A hallmark of many unconventional superconductors is the presence of many-body interactions that give rise to broken-symmetry states intertwined with superconductivity. Recent resonant soft X-ray scattering experiments report commensurate 3a 0 charge density wave order in infinite-layer nickelates, which has important implications regarding the universal interplay between charge order and superconductivity in both cuprates and nickelates. Here we present X-ray scattering and spectroscopy measurements on a series of NdNiO 2+x samples, which reveal that the signatures of charge density wave order are absent in fully reduced, single-phase NdNiO 2 . The 3a 0 superlattice peak instead originates from a partially reduced impurity phase where excess apical oxygens form ordered rows with three-unit-cell periodicity. The absence of any observable charge density wave order in NdNiO 2 highlights a crucial difference between the phase diagrams of cuprate and nickelate superconductors.

The discovery of superconductivity in infinite-layer nickelates 1 , and its analogy to its cuprate antecedents, offers a unique opportunity to better understand the key ingredients for high-temperature superconductivity. Although nickelates and cuprates share many commonalities, including a broadly similar crystal and electronic structure 1 , strong correlations, antiferromagnetic excitations 2,3 and a superconducting dome [4][5][6] , there are also many distinctions between the two families. These include the very different transition temperatures 1,4,5 , the relative oxygen and 3d character of the doped holes 7,8 , and the hybridization between the 3d and rare-earth states 9 . One important apparent similarity between the two families is the report of charge density wave order in a variety of infinite-layer nickelates by resonant soft X-ray scattering (RSXS) [10][11][12][13] . This discovery, if correct, would suggest a ubiquitous interplay between charge order and superconductivity in the phase diagram of both cuprates and nickelates, with important implications for a universal theory of high-temperature superconductivity.

Nevertheless, there are clear distinctions between the charge order reported in cuprates and nickelates. In cuprates, the wavevector is typically incommensurate and strongly doping dependent [14][15][16] , whereas in NdNiO 2 and PrNiO 2 , it is locked to q = (1/3, 0) (refs. 11-13). Additionally, charge ordering in cuprates is strongly temperature dependent [15][16][17][18][19][20] , whereas reports in nickelates exhibit a weak temperature dependence with no clear transition or onset 10,12,13 . To better understand the nature of putative charge ordering in the infinite-layer nickelates, we have investigated the q = (1/3, 0) superlattice peak in a series of samples with varying levels of reduction. We discover that the superlattice peak is entirely absent in fully reduced, single-phase NdNiO 2 samples, and instead arises from partially reduced impurity phases, namely, NdNiO

2.33 (Nd 3 Ni 3 O 7 ) or NdNiO 2.67 (Nd 3 Ni 3 O 8 ), produced

https://doi.org/10.1038/s41563-024-01797-0 estimated in-plane correlation length of ξ = 12-20 nm is also qualitatively similar to earlier reports [10][11][12][13] . Finally, a weak dependence on the out-of-plane momentum transfer L is observed with a maximum at roughly L = 0.31(2) reciprocal lattice units (r.l.u.), again consistent with earlier measurements (all the momenta in this text are quoted in r.l.u. with reference to the NdNiO 2 lattice; a = 3.905 Å, c = 3.286 Å). In contrast, the superlattice peak was not observed in samples E and F, which is notable since sample E exhibited the lowest resistivity of all the samples. Additional measurements on samples G and H were measured at a separate beamline and exhibited an extremely weak q = (1/3, 0) feature at the sample centre, whereas sample I showed no superlattice features whatsoever. Supplementary Information provides further details of the measurements on samples E-I.

This discrepancy across nominally similar NdNiO 2 samples suggests that some unknown variable, potentially associated with sample disorder or oxygen non-stoichiometry, influences the presence of the 3a 0 superlattice peak. One possibility is that the charge density wave order is intrinsic to infinite-layer nickelates, but triggered by the presence of atomic-scale disorder within the lattice. Alternatively, the peak could originate from an impurity phase produced during the reduction process, which would imply that the charge order observed to date does not play a role in the phase diagram of infinite-layer nickelates.

