# Electron-phonon mediated superconductivity in La$_{6}$Ni$_{5}$O$_{12}$ nickel oxides

Álvaro Adrián Carrasco Álvarez, Sébastien Petit, Lucia Iglesias, Manuel Bibes, Wilfrid Prellier, Julien Varignon

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Álvaro Adrián Carrasco Álvarez, Sébastien Petit, Lucia Iglesias, Manuel Bibes, Wilfrid Prellier, et al. Electron-phonon mediated superconductivity in La_6Ni_5O_12 nickel oxides. Physical Review B, 2024, 110 (23), pp.235107. 10.1103/PhysRevB.110.235107. hal-05393736

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Electron-phonon mediated superconductivity in La<sub>6</sub>Ni<sub>5</sub>O<sub>12</sub> nickel oxides

Álvaro Adrián Carrasco Álvarez,¹,² Sébastien Petit,¹ Lucia Iglesias,² Manuel Bibes,²

Wilfrid Prellier,¹ and Julien Varignon¹

¹Laboratoire CRISMAT, CNRS UMR 6508, ENSICAEN, Normandie Université, 6 boulevard Marechal Juin, F-14050 Caen Cedex 4, France

²Unite Mixte de Physique, CNRS, Thales, Université Paris Sud, Université Paris-Saclay, F-91767 Palaiseau, France

<figure><img src="image_2.png" /></figure>

(Received 22 November 2023; revised 24 September 2024; accepted 5 November 2024; published 2 December 2024)

Nickel oxide superconductors offer an alternative playground for understanding the formation of Cooper pairs in correlated materials such as the famous cuprates. By studying the La<sub>n+1</sub>Ni<sub>n</sub>O<sub>2n+2</sub> phase diagram on the basis of hybrid and spin-polarized density functional theory simulations, we reveal the existence of charge and bond ordered (CBO) insulating phases that are quenched by doping effects, ultimately resulting in a metallic phase at the *n* = 5 member. Nevertheless, the phonons associated with the CBO identified in the phase diagram remain sufficiently large to mediate Cooper pairs in La<sub>6</sub>Ni<sub>5</sub>O<sub>12</sub>, yielding a computed critical temperature between T<sub>c</sub> = 11 and 19 K consistent with the 13 K observed experimentally in Nd<sub>6</sub>Ni<sub>5</sub>O<sub>12</sub>. Hence, an electron-phonon mechanism cannot be ruled out in nickelates. Finally, in order to identify the superconducting mechanism, extracting the relevant instabilities in the doping phase diagram of superconductors appears critical.

DOI: 10.1103/PhysRevB.110.235107

# I. INTRODUCTION

The absence of analogous compounds to cuprates has been detrimental to understanding the underlying mechanism behind high-*T*<sub>c</sub> superconductivity. From that perspective, nickel-based superconductors were proposed decades ago [1,2] to host superconductivity (SC), but they were only realized in 2019 with the discovery of SC in infinite layered (IL) nickelates [3–8]. These materials possess the generic R<sub>n+1</sub>Ni<sub>n</sub>O<sub>2n+2</sub> chemical formula, also called the reduced Ruddlesden-Popper (RRP) compounds, where R = La, Pr, or Nd. They are based on *n* RNiO<sub>2</sub> building blocks stacked on top of each other and separated by a fluorite RO<sub>2</sub> spacing layer [see Fig. 1(a)]. In these systems, the formal oxidation state (FOS) *x* of the Ni cations depends on the number of layers *n* present in the structure with *x* = (n + 1)/n and hence it varies between *x* = 2 (n = 1) and *x* = 1 (n = ∞), as discussed in Ref. [9].

To date, most studies focus on the IL phase [10–18], and a few of them discuss the RRP series, where either they focus on specific cases of the series [19–32] or they study the whole series, albeit neglecting the spin and/or structural degrees of freedom, leading to the prediction that all compounds are either metallic [33,34] or insulating if long-range antiferromagnetic (AFM) order is imposed [35]. This behavior is at odds with experiments where the *n* = 2 (*x* = 1.5) compounds are insulating [25,27,30], the *n* = 3 (*x* = 1.33) case is found to be insulating at low temperatures [19,21–23], but a small pressure effect can turn it into a metal [20,24], and the *n* = 5 (*x* = 1.2) case is found to be superconducting [6]. This collection of experimental results shows that there has to be a transition between insulating to metallic character with increasing *n* in the RRP series of compounds. This is reminiscent of the results of Ref. [9], where the transition to SC appears at *x* = 1.2 for the doped infinite layered phase.

Nickelates are usually classified as nonconventional superconductors, and several theoretical studies have indeed ruled out the existence of an electron-phonon coupling (EPC) to explain the Cooper pair formation [36,37]. However, these studies are performed with dramatic shortcuts, such as (i) using highly symmetric primitive cells that prevent structural lowering events potentially coupled to electronic features, (ii) involving an inappropriate description of exchange and correlation (xc) phenomena in density functional theory calculations along with using the local density approximation (LDA) or the generalized gradient approximation (GGA), and/or (iii) forcing similar occupancies between up and down electrons locally on each site [nonmagnetic (NM) solution] despite the obvious presence of open shells on nickel atoms—experiments suggesting the presence of spins in the SC region [38]. All of these shortcuts are well known nowadays to be irrelevant to model properties of transition-metal oxides [39–42], including the trend in doping effects and the emergence of superconductivity [9]. Furthermore, several experimental data analysis suggest an electron-phonon mechanism in these nickelates [43–45]. It is then natural to wonder, “what is the trend in electronic structure upon doping the RRP nickelates if one performs density functional theory (DFT) calculations with including all relevant degrees of freedom,” “is an electron-phonon coupling really ruled out at this DFT level,” and “can we identify a common mechanism with other (oxide) superconductors?”

In this article, we present a systematic study of the phase diagram of the La<sub>n+1</sub>Ni<sub>n</sub>O<sub>2n+2</sub> family of compounds by using DFT calculations that involve (i) local Ni spin formation; (ii) all relevant structural degrees of freedom; and (iii) one of the highest levels of DFT functional for describing xc phenomena before explicitly including dynamical correlation effects. We reveal the existence of multiple charge and bond orderings (CBOs) as well as insulating states in the phase diagram, which disappears at the *n* = 5 case, finding a metallic com-
<figure><img src="image_3.png" /><figcaption>FIG. 1. Sketches of the high-symmetry cell and related struc-
tural distortions exhibited by $R_{n+1}Ni_nO_{2n+2}$ compounds. (a) High-
symmetry cell of reduced Ruddlesden-Popper compounds based on
$n$ NiO$_2$ planes stacked along the $c$ axis and separated by $RO_2$ layers.
(b)–(d) Disproportionation distortions exhibited by $n=2$ (b), $n=3$
(c), and $n=4$ (d) reduced Ruddlesden-Popper compounds.</figcaption></figure>

pound. We thus reproduce the experimental crossover from
insulating (n = 1, 2, 3, and 4) to metallic (n = 5). Within
the metallic compound, we demonstrate that the bond dis-
proportionation vibrations associated with the CBO identified
for n = 2 and 3 compounds are sufficient to reproduce an
electron-phonon coupling constant λ = 0.61, yielding a crit-
ical temperature T_c estimated between 19 and 11 K. This is
in very good agreement with the experimental T_c of 13 K
measured in Nd<sub>6</sub>Ni<sub>5</sub>O<sub>12</sub> [6], further supporting the phonon-
mediated mechanism identified in infinitely layered nickelates
[9] and proposed on the basis of experimental data analyses
[43,44]. These results highlight that the phase diagram of
nickelates and their SC properties must be understood start-
ing from the n = 1 member and not n = ∞, allowing us to
identify the existence of a CBO phase as a common root with
bismuthate and antimonate superconductors [46–50].

