# Doping-Dependent Structural Phase Transitions, Magnetic Order, and Excess Oxygen Behavior in Nd2−xSr2− xNiO4+δ

Sumit Ranjan Maity, Monica Ceretti, Lukas Keller, Jürg Schefer, Uwe Stuhr, Antonio Cervellino, Werner Paulus

► To cite this version:

Sumit Ranjan Maity, Monica Ceretti, Lukas Keller, Jürg Schefer, Uwe Stuhr, et al., Doping-Dependent Structural Phase Transitions, Magnetic Order, and Excess Oxygen Behavior in Nd2−xSr2− xNiO4+δ. Inorganic Chemistry, In press, 10.1021/acs.inorgchem.5c03882. [https://doi.org/10.1021/acs.inorgchem.5c03882](https://doi.org/10.1021/acs.inorgchem.5c03882). [arXiv:2106.09374](https://arxiv.org/abs/2106.09374)

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Distributed under a Creative Commons Attribution 4.0 International License
Doping-Dependent Structural Phase Transitions,
Magnetic Order, and Excess Oxygen Behavior in
Nd$_{2-x}$Sr$_x$NiO$_{4+δ}$

Sumit Ranjan Maity,¹,²,³* Monica Ceretti,⁴ Lukas Keller,² Jürg Schefer,² Uwe Stuhr,²

Antonio Cervellino,⁵ Werner Paulus⁴

¹Japan Synchrotron Radiation Research Institute (JASRI), SPring-8, Hyogo 679-5198, Japan

²PSI Center for Neutron and Muon Sciences, Villigen CH-5232 PSI, Switzerland

³Department of Quantum Matter Physics, University of Geneva, 24, Quai Ernest Ansermet, CH-1211 Genève 4, Switzerland

⁴ICGM, Univ Montpellier, CNRS, ENSCM, 34293 Montpellier, France

⁵Swiss Light Source, Paul Scherrer Institut, Villigen CH-5232, Switzerland

*Corresponding Author: sumit050491@spring8.or.jp
**Abstract**

Chemical doping in transition metal oxides drives a complex interplay among lattice, charge, and spin degrees of freedom. Herein, we investigate the structural and magnetic evolution of Nd₂ₓSrₓNiO₄+δ (0 ≤ x ≤ 0.7), a layered nickelate system with mixed-valence Ni²⁺/Ni³⁺ ions and tunable oxygen non-stoichiometry (δ), using high-resolution synchrotron X-ray and neutron diffraction, complemented by macroscopic measurements over a broad temperature range. High-temperature solid-state synthesis in air promotes oxygen interstitial incorporation, yielding hyper-stoichiometric compounds (δ > 0). For Nd₂NiO₄.23 (x = 0), excess oxygen atoms exhibit long-range ordering, giving rise to incommensurate structural modulations and a sequence of order-order transitions persisting up to 800 K. Sr substitution disrupts this ordering via Coulombic effects and suppresses both oxygen uptake and NiO₆ octahedral tilting. Stripe-like spin ordering emerges at x = 0.25, 0.33, and 0.5. The x = 0.5 sample exhibits magnetic incommensurability (ε ≈ 0.44) below 95 K, indicative of charge discommensuration in NiO₂ planes. Unlike the oxygen-doped (x = 0) phase, Sr-doped samples show quasi-two-dimensional magnetism, with in-plane and out-of-plane magnetic correlation lengths of 114(2) Å and 16(3) Å, respectively. These results reveal strong coupling among chemical doping, crystal structure, oxygen ordering, and magnetism in layered nickelates.
Introduction

The discovery of high-temperature superconductivity in hole-doped cuprates has attracted
extensive interest in structurally related Ruddlesden-Popper rare-earth nickelates, particularly
those with K₂NiF₄-type frameworks ¹⁻¹³. These layered compounds, comprising NiO₂ planes
separated by Ln₂O₂ (Ln = La, Pr, Nd) rock salt bilayers, exhibit a rich interplay among lattice,
charge, and spin degrees of freedom (cf. Figure 1)¹⁴. The structural strain originating from the
mismatch between Ln-O and Ni-O bond lengths is accommodated by the high-temperature
tetragonal (HTT) phase, mostly observed at elevated temperatures. Upon cooling, the strain is
relieved via cooperative tilting of the NiO₆ octahedra, giving rise to a sequence of low-temperature
phases including orthorhombic (LTO), less orthorhombic (LTLO), and low-temperature tetragonal
(LTT) structures¹⁵,¹⁶. In disordered systems or those with short-range octahedral tilt correlations,
the average structure is described by an orthorhombic LTO' phase¹⁷. These structural transitions
are intimately coupled to the electronic and magnetic properties, making these systems an excellent
platform for exploring correlated electron phenomena.

Undoped layered nickelates exhibit commensurate antiferromagnetic ordering within the NiO₂ planes, characterized by long-range alignment of the Ni²⁺ (S = 1) spins¹⁸. Upon hole doping, mixed-valence Ni²⁺/Ni³⁺ states are introduced, fundamentally altering the electronic interactions. This can be achieved through the substitution of rare-earth ions (Ln³⁺) with larger alkaline earth metals such as Sr²⁺, the incorporation of excess oxygen, or a combination of both. These doped holes tend to segregate into periodically spaced, quasi-one-dimensional domains within the NiO₂ planes, forming linear regions enriched in Ni³⁺-like character and separating regions predominantly composed of Ni²⁺ ions²,⁴,⁶. At low temperatures, these Ni²⁺ spins establish long-range antiferromagnetic order. Still, the hole-rich stripes interrupt this order and act as antiphase
domain boundaries, inducing a π-phase shift in spin orientation across each stripe. This interplay
results in a coupled spin-charge stripe pattern, with the periodicity of both charge and spin
modulations evolving systematically with hole concentration¹⁹,²⁰. As doping increases, the stripe
pattern becomes denser and evolves toward a two-dimensional checkerboard charge ordering near
$x \approx 0.5$, where the system can achieve a more isotropic arrangement of charges. Notably, Sr
substitution and oxygen intercalation influence structural and electronic ordering in distinct ways:
Sr²⁺ randomly replaces Ln³⁺ and introduces relatively subtle lattice perturbations. In contrast,
interstitial oxygen occupies tetrahedral sites within the Ln₂O₂ layers. The presence of these excess
oxygen atoms exerts strong Coulomb repulsion on adjacent apical oxygen atoms, significantly
distorting the NiO₆ octahedra and contributing to the emergence of complex structural phase
diagrams together with oxygen ordering²¹⁻²³.

In La₂NiO₄+δ, interstitial oxygen atoms were reported to form a one-dimensional (1D) staging
order at δ ≈ 0.11, transitioning to three-dimensional (3D) ordering near δ ≈ 0.125². A correlation
has been proposed between the emergence of 3D oxygen order and charge stripe formation. In
contrast, Nd₂NiO₄+δ exhibits markedly different behavior. Both moderately (δ ≈ 0.1) and highly (δ
≈ 0.23) oxygen-doped samples display 3D incommensurate oxygen ordering already above room
temperature, yet no evidence of charge stripe order was found down to 2 K²²⁻²⁴. Interestingly, a
long-range incommensurate magnetic order was reported in Nd₂NiO₄·₂₃ with a propagation vector
of (0.64, 0, 0.5)²³. This doubling of the magnetic unit cell along the c-axis points to enhanced inter-
plane magnetic correlations induced by the presence of excess oxygen atoms. In moderately doped
Nd₂NiO₄·₁₀, the periodicities of magnetic and oxygen orderings perfectly coincide below the Néel
temperature (T_N ≈ 50 K), highlighting a strong coupling between lattice and spin degrees of
freedom. These differences underscore the importance of A-site chemistry and oxygen
intercalation in dictating structural correlations and electronic phase behavior. Moreover, the smaller ionic radius of Nd³⁺ compared to La³⁺ facilitates higher interstitial oxygen uptake, leading to structural and magnetic behaviors distinct from those in La-based analogs.