To distinguish between these scenarios, we have investigated partially reduced, oxygen-deficient perovskite phases NdNiO 2+x (samples B and C). Sample B was only lightly reduced, with its XRD data nearly indistinguishable from the perovskite (sample A), but with a substantially broadened metal-insulator transition and increased sample resistance (Fig. 1a,b). Sample C was reduced further and shows an intermediate out-of-plane lattice constant of c = 3.66 Å and highly insulating electrical transport as expected for a predominantly oxygen-deficient perovskite phase 27 . The XAS spectra of sample B exhibit the typical two-peak structure observed in perovskite nickelates on the Ni L 3 edge 28 , as well as a strong prepeak at the O K edge; sample C is markedly different, with a sharp peak at 852.7 eV, shoulder at 852.0 eV and weak secondary peak at 854.3 eV, indicative of an oxygen-deficient perovskite phase 29,30 .

In Fig. 2a, we show the RSXS measurements of sample C, which exhibit a peak at q = (1/3, 0) virtually identical to the superlattice peak in sample D and the literature [10][11][12][13] in nearly all respects: wavevector, energy dependence, polarization dependence, temperature dependence and correlation length, with the exception that it is extremely during the reduction process, where excess apical oxygen atoms form ordered rows with three-unit-cell periodicity. This reveals that charge ordering with 3a 0 periodicity is not intrinsic to the infinite-layer nickelates-a discovery with important implications for our understanding of the phase diagram of nickelates and its relationship to cuprates.

To produce a sequence of NdNiO 2 samples with nominally identical Nd:Ni stoichiometry 21 and variable oxygen content, we have employed a combination of reactive-oxide molecular-beam epitaxy to synthesize the precursor perovskite and atomic hydrogen reduction to access the oxygen-deficient phases. A consistent series of perovskite films of NdNiO 3 (20 pseudocubic unit cells thick) with excellent crystallinity and sharp metal-insulator transitions were synthesized on SrTiO 3 . Following synthesis, all the films were capped by 2-3 unit cells of SrTiO 3 and exposed to a beam of >50% atomic hydrogen produced by a thermal cracker 22 . Although the reduction of perovskite nickelates has typically been achieved using CaH 2 and NaH powder 1,5,23 , atomic hydrogen offers the benefit of independent control over the sample temperature and reducing environment, as well as fast reaction times (<20 min).

Using this approach, we have synthesized a series of samples ranging from the pristine parent perovskite NdNiO 3 (sample A) to oxygen-deficient intermediate phases NdNiO 2+x (samples B and C) and infinite-layer NdNiO 2 (samples D-I), all of which were characterized by conventional X-ray diffraction (XRD) and transport measurements. All the NdNiO 2 samples appear highly crystalline and single phase by XRD, with low-temperature resistivities of 350-700 μΩ cm, comparable with or lower than undoped films on SrTiO 3 (refs. 4,5,24) and LSAT 25 (Fig. 1a,b; Supplementary Information shows the data for samples G-I). Further information about the growth and reduction procedures are available in Methods.

X-ray absorption spectroscopy (XAS) measurements on all the samples are shown in Fig. 1c, and RSXS measurements on three representative infinite-layer NdNiO 2 samples (samples D-F) are presented in Fig. 1d,e. The XAS spectra of all three NdNiO 2 samples appear similar and closely match with the published measurements 9,26 , with a single peak at the Ni L 3 edge (852.4 eV) with no visible prepeak at the O K edge. Despite the apparent similarity between these samples, only sample D exhibits a superlattice peak at the putative charge order wavevector of q = (1/3, 0). As shown in previous reports, this peak is observed on the Ni L edge (Fig. 1d) as well as the Nd (rare-earth) M edge (Fig. 1e), and exhibits a strong polarization dependence, namely, I σ /I π ≈ 4 at hν = 852.3 eV, consistent with prior measurements 11,13 . The intense (100-400 times stronger than that observed in sample D). This peak also displays a strong L dependence, with a maximum at 0.

