## Abstract

Rare-earth nickelates form an intriguing series of correlated perovskite oxides. Apart from LaNiO_{3}, they exhibit on cooling a sharp metal-insulator electronic phase transition, a concurrent structural phase transition, and a magnetic phase transition toward an unusual antiferromagnetic spin order. Appealing for various applications, full exploitation of these compounds is still hampered by the lack of global understanding of the interplay between their electronic, structural, and magnetic properties. Here we show from first-principles calculations that the metal-insulator transition of nickelates arises from the softening of an oxygen-breathing distortion, structurally triggered by oxygen-octahedra rotation motions. The origin of such a rare triggered mechanism is traced back in their electronic and magnetic properties, providing a united picture. We further develop a Landau model accounting for the metal-insulator transition evolution in terms of the rare-earth cations and rationalizing how to tune this transition by acting on oxygen rotation motions.

## Introduction

First synthesized in 1971^{1}, rare-earth nickelates (*R*NiO_{3}, with *R* = rare earth) are appealing for various applications^{2, 3}, and the possibility to tune their properties in epitaxial films and heterostructures^{4} has recently fueled an even larger interest^{5,6,7}. *R*NiO_{3} compounds belong to the family of perovskite oxides with a reference \(Pm\bar 3m\) cubic structure (Fig. 1a), nevertheless not directly observed. Apart for LaNiO_{3}, which always keeps a metallic \(R\bar 3c\) phase and will not be further discussed here, all *R*NiO_{3} adopt at reasonably high temperature a metallic *Pbnm* orthorhombic phase^{8}. This phase, rather ubiquitous^{9} amongst perovskites with small Goldschmidt tolerance factor^{8}, *t*, corresponds to a distortion of the cubic structure arising from the appearance of combined anti-phase rotations of the oxygen octahedra along the *x* and *y* directions, *R*
_{
xy
} (Fig. 1b), with the same amplitude *Q*
_{R} and in-phase rotations of the oxygen octahedra along *z*, *M*
_{
z
} (Fig. 1c), with amplitude *Q*
_{M}.

In this phase, all Ni atoms are equivalent and formally in a Jahn-Teller active *d*
^{7} (likely \(t_{{\mathrm{2g}}}^6e_{\mathrm{g}}^1\) low spin) configuration. Surprisingly, although compatible with the *Pbnm* symmetry, cooperative Jahn-Teller distortions do not appear. Instead, at a temperature *T*
_{MI}, which strongly evolves with the *R* cation (i.e., with *t*)^{10}, *R*NiO_{3} compounds exhibit an electronic metal-insulator transition (MIT), which was shown to be concurrent with a structural transition from *Pbnm* to monoclinic *P*2_{1}/*n* symmetry^{11}. This symmetry lowering is accompanied with the appearance of a breathing distortion of the oxygen octahedra, *B*
_{OC} (Fig. 1d), which alternatively expand and contract with amplitude *Q*
_{B}, according to a rock-salt pattern. This gives rise to two types of Ni sites with long and short Ni–O bonds, respectively.

At low temperature (100–200 K), *R*NiO_{3} compounds finally show a magnetic phase transition toward an unusual E′-type antiferromagnetic (AFM) spin order identified by a Bragg vector **q** = (1/4, 1/4, 1/4) in pseudocubic notation^{11,12,13}. For large cations (*R* = Nd and Pr), the Néel temperature *T*
_{N} = *T*
_{MI} and the system goes directly from paramagnetic metal (PM-M) to AFM insulator (AFM-I). For smaller *R* cations, *T*
_{N} is much lower than *T*
_{MI}; the two phase transitions are decoupled and the system goes through an intermediate paramagnetic insulating phase (PM-I).

