons and phonons. The same sample was cleaved at an ultra-high vacuum and immediately loaded into the STM stage. As displayed in Fig. 1d, we succeeded in performing single-particle spectroscopic and atomically-resolved topographic measurements in Dy-Bi2212 with p = 0.04, by exploiting the reliable methodology we have previously developed (Methods). As the sample is spatially inhomogeneous, the spectra evolve from a charge-transfer gap (black curve) to a PG lineshape (red curve), similar to that exhibited in superconducting samples20.

Observation of charge order

We now focus on the lower-energy range of our high-resolution RIXS and STM data, which is most relevant to CO, PG, EPC, and SC. Intriguingly, in this insulating Dy-Bi2212 we observe an evident bulk CO through careful examination of the energy-momentum RIXS intensity maps (Fig. 2a–c), which has never been reported before. It is directly visualized that at the quasi-elastic region a broad peak along the (H, 0) direction (i.e., Cu-O bond direction) associated with CO has already formed at 273 K and gradually becomes stronger as the temperature decreases (along path #1 of Fig. 1b). The CO intensity profile can be well-fitted by a Gaussian peak (blue shaded) plus a Lorentzian background (Methods), which is centered at an incommensurate QCO  = 0.31 r.l.u. at 17 K. The elastic intensity peak is absent along the diagonal (H, H) direction (gray squares in Fig. 2c), consistent with CO characteristics in other cuprate materials2. Previous studies have only proposed CO down to p = 0.07 using surface-specific STM7,8,21, while a bulk-form CO was only identified in superconducting Bi2212 down to p = 0.09 (Tc = 45 K)16. Our findings substantially extend the lower doping limit of CO to p = 0.04 in the insulating regime, suggesting that CO in insulating cuprates is incommensurate with the underlying lattice, contrary to some previous reports7,8.

Fig. 2: Observation of charge order and the corresponding electronic structure in the insulating sample (p = 0.04).
figure 2

a–c RIXS intensity maps with decreasing temperature, revealing the existence of CO in the quasi-elastic region centered around zero energy loss (indicated by the blue arrow). The gray circles in the lower panel of each figure represent the in-plane H-dependence of RIXS intensity, which is summed over the energy range between the two white dashed lines in the upper panel. The black curve and blue shaded areas are the total fits and the Gaussian component characterizing the CO peak, respectively. The CO is observed only along the (H, 0) or antinodal direction, and it is absent along the (H, H) or nodal direction (gray squares in c). d dI/dV map at V = 30 mV, displaying a very weak CO pattern along the Cu-O or antinodal directions (white arrows) at 77 K. e, Representative dI/dV spectra at points indicated in the inset, which is a zoomed-in view of the white dashed box in d. f, dI/dV map at V = 30 mV, displaying the CO pattern at 5 K. g The PG map at the same area as f, which displays the spatial distribution of the fitted PG magnitude (see Methods) from the tunneling spectra. h The spatial evolution of dI/dV spectrum along a line cutting through CO-poor and CO-rich regions is indicated by the red arrows in f and g, which reveals the particle-hole asymmetric PG by subtracting the CT gap background. The non-dispersive nature of the dI/dV maps in a wide energy range confirms the existence of static CO (see Supplementary Fig. 3 and Fig. 5). Scale bar: 5 nm.

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From a real-space perspective, the checkerboard CO feature is particularly weak at a high temperature of 77 K as revealed by the dI/dV map (see Methods) in Fig. 2d. At this temperature, only a small portion of space with large local density-of-states (bright) holds charge ‘puddles’ that are poorly connected along the Cu-O bond directions (white arrows). This observation aligns with the broad full-width at half-maximum (FWHM) of the CO peak revealed by RIXS (e.g., 0.072 r.l.u. at 90 K, corresponding to a very short correlation length ξ â‰ˆ 4.4 a0). A closer examination in Fig. 2e reveals that the dark region, where no CO develops, is characterized by a full charge-transfer gap. In contrast, the CO is accompanied by a density-of-states modulation, especially near the energy scale of the ‘V-shaped’ PG. Figure 2f, g are the low-temperature (T = 5 K) real-space patterns of the CO and PG magnitude (Methods) in the same area. It becomes immediately apparent that the CO is spatially correlated with smaller, or more precisely, sharper PG regions. The spectral linecut in Fig. 2h illustrates a smooth transition from the charge-transfer gap (which has been subtracted and shown as the red curve) in CO-poor locations to PG (CO-rich). Interestingly, the PG size is asymmetric at positive and negative bias, distinct from SC. This asymmetry is not unexpected, as a broken translational-symmetry phase, such as the CO here or the PG itself20,22, generally breaks particle-hole symmetry. Our result has, for the first time, demonstrated such asymmetry in both the occupied and unoccupied states.

