# Many-body electronic structure of $d^{9-\delta}$ layered nickelates

Harrison LaBollita<EMAIL_ADDRESS>Department of Physics, Arizona State
University, Tempe, AZ 85287, USA  Myung-Chul Jung  Department of Physics,
Arizona State University, Tempe, AZ 85287, USA  Antia S. Botana  Department of
Physics, Arizona State University, Tempe, AZ 85287, USA

###### Abstract

The recent observation of superconductivity in an infinite-layer and
quintuple-layer nickelate within the same $R_{n+1}$NinO2n+2 series ($R$ =
rare-earth, $n=2-\infty$, with $n$ indicating the number of NiO2 layers along
the $c$-axis), unlocks their potential to embody a whole family of
unconventional superconductors. Here, we systematically investigate the many-
body electronic structure of the layered nickelates (with $n=2-6,\infty$)
within a density-functional theory plus dynamical mean-field theory framework
and contrast it with that of the known superconducting members of the series
and with the cuprates. We find that many features of the electronic structure
are common to the entire nickelate series, namely, strongly correlated
Ni-$d_{x^{2}-y^{2}}$ orbitals that dominate the low-energy physics, mixed
Mott-Hubbard/charge-transfer characteristics, and $R$($5d$) orbitals acting as
charge reservoirs. Interestingly, we uncover that the electronic structure of
the layered nickelates is highly tunable as the dimensionality changes from
quasi-two-dimensional to three-dimensional as $n\rightarrow\infty$.
Specifically, we identify the tunable electronic features to be: the charge-
transfer energy, presence of $R(5d)$ states around the Fermi level, and the
strength of electronic correlations.

##  I Introduction

The search for materials with analogous structural, magnetic, and electronic
motifs to the high-$T_{c}$ cuprates has been an active field of research for
more than 30 years Bednorz and Müller (1986); Keimer  _et al._ (2015). Nickel
oxide (nickelate) materials have been an obvious class of candidate materials
as nickel sits next to copper in the periodic table, and it can realize a Ni+:
$d^{9}$ oxidation state, which is isoelectronic with Cu2+ Anisimov  _et al._
(1999); Lee and Pickett (2004).

The promise of nickelates in this context was realized in 2019 with the
observation of superconductivity in Sr-doped NdNiO2, referred to as the
‘infinite-layer’ ($n=\infty$) nickelate Li  _et al._ (2019), with a Ni+
($d^{9}$) oxidation state and NiO2 planes akin to the CuO2 planes of the
cuprates. Superconductivity in the infinite-layer material displays a dome-
like dependence with a maximum $T_{c}$ near 20% hole-doping or a $d^{8.8}$
electron filling Li  _et al._ (2020a). Since this initial discovery,
superconductivity has been observed in other rare-earth flavors of the
infinite-layer nickelate, namely hole-doped PrNiO2 Osada  _et al._ (2020) and
LaNiO2 Osada  _et al._ (2021); Zeng  _et al._ (2022), all with similar
$T_{c}$’s and superconducting domes.

Figure 1: Crystal structures of the layered nickelates Lan+1NinO2n+2 for
$n=2-6,\infty$ (from left to right). Each crystal structure contains $n$ NiO2
planes interleaved between R-O fluorite slabs. The stoichiometries for the
nickelate series are: La3Ni2O6 ($n=2$), La4Ni3O8 ($n=3$), La5Ni4O10 ($n=4$),
La6Ni5O12 ($n=5$), La7Ni6O14 ($n=6$), and LaNiO2 ($n=\infty$). Note for even
$n$ there is no mirror plane. Ni(i), Ni(m), and Ni(o) denote the inner,
middle, and outer inequivalent Ni sites, respectively.