II. METHOD

The choice of the exchange-correlation functional. To sufficiently amend self-interaction errors (SIEs) inherent to the practical implementation of DFT, the choice of the xc functional is a critical step, especially for modeling transition-metal oxides (TMOs). It is now well established that local and semilocal xc functionals such as LDA and GGA (i.e., rung 1 and 2 on Jacob's ladder [51]) cannot capture the trends in structural distortions and the metallic or insulating character of TMOs, and hence they cannot be employed to model their doping effects and superconducting properties [39–42]. One can employ meta-GGA xc functionals (rung 3, semilocal functional) such as the strongly constrained and appropriately normalized (SCAN) [52] or DFT+U/hybrid methods (rung 4) that are nonlocal operators of the density matrix. The DFT+U method has a disadvantage for doped compounds in which cations can exhibit multiple formal oxidation states (FOSs): the U parameter, preventing the delocalization of electrons, is FOS-dependent. The meta-GGA xc functionals offer a parameter-free technique, but it can fail to capture (or strongly underestimate) the band

splitting and band gap of compounds, especially for com-
pounds with low FOS [9,39,53]. In this study, we have
employed the hybrid Heyd-Scuseria-Ernzerhof HSE06 func-
tional [54], which is one of the highest levels of DFT before
explicitly treating dynamical correlation effects. It has two
advantages: (i) calculations remain at a parameter-free level,
and (ii) it offers a correct estimation of band splitting that is a
critical aspect to evaluate the electron-phonon coupling as we
discuss below. We have used the standard HSE06 parametriza-
tion with 25% of exact Hartree-Fock exchange and a range
separation of 0.2 Å. We emphasize that the HSE06 functional
involves a Perdew-Burke-Ernzerhof (PBE) correlation term
(GGA functional) [55], and hence it does not improve that
description. Instead, it corrects the exchange part with exact
Hartree-Fock exchange for amending the spurious SIE term.
Although the HSE06 can sometimes overlocalize electrons
and exaggerate the insulating state such as in cuprates [56,57],
the HSE06 correctly captures the insulating state of the $n=1$
RRP nickelate member (as shown later in the manuscript) as
well as the metallic nature of the $n = \infty$ member [53]. Hence
one expects that the trend in structural distortions and insulat-
ing or metallic character of the intermediate RRP members is
correctly captured.

The critical role of local spin formation. As discussed in the Introduction, there are many studies in the literature ruling out the existence of an EPC in nickelate superconductors [36,37]. However, these studies are performed with non-spin-polarized calculations (NM), preventing local spin formation by construction. This is rather counterintuitive, notably for Ni cations adopting a $1+(3d^9)$ to $2+(3d^8)$ formal oxidation state (FOS) for which there are obviously unpaired electrons. Such a shortcut was previously shown to be irrelevant to capture the physical properties of TMOs [39–42], including the superconducting mechanism in infinite layered nickelates [9]. As discussed in Ref. [9], the NM approximation for doped infinite layered nickelates produces solutions at least 1800 K above any spin-polarized solutions. Even though one involves the highest level of description of correlation effects with the Green’s function, it remains an improvement of an erroneous/irrelevant solution [37,58].

To circumvent this shortcut, we have employed a ferromag-
netic (FM) order since it is the lowest spin-order state allowing
coupled electronic-structural instabilities to be captured, such
as disproportionation and Jahn-Teller effects [39]. We again
emphasize that the HSE06 functional may sometimes over-
estimate magnetic moments and band gaps such as in the
AFM ground state of cuprates [56], inherently missing the
transition from an insulator to a metal with doping effects.
The use of the FM order has the advantage to avoid addi-
tional overlocalization introduced with an AFM order that
would promote more compact electronic states and tenden-
cies toward disproportionation effects [59]. A full magnetic
order study of the whole $La_{n+1}Ni_nO_{2n+2}$ phase diagram would
ideally require fully correlated techniques such as interaction
configuration methods that accurately estimate the magnetic
exchange interactions in oxides [60]. We have nevertheless
performed additional checks using various AFM orders, and
the overall results are unchanged.

*The choice of the unit cells.* Several studies restrict calculations to highly symmetric primitive unit cells preventing
structural lowering events from occurring. Again, these shortcuts cannot account for potential structural distortions altering the band structure and bands splitting [42]. We have therefore employed sufficiently large supercells allowing potential localization of added electrons (i.e., a polaronic states). At this stage, it is critical to use cells with an integer number of electrons to allow such phenomena, instead of using a simple virtual crystal approximation (VCA) with a fractional number of electrons per primitive cell—hence preventing polaronic states from forming by construction. Our supercells correspond to $(\sqrt{2}, \sqrt{2}, 1)$ ($n=1$), $(\sqrt{2}, \sqrt{2}, 1)$ ($n=2$), $(3, 1, 1)$ ($n=3$), $(2\sqrt{2}, 2\sqrt{2}, 1)$ ($n=4$), and $(5, 1, 1)$ ($n=5$) cells with respect to the high-symmetry 14/mmm cell of RRP compounds.

Estimation of the electron-phonon coupling and critical temperature. To date, there has been no practical implementation of the calculation of electron-phonon coupling with a local spin degree of freedom [61]. Thus, one has to circumvent the problem and compute manually the electron-phonon coupling. To that end, we have considered the phonons that are intrinsically strongly coupled to electronic features such as bond disproportionation [9,39,47–49]. The electron-phonon coupling constant $\lambda$ can be evaluated as the sum of the mode $\nu$ and momentum $\vec{q}$-dependent coupling constant $\lambda_{\vec{q},\nu}$ [62,63]:

$$\lambda = \frac{1}{N_{\vec{q}}} \sum_{\vec{q},\nu} \lambda_{\vec{q},\nu}, \quad (1)$$

where $N_{\vec{q}}$ is the number of $q$ points. Additionally, $\lambda_{\vec{q},\nu}$ in the double $\delta$ approximation is given by [64–67]

$$\lambda_{q,\nu} = \frac{2}{N(E_F)} \sum_{\vec{k}} \frac{1}{\omega_{\vec{q},\nu}} |M_{\vec{k},\vec{k}+\vec{q}}^{\nu}|^2 \delta(E_{\vec{k}} - E_F) \delta(E_{\vec{k}+\vec{q}} - E_F), \quad (2)$$

where $N(E_F)$ is the density of states at the Fermi level $E_F$, the electron energies are denoted by $E_{\vec{k}}$ and $E_{\vec{k}+\vec{q}}$, with momentum $\vec{k}$ and $\vec{k}+\vec{q}$, respectively, $\omega_{\vec{q},\nu}$ is the phonon mode frequency at a given momentum $\vec{q}$, and $M_{\vec{k},\vec{k}+\vec{q}}^{\nu}$ is the electron phonon matrix element, which can be defined as

$$M_{\vec{k},\vec{k}+\vec{q}}^{\nu} = \sum_{j} \left( \frac{\hbar^2}{2M_j \omega_{\vec{q},\nu}} \right)^{1/2} e_{\vec{q},j}^{\nu} \langle \vec{k} + \vec{q} | \frac{\delta V}{\delta u_{\vec{q},j}^{\nu}} | \vec{k} \rangle \quad (3)$$

with $M_j$ the mass of the moving atom, $e_{\vec{q},j}^{\nu}$ is the jth component of the polarization vector, and $\langle \vec{k} + \vec{q} | \frac{\delta V}{\delta u_{\vec{q},j}^{\nu}} | \vec{k} \rangle$ is the reduced electron-phonon matrix element (REPME). Since we are interested only in the electronic states close to the Fermi level, we can change the $\delta$ functions in Eq. (2) by the density of states at the Fermi level $N(E_F)$, obtaining

$$\lambda_{\vec{q},\nu} = 2N(E_F) \sum_{\vec{k}} \frac{1}{\omega_{\vec{q},\nu}} |M_{\vec{k},\vec{k}+\vec{q}}^{\nu}|^2. \quad (4)$$

Introducing Eq. (4) back into Eq. (1), we obtain that the electron-phonon coupling constant can be expressed as

$$\lambda = \frac{2N(E_F)}{N_{\vec{q}}} \sum_{\vec{k},\vec{q},\nu} \frac{1}{\omega_{\vec{q},\nu}} |M_{\vec{k},\vec{k}+\vec{q}}^{\nu}|^2. \quad (5)$$

For modes that involve only a single type of atomic species, such as the bond disproportionation modes that involve only oxygen anions with mass $M_O$, the electron phonon matrix element in Eq. (3) can be simplified into

$$M_{\vec{k},\vec{k}+\vec{q}}^{\nu} = \left( \frac{\hbar^2}{2M_O \omega_{\vec{q},\nu}} \right)^{1/2} e_{\vec{q},O}^{\nu} \langle \vec{k} + \vec{q} | \frac{\delta V}{\delta u_{\vec{q},O}^{\nu}} | \vec{k} \rangle. \quad (6)$$