To comprehend the evolution of a charge-ordered insulator into a high-temperature superconductor (along path #2 of Fig. 1b), we next move to the p = 0.06 sample with Tc = 20 K. The dI/dV map at 5 K (Fig. 3a) shows that the CO pattern is stronger than that in the insulator (Fig. 2f), which is also visualized by comparing the CO peak intensity and area from the RIXS data (Figs. 3c and 2c). This observation appears inconsistent with the notion that CO competes with SC, but rather plays a reinforced role. Remarkably, both STM (Fig. 3b) and RIXS (Fig. 3c) provide virtually identical QCO = 0.30 r.l.u. (≈3.3 a0 periodicity) along the (H, 0) directions, reconciling the charge modulations at the surface and in the bulk. The QCO of Bi2212 extended to p = 0.04 and p = 0.06 follows the extrapolated trend shown in Fig. 1a, in agreement with the behavior observed in most cuprates2.

Fig. 3: Charge order and phonons in the superconducting sample (p = 0.06).
figure 3

dI/dV map at V = 30 mV (a) and corresponding Fourier transformation (b), revealing a CO with QCO = 0.30 Â± 0.01 r.l.u. along the antinodal direction (blue arrow). Scale bar: 5 nm. c The RIXS intensity map (upper) and integrated quasi-elastic CO peak (lower), with the color scale identical to that in Fig. 2. The red, blue, and black arrows indicate the characteristic energy scales of CO, bond buckling (BB) phonon and bond stretching (BS) phonon, respectively. d Joint fit of the RIXS spectrum at H = 0.12. e Subtracted spectra highlighting the BB (red shaded) and BS (black shaded) phonon modes. The black and red triangles indicate the fitted energies of BS and BB phonons, respectively.

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Interplay between CO and phonons

Regarding the SC, it has been proposed that EPC plays a significant role in realizing high Tc, irrespective of the exact pairing mechanism23,24. However, the phonon modes are inaccessible by STM in the strongly underdoped regime, since the prominent PG prevents the extraction process, such as a strong-coupling Eliashberg approach25. In contrast, the RIXS cross-section offers the capability to simultaneously map the dispersion and EPC strength for specific phonon branches26, which also have an intricate interplay with CO16. Consequently, we conduct a more in-depth investigation to explore the potential connection between CO, EPC, and SC, utilizing the inelastic spectral weight of RIXS.

A previous study on antiferromagnetic NBCO demonstrated that Cu L3-edge RIXS can simultaneously detect the Cu-O bond stretching (BS) mode at approximately 70 meV and the out-of-plane A1g bond buckling (BB) phonon at about 35 meV27. When there is no charge order, the electron coupling to the BS phonon follows a sin2(πH) trend, while the BB phonon follows a cos2(πH) trend27, consistent with the model calculations of momentum-dependent EPC26. Figure 3d, e shows a raw RIXS spectrum and the inelastic parts after subtracting a resolution-limited elastic peak, respectively. The highly asymmetric lineshapes in the spectra indicate the necessity of considering two phonons in data fitting. With improved energy resolution and superb momentum accuracy in our experiment, we use two Gaussian functions (Methods) to extract the BS mode (shown in black) and the BB mode (in red). Our analyses show that both modes are nearly dispersionless within the energy resolution, which is distinct from previous one-phonon fitting models that presented a BS mode “softening”16,17,18. However, it is important to note that our present study is insufficient to rule out the softening behavior of lower-energy phonons. As energy resolution is further improved, softened phonons have been revealed by IXS on YBCO28 and higher-resolution RIXS in LSCO29.

