## Picking out the elusive stripes

Copper-oxide superconductors have periodic modulations of charge density. Typically, the modulation is not the same for the whole crystal, but breaks up into small nanosized domains. Bulk experiments show that the density is modulated along both axes in the copper-oxide plane, but it is not clear whether this is true only on the scale of the whole crystal or also locally, for each domain. Comin *et al.* analyzed the charge order in the compound YBa_{2}Cu_{3}O_{6+y}, using resonant x-ray scattering, and found that it was consistent with a local unidirectional, so-called stripy, ordering.

*Science*, this issue p. 1335

## Abstract

After the discovery of stripelike order in lanthanum-based copper oxide superconductors, charge-ordering instabilities were observed in all cuprate families. However, it has proven difficult to distinguish between unidirectional (stripes) and bidirectional (checkerboard) charge order in yttrium- and bismuth-based materials. We used resonant x-ray scattering to measure the two-dimensional structure factor in the superconductor YBa_{2}Cu_{3}O_{6+}* _{y}* in reciprocal space. Our data reveal the presence of charge stripe order (i.e., locally unidirectional density waves), which may represent the true microscopic nature of charge modulation in cuprates. At the same time, we find that the well-established competition between charge order and superconductivity is stronger for charge correlations across the stripes than along them, which provides additional evidence for the intrinsic unidirectional nature of the charge order.

Recent studies of Y-based copper oxides have highlighted the importance of a charge-ordered electronic ground state, also termed a charge density wave (CDW), for the phenomenology of high-temperature superconductors (*1*–*14*). Experiments on the family of YBa_{2}Cu_{3}O_{6+}* _{y}* (YBCO) compounds have yielded a wealth of experimental results that have enabled advancements in our understanding of CDW instabilities and their interplay with superconductivity (

*9*–

*11*,

*15*–

*22*).

YBCO is a layered copper oxide–based material in which hole doping is controlled by the oxygen stoichiometry in the chain layer, characterized by uniaxial CuO chains running along the crystallographic **b** axis. In addition to ordering within the chain layer—attained via the periodic alternation of fully oxygenated and fully depleted CuO chains—recent experiments have extensively shown the presence of charge ordering in the CuO_{2} planes, with an incommensurate wave vector *Q* ≈ 0.31 reciprocal lattice units (*23*), corresponding to a period of approximately three unit cells in real space (*9*–*11*). Although the stripy nature of La-based cuprates is long established (*1*–*3*), the local symmetry of the CDW in YBCO has not yet been resolved. Both charge stripes (in the presence of 90° rotated domains) and a checkerboard pattern are consistent with the globally bidirectional structure of the CDW, which is characterized by wave vectors along both the **a** and **b** axes, at **Q*** _{a}* ≈ (0.31, 0) and

**Q**

*≈ (0, 0.31), respectively (*

_{b}*10*,

*11*,

*24*–

*26*). This leaves open the fundamental question of whether stripes are the underlying charge instability in the whole class of hole-doped cuprates.

We used resonant x-ray scattering (RXS) to study the local density correlations of the charge-ordered state and the interaction of this state with superconductivity (SC) in underdoped YBCO. The RXS technique, which is now at full maturity, represents a unique combination of diffraction (to probe reciprocal space) and resonant absorption (allowing element specificity and therefore site selectivity). RXS directly measures the structure factor *S*(*Q _{x}*,

*Q*), where

_{y}*Q*and

_{x}*Q*represent the momenta along the reciprocal axes

_{y}*H*and

*K*, respectively. The structure factor is linked to the density-density correlation function and therefore to the CDW order parameter in momentum space (

*27*). To reconstruct the two-dimensional (2D) structure factor with high resolution, we used a specifically devised RXS probing scheme whereby a charge-ordering peak is sliced along different directions in the (

*Q*,

_{x}*Q*) plane, parameterized by the azimuthal angle α (Fig. 1). The resulting 2D shape of the CDW peaks rules out checkerboard order and is instead consistent with a stripy nature of charge modulations in YBCO (

_{y}*28*). We carried out RXS measurements in the vicinity of the CDW wave vectors

**Q**

*≈ (0.31, 0) and*

_{a}**Q**

*≈ (0, 0.31) for three detwinned, oxygen-ordered YBCO samples: YBa*

_{b}_{2}Cu

_{3}O

_{6.51}(Y651, with hole doping

*p*≈ 0.10), YBa

_{2}Cu

_{3}O

_{6.67}(Y667,

*p*≈ 0.12), and YBa

_{2}Cu

_{3}O

_{6.75}(Y675,

*p*≈ 0.13).

