# Electronic structure of higher-order Ruddlesden-Popper nickelates

Myung-Chul Jung<EMAIL_ADDRESS>Department of Physics, Arizona State
University, Tempe, AZ 85287, USA  Jesse Kapeghian  Department of Physics,
Arizona State University, Tempe, AZ 85287, USA  Chase Hanson  Department of
Physics, Arizona State University, Tempe, AZ 85287, USA  Betül Pamuk  School
of Applied and Engineering Physics, Cornell University, Ithaca, NY, 14853, USA
Antia S. Botana  Department of Physics, Arizona State University, Tempe, AZ
85287, USA

###### Abstract

We analyze the electronic structure of the recently synthesized higher-order
nickelate Ruddlesden-Popper phases Lan+1NinO3n+1 ($n=4-6$) using first-
principles calculations. For all materials, our results show large holelike
Fermi surfaces with $d_{x^{2}-y^{2}}$ character that closely resemble those of
optimally hole-doped cuprates. For higher values of $n$, extra non-cuprate-
like bands of $d_{z^{2}}$ orbital character appear. These aspects highlight
that this Ruddlesden-Popper series can provide a means to modify the
electronic ground states of nickelates by tuning their dimensionality. With
their similarities and differences to the cuprates, this new family of
materials can potentially shed light on the physics of copper-based oxides.

##  I Introduction

Nickelates have long been considered as cuprate analogs and have been
extensively studied for their potential for high-temperature superconductivity
[1, 2, 3, 4, 5]. The realization of this promise came in 2019, with the
observation of superconductivity in hole-doped NdNiO2 [6]. This material
contains infinite NiO2 planes, in analogy to the CuO2 planes of the cuprates,
and realizes the difficult to stabilize Ni1+ oxidation state, isoelectronic
with Cu${}^{2+}:d^{9}$ [7]. Subsequently, PrNiO2 and LaNiO2 were also found to
be superconducting upon doping [8, 9, 10, 11]. This series of discoveries led
to an immense amount of experimental [12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 23, 24] and theoretical research [25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. $R$NiO2 ($R$ = rare-
earth) materials are the $n=\infty$ members of a larger structural series
represented by the general formula $R$n+1NinO2n+2 where $n$ is the number of
NiO2 planes along the $c$-axis. Recently, the Nd-based $n=5$ material
(Nd6Ni5O12) has also been shown to be superconducting, unlocking the
possibility to discover a whole new family of nickelate superconductors [47].

All materials in the reduced layered $R$n+1NinO2n+2 family are synthesized by
removing the apical oxygen atoms from a parent $R$n+1NinO3n+1 perovskite
($n=\infty$) or Ruddlesden-Popper (RP) phase ($n$ $\neq\infty$) [48, 49, 50,
51, 52]. These $R$n+1NinO3n+1 compounds are formed by $n$-perovskite layers
separated by rocksalt $R$-O spacer layers (see Fig. 1). The $R$-O layer tunes
the dimensionality of the materials as $n$ decreases: from three-dimensional
(3D)-like in the $n=\infty$ case, to quasi-two-dimensional in the $n=1$
compound. The parent RP phases, in particular the $n=3$ counterparts
($R$4Ni3O10) at $d^{7.33}$ filling, are interesting in their own right as they
present remarkable physics and notable analogies to the cuprates [53, 54].
Specifically, angle-resolved photoemission spectroscopy (ARPES) experiments
have shown that the metallic trilayer nickelate La4Ni3O10 exhibits a holelike
Fermi surface of $d_{x^{2}-y^{2}}$ orbital character that closely resembles
optimally hole-doped copper oxides even though, in contrast to cuprates, an
extra band of $d_{z^{2}}$ orbital character arises [53]. In addition,
La4Ni3O10 exhibits an unusual metal-to-metal transition that has been
attributed to an incommensurate density wave with both charge and spin
characters [54]. In this context, it becomes interesting to analyze the
electronic structure of higher-$n$ RP nickelates and scrutinize their
similarities with and differences to the cuprates. La-based $n=4$ and $5$ RP
phases (La5Ni4O13 and La6Ni5O16, respectively) have already been synthesized
[51, 52]. These materials, with formal oxidation states between 2+ and 3+, are
correlated metals and as such should offer a playground for new physics to be
discovered [51]. The $n=6$ material La7Ni6O19, even though not yet
experimentally reported, should also be interesting to analyze trends towards
the three-dimensional limit.

