Introduction

The cuprate phase diagram illustrates a quintessential example of a low-dimensional correlated quantum system: a multitude of fascinating electronic phases, including spin-density waves, charge and nematic order, and high-temperature superconductivity1 (SC), emanating from the combination of complex interactions that govern a simple two-dimensional (2D) chemical structure — the CuO2 planes. These interactions within the CuO2 planes are largely influenced by the underlying characteristics of the anisotropic, planar Cu 3\({d}_{{x}^{2}-{y}^{2}}\) orbitals, which dominate the density of states near the Fermi surface due to a sizeable energy splitting of the eg orbitals2. The 2D character of the surviving \({d}_{{x}^{2}-{y}^{2}}\) orbital is directly observable in transport measurements, evidenced by its anisotropic electrical and thermal conductances3, for example. The 2D character of the system is also observable in scattering measurements, where the lack of coupling pathways between adjacent planes can yield overwhelmingly broad scattering peaks along L in reciprocal space4. Understanding how the dimensionality of the orbital degrees of freedom affect the stability and interplay of these phases could reveal important information about the mechanism of superconductivity with broader applications for modifying the characteristics of correlated oxides and layered materials via orbital engineering.

Among the most important cases of this interplay is the phenomenon of charge order (CO), a phase that is closely interconnected with SC5,6,7,8,9,10,11,12,13,14,15,16,17,18. Incommensurate CO exists in all superconducting cuprates as a 2D electronic phenomenon hosted in the CuO2 planes, reflecting the weak interplanar coupling of the planar Cu 3\({d}_{{x}^{2}-{y}^{2}}\) orbitals. In diffraction experiments, the 2D character is evidenced by a reciprocal space ‘rod’ that is broad along the out-of-plane direction (Miller index L), and maximized at half-integer values of L due to a weak, but out-of-phase, coupling between adjacent planes19. The strength of this interplanar coupling can be further quantified by extracting the correlation lengths from the widths of the scattered CO peaks along L. In YBa2Cu3O6+δ (YBCO), the highest reported out-of-plane correlation length (10 Å) is nearly an order of magnitude smaller than the highest reported in-plane correlation length (95 Å)5, highlighting the 2D nature of the CO phase. It is not clear whether disorder20,21,22 or the low dimensionality of the underlying Cu 3\({d}_{{x}^{2}-{y}^{2}}\) orbitals intrinsically limits the out-of-plane correlation length, or if the CO could, in principle, develop into a truly long-range order, as suggested by recent experiments20,21,22.

It has since been observed that the application of certain perturbations — high magnetic fields4,23,24,25, epitaxial strain in thin films26, or uniaxial strain27,28 — can induce a CO phase with three-dimensional (3D) coherence. Upon the application of these external influences, a second CO peak emerges, this time centered at integer L-values, evidencing an out-of-plane coupling that locks the phase of adjacent CuO2 planes. The 3D CO peaks have significantly increased out-of-plane correlation lengths, achieving up to 55 Å24, 61 Å26, and 94 Å27, respectively. All of these 3D CO correlation lengths are still considerably shorter than the typical crystalline c-axis correlation lengths found in this compound. Furthermore, the 2D rod centered at half-integer L-values gets enhanced upon applying the external influences, showing a persistent coexistence of the 3D and 2D COs. While it is easy to discern the 2D nature of the unperturbed CO upon consideration of the underlying planar Cu 3\({d}_{{x}^{2}-{y}^{2}}\) orbitals, the mechanisms by which these external perturbations are able to induce a 3D CO peak remain unclear. Moreover, the in situ application of these perturbations presents complicated technical challenges that preclude many experimental techniques altogether, making it difficult to systematically investigate how the dimensionality of the CO can be tuned and obscuring its connection with SC. Taking an orthogonal route, we hypothesized that 3D CO could instead be stabilized by virtue of tuning the underlying orbital character via hybridization to more directly enhance the out-of-plane coupling between adjacent CuO2 plane layers.