30 r.l.u (d/3 = 3.65 Å). The strong similarity between the superlattice features in samples C and D suggests a common origin: ordered oxygen-deficient phases arising from an incomplete reduction process. In reduced nickelates, excess apical oxygens can form ordered phases, such as the brownmillerite structure of La 2 Ni 2 O 5 (refs. 31-33), where the apical oxygen atoms form alternating rows. Other related structures also exist with different periodicities, such as Nd 3 Ni 3 O 7 (refs. 34-36), Pr 3 Ni 3 O 7 (refs. 34,35), La 3 Ni 3 O 8 (ref. 37) and (Pr,Ca) 4 Ni 4 O 11 (ref. 38). Recent in situ XRD studies indicate that the reduction pathway from bulk NdNiO 3 to NdNiO 2 occurs first via the formation of an intermediate-phase Nd 3 Ni 3 O 7 (ref. 36), where one-third of the apical oxygen sites are occupied and are ordered into chains with 3a 0 periodicity (Fig. 2c). This 3 × 1 × 3 superstructure of the original pseudocubic unit cell would naturally give rise to a superlattice peak at q = (1/3, 0, 1/3). This suggests that sample C is probably predominantly Nd 3 Ni 3 O 7 , and that an incomplete reduction of any NdNiO 2 samples would also leave traces of the Nd 3 Ni 3 O 7 phase behind (sample D).

To further investigate this hypothesis, we have also performed RSXS measurements on sample B, which should only be in the initial stages of the conversion from NdNiO 3 into a brownmilleritelike Nd 3 Ni 3 O 7 phase. The XRD and XAS measurements of this sample (Fig. 1a,c) are indistinguishable from the data for pristine NdNiO (sample A). Nevertheless, RSXS measurements (Fig. 2b) also show the q = (1/3, 0) peak with identical characteristics (L, energy and polarization dependence) and comparable intensity with sample D. The fact that sample B exhibits no obvious trace of the infinite-layer phase and still exhibits a clear q = (1/3, 0) superlattice peak demonstrates that this feature does not originate from intrinsic charge ordering within the infinite-layer phase itself. In fact, the q = (1/3, 0) peaks appear to be more reminiscent of Bragg peaks observed in resonant scattering from cuprates with oxygen ordering (for example, ortho-YBa 2 Cu 3 O 6+δ ), as opposed to intrinsic charge density wave order 39,40 . There, the strong resonant enhancement on the Cu L edge arises from the local oxygen environment strongly modifying the electronic structure of the Cu atoms, and a similar resonant enhancement at the Ni L edge should likewise occur for oxygen ordering in nickelates.

In Fig. 3a, we show the energy dependence of the q = (1/3, 0, 1/3) peak intensity in samples C and D (shaded), together with the background fluorescence measured off the superlattice peak (white). Because the superlattice peak intensity is far stronger in sample C, the relative strength of the background fluorescence in sample D is much larger than in sample C. Nevertheless, a two-peak resonance profile is apparent in both samples at the Ni L 3 edge with only a weak response at the L 2 edge, similar to prior measurements on NdNiO 2 (ref. 11). Assuming a structural origin arising from oxygen ordering, the superlattice peak should also be observable in the off-resonance case. In Fig. 3b,c, we show a series of scans across q = (1/3, 0) for samples C and D, spanning a 235 eV range about the Ni and Nd resonances. Although the intensity is weaker than the on-resonance case, the persistence of the peak strongly supports a structural origin. The temperature dependence of the scattering peak is shown for samples B-D (Fig. 3d). Similar to prior reports 10,12,13 , the peak shows a smooth decrease in intensity by 15-20% between 20 and 370 K, with no obvious change in the correlation length. This weak dependence-without a transition-is fairly similar to the temperature dependence of superlattice peaks resulting from the oxygen ordering in YBa 2 Cu 3 O 6+δ (refs. 40,41), as opposed to the more dramatic temperature dependence of the CDW order 18,40 .

