Introduction

High-Tc cuprates are the prominent example of a strongly correlated electronic system, featuring a rich phase diagram marked by novel types of order1. Tremendous effort has been devoted to studying these orders to understand the unconventional normal state and high-Tc superconductivity2. It is recognized that such complex systems are susceptible to electronic inhomogeneity3, which may arise intrinsically due to electronic interactions4 or for extrinsic reasons, such as chemical disorder5. Indeed, nanoscale spatial variations in electronic properties of the cuprates have been observed and may be relevant to their physics6,7. This issue is manifest in underdoped La2CuO4-based cuprates such as La2−xSrxCuO4 (LSCO), where long-range SDW (with correlation length of hundreds of unit cells) coincides with bulk superconductivity8. Consensus has yet to be reached regarding whether these orders coexist uniformly, or exist within distinct regions. Both scenarios have been suggested in theoretical studies9,10,11, while it has been difficult to determine experimentally12,13,14.

Recent observations of CDW order across cuprate families shed new perspective on the interplay between density wave orders and superconductivity1,2,8. When SDW order is absent, such as in Y-based cuprates (YBCO) around 1/8 doping, clear competition between CDW and superconductivity has been observed15,16. When both SDW and CDW are present in the La2CuO4-based cuprates, they appear to form a spin-charge stripe order pattern, indicated by nearly commensurate wave vectors (qcdw ~ 2qsdw)8. There are theoretical and experimental evidences of mutual cooperation between CDW and SDW order parameters when such commensuration is satisfied17,18. Here the interaction between density waves and superconductivity appears more than simple competition. For example, the SDW onset temperature (Tsdw as measured by neutron scattering) in LSCO around 1/8 doping is similar to the superconducting Tc8, and a putative two-dimensional superconductivity called pair density wave has been associated with the stripe order2,19. To elucidate the intrinsic behavior of these orders and distill universal features among different cuprates, it is important to carefully interpret observations in light of the inhomogeneity.

X-ray scattering combined with high magnetic fields provides a unique window into the nature of the coexistence of CDW, SDW, and superconductivity. If there is inhomogeneity on length scales larger than the CDW correlation length (typically tens of unit cells15,16,18,20,21,22), it should be manifest in the evolution of the CDW peaks when the strength of the various orders is tuned by T and H. We investigate La1.885Sr0.115CuO4 where the stripe order is robust and of comparable strength to superconductivity (Tsdw ~ Tc)23. Our state-of-the-art x-ray free electron laser measurements (see Methods for details) reveal evidence for two types of CDW orders in the same sample, distinguished by different correlation lengths and distinct T- and H- dependences. Most importantly, we further uncover a strong connection between the short-range CDW correlations and mobile superconducting vortices, which bears important implications regarding the nature of the superconducting transition in the cuprates.

Results

We first address the issue of inhomogeneity by examining the CDW T-dependence in zero magnetic field. As shown in Fig. 1a, the CDW peak appears as a rod of intensities along the l direction, which demonstrates its quasi-two-dimensional nature15,16,20,22 (more in Supplementary Figs. 5, 6). To focus on the CDW correlations within the CuO2 planes, the CDW intensities are integrated and projected along h. As a first analysis, a single Gaussian peak is found to fit the data well (Fig. 1b). Consistent with previous hard x-ray measurements20,22, the CDW intensities become appreciable below ~80 K (Fig. 1c). The onset with upward concavity is a common feature of the CDW in the cuprates, which indicates a lack of a long-range CDW transition24. While our hard x-ray scattering measurements are sensitive to the quasi-static lattice distortions associated with the CDW, other resonant x-ray scattering studies have detected CDW in LSCO up to higher temperatures18,21. This may be related to dynamical CDW fluctuations which persist over a much larger temperature range and can have important implications for high-Tc cuprates25,26,27,28.

Fig. 1: Temperature dependence of CDW in LSCO.
figure 1

a CDW intensity map measured at 6.5 K, 0 T, projected onto hl plane, integrated over k [−0.02,0.02] r.l.u. The dashed black rectangle encircles the CDW peak. b Temperature-dependent h-cuts through the CDW peak. Solid lines are single-peak fits to the data. A linear background has been subtracted, and data are shifted for clarity. The horizontal bar represents instrumental resolution. c Integrated CDW intensities extracted from the single-peak fits. The inset shows the corresponding peak width. The orange dashed line is a linear extrapolation of the CDW intensities at 28 K, 32 K, and 36 K to lower temperatures T < Tc, as described in the main text. The open circle shows the CDW intensity measured at 6.5 K, 24 T. Error bars represent one standard deviation.