Here, we investigate how structural and magnetic correlations, along with charge ordering, evolve with Sr doping in the Nd₂ₓSrₓNiO₄+δ system over a wide temperature range. Using synchrotron X-ray and neutron diffraction, we demonstrate that Sr substitution disrupts the long-range oxygen order present in undoped Nd₂NiO₄.₂₃, driving a transition from three-dimensional magnetic order to quasi-two-dimensional stripe magnetism in the isoelectronic Nd₁.₅Sr₀.₅NiO₄.₀₆ system. Magnetic susceptibility and specific heat measurements, supported by crystal field calculations, reveal the significant contribution of Nd³⁺ ions and emphasize the crucial role of A-site chemistry and oxygen content δ in determining the magnetic properties of the system at low temperatures.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** (a) Schematic illustration of the K₂NiF₄-type layered structure of Nd₂ₓSrₓNiO₄+δ, highlighting the NiO₆ octahedra and the rock salt-type Nd₂O₂ layer. The positions of apical (Oₐp),</figcaption></figure>
equatorial (O<sub>eq</sub>), and interstitial (O<sub>int</sub>) oxygen atoms are indicated. O<sub>int</sub> atoms are coordinated in a tetrahedral interstitial site inside the rock-salt layer. Octahedral tilt patterns associated with various structural phases: (b) high-temperature tetragonal (HTT, *F*4/*mmm*), (c) low-temperature orthorhombic (LTO, *B*mab), and (d) low-temperature tetragonal (LTT, *P*4<sub>2</sub>/ncm), with arrows showing octahedral tilting directions.

## Methods

Polycrystalline Nd<sub>2-x</sub>Sr<sub>x</sub>NiO<sub>4+δ</sub> (NSNO) samples were synthesized via the conventional solid-state reaction method at high temperatures in air, following procedures reported previously<sup>25</sup>. Stoichiometric amounts of high-purity Nd<sub>2</sub>O<sub>3</sub>, SrCO<sub>3</sub>, and NiO powders (99.9%, Alfa Aesar) were used as starting materials, with a 2-mol% excess of NiO added to compensate for its volatility during high-temperature processing. Before use, Nd<sub>2</sub>O<sub>3</sub> was pre-annealed at 1173 K overnight to remove moisture and hydroxide impurities. The powders were thoroughly mixed, ground, and pressed into pellets, then calcined in air at 1523 K for 12 hours in alumina crucibles. This process was repeated 3-4 times with intermediate grinding and pelletizing to ensure phase purity and homogeneity. The crystalline quality of the resulting powders was verified by laboratory-based X-ray powder diffraction (XRPD). Additionally, a single crystal of the x = 0.5 composition was grown using the optical floating zone method, as described in an earlier report<sup>25</sup>. Thermogravimetric analyses (TGA) were conducted to determine the overall oxygen non-stoichiometry (δ) of the NSNO samples. A small amount of powder was reduced in a 5% H<sub>2</sub> / 95% He gas mixture using a NETZSCH Jupiter STA 449C thermobalance over the temperature range of 300-1400 K. All samples were stored under ambient conditions for over six months before diffraction measurements at large-scale facilities. No unexpected or unusually high safety hazards
were encountered in the course of this work. Standard laboratory procedures and safety protocols
were followed throughout all experimental activities.

Synchrotron X-ray powder diffraction (SXRPD) measurements were carried out between 5 and
800 K at the Materials Sciences (MS) beamline X04SA of the Swiss Light Source (SLS), Paul
Scherrer Institute (PSI), using a short wavelength of 0.6208(1) Å<sup>26</sup>. Powdered samples were loaded
into 0.3 mm diameter glass capillaries and continuously rotated during data acquisition to
minimize preferred orientation effects. A hot gas blower was used for high-temperature
measurements, while liquid helium cooling enabled low-temperature studies. Temperature-
dependent powder neutron diffraction (PND) patterns were collected on the DMC diffractometer
at SINQ, PSI, using a neutron wavelength of λ = 2.458(1) Å<sup>26,27</sup>. Approximately 5 grams of powder
were loaded into vanadium cans and cooled in a <sup>4</sup>He cryostat. Data were collected at various
temperatures between 1.7 and 300 K, with measurement durations of 6-8 hours, covering a Q-
range of 0.44-3.7 Å<sup>-1</sup>. All powder patterns were analyzed by the Rietveld refinement method using
the FullProf Suite<sup>28,29</sup>. Single-crystal neutron diffraction reciprocal plane maps on the
Nd<sub>1.5</sub>Sr<sub>0.5</sub>NiO<sub>4.06</sub> sample were performed at different temperatures on DMC at the SINQ, PSI. X-
ray absorption was estimated with a packing fraction of ~0.6, while neutron absorption was
considered negligible. Crystal structures were visualized using VESTA<sup>30</sup>. Temperature-dependent
single-crystal neutron diffraction measurements were performed using the thermal triple-axis
spectrometer EIGER at SINQ, PSI<sup>31,32</sup>. Effective horizontal collimations of 40'-40'-40'-40' from
the reactor to the detector were used together with a pyrolytic graphite filter to eliminate neutrons
at shorter harmonic wavelengths. Neutrons with an energy of 14.68 meV were selected.

Zero-field-cooled (ZFC) DC magnetic susceptibility measurements were carried out using a
superconducting quantum interference device (SQUID) magnetometer (MPMS, Quantum Design)
at the Laboratory for Scientific Developments and Novel Materials (LDM), PSI. Measurements
were performed upon slow warming from 2 to 330 K under an applied magnetic field of 1 T.
Specific heat was measured on a ~10 mg pellet between 2 and 300 K using a Physical Property
Measurement System (PPMS, Quantum Design), also at LDM, PSI.

**Results and Discussions**

**Thermogravimetric analysis**

Overall oxygen stoichiometry was analyzed by thermogravimetric analysis (TGA), as shown in
Figure 2. All as-prepared NSNO samples contain excess oxygen accommodated in the interstitial
sites of the rock-salt layer (cf. Figure 1(a)), but their reduction behavior depends strongly on Sr
content. The complete reduction corresponds to decomposition of the primary phase into (1 -
x/2)Nd₂O₃, (x)SrO and Ni. The oxygen content δ was determined using the relation: δ =
$$ \frac{wM-16(1+x/2)}{16(1-w)} $$
where M = (2-x)Mₙd + xMₖr + Mₙi + 4MO and w is the fractional mass loss between
the starting and final points of the TGA curve. For compositions with x < 0.5, oxygen loss proceeds
in two steps. In the undoped (x = 0) and lightly doped (x = 0.1) samples, the first reduction step
(578-580 K) corresponds to the removal of interstitial oxygen and the formation of stoichiometric
Nd₂NiO₄.₀ and Nd₁.₉Sr₀.₁NiO₄.₀ phases, followed by complete decomposition at higher
temperatures (790-830 K). By contrast, the nominally oxygen-rich samples with x = 0.25 and x =
0.33 exhibit larger mass changes in the first step (602-645 K), which lead not to stoichiometric
214 phases but to oxygen-deficient compositions Nd₁.₇₅Sr₀.₂₅NiO₃.₈₆ and Nd₁.₆₇Sr₀.₃₃NiO₃.₇₸, prior
to final decomposition at 877-912 K. In these reduced phases, oxygen vacancies are created at the
equatorial sites of the NiO₆ octahedra as reported previously using neutron diffraction experiments
for Nd₁.₈Sr₀.₂NiO₃.₇₂³³,³⁴. For Sr-rich samples (x ≥ 0.5), the TGA curves display a nearly continuous
oxygen release beginning at 635 K (x = 0.5) and 656 K (x = 0.7), with no stable plateau, leading
to strongly oxygen-deficient metastable phases Nd₁.₅Sr₀.₅NiO₃.₅₆ and Nd₁.₃Sr₀.₇NiO₃.₄₂ before
decomposition at 946-975 K. The corresponding δ values derived from the total weight loss are
(cf. Table 1) 0.23(2) for x = 0, 0.09(2) for x = 0.1, 0.08(2) for x = 0.25, 0.05(2) for x = 0.33, 0.04(2)
for x = 0.5, and 0(2) for x = 0.7. The inset of Figure 2 illustrates the evolution of δ as a function of
x. The trend shows a monotonic decrease in δ with increasing x, indicating that Sr doping
suppresses excess oxygen uptake and stabilizes the oxygen stoichiometry. Furthermore, the
observed shift of the oxygen release to higher temperatures with increasing x suggests enhanced
thermal and structural stability of the K₂NiF₄-type tetragonal phase at room temperature in the Sr-
rich regime.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** TGA curves showing weight loss (%) of Nd₂-xSrxNiO₄+δ powder samples (x = 0-0.7) as a function of temperature in a reducing atmosphere (5% H₂ / 95% He) with a heating rate of 5 K/min. Arrows indicate the onset temperatures of significant weight loss events, attributed to the</figcaption></figure>
release of interstitial oxygen. The inset shows the monotonic decrease of excess oxygen content ($\delta$) with increasing Sr content, estimated from total weight loss. No correction was applied; thus, minor apparent weight increase below ~600 K may not be related to actual oxygen uptake.

