Charge disproportionation and nano phase separation in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{R}\mathrm{SrNiO}_{4}$$\end{document}RSrNiO4

We have successfully grown centimeter-sized layered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{R}\mathrm{SrNiO}_{4}$$\end{document}RSrNiO4 single crystals under high oxygen pressures of 120–150 bar by the floating zone technique. This enabled us to perform neutron scattering experiments where we observe close to quarter-integer magnetic peaks below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 77~\mathrm{K}$$\end{document}∼77K that are accompanied by steep upwards dispersing spin excitations. Within the high-frequency Ni–O bond stretching phonon dispersion, a softening at the propagation vector for a checkerboard modulation can be observed. We were able to simulate the magnetic excitation spectra using a model that includes two essential ingredients, namely checkerboard charge disproportionation and nano phase separation. The results thus suggest that charge disproportionation is preferred instead of a Jahn–Teller distortion even for this layered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Ni}^{3+}$$\end{document}Ni3+ system.

Scientific Reports | (2020) 10:18012 | https://doi.org/10.1038/s41598-020-74884-2 www.nature.com/scientificreports/ view of ab-initio theories 10,11 . Note that the tendency to a charge, or valence bond disproportionation is a local property. However, there exists also an alternative picture 12,13 , relying mainly on the Fermi-surface properties of metallic nickelates, which is actually a collective, not local physics. The interesting question now arises whether the charge or valence bond disproportionation still wins over the Jahn-Teller distortion for Ni 3+ oxides where the local coordination is no longer O h 14,15 . For example, the layered system RSrNiO 4 has a K 2 NiF 4 crystal structure with NiO 6 octahedra that are elongated along the tetragonal axis.
In such a case would the charge disproportionation be absent and instead the Jahn-Teller t 6 2g e 1 g be stabilized? Electronic structure calculations in the LDA+NMTO downfolding scheme predict the 3z 2 −r 2 orbital to be more stable than the x 2 −y 2 orbital by about ∼ 0.26 eV [16][17][18] , which is a large energy difference regarding an optical gap of 0.2 eV . However, until now one could not prepare large high quality single crystals of these layered Ni 3+ 2-1-4-nickelates. R 2−x Sr x NiO 4 have so far been studied only for doping levels x away from the x = 1 pure Ni 3+ situation. It is known, for example, that for x ≪ 1 there is Ni 2+ /Ni 3+ charge order which modulates the spin structure [19][20][21] . At half-doping, i.e. for x = 0.5, a Ni 2+ /Ni 3+ checkerboard charge ordering occurs 22 similar to the isostructural cobaltates [23][24][25][26][27][28] . This checkerboard charge order survives up to ~70% of hole-doping 29 and the materials stay insulating.
Some initial studies on single crystals of highly hole-doped R 2−x Sr x NiO 4 exist and show an about equal population of four different kind of states around x ∼ 1 : Ni 2+ , Ni 3+ with 3z 2 -r 2 orbital occupation, Ni 3+ with x 2 -y 2 orbital occupation and Ni 4+30 . Moreover, also ARPES studies on these materials exist 16,17 .
However, so far, no detailed study of the physical properties of the compounds with the pure Ni 3+ oxidation state have been carried out and it is unknown whether the Jahn-Teller orbital occupation can be stabilized by the strongly distorted Ni-oxygen environment instead of the Ni 2+ /Ni 4+ or Ni 2+ /Ni 2+ L 2 charge disproportionation. We note that the mechanism for such a charge disproportionation is quite distinct from that for a Ni 2+ /Ni 3+ charge ordering since a disproportionation in a stoichiometric system involves an energy term related to the onsite Coulomb interaction U (which needs to be compensated by another interaction energy), while for ordering of charge in a non-stoichiometric or doped material the term U is not operative.

Results
Powder X-ray diffraction shows that our NdSrNiO 4 samples are impurity-free. From the refinement of the powder X-ray diffraction data we obtain a c/a ratio that amounts to ∼3.26 and apical and basal Ni-O distances which amount to 2.058(8) Å and 1.895(1) Å respectively, thus, indicating a strongly tetragonaly distorted oxygen environment with an apical-to-basal distance ratio of 1.09.
The magnetic susceptibility in NdSrNiO 4 is dominated by the Nd 3+ moments, see Fig. 1. For La 2/3 Y 1/3 SrNiO 4 with non-magnetic R-ions the effective moment of Ni can be determined and amounts to ∼ 0.23 µ B /Ni according to Curie-Weiss fits.