Figure 1: (Top panel) Crystal structures of the high-order RP nickelate phases
($n=3-6$) and of the perovskite nickelate ($n=\infty$). Green, cyan, and red
represent the La, Ni, and O atoms, respectively. The colors of the octahedra
indicate inner- (yellow), mid- (magenta), and outer-layer character (cyan) for
the corresponding Ni ions. (Bottom panel) Schematic of the Ni $3d$ electron
count with increasing $n$ in these materials.

Here, using first-principles calculations, we study the electronic structure
of La-based higher-order RP nickelates ($n=4-6$) and perform a comparison to
the known $n=3$ counterpart. The organization of our paper is as follows. In
Sec. II, the structures for the higher-order RP phases ($n=4-6$) are
addressed. In Sec. III, we revisit the electronic structure of the $n=3$
material and present our main results on the electronic structure of the
$n=4-6$ RP phases. We find that the similarities with cuprates in the
fermiology of the trilayer material are kept in the $n=4-6$ nickelate RPs
while their electronic structure becomes more 3D-like for increasing $n$
values, as expected.

##  II Structures and computational details

As mentioned above, in the RP nickelate structures (Fig. 1), a basic building
block is formed by $n$ layers of corner-shared Ni-O perovskite octahedra. Each
of these $n$-layer-groups is separated from the neighboring one along the $c$
axis by a rocksalt $R$-O unit. This gives rise to a very small interplanar
coupling between Ni-O perovskite blocks. In their corresponding reduced
layered phases, obtained after oxygen reduction, the spacer rocksalt $R$-O
block in the RP nickelates turns into a fluorite-like one while the Ni-O
perovskite slabs turn into NiO2 layers. As $n$ increases in the $R$n+1NinO3n+1
series, the nominal Ni valence state changes from 2+ to 3+ (i.e., from $d^{8}$
to $d^{7}$ filling with increasing $n$). Explicitly, the RP $n=3$ compound
($R$4Ni3O10) has an average Ni valence of Ni${}^{2.67+}:d^{7.33}$. The $n=4$
compound ($R$5Ni4O13) corresponds to Ni${}^{2.75+}:d^{7.25}$, the $n=5$
material ($R$6Ni5O16) to Ni${}^{2.8+}:d^{7.2}$, and the $n=6$ one ($R$7Ni6O19)
to Ni${}^{2.83+}:d^{7.17}$ (see Fig. 1).

Given that the structures of the higher-order RP nickelates have not yet been
experimentally resolved, we construct the La-based $n=4-6$ phases using the
structure of La4Ni3O10 as a reference. La4Ni3O10 crystallizes in an
orthorhombic ($Bmab$) or monoclinic ($P2_{1}/a$) $\sqrt{2}$$\times$$\sqrt{2}$
supercell of the tetragonal ($I4/mmm$) parent phase [53, 55, 54, 56], the
latter containing two formula units. In this work, for the $n=4-6$ RP
compounds, we construct the parent tetragonal cells and we neglect any
possible octahedral tiltings and further distortions for simplicity. We adopt
this procedure based on the good agreement achieved in La4Ni3O10 ($n=3$)
between the electronic structure of the tetragonal ($I4/mmm$)
$\sqrt{2}$$\times$$\sqrt{2}$ doubled cell without octahedral tilts and the
experimental orthorhombic ($Bmab$) and monoclinic ($P2_{1}/a$) cells with
octahedral tilts (see below, Sec. III).