## Synchrotron X-ray powder diffraction measurements

**Previously reported structures.** Before presenting our synchrotron X-ray powder diffraction (SXRPD) data, we briefly summarize the known space group symmetries associated with various structural phases in $K_2NiF_4$-type compounds as a function of Sr doping ($x$), oxygen non-stoichiometry ($\delta$), and temperature ($T$). For $Ln_2NiO_{4.00}$ phases, the high-temperature (HTT) phase is tetragonal with space group $F4/mmm$ (cf. Figure 1(b)), primarily observed at elevated temperatures, and characterized by the absence of both structural strain and long-range octahedral tilt ordering. Upon cooling, structural strain arising from lattice mismatch between the $NiO_2$ and $Ln_2O_2$ layers induces cooperative octahedral tilting, resulting in symmetry-lowering phase transitions. These low-temperature phases are typically described by subgroups of the HTT phase: *Bmab*, *P42/ncm*, *Pccn*, and *Fmmm*$^{15,16,35}$. In the low-temperature orthorhombic (LTO) phase (*Bmab*), the $NiO_6$ octahedra tilt about the [100] axis (cf. Figure 1(c)) while adjacent rows along [010] exhibit opposite tilt senses, resulting in an antiferrodistortive pattern. Such a tilt pattern leads to the splitting of (hkl) reflections with $h \neq k$ (e.g., (200)/(020), (515)/(155)) and the appearance of superlattice peaks such as (212) and (012), indicating coherent tilt ordering. In the low-temperature tetragonal (LTT) phase (*P42/ncm*), the tilt axis alternately rotates between [110] and [-110] directions in adjacent layers along the c-axis (cf. Figure 1(d)), resulting in the absence of reflection splitting for $h \neq k$ but retaining superlattice peaks like (212) and (012). The less orthorhombic (LTLO) phase (*Pccn*) is characterized by a tilt axis between [100] and [110], yielding reduced
orthorhombicity compared to the LTO phase. When the long-range coherence of octahedral tilting is suppressed by A-site disorder or by long-range excess oxygen ordering, the average structure can be described as low-temperature orthorhombic (LTO') with space group *Fmmm*. However, recent studies have clarified that the true average symmetry of this phase is better represented by a low-temperature monoclinic (LTM) phase with space group *F*112/*m*, as observed for highly oxygen-doped samples¹⁷. The LTM phase closely resembles the LTO' phase, with the only distinguishing feature being the splitting of the (*hhl*)-type reflections.

**Synchrotron diffraction results.** Figure 3(a) displays SXRPD patterns collected at 300 K, while selected reflections in Figures 3(b)-(c) highlight features indicative of structural phase transitions. All samples exhibit a dominant single phase, with minor NiO impurities (<1 mol%) observed for *x* = 0.1, 0.25, and 0.33. No macroscopic phase separation is detected across compositions or temperatures. Rietveld refinements of all SXRPD data were performed with *δ* content fixed as determined from TGA data.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** (a) Representative room temperature SXRPD patterns of Nd₂ₓSrₓNiO₄+δ samples showing the structural evolution with increasing Sr content. (b) Magnified view of the (200)/(020)</figcaption></figure>
and (c) (220) reflections, illustrating the splitting observed for low *x*, consistent with orthorhombic or monoclinic symmetry, and their merging at higher *x*, indicative of a high-temperature tetragonal (HTT) phase.

*Nd₂NiO₄.₂₃*. The SXRPD pattern collected at room temperature for *x* = 0 is shown in the bottom panels of Figures 3(a)-(c), revealing clear splitting of the (200) and (020) Bragg reflections, along with subtle splitting of (220)-type peaks. These features confirm that the average crystal structure of *Nd₂NiO₄.₂₃* is monoclinic, well described by the *F*112/*m* LTM phase, which accounts for both peak positions and the observed splitting¹⁷,³⁶. In addition, superstructure reflections, approximately three orders of magnitude weaker than the main Bragg peaks, are observed (Figure S1 in the supporting information). These are not due to a simple expansion of the unit cell but indicate a complex, three-dimensional modulation of excess oxygen ordering as discussed in ref. ²². Rather than being randomly distributed, the interstitial oxygen atoms adopt periodic arrangements within the rock salt layers, minimizing electrostatic repulsion and lattice strain. This ordering induces local structural distortions, yielding displacements of nearby apical oxygen atoms and tilting of *NiO*₆ octahedra as shown in Figure S1(b), giving rise to superstructure reflections at positions not indexed by the basic lattice. As we have previously shown, the excess oxygen ordering remains unchanged down to 5 K for this compound, with all superstructure reflections fixed in position, indicating a stable oxygen-ordering pattern²². In contrast, upon heating above 300 K, the compound undergoes a series of first-order structural transitions involving distinct 3D modulated phases related to oxygen ordering, while retaining monoclinic symmetry up to 800 K²². At least five different modulated phases were identified in this temperature range, each characterized by well-defined translational periodicities exceeding 100 Å. These transitions exhibit sharp onset
temperatures and fast-ordering kinetics. The corresponding modulation vectors, referenced to the LTM cell, were determined using combined powder and single-crystal X-ray diffraction techniques²². Despite these *T*-dependent changes, the underlying Nd₂NiO₄+δ framework remains preserved. The results suggest a series of topotactic rearrangements of the oxygen sublattice, with multiple energetically similar configurations that the system can readily access. Above 800 K, a transition to the HTT phase occurs, marked by the disappearance of (*h*00)/(*0*k0) peak splitting and the loss of all superstructure reflections. Nonetheless, some broad features persist beyond the transition, indicating the presence of short-range structural correlations, even in the nominally disordered high-temperature phase.