Here we show that, by substituting Pr on the Y sites in PrxY1−xBa2Cu3O7 (Pr-YBCO) (Fig. 1a), a highly correlated 3D CO state can be stabilized with an out-of-plane correlation length of ~ 364 Å (Fig. 1b), a number that is bound by the crystalline correlation length, within our experimental resolution. This material was chosen because substitution by Pr, which is the largest trivalent rare-earth ion, except for Ce which does not form the YBCO structure29, results in the emergence of hybridization between the Pr 4f orbitals and planar CuO2 states30 that yields an electronically relevant, hybridized orbital31 with spatial extension in three dimensions, in stark contrast to the planar Cu 3\({d}_{{x}^{2}-{y}^{2}}\) orbitals that dominate the physics of the parent compound. Unlike substitution by other rare-earth elements, such as Dy, which do not significantly alter the parent YBCO phase diagram32, increasing Pr substitution in the PrxY1−xBa2Cu3O7 system continually reduces the superconducting Tc, yielding a pseudogap regime30,33,34 and eventually an antiferromagnetic insulating phase30,35,36,37,38,39,40. Furthermore, the in- and out-of-plane zero-temperature superconducting coherence lengths are substantially longer in Pr-YBCO than in YBCO and increase monotonically with Pr concentration41,42,43. This indicates that SC gains additional 3D character with increasing Pr substitution and has been attributed to increased coupling between CuO2 planes through the bonding with the substituted Pr ions. Various results suggest that localized Pr 4f states are appreciably hybridized with the valence band states associated with the conducting CuO2 planes, specifically the O 2p level44,45. We present density-functional calculations showing that, through this hybridization which is unique to Pr, the CO on adjacent CuO2 planes can couple to yield a stable 3D CO phase. Altogether, our results constitute the first detection of a fully stabilized, long-range 3D CO that competes with SC, achieved by intrinsically engineering the orbital character of the electronic structure.

Fig. 1: Stabilizing three-dimensional charge order.
figure 1

a The extended unit cell (dashed box) of PrxY1−xBa2Cu3O7 illustrating the charge order coupling between adjacent CuO2 planes that arises when introducing Pr at the Y sites. The Pr 4\({f}_{z({x}^{2}-{y}^{2})}\) orbital is shown which hybridizes with the 2pπ states of the planar O. b A schematic depiction of the 3D CO out-of-plane correlation length in PrxY1−xBa2Cu3O7 which spans ~31 sets of CuO2 planes (~364 Å).

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Results

We used resonant soft x-ray scattering (RSXS) at the Cu L3 and Pr M5 edges to investigate the CO properties in a PrxY1−xBa2Cu3O7 sample with x ≈ 0.3 and a superconducting Tc = 50 K; a concentration value chosen because it features pseudogap behavior, as measured by various probes30,33,34, and because it yields a Tc similar to underdoped YBa2Cu3O~6.67, a doping level where the CO phase is maximal. Due to not having detwinned samples (Methods section), we cannot determine whether the 3D CO peak is biaxial or uniaxial. If the 3D CO is uniaxial, we cannot determine whether it is located along the H or K reciprocal axis. The location of the 3D CO peak is thereby referred to as being along H or K to reflect this, except for places where the position is labeled simply by K for the sake of readability.

Reciprocal space dependence

A reciprocal space map of the HL or KL-plane in reciprocal lattice units (r.l.u.), measured at Tc = 50 K at 932.4 eV is shown in Fig. 2a. In eminent contrast to all other reports of 3D CO, no scattered intensity was detected in the vicinity of L ≈ 1.5, indicating the apparent absence of 2D CO (see Supplementary Methods). This represents the first unique aspect of our work: to within the limits of our instrumental resolution, we only observe a peak at L = 1, suggesting an effective isolation of the CO phase with an out-of-plane coupling.

Fig. 2: The reciprocal space structure of the 3D CO at T = 50 K.
figure 2

a An HL or KL reciprocal space map collected at 932.4 eV shows a diffraction feature centered at (0 -0.335 1) reciprocal lattice units (r.l.u.). b A cut along H or K at L = 1 shows the peak is centered around a value of H or K = -0.335 (r.l.u.). c A comparison of the out-of-plane L-widths of the known 3D CO peaks stabilized by magnetic field4, epitaxial strain in YBCO films26, uniaxial strain27, and the present work. For comparison, an L-cut of the structural (002) peak from Pr-YBCO is shown.

