# Investigation of spin excitations and charge order in bulk crystals of the
infinite-layer nickelate LaNiO2

S. Hayashida<EMAIL_ADDRESS>Max-Planck-Institute for Solid State
Research, Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  V. Sundaramurthy
Max-Planck-Institute for Solid State Research, Heisenbergstra$\beta$e 1, 70569
Stuttgart, Germany  P. Puphal  Max-Planck-Institute for Solid State Research,
Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  M. Garcia-Fernandez
Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom  Ke-Jin
Zhou  Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
B. Fenk  Max-Planck-Institute for Solid State Research, Heisenbergstra$\beta$e
1, 70569 Stuttgart, Germany  M. Isobe  Max-Planck-Institute for Solid State
Research, Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  M. Minola  Max-
Planck-Institute for Solid State Research, Heisenbergstra$\beta$e 1, 70569
Stuttgart, Germany  Y.-M. Wu  Max-Planck-Institute for Solid State Research,
Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  Y. E. Suyolcu  Max-Planck-
Institute for Solid State Research, Heisenbergstra$\beta$e 1, 70569 Stuttgart,
Germany  P. A. van Aken  Max-Planck-Institute for Solid State Research,
Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  B. Keimer
<EMAIL_ADDRESS>Max-Planck-Institute for Solid State Research,
Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany  M. Hepting
<EMAIL_ADDRESS>Max-Planck-Institute for Solid State Research,
Heisenbergstra$\beta$e 1, 70569 Stuttgart, Germany

###### Abstract

Recent x-ray spectroscopic studies have revealed spin excitations and charge
density waves in thin films of infinite-layer (IL) nickelates. However,
clarifying whether the origin of these phenomena is intrinsic to the material
class or attributable to impurity phases in the films has presented a major
challenge. Here we utilize topotactic methods to synthesize bulk crystals of
the IL nickelate LaNiO2 with crystallographically oriented surfaces. We
examine these crystals using resonant inelastic x-ray scattering (RIXS) at the
Ni $L_{3}$-edge to elucidate the spin and charge correlations in the bulk of
the material. While we detect the presence of prominent spin excitations in
the crystals, fingerprints of charge order are absent at the ordering vectors
identified in previous in thin-film studies. These results contribute to the
understanding of the bulk properties of LaNiO2 and establish topotactically
synthesized crystals as viable complementary specimens for spectroscopic
investigations.

##  I Introduction

Unconventional superconductors, including cuprates, iron pnictides, and heavy
fermion systems, are at the forefront of condensed matter research. While the
microscopic mechanisms behind their superconductivity remain a subject of
debate, spin and charge instabilities are recurrent features in their phase
diagrams [1, 2, 3, 4, 5]. The magnetic phase is typically most pronounced in
the parent materials, while charge carrier doping and/or external pressure
suppress the long-range magnetic order and facilitate the emergence of
superconductivity [6]. This suppression of long-range magnetic order
transforms well-defined spin wave excitations (magnons) of the parent
materials into heavily damped excitations (paramagnons) [7, 8, 9], which can
persist over wide regions of the phase diagrams [10, 11, 12, 13]. In cuprates,
the spin excitations arise from the two-dimensional CuO2 planes, exhibiting a
notable bandwidth of several hundred meV [14, 15, 16, 17] due to strong
exchange couplings ($J$), and are a prime candidate for the superconducting
pairing glue [1].

In addition, the ubiquitous charge density wave (CDW) ordering phenomenon
might have a profound impact on the emergence of high-temperature conductivity
in cuprates. These density waves are characterized by periodic electron
density modulations within the CuO2 planes, which compete with the
superconducting state, at least in the underdoped regime [18, 19, 20, 21]. A
more complex interplay between superconductivity and various types of short-
range and dynamic charge fluctuations has been observed at higher doping
levels, and is a current focus of research on the cuprates [22, 23, 24, 25].

In this context, recent reports of charge order in the undoped to lightly
doped regime [26, 27, 28, 29] of the phase diagram of infinite-layer (IL)
nickelates superconductors with composition (La,Pr,Nd)1-xSrxNiO2 [30, 31, 32,
33, 34, 35] were not entirely unexpected, given their numerous shared basic
characteristics with cuprate superconductors [36]. In particular, their
structural motif of square-lattice NiO2 planes along with a nominal 3$d^{9}$
electronic configuration draws close parallels to cuprates [37]. However, this
class of materials also features significant distinctions [38], such as a lack
of the cuprate-typical $3d$-$2p$ orbital hybridization between the transition
metal ion and oxygen. Instead, rare-earth $5d$ states contribute to the low-
energy electronic structure of IL nickelates [39, 40, 41]. Moreover, long-
range antiferromagnetic order is absent in the parent compounds of IL
nickelates [42, 43, 44, 45, 46, 47, 48, 49], although they display dispersive
magnetic excitations [50, 26, 27, 36, 51] that are reminiscent of paramagnons
in doped cuprates [10, 11]. As a consequence, fundamental questions about the
nature of spin excitations in nickelate superconductors have remained open.