Regarding the REPME, one can evaluate its magnitude for a given phonon branch and $\vec{q}$ vector by finite displacements and measuring the difference in the Kohn-Sham eigenenergies [62–64,68],

$$\langle \vec{k} + \vec{q} | \frac{\delta V}{\delta u_{\vec{q},O}^{\nu}} | \vec{k} \rangle = \frac{E_{\vec{k}+\vec{q}} - E_{\vec{k}}}{\Delta u_{\vec{q},O}^{\nu}}, \quad (7)$$

with $\Delta u_{\vec{q},O}^{\nu}$ being a finite amplitude of the given phonon mode frozen in the material. In the case of a mode involving a single atomic species, one can write the following identity:

$$e_{\vec{q},O}^{\nu}(\vec{k} + \vec{q}) |\frac{\delta V}{\delta u_{\vec{q},O}^{\nu}}|\vec{k}\rangle = \frac{E_{\vec{k}+\vec{q}} - E_{\vec{k}}}{|Q_{\vec{q},\nu}|}, \quad (8)$$

where $Q_{\vec{q},\nu}$ is the phonon mode amplitude. Now since we are interested mainly in the physics around the Fermi level, we can evaluate the change in the eigenstates as the band splitting $\Delta E_b = (\frac{E_{\vec{k}+\vec{q}} - E_{\vec{k}}}{2})$ and the REPME $D_{\vec{q}}^{\nu} = \frac{\Delta E_b}{2|Q_{\vec{q},\nu}|}$, Assuming an isotropic coupling (i.e., bands splitting are homogeneous through the entire Brillouin zone), we can drop the $\vec{k}$ dependence on the sum in Eq. (5), and we would obtain

$$\lambda = \frac{2N(E_F)}{N_{\vec{q}}} \sum_{\vec{q},\nu} \frac{\hbar^2}{2M_O \omega_{\vec{q},\nu}^2} |D_{\vec{q}}^{\nu}|^2. \quad (9)$$

In practice, the DFT calculations are used to extract (i) the bands splitting $\Delta E_b$ associated with the introduction of a finite $(\vec{q}, \nu)$ phonon mode, and (ii) the frequency $\omega_{\vec{q},\nu}$ of the mode. Point (i) is highly critical as one has to accurately estimate the bands splitting. It is well known that local and semilocal xc functionals underestimate band gaps—and hence bands splitting—of materials, including those of highly uncorrelated semiconductors such as Si and Ge. Hence one has to use a DFT functional accurately capturing this quantity. At this stage, hybrid DFT represents the best compromise between accuracy and a parameter-free technique. The evaluation of point (ii) is rather straightforward: by fitting the total energy $E$ versus mode amplitude $Q_{\vec{q},\nu}$ of a $(\vec{q}, \nu)$ phonon starting from the ground-state structure with a polynomial expression of the form $E(Q_{\vec{q},\nu}) = aQ_{\vec{q},\nu}^2 + bQ_{\vec{q},\nu}^4$, one directly extracts the mode frequency $\omega_{\vec{q},\nu}$ as

$$\omega_{\vec{q},\nu} = \sqrt{2a/M_O} \quad (10)$$

for $(\vec{q}, \nu)$ modes involving only O motions—recalling that the energy of a harmonic oscillator is given by $\frac{1}{2}M_O\omega_{\vec{q},\nu}^2Q_{\vec{q},\nu}^2$.

Finally, one estimates the critical temperature $T_c$ of the material by using the McMillan-Allen equation

$$T_c = \frac{\hbar\omega_{\text{log}}}{1.2} \exp\left(-\frac{1.04\lambda}{\lambda - \mu^*(1 + 0.62\lambda)}\right), \quad (11)$$
<table><caption>TABLE I. Key properties of relaxed RRP compounds with a non-spin-polarized (NM) and ferromagnetic (FM) solution for the *n* = 1 – 5 members. Experimental data when available are provided. Results entail the formal oxidation state (FOS), the space group, lattice parameters (in Å), the band gap *E*<sub>g</sub> (eV), the energy difference between the FM and NM solutions (in meV/f.u.), the magnetic moments of the different types of Ni cations (in µ<sub>B</sub>), and the amplitude (in Å/f.u.) of disproportionation modes *Q*<sub>Boc</sub> extracted from a symmetry mode analysis with respect to a high-symmetry 14/mmm cell.</caption><thead><tr><th>n</th><th>FOS</th><th>phase</th><th>Space group</th><th>a (Å)</th><th>b (Å)</th><th>c (Å)</th><th>E<sub>g</sub> (eV)</th><th>ΔE (meV/f.u.)</th><th>μ<sub>NiL</sub></th><th>μ<sub>NiS</sub></th><th>Q<sub>Boc</sub> (Å/f.u.)</th></tr></thead><tbody><tr><td rowspan="3">1</td><td rowspan="3">2+</td><td>NM</td><td>I4/mmm</td><td>3.70</td><td>3.70</td><td>13.45</td><td>1.36</td><td>0</td><td></td><td></td><td>0</td></tr><tr><td>FM</td><td>Aema</td><td>5.57</td><td>5.46</td><td>12.52</td><td>2.80</td><td>-28</td><td>1.73</td><td>1.73</td><td>0</td></tr><tr><td>exp.</td><td>Aema [79]</td><td>5.53</td><td>5.43</td><td>12.49</td><td>1.51 [81] -4 [82]</td><td></td><td>1.48 [78,83]</td><td></td><td>0</td></tr><tr><td rowspan="3">2</td><td rowspan="3">1.5+</td><td>NM</td><td>I4/mmm</td><td>3.95</td><td>3.95</td><td>18.93</td><td>0</td><td>0</td><td></td><td></td><td>0</td></tr><tr><td>FM</td><td>Ammm</td><td>5.62</td><td>5.62</td><td>18.88</td><td>1.24</td><td>-263</td><td>0.93</td><td>0.09</td><td>0.09</td></tr><tr><td>exp.</td><td>I4/mmm [25]</td><td></td><td></td><td></td><td>Ins. [30,84]</td><td></td><td></td><td></td><td>>0 [25]</td></tr><tr><td rowspan="3">3</td><td rowspan="3">1.33+</td><td>NM</td><td>I4/mmm</td><td>3.96</td><td>3.96</td><td>25.36</td><td>0</td><td>0</td><td></td><td></td><td>0</td></tr><tr><td>FM</td><td>Fmmm</td><td>16.88</td><td>5.63</td><td>23.31</td><td>1.00</td><td>-313</td><td>0.93</td><td>0.15</td><td>0.07</td></tr><tr><td>exp.</td><td>I4/mmm [85]</td><td>3.97</td><td>3.97</td><td>26.09</td><td>Ins. [22,24]</td><td></td><td>>0 [23,86]</td><td>>0</td><td>>0 [22,23]</td></tr><tr><td rowspan="3">4</td><td rowspan="3">1.25+</td><td>NM</td><td>I4/mmm</td><td>3.96</td><td>3.96</td><td>31.84</td><td>0</td><td>0</td><td></td><td></td><td>0</td></tr><tr><td>FM</td><td>Cmcm</td><td>11.27</td><td>11.27</td><td>31.71</td><td>1.49</td><td>-398</td><td>0.93</td><td>0.17</td><td>0.05</td></tr><tr><td>exp.</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">5</td><td rowspan="3">1.2+</td><td>NM</td><td>I4/mmm</td><td>3.96</td><td>3.96</td><td>38.33</td><td>0</td><td>0</td><td></td><td></td><td>0</td></tr><tr><td>FM</td><td>I4/mmm</td><td></td><td></td><td></td><td>0</td><td>0</td><td>0.77</td><td>0.77</td><td>0</td></tr><tr><td>exp.</td><td>I4/mmm</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

where μ* is the screened Coulomb potential with typical values between 0.1 and 0.15, and ω<sub>log</sub> is the logarithmic frequency, which can be estimated for a finite number of frequencies as

$$ \omega_{\text{log}} = \left( \prod_{i}^{n} \omega_{i} \right)^{1/n} . \qquad (12) $$

*Other details of the method.* First-principles DFT simulations are performed with the CRYSTAL17 software package [69]. Structural optimizations (lattice parameters plus atomic positions) are performed until the root mean square of forces and estimated displacements are lower than $3 \times 10^{-4}$ Ha Bohr⁻¹ and 0.0012 Bohr, respectively. The convergence on the total energy is set to $10^{-9}$ Hartree. Gaussian basis sets with core electrons treated in pseudopotentials are used for Ni and La cations, extracted from Refs. [70–72], and an all-electron basis set is used for O anions and taken from Ref. [73]. Brillouin zones are sampled with $8 \times 8 \times 8$ ($n = 1$), $4 \times 4 \times 8$ ($n = 2$), $3 \times 3 \times 3$ ($n = 3$), $4 \times 4 \times 2$ ($n = 4$), and $4 \times 4 \times 4$ ($n = 5$) points. All calculations are performed within the primitive cell as implemented in CRYSTAL17. The mesh was increased to $32 \times 32 \times 32$ points for density-of-state calculations for the *n* = 5 member. Symmetry mode analysis is performed with the AMPLIMODES application [74,75] implemented in the Bilbao Crystallographic Server and ISODISTORT from the ISOTROPY software packages [76,77].