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Further inspection of the 3D CO signal displays a reciprocal space structure that is broad along H or K but narrow along L. X-ray absorption fine structure measurements indicate that, while the Pr ions are relatively well-ordered at the Y sites, there is clear disorder in the CuO2 planes and in the oxygen environment around the Pr29, which makes the enhancement of the correlation length along the c-axis even more striking. The broad shape of the peak along H or K at L = 1, shown in Fig. 2b, is consistent with the broad feature observed in many previous RSXS measurements of CO in cuprates12,14,15,46 that has been attributed to a fluctuating component in YBCO47, suggesting that the actual static contribution may be narrower than it appears.

Another important feature of our discovery is shown in Fig. 2c, which compares reciprocal space cuts along L close to integer values with K centered at the in-plane CO wavevector. The broadest peak (dark green triangles) displays the data reported for 3D CO induced by high magnetic field4. The next broadest peak (red stars) displays the data for 3D CO induced by epitaxial strain in a thin film26. The next broadest peak (lime green squares) displays the data measured under the application of 1.0% uniaxial strain27, which has yielded 3D CO with the previously highest reported out-of-plane correlation length. The Pr-YBCO 3D CO peak (purple circles) is considerably narrower, yielding a correlation length of ~364 Å. This value is found to be similar to the absorption length for this compound, photon energy (~930 eV), and angle of incidence (~10∘). The observable correlation length of the 3D CO may thus be limited by the finite penetration depth being of similar magnitude. We believe this is not a significant factor, however, due to the (002) structural reflection (blue diamonds) having a correlation length that is within the experimental uncertainty of the 3D CO, even though it was measured at higher energy (~1750 eV) and angle of incidence (~38°), both of which contribute to a significantly longer absorption length. This suggests that, in this Pr-YBCO system, the 3D CO peak has a width that is limited by the width of the crystallographic Bragg peaks. The measured ~364 Å thus represents a lower bound on the out-of-plane correlation length.

Energy dependence

The energy dependence of the scattered intensity at Q = (0 -0.335 1) is shown in Fig. 3a, overlaid with the corresponding x-ray absorption spectrum (XAS) measured with the electric field of the x-rays parallel to the bond directions in the CuO2 planes. The XAS reveals two resonances that correspond to the Pr M5 (930.9 eV) and the Cu L3 (932.6 eV) edges. There are also two peaks observed in the energy dependence of the 3D CO (930.3 eV and 932.8 eV), which most likely correspond to contributions from Pr and Cu, respectively. However, due to the energetic overlap of the Pr M5 and Cu L3 edges, the energy dependence of the scattering is unavoidably complex; as such, we refer to them simply as peaks A and B (see Supplementary Discussion). Unlike in YBCO films with 3D CO26, we do not observe a significant shift in spectral weight to higher energy that would indicate CO coupling through the CuO chains. Furthermore, we observe in Fig. 3b that the 3D CO peak can still be detected at energies far below the resonance (850 eV) albeit much more weakly, which is in contrast to all other cuprates where the CO peaks studied by RSXS lack sufficient scattering strength to be observed off-resonance. This indicates a sizeable lattice distortion rarely seen7 in other cuprate systems, highlighting that in Pr-YBCO the 3D CO becomes more structurally stable than previously reported.

Fig. 3: The nature of the 3D CO.
figure 3

a The x-ray absorption spectrum (XAS) (gray) shows the Cu L3 edge with a shoulder at lower energies corresponding to the Pr M5 edge. The two dipole-allowed transitions are labeled as insets. The purple data show the energy dependence of the scattered intensity of the 3D CO at Tc = 50 K with the two most prominent features labeled as peaks A and B. b A semi-log plot of rocking curve scans of the 3D CO peak taken on resonance (930.3 eV) and well below the resonance (850 eV); the latter displays a weaker, but detectable, off-resonant intensity. c The temperature dependences measured at energies corresponding to the two peaks observed in the 3D CO energy dependence (A: red triangles; B: blue squares), showing detectable peaks at room temperature and drops below the superconducting transition (Tc).