The origin of the MIT has been widely debated in the literature^{14,15,16,17,18,19}. It was sometimes interpreted as a charge disproportionation at Ni sites^{20}: 2(*d*
^{7}) → *d*
^{8} + *d*
^{6}. However, the importance of Ni 3*d*-O 2*p* hybridization—i.e., transfer of electrons from O to Ni and formation of oxygen holes \(\left( {\underline L } \right)\) keeping Ni in a *d*
^{8} configuration (i.e., \(d^{8 - n} \approx d^8\underline L ^n\))—was evoked early on^{15}. Nowadays, the MIT is usually viewed as a charge ordering of the type \(2\left( {d^8\underline L ^1} \right) \to \left( {d^8} \right) + \left( {d^8\underline L ^2} \right)\)
^{17, 21, 22}. In this scenario, *B*
_{OC} appears important to stabilize the charge ordering and open the gap. As suggested in ref. ^{23} and confirmed from statistical correlation techniques^{24}, *R*
_{
xy
} and *M*
_{
z
} are also expected to play an active role. However, a complete picture linking electronic, structural and magnetic properties is yet to emerge.

Unlike recent theoretical studies, which were focusing specifically on the electronic properties^{17,18,19, 21}, we investigate here self-consistently the electronic and structural degrees of freedom of *R*NiO_{3} compounds from density functional theory calculations (DFT, see Methods). Specific care was given to the validation of our approach, which turns out to provide an unprecedented agreement with experimental data. Focusing on YNiO_{3}, we show (Supplementary Notes 1–4) that not only the atomic structure but also the AFM-E′ ground state, the estimated *T*
_{N} and the electronic bandgap of the insulating phase are very accurately reproduced, making therefore our approach a method of choice to shed light on the interlink between these different features.

## Results

### Energetics of individual lattice distortions

We start from the reference \(Pm\bar 3m\) cubic structure. Inspection of the phonon dispersion curves (Supplementary Note 5) reveals dominant structural instabilities at R (*ω*
_{R} = 310i cm^{−1}) and M (*ω*
_{M} = 278i cm^{−1}) points of the Brillouin zone (BZ), which are associated, respectively, to the *R*
_{
xy
} and *M*
_{
z
} distortions responsible for the *Pbnm* phase. These imaginary frequencies *ω*
_{i} are linked to a negative energy curvature *α*
_{i} at the origin \(\left( {\alpha _{\mathrm{i}} \propto \omega _{\mathrm{i}}^2 < 0} \right)\) and to a typical double-well (DW) shape of the energy when freezing *R*
_{
xy
} and *M*
_{
z
} distortions of increasing amplitude within the cubic structure, as illustrated in Fig. 1. These wells are nearly independent of the spin order but strongly evolve with the *R* cation: they become shallower when *R* size increases, consistently with a reduction of the related distortion amplitudes in the *Pbnm* phase.

In contrast, the *B*
_{OC} motion, corresponding to another phonon at R, is stable and extremely stiff (in fact the stiffest mode with *ω*
_{B} = 700 cm^{−1}), in line with the single-well (SW) shape illustrated in Fig. 1. Decreasing *R* cation size tends to stabilize slightly further *B*
_{OC}, in apparent contradiction with the observed increase of *T*
_{MI}. As illustrated for YNiO_{3}, switching from ferromagnetic (FM) to AFM-E′ spin order reduces slightly the curvature but does not reverse it; instead it shifts the SW to lower energy^{13}, yielding a finite *Q*
_{B} at the minimum. Although *B*
_{OC} tends to make the system insulating, this amplitude (25% of ground state’s value) is however not large enough to open a gap (more than 75% would be required). This shows that *B*
_{OC} and the magnetic order alone cannot explain the MIT by themselves.