### III. RESULTS AND DISCUSSION

#### A. The failure of the NM approximation

We first evidence the need to include the spin degree of freedom by performing a study of the phase diagram (*n* = 1 – 5), using the non-spin-polarized (NM) approximation. The

results of structural relaxations are provided in Table I. We find that (i) all compounds present conducting behavior except the first member (*n* = 1) that is insulating, and (ii) all ground states are achieved with a high-symmetry 14/mmm crystal structure [Fig. 1(a)] as found in previous studies [33,34]. This is in contrast to experimental data available for the *n* = 1 compound and with a spin-polarized DFT performed here [78–80] showing an Aema space group with octahedral rotations.

The *n* = 1 (Ni²⁺) compound is identified as a band insulator with a band gap of 1.36 eV, originating from a $3d^8$ low-spin (*S* = 0) electronic configuration enforced by the NM approximation (see Fig. 2). This means that Hund's coupling is neglected, and the crystal-field splitting of a doubly occupied 3d shell for the Ni²⁺ cation is responsible of a band-gap opening [see Fig. 2(a)]. However, this is in contrast with experiments that reveal the existence of a magnetic moment of 1.48 µ<sub>B</sub> per Ni cation in La₂NiO₄ [78,83], thus suggesting a high spin configuration as depicted in Fig. 2(b). The high-symmetry cell identified with DFT is also at odds with the experimental structure exhibiting $a^{-}a^{-}c^{+}$ octahedral rotations with Aema space group [78–80]. Even though we start from this structure, it relaxes back to a 14/mmm cell with no distortion. Regarding higher members of the series, the observation of a metallic character for *n* = 2 or 3 disagrees with the experimental observation of an insulating state at low temperatures [22,24,30,84].

This general failure of the NM approximation is grounded in the fact that Hund's rule is broken, and important terms in the energy are not well accounted. We evidence our claims by performing spin-polarized FM calculations in the high-symmetry 14/mmm cell, and we compute the total energy difference between the two types of calculations. This allows us to reveal the sole effect of the spin degree of freedom in the energy of the system without including any lattice distortions that may appear in the global ground state. We report the
<figure><img src="image_4.png" /><figcaption>FIG. 2. Diagram of the crystal-field splitting in Ni<sup>2+</sup> cations with a 3d<sup>8</sup> electronic configuration in the case of La<sub>2</sub>NiO<sub>4</sub> for two different situations: (a) Hund's coupling is neglected and a NM solution is employed in the DFT calculations producing a *S* = 0 spin state; (b) Hund's coupling is included and the spin degree of freedom is allowed in the DFT calculations producing a *S* = 1 spin state.</figcaption></figure>

energy difference Δ*E*<sub>NM-FM</sub> between the NM and FM solu-
tions as a function of *n* in Fig. 3. There is a massive energy
gain Δ*E*<sub>NM-FM</sub> of at least 375 meV/NiO<sub>2</sub> for all *n* = 1–5
RRP compounds, signaling that a huge part of the energy of
the system is not well accounted for if one ignores the local
spin formation. Thus, we conclude here that (i) local Ni spin
formation in these nickelates is a key factor that should not
be overlooked in the DFT simulations, and (ii) even though
we treat exchange correlation phenomena at one of the high-
est levels in DFT simulations taking into account hybrid xc
functionals, the NM solutions remain irrelevant for modeling
the electronic properties of oxide superconductors.

<figure><img src="image_5.png" /><figcaption>FIG. 3. Energy difference ΔE$_{NM-FM}$ between a non-spin-
polarized (NM) and ferromagnetic (FM) order (in meV/NiO$_{2}$) as a
function of the formal Ni 3d occupancy in the reduced Ruddlesden-
Popper compound. Red dashed lines indicate the position of each
member *n* of the RRP series.</figcaption></figure>

B. The ground-state properties with a spin-polarized solution

Key results of structural relaxations for *n* = 1–5 members
with an FM order are reported in Table I. For all members,
we identify structural lowering events producing large energy
gains up to 400 meV/f.u., with the exception of the *n* = 5
member, which keeps its high-symmetry *I*4/mmm cell.

The *n* = 1 is found to be an insulator with a gap lo-
cated between majority- and minority-spin channels of Ni
cations associated with a 3d⁸ (Ni²⁺) high-spin electronic
configuration [see Fig. 4(a)]. Unlike the NM solution, the
spin-polarized solution can stabilize the *Aema* space group
that is observed experimentally [78–80]. The computed
magnetic moments as well as band-gap values and the crystal-
lographic cell are in good agreement with experiments (μNi =
1.48 μB [78,80] and E<sub>g</sub> = 1.51–4 eV [81,82]), confirming the
quality and reliability of the HSE06 functional for studying
doping effects.

For the *n* = 2–4 members, we observe large energy gains
related to the appearance of sizable bond disproportionation
distortion amplitude Q<sub>Boc</sub> producing different local environ-
ments for Ni cations [see Figs. 1(b)–1(d)]. It produces two
types of Ni cations sitting in a compressed and extended
O<sub>4</sub> complex, and labeled Ni<sub>S</sub> and Ni<sub>L</sub>, respectively, with a
ratio between large and small complexes able to accommodate
the Ni mixed valence states. The disproportionation effect
produces different magnetic moments with Ni<sub>S</sub> approaching
0μ<sub>B</sub> and Ni<sub>L</sub> bearing a magnetic moment of roughly 1μ<sub>B</sub>. It
points toward Ni cations adopting a 3d<sup>8</sup> low spin (Ni<sub>S</sub><sup>2+</sup>)
and 3d<sup>9</sup> (Ni<sub>L</sub><sup>+</sup>) electronic configurations, respectively. The
resulting effect of the B<sub>oc</sub> modes is to open a band gap
in the materials with band edges formed between Ni<sub>L</sub><sup>+</sup> (d<sup>9</sup>)
and Ni<sub>S</sub><sup>2+</sup> (d<sup>8</sup>) states the *n* = 2, 3, and 4 RRP compounds
[see Figs. 4(b)–4(d)]. These results are compatible with ex-
periments that show an insulating character for *n* = 2 and
3 members—the *n* = 4 compound has not been synthesized
to the best of our knowledge. Furthermore, the existence of
charge orderings between Ni<sub>L</sub><sup>+</sup> and Ni<sub>S</sub><sup>2+</sup> for *n* = 2 and 3 RRP
is confirmed experimentally [19–22,25,30]. Their existence
was also predicted theoretically for *n* = 2–3 [31,87,88]. The
propensity of Ni cations to undergo disproportionation effects
in an infinite layered phase was also proposed on the basis of
more elaborate dynamical mean field theory (DMFT) calcu-
lations [14,89], confirming the reliability of the DFT results,
albeit not including dynamical correlation effects. It was also
suggested experimentally in O-rich doped *n* = ∞ members
[90]. Finally, the existence of disproportionation effects of
the 1.5+ formal oxidation state of Ni cations in La<sub>3</sub>Ni<sub>2</sub>O<sub>6</sub>
is in agreement with our previous simulations on half-doped
infinite layered nickelates [9]. However, we should point out
that we could not find an experimental structure able to adopt
a charge ordering—a primitive *I*4/mmm cell reported for the
*n* = 2 and 3 situations is unable to account for the Ni<sub>L</sub><sup>+</sup>/Ni<sub>S</sub><sup>2+</sup>
charge- and bond-ordered character and *de facto* its insulating
state [25]. The *I*4/mmm space group has also been precisely
challenged for the *n* = 3 situation by X-ray diffraction that
reveals a stripe charge order of 2 Ni<sub>L</sub><sup>+</sup> and 1 Ni<sub>S</sub><sup>2+</sup> in (ab) planes
stacked along the *c* axis—as we observe in our simulations
[22]—hence it is incompatible with an *I*4/mmm primitive cell
symmetry.
<figure><img src="image_6.png" /><figcaption>FIG. 4. Electronic properties of the reduced Ruddlesden-Popper material from *n* = 1 to 4. Projected identity of states on Ni<sub>L</sub> (filled gray area), Ni<sub>S</sub> (blue line), and O (red line) states for the *n* = 1 (a), *n* = 2 (b), *n* = 3 (c), and *n* = 4 (d) RRP members. The vertical dashed line represents the valence-band maximum (VBM).</figcaption></figure>

The *n* = 5 member is found to be metallic with an I4/mmm structure. Even though we start from a charge-ordered cell, it spontaneously vanishes and relaxes back to a high-symmetry metallic state. This observation of a metallic state for *n* = 5 is in agreement with the insulating to metallic crossover between *n* = 1–3 and 5 members observed experimentally. We conclude here that *n* = 2–4 RRP compounds are prone to exhibit CBO, resulting in an insulating phase.

