# Soft X-ray Spectroscopy of Low-Valence Nickelates

Matthias Hepting<EMAIL_ADDRESS>Max-Planck-Institute for Solid State
Research, Stuttgart, Germany
Mark P. M. Dean  Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, NY, USA
Wei-Sheng Lee  Stanford Institute for Materials and Energy Sciences, SLAC
National Accelerator Laboratory and Stanford University, Menlo Park, CA, USA

###### Abstract

Low-valence nickelates — including infinite-layer (IL) and trilayer (TL)
compounds — are longstanding candidates for mimicking the high-temperature
superconductivity of cuprates. A recent breakthrough in the field came with
the discovery of superconductivity in hole-doped IL nickelates. Yet, the
degree of similarity between low-valence nickelates and cuprates is the
subject of a profound debate for which soft x-ray spectroscopy experiments at
the Ni $L$\- and O $K$-edge provided critical input. In this review, we will
discuss the essential elements of the electronic structure of low-valance
nickelates revealed by x-ray absorption spectroscopy (XAS) and resonant
inelastic x-ray scattering (RIXS). Furthermore, we will review magnetic
excitations observed in the RIXS spectra of IL and TL nickelates, which
exhibit characteristics that are partly reminiscent of those of cuprates.

##  I Introduction

Spectroscopy has proven to be a versatile tool for studying the charge, spin,
lattice, and orbital degrees of freedom in quantum materials. Arguably, one of
the most fascinating families of quantum materials are cuprate high-
temperature superconductors [1, 2, 3]. Critical insights into cuprates have
been provided by electron, neutron, and photon spectroscopy techniques —
including photoemission and electron spectroscopy, scanning tunneling
microscopy, optical and Raman spectroscopy, as well as inelastic x-ray and
neutron scattering [4, 5, 6, 7, 8, 9, 10]. More recently, fresh perspectives
on the enigmatic ground states of cuprates opened up especially due to
resonant soft x-ray spectroscopy studies. For instance, resonant inelastic
x-ray scattering (RIXS) [11] revealed that damped spin excitations persist
even for high hole-doping levels far away from the parent
antiferromagnetically (AFM) ordered phase [12, 13]. Furthermore, RIXS allowed
to probe the three-dimensional (3D) dispersion of low-energy plasmons [14, 15,
16], which arise due to the characteristic quasi-2D layered crystal structure
of cuprates. Moreover, resonant elastic x-ray scattering (REXS) and RIXS
studies found that different types of static and dynamic charge orders emerge
ubiquitously in cuprates [17, 18, 19, 20, 21]. Nevertheless, a comprehensive
understanding of the most prominent phases in cuprates — including the
pseudogap, strange metal phase, and superconductivity — remains elusive.

One approach to gain a deeper understanding of cuprates involves the targeted
design of materials that mimic cuprate-typical properties, such as their
layered quasi-2D crystal structure, 3$d^{9}$ electronic configuration, spin
$S=1/2$ magnetic moments with antiferromagnetic (AFM) coupling, strong ligand-
oxygen hybridization, and a lifted degeneracy of the active $e_{g}$ orbitals
[22]. In principle, the discovery of superconductivity in a material that
emulates at least a subset of these properties could allow to identify the
hallmarks of cuprates that are crucial for invoking their exceptional high-
temperature superconductivity.