In our experimental scheme, the CDW peaks are scanned in a radial geometry via control of the azimuthal angle α (*29*) (Fig. 1A). At the Cu-*L*_{3} edge, the measured signal is mainly sensitive to periodic variations in the Cu-2p → 3d transition energy (*30*, *31*), which is a scalar quantity, even though the detailed contribution of a pure charge modulation versus ionic displacements to the RXS signal cannot be decoupled (*32*). In addition, the poor coherence of the CDW across the CuO_{2} planes (*11*, *24*, *26*) qualifies this electronic ordering as a 2D phenomenon, thus motivating our focus on the structure factor in the (*Q _{x}*,

*Q*) plane. Representative scans of the CDW peak for different α values and at the superconducting critical temperature

_{y}*T*≈

*T*

_{c}are shown in the insets to Fig. 1A, for the

**Q**

*and*

_{b}**Q**

*CDW peaks of a Y667 sample. A change in the peak half width at half maximum (HWHM) Δ*

_{a}*Q*between α = 0° and α = 90° is already apparent but is even better visualized in the color map of Fig. 1B, which shows the sequence of Q-scans versus azimuthal angle and the corresponding variation of Δ

*Q*for

**Q**

*in the range α = –90° to 90°. This same procedure is applied to all three YBCO doping levels, for both the*

_{b}**Q**

*and*

_{a}**Q**

*CDW peaks; polar plots of Δ*

_{b}*Q*versus α are shown in Fig. 1, C to E, for Y651, Y667, and Y675, respectively. With the aid of the ellipse fits to the CDW profiles (continuous lines), four key aspects of these data stand out: (i) All peaks show a clear anisotropy between the two perpendicular directions α = 0° and α = 90°; (ii) for each doping, the

**Q**

*and*

_{a}**Q**

*peaks have different shapes, and in the case of Y651 and Y667 this is more evident as the peaks are elongated along two different directions; (iii) the departure from an isotropic case, quantified by the elongation of the CDW ellipses, increases toward the underdoped regime and is opposite to the evolution of orthorhombicity, which is instead maximized at optimal doping [see (*

_{b}*27*) and fig. S4 for a more detailed discussion]; and (iv) the peak elongation at

**Q**

*and*

_{a}**Q**

*, evolving from biaxial (Y651 and Y667) to uniaxial (Y675), is inconsistent with the doping independence of the uniaxial symmetry of the CuO chain layer, which rules out the possibility that the observed CDW peak structure is exclusively controlled by the crystal’s orthorhombic structure [however, the uniaxial anisotropy observed for Y675 might reflect a more pronounced interaction between the Cu-O planes and chains in this compound, possibly caused by the increase in orthorhombicity upon hole doping (*

_{b}*27*)].

The observed 2D peak shape indicates that the four-fold (*C*_{4}) symmetry is broken at both the macro- and nanoscale, which is consistent with the emergence of a stripe-ordered state. In fact, under *C*_{4} symmetry the electronic density would be invariant under a 90° rotation in real space (*x* → *y*, *y* → –*x*), which is equivalent to a 90° rotation in momentum space (*Q _{x}* →

*Q*,

_{y}*Q*→ –

_{y}*Q*). Instead, the CDW structure factor

_{x}*S*(

*Q*,

_{x}*Q*) is clearly not invariant under such operation, as shown in the insets of Fig. 1, C to E, which compare the original

_{y}*S*(

*Q*,

_{x}*Q*) factors to their 90° rotated versions

_{y}*S*(

*Q*, –

_{y}*Q*). This finding demonstrates an unambiguous breaking of global

_{x}*C*

_{4}symmetry in all investigated samples and might elucidate the origin of the anisotropy observed in the Nernst effect (

*20*) and in optical birefringence measurements (

*33*).

