Introduction

The cuprate high-Tc superconductors are often conceptualized as doped Mott insulators, in which the electronic ground state spontaneously breaks rotational and/or translational symmetry1,2,3,4. While cuprate CDW correlations were discovered over two decades ago5, their possible contribution to the material’s anomalous electronic properties remains a matter of vigorous debate1,6,7,8,9,10. This issue has gained increasing attention in light of the ubiquity of CDW order in different cuprate families5,11,12,13,14,15,16,17,18,19. The cuprate phase diagram, shown in Fig. 1a, shows that pseudogap, strange metal, and superconducting phases exist over an extensive doping range below a critical doping level of xc ≈ 0.19, above which the cuprate electronic properties become gradually more Fermi-liquid-like20,21,22,23,24,25,26,27. If CDW correlations are confined to underdoped cuprates, as previously suggested11,12,13,14,15,16,17,18,19,20,21, that would preclude the possibility of CDW correlations having an important role in the anomalous electronic properties. For instance, it has been argued that since CDW correlations disappear at x ≪ xc, the quantum critical point (QCP) at x = xc must be magnetic in nature8. Tunneling spectroscopy studies have suggested a vestigial nematic QCP on a similar basis28,29. Very recent nuclear magnetic resonance results have reported the disappearance of spin glass behavior near xc30. Whether this disappearance is associated with the loss of stripe correlations (i.e., coupled spin and charge density waves) remains unresolved. Moreover, the existence of CDW correlations is also crucial for the relevance of intertwined order. Many theoretical models for pair density wave superconducting states, for example, require the presence of CDW correlations1,31.

Fig. 1: Doping-dependent electronic structure of LSCO.
figure 1

a Phase diagram of the hole-doped cuprates, constructed from magnetization, Nernst effect, and resistivity data for LSCO20,52. T* is the extracted pseudogap onset temperature20,52. b Schematic band structure of LSCO. The Fermi energy, EF, crosses the anti-nodal saddle point (labeled as SP) near xc ≈ 0.19 triggering a Lifshitz transition. c–e Fermi surface topology of LSCO12, LSCO17, and LSCO21. The intensity plots are obtained by integrating the spectra within ±10 meV of EF. Orange dashes outline the antiferromagnetic Brillouin zone. Red dashed contours represent a tight-binding fit of the Fermi surface (see Supplementary note 1). The data shown in c–e were collected at 11 K.

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Studies of underdoped and optimally doped cuprates have shown that CDW correlations exist up to temperatures well above the nominal CDW transition temperature32,33. More recently, re-entrant charge order, disconnected from the CDW at lower doping, was observed in overdoped (Bi, Pb)2.12Sr1.88CuO6+δ19. These results motivate a reconsideration of the cuprate phase diagram, in which CDW correlations may extend up to higher dopings than previously thought33. Herein, we address this issue by focusing on La2−xSrxCuO4 (LSCOx) (x = 0.12, 0.17, 0.21, and 0.25) single crystals in view of its particularly well-characterized transport properties and the feasibility of synthesizing high-quality samples across the entire phase diagram21,24,25,26,27 (see Methods section).

Results

Electronic structure

Previous measurements of LSCO established the presence of a critical doping xc ~ 0.19, which was defined as the doping above which the electronic transport acquires partial Fermi-liquid-like properties21,24. This point coincides with, and is perhaps related to, the doping where the Fermi surface (FS) topology undergoes a Lifshitz transition34,35. To prove the Sr doping, x, is consistent with previous studies, and that we indeed access the x > xc region of the overdoped phase diagram, we show the electronic structure evolution with doping in Fig. 1b–e. The angle-resolved photoemission spectroscopy (ARPES) methods used are described in the Methods section. Figure 1b illustrating the two-dimensional electronic structure. At low doping, to the extent that a FS exists, it is hole-like and centered at the Brillouin zone corner. With increasing hole concentration, the chemical potential drops and eventually passes through the saddle point. This results in a Lifshitz transition to an electron-like FS at the Brillouin zone center. In LSCO, the saddle point is three-dimensional with small kz dispersion. Near xc, the saddle point coincidentally crosses the Fermi level. It should be noted that the carrier concentration determined by the FS area is significantly larger than the nominal Sr doping. The origin of this effect remains unresolved34,35. Nevertheless, the well-established FS evolution in LSCO can be used to confirm the Sr doping in this study. Figure 1c–e shows ARPES measurements for LSCO12, LSCO17, and LSCO21. An electron-like FS is observed in LSCO21, consistent with xc = 0.19 and in agreement with previous ARPES studies34,35.