In this context, long-standing candidates are Ni-based compounds, as Ni is a
direct neighbor of Cu in the periodic table. In early works, it was speculated
that doped $RE_{2}$NiO4 ($RE$ = rare-earth ion), which is the $n=1$ member of
the Ruddlesden-Popper (RP) homologous series REn+1NinO3n+1 [23], could become
superconducting [24], as it is isostructural to the cuprate La2CuO4 and
possesses similar charge and spin stripe ordered states [25]. The formal
electronic configuration of La2NiO4, however, is 3$d^{8}$ (Ni2+) with $S=1$
[26, 27], providing a possible rationalization for the observed absence of
superconductivity. Perovskite nickelates $RE$NiO3 are the $n=\infty$ member of
the RP series with a formal 3$d^{7}$ (Ni3+) configuration and $S=1/2$,
although x-ray spectroscopic experiments indicated a 3$d^{8}\underline{L}$
configuration [28], with $\underline{L}$ denoting an O ligand hole. In their
seminal work, Chaloupka and Khaliullin proposed a significant modification of
the electronic structure of perovskite nickelates via tensile strain and
incorporation into thin epitaxial heterostructures [29], breaking of the
degeneracy of the $e_{g}$ orbitals, which in the extreme case could yield a
half filled 3$d_{x^{2}-y^{2}}$ band and a single-sheet, cuprate-like Fermi
surface [30]. In fact, spectroscopic studies detected new electronic and
magnetic ground states in perovskite nickelate heterostructures, distinct from
those of bulk nickelates [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
Nevertheless, superconductivity has not been found to date [42, 43, 44],
possibly due to the insufficient splitting of the orbital energy levels in the
realized heterostructures [45, 46]. More recently, oxygen-reduced variants of
the $n=3$ members of the RP series have attracted intense attention. These
RE4Ni3O8 trilayer (TL) nickelates are composed of three closely stacked
square-planar Ni-O layers with intervening $RE$ ions, which are separated by
rocksalt type $RE$-O blocking structures [47]. The electronic configuration of
TL nickelates is 3$d^{8.67}$ per Ni on average (Ni1.33+). It was found by
x-ray spectroscopy that metallic Pr4Ni3O8 exhibits low-spin configuration and
a significantly lifted orbital degeneracy, suggesting a close analogy to
cuprates [48]. Notably, the close parallel between TL nickelates and cuprates
was further corroborates by RIXS, revealing the presence of strong AFM
exchange coupling $J$ in La4Ni3O8 and Pr4Ni3O8 [49]. Nonetheless,
superconductivity has not been detected in TL nickelates. This could be due to
their electronic configuration, which corresponds to 1/3-hole-doping of a Ni1+
background and therewith falls into the overdoped regime of cuprates [48, 50,
51]. Experimental efforts to stabilize superconductivity via lowering the
Ni1.33+ valence by electron-doping are ongoing. Along these lines, also the
$n=5$ oxygen-reduced RP variants RE6Ni5O12 are promising candidates for
superconductivity, as these quintuple layer nickelates exhibit a 3$d^{8.8}$
(Ni1.2+) configuration — analogous to optimally doped cuprates — already
without additional electron-doping.

A breakthrough in the field came with the discovery of superconductivity in
hole-doped nickelates with the IL crystal structure [52]. In more detail,
epitaxial thin films of Sr- or Ca-substituted $RE$NiO2 obtained via topotactic
oxygen reduction of the perovskite phase show superconductivity below 9 - 15 K
[52, 53, 54, 55, 56, 57, 58]. The parent compounds of these nickelates
formally exhibit the 3$d^{9}$ (Ni1+) configuration with $S=1/2$, which
qualifies them as isostructural and isoelectronic to the parent cuprates.
Early neutron powder diffraction studies of parent IL nickelates, however,
indicated absence of long-range AFM order [59, 60] and electrical transport
measurements of films show weakly metallic behavior [61]. This is in stark
contrast to parent cuprates, which are AFM Mott (charge-transfer) insulators
[1]. Moreover, whereas first theoretical studies proposed that the electronic
and magnetic correlations of IL nickelates and cuprates share close
similarities [62], other theoretical works suggested significant distinctions,
including a multiband character of nickelates [63]. Along these lines,
insights from experiments can help to resolve the controversy about
similarities and differences between IL nickelates and cuprates. In
particular, recent x-ray and electron energy-loss spectroscopic studies [64,
65, 66] unveiled a reduced Ni-O hybridization, presence of a weakly
interacting $RE$ 5$d$ metallic band, and overdamped spin excitations with a
bandwidth as large as 200 meV in IL nickelates.

