Charge density waves in cuprate superconductors beyond the critical doping

The cuprate high-Tc superconductors are often conceptualized as doped Mott insulators, in which the electronic ground state spontaneously breaks rotational and/or translational symmetry. While cuprate CDW correlations were discovered over two decades ago, their possible contribution to the material’s anomalous electronic properties remains a matter of vigorous debate. This issue has gained increasing attention in light of the ubiquity of CDW order in different cuprate families. The cuprate phase diagram 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-like. If CDW correlations are confined to underdoped cuprates, as previously suggested, 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 xc, the quantum critical point (QCP) at xc must be magnetic in nature. Tunneling spectroscopy studies have suggested a vestigial nematic QCP on a similar basis. Very recent nuclear magnetic resonance results have reported the disappearance of spin glass behavior near xc. 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 correlations.

Studies of underdoped and optimally doped cuprates have shown that CDW correlations exist up to temperatures well above the nominal CDW transition temperature. More recently, re-entrant charge order, disconnected from the CDW at lower doping, was observed in overdoped (Bi, Pb)2Sr1.88CuO6+δ. These results motivate a reconsideration of the cuprate phase diagram, in which CDW correlations may extend up to higher dopings than previously thought. 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 diagram (see Methods section).

Prior work established that CDW correlations are ubiquitous across many cuprate families and were first reported over two decades ago. The cuprate phase diagram shows anomalous regimes (pseudogap, strange metal, superconductivity) below a critical doping xc ≈ 0.19, with more Fermi-liquid-like behavior above this level. Earlier studies largely confined robust CDW signatures to the underdoped regime, suggesting limited relevance to phenomena near xc and motivating alternate proposals for a QCP (e.g., magnetic or vestigial nematic). CDW correlations can persist to high temperatures above nominal ordering temperatures in several cuprates, and re-entrant charge order was observed in overdoped Bi-based cuprates (Bi2201), apparently disconnected from underdoped CDW and potentially linked to a van Hove singularity. Theoretical frameworks including strong-coupling stripe physics and Hubbard/t−J model studies support doping-insensitive CDW wavevectors and fluctuating stripe states, while intertwined orders such as pair-density waves often require underlying CDW correlations. These backgrounds frame the question of whether CDW correlations persist beyond xc and how they interact with superconductivity.

Samples: Single crystals of La2−xSrxCuO4 with x = 0.12, 0.17, 0.21, and 0.25 were grown by the traveling-solvent floating-zone method. The first few centimeters were removed post-growth; rods were annealed in flowing O2 at 980 °C for 1 week. Superconducting transition temperatures (from dc magnetization at 1 mT, ZFC) were 28 K (x=0.12), 37 K (x=0.17), 30 K (x=0.21), and 10 K (x=0.25). Tight-binding fits to ARPES confirmed hole concentration matches Sr content x.

ARPES: Conducted at NSLS-II 21-ID-1 using a Scienta-DA30 analyzer. Small beam spot (<10×10 µm²) with fixed sample position and light angle; mapping mode covered a 30° cone without rotation. Samples were cleaved in-situ and measured at 11 K under vacuum better than 7×10−11 mbar. Photon energies: 60 eV for x=0.12 and 0.17 (18 meV energy resolution); 195 eV for x=0.21 (90 meV resolution) to capture a closed electron-like Fermi surface in the second Brillouin zone. Chemical potential was calibrated with Ag reference before/after measurements.

Non-resonant hard X-ray scattering: High-precision X-ray diffraction at NSLS-II 4-ID and APS 4-ID-D. Incident energy 8.98 keV (slightly below Cu K-edge) minimized fluorescence background. NSLS-II used an avalanche photodiode detector with a LiF(004) crystal analyzer to suppress background; APS used a Vortex Si drift detector without analyzer. Reciprocal space defined as Q=(H,K,L) with effective tetragonal lattice constants a=b=3.8 Å, c≈13.2 Å.

Data acquisition and analysis: Reciprocal space scans were performed along H, K, and L to locate superlattice peaks and characterize in-plane and out-of-plane correlations. CDW superlattice peaks were searched at wavevectors near (±0.235, 0, L) and (0, ±0.235, L), across multiple Brillouin zones (e.g., L=8.5, 12.5). L-scans probed c-axis stacking. Temperature-dependent measurements (down to ~16 K and up to 90 K) were made for x=0.12, 0.17, and 0.21 at fixed L (typically 8.5). Peak profiles were fit using Lorentzian-squared functions; where necessary, two Lorentzian-squared functions displaced in the transverse K direction were used to account for domain-induced peak splitting. The in-plane correlation length ξ(T) was obtained from the longitudinal (H) half-width-at-half-maximum (HWHM) via ξ=1/HWHM. The CDW amplitude I_CDW(T) and integrated intensities were extracted to assess order parameter evolution. Measurements did not have energy resolution sufficient to distinguish static from dynamic CDW directly. Sensitivity checks were performed at x=0.25, where no CDW signal was detected.