The temperature dependence of the in-plane resistivity is shown in Fig. 2. A bad metallic behavior can be observed at high temperatures followed by a semiconducting temperature dependence at lower temperatures, thus, indicating a (gradual) metal-semiconductor transition around ∼ 150 K.
The Ni − L 2,3 X-ray absorption spectrum of our NdSrNiO 4 single crystal is shown in Fig. 3 and is compared to that of our Nd 2 NiO 4 single crystal serving as a Ni 2+ reference compound. It is well known that the XAS spectra at the L 2,3 edge of transition metals are highly sensitive to the valence state 31-33 -an increase of the valence state of the transition metal ion by one causes a shift of the XAS L 2,3 spectra by one or more eV towards higher energies. The more than one eV higher energy shift from Nd 2 NiO 4 to NdSrNiO 4 indicates the formal Ni 2+ and www.nature.com/scientificreports/ Ni 3+ valence states for the former and the latter compound respectively. Here we would like to note that the NdSrNiO 4 spectrum cannot be interpreted in terms of an ionic Ni 3d 7 configuration, but, rather by a coherent mixture of 3d 8 and 3d 8 L 2 configurations 3,4,7,34 , where each L denotes a hole in the oxygen ligand. We can exclude any Ni 2+ impurities in our NdSrNiO 4 single crystal, otherwise the sharp main peak of Ni 2+ impurity spectrum would have been visible as a sharp shoulder at the leading edge. Due to the availability of large single crystals we are now able to perform (polarized) neutron scattering experiments on RSrNiO 4 . As shown in Fig. 4, at low temperatures (i.e. below T N ∼ 80 K ) magnetic peaks could be detected around quarter-integer positions within the H K 0 plane of reciprocal space. The magnetic origin of these peaks can be confirmed by polarization analysis and the extracted M y and M z components are shown in Fig. 4. Gaussian functions are fitted to the data. In NdSrNiO 4 the Ni moments start ordering below ∼ 77 K , see Additionally, we measured also the temperature dependence of the Nd-and Ni-contribution of the antiferromagnetic reflections by means of resonant soft X-tray diffraction 35 , see Fig. 6. These element selective  www.nature.com/scientificreports/ measurements corroborate our findings that the Nd-moments start to order at distinctly lower temperatures than the Ni-moments. Note, that resonant X-rays are element specific and not probing the same magnetic intensity as neutrons are doing. Also short range correlations will be less integrated in this measurement due to the higher resolution which might result in a different temperature dependence compared to the neutron results. As shown in Fig. 7a and b, the magnetic fields suppress the magnetic reflection intensities drastically at 2 K, while having almost no influence at 40 K, which can be seen more clearly from the extracted magnetic field dependence of the integrated intensities using a Gaussian fit to the data, as shown in Fig. 7c and d. These results reflect the full polarization of the Nd 3+ moments below the Nd-magnetic ordering temperatures in fields above 6 T whereas the (much smaller) Ni magnetic moments (that are observable at 40 K) are much less affected.
Summarizing, distinctly below T N (e.g. around 40 K) the Ni magnetic moments are aligned within the ab planes. For NdSrNiO 4 also the Nd moments start ordering below roughly 20 K with the larger Nd moments being aligned in the c-direction. The observation of quarter integer magnetic peaks in RSrNiO 4 is compatible with a charge disproportionation of the nominal Ni 3+ ( 3d 7 ) ions into a 3d 8 /3d 8 L 2 configurations. The ordering of these charges is in a checkerboard pattern.
This kind of charge ordering is also in agreement with our inelastic neutron measurements of the longitudinal Ni-O bond stretching phonon mode, compare also Ref. 23 . For high − T C superconducting cuprates it is known that such phonon softening of the Cu-O bond stretching phonon modes has been observed at the propagation vector of the underlying charge (stripe) order 36 , i.e. one would expect a bond-stretching phonon anomaly at the propagation vector of the underlying charge ordering propagation vector. As can be seen in Fig. 8a, this highfrequency phonon dispersion softens for NdSrNi 3+ O 4 at half-integer propagation vectors similar to that in half-doped cobaltates with very robust checkerboard charge order 23 . In contrast to that, the same measurement of the pure Co 3+ reference material LaSrCo 3+ O 4 -see Fig. 8b-reveals no such phonon softening at the zone boundary i.e. towards Q = (2.5 2.5 0).
