Engineering Phase Competition Between Stripe Order and Superconductivity in La$_{1.88}$Sr$_{0.12}$CuO$_4$

Unconventional superconductivity often couples to other electronic orders in a cooperative or competing fashion. Identifying external stimuli that tune between these two limits is of fundamental interest. Here, we show that $c$-axis strain couples directly to the competing interaction between charge stripe order and superconductivity in La$_{1.88}$Sr$_{0.12}$CuO$_4$ (LSCO). Compressive $c$-axis pressure amplifies stripe order within the superconducting state, while having no impact on the normal state. By contrast, strain dramatically diminishes the magnetic field enhancement of stripe order in the superconducting state. This observation hints to the existence of a ground state without phase competition. As such, c-axis pressure is a promising tuning parameter in the search for ground states where superconductivity adopts broken translational symmetry from charge ordering phenomena.

have been devoted to study this novel superconducting state [13].In the cuprates, pair-density-waves have been predicted to exist in the limit where superconductivity is sufficiently weakened [14,15].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 enhances stripe order inside the superconducting state of La 1.88 Sr 0.12 CuO 4 (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.Tuning of the phase competition between charge order and superconductivity may be a route to pair-densitywave ground states.

Diffraction and c-axis strain application.
Stripe charge order in La-based cuprates manifests itself by weak reflections at Q co = τ +(δ, 0, 0.5) where τ represents fundamental Bragg peaks and δ ≈ 1/4 is the stripe incommensurability [6,7,16,17].We adopted an x-ray transmission geometry with crystalline a-and c-axis spanning the horizontal scattering plane as illustrated in Fig. 1a.Magnetic field and uniaxial pressure were applied along the c-axis direction.
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 τ = (0, 0, ±2n) with n being an 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 T c [18,19].Using polarized neutron scattering we confirmed the decrease of T c with compressive c-axis strain [20][21][22][23] -see Fig. 1d and Method section.
Exploiting that H c1 is low, we track the excess depolarisation of the neutron beam due to flux trapped along the c-axis.The in-plane polarized neutrons are depolarized by this trapped flux.Upon crossing T c , the flux is released.The reduced T c stems either from a lower superfluid density or a weaker Cooper pairing strength.
Uniaxial c-axis strain effects on charge order.X-ray diffraction intensity was collected using a twodimensional single-photon detector.Detector regions-ofinterest (ROI) are defined such that the signal or background of interest is covered (see Supplementary Fig. 2).We constructed standard one-dimensional rocking curves (see Fig. 1c for Q co = (δ, 0, 12.5)).An advantage of 2Ddetectors (over point detectors) is that a background can be estimated by slightly shifting the ROI (grey data in Fig. 1c).
In Fig. 2, we show scans through Q co =(δ, 0, 12.5) and (δ, 0, 16.5), with and without uniaxial c-axis pressure.Data for a La 2−x Sr x CuO 4 crystal with slightly different doping are shown in Supplementary Fig. 3.In the normal state (T > T c ), no pressure effect is observed.By contrast, 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.Ob- served 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 I co is extracted by fitting intensity profiles with a split-normal distribution on a linear background -see Fig. 1c, 2, Supplementary Fig. 5 and Methods.In Fig. 3, the intensities and fits are presented after subtracting the background.
The temperature dependence of the charge order amplitude is shown in Fig. 3 for strained and unstrained conditions.In the absence of a magnetic field, compressive caxis pressure enhances the charge order inside the superconducting state.The charge order peak amplitude, due to phase competition [17], displays a cusp at T c .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. 1d.At our base temperature (T = 10 K), the relative peak amplitude, I co (10 K)/I co (30 K), scales approximately linearly with the applied strain ϵ c .Within the examined range of ϵ c , the charge order peak amplitude increases by about 25 %.

Magnetic field effect.
Without strain, magnetic field effects on charge and spin order inside the superconducting state have already been studied [24][25][26][27][28][29].Consistent with previous studies, we find an increase of the charge order amplitude for T < T c (see Fig. 3b).At base temperature, the charge order intensity scales approximately linearly with magnetic field.At 10 T, the peak amplitude is more than doubled.This is in strong contrast to the ground state reached through application of c-axis strain.Here, as depicted in Fig. 3c a much weaker field effect is observed.The magnetic field effect for strained and unstrained conditions is shown in Fig. 3e.Upon increasing strain ϵ c , a decreasing magnetic field effect on the charge order intensity is observed inside the superconducting state.