### Mode coupling and triggered mechanism

Our central result is presented in Fig. 2 where we report in panel a the evolution of the *B*
_{OC} energy well of YNiO_{3} at various fixed amplitudes of oxygen rotation motions. It highlights that although initially stable (SW), *B*
_{OC} is progressively destabilized (DW) by the appearance of *R*
_{
xy
} and *M*
_{
z
}. As illustrated in panel b, *α*
_{B} is renormalized into \(\tilde \alpha _{\mathrm{B}}\), which evolves linearly with \(Q_{\mathrm{R}}^2\) and \(Q_{\mathrm{M}}^2\). The slope associated to *Q*
_{R} is twice as large as that related to *Q*
_{M}, attesting that each of the three individual rotations similarly affects *B*
_{OC}. This behavior arises from the presence in the energy expansion of cooperative (*λ* < 0) bi-quadratic coupling terms between *B*
_{OC} and oxygen rotations (\(E \approx \lambda _{{\mathrm{Bi}}}Q_{\mathrm{B}}^{\mathrm{2}}Q_{\mathrm{i}}^{\mathrm{2}}\), i = M,R), which, being the lowest-order couplings allowed by symmetry, should give rise to the appearance of *B*
_{OC} through a triggered phase transition according to Holakovsky^{25}. The same behavior is observed independently of the magnetic order (Supplementary Note 8). From now we focus on representative FM results while coming back to the role of magnetism later.

### Landau model and phase diagram

To further assess the relevance of such a triggered mechanism in nickelates, we built a Landau model, including *R*
_{
xy
}, *M*
_{
z
}, and *B*
_{OC} degrees of freedom^{24}, restricting ourselves to lowest coupling terms and assuming temperature dependence of the oxygen rotations only:

Within this model, *R*
_{
xy
} and *M*
_{
z
} appear at *T*
_{0R} and *T*
_{0M}, respectively. On cooling, they progressively develop within the *Pbnm* phase and renormalize the energy curvature *α*
_{B} of *B*
_{OC} as made clear when grouping the \(Q_{\mathrm{B}}^2\) terms in Eq. (1):

When reaching a critical amplitude at which \(\tilde \alpha _{\mathrm{B}} = 0\), they trigger the appearance of *B*
_{OC} and produce concurrent structural and metal-insulator phase transitions. The phase transition appears to be second order within this model, which is however too simple to be conclusive on this point (Supplementary Note 10).

All parameters and their evolution with *R* were directly fitted from first principles; only Curie temperatures were uniformly scaled to reproduce the experimental *T*
_{MI} of YNiO_{3} (Supplementary Note 8). The phase diagram of nickelates as predicted within this model is reported in Fig. 3. This figure demonstrates that the cooperative coupling of *B*
_{OC} with *R*
_{
xy
} and *M*
_{
z
} is a key mechanism that, by itself, accounts for the experimentally observed evolution of *T*
_{MI} with the tolerance factor.

Hence, the MIT in nickelates turns out to be a concrete example of triggered phase transition^{26, 27}, a kind of transition never identified before in simple perovskites. Indeed, although bi-quadratic interactions are generic in this class of compounds, different distortions usually compete and exclude each other^{9}. The cooperative coupling of *B*
_{OC} with oxygen rotations pointed out here is therefore an unusual and intriguing feature, whose origin is now traced back in the electronic band structure.