The discovery of superconductivity in the infinite-layer nickelates has
attracted a great deal of theoretical Wu  _et al._ (2020); Hu and Wu (2019);
Jiang  _et al._ (2019); Nomura  _et al._ (2019); Choi  _et al._ (2020a); Ryee
_et al._ (2020); Gu  _et al._ (2020a); Karp  _et al._ (2020a); Botana and
Norman (2020); Leonov  _et al._ (2020); Kapeghian and Botana (2020);
Lechermann (2020a, b); Choi  _et al._ (2020b); Kitatani  _et al._ (2020);
Petocchi  _et al._ (2020); Zhang  _et al._ (2020a); Sakakibara  _et al._
(2020); Jiang  _et al._ (2020); Werner and Hoshino (2020); Zhang  _et al._
(2020b); Zhang and Vishwanath (2020); Wang  _et al._ (2020a); Bandyopadhyay
_et al._ (2020); Been  _et al._ (2021) and experimental attention Li  _et al._
(2019); Fu  _et al._ (2020); Osada  _et al._ (2020); Lee  _et al._ (2020);
Goodge  _et al._ (2021); Hepting  _et al._ (2020); Fu  _et al._ (2020); Li
_et al._ (2020b, a); Wang  _et al._ (2020b); Gu  _et al._ (2020b); Cui  _et
al._ (2021); Liu  _et al._ (2020); Osada  _et al._ (2021); Zeng  _et al._
(2022). Importantly, the infinite-layer nickelate belongs to a large series of
layered nickelate materials with the general chemical formula,
$R_{n+1}$NinO2n+2 ($R$ = La, Pr, Nd; $n=2,3,\dots,\infty$). The finite-layer
($n\neq\infty$) nickelates have $n$-quasi-2D NiO2 planes (like the $n=\infty$
material) but display an additional blocking $R$-O fluorite slab interleaved
between neighboring blocks of NiO2 planes (see Fig. 1). The $R$-O layer tunes
the dimensionality of the materials as $n$ decreases: from three-dimensional-
like in the $n=\infty$ compound to quasi-two-dimensional as $n$ decreases. In
addition, the number of NiO2 planes along the $c$-axis ($n$) tunes the average
nominal Ni filling that can be defined as $d^{9-\delta}$, where $\delta=1/n$.
Notably, the quintuple-layer $(n=5)$ nickelate has a nominal $d^{8.8}$
($\delta=0.2$) Ni($3d$) filling that matches the optimal doping level of the
infinite-layer nickelates and the cuprates. Indeed, recently,
superconductivity has been observed in the quintuple-layer nickelate Nd6Ni5O12
without the need for chemical doping Pan  _et al._ (2022a). The fact that
superconductivity has now been demonstrated in two members of the layered
nickelate series (the only ones so far where an optimal $d^{8.8}$ filling has
been attained) suggests that a new family of superconductors has been
uncovered.

In this paper, we systematically investigate the many-body electronic
structure of the layered nickelates ($n=2-6,\infty$) by including electronic
correlations beyond density-functional theory (DFT) within a DFT plus
dynamical mean-field theory (DMFT) framework and contrast it with that of the
known superconducting members of the series and with the cuprates. We find
that there are many electronic features that span across the entire nickelate
series, namely, strongly correlated Ni-$d_{x^{2}-y^{2}}$ orbitals that
dominate the low-energy physics similar to the Cu-$d_{x^{2}-y^{2}}$ orbitals
of the cuprates, as well as mixed Mott-Hubbard/charge-transfer characteristics
and La($5d$) orbitals acting as charge reservoirs. We uncover certain features
of the electronic structure of layered nickelates are tunable with $n$,
specifically, the charge-transfer energy, involvement of $R(5d)$ states around
the Fermi level, and the strength of electronic correlations. All in all, with
the dimensionality (via the number of layers, $n$) controlling the electronic
structure of this nickel oxide family, we anticipate that other layered
nickelates should be able to host a superconducting instability provided that
the proper Ni electron filling (near $d^{8.8}$) can be achieved.

##  II Methodology

The charge self-consistent (CSC) combination of density-functional theory plus
dynamical mean-field theory (DFT+DMFT) is employed to calculate the many-body
electronic structure of the layered nickelates ($n=2-6,\infty$). The DFT
problem is solved using the all-electron, full-potential code wien2k Blaha
_et al._ (2020), which is built on the augmented plane wave plus local orbital
basis set (APW+lo). The Perdew-Burke-Ernzerhof (PBE) version Perdew  _et al._
(1996) of the generalized gradient approximation (GGA) is used as the
exchange-correlation functional. We choose $R$ = La to avoid any ambiguity in
the treatment of $4f$ electrons that would arise from $R$ = Nd or Pr. It seems
reasonable to choose $R$ = La for a general description of the many-body
electronic structure, as the electronic structure of the infinite-layer
compounds has been shown not to significantly change with $R$ Kapeghian and
Botana (2020); Bandyopadhyay  _et al._ (2020); Been  _et al._ (2021). A dense
$k$-mesh of 17$\times$17$\times$17 is used for integration in the Brillioun
zone. We used $R_{\mathrm{MT}}K_{\mathrm{max}}=7$ and muffin-tin radii of
2.35, 1.97, and 1.75 a.u. for La, Ni, and O, respectively.

To gain quantitative insights into the electronic structure of the layered
nickelates, we construct maximally localized Wannier functions (MLWFs) from
the DFT spectrum. We employed wannier90 Mostofi  _et al._ (2014) and
wien2wannier Kunes  _et al._ (2010) to obtain the MLWFs. We compute the MLWFs
within a wide energy window, including the Ni($3d$), O($2p$), and La($5d)$
orbitals, which provides orbital energies and allows us to estimate the
charge-transfer energy. We obtained well-localized (albeit not unique) Wannier
functions that correctly reproduce the DFT band structure.