Yet, the CDW phenomenon in IL nickelates is also debated controversially [52,
53, 54, 55]. One set of studies reported charge order in NdNiO2 thin films
without a capping layer [26, 27], while the corresponding signal at
approximately $\mathbf{q}=(\pm 1/3,0)$ was absent in capped films. By
contrast, spin wave excitations materialized in films with capping layer [27,
50], but were elusive in uncapped films [26, 27]. On the other hand, a
different study reported the simultaneous presence of both charge order and
spin excitations in LaNiO2 films with capping [28], while charge order signal
was not detected in PrNiO2 and NdNiO2 films [29]. Along these lines, recent
experiments using scanning transmission electron microscopy (STEM) and x-ray
scattering have cast doubt on the intrinsic origin of the CDW [54, 55]. One
proposed explanation for the observed charge ordering signal involves an
impurity phase, in which excess oxygen ions fill certain vacant apical
positions in the IL structure, aligning in a pattern that is commensurate with
a $\mathbf{q}=(\pm 1/3,0)$ ordering vector.

To examine if previously reported spin and charge instabilities in thin-film
samples originated from external factors, including interfacial effects with
capping layers, oxygen off-stoichiometries, and/or damages from the invasive
topochemical synthesis, we focus here on bulk crystals of the IL nickelate
LaNiO2. These bulk specimens with mechanically polished surfaces offer a
contrasting approach to thin films, as the material exposed after polishing
was neither affected by the invasive topotactic reduction atmosphere nor by
contact with a capping layer. Using resonant inelastic x-ray scattering (RIXS)
on a polished LaNiO2 surface, we detect low-energy spectral features that are
attributable to spin excitations. We demonstrate that the observed spin
excitation spectrum can be qualitatively reproduced with linear spin wave
theory and the magnetic exchange parameters known from thin films and that the
lack of apparent dispersion is a consequence of the presence of three twin
domains in our crystals. Hence, we conclude that the underlying spin
excitation phenomenology in LaNiO2 crystals aligns with that of thin films.
Conversely, a CDW signal at the ordering vectors reported in previous thin
film studies is not observed in our RIXS experiment on LaNiO2 crystals.

##  II Materials and methods

Single crystals of perovskite LaNiO3 were grown by the high-pressure optical
floating zone (OFZ) method, as described in Refs. [56, 57]. The as-grown
centimeter-sized crystals were oriented with x-ray Laue diffraction and cut
into smaller cube-shaped crystals with a wire saw, such that each surface of a
cube corresponds to a pseudocubic (100) plane. The lateral dimensions of the
surfaces of the cut crystals were approximately $0.8\times 0.8$ mm2.

For the topotactic oxygen deintercalation, several cube-shaped LaNiO3 crystals
with a total mass of about 100 mg were placed in direct contact with roughly
250 mg of powder of the reducing agent CaH2 in a DURAN glass tube. The glass
tube was evacuated to $\sim 10^{-7}$ mbar and sealed. The reduction process
was initiated by heating to 300∘C in a low-temperature furnace. The
temperature was maintained for a duration of approximately 13 days, which is
comparable to the conditions used for previous reductions of nickelate single
crystals [58, 56]. The surfaces of the obtained LaNiO2 crystals were polished
using a diamond suspension, removing the uppermost layers ($\sim 30~{}\mu$m)
to expose a pristine surface free of residues and secondary phases that formed
during the invasive topotactic reduction.

The reduced crystals were examined by STEM, which is a sensitive method for
detecting impurity phases, stacking faults in the crystal lattice, and
residual oxygen in nickelates [59, 58, 60, 61]. Electron-transparent specimens
were prepared on a Thermo Fisher Scios I focused ion beam (FIB) using the
standard liftout method. The lateral dimensions of the specimens were 20
$\mu$m by 1.5 $\mu$m with thicknesses between 50 and 100 nm. The STEM studies
were performed with a JEOL JEM-ARM200F scanning transmission electron
microscope equipped with a cold-field emission electron source, and a probe
abberation-corrected Cs corrector (DCOR, CEOS GmbH) at 200 kV. STEM imaging
was performed with probe semiconvergence angles of 20 mrad, resulting in probe
sizes of 0.8 Å. The collection angles for high-angle annular dark-field
(HAADF) and annular bright field (ABF) STEM images were 75 to 310 and 11 to 23
mrad, respectively. To improve the signal-to-noise ratio of the data while
minimizing sample damage, a high-speed time series (2 $\mu$s per pixel) was
acquired and then aligned and summed.