We conducted the structural optimization of the La-based $n=3-6$ RP phases
with $I4/mmm$ symmetry using the Vienna $ab$ $\it{initio}$ Simulation Package
(vasp) [57, 58], having both lattice parameters and internal coordinates
relaxed. We used La to avoid problems associated with the $4f$ electrons of Pr
or Nd but no large changes can be anticipated from the $R$ substitution in the
electronic structures, as shown for the $R$NiO2 materials in a number of works
[5, 32, 33, 37, 38]. The generalized gradient approximation (GGA) as
implemented in the Perdew-Burke-Ernzerhof (PBE) functional was used as the
exchange-correlation functional [59]. An energy cutoff of 650 eV and a $k$
mesh of 8$\times$8$\times$2 were adopted with a force convergence criterion of
1 meV/Å. The tetragonal La-based $n=3-6$ RP structures were relaxed in the
nonmagnetic state. We perform calculations in the nonmagnetic state since for
the known $n=3$ material Pauli paramagnetism has been reported in a wide
temperature range (4-300 K)[60] and the first-principles nonmagnetic
description appropriately accounts for the existing ARPES data [53]. The
derived lattice parameters for our $n=3-6$ RP phases and the bond lengths for
all structures are listed in Table 1. Good agreement is obtained between the
bond lengths of our relaxations in $I4/mmm$ symmetry and the experimental
structural data with $P2_{1}/a$ and $Bmab$ symmetry for the $n=3$ counterpart
[56, 55], serving as a benchmark for our structural optimizations.

Table 1: Lattice parameters and bond lengths of the relaxed La-based RP
tetragonal structures (in Å). $p$ and $a$ denote the planar and apical
oxygens, respectively, while $i$, $m$, and $o$ denote Ni atoms in the inner,
mid, and outer layers, respectively. For the $n=3$ nickelate (La4Ni3O10), the
bond lengths in the experimental $Bmab$ structure are 1.92 Å(Ni($i$)-O($p$)),
1.94 Å(Ni($i$)-O($a$)), 1.92, 1.94 Å(Ni($o$)-O($p$)), and 2.00, 2.17
Å(Ni($o$)-O($a$)) [56]. higher-order RP phases
---
|  $n$=3 |  $n$=4 |  $n$=5 |  $n$=6
d (Å) |  $a$ (Å)=3.84 $c$ (Å)=27.96 |  $a$=3.84 $c$=35.75 |  $a$=3.84 $c$=43.42 |  $a$=3.84 $c$=51.14
Ni($i$)-O($p$) | 1.92 | 1.92 | 1.92 | 1.92
Ni($i$)-O($a$) | 1.98 |  1.94 1.97 | 1.93 | 1.93
Ni($m$)-O($p$) |  |  | 1.92 | 1.92
Ni($m$)-O($a$) |  |  |  1.93 1.97 |  1.94 1.97
Ni($o$)-O($p$) | 1.92 | 1.92 | 1.92 | 1.92
Ni($o$)-O($a$) |  1.93 2.14 |  1.93 2.15 |  1.93 2.14 |  1.93 2.14
Figure 2:  Calculated band structure of La4Ni3O10 ($n=3$ RP nickelate) using
(a) a doubled ($\sqrt{2}$ $\times\sqrt{2}$) tetragonal structure with $I4/mmm$
symmetry, without octahedral tilts and (b) the experimental orthorhombic
structure with $Bmab$ symmetry (the band structure of the monoclinic cell with
$P2_{1}/a$ symmetry is almost identical, see Appendix A, Fig. 5). The bands
are shown along a high-symmetry path: Y (0,1/2,0)- $\Gamma$\- M (1/2,1/2,0) -
Y (0,1/2,0). The labels $\alpha$, $\beta$, and $\gamma$ are used to mark the
different bands contributing to the Fermi surface, following the notation of
ARPES experiments reported in Ref. [53]. The gray line indicates the blurry
area from the $\gamma$ band spectral weight in the Fermi surface observed in
ARPES. (c) Calculated Fermi surface of La4Ni3O10. Blue color corresponds to
the Fermi surface of the tetragonal (1$\times$1) $I4/mmm$ cell. Magenta color
corresponds to the folded Fermi surface for the doubled ($\sqrt{2}$
$\times\sqrt{2}$) $I4/mmm$ cell. Primed labels correspond to high-symmetry
points in the 1$\times$1 tetragonal cell.