In NdSrNiO 4 the static distortions implied by this kind of charge distribution could not be detected in analogous elastic neutron measurements. However, these distortions might be too weak to be detectable possibly because of small distortions that are associated with the charge disproportionation, since the charges are delocalized towards the oxygen ions. A similar situation might occur in LaNiO 3 2 . Note that the soft X-rays are unable to reach the required points in reciprocal space and that resonant hard X-ray scattering is unfavorable since any gain in intensity at the K-edge is marginal and will be overcompensated by the effects of fluorescence.
Finally, we measured the magnetic excitation spectrum of La 2/3 Y 1/3 SrNiO 4 at 6 K (at the MERLIN time-offlight spectrometer with an incident energy of 41 meV), see Fig. 9. These measurements show that these nickelates are different from the usual magnets-an upward dispersion becomes apparent which strongly resembles the one in highly hole-doped cobaltates 26 , see Fig. 9c. In the cobaltates this excitation spectrum could be explained by a nano phase separation model 24 with two relevant exchange interactions-one between two Co 2+ -ions accross a hole ( J ′ ) and a much weaker one across two or more Co 3+ -ions ( J ′′ etc.). Due to the similarity with the cobaltates, one might think of a similar model for the nickelates, but now with larger values of the exchange interactions that will scale the entire spectrum to higher energies. So we assume a fully charge disproportionated nickelate system with disorder, where we take the nearest neighbor exchange interactions between the magnetic Ni ions to be J = J   Fig. 10b,c. The incommensurate magnetic peaks appear at almost the same positions as observed in the experiment. Furthermore, also the magnetic excitation spectrum strongly resembles the experimental data, compare Figs. 9 and 10c. Hence, a nano phase separation model for a charge disproportionated sample is able to simulate the elastic and inelastic neutron scattering measurements in RSrNiO 4 . In our model, nano phase separation is not driven by carrier doping but by disorder within a completely charge disproportionated sample which inevitably leads to the creation of nanoscopic regions with large exchange interactions J (red areas in Fig. 10a) interspersed with regions with small exchange interactions J ′ (blue areas in Fig. 10a) and regions with small exchange interactions that can be neglected (black areas in Fig. 10a). Such a scenario might be also applicable to other systems with checkerboard charge order with a certain degree of disorder. Indications for nanoscale phase separation have recently also been reported for the high-temperature superconducting cuprates 37,38 which points to a possibly significant role of these effects for the physical properties of these systems.

Conclusion
We succeeded in synthesizing stoichiometric Ni 3+ 2-1-4 nickelate single crystals by the high-pressure floating zone technique. We found that this system prefers to undergo a charge disproportionation, although it crystallizes in a tetragonal structure with strongly distorted oxygen environment of the Ni ions that would allow for a Jahn-Teller effect with an occupation of the 3z 2 − r 2 orbital. This is the more remarkable when one considers the fact that the effective crystal field from theory places the 3z 2 − r 2 orbital ∼ 0.26 eV below the x 2 − y 2 which is a large energy difference. Hence, the experimental results suggest that the negative charge transfer character of the Ni 3+ ion in these oxides provides a very strong drive for charge disproportionation to occur of the type 2 · Ni 2+ L → Ni 2+ + Ni 2+ L 2 , leading to the formation of the observed checkerboard magnetic superstructure and overcoming the tendency to Jahn-Teller distortion. Alternatively, one might also think in terms of Fermi surface effects in view of the close proximity to a metallic state in this 2-1-4 material. In any case, it is now fully understandable that the 1-1-3 nickelates undergo charge disproportionation since the cubic crystal structure does not provide a priori a 3z 2 −r 2 /x 2 −y 2 crystal field splitting that would have helped the Jahn-Teller stabilization. RSrNiO 4 can thus serve as a bench mark system for further theory development to describe properly the role of holes in the oxygen band. Finally, the magnetic correlations and the magnetic excitation spectra in these a b c d www.nature.com/scientificreports/ nickelates can be explained by a nano phase separation model which is different from usual magnets with long range magnetic order and points to the possible significance of such nano phase separation for understanding the physical properties of these materials.