To investigate the microstructural properties of the LaNiO2 crystals, such as
the twin domain distribution, we utilized a Zeiss Merlin electron microscope
equipped with electron energy backscatter diffraction (EBSD) capabilities from
Oxford Instruments (Symmetry EBSD detector). The AZtec and AZtecCrystal
software were used for data acquisition and analysis, respectively. For the
EBSD measurements, the crystal was tilted to a high angle ($70^{\circ}$).
Probe currents ranged from 1 to 5 nA, and the accelerating voltage was 20 kV
for most measurements. Further details about the EBSD measurements are given
in the Supplemental Material.

X-ray absorption spectroscopy (XAS) and RIXS measurements were conducted at
the I21-RIXS beamline of the Diamond Light Source, UK [62]. All RIXS spectra
were collected at temperatures between 22 and 28 K with an effective energy
resolution of $\Delta E=37$ meV and a scattering angle of $154^{\circ}$.
Incident photons were linearly polarized in either the vertical ($\sigma$) or
horizontal ($\pi$) direction with respect to the scattering plane.

Figure 1: (a) Electronic transport of a LaNiO2 crystal. The inset is a side-
view SEM-SE image of the polished surface of the LaNiO2 crystal used in the
RIXS experiment. (b) XRD pattern acquired from the polished surface of a
LaNiO2 crystal. The Bragg peaks are indexed and the inset shows the tetragonal
$P4/mmm$ unit cell of LaNiO2. The asterisk symbol indicates an artifact
originating from a carbon-based paste used to mount the LaNiO2 crystal on the
sample holder.  Figure 2: (a) Top-view SEM-SE image of the polished surface of
a LaNiO2 crystal. (b) Cross-sectional view of the crystal in a low-
magnification STEM-HAADF image. The yellow arrow marks the same dark line as
the arrow in panel (a). (c),(d) Magnified view of the area indicated by the
green (orange) box in panel (b), which consists of non-crystalline
(crystalline) LaNiO2. The inset is the fast Fourier transformation. (e) High-
magnification STEM HAADF and (f) STEM ABF images acquired within a single-
domain of a LaNiO2 crystal. The inset in (f) zooms into the STEM ABF image,
revealing the ideal infinite-layer structure without visible traces of apical
oxygen (above and below the Ni atoms) remaining from the parent LaNiO3
structure.

##  III Results

###  III.1 Properties of LaNiO2 bulk crystals

In a previous study, millimeter-sized LaNiO2 crystals with randomly oriented
surfaces were obtained via the topotactic reduction of LaNiO3 single crystals,
utilizing a direct-contact method with CaH2 as the reducing agent [56]. Here
we employ a similar topotactic method to obtain LaNiO2 bulk crystals with
crystallographically oriented surfaces (for details see Sec. II). In addition,
we polish the LaNiO2 crystals to expose fresh surfaces, which were not
affected by the topotactic reduction. Figure 1(a) shows the electronic
transport of a LaNiO2 crystal measured in a four-point probe geometry on a
polished surface [see inset in Fig. 1(a)]. The transport behavior is metallic
with a subtle upturn below 50 K, which is consistent with that reported for
high-quality LaNiO2 thin films [63]. In contrast, the onset of the upturn in
LaNiO2 crystals with randomly oriented and unpolished surfaces was
approximately 100 K [56].

In the x-ray diffraction (XRD) pattern from an oriented and polished surface
we observe the characteristic Bragg peaks of LaNiO2 in the $P4/mmm$ space
group, while no additional peaks due to impurities or insufficiently reduced
phases are detected [Fig. 1(b)]. However, the simultaneous presence of
$(H,0,0)$ and $(0,0,L)$-type Bragg peaks suggests that the LaNiO2 crystals
contain three twin domains of the tetragonal $P4/mmm$ unit cell.

A closer inspection of the LaNiO2 crystals reveals a distinctive texture on
the polished surfaces, characterized by parallel or perpendicular dark lines
[Fig. 2(a)] that mostly align with the directions of the crystal edges. This
texture is both visible in secondary electron (SE) images acquired with a
scanning electron microscope (SEM) [Fig. 2(a)] and optical microscopy images
(see Supplemental Material). A low-magnification STEM-HAADF image showing a
cross-sectional view across a dark line and its vicinity is presented in Fig.
2(b). A zoom into the dark area [Fig. 2(c)] indicates a porous structure in
this region, and the fast Fourier transformation (FFT) of the image suggests
that crystalline order is absent [inset in Fig. 2(c)]. In contrast, a
magnified view of a representative area surrounding the dark lines [Fig. 2(d)]
reveals a well-ordered crystal lattice. The corresponding FFT image exhibits a
pattern that is consistent with the IL structure [58]. Further magnification
of a crystalline region is provided by high-resolution STEM-HAADF [Fig. 2(e)]
and ABF imaging [Fig. 2(f)], confirming high crystalline quality on a local
scale, without any stacking faults or secondary phase inclusions in the
lattice. Moreover, the inset in Fig. 2(f) illustrates the absence of residual
apical oxygen from the LaNiO3 phase, indicating that the topotactic
transformation to the IL phase is complete.