Using the vasp-optimized structures, we performed further electronic structure
calculations using the full-potential all-electron code wien2k [61]. We
continued to use GGA-PBE as our exchange-correlation functional [59]. Muffin-
tin radii of 2.41, 1.94, and 1.67 a.u. were used for La, Ni, and O,
respectively, and an $R_{MT}K_{max}$=7.0 was chosen in these simulations. A
fine 22$\times$22$\times$22 $k$-mesh was used for all RP phases. Additional
data related to the theory is available in Ref. [62]

##  III Electronic structure

We start by drawing a direct comparison between the known electronic structure
of La4Ni3O10 ($n=3$) in the experimentally resolved orthorhombic structure
with $Bmab$ symmetry [53, 63] and that for a $\sqrt{2}$ $\times\sqrt{2}$
supercell of our calculated tetragonal structure with $I4/mmm$ symmetry [see
Figs. 2 (a,b)]. The two band structures are very similar around the Fermi
level, which serves as a further benchmark of the realistic description
provided by our constructed tetragonal structures with $I4/mmm$ symmetry (see
Appendix B for further details). The different bands contributing to the Fermi
surface are marked as $\alpha$, $\beta$, and $\gamma$ following the notation
of Ref. [53]. As shown in Fig. 2(c), La4Ni3O10 displays a large hole pocket
centered at the corner of the original tetragonal Brillouin zone ($\alpha$
band) of dominant $d_{x^{2}-y^{2}}$ character. In addition, there is an extra
electron pocket of mixed orbital character around $\Gamma$ ($\beta$ band), and
a holelike pocket of dominant $d_{z^{2}}$ character at the corner of the zone
[$\gamma$ band in the gray area in Fig. 2a]. We note that the calculated Fermi
surface agrees well with the ARPES data of Ref. [53].

Figure 3:  Band structures of the $n=3-6$ RP nickelates calculated in the
tetragonal (1$\times$1) $I4/mmm$ cells: (a),(b) $n=3$ (La4Ni3O10), (c),(d)
$n=4$ (La5Ni4O13), (e),(g) $n=5$ (La6Ni5O16), and (h-j) $n=6$ (La7Ni6O19). The
bands (shown along $\Gamma$\- X′ (1/2,0,0)- M′ (1/2,1/2,0)- $\Gamma$\- Z
(0,0,1/2)) are projected onto Ni-$d_{z^{2}}$ (orange) and Ni-$d_{x^{2}-y^{2}}$
(blue) orbitals. The (a),(c),(e), and (h) panels highlight the inner-layer Ni
atoms, and the (b),(d),(g), and (j) panels the outer-layer Ni atoms for the
$n=3-6$ compounds. The (f) and (i) panels highlight the mid-layer Ni atoms for
the $n=5$ and $6$ compounds. The labels AB, NB, and B denote antibonding,
nonbonding, and bonding Ni-$d_{z^{2}}$ orbitals, as described in the main
text. Note that the line thickness is proportional to the contribution of each
orbital.

Considering the good agreement achieved between the band structures of the
calculated and experimental structures for the $n=3$ RP nickelate, we now
present the electronic structure for the $n=4-6$ compounds in the tetragonal
(1$\times$1) $I4/mmm$ cells. The band structures in the nonmagnetic state are
shown in Fig. 3 and they resemble that of La4Ni3O10 (the corresponding
densities of states are presented in Appendix C). All the band structures
($n=4-6$) show the involvement of bands of both $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ character around the Fermi level, as expected from their
$d^{7+\delta}$ filling. The number of $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ bands
crossing the Fermi level increases with $n$, to account for the increasing
number of Ni atoms in the structure (the band structure for LaNiO3 is shown in
Appendix D as a reference for the 3D limit). We note that metallic behavior
has been experimentally found in $n=3-5$ La-based RP nickelates, as mentioned
above [51]. Whereas the $n=3$ system shows a kink in the $T$-dependent
resistivity (a signature of density wave formation) [64, 65, 53, 54], no
upturn or kink has been experimentally found in the $n=4-5$ materials to date
[51].