*Nd*₁.₉*Sr*₀.₁*NiO*₄+δ. At room temperature, the SXRPD pattern of the *x* = 0.1 (δ = 0.11) sample reveals a clear splitting of the (*h*00) and (*0*k0) Bragg reflections, while the (*hh*0) peaks remain unsplit. This diffraction signature, along with a complete absence of any superstructure reflections such as (012) and (212), is characteristic of the *F*mmm LTO'*-type phase, which features incoherent tilting of NiO₆ octahedra. Importantly, no superstructure peaks related to the excess oxygen ordering are observed down to 5 K (Figure S1 in the supporting information), and all reflections remain indexable with a K₂NiF₄-type *F*mmm unit cell, indicating the absence of long-range interstitial oxygen ordering across the entire temperature range studied. Such a structural feature suggests an increased disorder in the excess oxygen ordering introduced by Sr-doping. This is particularly striking because the *δ* content in this sample is comparable to that in oxygen-rich La₂NiO₄+δ (0.07 ≤ *δ* ≤ 0.11), isostructural La₂-xSrxCuO₄+δ (0 ≤ *x* ≤ 0.09), and Nd₂NiO₄.₁₁, where both 1D and 3D interstitial oxygen ordering have been previously reported at low temperatures²,²⁴,³⁷. The suppression of oxygen ordering in NSNO, despite similar *δ*, likely arises from the Coulomb repulsion between O²⁻ interstitials and the less positively charged Sr²⁺ ions
(compared to Nd³⁺), which destabilizes ordered arrangements, an effect consistent with prior
observations in the LSNO system¹⁶. Thus, the x = 0.1 sample presents a unique case in which even
modest Sr doping is sufficient to disrupt long-range excess oxygen ordering through electrostatic
effects, while simultaneously promoting incoherent NiO₆ octahedral tilting due to the presence of
disordered excess oxygen. Upon heating, the compound undergoes a first-order structural phase
transition from the LTO' phase to the HTT phase at approximately 673 K (Figures S2 and S3 in
the Supporting Information). A broad biphasic region characterized by the coexistence of LTO'
and HTT phases is observed in the temperature range 473 K ≤ T ≤ 623 K. Representative Le Bail
fits of the SXRPD data collected between 5 and 673 K are shown in Figures S4(a) and S5(a). The
coexistence of both phases is more clearly evidenced by the simultaneous presence of
orthorhombic and tetragonal (h00)/(0k0) reflections at nearly identical 2θ positions (Figures S4(b)
and S5(b)). Figure S6(a) displays the temperature evolution of the (200) and (020) reflections,
revealing the emergence of the HTT phase at T ≈ 473 K. Although a small, unmodeled intensity
valley persists between the orthorhombic peaks below 473 K down to 5 K, it is likely not indicative
of a residual HTT phase since it remains essentially temperature-independent. The refined
temperature-dependent weight fraction of the tetragonal phase is presented in Figure S6(b). The
transition may be accompanied by a gradual loss of excess oxygen at elevated temperatures,
consistent with prior observations in Nd₂NiO₄.₂₃. The temperature dependence of lattice
parameters, obtained from Rietveld refinements, is shown in Figure S7 of the supporting
information. From 5 K to 473 K, the orthorhombic lattice parameters (a, b, c) and unit cell volume
(V) increase smoothly with temperature. Concurrently, the orthorhombicity, defined as
2(b - a)/(b + a), gradually decreases. As the system enters the biphasic regime, both a and b
parameters continuously converge with temperature, and the orthorhombicity sharply drops to zero
near the transition temperature, evidencing a tetragonal symmetry. Interestingly, the orthorhombic
and tetragonal phases have nearly identical c-axis length within the coexistence region.

<figure><img src="image_5.png" /></figure>

**Figure 4.** Structural analysis of samples with Sr content *x* = 0.5. (a) Selected regions of the refined SXRPD data at 5 K, fitted using the *Bmab* structural model (*R*<sub>p</sub>=7.14%, *R*<sub>wp</sub>=8.12%). Observed, calculated, and difference profiles, along with Bragg peak positions, are shown. Insets display the temperature dependence of the (012) superstructure peak intensity fit with a power law (red line, see text) and a close-up of the orthorhombic (200) and (020) peak splitting. (b) Refined neutron powder diffraction data using the identical structural model as in (a) at 1.5 K, highlighting a few incommensurate stripe magnetic reflections (marked with stars). (c) Evolution of lattice parameters *a*, *b*, and *c* as a function of *x* at room temperature, illustrating a transition from orthorhombic distortion at low *x* to tetragonal symmetry above *x* ≈ 0.25. (d) Sr-dependence of selected bond lengths: Ni-O<sub>eq</sub>, Ni-O<sub>ap</sub>, and Nd/Sr-O<sub>ap</sub>.
*Nd₂ₓSrₓNiO₄+δ* (0.25 ≤ x ≤ 0.7). For compositions with x ≥ 0.25, the room-temperature SXRPD patterns show no splitting of the (h00) and (0k0) reflections and no superstructure peaks, indicating the stabilization of the *F*4/*mmm* HTT phase even at ambient conditions (cf. Figures 3(a)-(c) and Figure S1 of the supporting information). This tetragonal stabilization is due to the complete release of structural strain at this composition. Similarly to the x = 0.1 composition, no long-range ordering of interstitial oxygen atoms is observed in these samples across the entire temperature range studied, despite a significant excess oxygen content. Among all compositions examined, only the x = 0.5 sample undergoes a structural phase transition from the HTT to the LTO phase upon cooling. This transition is evidenced by the splitting of the (h00)/(0k0) reflections (inset of Figure 4(a)) and by the emergence of superstructure peaks such as (012) (Figures S8(b) and S8(c)), indicative of coherent tilting of the NiO₆ octahedra. The temperature dependence of the superlattice reflection intensity (inset of Figure 4(a)) was analyzed using the critical form $A*[(T_c/T)/T_c]^{2β}$ to determine the transition temperature and critical behavior. The fit, restricted to the 120-210 K range to focus on the critical regime while excluding the low-temperature saturation region, yields a critical temperature T_c = 221(8) K and a critical exponent β = 0.34(2). The obtained β value is consistent with three-dimensional universality classes (3D Ising/XY), in agreement with a second-order HTT–LTO structural transition similar to that reported for related layered cuprates³⁸⁻⁴⁰. A representative Rietveld refinement of the SXRPD data collected at 5 K for x = 0.5, using the LTO structural model, is shown in Figure 4(a). Corresponding refined structural parameters at both 300 K and 5 K are provided in Table S1 of the Supporting Information. The emergence of the LTO phase in this Sr-doped system is attributed to an increasing lattice mismatch between the Nd₂O₂ and NiO₂ layers at low temperatures, which is relieved through octahedral distortions, analogous to the behavior in undoped Nd₂NiO₄.₀¹⁸. As shown in Figure S9 of the
supporting information, the lattice parameters for $x = 0.5$ evolve distinctly with temperature: between 5 K and 50 K, both *a* and *b*-axes, along with orthorhombicity increase, while the *c*-axis remains nearly constant. From 50 K to 210 K, both *a* and *c*-axes increase while the *b*-axis decreases sharply, resulting in a rapid suppression of orthorhombicity as the system approaches the HTT phase.

Our findings reveal a clear composition-dependent structural phase evolution in Nd$_{2-x}$Sr$_x$NiO$_{4+δ}$ at room temperature. For $0 < x < 0.1$, the structure undergoes a transition from the low-temperature monoclinic (LTM) phase to the *Fmmm* LTO' phase. With increasing *x*, a subsequent transition from LTO' to the high-temperature tetragonal (HTT) phase is observed, which remains stable for $x \ge 0.25$. This trend, along with the suppression of long-range interstitial oxygen ordering, aligns well with previous reports¹⁶. The refined room-temperature lattice parameters and bond lengths as a function of *x* are presented in Figures 4(c) and 4(d), respectively. The composition dependent lattice changes are very similar to those reported for the Ca-doped La$_{2-x}$Ca$_x$CoO$_{4+δ}$⁴¹. In the low-doping regime ($x \le 0.33$), the contraction of the in-plane lattice parameters *a* and *b* is governed primarily by the removal of interstitial oxygen rather than by Ni oxidation. The decrease in *δ* relaxes octahedral tilts and reduces local strain, allowing the Ni-O$_{eq}$ bond lengths to contract. Between $x = 0.33$ and 0.50, *δ* remains nearly constant, and further contraction is instead driven by oxidation of Ni²⁺ toward Ni³⁺, which pulls the in-plane oxygen inward and shortens Ni-O$_{eq}$ bonds. For $x \ge 0.50$, the system stabilizes due to competing effects: expansion from Sr²⁺ substitution on the Nd³⁺ site, contraction from the change in Ni average valence (from ~2.62+ to ~2.72+), and the complete loss of interstitial oxygen. The continuous increase in the Nd-O$_{ap}$ bond length across the series can be attributed to the progressive replacement of Nd³⁺ (1.16 Å in 9-fold coordination) with the larger Sr²⁺ (1.31 Å in 9-fold coordination) cation, which weakens the Nd/Sr-O$_{ap}$ attraction. The
observed increase in Ni-O<sub>ap</sub> bond length up to x ≈ 0.5 arises from removal of oxygen interstitials, which otherwise exert steric and Coulombic strain on the apical oxygen, forcing it slightly inward toward Ni. The continuous increase of the c-axis up to x = 0.5 is therefore a combined result of Sr<sup>2+</sup> substitution and outward relaxation of O<sub>ap</sub> as δ decreases. The sudden decrease in c observed at x = 0.7 correlates with the complete loss of interstitial oxygen atoms and the subsequent inward shift of apical oxygen toward Ni, leading to shorter Ni-O<sub>ap</sub> and longer Nd/Sr-O<sub>ap</sub> bonds. Thus, the structural response of Nd<sub>2-x</sub>Sr<sub>x</sub>NiO<sub>4+δ</sub> can be understood as a sequential interplay of oxygen interstitial removal, progressive Ni oxidation, and A-site substitution, highlighting how different mechanisms govern different regions of the phase diagram.