### Electronic origin of the triggered mechanism

In the cubic phase, as expected from the formal Ni 3*d*
^{7}
\(\left( {t_{{\mathrm{2g}}}^6e_{\mathrm{g}}^1} \right)\) occupancy, the Fermi energy *E*
_{f} crosses levels of dominant Ni 3*d*-*e*
_{g} character (i.e., anti-bonding Ni 3*d*-O 2*p* states); such levels form an isolated and rather dispersive set of two *e*
_{g} bands, shifted above the *t*
_{2g} levels by the crystal field (Fig. 4). Forcing into this cubic structure a *B*
_{OC} distortion, associated to a phonon at **q**
_{R} = (1/2, 1/2, 1/2), can open a gap at **q**
_{c} = (1/4, 1/4, 1/4) within the *e*
_{g} bands but well above *E*
_{f} and without any direct impact on the metallic character and the occupied states. Nevertheless, the oxygen rotation motions substantially affect the *e*
_{g} bands (Fig. 4), reducing their bandwidth and yielding a progressive down shift of the *e*
_{g} levels at **q**
_{c}. As the rotations gradually increase and the bandwidth decreases, *B*
_{OC} more substantially pushes down the electronic states around *E*
_{f}, providing a progressive gain of electronic energy, which, in turn, results in the softening of *ω*
_{B}. The critical rotation amplitude at which *B*
_{OC} becomes unstable \(\left( {\tilde \alpha_{\mathrm{B}} = 0} \right)\) is precisely that at which the *e*
_{g} levels at **q**
_{c} cross *E*
_{f}. At these amplitudes, the electronic system itself becomes unstable; the appearance of *B*
_{OC} is favored and opens a gap at *E*
_{f}, making the system insulating. As such, the MIT can therefore be interpreted as a Peierls instability but one that is not initially present and has been triggered by oxygen rotations.

## Discussion

For compounds with small *R* cations, *Q*
_{R} and *Q*
_{M} are large and able to produce the MIT at relatively high temperatures, well above *T*
_{N}. For large cations (*R* = Nd, Pr), oxygen rotations are reduced and, from our Landau model (built on FM results), no more sufficient to trigger the MIT (Fig. 3). However, as previously mentioned, the AFM-E′ spin order is compatible by symmetry with *B*
_{OC} and induces its appearance as an improper order (linear shift of *B*
_{OC} SW, Fig. 1d). Hence, although not opening a gap in the cubic phase, the onset of the AFM-E′ order in the *Pbnm* phase of NdNiO_{3} and PrNiO_{3} promotes the occurrence of the MIT almost triggered by the rotations. In these compounds, we have therefore *T*
_{MI} = *T*
_{N}; the transition is more abrupt and first-order^{8}. Such active role of magnetism for large cations is supported by the experimental results and discussion in ref. ^{28}. It is also confirmed by our first-principles calculations on NdNiO_{3}, showing that the predicted *T*
_{MI} is rescaled when including the change of magnetic order: while the system prefers to stay in the *Pbnm* metallic phase when imposing a FM order, it switches to the *P*2_{1}/*n* phase when adopting a AFM-E′ order. The cooperative effect of the magnetic order remains true for small cations but without any impact on *T*
_{MI} (>*T*
_{N}).

In conclusion, the concurrent electronic and structural transitions at *T*
_{MI} in *R*NiO_{3} compounds take the form of a Peierls instability, which, primarily, is structurally triggered by the oxygen rotation motions *R*
_{
xy
} and *M*
_{
z
} and, eventually, is further assisted by the appearance of the E′-type AFM magnetic ordering. Our Landau model, and its possible extension to incorporate explicitly strain degrees of freedom neglected here for simplicity, provides a simple and useful quantitative tool to estimate and interpret how *T*
_{MI} can be tuned toward the monitoring of oxygen rotation motions *R*
_{
xy
} and *M*
_{
z
} when making solid solutions^{10}, applying pressure^{8}, or playing with the epitaxial strain and the orientation of the substrate in thin films^{29}. Our findings are relevant to other families of perovskites like A^{2+} Fe^{4+} O_{3} compounds^{30}. For instance, they can explain why CaFeO_{3}, which exhibits oxygen rotations, undergoes a MIT while SrFeO_{3} and BaFeO_{3}, which stay cubic, remain metallic. In addition, the same physics is also inherent to manganites like LaMnO_{3}, suggesting a close competition between charge and orbital orderings in this family compounds. However, the situation is slightly different in a bismutate like BaBiO_{3}, in which *B*
_{OC} is intrinsically unstable in the cubic phase^{31}.