In order to clearly understand the band fillings in each material, the band
structures in Fig. 3 not only have their Ni-$d_{z^{2}}$ and $d_{x^{2}-y^{2}}$
orbital character highlighted, but they are also presented separately for each
inequivalent Ni atom in the structure (inner, mid, and outer layers). It
should first be noted that the $d_{z^{2}}$ bands are filled quite differently
for different Ni atoms in the structure. In contrast, no relevant distinction
is observed for the filling of the $d_{x^{2}-y^{2}}$ orbitals. We invoke the
molecular subband picture used in Ref. [66] to explain how the differences in
$d_{z^{2}}$ filling come about. In Ref. [66], this model was applied to the
electronic structure of the layered $n=3$ material, La4Ni3O8. The $n$
molecular subbands arise as the $d_{z^{2}}$ orbitals hybridize strongly along
the $c$ axis due to the natural quantum confinement in the structure provided
by the blocking fluorite slabs between NiO2 trilayers. As such, for the $n=3$
reduced compound, for each Ni triple along the $c$ axis with neighbors coupled
by the hopping integral $t$, the eigenvalues and eigenvectors are [66]

$\displaystyle E_{z^{2}}$ $\displaystyle=0,~{}\pm\sqrt{2}t,$ (1)
$\displaystyle\ket{E_{z^{2}}}$
$\displaystyle=\frac{1}{\sqrt{2}}(1,0,-1),~{}\frac{1}{2}(1,\mp\sqrt{2},1)$

The odd symmetry nonbonding (NB) state, which does not involve the inner Ni
site, as shown in the eigenvector coefficients, is surrounded below and above
by even symmetry bonding (B) and antibonding (AB) states.

Table 2: Eigenvalues and eigenvectors of a simple molecular orbital problem with strong $d_{z^{2}}$ coupling for the $n=4-6$ RP nickelates. The order for the eigenvector coefficients is lowermost-outer, lowermost-mid, inner, uppermost-mid, and uppermost-outer Ni (see Fig. 1). | Eigenvalue $E_{z^{2}}$ | Eigenvector $\ket{E_{z^{2}}}$
---|---|---
$n=4$ |  |
| $\frac{1}{2}(-1-\sqrt{5})t$ | $\frac{1}{\sqrt{5+\sqrt{5}}}(1,1+\sqrt{5},1+\sqrt{5},1)$
| $\frac{1}{2}(1-\sqrt{5})t$ | $\frac{1}{\sqrt{5+\sqrt{5}}}(1,-1+\sqrt{5},1-\sqrt{5},-1)$
| $\frac{1}{2}(-1+\sqrt{5})t$ | $\frac{1}{\sqrt{5+\sqrt{5}}}(1,1-\sqrt{5},1-\sqrt{5},1)$
| $\frac{1}{2}(1+\sqrt{5})t$ | $\frac{1}{\sqrt{5+\sqrt{5}}}(1,-1-\sqrt{5},1+\sqrt{5},-1)$
$n=5$ |  |
| $-\sqrt{3}t$ | $\frac{1}{2\sqrt{3}}(1,\sqrt{3},2,\sqrt{3},1)$
| $-t$ | $\frac{1}{2}(1,1,0,-1,-1)$
| 0 | $\frac{1}{2}(1,0,-1,0,1)$
| $t$ | $\frac{1}{2}(1,-1,0,1,-1)$
| $\sqrt{3}t$ | $\frac{1}{2\sqrt{3}}(1,-\sqrt{3},2,-\sqrt{3},1)$
$n=6$ |  |
| $-1.8t$ | $\frac{1}{4.31}(-1,1.80,-2.25,2.25,-1.80,1)$
| $-1.25t$ | $\frac{1}{2.39}(1,-1.25,0.55,0.55,-1.25,1)$
| $-0.44t$ | $\frac{1}{1.92}(-1,0.44,0.80,-0.80,-0.44,1)$
| $0.44t$ | $\frac{1}{1.92}(1,0.44,-0.80,-0.80,0.44,1)$
| $1.25t$ | $\frac{1}{2.39}(-1,-1.25,-0.55,0.55,1.25,1)$
| $1.8t$ | $\frac{1}{4.31}(1,1.80,2.25,2.25,1.80,1)$

In Fig. 3(a) and 3(b) the Ni-$d_{z^{2}}$ characters are emphasized for inner-
and outer-layer Ni atoms, respectively, in the $n=3$ RP compound, La4Ni3O10.
Even though this material does have apical oxygens (unlike its La4Ni3O8
layered counterpart), the molecular subband picture survives enabled by the
small interplanar coupling between cells due to the blocking $R$-O perovskite
slab. The bands display both the spectrum and the inner-outer character – the
midenergy $d_{z^{2}}$ band (NB) has purely outer-layer Ni atom character,
while the other two bands above and below (AB and B) have both inner- and
outer-layer Ni atom character.