To evaluate the thermal expansion behavior of the prepared samples, the temperature dependence of the unit-cell volume was fitted with a quadratic function V(T) = V₀ + aT + bT², using fitting ranges of 5-873 K for x = 0, 5-723 K for x = 0.1, and 5-300 K for the other compositions. From the derivative dV/dT = a + 2bT, the volumetric thermal expansion coefficient was obtained as α<sub>v</sub>(T)=(1/V)(dV/dT). The determined values are summarized in Table S2 of the Supporting Information. The average α<sub>v</sub> values over the respective temperature ranges lie between 4.94×10<sup>-5</sup> and 2.08×10<sup>-5</sup> K<sup>-1</sup>, with a systematic reduction up to x = 0.5 followed by a slight increase at x = 0.7. The larger coefficients obtained for x = 0 and 0.1 reflect the inclusion of higher temperatures, where enhanced anharmonicity and possible contributions from oxygen non-stoichiometry become significant. A similar analysis of the lattice parameters provided the in-plane and out-of-plane (along the c-axis) thermal expansion coefficients. The in-plane expansion is relatively small, typically 2-10 × 10<sup>-6</sup> K<sup>-1</sup>, whereas the c-axis exhibits much stronger expansion, reaching 25-50 × 10<sup>-6</sup> K<sup>-1</sup>. This pronounced anisotropy reflects the layered nature of the K<sub>2</sub>NiF<sub>4</sub> structure, in which weak interlayer bonding along c-axis allows larger thermal displacements compared to the more
rigid NiO₂ planes. With increasing Sr content, both directions display a reduction in expansion up to *x* = 0.5, followed by a slight recovery at *x* = 0.7, consistent with changes in bonding strength and lattice stiffness across the series. The magnitude and anisotropy of these values are qualitatively consistent with those reported for related nickelates and cuprates in the literature<sup>42-44</sup>.

**Oxygen diffusion.** A phonon-assisted oxygen diffusion mechanism, mediated through apical and interstitial sites, has been proposed for K₂NiF₄-type structures near room temperature<sup>9-11,22,36,45,46</sup>. Within this framework, an interstitial oxygen atom "pushes" a neighboring apical oxygen atom out of its regular position, taking its place. In contrast, the displaced apical oxygen atom moves to a new interstitial site. Such migration predominantly occurs within the *ab*-plane and primarily along the [110] direction inside the rock salt layer, on a picosecond timescale<sup>11,47-49</sup>. Excess oxygen atoms in interstitial sites play a crucial role in pushing nearby apical oxygen atoms toward vacant positions, facilitating oxygen migration. As a result, the in-plane thermal displacement parameters of apical oxygen atoms serve as sensitive indicators of oxygen mobility at room temperature. As shown in Figure S10(a), the thermal ellipsoids of O<sub>ap</sub> atoms exhibit significant anisotropy, with displacements primarily confined to the *ab*-plane. The composition-dependent in-plane (*U*<sub>11</sub>, *U*<sub>22</sub>) and out-of-plane (*U*<sub>33</sub>) thermal displacement parameters of apical oxygen atoms are displayed in Figure S10(b). The progressive suppression of in-plane thermal motion with increasing *x* correlates with a decreased concentration of interstitial oxygen upon Sr doping, effectively pinning the apical oxygen atoms to their lattice positions. These findings suggest that oxygen-rich compositions, which retain higher levels of interstitial oxygen, exhibit enhanced oxygen mobility at room temperature and are therefore more promising for oxide-ion conduction compared to Sr-doped variants.
Magnetic susceptibility

The temperature dependence of macroscopic DC magnetic susceptibility for the NSNO
compounds is shown in Figure 5, measured between 2 K and 330 K. With increasing x, a gradual
decrease in the overall susceptibility is observed, which can be attributed to the progressive
substitution of paramagnetic Nd³⁺ ions with non-magnetic Sr²⁺ ions. While previous studies
suggest that hole doping suppresses three-dimensional antiferromagnetic (AFM) order and
promotes stripe-type spin correlations within the Ni sublattice¹⁸, such effects are suppressed
mainly in our powder-averaged data due to the dominant paramagnetic response of Nd³⁺.
Consequently, no apparent magnetic anomalies associated with Ni spin ordering are observed. The
high-temperature (220-330 K) susceptibility data of all Nd₂-xSrxNiO₄+δ were analyzed using the
Curie-Weiss law, χ = χ₀ + C/(T + Θ), where C and Θ represent the Curie constant and Weiss
temperature, respectively. The χ₀ term accounts for temperature-independent contributions such as
Pauli paramagnetism, core diamagnetism, and Van Vleck susceptibility. Deviations from the
Curie-Weiss behavior below 100 K could be attributed to the crystal field splitting of the Nd³⁺ ⁴I₉/₂
ground state. The fitting parameters, listed in Table 1, show negative Weiss temperatures for all
compositions, indicating dominant AFM interactions at low temperatures. Moreover, the
magnitude of Θ slightly increases with x, suggesting an enhancement of AFM exchange
interactions as excess oxygen is removed upon Sr doping. The Θ values obtained in this study are
somewhat lower than those reported for similar Sr-doped compounds in the literature³⁵, likely due
to differences in δ content. The effective magnetic moments (μeff) derived from the Curie constants
are summarized in Table I. Assuming that Nd³⁺ (3.60 μB), Ni²⁺ (S = 1, HS, μeff = 2.83 μB) and
Ni³⁺ (S = 1/2, μeff = 1.73 μB) ions contribute independently to the total moment, the effective
moment can be calculated as $\mu_{calc}^2 = (2-x)\mu_{Nd^{3+}}^2 + f(Ni^{2+})\mu_{Ni^{2+}}^2 + f(Ni^{3+})\mu_{Ni^{3+}}^2 \cdot f(Ni^{2+})$ and $f(Ni^{3+})$ respectively denote the fractions of $Ni^{2+}$ and $Ni^{3+}$ in our prepared samples. The experimental and calculated values are listed in Table 1 and plotted in the inset of Figure 5(a), showing a significant mismatch at low Sr content. To reconcile this, we refined the $Nd^{3+}$ magnetic moment using the above relation and obtained the best agreement for $\mu_{eff}(Nd^{3+}) = 3.19 \mu_B$. This value is ~11% smaller than the free-ion value ($3.62 \mu_B$), which can be attributed to crystal-field effects and partial quenching of the $Nd^{3+}$ moment. At low temperatures, a broad anomaly in the susceptibility data around 16 K is observed in samples with $0.33 \le x \le 0.7$, while an additional sharp upturn appears at $x \approx 5$ K for $x \ge 0.25$ [Fig. 3(a), inset]. The 16 K anomaly was previously attributed to AFM ordering of $Nd^{3+}$ spins$^{35,50}$, but recent neutron scattering results$^51$, along with our data, demonstrate it as a Schottky-type anomaly. This feature arises from the lifting of degeneracy in the ground-state doublet of $Nd^{3+}$ ions in internal magnetic fields created by ordered $Ni^{2+}/Ni^{3+}$ spins, as discussed below. In comparison, the 5 K upturn corresponds to the long-range magnetic ordering of the $Nd^{3+}$ sublattice, which becomes observable once Nd-Nd exchange interactions ($J_{Nd-Nd}$) dominate over Ni-Nd coupling ($J_{Ni-Nd}$). These magnetic interactions are visualized in Figure S12. At higher temperatures, $J_{Ni-Nd} > J_{Nd-Nd}$, so the $Nd^{3+}$ spins remain disordered due to strong polarization effects from $Ni^{2+}$. However, as the temperature decreases, $J_{Nd-Nd}$ interactions become dominant, enabling the formation of static magnetic order. This rare earth ordering has been reported at $T_{Nd} \approx 11$ K in undoped $Nd_2NiO_{4.0}^{18}$. In the present Sr-doped compounds, this transition shifts to lower temperatures (~5 K) and is only observed in samples with relatively low $\delta$ content.