### C. Origin of the various bond disproportionation modes

*Potential energy surfaces for the FM order.* To understand the origin of the charge orderings observed in the different nickelate members, we plot the potential energy surface (PES) associated with each type of disproportionation mode starting from the highly symmetric undistorted I4/mmm cell with identical O4 complexes for each member of RRP *n* = 2–4. Results are displayed in Fig. 5. We start by analyzing the PES for the FM state. The *n* = 2 type of distortion shows a shifted single-well potential to nonzero values of the mode and produces a large energy gain of –117 meV/NiO2. It signals the presence of an electronic instability associated with the 1.5+ formal oxidation state that prefers to transform to more stable 1+ and 2+ in the ground state. The potentials for *n* = 3 and 4 are slightly different with respect to the *n* = 2 mode by showing a double-well potential shape and lower energy gains, reaching at most –40 meV/NiO2. The existence of a double-well potential and lower energy gains suggest a partial screening of the electronic instability. Such screening originates from doping effects that alter the bandwidth of the

Ni *d* states, thereby weakening the electronic instability as explained in Ref. [54]. We indeed observe a global increase of the bandwidth *W* from *W* = 1.5 eV (*n* = 2) to *W* = 2.2 eV for *n* = 3 and 4 in the starting I4/mmm undistorted cell.

*Results are independent of magnetic orders.* By using AFM orders (Fig. 5), the extra correlations can shrink the bandwidth of Ni *d* states with respect to FM order [59]. The resulting effect is to shift the single-well potential to lower energies, signaling a stronger tendency to develop disproportionation effects for *n* = 2, 3, and 4 potential energy surfaces.

*Local spin formation is a critical ingredient.* We performed additional calculations of the PES using the NM approximation. Unlike any of the tested spin-polarized solutions, the PESs for the *n* = 2, 3, and 4 compounds exhibit single-well potentials centered at 0, indicating that the materials are not willing to undergo any disproportionation effects. This is in contrast with experiments and other theoretical calculations that show the stabilization of Ni<sub>L</sub><sup>+</sup> and Ni<sub>S</sub><sup>2+</sup> in the compounds [9, 19, 20, 21, 22, 23, 24, 25, 30, 31]. This observation is reminiscent of the RNiO<sub>3</sub> situation in which non-spin-polarized DFT calculations could not predict the stabilization of the disproportionation effects, at odds with experiments and spin-polarized DFT calculations [39].

*Check of the existence of an electronic instability.* The existence of an electronic instability can be confirmed by performing two sets of calculations in the high-symmetry cell for the *n* = 2 compound without any lattice distortion and with an FM order in which (i) we initially force an equal occupancy of all neighboring Ni cations, and (ii) we perform a calculation in which we initially force neighboring Ni cations
<figure><img src="image_7.png" /><figcaption>FIG. 5. Potential energy surface associated with the various bond disproportionation modes. Evolution of the total energy $ΔE$ (in meV/f.u.) as a function of the $B_{oc}$ mode amplitude $Q_{Boc}$ (in Å/f.u.) in the $n = 2$ (left panel), $n = 3$ (middle panel), and $n = 4$ (right panel) RRP compounds. The reference energy is taken at $Q_{Boc} = 0$ for each magnetic order, and the depth of the curves is actually not representative of the stability of each magnetic state with respect to each other. Calculations are performed for a ferromagnetic interaction (FM, gray filled circles), (ab) plane FM interactions with antiferromagnetic interactions between consecutive planes along the *c* axis (AFMA, brown filled squares), pure AFM interaction between all nearest neighbors in all directions (AFM, red filled diamonds), AFM interaction between second nearest neighbor in the (ab)-plane with FM couplings along *c* (AFME, orange filled down triangles), and non-spin-polarized solutions (NM, green filled triangles).</figcaption></figure>

to have either a $d^8$ or a $d^9$ configuration. After the variational self-consistency, we observe that point (ii) produces an energy gain of 35 meV/NiO$_2$, confirming the existence of the electronic instability of the Ni $3d^{8.5}$ configuration. The material then develops a lattice distortion to accommodate the charge ordering with a ratio 1:1 of $Ni_L^{2+}:Ni_S^{2+}$ and produces an insulating state.

The disproportionation effects are independent of the *xc* functional. Although one may argue that the HSE06 functional may overlocalize electrons and enforce insulating states instead of metals, such as in doped cuprates, we have checked that the tendency toward disproportionation effects is independent of the DFT *xc* functional. Reference [9] already identified the tendency toward disproportionation effect of La$_3$Ni$_2$O$_6$ using the meta-GGA SCAN functional. It signals the presence of a strong electronic instability toward disproportionation even though one implies a “lower level” DFT *xc* functional on the Jacob’s ladder. We conclude here that these nickelates are governed by a large electron-phonon coupling able to produce a charge ordering, electron localization, and open a band gap. This is observed only if relevant degrees of freedom such as the local Ni spin formation are included in the simulation.

### D. The metallic state is in the vicinity of a charge-ordered phase

The weakening of the tendency toward disproportionation effects of Ni cations upon doping the compounds can also be tracked by quantifying the amplitude associated with the relevant disproportionation $B_{oc}$ modes in the ground-state structures (Fig. 6). We observe that upon increasing the number of layers $n$, from $n = 2$ to $5$, there is a clear weakening of the $B_{oc}$ modes amplitude until it vanishes at $n = 5$. Therefore, all types of disproportionation effects are quenched by electron doping the nickelates. Thus, the metallic region and the observed superconducting RRP compound (i.e., $n = 5$) sit in the vicinity of a charge- and bond-ordered phase. This situation is reminiscent of antimonate (bismuthate) superconductors Ba$_{1-x}$K$_x$SbO$_3$ (Ba$_{1-x}$K$_x$BiO$_3$) in which hole doping

(i) suppresses disproportionation instabilities associated with Sb$^{4+}$ (Bi$^{4+}$) cations at $x = 0$, and (ii) enables a superconducting regime once the material is sufficiently doped [46,48,50]. Although one may think that the insulator-to-metal transition could be related to dimensionality crossover as in iridates SrIrO$_3$ (metal) and Sr$_2$IrO$_4$ (insulator) [91,92]—going from $n = 2$ to $n = \infty$ members allows us to tune the distance between Ir and apices O and therefore alter the crystal-field splitting of Ir *d* states—the nickelates systems are different since there are no apices oxygen. Hence the crystal field is not sizably altered by increasing the number of layers in the RRP series. We thus conclude here that the electronic instability and its subsequent lattice distortion in nickelates are sufficiently weakened by electron doping in order to prevent electron localization, consequently resulting in a metallic state.

<figure><img src="image_8.png" /><figcaption>FIG. 6. Amplitude of the breathing $B_{oc}$ mode (in Å/f.u.) extracted from a symmetry mode analysis as a function of the Ni 3d electron count.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 7. Superconducting quantities in the metallic state of La₆Ni₅O₁₂. (a) Energy difference ΔE (in meV/f.u.) upon freezing amplitude (in Å) of the *n* = 2 (blue filled circles) and *n* = 3 (green filled diamonds) B<sub>oc</sub> modes in the ground-state structure. The zero energy is set to the ground-state energy. (b) Bands splitting ΔE<sub>b</sub> (in eV) induced by freezing amplitude (in Å/f.u.) of the *n* = 2 (blue filled circles) and *n* = 3 (green filled diamonds) B<sub>oc</sub> modes in the ground-state structure. (c) Density of states (in states/eV/f.u.) associated with the ground-state structure. A finer kmesh made of 32 × 32 × 32 points is used to sample the Brillouin zone.</figcaption></figure>

E. Superconductivity can be mediated by
electron-phonon coupling

Although the electron-phonon coupling (EPC) is suffi-
ciently weakened by doping effects so that it cannot produce
electron localization or an insulating state, the disproportion-
ation modes can still be highly coupled to the electronic
structure of the metallic n = 5 case. This enables the possi-
bility of phonon-mediated superconductivity in this metallic
compound. We then evaluate the EPC of the Boc modes using
Eq. (9). This is performed by freezing the n = 2 and 3 dis-
proportionation modes in the n = 5 structure—we could not
freeze the n = 4 disproportionation mode in the n = 5 case
due to the required supercell size that becomes too large.