CDW order

Having confirmed the electronic structure, we now present our main experimental finding of CDW correlations beyond xc. Figure 2 plots X-ray reciprocal space scans for LSCO21 at T = 16 K, where reciprocal space is defined in terms of scattering vector Q = (H, K, L) using effective tetragonal lattice constants a = b ≈ 3.8 Å and c ≈ 13.2 Å. High sensitivity is achieved by exploiting the high brightness of the National Synchrotron Light Source II (NSLS-II) and by careful configuration of the detection system to suppress background signal (see Methods section). Superlattice peaks are observed at (0.235, 0, 12.5), and equivalent locations, along both the H and K directions [Fig. 2a, b]. The observed H = 0.235 matches the CDW wavevector in underdoped LSCO16,17,18,36 and is consistent with the charge stripe picture5. The peaks are symmetric with respect to ±H and K and are observed in multiple Brillouin zones including (±0.235, 0, L) for L = 8.5 and 12.5. An L-scan along Q = (−0.235, 0, L) [Fig. 2d] reveals that the CDW intensity is broadly peaked at half-integer L similar to underdoped LSCO17,18,37. These results demonstrate the presence of CDW correlations beyond xc. Subsequent inelastic X-ray scattering studies show that the CDW is associated with phonon softening even in the overdoped regime38.

Fig. 2: Discovery of a CDW beyond xc.
figure 2

a, b X-ray diffraction measurements of LSCO21 at T = 16 K along (0, K, 12.5) and (H, 0, 12.5). Supperlattice peaks are observed at (0, 0.235, 12.5) and (0.235, 0, 12.5). The H-scans in c reveal further CDW peaks at (±0.235, 0, L) for L = 8.5 and 12.5. The data at L = 8.5 are offset by −10 counts s−1 for visibility. d The L-dependence of the intensity along (−0.235, 0, L) demonstrates poorly correlated out-of-phase CDW stacking along the c-axis. Solid lines are fits to the experimental data as described in the text and Supplementary note 2. Error bars are one standard deviation based on Poissonian statistics.

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CDW temperature dependence

Figure 3 summarizes the doping and temperature dependence of the CDW correlations. In Fig. 3a, Lorentzian-squared fits to the data are shown, which are parameterized in terms of amplitude, ICDW(T), and in-plane correlation length, ξ∥(T) = 1/HWHM (where HWHM is half-width at half-maximum) (see Supplementary note 2). Since domain formation can lead to transverse peak splitting in LSCO [c.f. refs. 16,18,36 and Fig. 3a], we scanned through the peaks in all three reciprocal space directions. Two Lorentzian-squared functions displaced in the K (transverse) direction were used, where necessary, to account for the full intensity distribution. Peak widths and correlation lengths are determined using the H (longitudinal) cut. ICDW(T) is found to be largest near TSC for all dopings [Fig. 3b]. Above TSC, both ICDW and ξ∥(T) decrease with increasing temperature but remain finite up to at least T = 90 K [Fig. 3a]. In agreement with previous X-ray diffraction studies of LSCO16,17,36, the correlation length can be separated into a marginally-ordered regime where ξ∥(T) is ~4-unit cells (about one period of the CDW order), and a strongly T-dependent regime where ξ∥(T) continues to expand until superconductivity intervenes. We refer to the CDW in the T-independent regime as “precursor” CDW correlations in the sense that they come before the emergence of a stronger, more correlated CDW at low temperatures. Note that for these measurements we do not have the energy resolution to directly distinguish between dynamic and static correlations. The short correlation length and quasi-temperature-independent nature of the percursor CDW indicates that it might be dynamic in nature. This phenomenology is consistent with resonant inelastic X-ray scattering (RIXS) scattering studies of La2−xBaxCuO4 (LBCOx) and YBa2Cu3O6+δ which show a similar two-stage CDW formation32,33,39,40. While the CDW evolves smoothly from LSCO12 to LSCO21, both ξ∥ and the onset temperature of the longer-range CDW, Tξ, are suppressed in the overdoped regime around xc [Figs. 3c and 4]. The Q-integrated scattering intensity, as estimated by \({I}_{\text{CDW}}{\xi }_{\parallel }^{2}\), shows minimal variation through TSC, indicating that while superconductivity alters the CDW correlation length, it does not strongly suppress the order parameter. We do not observe any CDW correlations in our high-sensitivity X-ray measurements at x = 0.25 (see Supplementary note 3).