Based on this procedure, we calculate the eigenvalues and eigenvectors for the
$n=4-6$ nickelates as shown in Table 2. The bands once again display the
calculated spectrum and inner-, mid-, and outer-layer character faithfully.
For the $n=4$ (La5Ni4O13) nickelate, Fig. 3(c) shows how the inner-layer Ni
atoms have less contribution to the mid-energy $d_{z^{2}}$ band, in agreement
with the derived coefficients for their eigenvectors. In the $n=5$ compound
(La6Ni5O16) the inner-layer Ni atoms do not contribute to the second and
fourth $d_{z^{2}}$ bands [Fig. 3(e)], whereas the midlayer Ni atoms do not
contribute to the $d_{z^{2}}$ midenergy band [Fig. 3(f)], all in agreement
with the zero coefficients in their eigenvectors. In the $n=6$ compound
(La7Ni6O19), the inner-layer Ni atoms have less contribution to the second
$d_{z^{2}}$ band [Fig. 3(h)], the midlayer Ni atoms to the third one [Fig.
3(i)] and the outer-layer ones to the highest-lying $d_{z^{2}}$ band [Fig.
3(j)], in agreement with the eigenvector coefficients as well. Based on this
good agreement between the band character projected onto the inner-, mid-, and
outer-layer Ni atoms and the eigenvalues and eigenvectors in our model,
natural quantum confinement effects (enabled by the blocking $R$-O perovskite
unit) can be anticipated in these nickelate materials. Further, this finding
stresses the importance of distinguishing the inner-, mid-, and outer-layers
of Ni atoms, since some charge differentiation among them will likely take
place, as happens in multilayered cuprates.

Figure 4: Fermi surfaces of the RP nickelate phases calculated in the
tetragonal (1$\times$1) $I4/mmm$ structure for (a) $n=4$ (La5Ni4O13), (b)
$n=5$ (La6Ni5O16), and (c) $n=6$ (La7Ni6O19).

Fig. 4 shows the Fermi surface plots for the $n=4-6$ RP nickelates calculated
in the tetragonal (1$\times$1) $I4/mmm$ cells. Similar to cuprates, all RP
nickelates display large hole pockets of $d_{x^{2}-y^{2}}$ character
(resembling cupratelike arcs), that are centered at the corner of the original
Brillouin zone. The number of these $d_{x^{2}-y^{2}}$ pockets is equal to the
number of inequivalent Ni atoms and the Fermi surface sheets are split due to
the interlayer hopping. In addition to the $d_{x^{2}-y^{2}}$ holelike sheets,
extra electronlike pockets appear at the zone center of mixed
$d_{z^{2}}$+$d_{x^{2}-y^{2}}$ character or pure $d_{z^{2}}$ character. There
is more contribution from these central pockets as $n$ increases, in agreement
with the increasing number of Ni atoms in the structure. Also, the fermiology
can be seen to become more 3D-like as $n$ increases with an extra outermost
arc of dominant $d_{z^{2}}$ character (associated with the outer-Ni atoms)
appearing in the $n=4-6$ phases (see Appendix E for more details on the Fermi
surfaces). The fact that holelike pockets resembling a cuprate Fermi surface
are found likely implies the possibility that more cuprate properties could be
achievable in this family of materials. Importantly, our results are
consistent with ARPES data in La5Ni4O13 [51] even though the electronlike band
around the zone center is not visible in these experiments (the authors argue
this is likely due to the strong $k_{z}$ dispersion of this band that cannot
be probed by He-I$\alpha$ photons). In the trilayer system, Fermi surface
nesting is a distinct possibility as shown in Ref. [54]. Testing this
possibility in higher-$n$ nickelates would require calculating the static
susceptibilities, which is left for future work, but is definitely an
interesting route to pursue.

##  IV Summary

In summary, we have presented the electronic structure of the newly
synthesized higher-order nickelate RP phases ($n=4-6$) using first-principles
calculations. Based on the known electronic structure of La4Ni3O10, we analyze
the electronic structure and Fermi surfaces of our calculated RP phases to
find their similarities with and differences to the cuprates. Our results
show, for all materials, large holelike Fermi surfaces of $d_{x^{2}-y^{2}}$
character that closely resemble the Fermi surface of optimally hole-doped
cuprates. However, extra $d_{z^{2}}$ orbitals appear with increasing
involvement (and consequently 3D-like character) for higher values of $n$.
These aspects indicate that nickelate physics could potentially shed light on
the origin of certain aspects of cuprate physics. We expect that our study
will stimulate further experiments to characterize the electronic structure of
higher-order nickelate RP phases.