DISCUSSION
There has been an interest to study the stripe order beyond the upper critical field (H c2 ) for superconductivity [30].The magnetic field dependence of the charge order amplitude is often modelled by I co (H) = I co (0)+I 1 H ln(1/H), where H = H/H c and I 1 is a fitting parameter.The critical field scale H c has been linked to the upper critical field H c2 of superconductivity [31].We however stress that these two field scales (H c and H c2 ) are not necessarily identical.As indicated in Fig. 3e, uniaxial c-axis strain reduces H c , driving H closer to the limit H → 1.We show in Fig. 3f that we could double the reachable parameter space in H by applying strain along c.By fitting the observed charge order peak amplitude versus magnetic field, we extract H c at base temperature (see fitted curves in Fig. 3e).In Fig. 4, we plot H c versus strain ϵ c .The field scale H c seems to scale ap-FIG.3. Phase competition of charge order and superconductivity.a Temperature dependence of amplitude Ico of the (0.231, 0, 12.5) charge order peak without and with strain measured in setup without magnetic field.b, c Temperature dependence for charge order peak at (0.231, 0, 12.5) in 0 T and 10 T without and with strain, respectively.d 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. 1.Filled (unfilled) points stem from the experimental setup with (without) a magnet.The grey shaded area is a guide to the eye.e Magnetic field dependence of the charge order peak amplitude at (0.231, 0, 12.5) (circles, left y-axis) and (0.231, 0, 16.5) (diamonds, right y-axis) for different strains as indicated.Solid lines are fits to the data -see main text.f Comparison of data obtained with and without strain at (0.231, 0, 12.5) (circles) and (0.231, 0, 16.5) (diamonds).By application of uniaxial strain, we roughly double the reachable parameter space in H/Hc.Lines in a-c are guides to the eye.
proximately linearly with c-axis strain: H c = H c (0)−αϵ c with α being a constant.By extrapolation, our results suggest that there exists a critical c-axis strain at which H c → 0. At this critical strain ϵ crit = H c (0)/α ≈ 0.29 %, the magnetic field effect is expected to vanish.
Outlining our findings, we illustrate schematically in Fig. 4b the charge order peak amplitude versus magnetic field and c-axis strain inside the superconducting state.The region with highest intensities is found at high magnetic fields and low strains.The strength of the magnetic field effect depends on the uniaxial strain along c.At the highest applied c-axis strain, the magnetic field effect is strongly reduced.Typically, magnetic field effects inside the superconducting state are interpreted in terms of (i) vortex physics [32,33] or (ii) phase competition.The vortex density and volume increase with increasing mag-netic field strength.It is however difficult to explain the strain-field effects in terms of vortices alone.Even in the high-magnetic field state where vortices are present, it is not obvious how to explain the strain effect without also involving phase competition.
Reduced (absence of) magnetic field effect would imply either that one phase is partially suppressed (absent) or that the competing interaction is weakened.As no pressure effect is found in the normal state, c-axis pressure is inert to charge order itself.On the other hand, superconductivity is known to be weakened by increasing compressive c-axis pressure [34].One commonly accepted explanation is that compressive c-axis pressure reduces the apical oxygen distance to the CuO 2 layers [18].This change of crystal field environment, in turn, boosts the hybridization of d x 2 −y 2 and d z 2 orbitals [35].Theoret-FIG.4. Magnetic field and strain effects.a The magnetic field scale Hc versus compressive c-axis strain at 10 K. Vertical error bars are Gaussian standard deviations from fits to the magnetic field dependence of the charge order reflection.Horizontal error bars reflect uncertainty of the strain estimation.The black dashed line is a linear guide to the eye b Schematic illustration of the magnetic field and strain effect on the charge order peak amplitude inside the superconducting state.In the grey region, we expect -by extrapolationthe magnetic field enhancement to fully vanish.Crosses display the phase space covered by our experiments.
ically, such hybridization is expected to be unfavorable for d-wave superconductivity [36,37].Therefore, c-axis pressure diminishes the superconducting order parameter.However, there is no evidence of superconductivity being completely suppressed upon moderate pressure along the c-axis [19,20].Superconductivity and charge order are therefore expected to coexist.One possible explanation for the diminished magnetic field effect is that c-axis strain induces a state in which superconductivity rather cooperates than competes with charge order.This may pave the way for pair-density-wave states in which superconductivity and charge order couple in a cooperative fashion [13].The superconducting order parameter would have a spatial modulation commensurate to the charge order.Search for such a superconducting modulation with c-axis pressure is an interesting experimental continuation.