For the DMFT part, we construct a basis of atomic-like orbitals from the Kohn-
Sham wavefunctions using a projector scheme Aichhorn  _et al._ (2016, 2009)
spanning a correlated subspace of size $-10$ eV to $10$ eV around the Fermi
level. Each inequivalent Ni site in the crystal structure is treated as its
own quantum impurity problem, where the full Ni($3d$) manifold is treated as
correlated and governed by a fully rotationally-invariant Slater Hamiltonian.
We parameterize the Slater integrals with parameters representative of the
nickelates: a Hubbard $U=F^{0}=7$ eV and Hund’s coupling
$J_{\mathrm{H}}=(F^{2}+F^{4})/14=0.7$ eV Nowadnick  _et al._ (2015); Karp  _et
al._ (2020b, a, 2021). The quantum impurity problem(s) are solved using a
continuous-time quantum Monte Carlo (QMC) algorithm based on the hybridization
expansion method as implemented in TRIQS/cthyb Parcollet  _et al._ (2015);
Seth  _et al._ (2016). To reduce high-frequency noise in the QMC data, we
represent both the Green’s function and self-energies in a basis of Legendre
polynomials and sample the Legendre coefficients directly within the
TRIQS/cthyb solver Boehnke  _et al._ (2011). The fully-localized limit (FLL)
formula is used for the double counting correction. All calculations are
performed at a system temperature of 290 K ($\beta=40$ eV-1) in the
paramagnetic state. Maximum entropy methods are used to analytically continue
the QMC data from Matsubara space to real-frequency space Kraberger  _et al._
(2017).

Figure 2: DFT band structures for the $n=2-6,\infty$ (left to right) layered
nickelates within the ‘fatband’ representation, which highlights the orbital
character of the DFT bands. The orbital character of the Ni-$d_{x^{2}-y^{2}}$
(blue), Ni-$d_{z^{2}}$ (pink), La-$d_{z^{2}}$ (green), and La-$d_{xy}$
(yellow) orbitals are shown.

##  III Crystal structure and ionic count

The layered nickelates are derived from either a perovskite ($n=\infty$) or
Ruddlesden-Popper ($n\neq\infty$) parent compound via oxygen deintercalation.
Currently, the $n=2-5$ parent Ruddlesden-Popper phases of the series have been
synthesized in thin film geometry for $R=$ La and Nd Li  _et al._ (2020c); Pan
_et al._ (2022b). These reduced $R_{n+1}$NinO2n+2 compounds crystallize in a
tetragonal structure belonging to either the I4/mmm ($n\neq\infty$) or P4/mmm
($n=\infty$) space group. As mentioned above, the series of layered nickelates
contains blocks of $n$ two-dimensional NiO2 planes (with square-planar
coordination) separated by rare-earth layers. For the finite-layer nickelates,
an extra rare-earth fluorite slab is present, separating each block of $n$
$R$-NiO2 planes (see Fig. 1). Furthermore, each $R$-NiO2 block is shifted by
half a lattice constant in the $ab$-plane. These two additional structural
features (that are not present in the infinite-layer nickelate) effectively
decouple neighboring $R$-NiO2 blocks along the $c$-direction.

As mentioned above, the average nominal occupation on the Ni sites for each
layered nickelate is $d^{9-\delta}$, where $\delta=1/n$. Specifically, the
average oxidation states for $n=2-6,\infty$ layered nickelates are: Ni1.5+
($d^{8.5}$) for $n=2$ (La3Ni2O6), Ni1.33+ ($d^{8.67}$) for $n=3$ (La4Ni3O8),
Ni1.25+ ($d^{8.75}$) for $n=4$ (La5Ni4O10), Ni1.2+ ($d^{8.8}$) for $n=5$
(La6Ni5O12), Ni1.17+ ($d^{8.83}$) for $n=6$ (La7Ni6O14), and Ni+ ($d^{9}$) for
$n=\infty$ (LaNiO2).

Each material contains either an inner (i), inner (i) and outer (o), or an
inner (i), middle (m), and outer (o) symmetry-inequivalent Ni site along the
$c$-axis. For odd $n$ materials, the inner Ni site acts as a mirror plane for
the outer layer(s) (see Fig. 1). We use the experimental crystal structures
for the $n=2,3,\infty$ layered nickelates Poltavets  _et al._ (2006, 2007);
Hayward  _et al._ (1999). As the structures of lanthanum-based $n=4-6$ layered
nickelates have not been experimentally resolved yet, we derive their crystal
structures from the experimental structure of the $n=3$ material La4Ni3O8. For
further details on the derivation of the crystal structures of the $n=4-6$
materials see Ref. LaBollita and Botana, 2021.

##  IV non-interacting electronic structure

Figure 2 summarizes the electronic structure of the $n=2-6,\infty$ layered
nickelates at the DFT level. Starting with the $n=2$ material with an average
Ni($3d$) occupation of $d^{8.50}$ ($\delta=0.5$), only two partially filled
$d_{x^{2}-y^{2}}$ bands (one per Ni) cross the Fermi level, in a cuprate-like
fashion. The splitting between these bands (highlighted at X) is a consequence
of the interlayer hopping within NiO2 blocks and is analogous to that of the
multi-layer cuprates Sakakibara  _et al._ (2014). Below
$\varepsilon_{\mathrm{F}}$, there is a complex of mostly filled Ni-$t_{2g}$
bands and particularly flat Ni-$d_{z^{2}}$ bands highlighted in pink. The flat
dispersion of the Ni-$d_{z^{2}}$ is a unique feature of the finite-layer
nickelates, dissimilar to the infinite-layer compound, which has a dispersive
band coming from the Ni-$d_{z^{2}}$ orbital along the $\Gamma-$Z direction.
The flatness of these bands is a consequence of the fluorite slab, present in
all finite-layer nickelates, which cuts the $c$-axis dispersion across the
neighboring NiO2 blocks, as described above. Above $\varepsilon_{\mathrm{F}}$,
there are a set of bands of mostly $d_{xy}$ and $d_{z^{2}}$ orbital character
stemming from the La sites within the NiO2 blocks.