**Table 1.** Average oxygen content, fraction of $Ni^{2+}$ and $Ni^{3+}$ and average Ni valence state along with the least-square results obtained from the fitting of the DC magnetic susceptibility data in the 200-330 K temperature range. $\mu_{eff}(Exp)$, $\mu_{eff}(Theory)$ and $\mu_{eff}(Fit)$ values respectively shows
experimentally obtained, calculated and fit values of magnetic moments in Bohr magneton. The splitting energy $\xi$ of the ground state doublet of the Nd$^{3+}$ ions is evaluated from the specific heat data. Magnetic susceptibility is given in $10^{-3}$ emu/mol. Oe.

<table><thead><tr><th>x</th><th>δ</th><th>Fraction of (Ni<sup>2+</sup>)</th><th>Fraction of (Ni<sup>3+</sup>)</th><th>Avg. Ni valence</th><th>χ<sub>0</sub></th><th>μ<sub>eff</sub> (Exp)</th><th>μ<sub>eff</sub> (Theory)</th><th>μ<sub>eff</sub> (Fit)</th><th>Θ (K)</th><th>ξ (K)</th></tr></thead><tbody><tr><td>0.00</td><td>0.23(2)</td><td>0.54</td><td>0.46</td><td>2.46</td><td>1.00</td><td>5.125</td><td>5.6233</td><td>5.113</td><td>-44.63</td><td>-</td></tr><tr><td>0.10</td><td>0.09(2)</td><td>0.72</td><td>0.28</td><td>2.28</td><td>1.18</td><td>4.963</td><td>5.5882</td><td>5.1014</td><td>-43.59</td><td>-</td></tr><tr><td>0.25</td><td>0.08(2)</td><td>0.59</td><td>0.41</td><td>2.41</td><td>1.27</td><td>4.770</td><td>5.3509</td><td>4.8825</td><td>-42.24</td><td>-</td></tr><tr><td>0.33</td><td>0.04(2)</td><td>0.59</td><td>0.41</td><td>2.41</td><td>1.48</td><td>4.771</td><td>5.2531</td><td>4.798</td><td>-43.74</td><td>16.56</td></tr><tr><td>0.50</td><td>0.06(2)</td><td>0.38</td><td>0.62</td><td>2.62</td><td>1.07</td><td>4.691</td><td>4.9335</td><td>4.4978</td><td>-37.99</td><td>16.51</td></tr><tr><td>0.70</td><td>0.00(2)</td><td>0.3</td><td>0.7</td><td>2.7</td><td>0.98</td><td>4.318</td><td>4.6201</td><td>4.2172</td><td>-53.34</td><td>16.56</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

**Figure 5.** (a) DC magnetic susceptibility of Nd$_{2-x}$Sr$_x$NiO$_{4+\delta}$ compounds as a function of temperature. Arrows indicate characteristic magnetic transition temperatures reported in previous studies, as depicted in Figure S11. The left inset highlights the low-temperature region, showing a Schottky-type anomaly around $T \approx 16$ K and the onset of Nd$^{3+}$ magnetic ordering at $T \approx 5$ K. The right inset presents the experimental (black squares), calculated (red circles) and fit (blue triangles) effective magnetic moments as a function of $x$, (b) Specific heat (C) as a function of temperature for Nd$_{2-x}$Sr$_x$NiO$_{4+\delta}$ samples. Reference data for La$_2$NiO$_{4.0}$ (from Ref. 52) are overlaid to isolate the magnetic contribution from the Nd site. The bottom inset shows C/T vs. T plots, revealing a
Schottky anomaly around 8 K and a broad anomaly near $T_i$. The top inset displays the HTT-to-LTO structural transition at $T_s \approx 210$ K for $x = 0.5$.

## Specific heat

The temperature-dependent specific heat data of selected NSNO compositions ($x = 0, 0.33, 0.5$) are presented in Figure 5(b), alongside La$_2$NiO$_4$$_{.}$$_{0}$ data from Ref. 52 to separate lattice and magnetic contributions from the rare-earth ion. For $x = 0.5$, a broad peak centered at 205 K, extending over the 175-225 K range. This temperature is slightly lower than the transition temperature $T_C$ obtained from diffraction. Because the available SXRPD measurements are sparse near the critical region, with data collected up to 210 K and then at 250 K, the limited temperature resolution likely results in a slight overestimation of the fitted $T_C$. Such a difference between the thermodynamic anomaly and the diffraction-derived $T_C$ is typical of continuous (second-order) transitions, where diffraction detects the onset of long-range structural distortion, while the specific-heat anomaly marks the temperature at which the order parameter changes most rapidly and the entropy change is greatest. The broad anomaly also indicates a gradual development of the octahedral tilt order parameter, possibly broadened by compositional variation or internal strain. In contrast, no such anomaly is detected in other samples down to 2 K, and no signatures of charge or spin ordering are observed in any of the specific heat curves. A secondary feature is identified in the $CT^{-1}$ vs. $T$ data for $x = 0.33$ and $0.5$, where a subtle anomaly appears at $T_i \approx 53$ K ($x = 0.33$) and shifts to 60 K ($x = 0.5$), as shown in the inset of Figure 5(b). Although this was previously attributed to a possible structural transition$^{51}$, our SXRPD data do not support this interpretation. The minimal specific heat change at $T_i$ also excludes the possibility of long-range ordering. Instead, this anomaly likely arises from a Ni$^{2+}$/Ni$^{3+}$ spin reorientation within the stripe-ordered phase, similar to that proposed for stripe
ordered La₂-xSrxNiO₄ in the temperature range of 50-57 K⁵³. Notably, this feature is absent for x > 0.5, where the checkerboard (CB) charge-ordered phase dominates, suggesting that it is an intrinsic characteristic of the stripe phase in the NSNO system. A distinct anomaly appears near 8 K in the C/T vs. T plot for the Nd-based samples, while it is absent in the La-based reference compound, indicating that it originates from a Schottky-type anomaly related to the Nd³⁺ magnetic moments. The associated entropy approaches Rln2 by 50 K, implying that the Kramers doublet ground state of Nd³⁺ splits into two singlets. The splitting energy (ξ) was quantitatively estimated from both specific heat and susceptibility data, and the results are listed in Table I. Assuming a simple two-level system with the energy gap of ξ between two singlets, the maximum Schottky peak in the specific heat data should appear at temperature T_m such that ξ = 2.399T_m. This relationship is described in detail in the supporting information. The corresponding peak in the magnetic susceptibility data will appear at T_mag, so that ξ = 1.543T_mag, while in the differential susceptibility data, it will appear at T'mag, so that ξ = 2.703T'_mag. The calculated values of ξ obtained from both specific heat and susceptibility data coincide well and are consistent with previously reported results⁵¹.

**Crystalline electric field effects.** Since the magnetic behavior of Nd³⁺ ions predominantly governs the low-temperature thermodynamic properties, we performed a quantitative assessment of the crystal electric field (CEF) levels and their evolution in the presence of interstitial oxygen to clarify the origin of the Schottky-type anomaly observed in our magnetic susceptibility and specific heat below 20 K. The single-ion CEF levels of Nd³⁺ were calculated using the program McPhase⁵⁴ employing a simplified point-charge model that considers only the nearest-neighbor O²⁻ ions. In particular, two structural configurations were considered: the stoichiometric Nd₂NiO₄₀ phase, where each Nd³⁺ ion is surrounded by nine O²⁻ ions, and the oxygen-intercalated Nd₂NiO₄₂₃ phase,
phase, where statistically four out of sixteen available Nd³⁺ sites have an additional interstitial O²⁻ ion among its nearest neighbors, while the remaining sites retain the original nine-fold coordination. Nd-O bond distances for Nd₂NiO₄.₂₃ were taken from our SXRPD refinements, while those for Nd₂NiO₄.₀ were adopted from reference ¹⁸. Full details of the computational procedure and results are provided in the Supporting Information and Tables S3-S4.