## Methods

### First-principles calculations

First-principles calculations were performed in the framework of DFT^{32, 33} using a Projected Augmented Wave (PAW) approach^{34} as implemented within the ABINIT package^{35,36,37,38}. The calculations relied on the Generalized Gradient Approximation using the PBEsol^{39} exchange-correlation functional. We worked within a collinear spin approximation. We included a Hubbard correction *U* = 1.5 eV^{40} on the 3*d* orbitals of Ni atoms. A special care was devoted to the determination of the appropriate *U* parameter (Supplementary Notes 1–
4).

We made use of JTH atomic potentials^{41}. For the wavefunctions, we used an energy cutoff of 24 Ha (38 Ha for the second grid in the PAW spheres), which guarantees a convergence better than 1 meV on the total energy. The BZ was sampled with *k*-point meshes equivalent to a 12 × 12 × 12 grid in the five-atom unit cell. During structural relaxations, thresholds of 10^{−5} Ha/bohr on the maximum force and of 10^{−7} Ha/bohr^{3} on the maximum stress have been considered.

### Structural analysis

The Goldschmidt tolerance factor^{42}, *t* = *d*
_{R−O}/\(\sqrt 2\)
*d*
_{Ni−O}, of *R*NiO_{3} compounds were determined using Nicole Benedek’s tool^{43} relying on a bond valence model^{44} to calculate *d*
_{R−O} and *d*
_{Ni−O}, respectively, the ideal *R*–O and Ni–O bond lengths in the cubic perovskite structure.

Symmetry-adapted mode analysis have been performed with AMPLIMODE^{45, 46}. The modes are normalized to their amplitude in the *P*2_{1}/*n* AFM-E′ ground state. This normalization is such that in cubic phase (volume of the ground state) *Q*
_{R} = 100% corresponds to rotation angles *ϕ*
_{
x
} = *ϕ*
_{
y
} = 11.33° (Ni–O–Ni angle of 157.33°), *Q*
_{R} = 100% corresponds to a rotation angle *ϕ*
_{
z
} = 12.12° (Ni–O–Ni angle of 155.75°) and *Q*
_{B} = 100% corresponds to oxygen displacements *d*
_{O} = 0.0358.

### Landau model

The Landau model parameters have been fitted for YNiO_{3}, GdNiO_{3}, and SmNiO_{3} on first-principles data using in each case a FM cubic phase (volume of the ground state) and interpolated for the other compounds. *T*
_{MI} was determined analytically (Supplementary Notes 6–9).

### Data availability

All relevant data are available from the authors.

## Additional information

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## Acknowledgements

We thank Y. Zhang, E. Bousquet, F. Ricci, M. Verstraete, J.-Y. Raty, S. Catalano, J.-M. Triscone, A. Georges and J. Varignon for fruitful discussions. This work was supported by FRS-FNRS project HiT4FiT and ARC project AIMED. J.I. acknowledges support form FNR Luxembourg Grant P12/4853155 “COFERMAT”. Calculations have been performed on the Céci facilities funded by F.R.S-FNRS (Grant No 2.5020.1) and Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the Walloon Region (Grant No 1117545). They also took advantage of HPC resources from the French Research and Technology Computing Center (CCRT) and from the PRACE project Megapasta.

## Author information

### Affiliations

#### Theoretical Materials Physics, Q-MAT, CESAM, University of Liège, B-4000 Liège, Belgium

- Alain Mercy
- , Jordan Bieder
- & Philippe Ghosez

#### CEA DAM-DIF, F-91297, Arpajon, France

- Jordan Bieder

#### Materials Research and Technology Department, Luxembourg Institute of Science and Technology (LIST), 5 avenue des Hauts-Fourneaux, L-4362, Esch/Alzette, Luxembourg

- Jorge Íñiguez

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### Contributions

P.G. conceived the study with A.M. and J.I. and directed the work. A.M. and J.B. did the first-principles simulations. A.M. and P.G. did the Landau modeling. P.G. wrote the manuscript with A.M. and J.B. All authors discussed the results at different stages of the study and commented on the manuscript.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to Philippe Ghosez.

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