Fig. 3: CDW temperature dependence.
figure 3

Doping dependence of the CDW peak intensity for temperatures T < TSC, T ≈ TSC, and T > TSC for a LSCO12, b LSCO17, and c LSCO21. The inset of each panel represents the respective cut in reciprocal space. All data were taken at L = 8.5. Temperature dependence of the CDW intensity in LSCO for d LSCO12, e LSCO17, and f LSCO21. The shaded area corresponds to Tξ where the in-plane CDW correlation length, ξ∥, starts to increase [as determined in g]. The main panels show peak height intensity and the insets show integrated intensity. g Temperature dependence of ξ∥. The colored shaded curves are phenomenological fittings, \(a\,+\,\frac{b}{1+{e}^{(T\,-\,{T}_{0})/4{{\Delta }}T}}\), of the temperature-dependent ξ∥ for different dopings. Here we define Tξ = T0 + ΔT. ξ∥ increases with decreasing temperature for TSC < T < Tξ. Two independent measurements of LSCO17 samples at different beamlines show consistent suppression of ξ∥ and Tξ, indicating that systematic errors are minimal. Due to the short correlation length, the uncertainty of Tξ may be larger in LSCO17 than other dopings. Error bars are one standard deviation from either Poissonian statistics or least-squares fitting.

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Fig. 4: Illustration of the extent of CDW correlations in the cuprate phase diagram suggested by this work.
figure 4

Green-yellow tones represent our main result—the presence of CDW correlations, from 0.12 < x < 0.21. Green denotes the precursor CDW, which appear at high temperature and which have a correlation length of approximately one CDW period32,33,39,40. At lower temperature, the correlations start to grow into larger CDW domains, as evidenced by the increased correlation length, which we denote by the yellow tone. Red points mark where the correlation length starts to increase. This should be considered an approximate cross-over and not over-interpreted as a well-defined phase transition. At lower temperatures still, bulk d-wave superconductivity intervenes at TSC whereupon both the CDW amplitude and the correlation length saturate or start to decrease. The doping dependence reveals an anticorrelation between Tξ and TSC, providing evidence for an interaction between the CDW and superconductivity. This is illustrated by the cartoon in the bottom of the diagram in which superconducting pairing (green spin pairs) suppress the CDW (yellow solid and dashed sinusoidal curves). The CDW intensity disappears in heavily overdoped LSCO25, where a Fermi-liquid-like state is recovered (Supplementary notes 3 and 5). The red diamonds reflect the present study. Pink squares and green circles are data from previous work16,17,36,52.

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Previous measurements of the same x = 0.12 sample allow us to compare the CDW order parameter, taken to be captured by the total Q-integrated scattering intensity, to other cuprate systems16. The CDW order parameter of LSCO12 is only four times weaker than La1.875Ba0.125CuO4 (which has the strongest zero-field CDW order). With increasing doping, the LSCO CDW becomes somewhat stronger for x = 0.17 and drops appreciably for x = 0.21 (see Supplementary note 4). Consequently, CDW correlations can have an appreciable effect on the physics of LSCOx for dopings through xc.

