Introduction

Electronic phases may coexist microscopically, either in a collaborative or competing manner. In elementary chromium, for example, spin and charge density wave orders collaboratively coexist with commensurate ordering vectors1,2,3. A similar spin-charge intertwined order is found in doped lanthanum-based (La-based) cuprate superconductors4,5,6,7,8. Competing interaction is often found in the context of unconventional superconductivity. For example, in kagome metals9,10, pnictides11,12,13, and heavy Fermion systems14, superconductivity can be optimized through the suppression of charge or spin density wave orders.

However, an interplay between density waves and superconductivity – at least theoretically – can lead to a collaborative state. This state would be characterized by a spatially modulated Cooper pair density with a commensurate wave vector. Extensive experimental and theoretical efforts have been devoted to study this novel superconducting state15. Theory works have predicted a connection between superconductivity and stripe order through a so-called pair density wave16. Signatures of these pair density waves have been reported by scanning tunneling microscopy17, but direct diffraction evidence is still missing. A general challenge is therefore to switch the coupling between superconductivity and charge order from competing to collaborative. Ideally, an external stimulus would tune the coupling between these two phases.

Here, using high-energy x-ray diffraction, we show how compressive c-axis uniaxial pressure, perpendicular to the copper-oxide planes, enhances stripe order inside the superconducting state of La1.88Sr0.12CuO4 (LSCO), while charge order remains unchanged in the normal state. We furthermore discover that the magnetic field enhancement of charge order inside the superconducting state is dramatically reduced upon compressive c-axis strain application. This observation suggests a correspondingly reduced phase competition. We thus demonstrate that c-axis pressure acts directly on the coupling between charge stripe order and superconductivity.

Results

X-ray methodology

Stripe charge order in La-based cuprates manifests itself by weak reflections at Qco = τ+(δ, 0, 0.5) where τ represents fundamental Bragg peaks and δ ≈ 1/4 is the stripe incommensurability6,7,18,19. We adopted an x-ray transmission geometry with crystalline a- and c-axes spanning the horizontal scattering plane as illustrated in Fig. 1a. Magnetic field and uniaxial pressure were applied along the c-axis direction.

Fig. 1: Uniaxial pressure application in the hard x-ray diffraction experiment on LSCO.
figure 1

a Schematic illustration of the scattering geometry for high-energy x-ray diffraction on LSCO. Uniaxial pressure and magnetic field are applied along the crystallographic c-axis direction on a cuboid-shaped crystal with dimensions La ⋅ Lb ⋅ Lc as sketched in the top left inset. Detector read-outs with exemplary region-of-interest (ROI, black box in detector images) for three different rotation angles ω around the vertical axis are shown. b c-axis lattice parameters, extracted from fits of (0, 0, ℓ) Bragg peaks, corresponding to c-axis strains εc. Solid lines are least-square fits to \(c=\lambda /(2\sin \omega )\cdot \ell\), where λ is the wavelength of the x-rays. Error bars are standard deviations obtained from the fitting procedure. c Spin flip intensities from polarized neutron scattering at τ = (−1, −1, 0) on LSCO (Tc = 27 K), normalized by the monitor and scaled to the integrated Bragg peak intensity of the unstrained measurement. The drop of the curves yields the superconducting transition temperature with and without c-axis uniaxial pressure (see Methods for detailed definition of Tc). d Charge order peak at 30 K obtained by integrating the intensities in a region-of-interest (ROI) around Qco = (0.231, 0, 12.5) (blue points). A background (bg, grey points) is estimated by a similar integration of a ROI slightly shifted off the (h, 0, ℓ) scattering plane. Error bars in c and d are dictated by counting statistics.

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Uniaxial pressure application

The strain as a result of uniaxial c-axis pressure can be directly estimated from lattice parameter measurements. Pressure-induced compression of the c-axis lattice parameter is evidenced by a shift of (0, 0, n) Bragg peaks to larger scattering angles. Precise strain characterization utilizes multiple such Bragg peaks with n being an even integer (see Method section and Fig. 1b). The resulting c-axis lattice parameter for strained and unstrained LSCO is exemplified in Fig. 1b. As expected, the uniaxial c-axis pressure reduces the c-axis lattice parameter that in turn lowers the superconducting transition temperature Tc20,21,22. Using polarized neutron scattering we confirmed the decrease of Tc with compressive c-axis strain23,24,25,26 – see Fig. 1c and Method section. Exploiting that the lower critical field for superconductivity Hc1 is low, we track the excess depolarisation of the neutron beam due to flux trapped along the c-axis after field-cooling through Tc. The in-plane polarized neutrons are depolarized by this trapped flux. Upon crossing Tc, the flux is released.