The real-space representation of charge order branches off into two possible scenarios: (i) a biaxial anisotropy, where both *x*- and *y*-elongated domains (*34*) are present (Fig. 2, A and C); (ii) a uniaxial anisotropy, where only *y*-elongated (or, equivalently, *x*-elongated) domains are found (Fig. 2, B and D). Note that these domains need not necessarily lie in the same layer, but they need to be present at the same time within the bulk of the material (e.g., they can be present in alternating layers while still leading to the same momentum structure). The momentum-space representation of the order parameter—and therefore of the electronic density fluctuations—is shown in the corresponding panels in Fig. 2, E to H, where *S _{a}*(

**Q**) (red ellipses) and

*S*(

_{b}**Q**) (blue ellipses) represent the structure factor associated to the charge modulations along

**a**and

**b**, respectively. The profile of a single structure-factor peak is the result of two contributions: the underlying CDW symmetry as well as its 2D correlation length, which can also be anisotropic.

As a net result, the anisotropy of a single peak in Q-space cannot be used to discriminate between different CDW symmetries. Instead, the latter can be resolved by probing the 2D CDW structure factor—that is, by comparing the CDW peak shapes for **Q*** _{a}* and

**Q**

*. Inspection of the diagrams in Fig. 2, E to H, reveals a common trait of checkerboard structures in momentum space, in that the following conditions (Fig. 2, G and H) must always hold by symmetry: = and = . That is, the peak broadening along a given direction must coincide for Δ*

_{b}*and Δ*

_{a}*(see bottom of Fig. 2, E to H, for case-specific conditions on the peak linewidths). Intuitively, this follows from the fact that for the checkerboard case, the charge modulations along*

_{b}**a**and

**b**axes must be subject to the same correlation lengths within each domain—irrespective of its orientation—and must therefore lead to an equivalent peak broadening along the same direction in reciprocal space, in contrast to our findings for the CDW linewidths (

*27*). From this symmetry analysis, we can conclude that for both uniaxial and biaxial anisotropy it is in principle possible (

*35*) to discriminate between a pure checkerboard and a pure stripe charge order, even in the presence of a distribution of canted domains (see tables S1 to S3 for a complete classification). Ultimately, the inequivalence of the peak broadening Δ

*Q*along different directions for all studied YBCO samples, combined with the macroscopic

*C*

_{4}symmetry breaking, provides clear evidence for the unidirectional (stripe) intrinsic nature of the charge order (

*28*).

Having established the underlying stripelike character of charge modulations in the CuO_{2} planes, we turn to the temperature dependence of the longitudinal and transverse correlation lengths, respectively parallel and perpendicular to the specific ordering wave vector. These can be extracted by inverting the momentum HWHM Δ*Q*, as illustrated in Fig. 3A. Longitudinal correlations are then given by ξ_{||} = Δ*Q*_{||}^{–1}, where Δ*Q*_{||} represents the momentum linewidth in the direction parallel to the ordering wave vector; transverse correlations are given by ξ_{⊥} = Δ*Q*_{⊥}^{–1}, where Δ*Q*_{⊥} represents the momentum linewidth in the direction perpendicular to the ordering wave vector.

We subsequently studied the temperature dependence of ξ_{||} and ξ_{⊥} for both the **Q*** _{a}* and

**Q**

*ordering wave vectors (Fig. 3, B to F). We observed a rise of correlation lengths below the CDW onset near 150 K, followed by their suppression below the SC transition temperature*

_{b}*T*

_{c}; this confirms the competition between these two orders, in agreement with recent energy-integrated as well as energy-resolved RXS studies (

*10*,

*11*,

*24*,

*25*,

*30*).

