Observation of two types of charge-density-wave orders in superconducting La2-xSrxCuO4

The discovery of charge- and spin-density-wave (CDW/SDW) orders in superconducting cuprates has altered our perspective on the nature of high-temperature superconductivity (SC). However, it has proven difficult to fully elucidate the relationship between the density wave orders and SC. Here, using resonant soft X-ray scattering, we study the archetypal cuprate La2-xSrxCuO4 (LSCO) over a broad doping range. We reveal the existence of two types of CDW orders in LSCO, namely CDW stripe order and CDW short-range order (SRO). While the CDW-SRO is suppressed by SC, it is partially transformed into the CDW stripe order with developing SDW stripe order near the superconducting Tc. These findings indicate that the stripe orders and SC are inhomogeneously distributed in the superconducting CuO2 planes of LSCO. This further suggests a new perspective on the putative pair-density-wave order that coexists with SC, SDW, and CDW orders.

S ince the 1986 discovery 1 of high-T c superconductivity (SC) in the cuprates, there has been an intense focus on understanding the essential physics of high-temperature SC. A major difficulty arises from the remarkably complex phase diagram. For example, in cuprates there are the so-called pseudogap and strange-metal regimes, and a variety of non-superconducting electronic orders (e.g., density-waves) associated with brokensymmetries [2][3][4] . Density wave order can occur on its own, or can coexist with superconducting order [5][6][7][8][9][10][11][12][13][14][15][16][17][18] . These complexities raise a fundamental question that has yet to be fully addressed-how do these orders/phases interact with SC [19][20][21] ? Regarding this question, the role of charge-density-wave (CDW) order, which has been observed universally across different families of cuprates [10][11][12][13][14][15][16][17][18] , is seemingly the most straightforward to interpret. For instance, in YBa 2 Cu 3 O 6+x (YBCO) the CDW order, which develops above the superconducting T c , is suppressed with the emergence of SC upon cooling 12,13 . Conversely, CDW order below T c is reinforced upon the suppression of the superconducting state by a magnetic field 22,23 . These phenomena clearly demonstrate a competitive relationship between CDW and SC. Another universally observed electronic order is spin-densitywave (SDW) order [5][6][7][8] , but its relationship with CDW and SC is less clear-cut.
An ideal candidate-material for this study should contain regions in the phase diagram where CDW, SDW, and SC coexist. In YBCO, one of the most-studied cuprates, the experimentally observed CDW and SDW orders appear to be mutually incommensurate, and there is little or no regime in its phase diagram in which these three orders coexist 24,25 . From a phenomenological perspective, this can be accounted for in terms of a strongly repulsive bi-quadratic coupling in Landau-theory 26,27 . On the other hand, in the La-based cuprates, La 2-x-y (Ba,Sr) x (Nd,Eu) y-CuO 4 , the CDW and SDW orders tend to satisfy a mutual commensurability condition on the ordering vectors, q cdw = 2q sdw 28 , which implies the importance of a special tri-linear term in the Landau theory 26,27 . Moreover, there is generally a range of doping in which all three orders are observed to coexist [29][30][31][32] . In La 2-x Ba x CuO 4 (LBCO), La 1.8-x Eu 0.2 Sr x CuO 4 (LESCO), and La 1.6-x Nd 0.4 Sr x CuO 4 (LNSCO), however, the phase diagram is further complicated by the existence of a low temperature tetragonal (LTT) phase that tends to stabilize the CDW and SDW orders, and depress SC 28,33 .
In this regard, we view La 2-x Sr x CuO 4 (LSCO), which does not undergo LTT transition, as the ideal platform for the present study. As summarized in the schematic phase diagram in Fig. 1, in this study we find a temperature (T) and doping (x) dependence of the CDW order that differs in significant ways from what has been previously conjectured. We reveal distinct behaviors in various ranges of doping: (i) For x < x cdw~0 .1 we observe no feature that can readily be identified with CDW order, although neutron scattering and NMR studies reveal the existence of stripe-like SDW order 9,34,35 . (ii) For x cdw ≤ x ≤ x sdw~0 .135 we observe CDW order with long correlation length that grows still longer even below superconducting T c ; from previous studies, x sdw is identified as the upper boundary of the regime in which (quasi) static SDW order persists at low T (with an onset in the neighborhood of T c ); 9 we identify the growth of CDW correlations below T c with its mutual commensuration with the SDW order. (iii) For x sdw < x < x*~0.18, where neutron scattering indicates the existence of a spin-gap at low T and NMR studies show no signatures of quasi-static magnetic order 9,35 , we still observe clear evidence of well-developed short-range CDW correlations, but these are significantly suppressed at temperatures below T c . (iv) For x ≥ x*, no clear evidence of CDW or SDW order has been observed. In the following sections, detailed experimental findings and their implications will be discussed.