Notably, the texture featuring dark lines is not present in our unreduced
LaNiO3 crystals. This suggests that the disordered regions develop during the
topotactic reduction, potentially acting as channels that facilitate the
oxygen transport from the crystal’s interior to its surface. Similar
disordered areas separating highly crystalline regions were also identified in
a previous STEM study on topotactically reduced Pr0.92Ca0.08NiO2.75 crystals
[60]. Elucidating the nature of these disordered regions and their role in the
topotactic transformation process presents a compelling subject for future
STEM investigations.

To further characterize our LaNiO2 crystals, we map the distribution of twin
domains across the surface using EBSD. The top panel of Fig. 3(a) displays an
SEM-SE image of a local area on the crystal surface after mechanical and low-
angle Ar ion polishing, which is necessary to prepare the surface for EBSD
measurements. The bottom panel of Fig. 3(a) shows the acquired EBSD map from
the same surface area, where we identify three twin domains of the tetragonal
space group $P4/mmm$ of LaNiO2, color-coded in green, purple, and brown. We
denote these domains as $A$, $B$, and $C$ in the following. Specifically, the
EBSD map reveals extended regions of single-domain character, with domain $C$
predominating. Around the center and the top left corner of the map,
significant areas of domain $A$ are observed, interspersed with smaller
islands of domain $B$. A mapping of other areas on the crystal surface (see
Supplementary Material), however, reveals varying domain distributions,
including regions where domain $A$ or $B$ are dominant. This variation in
domain population across areas and length scales of tens of micrometers has
important implications for the RIXS experiment, considering that the x-ray
beam’s footprint on the sample surface is of a comparable size. This aspect
will be elucidated later in the text.

Interestingly, we find that the non-indexed pixels (white color) in the EBSD
maps mostly coincide with dark lines in the associated SE-SEM images, which
according to the analysis in Fig. 2 correspond to disordered regions. The
orientation of the twin domains is often identical on both sides of streaks of
non-indexed pixels, hence suggesting that these regions are not corresponding
to twin domain boundaries. Instead, the boundaries between domains [see
different colors in Fig. 3(a)], exhibit a more complex structure varying on
finer length scales, which also presents a subject for future STEM
investigations.

Figure 3: (a) SEM-SE image (top panel) and EBSD map (bottom panel) of the same
surface region. The color code (green, purple, and brown) indicates the three
twin domains ($A$, $B$, and $C$) of LaNiO2 with the tetragonal $P4/mmm$ unit
cell. White areas correspond to pixels where the EBSD pattern could not be
indexed (see Supplemental Material). (b) Schematic of the LaNiO2 crystal
structure with three twin domains, along with the scattering geometry of the
RIXS experiment. The $a$, $b$, and $c$ axes of each domain are indicated, and
the square-lattice coordination of Ni (gray) and O (red) is shown for two NiO2
planes in each domain, while La ions between the planes are omitted for
clarity. The incident photons ($\mathbf{k}_{\rm i}$) are linearly polarized
either parallel ($\pi$-pol.) or perpendicular ($\sigma$-pol.) to the
scattering plane (yellow). A polarization analysis of the the outgoing photons
($\mathbf{k}_{\rm f}$) was not performed. The incident angle $\theta$ is
defined as the angle between $\mathbf{k}_{\rm i}$ and the sample surface.

###  III.2 RIXS experiment

The scattering geometry of the RIXS experiment together with the orientation
of the three twin domains is depicted in Fig. 3(b). We denote the momentum
transfer $(H,K,L)$ in reciprocal lattice units $(2\pi/a,2\pi/b,2\pi/c)$, with
the lattice parameters $a,b=3.9642(3)$ Å, and $c=3.3561(3)$ Å determined from
the powder XRD refinement of a crushed LaNiO2 crystal (see Supplemental
Material). For simplicity, we mostly focus on domain $A$ in the following,
where the scattering plane is parallel to the crystallographic $a$-$c$ plane
[see Fig. 3(b)].

Figure 4: (a) RIXS intensity map of the LaNiO2 crystal for incident photon
energies varied across the Ni $L_{3}$ edge, taken at $\theta=51.2^{\circ}$.
The corresponding XAS measured in total electron yield mode is superimposed as
the solid white line. The increase of the XAS signal towards low incident
energies is due to the tail of the strong La $M_{4}$-line centered below 851
eV. (b) RIXS spectra for three selected incident photon energies. Arrows at
$\sim 0.6$ and $\sim 0.15$ eV indicate characteristic low energy excitations,
as described in the text.