Figure 4 a, b compare the decomposed CO (lower panels), BB phonon (middle panels), and BS phonon (upper panels) in the insulating and superconducting Dy-Bi2212 samples (Methods and Supplementary Fig. 8). The corresponding energy-integrated intensity profiles are shown in Fig. 4c. Intriguingly, the BB phonon exhibits a broad peak centered around 0.32 r.l.u., which is likely due to an interaction between the phonon and the dispersive CO excitation16. Both the CO and BB phonon intensities are higher in the superconducting sample (see also Fig. 4d). In contrast, the momentum-dependent EPC of the BS phonon lacks a peak structure and is well captured by a sin2(πH) profile within a non-interacting model27. Therefore, our new analysis suggests that the intensity anomaly observed in previous studies should be attributed to the BB phonon mode, which varies with material, doping, and temperature.

Fig. 4: The doping evolution of RIXS spectra near the entrance of high-temperature superconductivity.
figure 4

The RIXS intensity map highlighting each fitted excitation is shown for the p = 0.04 insulating (a) and p = 0.06 superconducting (b) samples for comparison. Upper panels are obtained by subtracting the fitted elastic and bond buckling (BB) phonon peaks from the raw data. Middle panels: subtracting the fitted elastic and bond stretching (BS) phonon peaks. Lower panels: subtracting the fitted BB and BS phonon peaks. White symbols superimposed on the maps indicate the fitted dispersion of the phonons, and the error bars represent the 95% confidence intervals of the fit. c, Comparison of RIXS intensity profiles integrated within the white dashed boxes in a and b. Note that a peak feature of the BS phonon branch reported at H = 0.36 r.l.u. on underdoped Dy-Bi2212 (ref. 16) is absent in our analysis. d Raw RIXS spectra (light-colored) at selective momenta indicated by the black dashed lines in c, highlighting the doping comparisons of different components (red and blue lines).

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Doping- and T-dependence of CO and EPC

To gain more insights into what happens when the system enters the superconducting phase, we consider both the doping and systematic temperature evolutions following the three paths depicted in Fig. 1b. The short-range charge correlation already exists at 273 K (Fig. 5a), which is accompanied by a very broad BB phonon intensity peak (Fig. 5b). Therefore, the weak CO and its interplay with EPC set the stage for the formation and condensation of Cooper pairs. Upon cooling along paths #1 and #3, both CO and BB phonon gradually gain more weight, but they follow very different trends (see Fig. 5c,d): the CO intensity grows faster in the superconducting sample than in the insulating one. In contrast, the BB phonon displays a concurrent and smooth increase, except for an abrupt jump (paths #2 and #3) when the SC appears. These results agree with previous reports on the separation of CO intensity and excitations6,17. The intensity profile of the BS phonon, on the other hand, maintains a sinusoidal trend at different dopings, temperatures (Fig. 5e), and materials (refs. 26,27), indicating a negligible correlation with CO excitations and SC.

Fig. 5: The temperature and doping evolutions of CO and EPC.
figure 5

a CO peak profiles obtained by integrating the quasi-elastic maps (Supplementary Fig. 8). The curves are fit to the peaks (Methods). b Normalized phonon intensity profiles of the p = 0.04 (upper panel) and p = 0.06 (lower panel) samples obtained by integrating the BB phonon maps (Supplementary Fig. 8 and Fig. 10). The colored curve is the fit to each profile using a Gaussian function plus a linear background. c, Elastic peak area in a retrieved by the fits. The error bars represent the 95% confidence intervals. d, BB phonon peak area obtained by integrating the spectral weight within the black dashed boxes in b. e, BS phonon intensity profiles at the two Bi2212 samples and all temperatures investigated, with a sinusoidal line (gray shaded) overlaid. The predicted sin2(πH) trend agrees with and substantially extends a recent RIXS result on La2−xSrxCuO4 cuprates18.

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Figure 6 delineates our idea of how the SC emerges below Tc. Our findings reveal that while the SC emerges from the CO phase (along paths #2 and #3 in Fig. 1b), the CO wavevector QCO continuously changes with doping, irrespective of the SC (Fig. 6a). In concert with this observation, previous studies have noted that the CO intensity could respond differently to SC3,30,31. It is the BB phonon intensity anomaly, or the dispersive CO excitations6 triggered by the underlying CO, that induces an abrupt enhancement in the SC state (Fig. 6b). The enhanced EPC strength of the BB phonon may drive or accompany the condensation of Cooper pairs, as evidenced by the coherent superconducting gap everywhere (Fig. 6c). Therefore, the CO and EPC play distinct roles but work together in establishing the macroscopic SC.