Uniaxial c-axis strain effects on charge order

X-ray diffraction intensity was collected using a two-dimensional single-photon detector. Detector regions-of-interest (ROI) are defined such that the signal or background of interest is covered (see Supplementary Fig. 1). We constructed standard one-dimensional rocking curves (see Fig. 1d for Qco = (δ, 0, 12.5)). An advantage of 2D-detectors (over point detectors) is that a background can be estimated by slightly shifting the ROI (grey data in Fig. 1d).

In Fig. 2, we show scans through Qco = (δ, 0, 12.5) and (δ, 0, 16.5), with and without uniaxial c-axis pressure. Data for a La2−xSrxCuO4 crystal with slightly different do** are shown in Supplementary Fig. 2. Intensities and fits are presented after subtracting the background. In the normal state (T > Tc), no pressure effect on charge stripe order is observed. Further, we find a significant pressure-induced enhancement of the charge order reflection inside the superconducting state. The correlation length and incommensurability δ remain virtually unaffected by uniaxial pressure. Observed shifts are within the error bars and thus negligible. From here, we therefore consider the charge order peak amplitude as a function of temperature, uniaxial c-axis pressure, and magnetic field. The peak amplitude Ico is extracted by fitting intensity profiles with a split-normal distribution on a linear background – see Figs. 1d and 2, Supplementary Fig. 3 and Methods.

Fig. 2: Charge order reflection in LSCO upon application of c-axis uniaxial strain.
figure 2

a–d Background subtracted charge order reflections for temperatures and momenta as indicated. Red (blue) points are recorded with (without) compressive c-axis pressure application. Error bars stem from counting statistics and solid lines are fits with a split-normal distribution including a linear background. The horizontal lines in the bottom of (a) - applying to all panels - represent a systematic error in h stemming from twinning of the sample28.

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The temperature dependence of the charge order amplitude is shown in Fig. 3a for strained and unstrained conditions. In the absence of a magnetic field, compressive c-axis pressure enhances the charge order inside the superconducting state. The charge order peak amplitude, due to phase competition19, displays a cusp at Tc. The cusp is shifted to slightly lower temperatures upon application of c-axis pressure. Assuming phase competition between charge order and superconductivity, this suggests a reduction of the superconducting transition temperature, in agreement with the measurements in Fig. 1c. At our base temperature (T = 10 K), the relative peak amplitude, Ico(10 K)/Ico(30 K), scales approximately linearly with the applied strain εc (see Fig. 3b). Within the examined range of εc, the charge order peak amplitude increases by about 25%.

Fig. 3: Phase competition of charge order and superconductivity.
figure 3

a Temperature dependence of the amplitude Ico of the (0.231, 0, 12.5) charge order peak without and with strain measured in a setup without magnetic field. b Low-temperature (10 K) charge order amplitude normalized to the normal state (30 K) plotted as a function of c-axis strain. For the black (grey) points, the strains are directly (indirectly) measured. Indirect measurements use the calibration curve in Supplementary Fig. 4. The grey shaded area is a guide to the eye. c, d Temperature dependence for charge order peak at (0.231, 0, 12.5) in 0 T and 10 T without and with strain, respectively measured in the cryomagnet. The horizontal dashed lines at Ico(Tc) in panels (a, c, d) serve to facilitate comparison between experiments. e Magnetic field dependence at 10 K of the charge order peak amplitude at (0.231, 0, 12.5) (circles) and (0.231, 0, 16.5) (diamonds) for strains as indicated. Solid lines are fits to the data with \({I}_{co}({{{{{{{\mathcal{H}}}}}}}})={I}_{co}(0)+{I}_{1}{{{{{{{\mathcal{H}}}}}}}}\ln (1/{{{{{{{\mathcal{H}}}}}}}})\)34, where \({{{{{{{\mathcal{H}}}}}}}}=H/{H}_{c},{I}_{1}\) is a fitting parameter, and Hc is a critical field scale. f Comparison of data obtained with and without strain at (0.231, 0, 12.5) (circles) and (0.231, 0, 16.5) (diamonds) (T = 10 K). By application of uniaxial pressure, we roughly double the reachable parameter space in H/Hc. Solid lines in a-c are guides to the eye. The dashed lines indicate the charge order amplitude at Tc and ambient pressure. Error bars are standard deviations from the respective fits.

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Magnetic field effect

Without strain, magnetic field effects on charge and spin order inside the superconducting state have already been studied\({p}_{0}\cdot \tanh (T-{T}_{c})+{p}_{1}\). The resulting Tc of the zero-pressure LSCO crystal is in accord with the transition temperature obtained from a magnetic susceptibility measurement of the rod (see Supplementary Fig. 8).