###  III.3 Orbital excitations

Figure 4(a) shows the RIXS spectra of the polished LaNiO2 crystal, with the
incident photon energy varied across the Ni $L_{3}$-edge. The major spectral
weight is centered around 1.2 eV energy loss for incident energies between
$E_{i}=852.4$ and 853.4 eV, which roughly coincides with the energy range of
the main Ni $L_{3}$-edge peak in the XAS [see solid white line in Fig. 4(a)].
In addition, a sequence of excitations at higher energy losses can be observed
in the RIXS intensity map, along with a fluorescence emission feature that
streaks across the map up to the highest energies. Overall, the RIXS features
in Fig. 4(a) are closely reminiscent of those reported for LaNiO2 thin films
[39], attributable to $dd$ excitations that emerge roughly between 1 and 3 eV
energy loss. Subtle differences between the relative intensities of individual
$dd$ excitation peaks are likely associated with the presence of three
tetragonal twin domains in our LaNiO2 crystals, whereas the RIXS map in Ref.
[39] was obtained from a single-domain LaNiO2 film.

Importantly, a close inspection of the region around 0.6 eV energy loss in
Fig. 4(a) reveals the presence of spectral intensity for incident energies
around $E_{i}=852.3$ eV. In prior thin film studies, this feature has been
identified as a hallmark of the infinite-layer nickelates, originating from
hybridized La-$5d$ and Ni-$3d$ orbital states [39, 41]. The 0.6 eV feature
becomes better discernible when the RIXS spectrum for $E_{i}=852.3$ eV is
plotted separately [see dark blue arrow in Fig. 4(b)]. In addition, the RIXS
spectra for $E_{i}=853.3$ and 853.9 eV are displayed in Fig. 4(b). While
spectral weight in the latter spectrum shifts towards energy losses above 1.5
eV, we observe a relatively sharp accumulation of spectral intensity below 0.3
eV in the former spectrum (see light blue arrow). The maximum of this spectral
weight is situated around $\sim$0.15 eV. This energy range is above that of
phonons, but it is compatible with the bandwidth of spin excitations reported
in previous thin film studies of infinite layer nickelates [50, 28, 27, 29].

###  III.4 Spin excitations

To elucidate the characteristics of the possible spin excitations, we
investigate the momentum-dependence of this spectral weight (Fig. 5). To
account for photon absorption effects in the crystal for different incident
angles, we perform a self-absorption correction of the RIXS data [16, 12].
Details about the employed procedure can be found in the Supplemental
Material. Notably, the RIXS spectra along the $(H,0)$ [Fig. 5(a)] and $(H,H)$
[Fig. 5(b)] directions reveal that the spectral weight is broadly distributed
and extends up to $\sim$300 meV for all measured momenta. This is in contrast
to the highly dispersive spin wave signal in the RIXS spectra of the nickelate
thin films [50, 28, 27], although the maximum bandwidth across the Brillouin
zone is comparable. A subtle increase in spectral intensity is discernible
towards higher momenta along both directions, mostly within the energy range
below 100 meV, which is close to the energy regime of phonons. In particular,
a prominent phonon peak centered around 60 meV has been reported in Ni
$L_{3}$-edge RIXS experiments on LaNiO2 thin films [28]. Nonetheless, when
comparing the RIXS spectra for representative momenta along the $(H,0)$
direction [Fig. 5(a)] it appears that the most significant change occurs
between 80 and 100 meV. Hence, it is plausible that the broad distribution of
low-energy spectral weight in Figs. 5(a) and 5(b) contains contributions from
elastic scattering mostly below 40 meV, a phonon around 60 meV, and spin
excitations or other contributions at higher energies up to 300 meV.

Figure 5: (a) RIXS spectra of the LaNiO2 crystal for varied momenta along the
$(-H,0)$ direction, acquired with $\pi$-polarized photons. The dashed curves
are Gaussian fits of the elastic line centered at zero-energy loss, using an
energy resolution of 37 meV. Substantial spectral weight extends up to $\sim$
300 meV, but shows essentially no dispersion as a function of momentum
transfer. (b) RIXS spectra along the $(-H,-H)$ direction.