*However, the drop in the correlation lengths below*

*T*

_{c}(Δξ) was in all instances larger for the longitudinal correlations, or Δξ

_{||}> Δξ

_{⊥}. In particular, the discrepancy between Δξ

_{||}and Δξ

_{⊥}, although small for Y675, was quite substantial for the more underdoped Y667 and Y651. This anisotropy provides additional evidence for the unidirectional nature of the charge ordering and thus the breaking of

*C*

_{4}symmetry, because a bidirectional order would be expected to exhibit an isotropic drop in correlation length across

*T*

_{c}. The anisotropy has an opposite doping trend from the crystal orthorhombicity, whose associated anomalies across

*T*

_{c}increase with hole doping [as revealed, for example, by lattice expansivity measurements (

*27*,

*36*)].

The inferred real-space representation of the evolution across *T*_{c} is schematically illustrated in Fig. 3, G and H, where nanodomains are used to pictorially represent a charge-ordered state with finite correlation lengths. We conclude that the largest change occurs along the direction perpendicular to the stripes. This reflects the tendency of the SC order to gain strength as temperature is lowered, primarily at the expense of longitudinal CDW correlations; the implication is that the mechanism responsible for the density fluctuations across the periodically modulated stripes might be the main one competing with the Cooper pairing process.

Our results may explain many common aspects between CDW physics in YBCO and the stripy cuprates from the La-based family, such as thermoelectric transport (*37*, *38*), strength of the order parameter (*39*), out-of-equilibrium response (*40*, *41*), and energy-dependent RXS response (*30*, *31*). The nanoscopic nature of the stripe instability and the presence of both **a**- and **b**-oriented domains also clarify why this broken symmetry has been difficult to disentangle from a native bidirectional order (*10*, *11*, *42*), therefore requiring a tailored experimental scheme to resolve the 2D CDW structure factor *S*(*Q _{x}*,

*Q*). The pronounced directionality in the competition between superconductivity and stripe order reveals the underlying interplay between particle-particle and particle-hole pairing in high-temperature superconductors and provides insights for an ultimate understanding of these materials.

_{y}## Supplementary Materials

www.sciencemag.org/content/347/6228/1335/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S9

Tables S1 to S3

## References and Notes

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The small but finite doping dependence of the ordering wave vector is here neglected, because it is not the focus of our investigation. For more details, see (
*24*,*25*,*43*). - ↵
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See supplementary materials on
*Science*Online. - ↵ Because of our inability to measure magnetic ordering, our concept of stripes only applies to the charge modulations, unlike the canonical definition used in La-based cuprates, which also implies concomitant magnetic order.
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However, x-ray diffraction experiments estimate the upper bound for ionic displacements to be around 7 × 10
^{–4}*a*, or ~2.7 × 10^{–3}Å (*11*). - ↵
- ↵ We use the concept of domain to indicate a coherence subregion (i.e., a portion of space where density modulations are largely in phase); in this sense, the size of the domains is intended to reflect the magnitude of correlation lengths.
- ↵ Unless = , in which case the patterns shown in Fig. 2, F and H, become indistinguishable. This situation, however, is not encountered in our study.
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- ↵ The data-fitting routines used for this analysis implicitly assume that the errors are normally distributed with zero mean and constant variance and that the fit function is a good description of the data. The coefficients and their sigma values (representing uncertainties) are estimates of what one would get if the same fit were performed an infinite number of times on the same underlying data (but with different noise each time) and then the mean and standard deviation were calculated for each coefficient.
**Acknowledgments:**We thank P. Abbamonte, E. Blackburn, L. Braicovich, J. Geck, G. Ghiringhelli, B. Keimer, S. Kivelson, M. Le Tacon, and S. Sachdev for insightful discussions. Supported by the Max Planck–UBC Centre for Quantum Materials, fellowships from the Killam, Alfred P. Sloan, and Alexander von Humboldt Foundations, and a Natural Sciences and Engineering Research Council (NSERC) Steacie Memorial Fellowship (A.D.); the Canada Research Chairs Program (A.D., G.A.S.); and NSERC, CFI, and CIFAR Quantum Materials. All of the experiments were performed at beamline REIXS of the Canadian Light Source, which is funded by the CFI, NSERC, NRC, CIHR, the Government of Saskatchewan, WD Canada, and the University of Saskatchewan. R.C. acknowledges the receipt of support from the CLS Graduate Student Travel Support Program. E.H.d.S.N. acknowledges support from the CIFAR Global Academy.