As the number of NiO2 layers increases with increasing $n$, we find many of
the same features: $n$ partially filled $d_{x^{2}-y^{2}}$ bands coming from
the Ni sites, flat Ni-$d_{z^{2}}$ bands, mostly filled Ni-$t_{2g}$ bands, and
La($5d$) bands above $\varepsilon_{\mathrm{F}}$. However, a key difference
with increasing $n$ is the location of La($5d$) bands as they experience a
significant downward shift in energy with increasing $n$. For the $n=4$
material, a band with La-$d_{xy}$ character already crosses
$\varepsilon_{\mathrm{F}}$ giving rise to an electron pocket at the zone
corners of the Brillouin zone that self-dopes the $d_{x^{2}-y^{2}}$ bands.
Moving onto the $n=5$ and 6 materials, these electron pockets continue to
increase in size as the bands with mostly La-$d_{xy}$ character shift even
deeper in energy. Across the entire $n=2-6,\infty$ series, the $n=2,3$
materials are then markedly different as they lack this self-doping effect. In
fact, because the trilayer ($n=3$) nickelate does not have La($5d$) bands
crossing the Fermi level, and given its close in proximity to the cuprate
superconducting dome in terms of Ni($3d$) electron count, previous works have
suggested this material to be one of the closest cuprate analogs to date
Botana  _et al._ (2017); Zhang  _et al._ (2017); Lin  _et al._ (2021); Shen
_et al._ (2022). Furthermore, as $n$ reaches the infinite-layer limit the
electronic structure changes from quasi-two-dimensional to three-dimensional
LaBollita and Botana (2021). This can be easily observed by looking at the
bands around the Fermi level in the $k_{z}=0$ ($\Gamma$-X-M-$\Gamma$) and
$k_{z}=1/2$ (Z-R-A-Z) planes for the infinite-layer material. The bands in
these two $k_{z}$ planes would generate distinctly different constant energy
surfaces and therefore a rather three-dimensional-like electronic structure
relative to the $n\neq\infty$ nickelates.

Figure 3: Top panels: ${\bf k}$-resolved spectral functions, $A({\bf
k},\omega)$, along high-symmetry lines in the Brillioun zone for
$n=2-6,\infty$ layered nickelates (from left to right). Middle panels:
Orbital-projected $A({\bf k},\omega)$ for the Ni($3d$) shell with: blue
denoting the $d_{x^{2}-y^{2}}$ orbitals, red denoting the $d_{z^{2}}$
orbitals, and green denoting the $t_{2g}$ orbitals. Bottom panels:
Corresponding interacting Fermi surfaces $A({\bf k},\omega=0)$ in the
$k_{z}=0$ plane. The high-symmetry points in the BZ are denoted.

According to the Zaanen-Sawatsky-Allen (ZSA) classification scheme Zaanen  _et
al._ (1985), cuprates sit well within the charge-transfer regime
($U\gg\Delta_{\mathrm{CT}}$) with the on-site Coulomb repulsion $U$ being
larger than the the charge-transfer energy ($\Delta_{\mathrm{CT}}$) that
measures the hybridization between the TM($d$) and ligand($p$) orbitals. In
this regard, the hybridization of the O($2p$) orbitals with the Cu($3d$)
orbitals is believed to be an essential ingredient for the formation of Zhang-
Rice (ZR) singlets and for cuprate superconductivity. At the level of DFT, the
hybridization between the Ni($3d$) and O($2p$) orbitals can be quantified by
$\Delta_{\mathrm{CT}}=\varepsilon_{d}-\varepsilon_{p}$, where
$\varepsilon_{d(p)}$ refers to the on-site orbital energy for the
Ni-$d_{x^{2}-y^{2}}$ (O-$p_{\sigma}$) orbital derived from fitting maximally
localized Wannier functions (MLWFs) to the DFT spectrum. For cuprates, a
typical value of $\Delta_{\mathrm{CT}}$ is $1-2$ eV. For the infinite-layer
nickelate, values of $4.4-5$ eV have been obtained with similar methods
Lechermann (2020a); Nica  _et al._ (2020). We note that the larger $p-d$
splitting in the infinite-layer nickelates has been identified as one of the
main differences with the cuprates Nica  _et al._ (2020); Lee and Pickett
(2004); Shen  _et al._ (2022); Goodge  _et al._ (2021); Hepting  _et al._
(2020). Here, we obtain charge-transfer energies of
$\Delta_{\mathrm{CT}}(n=2)=3.3$ eV, $\Delta_{\mathrm{CT}}(n=3)=3.7$ eV,
$\Delta_{\mathrm{CT}}(n=4)=3.8$ eV, $\Delta_{\mathrm{CT}}(n=5)=3.9$ eV,
$\Delta_{\mathrm{CT}}(n=6)=4.0$ eV, and $\Delta_{\mathrm{CT}}(n=\infty)=4.5$.
As such, as $n$ increases from 2 to $\infty$, the charge-transfer energy
increases by $\sim 1$ eV indicating the increasing hybridization between the
Ni($3d$) and O($2p$) orbitals for smaller $n$ (larger $\delta$) nickelates
i.e. when going towards the 2D limit. This trend in charge-transfer energies
(increasing with $n$) qualitatively agrees with the available experimental
data for the $n=3$, $n=5$, and $n=\infty$ materials Pan  _et al._ (2022a).
Assuming a reasonable Coulomb repulsion of $U=5-7$ eV Nowadnick  _et al._
(2015), this places the layered nickelates at the boundary between the charge-
transfer and Mott-Hubbard regimes in the ZSA scheme ($U\sim\Delta$).
Interestingly, even though these energies are larger than in the cuprates,
they still lead to sizeable superexchanges Lin  _et al._ (2021); Lu  _et al._
(2021). Therefore, this suggests that the O($2p$) states remain an appreciable
ingredient for the electronic structure of the layered nickelates Karp  _et
al._ (2020b); Shen  _et al._ (2022).