Fig. 6: Intimate correlation between CO, EPC and SC.
figure 6

a Summarized plot of QCO as a function of hole concentration, displaying a monotonously decreasing trend with increasing doping. The black symbols are the same as Fig. 1a, adapted from refs. 5,6,16. Note that this decreasing trend may only be at work below p = 0.1653,54. b In the superconducting Dy-Bi2212 (p = 0.06), the dispersive excitations (white dash-dotted) of the short-range CO cross the BB phonon dispersion and induce a strong peak at this branch. The intensity map is obtained by first subtracting the BS phonon mode from Fig. 3c and is then smoothed. c A superconducting gap of  ~ 20 meV is found in different locations on the p = 0.06 sample, indicated by different colors (crosses in Fig. 3a). The shaded blue and red arrows indicate the intimate connections between RIXS and STM experiments. On one hand, the CO branch revealed by the quasi-elastic peak of RIXS gives the same QCO and its doping evolution as STM. On the other hand, the inelastic BB phonon strength abruptly increases once the Cooper pairs condense, corresponding to the SC gap revealed by STM.

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Discussion

The combined RIXS and STM results here touch on the important issues regarding the relations between PG, CO, EPC, and SC, all of which are at play in the present study. While a short-range CO has been found pervading the hole-doped phase diagram, where SC acts as the ground state, its extension to the insulating regime remains elusive, with only two STM works (refs. 7 and 8) reporting surface CO. Our work, for the first time, has not only provided concrete evidence of CO in an insulating cuprate through a combined bulk and surface study but also substantially extended the lower doping limit of CO to p = 0.04 with a broad temperature range. Our observations reveal an incommensurate CO wave vector surrounding  ~ 0.31 r.l.u. However, our RIXS results and traditional Fourier-transformed dI/dV (proportional to the local density-of-states) channel cannot differentiate between truly incommensurate charge order and discommensurations within a locally commensurate phase resulting from disorder or topological defects29,32,33. Therefore, the true nature of the CO, whether it is genuinely incommensurate or merely appears to be incommensurate due to certain factors, remains a topic of controversy.

Another hotly debated issue pertains to the origin of CO. In a weak coupling k-space Fermi-surface nesting scenario, the CO wave vector would continuously decrease with increased doping4,10. Alternatively, in a strong coupling r-space scenario, the charge order would be commensurate with the underlying lattice. While we have noted a continuous decrease in QCO with doping, the nesting scenario is not applicable in the heavily underdoped regime where the Fermi surface is entirely gapped13. However, our findings can be reconciled through discommensuration in the r-space strong correlation scenario, where smaller random commensurate domains can cause a larger mismatch of commensurate wave vectors at low doping, as proposed by phase-resolved STM studies32. One may also suspect if this incipient CO is caused by antiferromagnetic excitations in the insulating regime, as suggested in other hole-doped cuprates34,35. However, our RIXS results reveal a smooth paramagnon dispersion and damping across QCO (Fig. 1c and Supplementary Fig. 11), showing an absence of intertwining between spin excitations and CO.

We move forward to the initial question of how the unconventional SC emerges. In the lightly doped cuprates, we are close to a non-Fermi-liquid Bose-Einstein condensate regime. Although our experiments do not give direct evidence on incoherent Cooper pairs, we do suggest that CO is indispensable for their formation. Signatures of “pre-formed” Cooper pairs have been widely observed by STM36,37, ARPES38, and transport measurements39,40. The coexistence of pre-formed Cooper pairs and CO outside the superconducting dome suggests that the charge-ordered electronic state is energetically favorable for the formation of Cooper pairs, akin to the description provided by the stripe model41. This agrees with the observation that two holes are distributed on the peak of the checkerboard plaquette, as visualized by a periodical superconducting coherence peak modulation atop the CO37,42.