For Nd³⁺ in its 4f³ configuration, the ground-state multiplet ⁴I₉/₂ arises from strong spin-orbit coupling, which dominates over CEF effects. In the crystalline environment provided by the surrounding O²⁻ ions in Nd₂NiO₄.₀, this tenfold-degenerate *J* = 9/2 state is split by the CEF into five Kramers doublets, following Kramers' theorem. The computed splitting energies span from a few meV to several tens of meV, resulting in characteristic features in the temperature dependence of the specific heat and susceptibility. Our calculated energy levels are in close agreement with the previous report⁵⁵, validating the model. Upon insertion of an interstitial oxygen atom near Nd³⁺, the local electrostatic potential is significantly altered, leading to a noticeable shift in the CEF level energies. As summarized in Figure S12, this produces enhanced CEF splittings, which in turn modify the low-temperature thermodynamic response. Furthermore, the inclusion of an internal magnetic field, arising from exchange coupling between Nd³⁺ and the ordered Ni²⁺ moments in the NiO₂ planes, leads to Zeeman splitting of the ground-state doublet. This internal magnetic field, estimated to be ≈ 5.2 T based on previous specific heat analysis⁵⁶, was applied along the *c*-axis in our model. The resulting Zeeman splittings are ≈ 1.19 meV for Nd₂NiO₄.₀ and ≈ 1.37 meV for Nd₂NiO₄.₂₅, confirming that the local structural distortion introduced by oxygen intercalation enhances the exchange field experienced by Nd³⁺ ions. These results collectively explain the Schottky-type anomaly observed in both specific heat and susceptibility data below 20 K. The temperature position and shape of this anomaly are shown to be highly sensitive to both the detailed
CEF scheme and the magnitude of the internal magnetic field, both of which are governed by the
local symmetry and bonding environment around Nd³⁺.

In our data, the Schottky anomaly is notably broadened for samples with $x \le 0.25$, and the Nd³⁺ magnetic ordering temperature is suppressed. Therefore, this behavior could be primarily attributed to the presence of a significant amount of interstitial oxygen atoms ($\delta \ge 0.09$), which locally distort the lattice environment surrounding the Nd³⁺ ions. These distortions introduce a distribution of local crystal fields, resulting in inhomogeneous CEF splittings, as well as a distribution in the internal magnetic fields acting on the Nd³⁺ ions. Consequently, the Zeeman splitting also varies across different sites. Since both specific heat and susceptibility are macroscopic averages over all Nd³⁺ environments, the combined effect of CEF disorder and inhomogeneous exchange interactions leads to a broadened and less well-defined Schottky anomaly, in contrast to the sharper features observed in highly Sr-doped systems.

<figure><img src="image_8.png" /><figcaption>**Figure 6.** (a) Reconstructed (*h*0*1*) reciprocal plane of Nd<sub>1.5</sub>Sr<sub>0.5</sub>NiO<sub>4.06</sub> at 2 K from single-crystal neutron diffraction. Yellow boxes highlight strongly diffuse stripe magnetic reflections only. These magnetic peaks are strongly diffuse along the reciprocal *l*-direction. The Al powder rings are from the sample environment. (b) A *q*<sub>*h*</sub>-cut through the stripe spin order peak at (*h*, 0, 1), revealing magnetic incommensurability of about *ε* = 0.44. (c) Temperature evolution of the</figcaption></figure>
integrated intensity of the spin stripe peak at (0.56, 0, 1), showing magnetic ordering around $T_N \approx$
95 K.

Neutron diffraction

To investigate magnetic ordering behavior in Sr-doped samples, temperature-dependent neutron powder diffraction measurements were performed for $x = 0.25$, 0.33, and 0.5 compositions. At room temperature, all these samples adopt the HTT phase, consistent with our SXRPD data, and show no magnetic features. Upon cooling, weak and diffuse magnetic reflections begin to emerge in all compositions, signaling the onset of magnetic correlations. These magnetic features are visible at 2 K but are notably weak in $x = 0.25$ and 0.33 (see Figure S13 of the supporting information), making quantitative analysis difficult. In contrast, for the $x = 0.5$ sample, two distinct diffuse magnetic reflections are observed at $Q \approx 0.84 \text{ Å}^{-1}$ and $1.29 \text{ Å}^{-1}$ as shown in Figure 4(b). The stripe spin ordering can be described by a modulation vector $Q_{so} = Q_{AF} \pm (\epsilon, 0, 0)$ relative to the pseudo-tetragonal chemical unit cell, where $Q_{AF} = (1, 0, 0)$ is the commensurate antiferromagnetic wave vector found for the undoped compound$^{6,57}$. Following this indexation, the two magnetic reflections observed for $x = 0.5$ can be assigned to the incommensurate positions (0.56, 0, 1) and (0.44, 1, 0), respectively. The incommensurability parameter $\epsilon$ is thus estimated to be approximately 0.44, consistent with previous reports$^{20,58}$. The broader width of the (0.56, 0, 1) reflection, compared to nearby structural Bragg peaks, suggests short-range inter-plane correlation, while the comparatively sharp presence of the (0.44, 1, 0) peak indicates stronger in-plane correlations. To further characterize the stripe order, single-crystal neutron diffraction reciprocal plane maps were conducted on the $x = 0.5$ sample at different temperatures. Fig. 6(a) shows a section of the reconstructed (h0l) reciprocal plane obtained at 2 K, revealing stripe magnetic peaks
at $(h \pm \epsilon, 0, l)$ positions with $h$ and $l$ being odd numbers. Figure 6(b) shows a $q_h$ cut through one of these spin-ordering peaks at $l=1$, yielding a magnetic incommensurability of 0.44, consistent with our PND results. The temperature dependence of the integrated intensity of this peak is shown in Figure 6(c), indicating a magnetic ordering temperature of $T_N \approx 95$ K. The exclusive appearance of magnetic peaks at odd $l$ values indicates antiferromagnetic coupling between adjacent $NiO_2$ layers. Notably, the intensity of the $l=3$ peak is significantly weaker than that of $l=1$, and the $l=5$ and $7$ reflections are barely discernible. Although the actual magnetic structure at 2 K is likely more complex, this trend primarily suggests weak interlayer coupling and that the ordered magnetic moments are significantly tilted along the c-axis, similar to what has been observed in $Nd_{1.7}Sr_{0.3}NiO_4^{59}$. This behavior is in stark contrast to that of the isoelectronic $Nd_2NiO_{4.23}$ system, which shows no evidence of stripe-type spin order but instead hosts a long-range incommensurate magnetic structure with a doubling of the magnetic c-axis, as previously established by both polarized and unpolarized single-crystal neutron diffraction, as shown in Figure S14 of the supplemental information²³. The presence of interstitial oxygen in $Nd_2NiO_{4.23}$ strengthens the interlayer antiferromagnetic coupling, resulting in a magnetic unit cell that is doubled along the c-axis relative to the chemical unit cell. These findings suggest that Sr doping drives a transition from a long-range incommensurate magnetic state to stripe-type spin ordering in $Nd_{2-x}Sr_xNiO_{4+\delta}$.