##  V DFT+DMFT results

###  V.1 Spectral properties and electronic correlations

With the non-interacting electronic structure of the layered nickelates
established, we now analyze the interacting problem to investigate the role of
electronic correlations. Figure 3 summarizes the ${\bf k}$-resolved spectral
functions, $A({\bf k},\omega)$ for the $n=2-6,\infty$ layered nickelates. The
low-energy spectrum exhibits coherent quasiparticle (QP) dispersions
corresponding to the Ni-$d_{x^{2}-y^{2}}$ states. Compared to the DFT
spectrum, the Ni-$d_{x^{2}-y^{2}}$ states are strongly renormalized. Mass
enhancements are derived from the electronic self-energies,
$m^{\star}/m_{\mathrm{DFT}}=1-\partial\mathrm{Im}\Sigma(i\omega_{n}\rightarrow
0)/\partial\omega_{n}$ shown in Fig. 4. We find the mass enhancement for the
Ni-$d_{x^{2}-y^{2}}$ orbital monotonically increases from
$m^{\star}/m_{\mathrm{DFT}}=2.5$ to $3.7$ as $n$ increases from 2 to $\infty$,
which is in agreement with previous DFT+DMFT calculations Karp  _et al._
(2020a, b, 2021); LaBollita and Botana (2022). Note that both the $e_{g}$
self-energies and mass enhancements have been averaged over inequivalent Ni
impurities. Thus, the strength of electronic correlations on the
Ni-$d_{x^{2}-y^{2}}$ states increases with $n$, which is the expected trend
for a mixed Mott-Hubbard/charge-transfer system as we approach the $d^{9}$
($\delta=0$) limit. This monotonic increase in mass enhancements with $n$
matches previous works studying the hole-doping effects on the electronic
structure of infinite-layer nickelates Leonov  _et al._ (2020); Wang  _et al._
(2020a). The mass enhancements on all other Ni($3d$) orbitals are much smaller
and remain $\sim 1.5-1.7$.

Figure 4: Imaginary part of the Ni self-energies (averaged over inequivalent
Ni sites) for the $e_{g}$ manifold in Matsubara space: $d_{x^{2}-y^{2}}$
(left) and $d_{z^{2}}$ (right).

Figure 3 (middle panels) also shows the orbital character of the spectral
weight using the ‘fatspec’ representation introduced in Ref. Lechermann, 2022,
where blue corresponds to the Ni-$d_{x^{2}-y^{2}}$ orbitals, red to the
Ni-$d_{z^{2}}$ orbitals, and green to the Ni-$t_{2g}$ orbitals. For the
finite-layer nickelates, apart from the Ni-$d_{x^{2}-y^{2}}$ states dominating
around the Fermi level, the spectral functions reveal non-dispersing spectral
weight corresponding to the Ni-$d_{z^{2}}$ states located just below these
metallic Ni-$d_{x^{2}-y^{2}}$ states. This flat Ni-$d_{z^{2}}$ spectral weight
retreats away from the chemical potential as $n$ increases. Similarly, the
spectral weight corresponding to the Ni-$t_{2g}$ states experiences a downward
shift in the energy spectrum as $n$ is increased. For the infinite-layer
material, the spectral weight corresponding to the Ni-$d_{z^{2}}$ orbital is
dispersive along the $k_{z}$ direction and becomes flat only in the
$k_{z}=1/2$ plane.