A difference to single-domain thin films [50, 28, 27] is that the RIXS signal
in our experiment contains the superposition of the three twin domains. To
test whether this superposition is responsible for the observed “washed-out”
character of the spin excitations in Figs. 5(a) and 5(b), we simulate spin-
wave spectra based on the linear spin-wave theory for the simultaneous
presence of three orthogonal crystallographic orientations. Specifically, we
employ a $J_{1}$-$J_{2}$ model for a square-lattice Heisenberg
antiferromagnet, which was also used to model the dispersion measured from an
IL nickelate film [50]. As an approximation, we assume that the RIXS intensity
is proportional to the dynamical structure factor of the spin-wave excitations
[64, 65]. The magnetic Hamiltonian is formulated as follows:

${\cal
H}=J_{1}\sum_{i,j}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}+J_{2}\sum_{i,k}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{k},$
(1)

where $\hat{\mathbf{S}}_{i}$ is a spin operator with spin $S=1/2$ at a site
$i$. $J_{1}$ and $J_{2}$ are the nearest and next-nearest neighbor exchange
interactions in the $ab$ plane, respectively. We adopt the reported values
$J_{1}=63.6$ and $J_{2}=-10.3$ meV from a NdNiO2 thin film study [50], where
the spin waves observed in the RIXS spectra exhibited close similarities to
those in square-lattice cuprates. Accordingly, we define a collinear magnetic
structure with an antiferromagnetic ordering vector $\mathbf{Q}=(1/2,1/2)$
[15, 66]. Using the linear spin wave theory, the dispersion relation is given
by:

$\hbar\omega_{\mathbf{q}}=2Z_{c}\sqrt{A_{\mathbf{q}}^{2}-B_{\mathbf{q}}^{2}},$
(2)

where

$\displaystyle A_{\mathbf{q}}$ $\displaystyle=$ $\displaystyle
J_{1}-J_{2}\left\\{1-\cos{(2\pi H)}\cos{(2\pi K)}\right\\}$ (3) $\displaystyle
B_{\mathbf{q}}$ $\displaystyle=$ $\displaystyle J_{1}\left\\{\cos{(2\pi
H)}+\cos{(2\pi K)}\right\\}/2.$ (4)

The renormalization factor $Z_{c}$ is fixed at 1.18 [67], which is expected
for a two-dimensional $S=1/2$ square-lattice antiferromagnet. Further, the
inelastic part of the dynamical structure factor is given by:

$S(\mathbf{q},\omega)\propto\sqrt{\frac{A_{\mathbf{q}}-B_{\mathbf{q}}}{A_{\mathbf{q}}+B_{\mathbf{q}}}}\delta\left(\omega-\omega_{\mathbf{q}}\right).$
(5)

In analogy to paramagnons in cuprates, which emerge in the absence of long-
range order, also the spin excitations in IL nickelates feature a large
linewidth [50]. Hence, we convolute our calculated spin spectra with a damped
harmonic oscillator function. We employ a damping factor of 74 meV, in accord
with the average of the momentum-dependent damping factors in Ref. [50].
Furthermore, our damped spectra are convoluted with Gaussian functions along
the energy and momentum directions according to the RIXS experimental energy
resolution of 37 meV and the $q$ resolution of 0.09 Å-1, respectively The $q$
resolution is estimated from the mosaicity of the LaNiO2 crystal according to
the full width at half maximum of the $(001)$ Bragg peak.

Figure 6: Spin wave simulation for a system with three twin domains and the
scattering geometry used in the RIXS experiment, with varying incident angle
$\theta$, fixed scattering angle at 154∘, and $E_{i}=853$ eV. (a)-(c) False-
color plots (top panels) of the simulated spin-wave spectra of the twin
domains $A$, $B$, and $C$, respectively. The intensity scale corresponds to
the dynamical structure factor $S(\mathbf{q},\omega)$ and the solid black
lines are the dispersion relations. The solid red lines (bottom panels)
indicate the corresponding momentum transfer for each domain within the $H-L$,
$L-H$, and $K-H$ planes, respectively. (d)-(g) Superposition of
$S(\mathbf{q},\omega)$ from the three domains assuming different populations
of the domains $A$, $B$, and $C$. For better comparability to the RIXS data in
Fig. 5(a), the momenta in panel (d) are indexed according to domain $A$ with
variation along the $(H,0)$ direction, although the calculated
$S(\mathbf{q},\omega)$ contains the contributions from all three domains. The
domain population ratios in panels (d)-(f) are $1:1:1$, $1:1:0.3$, and
$0.4:1:0.1$, respectively. In panel (f), an almost non-dispersive behavior of
the spin wave is reproduced, which can be further enhanced by taking into
account a 25% volume fraction with randomly oriented grains or disordered
crystal structure (powder).

Figures 6(a)-6(c) present the simulated spin wave spectra for three orthogonal
crystal orientations, each contributing a distinct spin-wave signal to the
RIXS spectrum. The three orientations represent the twin domains illustrated
in Fig. 3(b). In our RIXS experiment, the momentum transfer is varied by
changing the incident angle $\theta$ [see Fig. 3(b)], while keeping the
scattering angle fixed at $154^{\circ}$. As a consequence, a change of the in-
plane momentum transfer also involves a change of the out-of-plane momentum.
For a single-domain sample, such as an IL nickelate thin film, the change in
the out-of-plane momentum is usually neglected since the dispersion of the
spin waves along the $L$ direction is flat for a quasi-two-dimensional square
lattice antiferromagnet. However, for the simultaneous presence of three twin
domains the situation is more involved and hence requires the separate
consideration of each domain [Figs.6(a)-6(c)]. The corresponding trajectories
in reciprocal space for the three domains when the incident angle is varied
from $\theta=20^{\circ}$ to 134∘ are shown in the bottom panels in Figs.
6(a)-6(c).