• ARPES confirmed the expected doping evolution of LSCO’s Fermi surface, including a Lifshitz transition: LSCO21 (x=0.21) exhibits an electron-like Fermi surface centered at the Brillouin zone center, consistent with crossing the anti-nodal saddle point near xc≈0.19.
• High-sensitivity X-ray diffraction revealed CDW correlations in overdoped LSCO up to at least x=0.21. Superlattice peaks were observed at Q≈(±0.235, 0, L) and (0, ±0.235, L) (e.g., L=8.5, 12.5), with symmetric peaks along H and K. The in-plane wavevector H=0.235 r.l.u. matches that in underdoped LSCO and is consistent with charge stripe correlations.
• L-dependence shows intensity broadly peaked at half-integer L, indicating poorly correlated, out-of-phase stacking along the c-axis, similar to underdoped LSCO.
• Temperature dependence (x=0.12, 0.17, 0.21): Above Tsc, both the CDW amplitude and in-plane correlation length decrease with temperature but persist to at least 90 K. A quasi–temperature-independent, short-range “precursor” CDW regime exists with ξ≈4 unit cells (~one CDW period), followed at lower T by a regime where ξ increases until superconductivity onsets, after which ξ and intensity saturate or slightly decrease.
• The onset temperature for increased correlation length (Tε) and the correlation length ξ are suppressed near xc in the overdoped regime; nonetheless, CDW evolves smoothly through xc with no discontinuity in wavevector or symmetry despite FS topology and transport changes at xc.
• Superconductivity strongly affects the CDW correlation length around Tsc, but the Q-integrated CDW scattering (order parameter proxy) shows minimal change through Tsc, indicating interaction rather than complete suppression.
• No CDW correlations were detected at x=0.25 under high-sensitivity conditions, suggesting the CDW dome terminates between x=0.21 and 0.25, coincident with recovery of Fermi-liquid-like behavior.
• Comparative intensity: The CDW order parameter at x=0.12 is only about four times weaker than in La1.875Ba0.125CuO4 (one of the strongest zero-field CDW systems). With doping, the CDW strengthens somewhat at x=0.17 and drops appreciably at x=0.21.
• Subsequent inelastic X-ray (RIXS) and phonon studies show the overdoped CDW is associated with phonon softening and unusual coupling to lattice vibrations.

The observation of robust CDW correlations up to x=0.21 demonstrates that CDW phenomena extend beyond the putative critical doping xc≈0.19 in LSCO, challenging views that restrict CDW relevance to the underdoped regime. The nearly doping-independent CDW wavevector (Q≈0.235 r.l.u.) across the Lifshitz transition argues against weak-coupling Fermi surface nesting or Friedel oscillation mechanisms and supports strong-coupling scenarios rooted in the balance of Coulomb interactions and kinetic energy in doped Mott insulators. Numerical studies of the Hubbard/t−J models and filled-stripe pictures are consistent with doping-insensitive wavevectors and fluctuating stripes.

The continuous evolution of CDW properties through xc is inconsistent with theories positing a CDW-driven (or coupled CDW/SDW) quantum critical point at xc, which would require disappearance or symmetry change at xc. Instead, the data show strong CDW–superconductivity interplay: superconductivity modifies the CDW correlation length near Tsc while leaving the integrated CDW intensity roughly unchanged, indicating competition or intertwining rather than a simple suppression. The inferred CDW dome likely terminates between x=0.21 and 0.25, where Fermi-liquid behavior recovers, consistent with increased screening weakening strong-coupling CDW mechanisms. The proximity of the structural LTO transition termination near x≈0.21 and the observed phonon anomalies suggest electron–phonon coupling contributes to stabilizing CDW correlations.

Comparisons with overdoped Bi2201 indicate distinct behaviors: re-entrant charge order in Bi2201 appears isolated in the overdoped region with longer correlation lengths and higher characteristic temperatures, and shows little interaction with superconductivity, possibly linked to a van Hove singularity. In contrast, LSCO’s overdoped CDW smoothly connects to underdoped behavior, with similar wavevectors and strong coupling to superconductivity and transport, favoring a strong-coupling origin.

These findings imply CDW correlations may play a more extensive, potentially fluctuating role across the cuprate phase diagram, influencing transport and enabling intertwined orders such as pair-density waves. Further studies in other cuprate families at high overdoping—though hampered by sample challenges—would test the generality of these conclusions and probe wavevector differences potentially arising from secondary couplings to spin or lattice degrees of freedom.

High-sensitivity X-ray diffraction, supported by ARPES, reveals that charge density wave correlations in LSCO persist into the overdoped regime up to at least x=0.21, with wavevector and temperature evolution continuous across the Lifshitz transition near xc≈0.19. CDW correlations interact with superconductivity by modifying correlation lengths near Tsc while maintaining substantial order parameter strength, and they disappear by x=0.25 as Fermi-liquid behavior returns. These results support strong-coupling mechanisms for CDW formation and suggest a broader role for CDW correlations in cuprate physics than previously recognized. Future work should extend similar measurements to other cuprates (e.g., YBCO) at high overdoping, refine energy-resolved probes to distinguish static versus dynamic CDW components, and explore the roles of electron–phonon coupling and structural distortions in setting CDW wavevectors and domes.

• Energy resolution in the X-ray diffraction measurements does not distinguish static from dynamic CDW correlations; the short, quasi–temperature-independent precursor CDW may be dynamic in nature.
• Determination of Tε (onset of increasing correlation length) carries uncertainty, particularly in x=0.17 due to short ξ.
• While ARPES confirms FS evolution, the carrier concentration inferred from FS area differs from nominal Sr content; although well-established in LSCO and used for consistency checks, the microscopic origin of this discrepancy remains unresolved.
• No CDW signal was detected at x=0.25 within the experiment’s sensitivity; while this supports termination of the CDW dome, absolute absence cannot be proven beyond experimental detection limits.
• Results are specific to LSCO; extension to other cuprate families is limited by challenges in obtaining high-quality heavily overdoped crystals.