From the PES associated with the *n* = 2 and 3 modes
starting from the relaxed *n* = 5 ground state [Fig. 7(a)], we
can fit the curves and use Eq. (10) to extract *ω*₂ = 69 meV
and *ω*₃ = 66 meV for the modes with the *q̄* vector of the
*n* = 2 and 3 *B*<sub>oc</sub> modes, respectively [i.e., (1/2,1/2,0) and
(1/3,1/3,0)]. The *n* = 2 mode is compatible with the fre-
quency computed for half-doped infinite layered nickelates
in Ref. [9] to *ω*₂ = 65 meV. This frequency is also very

similar to disproportionation modes appearing in bismuthates
and antimonates [47–49,93]. Regarding the evaluation of the
reduced electron-phonon matrix element (REPME) D̃<sub>q̄,v</sub>, we
have frozen finite amplitudes Q of *n* = 2 or 3 phonon modes
in the *n* = 5 ground-state structure. We observe a splitting of
bands dispersing around the Fermi level (Fig. 8). The splitting
is homogeneous along the Γ – X–M–Γ – Z path, meaning
that the effect of modes on the band dispersion is isotropic.
We further check that the bands splitting ΔE<sub>b</sub> are linear with
the amplitude of the displacements [Fig. 7(b)]—thereby one
is still in the harmonic regime—and from a linear fit of the
curves, we extract that D<sub>2</sub> = 7.8 eV/Å and D<sub>3</sub> = 8.6 eV/Å
for the *n* = 2 and 3 modes, respectively. Finally, we extract the
ground-state density of states at the Fermi level N(E<sub>F</sub>) of the
*n* = 5 material by sampling the Brillouin zone with a kmesh
made of 32 × 32 × 32 points, and we extract N(E<sub>F</sub>) = 0.607
states/eV/f.u./spin channel.

We then proceed to evaluate the EPC. Since modes are
optics and isotropic, we use Eq. (9) and obtain an EPC
λ = 0.61. This value is reminiscent of λ = 0.51 computed in
infinite layered nickelates phases [9] but also with experimen-

<figure><img src="image_10.png" /><figcaption>FIG. 8. Band structure of La₆Ni₅O₁₂ ground state (a) and with the addition of a finite amplitude *Q* = 0.15 Å/f.u. of the *n* = 2 mode (b) in the ground-state structure. A ferromagnetic order is used.</figcaption></figure>
tal data analysis suggesting an EPC oscillating between 0.56 and 0.61 [43]. Finally, we use the McMillan-Allen equation [Eq. (11)] for computing the critical temperature. The log average frequency $\omega_{\text{log}}$ is estimated as $\omega_{\text{log}} = (\omega_2\omega_3)^{1/2} = 67$ meV [Eq. (12)]. Within this approximation, and taking typical values of screening parameter $\mu^* = 0.1-0.15$, we estimate a critical temperature of $T_c$ between 11 and 19 K. These values are close to the experimental $T_c$ of Nd₆Ni₅O₁₂ ($T_c$ = 13 K [6]) and La₀.₈Sr₀.₂NiO₂ ($T_c$ = 9-14 K [5,8]), where both systems present an effective Ni¹.²⁺ 3d⁸.⁸ configuration.

## IV. CONCLUSIONS

We explain the electron doping phase diagram of RRP nickelate compounds, and we show that superconductivity in the appropriately doped nickelate can emerge through an EPC associated with charge and bond orderings. This agrees with our previous DFT study on infinite layered nickelates, confirming that there is a common root between all these

nickelate superconductors. Although Ni cations present correlation effects inherent to any transition-metal element with partly filled *d* states, the origin of superconductivity is in fact very similar to nonmagnetic oxide superconductors such as antimonates and bismuthates once the relevant Ni spin degree of freedom is included in the simulations. This opens the way to reinvestigating superconductivity in oxide superconductors where the role of local spin formation has been previously overlooked.

## ACKNOWLEDGMENTS

This work has received financial support from the CNRS through the MITI interdisciplinary programs under the project SuNi. The work received support from the French ANR Supernickel. The authors acknowledge access granted to HPC resources of Criann through the projects 2020005 and 2007013 and of Cines through the DARI project A0080911453.

[1] V. I. Anisimov, D. Bukhvalov, and T. M. Rice, Electronic structure of possible nickelate analogs to the cuprates, Phys. Rev. B **59**, 7901 (1999).

[2] J. Chaloupka and G. Khaliullin, Orbital order and possible superconductivity in LaNiO₃/LaMO₃ superlattices, Phys. Rev. Lett. **100**, 016404 (2008).

[3] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Superconductivity in an infinite-layer nickelate, Nature (London) **572**, 624 (2019).

[4] M. Osada et al., A superconducting praseodymium nickelate with infinite layer structure, Nano Lett. **20**, 5735 (2020).

[5] M. Osada, B. Y. Wang, B. H. Goodge, S. P. Harvey, K. Lee, D. Li, L. F. Kourkoutis, and H. Y. Hwang, Nickelate superconductivity without rare-earth magnetism: (La, Sr)NiO₂, Adv. Mater. **33**, 2104083 (2021).

[6] G. A. Pan et al., Superconductivity in a quintuple-layer square-planar nickelate, Nat. Mater. **21**, 160 (2022).

[7] S. Zeng et al., Superconductivity in infinite-layer nickelate La₁₋ₓCaₓNiO₂ thin films, Sci. Adv **8**, 9927 (2022).

[8] M. Osada, K. Fujiwara, T. Nojima, and A. Tsukazaki, Improvement of superconducting properties in La₁₋ₓSrₓNiO₂ thin films by tuning topochemical reduction temperature, Phys. Rev. Mater. **7**, L051801 (2023).

[9] Á. A. Carrasco Álvarez, L. Iglesias, S. Petit, W. Prellier, M. Bibes, and J. Varignon, Charge ordering as the driving mechanism for superconductivity in rare-earth nickel oxides, Phys. Rev. Mater. **8**, 064801 (2024).

[10] Á. A. Carrasco Álvarez, S. Petit, L. Iglesias, W. Prellier, M. Bibes, and J. Varignon, Structural instabilities of infinite-layer nickelates from first-principles simulations, Phys. Rev. Res. **4**, 023064 (2022).

[11] Y. Gu, S. Zhu, X. Wang, J. Hu, and H. Chen, A substantial hybridization between correlated ni-d orbital and itinerant electrons in infinite-layer nickelates, Commun Phys **3**, 84 (2020).

[12] M. Y. Choi, W. E. Pickett, and K. W. Lee, Fluctuation-frustrated flat band instabilities in NdNiO₂, Phys. Rev. Res. **2**, 033445 (2020).

[13] B. H. Goodge, D. Li, K. Lee, M. Osada, B. Y. Wang, G. A. Sawatzky, H. Y. Hwang, and L. F. Kourkoutis, Doping evolution of the mott-hubbard landscape in infinite-layer nickelates, Proc. Natl. Acad. Sci. USA **118**, e2007683118 (2021).

[14] K. G. Slobodchikov and I. V. Leonov, Spin density wave, charge density wave, and bond disproportionation wave instabilities in hole-doped infinite-layer RNiO₂, Phys. Rev. B **106**, 165110 (2022).

[15] Y. Wang, C. J. Kang, H. Miao, and G. Kotliar, Hund's metal physics: From SrNiO₂ to LaNiO₂, Phys. Rev. B **102**, 161118(R) (2020).

[16] Y. Zhang, J. Zhang, X. He, J. Wang, and P. Ghosez, Rare-earth control of phase transitions in infinite-layer nickelates, PNAS Nexus **2**, 1 (2023).

[17] Y. Zhang, X. He, J. Zhang, J. Wang, and P. Ghosez, Incommensurate magnetic order in hole-doped infinite-layer nickelate superconductors due to competing magnetic interactions, Adv. Funct. Mater. **33**, 2304187 (2023).

[18] R. Zhang, C. Lane, B. Singh, J. Nokelainen, B. Barbiellini, R. S. Markiewicz, A. Bansil, and J. Sun, Magnetic and F-electron effects in LaNiO₂ and NdNiO₂ nickelates with cuprate-like 3dₓ₂⁻ᵧ₂ band, Commun. Phys. **4**, 118 (2021).

[19] J. Zhang, A. S. Botana, J. W. Freeland, D. Phelan, H. Zheng, V. Pardo, M. R. Norman, and J. F. Mitchell, Large orbital polarization in a metallic square-planar nickelate, Nat. Phys. **13**, 864 (2017).

[20] X. Chen, H. Zheng, D. P. Phelan, H. Zheng, S. H. Lapidus, M. J. Krogstad, R. Osborn, S. Rosenkranz, and J. F. Mitchell, Competing charge/spin-stripe and correlated metal phases in trilayer nickelates (Pr₁₋ₓLax)₄Ni₃O₈, Chem. Mater. **34**, 4560 (2022).