In cuprates, another intriguing correspondence is between the short-range nature of CO and nanoscale electronic inhomogeneity43, indicative of the weakly-overlapped Cooper pairs in a real-space picture. The CO, though still short-ranged, gains more amplitude with cooling (along paths #1 and #3) or doping (along path #2). Working in conjunction with the CO, it is the enhanced EPC strength that accompanies the eventual condensation of Cooper pairs (Fig. 6b, c). Remarkably, the BS phonon is invariable against temperature and doping (Fig. 5e), agreeing well with the model calculations26, RIXS measurements in Nd1+xBa2−xCu3O7−δ27, and ARPES measurements in Bi221244. In contrast, the coupling to the BB phonon is a more relevant process, which has been proposed to promote the magnitude of Tc when the long-range Coulomb interaction is plausible24. Similar observations sensitive to the Cooper pairs condensation include the antinodal kink of the buckling B1g phonon observed by ARPES45, and the spin resonance mode observed by inelastic neutron scattering46. Whether these are the driving forces or natural consequences of the itinerant Cooper pairs remains an open question.

Methods

Sample growth and characterizations

High-quality Bi2Sr2Ca0.6Dy0.4Cu2O8+δ single crystals were grown by the traveling solvent floating zone method. The high Dy concentration was intentionally introduced during the growth to achieve the highly underdoped regime. The as-grown sample displays a superconducting transition with \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\) = 40 K as revealed by the magnetic susceptibility curve, corresponding to a hole doping level of p = 0.067 (0.067 Â± 0.008). The doping and error bar were estimated using the empirical formula from ref. 47, where the \({T}_{{{\rm{c}}}}^{\max }\) was set to 96 K for the Bi2212 system. One as-grown sample was divided into two crystals, which were used in the present study after different post-annealing processes in a vacuum atmosphere. Notably, the STM measurements were carried out in the very same samples after the RIXS experiments. The first annealing condition (Ann1) was T = 350 ∘C for 2 days, which yields a superconducting transition with Tc ~ 20 K and p = 0.062 (0.062 Â± 0.006). The second one (Ann2) was T = 450 ∘C for 3 days, which completely suppressed SC showing no evidence of the Meissner effect. We estimate the doping level of Ann2 to be p ~ 0.04 (0.04 Â± 0.01) for two reasons: (i) It is below the onset doping of the superconducting dome (p ~ 0.05) and is highly insulating as revealed by the resistivity measurement, and (ii) Previous reports have proposed that it is impossible to carry out liquid-helium-temperature STM measurements on cuprates at or below p = 0.037, even with our delicate methodology (see below). Therefore, the sample prepared by Ann2 is located somewhere below but in the vicinity of the SC onset. The magnetization and resistivity curves are shown in Supplementary Fig. 1.

RIXS measurements

The RIXS experiments were performed at the I21 RIXS beamline of the Diamond Light Source19. The momentum resolution is estimated to be 0.01 Å−1 based on the scattering geometry employed in our study19. For Bi2212, the lattice parameters are a = b = 3.83 Å, corresponding to ΔQ ~ 0.006 r.l.u., which is sufficient for the CO analysis. The Dy-Bi2212 samples were cleaved to expose a fresh surface and then immediately transferred into the ultrahigh vacuum (UHV) chamber. Before the RIXS measurements, we did careful alignments for our samples utilizing the (002) Bragg peak and the (1-10) superstructure peak, which are critical for an accurate determination of the scattering vector q. All RIXS spectra in our study were collected with the incident photon energy tuned to the Cu L3 resonant edge (see Supplementary Fig. 1), where we can achieve an ultra-high energy resolution ΔE ~ 36 meV. To enhance the charge-related excitations, such as CO and phonon branches, all data, except for those shown in Supplementary Fig. 9, were taken with the vertical (σ) polarization of the incident X-ray. To maximize the in-plane momentum-transfer H along the (H, 0) direction, a large scattering angle 2θ = 154° was chosen.

STM measurements

The STM experiments were performed on the same crystals immediately after their return from the RIXS measurements at the Diamond Light Source. We used a commercial (CreaTec) STM system working at low-temperature and UHV environments. Each Dy-Bi2212 single crystal was cleaved in situ at T = 77 K and then immediately transferred into the STM stage stabilized at 77 K or 5 K.