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Temperature dependence

The interplay of the 3D CO with superconductivity can be investigated by measuring the temperature dependence of the former. In Fig. 3c, we plot the scattered intensity at Q = (0 -0.335 1) at energies corresponding to peaks A and B in the energy dependence as a function of temperature. While the overall scattering intensity is higher at the peak A energy than the peak B energy, which is consistent with the measured energy dependence, it is notable that the 3D CO scattering signal is still detectable at room temperature for both energies. Upon cooling from T = 300 K, the temperature dependence at both energies maintain roughly equivalent slopes until within the vicinity of Tc = 50 K. Cooling below Tc produces a cusp-like maximum, indicating a competition between SC and the isolated 3D CO phase. This signature behavior confirms that CO is at least a major contributor to the observed scattering, regardless of whether additional structural contributions exist. This is in contrast to the 3D CO induced by very high magnetic fields, where any competition between 3D CO and SC is obscured by the very presence of the magnetic field which, while necessary to induce 3D coherence, comes with the unavoidable expense of greatly suppressing the SC phase.

Discussion

Having established experimentally that 3D CO can be stabilized with long out-of-plane correlation length, we turn to discuss the possible origin of the c-axis coupling in the Pr-YBCO system. It is already well known that, unlike any other rare earth, Pr substitution uniquely suppresses SC in YBCO by localizing holes via orbital hybridization44,45,48,49. To this end, we performed density- functional theory plus Hubbard U (DFT+U) calculations for both PrBa2Cu3O6 (PrBCO) and DyBa2Cu3O6 (DyBCO) structures50 to understand the role of this hybridization within the context of 3D CO and its competition with SC (see Supplementary Methods). Figure 4a schematically depicts the orbital character of the electronic states near the Fermi level (EF) in PrBCO. In addition to the characteristic pdσ bands of the CuO2 planes which host all the 2D electronic phenomena, another band crosses the Fermi level. From prior calculations49, it is clear that the effective doping is affected, as this band above the Fermi level takes holes from the superconducting band, which is consistent with the observation that Tc is suppressed with increasing Pr concentration. This results from the antibonding coupling between the Pr 4\({f}_{z({x}^{2}-{y}^{2})}\) state and its nearest-neighbor O 2pπ states in adjacent CuO2 planes48,49 (Fig. 4c, d). We speculate that this orbital coupling with an out-of-plane component locks together the phase of the CO on adjacent CuO2 planes, resulting in a diffraction peak at L = 119. For later rare-earth elements with lower 4f energy, the 4\({f}_{z({x}^{2}-{y}^{2})}\)–2pπ antibonding band is expected to be lowered and removed from the Fermi level. Figure 4b depicts the calculated electronic level structure for DyBCO, where the top of this band is deep below the Fermi level at around −0.8 eV. This leaves the charge carriers in the two adjacent CuO2 planes essentially decoupled. This unique aspect of Pr makes it the appropriate rare-earth to substitute into Y to stabilize and isolate 3D CO, which occurs concomitantly with a lattice distortion, according to our data. The exact structural mechanism of stabilization, e.g., phonons, is a subject of future research.

Fig. 4: DFT-calculated band structures showing hybridization.
figure 4

a A schematic representation of the orbital character of the electronic levels near the Fermi energy (EF) in PrBa2Cu3O6 (PrBCO), showing a mixed 4f (purple) and 2p (green) antibonding band crossing EF. b The equivalent schematic for DyBa2Cu3O6 (DyBCO). c A schematic depicting the 4f (purple) and 2p (green) orbitals within the crystal structure of PrBCO in three dimensions. d The hybridization that occurs between the 4\({f}_{z({x}^{2}-{y}^{2})}\) (purple) and a linear combination of 2px and 2pz orbitals (green) oriented towards each other, shown within the plane represented in panel c.