Results
Extending CDW phase diagram of LSCO. The CDW order in LSCO has been observed previously by using X-ray scattering within a limited doping range in the underdoped side (x = 0.11, 0.12, and 0.13, also see Fig. 1) 31,[36][37][38] . From a detailed analysis of the thermal evolution of the Seebeck coefficient, it was suggested that CDW order is confined to x smaller than a critical doping 0.15 39 . In this context, we first aimed to complete the CDW phase diagram in a wide doping range that extends from the underdoped to the overdoped side (x = 0.075, 0.10, 0.115, 0.12, 0.13, 0.144, 0.16, and 0.18). For this purpose, we employed a novel resonant soft X-ray scattering (RSXS) approach that significantly mitigates fluorescence background (see "Methods" section). Figure 2a shows a schematic of how the fluorescence rejection works during the RSXS measurement. A large area detector was used to measure both the signal of interest (near CDW area) and the background signal (away from CDW) simultaneously. By subtracting the background (Fig. 2b, c), we achieve a significantly improved detecting sensitivity in the RSXS, allowing us to explore weak CDW signals. Figure 2d shows the scattering intensity maps along the h-/k-direction centered at q cdw~( −0.23, 0, l) r.l.u., which were measured at respective T c . Clear CDW peaks are observed for 0.1 < x < 0.18 (x = 0.115, 0.12, 0.13, 0.144, and 0.16). Furthermore, in the low dopings we can clearly resolve a CDW peak splitting along the k-direction, which is consistent with previous reports of CDW and SDW coexistence in x = 0.12 37,40 . This splitting becomes rather unclear for x ≥ 0.144, a behavior that seems to be correlated with the disappearance of SDW stripe order above x sdw~0 .135 9 .
Two types of CDW orders. As a next step, we explore the T dependence of the CDW orders. Figures 3a, b show the projected CDW signals along the h-direction for x = 0.144 and 0.13 LSCO, respectively (See Supplementary Fig. 1 for corresponding data for x = 0.115, 0.12, and 0.16). Upon cooling, the CDW peak in x = 0.144 (x > x sdw ) increases, reaching the maximum around T c , and then decreases as SC emerges below T c . This is consistent with the expectation that CDW competes with SC 12,41,42 . For a slightly The open and filled diamonds denote T w and the onset temperature of CDW stripe order determined in this study, respectively. The error bar of T w is estimated to be 10 K. The filled squares are the CDW onset temperature reported in previous X-ray scattering studies 31,39 . T sdw is the SDW onset temperature determined from neutron scattering measurements 9,39 . T c is the superconducting transition temperature 39 and T* is the pseudogap temperature 43 . The black and red cross symbols indicate the doping levels where no CDW-SRO or CDW stripe order have been detected in this study, respectively less doped sample of x = 0.13 (x < x sdw ), while the high-T behavior is quite similar, the evolution of the CDW correlations through T c is surprisingly different. The CDW peak continues to increase with decreasing temperature even below T c , which has not been unambiguously reported in the previous studies on LSCO 31,[36][37][38] (Supplementary Discussion and Supplementary  Fig. 2). For quantitative investigation of the T dependence, we fitted the CDW peaks with the Lorentzian function ( Fig. 3c-l). For temperatures above T c , the CDW correlations in all dopings follow similar trend. The peak intensities in all dopings decrease continuously with increasing temperature, showing no clear indication of an onset-T. Notably, the CDW peak intensity in our study persists even above recent estimates of the pseudogap temperature (T * ) (see Supplementary Fig. 3) 43 . Instead of an onset-T, we could extract a characteristic temperature (T w ) from the T-dependent full width at half maximum (FWHM) plots. The T w is the temperature above which the CDW correlation length ξ cdw (calculated as 2/FWHM) becomes approximately T-independent and below which the ξ cdw increases with decreasing temperature. Such extracted T w matches previously reported CDW intensity onset-T as shown in Fig. 1. The T-independent ξ cdw at high temperatures is reasonably short at~24 Å, but is still longer than the CDW wave-length λ cdw~4 a~15 Å. Note that we view ξ cdw ≥ λ cdw or equivalently FWHM ≤ q cdw /π, as a minimal condition for unambiguously identifying a diffraction peak as Max.