[21] J. Hao, X. Fan, Q. Li, X. Zhou, C. He, Y. Dai, B. Xu, X. Zhu, and H. H. Wen, Charge-stripe fluctuations in Nd₄Ni₃O₈ as evidenced by optical spectroscopy, Phys. Rev. B **103**, 205120 (2021).

[22] J. Zhang, Y. S. Chen, D. Phelan, H. Zheng, M. R. Norman, and J. F. Mitchell, Stacked charge stripes in the quasi-2d trilayer
nickelate La₄Ni₃O₈, *Proc. Natl. Acad. Sci. USA* **113**, 8945 (2016).

[23] V. V. Poltavets *et al.*, Bulk magnetic order in a two-dimensional Ni¹⁺/Ni²⁺ (*d*⁹/*d*⁸) nickelate, isoelectronic with superconducting cuprates, *Phys. Rev. Lett.* **104**, 206403 (2010).

[24] J. G. Cheng, J. S. Zhou, J. B. Goodenough, H. D. Zhou, K. Matsubayashi, Y. Uwatoko, P. P. Kong, C. Q. Jin, W. G. Yang, and G. Y. Shen, Pressure effect on the structural transition and suppression of the high-spin state in the triple-layer T'-La₄Ni₃O₈, *Phys. Rev. Lett.* **108**, 236403 (2012).

[25] V. V. Poltavets, K. A. Lokshin, S. Dikmen, M. Croft, T. Egami, and M. Greenblatt, La₃Ni₂O₆: A new double T'-type nickelate with infinite Ni¹⁺/²⁺O₂ layers, *J. Am. Chem. Soc.* **128**, 9050 (2006).

[26] N. apRoberts-Warren, J. Crocker, A. P. Dioguardi, K. R. Shirer, V. V. Poltavets, M. Greenblatt, P. Klavins, and N. J. Curro, NMR evidence for spin fluctuations in the bilayer nickelate La₃Ni₂O₆, *Phys. Rev. B* **88**, 075124 (2013).

[27] K. A. Lokshin, D. Mitchell, M. V. Lobanov, V. Struzhkin, and T. Egami, Synthesis and characterization of pure infinite layer Ni⁺ nickelates: LnNi₂O₇ (Ln = La, Nd, Pr) and La₃Ni₂O₆, *ECS J. Solid State Sci. Technol.* **11**, 044008 (2022).

[28] R. Dutta, A. Maity, A. Marsicano, J. R. Stewart, M. Opel, and W. Paulus, Revealing the effect of interstitial oxygen on the low-energy crystal electric field excitations of Pr³⁺ in 214 -nickelates, *Phys. Rev. B* **105**, 195147 (2022).

[29] Y. Shen *et al.*, Charge condensation and lattice coupling drives stripe formation in nickelates, *Phys. Rev. Lett.* **126**, 177601 (2021).

[30] Z. Liu *et al.*, Evidence for charge and spin density waves in single crystals of La₃Ni₂O₇ and La₃Ni₂O₆, *Sci. China: Phys., Mech. Astron.* **66**, 217411 (2023).

[31] A. S. Botana, V. Pardo, W. E. Pickett, and M. R. Norman, Charge ordering in Ni¹⁺/Ni²⁺ nickelates: La₄Ni₃O₈ and La₃Ni₂O₆, *Phys. Rev. B* **94**, 081105(R) (2016).

[32] H. Li, P. Hao, J. Zhang, K. Gordon, A. G. Linn, X. Chen, H. Zheng, X. Zhou, J. F. Mitchell, and D. S. Dessau, Electronic structure and correlations in planar trilayer nickelate Pr₄Ni₃O₈, *Sci. Adv.* **9**, eade4418 (2023).

[33] H. LaBollita, M. C. Jung, and A. S. Botana, Many-body electronic structure of $d^{9-6}$ layered nickelates, *Phys. Rev. B* **106**, 115132 (2022).

[34] H. LaBollita and A. S. Botana, Electronic structure and magnetic properties of higher-order layered nickelates: La$_{n+1}$Ni$_n$O$_{2n+2}$ (n = 4–6), *Phys. Rev. B* **104**, 035148 (2021).

[35] M. C. Jung, H. LaBollita, V. Pardo, and A. S. Botana, Antiferromagnetic insulating state in layered nickelates at half filling, *Sci. Rep.* **12**, 17864 (2022).

[36] Y. Nomura, M. Hirayama, T. Tadano, Y. Yoshimoto, K. Nakamura, and R. Arita, Formation of a two-dimensional single-component correlated electron system and band engineering in the nickelate superconductor NdNi₂O₄, *Phys. Rev. B* **100**, 205138 (2019).

[37] Q. N. Meier, J. B. de Vaulx, F. Bernardini, A. S. Botana, X. Blase, V. Olevano, and A. Cano, Preempted phonon-mediated superconductivity in the infinite-layer nickelates, *Phys. Rev. B* **109**, 184505 (2024).

[38] J. Fowlie *et al.*, Intrinsic magnetism in superconducting infinite-layer nickelates, *Nat. Phys.* **18**, 1043 (2022).

[39] J. Varignon, M. Bibes, and A. Zunger, Mott gapping in 3d ABO₃ perovskites without mott-hubbard interelectronic repulsion energy U, *Phys. Rev. B* **100**, 035119 (2019).

[40] G. Trimarchi, Z. Wang, and A. Zunger, Polymorphous band structure model of gapping in the antiferromagnetic and paramagnetic phases of the mott insulators MnO, FeO, CoO, and NiO, *Phys. Rev. B* **97**, 035107 (2018).

[41] O. I. Malyi and A. Zunger, False metals, real insulators, and degenerate gapped metals, *Appl. Phys. Rev.* **7**, 041310 (2020).

[42] J. Varignon, M. Bibes, and A. Zunger, Origin of band gaps in 3d perovskite oxides, *Nat. Commun.* **10**, 1658 (2019).

[43] E. F. Talantsev, Quantifying interaction mechanism in infinite layer nickelate superconductors, *J. Appl. Phys.* **134**, 113904 (2023).

[44] E. F. Talantsev, Classifying superconductivity in an infinite-layer nickelate Nd₀.₈Sr₀.₂NiO₂, *Results Phys* **17**, 103118 (2020).

[45] E. F. Talantsev and V. V. Chistyakov, Debye temperature, electron-phonon coupling constant, and three-dome shape of crystalline strain as a function of pressure in highly compressed La₃Ni₂O₇₋₈, *Lett. Mater.* **14**, 262 (2024).

[46] A. W. Sleight, Bismuthates: BaBiO₃ and related superconducting phases, *Physica C* **514**, 152 (2015).

[47] Z. P. Yin, A. Kutepov, and G. Kotliar, Correlation-enhanced electron-phonon coupling: Applications of GW and screened hybrid functional to bismuthates, chloronitrides, and other high-$T_c$ superconductors, *Phys. Rev. X* **3**, 021011 (2013).

[48] J. Varignon, Origin of superconductivity in hole doped SrBiO₃ bismuth oxide perovskite from parameter-free first-principles simulations, *npj Comput. Mater.* **9**, 30 (2023).

[49] Z. Yuan, P. Zheng, Y. Peng, R. Liu, X. Ma, G. Wang, T. Yu, and Z. Yin, Correlation-enhanced electron-phonon coupling and superconductivity in (Ba, K)SbO₃ superconductors, *Phys. Rev. B* **105**, 014517 (2022).

[50] M. Kim *et al.*, Superconductivity in (Ba, K)SbO₃, *Nat. Mater.* **21**, 627 (2022).

[51] R. Car, Density functional theory: Fixing jacob's ladder, *Nat. Chem.* **8**, 820 (2016).

[52] J. Sun, A. Ruzsinszky, and J. Perdew, Strongly constrained and appropriately normed semilocal density functional, *Phys. Rev. Lett.* **115**, 036402 (2015).

[53] A. A. Carrasco Alvarez, M. Bibes, W. Prellier, and J. Varignon, Strongly constrained and appropriately normed density functional theory exchange-correlation functional applied to rare earth oxides, *Phys. Rev. B* **107**, 115109 (2023).

[54] A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, Influence of the exchange screening parameter on the performance of screened hybrid functionals, *J. Chem. Phys.* **125**, 224106 (2006).

[55] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, *Phys. Rev. Lett.* **77**, 3865 (1996).