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Our discovery of a fully stable 3D CO without a 2D signal has important implications to our understanding of CO and its interplay with SC. First, we confirm that a fully coherent, isolated 3D CO can be stabilized despite the intrinsic disorder inevitably present in cuprates. We note here that our Pr-substituted samples are expected to host at least as much structural and chemical disorder than in pristine YBCO, if not more so, due to the additional defect channel. This result may further elucidate the complex relationship between CO and SC, both of which have now been shown to substantially gain 3D character with increasing Pr concentration in the Pr-YBCO system41,42,43. Second, we confirm that a stable 3D CO still coexists and competes with SC, implying that the system’s ground state can comprise two long-range, static, coexisting orders. Third, since the 3D coupling does not rely on the CuO chains that are unique to YBCO, perhaps other forms of hybridization can be used to stabilize 3D CO in other cuprate families, which has not yet been observed. Finally, we show that controlling the orbital content of the Fermi surface by assigning it a 4f character with an out-of-plane component can yield a sizable impact on the electronic ordering tendencies of the CuO2 plane. It can be used as a tuning knob to study the validity of 2D models to describe layered systems, like the cuprates or intercalated graphitic systems51,52,53,54,55.

In summary, we have shown how utilizing the hybridization between the 4f states of Pr and planar CuO2 orbitals to tune the underlying orbital character can significantly enhance the out-of-plane coupling, phase-locking the CO across adjacent planes and rendering a stable CO phase that is fully correlated along the out-of-plane direction without the 2D version. The c-axis correlation length has a lower bound matching that of the crystal itself, showing that Pr substitution is the most efficient way of stabilizing 3D CO compared to using external perturbations, like magnetic fields and strain, and uniquely does not suffer from experimental complications arising from in situ application. Furthermore, through resonant spectroscopy, we attribute the formation of 3D coupling to the role of the Pr ions located between CuO2 planes. To understand the mechanism of this out-of-plane coupling, we turned to DFT+U calculations that show a hybridized 4f-2p band crossing the Fermi level, a feature that is unique to Pr-substituted YBCO. Since our system does not rely on external perturbations, other techniques can be employed to investigate this material and shed light on the connection between CO and SC. Moreover, this demonstrates how the influence of underlying orbital character on an electronic phase can be tuned via orbital hybridization, which can be generalized to other correlated transition metal oxides and layered systems.

Methods

Sample preparation

Single crystals of PrxY1−xBa2Cu3O745 were grown according to the method described in reference56. The starting materials used in the crystal growth consisted of 99.99% pure Y203, Pr6O11, BaCO3, and CuO powders. The crystals were annealed in flowing oxygen to maintain full oxygenation and optimize their superconducting properties. The Pr0.3Y0.7Ba2Cu3O~7 sample we studied has an orthorhombic crystal structure that is not detwinned with lattice parameters c = 11.67 Å and a = b = 3.87 Å. The superconducting transition temperatures of the crystals were determined from magnetization measurements performed with a vibrating sample magnetometer in a Quantum Design DynaCool Physical Property Measurement System.

RSXS measurement

The data shown in this manuscript were collected from scattering experiments carried out at beam line 13-3 of the Stanford Synchrotron Radiation Lightsource (SSRL). Crucial measurements and insights where gained through scattering experiments carried out at Sector 29 of the Advanced Photon Source (APS). The sample was mounted using silver paint on an in-vacuum multiple-circle diffractometer. The sample temperature was controlled by an open-circle helium cryostat. The incident photon polarization was fixed as σ (vertical linear) polarization. The (0 K L) scattering plane was determined by aligning the (0 0 2), (0 -1 1), and (0 1 1) structural Bragg reflections at 1746 eV photon energy. We note that we have also observed this phenomenon in a second sample with a similar Pr concentration (see Supplementary Note).

A 256 × 1024 pixel (26 μm × 26 μm pixel size) CCD detector was used. The scattering intensity data were collected within a region-of-interest in the center of the CCD detector. Dark images and data measured by the CCD detector outside of this region-of-interest were used to subtract any background fluorescence contributions, which were generally very small compared to the 3D CO scattered intensity, except for when off-resonance or at high temperature. 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. A 100 nm Parylene/100 nm Al filter was placed in front of the CCD to stop photoelectrons emitted from the sample from contributing to the signal on the CCD. Further details about the data collection and analysis methods used may be found in the Supplementary Methods.