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The CDW intensity reaches a maximum near the superconducting Tc before getting suppressed at lower temperatures, indicating competition between CDW and superconductivity. However, unlike a homogeneously weakened order, the CDW peak width keeps decreasing for T < Tc (Fig. 1c inset), which implies a growing CDW correlation length. Such contradictory behavior between the CDW intensity and correlation length has been observed, though with varying clarity, in previous x-ray measurements for LSCO at similar doping levels20,21,22 (also see Supplementary Fig. 7), and is in stark contrast with YBCO, where both the CDW intensity and correlation length decrease for T < Tc15,16. This indicates that mere competition between CDW and superconductivity is insufficient to explain the CDW behavior in LSCO.

As alluded to above, the cause here is the SDW order which onsets at Tsdw ~ Tc in LSCO and enhances CDW18,23, but does not coexist with CDW in YBCO29. To further elucidate the thermal evolution of the CDW, we decompose the CDW peak into two components: CDWstripe to describe the SDW-enhanced component, and CDWSRO (short-range order) to account for the competition with superconductivity. CDWSRO is expected to behave similarly to the CDW in LSCO near optimal doping (x ~ 0.145) where the SDW order is absent18. Since the CDW peak width there is weakly T-dependent at low temperatures18, we fix the CDWSRO peak width for T < Tc to that extracted at Tc in the single-peak fit, where the SDW order is just about to develop (Fig. 2b). This CDWSRO component has correlation length of ξSRO = 2/FWHM = 56(2) Å. As for CDWstripe, to a first approximation we also assume a T-independent peak width. To extract this width, we fit the T = 6.5 K data with the sum of the aforementioned CDWSRO peak and a second peak. The best fit yields a sharper component (CDWstripe) with correlation length ξstripe = 80(11) Å. Interestingly, ξSRO is comparable to that in YBCO at low temperatures15,30, while ξstripe is comparable to that in the canonical stripe ordered cuprates La1.48Nd0.4Sr0.12CuO4 and La2−xBaxCuO4 (LBCO) for which the CDW correlation length is ~1/4 of the SDW correlation length31,32 (here, the SDW correlation length in LSCO is ~300 Å33). Fixing the widths for these two components and allowing the respective intensities to vary, such constrained two-component fitting provides an alternative and better account of the data (Fig. 2d–f, Supplementary Fig. 8). Consistent with our hypothesis, the fitting results show that the CDW T-dependence for T < Tc can be described in terms of the weakening of CDWSRO and the enhancement of CDWstripe (Fig. 2a). In particular, the CDWstripe intensity follows the SDW intensity measured by neutron scattering23 (Fig. 2a), consistent with their cooperative interactions.

Fig. 2: Two-component decomposition of the CDW order in LSCO.
figure 2

a Integrated intensities for CDWSRO and CDWstripe extracted from the two-component fits. The brown dashed line shows the scaled SDW intensities measured by neutron scattering23. b Corresponding peak width for CDWSRO and CDWstripe. The widths are fixed for T < Tc for the two-component fits, as described in the main text. c A pictorial illustration of the mixed phases in LSCO at low temperatures, which consist of superconducting regions with suppressed CDWSRO, and separate regions dominated by spin-charge stripe order. d–f Representative data and corresponding fits at 6.5 K, 23 K, and 32 K, respectively. Orange, blue, and red dashed lines show the total, CDWSRO, and CDWstripe of the two-component fits, respectively. For comparison, the single-peak fits are shown in solid black lines. Blue and red shadings illustrate contributions from CDWSRO and CDWstripe, respectively. Error bars represent one standard deviation.

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Such a decomposition provides a natural explanation of the seemingly contradictory behavior of the intensities and widths in the single-peak analysis. The combined effect of a weakened CDWSRO (broad peak) and an enhanced CDWstripe (sharp peak) results in the net reduction of the integrated intensity simultaneously with a narrowing of the width. In fact, previous local probe measurements (such as μSR12 and NMR34) have indicated heterogeneous phases, and scattering studies which measure average correlations in the bulk have also provided evidence for charge inhomogeneities35. The high statistical quality of the zero-field data and resulting success of the two-component analysis to the CDW T-dependence strongly support the assertion that the CDW order is indeed heterogeneous for T < Tc in LSCO, with coexisting CDWSRO and CDWstripe components. This further indicates superconductivity and SDW does not coexist in a uniform phase: the CuO2 plane segregates into superconducting regions with suppressed CDWSRO, and separate regions dominated by spin-charge stripe order (CDWstripe commensurate with SDW), as illustrated in Fig. 2c.