We also observe a strong dependence of the q = (1/3, 0) peak intensity in sample G for the measurement location on the 10 × 10 mm sample (Supplementary Information). Due to the spread of the atomic hydrogen beam and thermal gradients, the edges of the sample are not as well reduced as the centre-correspondingly, we observe a clear superlattice peak near the edge of the sample, which decreases in intensity approaching the sample centre. Spatially resolved synchrotron hard XRD measurements at the same locations confirm the prevalence of intermediate reduction products at the edge of sample G, versus fully reduced NdNiO 2 near the centre, thereby correlating the presence of the resonant feature with that of intermediate phases on the same sample.

Prior powder XRD studies of bulk Nd 3 Ni 3 O 7 reveal that the excess apical oxygens form octahedrally coordinated chains that run perpendicular to the NiO 2 planes and which are separated by 3a 0 (ref. 34). Another potential arrangement is to have the apical oxygens form pyramidal chains that lie within the NiO 2 plane and are oriented along the a axis (Fig. 4a). In addition, Nd 3 Ni 3 O 8 (NdNiO 2.67 ), where two rows of apical oxygens are occupied followed by a row of vacancies, would likewise generate the same superlattice peak at q = (1/3, 0, 1/3), although this phase has not been previously reported. Scanning transmission electron microscopy (STEM) measurements on a separate, partially reduced sample (sample J) are detailed in Fig. 4b,c,d. The Fourier transform of a high-angle annular dark-field image reveals 1/3-order diffraction peaks corresponding to 3a 0 ordering along both a and c, consistent with other recent transmission electron microscopy measurements on uncapped NdNiO 2 films 42 . Additionally, annular bright-field data show regions with two filled apical oxygen chains alternating with a single vacant chain, consistent with the Nd 3 Ni 3 O 8 structure shown in Fig. 4a, where the apical oxygen chains are directed into the page. Sample J was prepared using the same conditions as samples D-I, but with a four-unit-cell SrTiO 3 cap. Although RSXS measurements were not performed on sample J, XRD and transport measurements (Supplementary Information) indicate that this sample is less reduced than sample D as both NdNiO 2 and intermediate-phase peaks are visible in lab-based XRD. Since the difference in formation energies of the various ordered oxygen structures is small, it is possible that multiple compositions or structures of Nd 3 Ni 3 O 7,8 could exist within our series of samples, depending on factors such as the reduction conditions or epitaxial strain 43 . A more detailed investigation of the precise structures of Nd 3 Ni 3 O 7 or Nd 3 Ni 3 O 8 in our thin films is currently the subject of further investigation. Nevertheless, the key point is that all the configurations of Nd 3 Ni 3 O 7 or Nd 3 Ni 3 O 8 will form a 3 × 1 × 3 supercell that will generate the q = (1/3, 0) superlattice peak at the observed wavevector. Finally, a macroscopic sample would be expected to possess equal domains of chains along both in-plane directions, in which case the superlattice peaks would then be observable along both H and K directions.