In the following, we will review recent soft x-ray absorption spectroscopy
(XAS) and RIXS studies at the O $K$-edge and Ni $L$-edge of IL and TL
nickelates. We will discuss the essential elements of their distinct
electronic structure. Furthermore, the spin excitation spectra of IL and TL
nickelates observed with RIXS will be reviewed.

##  II Electronic structure

XAS at the O $K$-edge measures core-hole excitations from O 1$s$ to unoccupied
O 2$p$ states and is also a sensitive probe of the covalent mixing between O
2$p$ and transition-metal $d$ states [67]. In particular the O-$K$ pre-edge
fine structure can provide valuable information about Ni-O hybridized states
and the associated electronic structure, for instance in the cases of NiO and
perovskite $RE$NiO3 [68, 67, 28]. In both materials, Ni 3$d$-orbitals strongly
hybridize with oxygen ligands, giving rise to a pre-peak in the absorption
spectra near the O $K$-edge (Fig. 1A). Due to different relative energy scales
between the charge-transfer energy $\Delta$ and the Coulomb interaction $U$,
according to the Zaanen–Sawatzky–Allen (ZSA) scheme [69], the former material
falls into the regime of charge-transfer insulators, whereas the latter is a
negative charge-transfer compound. In contrast, the O $K$-edge absorption
spectra of the IL nickelates LaNiO2 and NdNiO2 lack a prominent pre-edge peak
(Fig. 1A), suggesting a substantially weaker effective mixing between oxygen
and the unoccupied 3$d$ states of the upper Hubbard band (UHB) of the Ni1+
cations [64]. In the case of cuprates, a prominent pre-peak feature is present
in O $K$-edge absorption spectra [70]. This is known to originate from the
charge-transfer nature of these materials, with $\Delta$ smaller than $U$, and
O 2$p$ states mixed with both the lower Hubbard band (LHB) and the UHB of the
Cu 3$d_{x^{2}-y^{2}}$ states. A sizable pre-peak has also been observed in TL
nickelates (Fig. 1C), indicating mixing between O 2$p$ and the Ni UHB [48,
49]. Upon hole-doping of cuprates, spectral weight shifts from the UHB pre-
peak to a lower-energy peak (Fig. 1C) that is associated with transitions into
the doped hole levels [70, 71]. These hole-states constitute Zhang-Rice
singlets (ZRS), which are plaquettes of two doped holes and four O atoms in
square planar coordination around a Cu atom, playing the same role as a fully
occupied or empty site in an effective single-band Hubbard model [72].
Notably, a recent scanning transmission electron microscopy electron-energy
loss spectroscopy (STEM-EELS) study suggested that a similar ZRS peak emerges
in IL nickelates upon hole-doping [65], although it carries significantly less
spectral weight (Fig. 1C). Overall, there is good evidence for reduced oxygen
hybridization in nickelates compared to cuprates, which likely comes from an
enhanced $\Delta$ value. Direct and quantivative determination of $\Delta$ and
$U$ represents an important issue for future soft x-ray studies.