To quantitatively evaluate the dimensionality of the stripe correlations, Lorentzian fits (convoluted with the Gaussian-type instrumental resolution) were applied to Q-scans of the (0.56, 0, 1) peak along both in-plane and out-of-plane directions. The resulting in-plane correlation length ($\xi_{ab}$) is 114(2) Å. In contrast, the out-of-plane correlation length ($\xi_c$) is 16(3) Å, further confirming the quasi-two-dimensional nature of the magnetic stripe phase.
The observed magnetic incommensurability $\epsilon \approx 0.44$ for $x = 0.5$ suggests a complex underlying charge distribution. In La$_{1.5}$Sr$_{0.5}$NiO$_{4.0}$, the coexistence of stripe and checkerboard-type charge orders has been proposed to give rise to charge discommensuration, a phenomenon in which the periodicity of the charge modulation becomes locally disrupted due to the competition between two distinct ordering tendencies$^{20}$. Stripe order is typically favored at lower doping levels, whereas checkerboard order becomes energetically favorable near half-doping. In intermediate regimes, where both orders are nearly degenerate in energy, spatial inhomogeneity arises, with stripe-ordered regions interrupted by checkerboard segments, or vice versa. These local disruptions, referred to as discommensurations, result in an overall incommensurate modulation wave vector. The incommensurability observed in this work can be quantitatively described using an empirical relation, $\epsilon = (n + m)/(3n + 2m)$, where n and m represent the numbers of stripe- and checkerboard-ordered units with $\epsilon = 1/3$ and $1/2$, respectively. Both types of magnetic and charge ordering cells are illustrated in Figures 7(a)-(d). In the schematic representations, no magnetic moment is assigned to the Ni$^{3+}$ sites. Although these ions are magnetically active, previous neutron-scattering studies have shown that the Ni$^{3+}$ sites exhibit only quasi-one-dimensional antiferromagnetic fluctuations along the charge stripes, with a short correlation length (~15 Å) and a low energy scale (~2.5 meV)$^{60,61}$. These fluctuations are well described by models of weakly coupled one-dimensional AFM Ni$^{3+}$ chains running along the stripe direction. If such static AFM order were present on the Ni$^{3+}$ sites, magnetic Bragg peaks would appear at different positions due to magnetic cell doubling along the stripe direction. However, no such reflections are observed in our diffraction data, confirming the absence of static magnetic order on the Ni$^{3+}$ sublattice. This model was initially proposed for La$_{2-x}$Sr$_x$NiO$_{4.0}$ in the incommensurability range $1/3 \le \epsilon \le 1/2$ $^{20}$. For the present case, inserting one stripe segment ($n = 1$) with $\epsilon = 1/3$ after every three checkerboard units
(m = 3) with ε = 1/2 yields ε ≈ 0.44, in excellent agreement with our experimental value. However, we note that in our neutron diffraction reciprocal plane map, we did not see any charge ordering peaks. Figures 7(e)-(f) illustrate a one-dimensional projection of this discommensuration model for both Ni-site-centered and bond-centered (O²-site-centered) stripe configurations. This framework effectively captures the resulting incommensurate spin structure of the x = 0.5 compound. Notably, for bond-centered stripes, a small ferromagnetic component is expected near the charged ordered domain walls because of Ni²⁺-Ni²⁺ superexchange interactions. Although a diffraction study alone cannot distinguish whether the hole resides on a Ni or O site in the discommensuration region, our field-dependent isothermal magnetization data on the powder sample show no evidence of a ferromagnetic component down to 2 K, as illustrated in Figure S15.
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

**Figure 7.** (a) Idealized sketch of the NiO₂ plane showing spin and charge ordering for Ni-site centered stripe order with ε = 1/3, corresponding to a magnetic unit cell 3 times larger (red dashed line) and a charge ordering cell 1.5 times larger (light blue dashed line) than the chemical unit cell (black dashed line). (b) Checkerboard type charge ordering with ε = 1/2, associated with a magnetic unit cell twice as large. In the stripe phase, both intra-stripe (*J*₀) and inter-stripe (*J*₁) superexchange interactions are possible, while in the checkerboard configuration, only *J*₁ is active. Orange-shaded regions represent charge-ordered domains, illustrating the unidirectional stripe-type and two-dimensional checkerboard-type charge orders. (c-d) One-dimensional projection (along the *a*-axis)
of the spin and charge-ordering unit cells of those shown in (a) and (b). (c) Checkerboard charge order with $\epsilon = 1/2$, (d) Stripe charge order with $\epsilon = 1/3$, (e) a charge discommensuration picture where 3 Ni-site centered checkerboard patterns (with $\epsilon = 1/2$) are repeated with 1 Ni-site centered stripe pattern (with $\epsilon = 1/3$) that gives $\epsilon = 4/9 \sim 0.44$ as observed for $x = 0.5$. (f) Same as (e) but with a bond-centered stripe pattern in which the hole is located in the oxygen site ($O_h^{2-}$). A small ferromagnetic component is expected in the case of bond-centered stripes, as illustrated by a bracket in (f). Note that spins are projected into the *ab*-plane for clarity, while they are significantly tilted along the *c*-axis at 2 K.

## CONCLUSIONS

In this comprehensive study, we explored the coupling between lattice, charge, and spin degrees of freedom in the Sr-doped Nd$_{2-x}$Sr$_x$NiO$_{4+δ}$ system. Our results reveal a complex interplay between oxygen non-stoichiometry, structural phase transitions, and magnetic order across the doping range ($0 \le x \le 0.7$). We demonstrated that Sr doping progressively suppresses excess oxygen uptake, inducing a composition-dependent transition from the LTM to LTO' and then to the HTT structure at room temperature. While the Sr-free sample exhibits complex incommensurate structural modulations related to long-range excess oxygen ordering and transitions with increasing temperature, even modest Sr doping disrupts these ordered configurations. Although Sr doping and excess oxygen content are closely related, the minimum Sr doping level at which excess oxygen is eliminated remains an interesting avenue for future investigation. Additionally, we identified quasi-two-dimensional stripe-like spin ordering for $x = 0.25-0.5$, with the $x = 0.5$ composition showing pronounced magnetic incommensurability ($\epsilon \approx 0.44$) and anisotropic magnetic correlation lengths, which are well described by a charge discommensuration model in
which both stripe and checkerboard charge configurations coexist in the NiO₂ planes. The magnetic behavior observed in the *x* = 0.5 compound contrasts sharply with that of the isoelectronic Nd₂NiO₄.₂₃, which does not exhibit stripe-type spin order. Instead, Nd₂NiO₄.₂₃ hosts a long-range incommensurate magnetic structure characterized by a doubling of the magnetic unit cell along the *c*-axis, attributed to enhanced interlayer magnetic correlations driven by excess oxygen ordering within the rock-salt layers. Our findings establish Nd₂-xSrxNiO₄+δ as a model system to study the evolution of lattice, spin, and charge order as a function of chemical doping and temperature. The sensitivity of structural and magnetic properties to subtle variations in oxygen content and doping highlights the importance of controlling both A-site chemistry and oxygen stoichiometry for tuning emergent electronic phases.

## ASSOCIATED CONTENT

**Supporting Information.** The Supporting Information is available free of charge.

Additional figures (S1-S15), tables (S1-S4), and references providing detailed structural, magnetic, and thermodynamic analyses of Nd₂-xSrxNiO₄+δ. Includes: Temperature- and doping-dependent synchrotron X-ray diffraction data. Structural refinements, lattice parameters, and thermal displacement parameters. Phase diagrams and crystal field calculations. Neutron diffraction reciprocal plane maps and powder pattern refinements at low temperature. Specific heat and magnetization analysis, including CEF and Zeeman splitting models. (PDF)

## AUTHOR INFORMATION

### Corresponding Author
Sumit Ranjan Maity

Japan Synchrotron Radiation Research Institute (JASRI), SPring-8, Hyogo 679-5198, Japan

Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, Villigen CH-5232,
Switzerland

orcid.org/0000-0002-8168-8024;

Email: sumit050491@spring8.or.jp

**Monica Ceretti**: orcid.org/0000-0001-9704-8251

**Lukas Keller**: orcid.org/0000-0002-8492-4117

**Jürg Schefer**: orcid.org/0000-0002-0981-4559

**Uwe Stuhr**: orcid.org/0000-0003-1532-2267

**Antonio Cervellino**: orcid.org/0000-0002-9086-2717

**Werner Paulus**: orcid.org/0000-0001-6472-8162

**Funding Sources**

The authors gratefully acknowledge the financial support from the Swiss National Science Foundation (SNF) through grant 200021L_57131 and the French National Research Agency (ANR) through grant 14-CE36-0006-01 of the SECTOR project.

**Notes**

Any additional relevant notes should be placed here.

**ACKNOWLEDGMENT**

The neutron diffraction measurements were performed at the Swiss Spallation Neutron Source (SINQ), Paul Scherrer Institute (PSI), Villigen, Switzerland. We gratefully acknowledge beamtime
allocation at the Swiss Light Source (SLS), Materials Science beamline X04SA. The authors thank
the Paul Scherrer Institute for access to both SINQ and SLS facilities.

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For Table of Contents Only

<figure><img src="image_14.png" /></figure>

Caption: Structural transitions, suppression of excess oxygen order and emergence of stripe magnetism in Sr-doped Nd$_{2-x}$Sr$_x$NiO$_{4+δ}$ revealed through X-ray and neutron diffraction signatures.