Our results suggest the scenario illustrated in Fig. 4a. The reduction of NdNiO 3 into NdNiO 2 occurs via the production of an intermediate, partially reduced Nd 3 Ni 3 O 7 and/or Nd 3 Ni 3 O 8 phase, where excess apical oxygen atoms (or vacancies) are ordered into rows with 3a 0 periodicity forming a 3 × 1 × 3 supercell with a superlattice peak at the putative charge order wavevector of q = (1/3, 0, 1/3) (sample C). Because of its extremely strong intensity in the on-resonance case, this peak is detectable even if very small amounts of the partially reduced phase are present at levels nearly undetectable by conventional techniques (XRD, transport and XAS; that is, samples B, D and H). This picture is supported by a multitude of our observations: (1) the superlattice peak is absent in the plurality of low-resistivity, fully reduced NdNiO 2 samples (samples E, F and I); The presence of an ordered oxygen impurity phase also explains numerous discrepancies and unusual features in the literature. The strong intensity on the rare-earth edge is surprising for a charge ordering scenario, since the coupling of the rare-earth 4f electrons to the Ni3d electrons should be relatively weak and indirect, but should naturally occur for a structural Bragg peak where the Nd ions are displaced due to oxygen ordering. In addition, prior reports claim the presence/ absence of a superlattice peak in uncapped versus capped samples of NdNiO 2 . Our measurements indicate that although the capping layer alone does not dictate the presence of the superlattice peak (all the samples were capped), even small variations in reduction conditions can result in residual amounts of Nd 3 Ni 3 O 7,8 . The doping dependence of the intensity of the superlattice peak, which is the strongest for the undoped parent compound and vanishing near x = 0.20, may also be naturally explained by the presence of an excess oxygen impurity phase. Bulk studies indicate that Sr doping, which increases the average targeted Ni valence of RE 1-x Sr x NiO 2 , makes the material easier to reduce 36 . Thus, the addition of Sr may naturally diminish the amount of residual RE 3 Ni 3 O 7,8 in a sample and suppress the intensity of the q = (1/3, 0) Bragg peak. Finally, although our study is limited to NdNiO 2+x , these findings should broadly apply to all infinite-layer nickelates, since Pr 3 Ni 3 O 7 and La 3 Ni 3 O 8 also form 3 × 1 × 3 superstructures and are known to be the reduction products of PrNiO 3 and LaNiO 3 , respectively 34,35,37 . Although the superlattice peak in LaNiO 2 shares many overall similarities with those in (Nd,Pr)NiO 2 , including its energy and temperature dependence, it does exhibit some subtle apparent differences, particularly a very slight displacement from the commensurate q = (1/3, 0) wavevector (Δq ≈ 0.01 r.l.u) (ref. 10) at x = 0. Future experiments will be important for conclusively determining the origins of the superlattice peak in the other members of infinite-layer nickelates. Although we focus here specifically on ordered excess oxygen phases, this highlights the role that excess oxygen may play in reduced films more broadly. The ordered excess oxygen phases appear to be highly insulating, and therefore, small inclusions may not substantially contribute to the electrical properties as a whole when averaging over an entire macroscopic sample. Nevertheless, if low concentrations of excess apical oxygen ions are randomly distributed, it is conceivable that they could potentially act as hole dopants, which could be consistent with traces of superconductivity observed in nominally undoped LaNiO 2 (ref. 6). Through a multimodal investigation of a large sequence of samples with varying levels of reduction, we conclude that charge ordering with 3a 0 periodicity is not intrinsic to the infinite-layer nickelates. The superlattice peak previously identified as charge ordering at q = (1/3, 0) originates from the three-unit-cell ordering of excess apical oxygen ions in small amounts of brownmillerite-like inclusions of Nd 3 Ni 3 O 7 or Nd 3 Ni 3 O 8 produced during the reduction process. Topotactically reduced complex oxides present an exciting new frontier in quantum materials 1,44 , but this work also highlights some of the materials challenges inherent in these systems. We demonstrate RSXS as a highly sensitive and powerful probe for investigating these reduced compounds, which can detect even trace amounts of impurity phases. Although the phase diagrams of nickelates and cuprates show many similarities, including a superconducting dome and strong antiferromagnetic fluctuations on the underdoped side, this work establishes a clear distinction between the two families, namely, the fact that charge ordering does not appear to be directly relevant to the phase diagram of nickelates. This finding should have important implications for understanding universal models of high-temperature superconductivity, and may help to explain some of the key differences between the two families of materials.

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41563-024-01797-0. (b) and corresponding high-angle annular dark-field image (c) of a defective region are shown. The positions of the third-order peaks in the Fourier transform are circled in green. An annular bright-field image showing two filled rows of apical oxygens (d), followed by one row of missing apical oxygen positions, corresponding to the schematic for Nd 3 Ni 3 O 8 in a. In this image, the apical oxygen chains run into the page.