## Acknowledgements

We thank V. Pardo and G.A. Pan for fruitful discussions. We acknowledge NSF
Grant No. DMR-2045826 and the ASU Research Computing Center for HPC resources.
This research was supported in part by the National Science Foundation
[Platform for the Accelerated Realization, Analysis, and Discovery of
Interface Materials (PARADIM)] under Cooperative Agreement No. DMR-2039380.

##  Appendix A Electronic structure of the $n=3$ $P2_{1}/a$ structure

The $n=3$ material La4Ni3O10 is known to crystallize in an orthorhombic
($Bmab$) or monoclinic ($P2_{1}/a$) structure that is a
$\sqrt{2}$$\times$$\sqrt{2}$ supercell of the tetragonal ($I4/mmm$) parent
phase [53, 55, 54, 56]. The electronic structure of the monoclinic
($P2_{1}/a$) structure for La4Ni3O10 has also been calculated. The band
structure is similar to that of the $Bmab$ structure shown in the main text
(See Appendix Fig. 5). For this calculation we used the wien2k [61] code with
the computational parameters described in Section II, and a $k$-mesh of
25$\times$24$\times$9.

Figure 5: Electronic structure for the monoclinic structure (with $P2_{1}/a$
symmetry) of La4Ni3O10. Structural data were taken from Ref. [55].

##  Appendix B Electronic structure of $\sqrt{2}\times\sqrt{2}$ RP phases

In the main text, we present only the electronic structure of the calculated
tetragonal (1$\times$1) $I4/mmm$ structures for the $n=4-6$ RP phases.
However, we also investigated the corresponding $\sqrt{2}\times\sqrt{2}$
supercells to be able to further draw comparisons to the known electronic
structure of the $n=3$ material (for which an orthorhombic ($Bmab$) and
monoclinic ($P2_{1}/a$) symmetry have been reported, as mentioned above [55]).
We performed the corresponding electronic structure calculations in the
$\sqrt{2}\times\sqrt{2}$ $I4/mmm$ cells using the wien2k code [61]. We
employed the computational parameters described in Section II and used a fine
$k$-mesh of 21 $\times$ 21 $\times$ 21\. The obtained band structures with
$d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ characters highlighted are shown in Appendix
Figs. 6-9.

Figure 6: Orbital-resolved band structure of La4Ni3O10 ($n=3$) in the
$\sqrt{2}$ $\times$ $\sqrt{2}$ supercell of the $I4/mmm$ tetragonal structure
with $d_{z^{2}}$ (orange) and $d_{x^{2}-y^{2}}$ (blue) characters highlighted
for inner (a) and outer (b) Ni atoms. As in Fig. 3(a), we denote the molecular
subbands for the AB(red arrow), NB(purple), and B(magenta) $d_{z^{2}}$ states.

Figure 7: Orbital-resolved band structure of La5Ni4O13 ($n=4$) in the
$\sqrt{2}$ $\times$ $\sqrt{2}$ supercell of the $I4/mmm$ tetragonal structure
with $d_{z^{2}}$ (orange) and $d_{x^{2}-y^{2}}$ (blue) characters highlighted
for inner (a) and outer (b) Ni atoms.

Figure 8: Orbital-resolved band structure of La6Ni5O16 ($n=5$) in the
$\sqrt{2}$ $\times$ $\sqrt{2}$ supercell of the $I4/mmm$ tetragonal structure
with $d_{z^{2}}$ (orange) and $d_{x^{2}-y^{2}}$ (blue) characters highlighted
for inner (a), mid (b), and outer (c) Ni atoms. Figure 9: Orbital-resolved
band structure of La7Ni6O19($n=6$) in the $\sqrt{2}$ $\times$ $\sqrt{2}$
supercell of the $I4/mmm$ tetragonal structure with $d_{z^{2}}$ (orange) and
$d_{x^{2}-y^{2}}$ (blue) characters highlighted for inner (a), mid (b), and
outer (c) Ni atoms.