Assuming an equal population $(1:1:1)$ of the $A$, $B$, and $C$ twin domains,
we obtain a composite spectrum shown in Fig. 6(d). The simulated peak
positions disperse significantly more weakly as a function of $H$ than the
spin waves of single-domain thin films [50]. However, a strong increase in
intensity for the highest momenta $H$ in Fig. 6(d) is not present in our RIXS
data in Fig. 5(a). Considering that the typical lateral dimensions of the
domains in the LaNiO2 crystal are between a few and several tens of
micrometers [Fig. 3(a) and Supplemental Material] while the footprint of the
x-ray beam on the sample surface is 2.5 $\mu$m in the vertical direction and
more than 40 $\mu$m in the horizontal direction (depending on the incident
angle $\theta$), it is plausible to assume that the population of the three
domains in the probed sample volume was not equal. Accordingly, we next
simulate unequal domain populations of $1:1:0.3$ and $0.4:1:0.1$ in Figs. 6(e)
and 6(f), respectively. A notable trend in these simulations is a further
reduction of the dispersive character and intensity enhancement, aligning more
closely with the RIXS data presented in Fig 5(a).

As a final step, we take into account the presence of the textures with dark
lines in our LaNiO2 crystals [Figs. 2(a)-2(c)], indicative of regions with a
disordered crystal structure. From Fig. 2 and optical microscope images (see
Supplementary Material) we estimate that these regions contribute at most 25%
of the the sample’s volume fraction, and we incorporate this sample
characteristic as powder-averaged spin wave spectra in our simulation.
Although the simulation result [Fig. 6(g)] does still not exactly match our
RIXS data, the trend is notably consistent. Considering that we used the
damping and magnetic exchange parameters from previous thin film studies as
sole input for our simulation, it suggests that the magnetic correlations
within a single domain of our crystals closely resemble those in single-domain
films. The remaining discrepancy observed in Fig. 6(g) might be due to
increased damping of the spin waves in the crystals or other factors,
highlighting the need for further investigation.

###  III.5 Charge order

Next, we examine whether the charge order phenomenology reported for IL
nickelate thin films similarly occurs in crystals. Specifically, a number of
previous RIXS studies on thin films performed under comparable experimental
conditions, reported the occurrence of a CDW signal in the form of enhanced
quasielastic scattering around $H=\pm 1/3$ [28, 27, 26].

Figure 7(a) shows a RIXS intensity map of the region of quasielastic
scattering of the LaNiO2 crystal for different momenta along the $(H,0)$
direction, utilizing $\sigma$-polarized incident photons to enhance the charge
signal. Note that these RIXS spectra were obtained from a different surface
spot on the LaNiO2 crystal compared to those in Fig. 5. Hence, the underlying
domain population is likely distinct, and for simplicity the momentum transfer
in Fig. 7 is indexed with respect to domain $A$. The self-absorption
correction [16, 12] was applied to all spectra in Fig. 7(a).

Remarkably, we find that the data from our LaNiO2 crystal lack clear charge
ordering signatures around $H=\pm 1/3$ [Fig. 7(a)]. This absence of an
enhanced quasielastic signal is corroborated when examining the integrated
RIXS intensity between $-50$ meV $\leq E\leq 50$ meV energy loss [red data
points in Fig. 7(b)]. For a direct contrast, we also plot LaNiO2 film data
from Ref. [28] in Fig. 7(b). Note that we have normalized both our RIXS data
and those of Ref. [28] by the integrated intensity of the respective $dd$
excitations to ensure direct comparability.

Figure 7: (a) RIXS intensity map of the LaNiO2 crystal for momenta varied
along the $(\pm H,0)$ direction, acquired with $\sigma$-polarized incident
photons. (b) Integrated intensity of the quasielastic scattering along the
$(\pm H,0)$ direction. The integration range is $-50$ meV $\leq E\leq 50$ meV
energy loss. Error bars are determined by the standard deviation of six RIXS
spectra measured at each momentum. In addition, the quasielastic intensity
from a RIXS experiment on a LaNiO2 thin film in Ref. [28] is superimposed as
gray symbols, exhibiting a distinct CDW peak around $H=0.344$. The RIXS
intensities of the crystal and the film are both normalized to the integrated
intensity of the $dd$ excitations (see Supplementary Material). In addition, a
vertical offset is added to the film intensity to match the baseline of the
crystal data.