To carry out STM measurements in insulators and to obtain high-quality tunneling spectra over a large bias range (e.g. Fig. 1d), it is crucial to have a stable metallic tip. In our experiment, an electrochemically etched tungsten tip was transferred into the UHV chamber and was then electron-beam bombarded for 15 min to remove the oxides. Before the measurements, the tip was carefully treated and calibrated on an Au(111) single crystal. The basic requirements include: (i) an atomically sharp apex, (ii) metallic DOS which reveals a well-defined surface state and quasi-particle interference pattern of the Au(111), and (iii) stable enough under a point-contact current as high as 10 uA between the tip and Au(111). The tip was then slowly approached onto the cleaved Dy-Bi2212 sample surface with a large magnitude of negative bias voltage, since the occupied state has a much larger DOS due to the tunneling asymmetry seen in CT gap and PG (e.g. Fig. 1d). More details of this methodology can be found in our previous paper48.

Set-point conditions for Fig. 1: tunneling current I = 30 pA, bias voltage V0 = âˆ’ 1.0 V, ac âˆ’ modulation Vm = 30 meV. To mitigate the effect of V0 on the low-energy density-of-states, which is known as the set-point effect, a higher V0 is utilized to reveal the CO signal. In our experiment, set-point conditions for Fig. 2 (the p = 0.04 sample): I = 100 pA, V = âˆ’ 0.5 V, Vm = 6 mV. Set-point conditions for Fig. 3 and Fig. 5 (the p = 0.06 sample): I = 150 pA, V = âˆ’ 0.3 V, Vm = 4 mV.

RIXS Data analysis

We first determined the zero-energy-loss of each RIXS spectrum by comparing it with that recorded on a nearby carbon tape with the same scattering geometry. After that, the data were normalized to the incident beamline flux and then corrected for the self-absorption effect using the procedures described in ref. 49. The data in Supplementary Fig. 8 were normalized to the areas of the dd-excitations.

The quasi-elastic CO peak profiles shown in Figs. 2a–c, 3c, and 5a were fitted to a Gaussian peak with a Lorentzian background centered at H = 0. From this analysis, the CO wavevector (e.g., Fig. 6a) and amplitude (e.g., Fig. 5c) were respectively characterized by the position and area of the Gaussian component.

The inelastic phonon branches were extracted through a fitting model to each RIXS spectrum, as exemplified in Fig. 3d. It involves resolution-restricted Gaussians for the quasi-elastic, BB phonon, and BS phonon peaks, and a higher-energy background of an anti-symmetric Lorentzian convoluted with the energy resolution. From this analysis, we decomposed the raw data to CO, BB phonon, and BS phonon intensity maps shown in Fig. 4 and Supplementary Fig. 8, which further generate Fig. 5a,b. The adequacy of our model and the comparison with other one-phonon fitting models are demonstrated in Supplementary Fig. 7 and discussed in the accompanying text.

STM data analysis

For characterizing the bi-directional checkerboard CO, the dI/dV map is widely employed for STM analysis4,7,8,10,21,37, which provides a direct comparison with the resonant X-ray scattering signal4,10. From the STM point of view, the defining feature of the CO is an in-phase density-of-states distribution between the particle (-E) and hole (+E) state, distinguishing it from other electronic modulations that are studied by the “Z-map"20,32,36,50,51. In practice, the “Z-map" has been effective in mitigating the CO signal for QPI and uni-directional modulation studies.

The PG map ΔPG(r) shown in Fig. 2g was obtained by fitting the tunneling spectrum at each pixel dI/dV(r, E). The fit was done on the positive-E side (unoccupied state) with an evident peak, where a s-wave Dynes formula works pretty well for the broad feature in this highly underdoped regime50. We note that the density-of-states, or tunneling rate, asymmetry seen in the PG lineshape is generic for cuprates20, which is due to the inequivalence between electron injection and extraction and reflects, thereby, the strong electron-electron correlation effect52.

The STM momentum resolution is estimated to be ΔQ = 2π/23.8 nm = 0.026 Å−1 ~ 0.016 r.l.u. (where 23.8 nm represents the real-space size of Fig. 3a), which is sufficient for the CO analysis. To directly compare the QCO between Fourier-transformed STM and RIXS (Fig. 3), we have corrected the thermal and piezoelectric drifts of the p = 0.06 dataset. The methodology used for this correction, Lawler-Fujita’s algorithm, has been well documented in ref. 51.