##  Appendix C Density of States of RP nickelate phases

The atom-resolved densities of states for the tetragonal $n=3-6$ La-based RP
phases are shown in Appendix Fig. 10. The centroids of the Ni and O states do
not significantly change with $n$.

Figure 10: Atom-resolved densities of states for Ni (top) and O (bottom) ions
in La4Ni3O10 ($n=3$), La5Ni4O13 ($n=4$), La6Ni5O16 ($n=5$), and La7Ni6O19
($n=6$) RP phases obtained using the tetragonal (1$\times$1) $I4/mmm$ cells.
In the density of states (DOS) plots for the oxygen atoms, spacer refers to
the oxygens in the LaO blocking layer.

##  Appendix D Electronic structure of LaNiO3

The electronic structure (band structure and Fermi surface) of LaNiO3 in a
tetragonal $P4/mmm$ and cubic $Pm\bar{3}m$ phase obtained using GGA and the
computational parameters described above are shown in Appendix Fig. 11. We
used a $k$-mesh of 27$\times$27$\times$27 for the cubic phase and
25$\times$25$\times$30 for the tetragonal phase.

Figure 11: (Top and middle panel) Orbital-resolved band structures of LaNiO3
in the cubic $Pm\bar{3}m$ (a) and the tetragonal $P4/mmm$ (b) phases,
respectively, with highlighted character for Ni-$d_{z^{2}}$ states in orange
and for Ni-$d_{x^{2}-y^{2}}$ states in blue. (Bottom) Fermi surface of LaNiO3
calculated in the cubic and the tetragonal cells. The employed high-symmetry
paths are the following: $\Gamma$ (0,0,0), X (1/2,0,0), M (1/2,1/2,0), R
(1/2,1/2,1/2) in the cubic phase and $\Gamma$ (0,0,0), X (1/2,0,0), M
(1/2,1/2,0), Z (0,0,1/2), R (1/2,0,1/2), A (1/2,1/2,1/2) in the tetragonal
phase.

##  Appendix E Fermi surface of RP nickelates: orbital characters

Here, we further analyze the character of the different Fermi surface sheets
shown in Fig. 4 of the main text for the $n=4-6$ RP nickelates. We define the
corner-centered Fermi surfaces as the arcs for convenience. The
$\Gamma$-centered Fermi surfaces and the corner-centered arcs are electron-
and hole-like, respectively. There are Fermi arcs of dominant
$d_{x^{2}-y^{2}}$ character in each material (one for each inequivalent Ni
atom in the structure). Interestingly, extra arcs of $d_{z^{2}}$ character
start showing up as one moves towards the 3D limit with increasing $n$, as
described below. As such, for the $n=4$ material, the outermost arc has
dominant outer-Ni $d_{z^{2}}$ character, the middle arc outer-Ni
$d_{x^{2}-y^{2}}$ character, and the inner arc inner-Ni $d_{x^{2}-y^{2}}$
character. For the electron pockets, the outer circular one has dominant
inner-Ni $d_{x^{2}-y^{2}}$ character, and the inner square-like one has outer-
Ni $d_{z^{2}}$ orbital character. The Fermi surface arcs from the corner to
$\Gamma$ in the $n=5$ phase show, in sequence, outer-Ni $d_{z^{2}}$ character,
outer-Ni, mid-Ni, and inner-Ni $d_{x^{2}-y^{2}}$ characters, respectively. The
two large electron pockets centered around $\Gamma$ have inner- and mid-Ni
$d_{x^{2}-y^{2}}$ character. For the $n=6$ material, from the zone corner to
$\Gamma$ the first arc shows a mixture of outer- and mid-Ni $d_{z^{2}}$
character, the subsequent two arcs have dominant outer-Ni $d_{x^{2}-y^{2}}$
character, followed by a mid-Ni $d_{x^{2}-y^{2}}$ arc, and finally one arc of
inner-Ni $d_{x^{2}-y^{2}}$ character. The electron pockets have mid-Ni
$d_{x^{2}-y^{2}}$, inner-Ni $d_{x^{2}-y^{2}}$, and inner- and outer-Ni
$d_{z^{2}}$ characters, respectively.

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