Methods
Single crystals of NdSrNiO 4 , La 2/3 Y 1/3 SrNiO 4 and Nd 2 NiO 4 were grown by the floating zone technique using a high pressure mirror furnace from Scidre. Here, we studied two Ni 3+ systems: NdSrNiO 4 and La 2/3 Y 1/3 SrNiO 4 . The former one because for R=Nd there is no undesirable mixture of Ni − L 3 and La − M 4 edges in the XAS spectra, the latter one because Nd has an undesirable large magnetic moment that might easily overshadow the Ni contribution. We note that the pure La system was more difficult to grow as large single crystals than the Y-substituted compound. The Nd 2 NiO 4 was grown to serve as a Ni 2+ reference system with the same crystal structure as the NdSrNiO 4 and La 2/3 Y 1/3 SrNiO 4 . In order to perform floating zone growth for Nd 2−x Sr x NiO 4 ( x = 0.9 , 1.0 and 1.1) and La 2/3 Y 1/3 SrNiO 4 starting materials of R 2 O 3 , SrCO 3 and NiO were mixed in appropriate ratios and ground thoroughly in an agate mortar followed by sintering steps at 1000 • C−1200 • C in air for several days with intermediate grindings. The obtained composition was then pressed under ∼ 100 MPa hydrostatic pressure into a rod of about 6-8 mm in diameter and 120(20) mm in length, which was subsequently sintered at 1000 • C in air. The floating zone growth was performed with a growth rate of 2 mm/h in a flowing O 2 atmosphere ( ∼ 0.2 l/min ) at a pressure of 100 bar, 120 bar and 150 bar for x = 0.9 , 1.0 and 1.1, respectively. Phase purity was confirmed by powder X-ray diffraction (XRD) measurements on ground single crystals with a 2θ step of 0.005 o using Cu K α1 radiation of a laboratory X-ray source. It was possible to describe the crystal structure properly with space group I4/mmm for all grown samples. The refined lattice parameters and the resulting unit cell volumes are listed in Table 1. The linear change of the lattice parameters as a function of x further confirms the successful substitution of the Nd atom by Sr atom within the Nd 2−x Sr x NiO 4 series. This is also corroborated by inductively coupled plasma optical emission spectrometry (ICP-OES) measurements, see the value of x(ICP) in Table 1. Moreover, thermogravimetric measurements confirmed that the oxygen content is close to the nominal value, see Table 1. The Ni − L 2,3 X-ray absorption spectroscopy (XAS) measurements have been performed at the 11A beamline of the National Synchrotron Radiation Research Center (NSRRC), Taiwan. A NiO single crystal was measured simultaneously for energy calibration. The photon energy resolution at the Ni L 2,3 edges was set at 0.3 eV. The spectra were recorded at 300 K using the total electron yield method.
Unpolarized elastic neutron scattering measurements were performed on the IN8 spectrometer at the Institut Laue Langevin (ILL) using PG monochromator and PG analyzer with fixed k f = 2.662Å −1 and two PG filters for the suppression of higher order contaminations. Unpolarized inelastic neutron scattering measurement were performed on the IN8 spectrometer at the ILL and on the MERLIN time-of-flight (TOF) spectrometer at ISIS 39,40 . For the IN8 measurement, doubly focused Cu monochromator and PG analyzer were used with two PG filters. The MERLIN experiment has been performed in repetition-rate multiplication (RRM) mode. Longitudinal polarized elastic neutron scattering measurements were performed on the IN12 spectrometer at the Institut Laue-Langevin (ILL) equipped with double focusing pyrolythic graphite (PG) monochromator and Heusler analyzer. The beam was polarized by a transmission polarizer in the neutron guide. The monochromator was set for a wave vector of 2.25Å −1 , and, a velocity selector was used for suppression of higher order contamination. The flipping ratio amounts to ∼ 22.2 . For the polarization analysis, the x axis is defined along the direction of Q, the y axis is perpendicular to Q and within the scattering plane, and the z axis is perpendicular to the scattering plane. Note, that we used the tetragonal setting for all our neutron measurements with lattice constant a = b ∼ 3.79 Å and c ∼ 12.4Å.
Resonant soft X-ray diffraction at the Nd 3d → 4f ( M 5 , 1000 eV) and Ni 2p → 3d ( L 3 , 853.2 eV) resonances have been measured at the UE56/2-PGM1 beam line at BESSY II. The data were recorded in horizontal scattering geometry with the X-rays linear polarized in the scattering plane ( π-polarization). The scattered photons were detected with an in-vacuum CCD camera.