Discussion

Figure 4 summarizes our main observations—that CDW correlations exist far into the overdoped regime of the cuprate phase diagram. This immediately yields three important consequences for LSCO. Firstly, very similar CDW properties are observed either side of the Lifshitz transition. This provides a vivid demonstration that CDW correlations cannot be explained within a weak coupling FS nesting picture nor Friedel oscillations. Instead, the nearly constant QCDW for dopings x ≥ 0.125 support strong coupling mechanisms, which date back to seminal work in the late 1980s2,3,4. In these mechanisms, one considers the balance between Coulomb interactions and kinetic energy. When doping a Mott insulator, holes can save energy by clustering together as this breaks fewer magnetic bonds than widely dispersed holes. At the same time, this clustering is disfavored by the increased Coulomb repulsion and kinetic energy reduction. Since these different interactions act on different lengthscales, the overall minimum energy solution is expected to involve a spatially modulated state. Modern numerical solutions of the Hubbard model further support this idea41,42,43 and models based on filled stripes can reproduce a doping-independent CDW wavevector from x = 1/8 to x = 1/444. We also note that precursor CDW correlations are emerging as a ubiquitous feature for many cuprates, including LSCO in this study, underdoped LBCO40, underdoped and optimally doped YBCO33, underdoped Bi221245, and HgBa2CuO4+δ46. In underdoped and optimally doped YBCO, the precursor correlations appear to exist at the same wavevector around 0.3 r.l.u. different to the doping-dependent low-temperature CDW33. It would consequently be interesting to consider a possible role for strong coupling mechanisms for all cuprates. An obviously desirable experiment would be to test whether other cuprates, such as YBCO, also exhibit CDW correlations up to similarly high dopings as LSCO. Such experiments are, however, currently held back by challenges in stabilizing high-quality heavily overdoped YBCO crystals. The robust presence of CDW correlations in LSCO seen here as a function of temperature and doping, as well as the fact that model Hamiltonian calculations reliably predict CDW correlations41,42,43, would point towards their likely presence. The issue of differing wavevectors in different cuprates would, however, not necessarily be solved by such an experiment. In this regard, it is important to point out the low-temperature ordering wavevector can be influenced by coupling between the CDW and spin correlations or coupling between the CDW and the lattice, as has been suggested theoretically47, so differences in CDW wavevectors could arise from secondary interactions rather than necessarily indicating a distinct origin for the correlations. Prior work has pointed towards this as a possible explanation for temperature-induced changes in CDW wavevector in LBCO32.

A second immediate conclusion is that the continuous evolution of the CDW correlations is inconsistent with the proposed QCP that is associated with xc arising from CDW or coupled CDW/spin density wave order6,10. Such theories can still be excluded even if one postulates a very narrow range of criticality around xc, since they require either a disappearance or a symmetry change of the CDW through xc.

Last but not least, the disappearance of CDW in LSCO25 suggests that the CDW dome in LSCO terminates between x = 0.21 and 0.25, where the Fermi-liquid behavior starts to recover21,24,25. This is, again, consistent with a strong coupling CDW mechanism as Coulomb repulsion is largely screened in the Fermi-liquid state. We note that in LSCO, the structural high-temperature tetragonal to low-temperature orthorhombic (LTO) phase transition also terminates near x = 0.2148. It has been argued that the local LTO distortion may help to stabilize the CDW49. The persistence of CDW correlations up to x = 0.21 observed in this study is consistent with this scenario and indicates that electron-phonon coupling might be an important ingredient for the CDW formation38,39.

The observations herein also urge a re-examination of the potential role of CDWs in the anomalous electronic properties of the cuprates. CDW correlations are a prerequisite (but not a proof) of several prominent theories of cuprate properties, which would be expected to apply across the phase diagram and not just in the underdoped region where CDW correlations have been studied extensively in the past. This include the possibility that CDW correlations play a key role in the electronic transport properties6,10. Theories of pair density wave order1,31,50,51, which predict competition between the CDW and uniform d-wave superconductivity, also fall into this category. As shown in Fig. 3, neither the CDW peak intensity nor the CDW correlation length shows the type of divergent-behavior associated with a typical phase transition. This behavior is consistent with a possible fluctuating CDW component, potentially influencing cuprate transport properties24,25,26,27,33.