Further insights into the electronic structure can be obtained from Ni
$L$-edge XAS, corresponding to 2$p$-3$d$ multiplet transitions, reflecting the
valence configuration of the Ni ions. In the cases of NiO and perovskite
$RE$NiO3 (2$p^{6}$3$d^{8}$–2$p^{5}$3$d^{9}$ and
2$p^{6}$3$d^{8}L^{n}$–2$p^{5}$3$d^{9}L^{n}$ transitions, respectively)
distinct multi-peak structures emerge across the $L_{3}$-edge (Fig. 1B) [28,
73, 64]. Conversely, the line shapes of the IL nickelates LaNiO2 and
especially of NdNiO2 (Fig. 1B) resemble rather the single-peak XAS spectrum of
IL cuprates with only one possible final XAS state
(2$p^{6}$3$d^{9}$-2$p^{5}$3$d^{10}$ transition). In more detail, the
$L_{3}$-edge XAS of LaNiO2 and NdNiO2 is dominated by a main peak A (Fig. 1B),
while LaNiO2 shows an additional minor low energy shoulder A’ at slightly
lower energies.111The A’ feature in XAS is only visible in LaNiO2 films
without a SrTiO3 (STO) capping layer. Based on our recent measurements on
LaNiO2 films with a STO capping layer, the A’ feature, which arises from the
resonance of the $\sim$ 0.6 eV feature in the RIXS map, coincides with the the
main XAS peak and becomes invisible. In other words, the XAS of La- and Nd-
based infinite layer nickelates (with a STO capping layer) are essentially the
same. Figure. 1D displays the RIXS intensity map of NdNiO2 as a function of
the incident photon energy and Fig. 1E shows RIXS spectra for selected
incident energies. Importantly, the RIXS spectra of LaNiO2 and NdNiO2 exhibit
a distinct feature around 0.6 eV energy loss (Fig. 1D,E), which is visible in
the RIXS spectra with incident energies coinciding with the XAS peak A (A’).
Furthermore, this feature emerges exclusively in the IL compounds and not the
perovskite nickelate LaNiO3 (Fig. 1E). Using exact diagonalization, the
general XAS and RIXS features can be reproduced (Fig. 1F) and the 0.6 eV
feature can be assigned to the hybridization between the Ni 3$d_{x^{2}-y^{2}}$
and La 5$d$ orbitals, involving a charge-transfer from Ni to the $RE$ cation
[64]. From LDA+$U$ it is found that a Fermi pocket of mainly La 5$d$ character
forms near the $\Gamma$ point, which is quite extended and three-dimensional.
On the other hand, the Ni 3$d_{x^{2}-y^{2}}$ states in the NiO2 planes are
quasi-2D and strongly correlated [64, 74]. Nevertheless, the relevance of the
rare-earth 5$d$ bands for the low-energy physics of IL nickelates is still
under debate and proposals range from effective single-band to multi-orbital
models, including various Ni 3$d$ and rare-earth 5$d$ as well as interstitial
orbitals [62, 63, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89].
Hence, future experiments probing the Fermi surface topology, such as angle
resolved photoemission (ARPES) and quantum oscillation measurements, are
highly desirable.

##  III Magnetic correlations

###  III.1 Magnetic excitations in infinite-layer nickelates

Despite of the involvement of rare-earth 5$d$ states, the fact that the
electronic structures of the Ni 3$d$ states resemble a cuprate-like 3$d^{9}$
system raises a curious question: whether the Mott-physics, a key ingredient
in the cuprate phenomenology [90], also play an important role in sculpting
the electronic structures in IL nickelates. Since a strong AFM interaction is
a consequence of Mott physics due to strong onsite Coulomb interaction,
information about the magnetic structures in IL nickelate is imperative to
gain further insight into this issue. Early investigations of bulk
polycrystalline LaNiO2 and NdNiO2 found no evidence of AFM order [59, 60],
which appeared to suggest a significantly weaker magnetic interaction than in
cuprates. On a different ground, theories have been debating the energy scale
of magnetic interactions in the IL nickelates. Some theories predict a small
AFM interaction ($\sim$ an order of magnitude smaller than that of cuprates)
because of the larger charge transfer energy $\Delta$ [91, 92, 93, 94].
Conversely, other theories argue that the magnetic interactions are comparable
to those in cuprates [74, 95, 92]. Experimental information about magnetic
excitations is crucial to clarify this important issue.