The impurity-resolved local spectral functions shown in Fig. 5 offer further
insights into the low-energy physics of the layered nickelates. The component
corresponding to the Ni-$d_{x^{2}-y^{2}}$ orbital is dominant at low-energy
and exhibits the characteristic three-peak structure of a mixed Mott-
Hubbard/charge-transfer system. There is some minor Ni-$d_{z^{2}}$ spectral
weight but this arises from the small hybridization between the
Ni-$d_{x^{2}-y^{2}}$ and Ni-$d_{z^{2}}$ states which can be seen from the
fatspec representation of the ${\bf k}$-resolved spectral function in Fig. 3.
The local spectral function highlights the evolution of the flat
Ni-$d_{z^{2}}$ band captured in Fig. 3 stemming from the innermost Ni site.
For the $n=2$ material, this flat band is closest to the Fermi level and
causes a large peak in the Ni-$d_{z^{2}}$ local spectral function. This peak
softens as this band becomes more dispersive with increasing $n$ as the
electronic structure becomes more three-dimensional-like. The Ni-$t_{2g}$
states remain mostly filled and inert by only shifting slightly downward in
the energy spectrum with increasing $n$.

For the $n=4-6,\infty$ nickelates, the La($5d$) states survive electronic
correlations and remain active crossing the chemical potential. Despite the
fact that these states exhibit some Ni-$d_{z^{2}}$ weight (see Fig. 3), the
Ni-$d_{z^{2}}$ states remain only weakly correlated meaning that a significant
correlation-induced shift is not likely to occur. Previous works on the $n=5$
material have shown that these La($5d$) states can be lifted away from the
chemical potential by either treating the La($5d$) orbitals as correlated Worm
_et al._ (2021) or within a DFT+sicDMFT framework with strong coupling
Lechermann (2022). Whether $R(5d)$ states do indeed contribute to the Fermi
surface is an open question and can only be settled with further experiments.

In this context, we calculate the interacting Fermi surface, $A({\bf
k},\omega=0)$ in the $k_{z}=0$ plane for each material as shown in Fig. 3
(bottom panels). For all $n$, there are large hole-like sheets (arcs) coming
from the Ni-$d_{x^{2}-y^{2}}$ states, whose van Hove singularity lies below
the chemical potential and a single electron-like sheet (square-like) coming
from the Ni-$d_{x^{2}-y^{2}}$ states whose van Hove singularity sits above the
chemical potential. As we increase $n$, the splitting between the
Ni-$d_{x^{2}-y^{2}}$ quasiparticle dispersions decreases and the topology of
this electron-like sheet transforms as $n$ approaches the infinite-layer
limit. Additionally, there are electron pockets at the zone corners of
La($5d$) character previously discussed for the $n=4-6$ materials (that
increase in size as $n$ increases). For the infinite-layer material, the
electron-like pocket at the zone center comes from the La($5d$) band crossing
at $\Gamma$ of $d_{z^{2}}$ character (see Fig. 3). The electron-like pockets
from the La($5d$) band crossing at A of $d_{xy}$ character is only visible in
the $k_{z}=1/2$ plane, see for example Ref. Karp  _et al._ , 2020b. Overall,
the fermiology matches the Fermi surfaces calculated within DFT LaBollita and
Botana (2021). One important difference between the fermiology of the finite-
layer nickelates when compared to the infinite-layer compound is the overall
dimensionality. The finite-layer nickelates have a two-dimensional-like
fermiology, very similar to that of the multi-layer cuprates, whereas the
fermiology of the infinite-layer nickelates is 3D-like Sakakibara  _et al._
(2014). This reduction in dimensionality for the finite-layer materials is a
consequence of the difference in structure described previously and further
strengthens the cuprate-like nature of the finite-layer nickelates LaBollita
and Botana (2021).

material | Ni site | $n_{d_{x^{2}-y^{2}}}$ | $n_{d_{z^{2}}}$ | $n_{t_{2g}}$ | $n_{\mathrm{tot}}$
---|---|---|---|---|---
La3Ni2O6 | Ni(i) | 0.96 | 1.67 | 5.84 | 8.47
La4Ni3O8 | Ni(i) | 1.00 | 1.62 | 5.82 | 8.44
– | Ni(o) | 1.03 | 1.66 | 5.83 | 8.52
La5Ni4O10 | Ni(i) | 1.06 | 1.61 | 5.81 | 8.49
– | Ni(o) | 1.06 | 1.65 | 5.81 | 8.52
La6Ni5O12 | Ni(i) | 1.10 | 1.60 | 5.80 | 8.50
– | Ni(m) | 1.08 | 1.60 | 5.80 | 8.49
– | Ni(o) | 1.05 | 1.64 | 5.80 | 8.50
La7Ni6O14 | Ni(i) | 1.11 | 1.60 | 5.80 | 8.51
– | Ni(m) | 1.09 | 1.61 | 5.81 | 8.51
– | Ni(o) | 1.06 | 1.65 | 5.81 | 8.52
LaNiO2 | Ni(i) | 1.14 | 1.58 | 5.78 | 8.51
Table 1: Impurity- and orbital- resolved Ni($3d$) occupations for the
$n=2-6,\infty$ layered nickelates computed from the impurity Green’s
functions. Figure 5: Left panels: Orbital-resolved k-integrated spectral
functions for the $n=2-6,\infty$ layered nickelates. The inset shows the
orbital-projected spectral weight corresponding to the O($2p$) and La($5d$)
states around the Fermi level. Right panels: Impurity-resolved local spectral
function(s) for the Ni($3d$) manifold.