With this picture in mind, we turn to the evolution of the CDW order under high magnetic field. Fig. 3a shows the intensity collected at 24 T. The persistence of the rod-like scattering shows that the CDW correlations remain two-dimensional (more in Supplementary Fig. 5). We note that we find no evidence of three-dimensional CDW order at integer l position (here l = 6) (see Supplementary Fig. 13), which has been observed in YBCO at a similar doping level and magnetic field30. This difference could be related to the different value for Hc2 and/or how the CDW order on neighboring CuO2 planes interacts36. Future measurements at even higher magnetic fields may provide more insight.

Fig. 3: Magnetic-field dependence of CDW in LSCO.
figure 3

a CDW intensity map measured at 6.5 K, 24 T, projected onto hl plane. b Magnetic-field-dependent h-cuts through the CDW peak. Solid lines are one-Gaussian fits to the data. A linear background has been subtracted, and data are shifted for clarity. The horizontal bar represents instrumental resolution. c Integrated CDW intensities as a function of magnetic field. Note here the magnetic-field projection perpendicular to the CuO2 planes H⊥ is plotted. The vertical dashed line marks the vortex-melting field Hm for LSCO at T ~ 6.5 K determined by in-plane resistivity measurements41. The red diamonds are field-dependent SDW intensities measured by neutron for LSCO14, scaled such to deduce the magnetic-field-induced enhancement of CDWstripe, as described in the main text. The red dashed line and red shading are guide to the eye. Error bars represent one standard deviation.

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Focusing on effects of field, we plot scans through the CDW peak in Fig. 3b. An increase in the CDW intensity with magnetic field is clearly observed. Although the statistical quality of the high-field data (constrained by magnet recovery time between magnetic pulses, see Methods) hinders a two-component analysis, the successful decomposition at zero field enables us to surmise the respective evolution of CDWSRO and CDWstripe in magnetic field. Assuming the proportionality between CDWstripe and SDW (Fig. 2a) holds in magnetic field, which is supported by previous measurements in LSCO and LBCO22,37,38 (see Supplementary Fig. 9 and corresponding discussion), we use the SDW magnetic-field dependence14 to infer the CDWstripe enhancement (with no adjustable parameters), δCDWstripe(H⊥) = CDWstripe(0) × [SDW(H⊥)/SDW(0) − 1], where H⊥ is the applied field projection perpendicular to CuO2 planes. As shown in Fig. 3c, δCDWstripe(H⊥) can account for the overall CDW enhancement at H⊥ ~ 10 T. Comparison to prior CDW measurements22 shows δCDWstripe(H⊥) also nicely describes the CDW evolution at smaller magnetic fields H⊥ < 10 T (see Supplementary Fig. 11), suggesting that in the low field regime < ~ 10 T, the enhancement to the overall CDW is mainly due to CDWstripe, while the CDWSRO intensity is relatively constant. Considering that CDWSRO is expected to be strengthened as the magnetic field suppresses superconductivity16, the volume fraction for CDWSRO has likely been reduced by the field. This is consistent with μSR measurement that suggests the magnetic field increases the stripe order volume fraction39.

At larger field H⊥ > 10 T, we observe a sudden increase in the overall CDW intensity that cannot be accounted for by δCDWstripe (Fig. 3c). We conclude this originates from CDWSRO. The field range where this enhancement occurs coincides with the vortex-melting field Hm ~ 10 T inferred in transport measurements40,41. Superconducting vortices which were pinned in the vortex solid at lower magnetic fields become mobile above Hm. This results in the loss of superconductivity due to the destruction of long-range phase coherence40. Considering the interplay between CDWSRO and superconductivity (Fig. 2), the sudden increase of the CDW amplitude δCDWSRO above Hm (or, equivalently, a suppression of CDWSRO upon entering the vortex-solid phase) implies a strong response of the CDW to the state of the vortices, whether pinned or mobile. Our result is complementary to recent theoretical and experimental studies that reveal an intricate interaction between superconductivity and topological defects in the coexisting CDW order42,43,44. We show that mobile topological defects in the superconducting order, the vortices, greatly enhances the CDW. It will be interesting to measure LSCO samples of different doping levels in the future to investigate how the detailed structure of the vortex solid affects the response of the CDW order at the vortex-melting transition45.

Discussion

Both sets of our x-ray measurements (the T-dependence in zero-field, and the H-dependence at low-T) reveal the presence of two regions: regions that favor static spin-charge stripe order and regions that favor superconductivity (where the latter also harbor short-range CDWSRO). The stripe phase competes with uniform superconductivity, consistent with model calculations which reveal a near degeneracy between the superconducting and the stripe states46. Interestingly, a small change in the Hamiltonian can drive the system between these two distinct phases through an intermediate state featuring phase separation47. In LSCO, the presence of dopant disorder or structural inhomogeneity may play a role in stabilizing both phases simultaneously within the same sample.