Recently, magnetic excitations in Nd1-xSrxNiO2 have been revealed using RIXS
at the Ni $L_{3}$-edge [66]. As shown in Fig. 2A, a branch of dispersive
magnetic excitations has been observed in NdNiO2, whose energy-momentum
dispersion resembles the spin wave excitations of AFM coupled spins in a
square lattice. Importantly, the bandwidth of the magnetic excitations is
approximately 200 meV, corresponding to a nearest neighbor spin interaction
$J_{1}\sim$ 65 meV. This is about half of the $J_{1}$ in cuprate
superconductors and similar to that in the TL nickelates [49], but is notably
higher than in the stripe-ordered single-layer ($n=1$) nickelates [25, 26,
27], perovskite nickelates ($n=\infty$) [96], and cubic NiO [97, 98].
Therefore, the observation of the high energy scale of $J_{1}$ in IL
nickelates confirms the presence of a strong onsite Coulomb interaction,
indicating that the strong correlation effect associated with the Mott-physics
is likely also at play in the nickelate superconductors. Notably, distinct
from the sharp magnetic modes observed in undoped cuprates, the magnetic
excitations in the undoped parent compound of IL nickelates are damped, which
is likely due to the coupling to the metallic Nd 5$d$ states.

Upon hole doping, the magnetic excitations become less dispersive as a
function of momentum and significantly damped. By fitting the spectrum to a
damped harmonic oscillator function, it is found that mode energies soften
accompanied by slightly reduced spectral weight (Fig. 2B). The observed doping
dependence is consistent with spin dilution in a Mott insulator. This is in
fact different from those observed in cuprates, in which the mode energy and
spectral weight do not decrease with increasing doping [99]. The doping
dependence of the magnetic excitations in cuprates has been attributed to the
longer-range charge dynamics emergent with increasing hole doping, for
example, the three site terms in a Hubbard model [100, 101]. Such dynamics
appear to be less prominent in the doped IL nickelates, likely due to the
larger charge transfer energy $\Delta$ and the presence of the rare-earth 5$d$
metallic state, calling for further investigation.

We note the next nearest-neighbour exchange interaction $J_{2}$ extracted from
the magnetic excitations dispersion possesses an opposite sign to the nearest
neighbor $J_{1}$, which should favor the formation of AFM ordering at (0.5,
0.5). Unfortunately, RIXS at the Ni $L_{3}$-edge cannot reach (0.5, 0.5) due
to insufficient momentum transfer of the photons, preventing a direct
scrutinization on the putative AFM order. Notably, recent susceptibility
measurement on bulk powder samples indicated spin glass behaviors, but
signatures of an AFM phase transition were still not observed. Thus, it would
be interesting to investigating why IL nickelates are a failed AFM. However,
one should be cautious about the difference between bulk and thin film
samples, as well as the disorders in both types of materials, which were
significantly reduced over time along with the optimization of material
synthesis protocols.

Interestingly, the IL nickelates add one more case in which the magnetic
correlations are in proximity to superconductivity in the phase diagram,
similarly to a number of unconventional superconductors, such as cuprates,
iron-based superconductors, and heavy fermion superconductors [102]. It might
be tempting to attribute magnetic fluctuations as a candidate mechanism of
superconductivity. However, among these superconducting compounds, including
the nickelate superconductors, there appears no clear correlation between the
energy scale of the magnetic excitation and the superconducting transition
temperature, casting doubt on this notion. In any case, the relationship
between magnetic fluctuations and the superconductivity remains an important
issue in nickelate superconductors.