Finally, we note that a charge Bragg peak has recently been observed in overdoped (Bi, Pb)2.12Sr1.88CuO6+δ (Bi2201), with a maximum doping comparable to that observed here19. This state, termed re-entrant charge order, has several properties that are different to CDW states in LSCO and other cuprates. Re-entrant charge order appears to exist only in an isolated region of the overdoped phase diagram, disconnected from the underdoped CDW order. The correlation length and temperature scale of this state are also far higher than other cuprates. Intriguingly, no interaction between re-entrant charge order and superconductivity is observed in Bi2201. In contrast, similarly well-correlated CDW states are associated with a strong suppression of superconductivity. All these behaviors are in strong contrast with the CDW in overdoped LSCO, where the CDW wavevectors, correlation length, and temperature dependence evolve smoothly from the properties of underdoped LSCO and strongly intertwine with superconductivity and low-temperature transport. Based on the electronic structure of Bi2201 and the wavevector of re-entrant charge order around 0.1 r.l.u., which extrapolates roughly linearly from the underdoped CDW wavevector, re-entrant charge order was proposed to arise from a van Hove singularity19. The overdoped CDW in LSCO appears to have no connection to this mechanism, since the CDW remains unchanged regardless of the proximity to the van Hove singularity at x = xc. Instead, our observations support strong coupling mechanisms.

In summary, high-sensitivity X-ray measurements have revealed that cuprate CDW correlations persist across almost the whole cuprate doping phase diagram, despite dramatic changes in the transport properties and FS topology, before disappearing when Fermi-liquid-like properties are restored. We have shown that these correlations impact superconductivity even in overdoped cuprates, suggesting that CDW correlations can have a far more extensive role in the cuprate phase diagram than previously envisaged, prompting investigations of CDW correlations in other overdoped cuprates. The discovery of CDWs beyond xc is confirmed by subsequent RIXS studies, which uncovered an unusual coupling between the CDW and lattice vibrations38.

Methods

Samples

Single crystals of La2−xSrxCuO4 (x = 0.12, 0.17, 0.21, and 0.25) were grown by the traveling-solvent floating-zone method. For each composition, a single feed rod of 20 – 25 cm long was used, the first few centimeters of which was removed and discarded after growth. The remaining rod was annealed in flowing O2 at 980 °C for 1 week. The superconducting transition temperatures, 28, 37, 30, and 10 K were determined by dc magnetization measurements in an applied field of 1 mT (after cooling in zero field). Our tight-binding fits to our ARPES measurements of these samples, described in Supplementary note 1, confirm the hole concentration matches the strontium content x.

ARPES

ARPES measurements were performed at the 21-ID-1 beamline of the NSLS-II using a Scienta-DA30 analyzer. Due to the small incident beam spot-size (less than 10 × 10 μm2), both the sample position and the incident light angle are fixed during the measurement. The ARPES intensity maps are obtained using the mapping-mode of the DA30 analyzer, which can cover 30° of cone acceptance without sample rotation. All samples were cleaved in-situ and measured at 11 K within a vacuum better than 7 × 10−11 mbar. The photon energy was set to 60 eV for LSCO12 and LSCO17 with 18 meV energy resolution. To confirm the FS of LSCO21 is a closed loop at the Γ point, we set the photon energy to 195 eV for LSCO21 with 90 meV energy resolution. At this energy, we were able to cover the second Brillouin zone without sample rotation. The chemical potential is calibrated based on the ARPES spectra on Silver that are recorded before and after the ARPES measurement.

Non-resonant hard X-ray scattering

High-precision X-ray scattering measurements were performed at the in-situ and resonant 4-ID beamline of NSLS-II and 4-ID-D beamline of the Advanced Photon Source (APS). The incident photon energy was set to 8.98 keV; slightly below the Cu K-edge to minimize the fluorescence background. The measurements at NSLS-II were carried out with an avalanche photodiode detector. A LiF(004) crystal analyzer was used to further suppress the background signal (Fig. 5). The measurements at the APS used a Vortex Si drift detector without any crystal analyzer.

Fig. 5: Experimental setup.
figure 5

Illustration of the experimental geometry at the 4-ID beamline of NSLS-II.

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