The observed behavior of the CDWSRO within the majority superconducting regions is particularly interesting. While CDW peaks have been observed in the normal and superconducting states of other cuprates, here, we find evidence that enhanced CDWSRO is linked to the high-field vortex-liquid state. Noting that similar CDW peaks are observed in zero field in a broad temperature range above Tc, one finds an interesting connection with Nernst effect measurements. Despite different interpretations regarding the origin of the Nernst signal at higher temperatures, there is consensus that for a temperature range ~30 K above Tc in LSCO the Nernst signal can be attributed to superconducting fluctuations40,48,49. This is the same temperature range where the CDW becomes significant (Fig. 1c). By plotting the CDW intensity in the vortex-liquid state (in a field of 24 T) in Fig. 1c, we find good agreement with an extrapolation of the zero-field intensity from above Tc to low temperatures. This connection between the CDW in the vortex liquid to that above Tc is consistent with the interpretation that the vortex-liquid state is continuously connected to the non-superconducting state above Tc, where both regions of the H − T phase diagram possess a large Nernst signal due to mobile vortices40. Our results indicate that mobile vortices and CDWSRO correlations are both inherent to the state where superconducting long-range phase coherence is lost. Within a phase-disordering scenario for the loss of superconductivity40, short-range two-dimensional CDW correlations appear to be compatible with local superconducting pairing.

These results reveal a distinction between the two types of charge order in the cuprates: CDWstripe and CDWSRO. CDWstripe and the associated static stripe state is most prominent in the La2CuO4-based cuprates, and is clearly competitive with uniform superconductivity. The ubiquitous CDWSRO, on the other hand, coexists with local superconductivity and may even aid the formation of vortices50,51. It is long range superconducting phase coherence that simultaneously suppresses CDWSRO and the presence of mobile vortices. The apparent sensitivity of CDW correlations to superconducting phase coherence further suggests a unified quantum description of the density waves and superconductivity in cuprate superconductors2,8,24.

Methods

Sample preparation

High-quality single crystalline La1.885Sr0.115CuO4 samples were grown by the traveling solvent floating zone method. The typical growth rate was 1.0 mm h−1 and a 50–60 mm-long crystal rod was successfully obtained. A 10 mm-long crystalline piece from the end part of the grown rod was annealed in oxygen gas flow to minimize oxygen deficiencies. The superconducting transition temperature Tc of the sample is determined to be 27.5(2) K (Supplementary Fig. 2). We focus on the CDW peak near (2.23,0,5.5) r.l.u. (reciprocal space is denoted using the tetragonal unit cell, a = b = 3.77 Å, c = 13.25 Å), where the CDW intensity is strong20 and the scattering geometry allows a large magnetic field projection ( ~ 82%) along the crystallographic c-axis (perpendicular to the CuO2 planes). The CDW peaks were initially confirmed by resonant soft x-ray scattering at SSRL18. A 1 × 0.5 × 0.5 mm3 sample was oriented using Laue x-ray diffraction and polished to the desired dimensions with [2.23,0,5.5] direction normal to the scattering surface. The vertical scattering plane is spanned by the nominal [2.23,0,5.5] and [0,1,0] directions. In this geometry, the magnetic field direction is ~ 35. 2∘ tilted away from the crystalline c-axis.

XFEL measurement

The x-ray scattering experiment was carried out on the X-ray Correlation Spectroscopy (XCS) instrument at the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory (Supplementary Fig. 1). Horizontally polarized x-ray beam with incident energy of 8.8 keV was used with the pink beam set up. The photon energy was chosen to be just below the Cu K-edge to reduce fluorescence background. A split coil pulsed magnet was used to provide large magnetic fields. Femtosecond x-ray pulses were synchronized with the magnetic field pulse (~1 ms duration) such that one photon pulse arrives on the sample at the maximum magnetic field strength. The orientation of the sample was determined by measuring the (204) and (206) nuclear Bragg peaks, which was then used to convert the pixel coordinate to the reciprocal-space coordinate (Supplementary Fig. 3). The CDW temperature dependence was probed upon warming. The magnetic-field-dependent measurements were carried out at the lowest temperature achievable of 6.5 K, where the largest field-induced effect is expected. The sample rotation was fixed at the CDW rocking scan peak center (Supplementary Fig. 4). For each magnetic-field run, 10 measurements at zero field were taken immediately before and after the field pulse to provide an accurate zero-field reference (Supplementary Fig. 1). While for zero-field measurements data can be taken continuously at 120 Hz x-ray pulse frequency, measurements in magnetic fields require extra time to cool down the magnet after each magnetic-field pulse. For example, at 24 T the cool-down time was typically ~15 minutes between magnetic pulses. This imposed a constraint on the statistical quality of the high-field data achievable during the finite beam time.