###  III.2 Magnetic excitations in trilayer nickelates

In parallel with the measurement of magnetic excitations in the IL material
Nd1-xSrxNiO2, magnetic excitations were also measured in the TL materials
La4Ni3O8 and Pr4Ni3O8 [49]. The crystal structure of La4Ni3O8 is shown in Fig.
2C. This material has some features that indicate that it might be especially
promising as a cuprate analog. The rock salt $RE$-O layers present in its
structure make it more two-dimensional than IL compounds and the rare-earth
orbitals that are populated in IL are predicted to have a less significant
role in TL systems [48, 103, 104]. As explained in the introduction, this
compound is naturally self-doped and has a nominal hole concentration of 1/3.
A disadvantage of the La4Ni3O8 series is that they have, to date, proven
difficult to chemically dope. Like cuprates, and some other complex oxides,
La4Ni3O8 has charge and spin order [47, 105]. This structure, illustrated in
Fig. 2D, features diagonal rows of Ni sites with enhanced hole character and
neighboring diagonal stripes of up and down spin-ordered sites with reduced
hole character. The overall magnetic dispersion, measured with Ni $L_{3}$ edge
RIXS, is plotted in Fig. 2E and features a bandwidth of $\sim 80$ meV with a
downturn near $(-\frac{1}{3},-\frac{1}{3})$, which is the charge and spin
stripe-ordering wavevector. La4Ni3O8 was modeled by solving a Heisenberg
Hamiltonian which accounts for stripes, and assumes complete charge
disproportionation into $d^{9}$ and $d^{8}$ sites, similar to prior studies of
other stripe-ordered cuprates and nickelates [106, 18] . This includes
$J_{1}$, which connects sites 1 and 2 in Fig. 2D, $J_{3}$, which couples atom
2, through the purple doped site, to atom 1 in the next unit cell and $J_{z}$,
which reflects interactions along the $c$-axis, for example, site 1 to site 3.
The diagonal $J_{2}$ interaction that was included in the analysis of NdNiO2
is not expected to be important here, as it couples to the sites with an
enhanced hole character and which would be spinless when hosting an extra hole
($S=0$) [48]. Solving this Hamiltonian in the spin-wave approximation yields
three modes. Since these modes could not be resolved separately, the RIXS
intensity of these modes was computed and summed to predict the intensity of
the magnetic feature in RIXS. Values of $J_{1}=69(4)$ meV and $J_{3}=17(4)$
meV were obtained by fitting a Hamiltonian of this type.222The notation used
here was been modified from the original work of Ref. [49] to facilitate
comparison with Ref. [66] Similar to the Nd1-xSrxNiO2 case, the effects of
$J_{z}$ were too small to be constrained by the experiment, so the theoretical
value of $J_{z}=13.6$ meV was used. Very similar dispersions were found in
Pr4Ni3O8, which may imply that dynamical stripes exist in this compound even
though long range stripe order has not been detected, as has been suggested
independently in muon spin rotation studies [107].

The leading value of $J_{1}=69(4)$ meV in La4Ni3O8 is strikingly close the
65(1) meV value obtained for NdNiO2. It should be noted that this similar
value arises from a much smaller magnetic bandwidth, as within the stripe-
ordered state, each Ni will have only two magnetic neighbors. An approximate
extrapolation of the magnetic dispersion of Nd1-xSrxNiO2 to a doping of $x\sim
1/3$ implies that it would have a bandwidth comparable to La4Ni3O8 at this
doping; yet, whether stripe order or fluctuations exist in the IL nickelates,
like those found in the TL nickelates, remains an important open question.
Overall, this suggests that the local correlated physics in these reduced RP
cousins is very similar provided they are compared at the same effective
doping, although their precise low-energy ground states might be more
different.

##  IV Conclusion

In summary, soft x-ray spectroscopic studies have provided valuable insights
into the physics of RP-phase and RP-derived nickelates. Nevertheless, for low-
valence nickelates there is still limited consensus on the essential
ingredients of their electronic structure. Along these lines, we anticipate
that advances in sample synthesis and the application of complementary
experimental techniques, including ARPES and quantum oscillation measurements,
will be helpful. Moreover, the role of disorder and capping layers, as well as
the the apparent differences between film and bulk samples need further
clarification. These insights could point the way towards improved low-valance
nickelate superconductors, including multilayer systems [108]. Finally, a
pertinent question is whether suitable sample preparation allows to realize
other cuprate-typical ground states in nickelates, such as antiferromagnetism,
pseudogap, as well as nematic, charge, and spin orders.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

The manuscript was written by MH, MPMD, and WSL.