While the pronounced $H=\pm 1/3$ peaks of Ref. [28] are clearly absent in our
data, we observe a broad peak-like feature around $H\sim 0.27$, which is
subtle but exceeds the range of our experimental error bars. Yet, for momentum
transfer in the opposite direction, a corresponding peak feature at
$H\sim-0.27$ is not visible. Instead, the quasielastic scattering intensity in
this direction increases continuously from $H\sim-0.15$ to -0.3. To verify
that these behaviors are not associated with CDW signals, we carried out
additional RIXS measurements on other locations on the surface of the same
crystal as well as on two additional crystals (see Supplementary Material),
which did not reproduce the peak at $H\pm 0.27$. In consequence, we attribute
such deviations from the baseline intensity of quasielastic scattering to
artifacts resulting from the partially uneven surface topography of our LaNiO2
crystals [see inset in Fig. 1(a) and Supplementary Material].

Nonetheless, due to the presence of three twin domains in our LaNiO2 crystals,
we cannot fully rule out that a CDW signal has remained undetected in our
experiment. For instance, one scenario could involve an unfavorable domain
distribution within the surface area probed by RIXS in Fig. 7 where the
domains $A$ and $B$ would be essentially unpopulated, while for the dominating
domain $C$ our applied momentum transfer range would not cover the nominal
$\mathbf{q}=(\pm 1/3,0)$ ordering vector. However, this scenario seems
implausible, as our EBSD analysis of various crystal surface areas (see
Supplemental Material) revealed that typically at least two twin domains
exhibit significant population within an area corresponding to the x-ray beam
footprint of approximately 2.5 $\mu$m $\times$ 40 $\mu$m. Moreover, our search
for a CDW signal encompassed different spots on multiple LaNiO2 crystals (see
Supplemental Material), making it even less likely that we consistently
encountered unfavorable domain distributions.

The absence of the $\mathbf{q}=(\pm 1/3,0)$ CDW peaks in our LaNiO2 sample is
particularly noteworthy when considered in conjunction with the STEM
characterization of our crystal [Fig. 2(f)]. Specifically, the absence of any
traces of residual apical oxygen indicates that the ideal IL structure is
realized in our crystals. This concomitant absence of apical oxygen and a CDW
signal is compatible with the notion that excess oxygen arranged in patterns
characterized by a $\mathbf{q}=(\pm 1/3,0)$ ordering vector is a prerequisite
for the emergence of charge order in nickelates [54, 55].

##  IV Conclusion

In summary, our results have several important implications for the ongoing
debate about spin and charge instabilities in the material class of IL
nickelates. In particular, our observation of spin excitations with a
bandwidth of more than 200 meV akin to those reported for thin films [50, 27,
28] suggests that the associated spin dynamics are an ubiquitous feature of
the pure IL phase. Nevertheless, the absence of a discernible spin wave
dispersion in our data, presumably due to a distribution of twin domains
within the probed region on the crystal surface, calls for further
investigations. In particular, the synthesis of single-domain LaNiO2 crystals
becomes imperative for future experimental work, potentially achievable by
applying strain or pressure during the topotactic reduction, akin to the
epitaxial strain conditions that yield single-domain IL thin films [68, 69,
70].

Furthermore, the lack of a $\mathbf{q}=(\pm 1/3,0)$ CDW signal in our data a
suggests that charge order is not intrinsic to the bulk phase of IL
nickelates. This notion is compatible with the scenario that a secondary phase
of residual apical oxygen ions hosts the CDW [54, 55], or that epitaxial
strain from a substrate plays a role in the emergence of charge order in IL
nickelates [29]. In both cases, further investigations are warranted to
clarify if analogies between the $\mathbf{q}=(\pm 1/3,0)$ nickelate charge
order phenomenology and that in cuprates [18, 19, 20, 21, 22, 23, 24, 25]
exist.

Given our focus on the undoped parent compound LaNiO2 and axial directions up
to wave vectors of $\mathbf{q}=(\pm 0.4,0)$, our results do not rule out the
existence of alternative charge ordering patterns in IL nickelates with
different vectors and/or an emergence upon hole doping. This highlights the
need for future studies on IL nickelate bulk crystals and thin films with
assured stoichiometric oxygen content, with a focus either on exploring the
phase space suggested by the CDW in cuprates, or undertaken as a comprehensive
survey of the entire experimentally accessible reciprocal space and doping
parameters.

###### Acknowledgements.

We thank W.-S. Lee and J. Weis for helpful discussions, and C. Busch, U.
Waizmann, and M. Hagel for technical support. We thank J. Deuschle for FIB
lamella preparation. We acknowledge Diamond Light Source for providing the
beamtime under the proposal MM33241-1, and are grateful to the technical staff
for assistance during the experiment.

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