Min.  (Fig. 3c-f), the increasing trend in the ξ cdw stops or even slightly reverses below T c , which is coincident with the suppression of the CDW peak intensity. This is similar to what is commonly observed in the superconducting YBCO and can be readily explained by the competition between the CDW and the SC 12,41,42 . Hereinafter, we refer to this as CDW short-range order (SRO). In stark contrast, for x < x sdw (Fig. 3g-l), both ξ cdw and peak intensity keep increasing below T c . Such contrast cannot be attributed in any naïve way to enhanced disorder-induced pinning of the CDW-SRO in x ≤ x sdw samples (i.e., x = 0.115, 0.12, and 0.13). Given that disorder in LSCO is mainly induced by Sr substitution, these less doped samples are expected, if anything, to have less pinning. This indicates that the enhanced CDW at low-T is a new type of CDW order.
Interplay between the two types of CDW and SDW. To scrutinize this new type of CDW order and the CDW-SRO, we replotted CDW intensities vs. T/T c for different doping levels with the intensities at T c normalized to unity (see Fig. 4a). This figure clearly shows two drastically different temperature regimes separated by T/T c = 1. For T < T c , the contrasting behavior of the CDW intensity between x < x sdw (x = 0.115, 0.12, and 0.13, where static SDW order develops below~T c ) and x > x sdw (x = 0.144 and 0.16, where static SDW order is absent) indicates that the enhancement of the CDW order is associated with the development of SDW order. This is further supported by the step-like increase of the low-T CDW correlation length for x < x sdw as shown in Fig. 4b. Furthermore, the CDW and the SDW in LSCO are found to follow the q cdw~2 q sdw relation (see Supplementary  Fig. 4), which is analogous to the prototypical stripe order in LBCO 30 . We therefore identify the new type of CDW as the CDW stripe order. For T > T c , the normalized CDW intensities for all doping levels track each other, suggesting that the CDW orders above T c are of the same short-range-order type. These observations, in conjunction with the fact that neither of the two CDW types exists at doping levels x = 0.1 and below even though clear SDW stripe order exists in these dopings 9,34 , imply that the CDW stripe order is not simply parasitic to the SDW stripe order, but rather is due to a cooperative interaction between SDW and the preexisting CDW-SRO. In short, these findings provide clear and direct evidence of the intertwining between CDW and SDW orders in LSCO.

Discussion
We now discuss the implications of our results on SC. Previous NMR, μSR and neutron studies of the magnetic correlations for x < x sdw have demonstrated that the static SDW order is inhomogeneously distributed in the superconducting CuO 2 plane at T < T c 9, 35,44,45 . The linkage between the CDW stripe order and the growth of the SDW order for x cdw < x < x sdw and T < T c implies that they are coincident in the same regions. On the other hand, the CDW-SRO, which is presumably uniformly distributed at high-T, is likely to continue to persist in the rest of the regions where SDW order is absent at low-T, but is suppressed by the development of SC. We illustrate such low-T state in Fig. 5, which represents a CuO 2 plane consisting of regions of uniform d-wave SC and regions with the stripe orders. This is the sort of structure expected when two-phase coexistence is frustrated either by disorder or long-range interactions 3,4 , and is consistent with recent numerical studies that find near degeneracy between SC and stripe state 46,47 . This picture also provides a plausible explanation for the seemingly contradictory CDW results on x = 0.12 LSCO, where the CDW peak intensity increases (ref. 36 and this study), decreases 31 , or levels off 37,38 upon cooling below T c . Such discrepancy can be due to the different volume fractions of CDW stripe order that were probed in different experiments (see "Supplementary Discussion" and Supplementary Fig. 2). The fact that SDW/CDW stripe order, and SC order all onset at roughly the same T, and that the SC T c remains sharp implies that these orders are intimately related. In particular, these findings suggest that substantial SC order permeates the stripe-ordered regions of LSCO. Given that where such stripe order occurs in LBCO, it has