## Funding

Work at SLAC National Lab was supported by U.S. Department of Energy, Office
of Science, Office of Basic Energy Sciences under Contract No. DE-
AC02-76SF00515. Work at Brookhaven National Laboratory was supported by the
U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences.
The use of resources at the SIX beamline of the National Synchrotron Light
Source II, a U.S. Department of Energy (DOE) Office of Science User Facility
operated for the DOE Office of Science by Brookhaven National Laboratory under
Contract No. DE-SC0012704, is acknowledged.

## Acknowledgments

The authors thank K. Fürsich for useful discussions and comments.

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Figure 1: Electronic structure of nickelates. (A) Upper panel: O $K$-edge XAS
of NiO, LaNiO3 and LaNiO2. Red arrows mark the pre-edge peaks indicative of
Ni–O hybridization. Lower panel: O $K$-edge XAS of NdNiO3 and NdNiO2. Dashed
vertical lines indicate features of the SrTiO3 (STO) substrate (solid grey
line) in the XAS spectra of NdNiO3 and NdNiO2 due to the film thickness being
thinner than that of the La-based films in the upper panel. Spectra are
vertically offset for clarity. (B) Ni $L_{3}$-edge XAS of NiO, LaNiO3, LaNiO2,
and NdNiO2. The La $M_{4}$-line was subtracted from the LaNiO3 and LaNiO2
spectra. (C) Comparison of the O $K$-edge pre-peak intensities (shaded areas)
of a TL nickelate (La4Ni3O8), a hole-doped cuprate (La1.85Sr0.15CuO4), and a
hole-doped IL nickelate (Nd0.775Sr0.225NiO2). (D) RIXS intensity map of NdNiO2
measured as a function of incident photon energy across the Ni $L_{3}$-edge.
(E) Representative RIXS spectra of LaNiO3, LaNiO2, and NdNiO2. Black arrows
highlight the 0.6 eV features of LaNiO2 and NdNiO2. (F) Calculated RIXS map
and XAS (solid black line) of LaNiO2 for a 3$d^{9}$ \+ 3$d^{8}R$ ground state,
with $R$ denoting a charge-transfer to the La cation. The dashed box
highlights the same feature as the box in panel D. Panels adapted from Refs.
[64, 49]. Figure 2: Magnetic excitations in nickelates. (A) Magnetic
excitations in NdNiO2. Solid and open symbols are mode energy and the damping
parameters, respectively, extracted from fitting RIXS spectra to a damped
harmonic oscillators (DHO). The dashed curve is the linear spin wave fit to
the extracted energy-momentum dispersion. The horizontal dashed line
represents the energy resolution of the RIXS measurement. (B) Doping
dependence of the spectral weight (upper) and mode energy (lower) deduced from
the DHO fitting. (C) The structural unit cell of La4Ni3O8 with Ni/O/La atoms
shown as purple/gray/green spheres. (D) The electronically active TL nickel-
oxide plane structures in La4Ni3O8, showing the diagonal stripe-ordered state
[47]. Ni sites with additional hole character are in purple, whereas spinful
Ni up(down) sites are depicted in red(blue). (E) Measured magnetic excitations
in La4Ni3O8. Black squares are the extracted energies of the magnetic
excitations. The dark gray line is the fit to the experimental dispersion,
which is composed of three modes plotted in blue, orange, and green,
respectively. The doubling of the modes from $(-\frac{1}{3},-\frac{1}{3})$ to
$(0,0)$ arises from magnetic twinning and the line thickness reflects the
predicted intensity of the modes. Panels adapted from Refs. [66, 49].