METHODS Crystal Growth and Characterization
La 2−x Sr x CuO 4 crystals were synthesised by the traveling-zone method.We used samples from three different batches -all with x ≈ 0.12.All x-ray diffraction data in the main text stem from a batch with x = 0.12 and T c = 27 K.The neutron experiment was carried out on a batch with x = 0.115 and T c = 27 K.X-ray data in the Supplementary Information were recorded on a x = 0.125 crystal with T c = 30 K. Crystal orientations were determined by x-ray Laue diffraction.In this fashion, samples were cut to enable application of c-axis uniaxial pressure in combination with x-ray transmission.

X-Ray Diffraction and Pressure Cell
Hard x-ray diffraction experiments were carried out at the P21.1 beamline at PETRA III (DESY, Hamburg) synchrotron using 101.6 keV photons.All measurements were performed with a Dectris Pilatus 100K CdTe detector.Uniaxial pressure and magnetic field were applied along the crystallographic c-axis.In the absence of a magnetic field, a standard (ex-situ) screw cell was used [25,38].An adaption was made to fit into a horizontal magnet [24,28] with pressure application parallel to the direction of the magnetic field.The designs of two uniaxial pressure cells are shown in Supplementary Fig. 4a, b.For both cells, the horizontal scattering plane is spanned by a copper-oxygen bond direction and the c-axis.This gives access to the (h, 0, ℓ) reciprocal plane.Throughout the manuscript, reciprocal space (h, k, ℓ) are given in units of ( 2π a , 2π b , 2π c ) with a = b = 3.78 Å and c ≈ 13.2 Å.The magnitude of applied strain is determined from measurements of Bragg reflections, similar to what is described in Ref. [25].We defined the zerostrain, low-temperature lattice parameter as c 0 and c-axis strain as ϵ c = (c 0 − c)/c 0 .Both c 0 and c are determined from Bragg peaks of the type (0, 0, ±2n) with n being an integer (see Fig. 1b).Using this methodology a calibration curve between screw turn and strain is established (See Supplementary Fig. 1).

Neutron Diffraction
Neutron diffraction data were measured at the TASP spectrometer at SINQ, PSI on LSCO x = 0.115 (T c ≈ 27 K).The sample in the (ex-situ) pressure cell was mounted in a coil (see Supplementary Fig. 4c), custom-built to fit inside a cryostat, which itself was mounted inside the Mu-metal polarization analysis device MuPAD [39].The sample was cooled from 40 K to 10 K in a magnetic field of 30 G, applied along the crystallographic c-direction, perpendicular to the scattering plane.At base temperature, the field was subsequently turned off.Using MuPAD's spherical polarimetry capabilities, a full polarization matrix was measured on the in-plane Bragg peak τ = (−1, −1, 0).The highest flipping ratio of 6.87 was found for neutron polarization along y (in the scattering plane but perpendicular to τ ).The temperature dependence of the intensity of τ = (−1, −1, 0) was tracked in the Py-y spin flip channel upon slowly heating.The flux trapped along the c-direction during the field-cooling procedure gives rise to an excess depolarization of the neutron beam, which is reflected in an increased spin-flip intensity.Upon crossing T c , the trapped flux is released.Intensities are normalized to the monitor and scaled by the integrated Bragg peaks of (−1, −1, 0) in unstrained LSCO.To extract the transition temperature, spin flip intensities were fitted with p 0 • tanh(T − T c ) − p 1 .

FIG. 1 .
FIG. 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 as depicted.Detector read-outs with exemplary regionof-interest for three different rotation angles ω around the vertical axis are shown.b Lattice parameters, extracted from fits of (0, 0, ℓ) Bragg peaks.Solid lines are least-square fits to c = λ/(2 sin ω) • ℓ. c Charge order peak obtained by integrating the intensities in a region-of-interest (ROI) around Q co = (0.231, 0, 12.5) (blue points).A background (bg, grey points) is estimated by a similar integration of an ROI slightly shifted off the (h, 0, ℓ) scattering plane.d Spin flip intensities from polarized neutron scattering at Q co = (−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).

FIG. 2 .
FIG.2.Charge order reflection in LSCO upon application of c-axis uniaxial strain.a-d Background subtracted charge order reflections for temperatures and momenta as indicated.Red (blue) points are recorded with (without) compressive c-axis strain application.Error bars stem from counting statistics and solid lines are fits with a split-normal distribution including a linear background.