###  V.2 The role of La$(5d)$ and O$(2p)$ states

Changing the number of layers along the $c$-axis ($n$) effectively tunes the
formal charge on the Ni sites: as $n$ decreases from $n=\infty$ to $n=2$, one
is essentially hole-doping the nickelate system by $\delta=1/n$, as mentioned
above. The consequences of this ‘hole-doping’ effect are revealed in the
overall ${\bf k}$-integrated spectral functions (see Fig. 5, left panel).
Starting with the bilayer ($n=2$) material, the Ni($3d$) states dominate the
low-energy physics. Most of the spectral weight around $\omega=0$ corresponds
to the Ni-$d_{x^{2}-y^{2}}$ orbitals, as revealed in the local spectral
function (see Fig. 5, right panel). In the addition spectrum ($\omega>0$), the
La($5d$) states are far above the chemical potential. The states corresponding
to the Ni-$t_{2g}$ orbitals contribute a large peak just below the chemical
potential. The O($2p$) states dominate the remainder of the removal spectrum
($\omega<0$) exhibiting a broad peak centered around $\omega=-3$ eV. Upon
increasing $n$, the complex of O($2p$) states shifts downward in the spectrum,
while the La($5d$) states become active around the chemical potential, giving
rise to the self-doping effect described above. The shift in O($2p$) states
qualitatively matches the increasing charge-transfer energy with $n$ discussed
in the previous section. Additionally, the qualitative trends in the
electronic spectrum as $n$ changes agree with hole-doping studies of the
infinite-layer nickelate, pointing to the importance of making comparisons at
the same Ni($3d$) filling Kitatani  _et al._ (2020); Karp  _et al._ (2020b);
LaBollita and Botana (2022).

While the metallic Ni-$d_{x^{2}-y^{2}}$ states are dominant around the
chemical potential, there is some non-zero spectral weight corresponding to
the O($2p$) states present at all $n$ (see insets shown in Fig. 5). This
spectral feature is reminiscent of the ZR physics of the cuprates and has been
observed in spectroscopic measurements of the hole-doped infinite-layer
nickelates Goodge  _et al._ (2021). Additionally, there is a small amount of
spectral weight corresponding to La($5d$) states, which increases with $n$.
For the cuprates, doped holes reside on the oxygen sites and act as a charge-
reservoir in addition to providing bandwidth to the Cu($3d$) states. In the
case of the nickelates, where both O($2p$) and La($5d$) states are present
near the chemical potential, the Ni($3d$) bandwidth is provided through
hybridization with the O($2p$) states, analogous to the cuprates and supported
by experiment Hepting  _et al._ (2020). However, unlike the cuprates, the
La($5d$) states act as a charge reservoir absorbing most of the dopants Karp
_et al._ (2020b).

Figure 6: Most probable spin configurations (left) and occupation number
(right) across the layered nickelate series $n=2-6,\infty$.

###  V.3 Spin and charge statistics

The DFT+DMFT builds on an atomic picture of a solid, thus places a high
importance on the atomic multiplets of the impurity problem being solved. We
can gain further insight into the electronic structure of the layered
nickelates by analyzing the probabilities of the atomic multiplets, which are
obtained directly from the TRIQS/cthyb solver. Figure 6 shows the evolution of
the three dominant spin states: $S=1/2$ ($d^{9}$), $S=1$ (high-spin $d^{8}$),
and $S=0$ (low-spin $d^{8}$). Throughout the entire series, the significant
charge fluctuations are between the $d^{9}$ and $d^{8}$ charge sectors with
essentially no dependence on $n$ (see Fig. 6). The dominant spin configuration
remains $S=1/2$, which is independent of $n$. The next two most probable
configurations correspond to the low-spin (LS) or high-spin (HS) states
associated with the $d^{8}$ charge sector. We find that the probability of
these two spin configuration does exhibit a dependence on $n$. For the $n=2$
material, low-spin $d^{8}$ and high-spin $d^{8}$ configurations are
essentially equally probable, while with increasing $n$ the high-spin $d^{8}$
configurations become dominant over the low-spin configurations Karp  _et al._
(2020b); Petocchi  _et al._ (2020); LaBollita and Botana (2022).

Table 1 summarizes how the Ni($3d$) occupations evolve across the series of
layered nickelates, which have been calculated from the impurity Green’s
function(s). We find the total Ni($3d$) occupation remains essentially
independent of $n$, in agreement with previous DFT+DMFT works for the $n=3$
and $n=\infty$ materials Karp  _et al._ (2020b); Petocchi  _et al._ (2020).
Within the Ni($3d)$ shell, the occupation of the $d_{x^{2}-y^{2}}$ orbital(s)
monotonically decreases with decreasing $n$. This overall reduction in
$d_{x^{2}-y^{2}}$ occupation is compensated by the filling of the remaining
Ni($3d$) orbitals. It is important to note that these occupations should be
considered on a qualitative level as they are dependent upon the projectors
(i.e., projectors or Wannier functions) used to create the correlated
subspace, as well as on the size of the correlated subspace Karp  _et al._
(2021).