been proposed based on transport anomalies that the SC order primarily takes the form of the pair-density-wave (PDW) 48,49 , it is natural to conjecture that the same is true of the stripe-ordered regions in LSCO. If this is the case, altering the balance between stripe (PDW) ordered and uniform SC regions in CuO 2 plane is expected to affect c-axis SC coherence much more dramatically than in-plane SC properties. Indeed, our conjecture is supported by the observation that for x~0.1 LSCO, an applied magnetic field much smaller than H c2 strongly enhances stripe order 50 and abruptly quenches c-axis SC coherence measured optically 51 . Thus the connection between stripe orders and PDW is likely universal in superconducting cuprates beyond LBCO 48,49 . Future experimental work using a spatiallyresolved probe, such as STM, will provide further evidence regarding the spatial distribution of SC order in LSCO. Finally, turning to high T and x, the observation that CDW-SRO persists above T*, without a change in the correlation length, opens the possibility that it might be directly correlated with other pseudogap phenomena. At minimum, the upper critical doping boundary of the CDW-SRO is strikingly close to the pseudogap critical doping, x* = 0.18 39,43 . Similar correlation has also been noticed in a previous Bi 2 Sr 2 CaCu 2 O 8+δ study 52 . For the RSXS measurements, we prepared all the samples with a typical dimension of 1.5 mm × 1.5 mm × 2.5 mm (a × b × c axis). To achieve a fresh c-axis normal surface, each sample was ex situ cleaved, before transported into the ultrahigh vacuum chamber (base pressure = 8 × 10 −10 Torr).

RSXS measurement.
All the experiments were carried out at beamline 13-3 of the Stanford Synchrotron Radiation Lightsouce (SSRL). The sample was mounted on an in-vacuum 4-circle diffractometer. The sample temperature was controlled by an open-circle helium cryostat. Incident photon polarization was fixed as sigma (vertical linear) polarization. Exact (h, 0, l) scattering plane was aligned by the measured (0, 0, 2), (−1, 0, 1), and (1, 0, 1) structural Bragg reflections at the photon energy~1770 eV. The energy of the Cu L 3 -edge was determined by X-ray absorption spectroscopy (see Supplementary Fig. 6). The typical energy resolution of incident X-ray at the Cu L-edge region is 0.1 eV.
A 256 × 1024 pixel CCD detector was used for the RSXS measurements. Each pixel is of size 26 μm × 26 μm. To minimize the geometric effect in the measurements (the background slope as a function of θ-angle), the vertically wide 2D-CCD detector was fixed at 2θ~154°which is the highest achievable 2θ position without blocking incoming X-rays. By rotating the sample (i.e., θ-scan) with 0.5°p er step, we obtained the h-dependence. In a typical CDW measurement, each CCD image was accumulated with an exposure time of 3-5 s at each θ. A beam shutter was used to cut the incoming X-ray beam between two consecutive CCD shots to prevent undesired collection of X-ray photons during read-out. In front of the CCD is mounted a 100 nm Parylene/100 nm Al filter to stop electrons emitted from the sample from contributing to CCD signals, ensuring our CCD measures purely X-ray photons. During the θ-scan, each CCD image covers scattering intensities from the well-aligned (h, 0, l) scattering plane near the detector center as well as from off-scattering planes (h, ±k, l) at the top/bottom area of the CCD. As described in the main text (Fig. 2a), the scattering intensity near the center of the detector corresponds to the signal of interest (i.e., CDW in this case), while the signals at the off-centered areas are dominated by the fluorescence background. This simultaneously recorded background signal (~256 × 50 pixels near the top and bottom of the CCD detector respectively) was used to subtract out the zeroth order fluorescence contribution near the CDW area. These background regions correspond to k~± 0.04 r.l.u., which is considerably far away from the center of the CDW peak, considering the finite width of the CDW peak. For data analysis, each CCD pixel was converted to an hkl reciprocal space index, and the resultant three-dimensional scattering intensity data set was projected onto different planes/directions for subsequent analysis.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.