[56] K. Pokharel, C. Lane, J. W. Furness, R. Zhang, J. Ning, B. Barbiellini, R. S. Markiewicz, Y. Zhang, A. Bansil, and J. Sun, Sensitivity of the electronic and magnetic structures of cuprate superconductors to density functional approximations, *npj Comput. Mater.* **8**, 31 (2022).

[57] J. K. Perry, J. Tahir-Kheli, and W. A. Goddard, Ab initio evidence for the formation of impurity $d_{3z^2-r^2}$ holes in doped La$_{2-x}$Sr$_x$CuO$_4$, *Phys. Rev. B* **65**, 144501 (2002).
[58] Z. Li and S. G. Louie, Two-gap superconductivity and the decisive role of rare-earth d electrons in infinite-layer nickelates, Phys. Rev. Lett. **133**, 126401 (2024).

[59] J. Varignon, O. I. Malyi, and A. Zunger, Dependence of band gaps in d-electron perovskite oxides on magnetism, Phys. Rev. B **105**, 165111 (2022).

[60] A. Gellé, J. Varignon, and M. B. Lepetit, Accurate evaluation of magnetic coupling between atoms with numerous open shells: An ab initio method, Europhys. Lett. **88**, 1 (2009).

[61] S. Poncé, E. R. Margine, C. Verdi, and F. Giustino, EPW: Electron-phonon coupling, transport and superconducting properties using maximally localized wannier functions, Comput. Phys. Commun. **209**, 116 (2016).

[62] K. J. Chang, M. M. Dacorogna, M. L. Cohen, J. M. Mignot, G. Chouteau, and G. Martinez, Superconductivity in high-pressure metallic phases of Si, Phys. Rev. Lett. **54**, 2375 (1985).

[63] M. M. Dacorogna, K. J. Chang, and M. L. Cohen, Pressure increase of the electron-phonon interaction in superconducting hexagonal silicon, Phys. Rev. B **32**, 1853(R) (1985).

[64] J. M. An and W. E. Pickett, Superconductivity of MgB$_2$: Covalent bonds driven metallic, Phys. Rev. Lett. **86**, 4366 (2001).

[65] A. B. Migdal, Interaction between electrons and lattice vibrations in a normal metal, Sov. Phys. JETP **7**, 996 (1958).

[66] G. M. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. JETP **11**, 696 (1960).

[67] F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. **89**, 015003 (2017).

[68] P. K. Lam, M. M. Dacorogna, and M. Cohen, Self-consistent calculation of electron-phonon couplings, Phys. Rev. B **34**, 5065 (1986).

[69] R. Dovesi et al., Quantum-mechanical condensed matter simulations with CRYSTAL, Wiley Interdiscip. Rev. Comput. Mol. Sci. **8**, e1360 (2018).

[70] J. K. Desmarais, A. Erba, and R. Dovesi, Generalization of the Periodic LCAO approach in the CRYSTAL code to G-type orbitals, Theor. Chem. Acc. **137**, 28 (2018).

[71] M. D. Towler, N. L. Allan, N. M. Harrison, V. R. Saunders, W. C. Mackrodt, and E. Apra, Ab initio study of MnO and NiO, Phys. Rev. B **50**, 5041 (1994).

[72] K. E. El-Kelany, C. Ravoux, J. K. Desmarais, P. Cortona, Y. Pan, J. S. Tse, and A. Erba, Spin localization, magnetic ordering, and electronic properties of strongly correlated Ln$_2$O$_3$ sesquioxides (Ln = La, Ce, Pr, Nd), Phys. Rev. B **97**, 245118 (2018).

[73] A. M. Ferrari and C. Pisani, An ab initio periodic study of NiO supported at the Pd(100) surface. Part 1: The perfect epitaxial monolayer, J. Phys. Chem. B **110**, 7909 (2006).

[74] D. Orobengoa, C. Capillas, I. Aroyo, and J. M. Perez, AMPLIMODES: Symmetry mode analysis on the bilbao crystallographic server research papers, J. Appl. Crystallogr. **42**, 820 (2009).

[75] J. M. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Mode crystallography of distorted structures, Acta Crystallogr. A **66**, 558 (2010).

[76] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODISPLACE: A web-based tool for exploring structural distortions, J. Appl. Crystallogr. **39**, 607 (2006).

[77] ISOTROPY Software Suite, Iso.Byu.Edu, https://iso.byu.edu/iso/isodistort_version5.6.1/isodistort.php.

[78] J. Rodriguez-Carvajal, M. T. Fernandez-Diaz, and J. L. Martinez, Neutron diffraction study on structural and magnetic properties of La$_2$NiO$_4$, J. Phys.: Condens. Matter **3**, 3215 (1991).

[79] M. Medarde and J. Rodríguez-Carvajal, Oxygen vacancy ordering in La$_{2-x}$Sr$_x$NiO$_{4-δ}$ (0.4 ≤ x4 ≤ 0.5): The crystal structure and defects investigated by neutron diffraction, Zeitschrift für Physik B Condensed Matter **102**, 307 (1997).

[80] A. Hayashi, H. Tamura, and Y. Ueda, Successive structural phase transitions in stoichiometric La$_2$NiO$_4$ observed by x-ray diffraction, Physica C **216**, 77 (1993).

[81] R. Laouici, S. Douafer, H. Lahmar, G. Rekhila, M. Trari, and M. Benamira, Elaboration and studies of physical and photo-electrochemical properties of La$_2$NiO$_4$ and its use with SnO$_2$ in photo-evolution of hydrogen under visible light irradiation, Optik (Stuttg) **236**, 166654 (2021).

[82] P. Kuiper, J. van Elp, G. A. Sawatzky, A. Fujimori, S. Hosoya, and D. M. de Leeuw, Unoccupied density of states of La$_{2-x}$Sr$_x$NiO$_{4+δ}$ studied by polarization-dependent x-ray-absorption spectroscopy and bremsstrahlung isochromat spectroscopy, Phys. Rev. B **44**, 4570 (1991).

[83] K. Yamada, T. Omata, K. Nakajima, S. Hosoya, T. Sumida, and Y. Endoh, Magnetic structure and weak ferromagnetism of La$_2$NiO$_{4+δ}$, Physica C **191**, 15 (1992).

[84] V. V. Poltavets, M. Greenblatt, G. H. Fecher, and C. Felser, Electronic properties, band structure, and fermi surface instabilities of Ni$^{1+}$/Ni$^{2+}$ nickelate La$_3$Ni$_2$O$_6$, isoelectronic with superconducting cuprates, Phys. Rev. Lett. **102**, 046405 (2009).

[85] V. V. Poltavets, K. A. Lokshin, M. Croft, T. K. Mandal, T. Egami, and M. Greenblatt, Crystal structures of Ln$_4$Ni$_3$O$_8$ (Ln = La, Nd) triple layer T'-Type nickelates, Inorg. Chem. **46**, 10887 (2007).

[86] N. apRoberts-Warren, A. P. Dioguardi, V. V. Poltavets, M. Greenblatt, P. Klavins, and N. J. Curro, Critical spin dynamics in the antiferromagnet La$_4$Ni$_3$O$_8$ from La$^{139}$ nuclear magnetic resonance, Phys. Rev. B **83**, 014402 (2011).

[87] V. Pardo and W. E. Pickett, Metal-insulator transition in layered nickelates La$_3$Ni$_2$O$_7-δ$ (δ = 0.0,0.5,1), Phys. Rev. B **83**, 245128 (2011).

[88] X. Fan and H. H. Wen, Antiferromagnetism, charge ordering and stretched Ni-O bond in Ln$_4$Ni$_3$O$_8$ (Ln = La, Nd), J. Phys. Condens. Matter **33**, 075503 (2020).

[89] H. Chen, Y. feng Yang, G. M. Zhang, and H. Liu, An electronic origin of charge order in infinite-layer nickelates, Nat. Commun. **14**, 5477 (2023).

[90] A. Raji, G. Krieger, N. Viart, D. Preziosi, J.-P. Rueff, and A. Gloter, Charge distribution across capped and uncapped infinite-layer neodymium nickelate thin films, Small **19**, 2304872 (2023).

[91] B. Kim, P. Liu, and C. Franchini, Dimensionality-strain phase diagram of strontium iridates, Phys. Rev. B **95**, 115111 (2017).

[92] A. Dhingra, T. Komesu, S. Kumar, K. Shimada, L. Zhang, X. Hong, and P. A. Dowben, Electronic band structure of iridates, Mater. Horiz. **8**, 2151 (2021).

[93] M. Braden, W. Reichardt, A. S. Ivanov, and A. Y. Rumiantsev, Anomalous dispersion of LO phonon branches in Ba$_{0.6}$K$_{0.4}$BiO$_3$, Europhys. Lett. **34**, 531 (1996).