The overall trend in the spin statistics can be understood from the interplay
between O$(2p$) and La$(5d$) orbitals. In cuprates, the ground state
electronic configuration is comprised of equal amounts of $|d^{9}\rangle$ and
$|d^{10}\underline{L}\rangle$ (where $\underline{L}$ denotes ligand hole),
which is expected of a charge-transfer material with small $p-d$ splitting.
The large amount of $|d^{8}\rangle$ in the ground electronic configurations of
the nickelates contrasts with that of the cuprates. In the case of the
nickelates, there is a larger $p-d$ splitting (large charge-transfer energy)
which increases the significance of the $d^{8}$ states, while decreasing the
relevance of the $d^{10}\underline{L}$ states. Furthermore, the larger $p-d$
splitting forces charge-transfer from the Ni sites to the La sites that
function as a charge reservoir Karp  _et al._ (2020b).

Figure 7: Key features of the electronic structure as a function of $n$: the
number of NiO2 layers. (a) Charge-transfer energy ($\Delta_{\mathrm{CT}}$),
(b) $A_{\mathrm{proj}}(\omega=0)$, and (c) $m^{\star}/m_{\mathrm{DFT}}$ for
the $e_{g}$ orbitals (averaged over the Ni impurities). 11footnotetext: We
note that the the charge-transfer energies reported in Fig. 7 were calculated
on the level of DFT. The charge self-consistent DFT+DMFT theory has the effect
to decreases the $p-d$ splitting (the charge-transfer energy) by a small
constant for each $n$. Therefore, the qualitative trend is the same at the
level of both DFT and DFT+DMFT.

##  VI Summary and Discussion

We have employed charge self-consistent DFT+DMFT to calculate the many-body
electronic structure of the layered nickelates $R_{n+1}$NinO2n+2 with
$n=2-6,\infty$ within the paramagnetic state. Despite the significant change
in formal valence on the Ni site, we find the general spectral properties
across the series of materials are remarkably similar not only to each other,
but also to the end member ($n=\infty$) which possesses a different crystal
structural (while finite-layer nickelates have a spacing $R$-O fluorite block,
the infinite-layer material does not). For all layered nickelates, the
metallic Ni-$d_{x^{2}-y^{2}}$ states are the most strongly correlated and
dominate the low-energy physics making the largest contribution to the density
of states around the Fermi level. We identify key features of the electronic
structure that are a function of $n$, the number of NiO2 layers along the
$c$-axis. Figure 7 summarizes how these key electronic features change with
$n$: we find that the number of NiO2 layers controls the magnitude of the
charge-transfer energy Note (1), the activity of the La($5d$) states around
the Fermi level, and the strength of Ni($3d$) electronic correlations.
Specifically, upon increasing $n$, the Ni($3d$)-O($2p$) hybridization
decreases, increasing the charge-transfer energy, and the La($5d$) states
shift closer to the Fermi level and begin to play an active role in the
fermiology for the $n=4-6,\infty$ materials. Furthermore, we find that the
electronic structure of the layered material transitions from quasi-two-
dimensional to three-dimensional in the $n\rightarrow\infty$ limit.

The layered nickelates have been classified as mixed charge-transfer-Mott-
Hubbard materials Karp  _et al._ (2020b); Shen  _et al._ (2022) with respect
to the ZSA classification scheme with our DFT+DMFT calculations further
supporting this classification. In this regard, the ground state of the
layered nickelate systems contains significantly more $d^{8}$ weight, which is
different than a pure charge-transfer system, like the cuprates. In spite of
the increased $d^{8}$ content, these materials only contain a single strongly
correlated orbital (Ni-$d_{x^{2}-y^{2}}$) and the local spectral function
exhibits strong Mott-Hubbard/charge-transfer character, akin to cuprates.
Finally, we uncover the largest difference between the finite-layer nickelates
and the infinite-layer nickelates is in the dimensionality of the electronic
structure. While the finite-layer nickelates have a two-dimensional-like
electronic structure (mostly due to the presence of a blocking fluorite slab),
the infinite-layer member has a marked three-dimensional character. This draws
the electronic structure of the finite-layer nickelates closer to that of the
cuprates than the infinite-layer material, even though we caution towards the
need of analyzing electronic structures at the same nominal Ni($3d$) filling
to be able to do meaningful comparisons. All in all, we find similar
electronic features at all $n$ suggesting the possibility that superconducting
instabilities may be present in other finite-layer nickelates (beyond the
$n=5$ material) if appropriate chemical doping can be achieved.

###### Acknowledgements.

H.L and A.S.B acknowledge the support from NSF Grant No. DMR 2045826. We
acknowledge the ASU Research Computing Center and the Extreme Science and
Engineering Discovery Environment (XSEDE) Towns  _et al._ (2014) through
research allocation TG-PHY220006, which is supported by NSF grant number
ACI-1548562 for HPC resources.

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