Introduction

Transition metal oxides with an AMO3 perovskite structure have attracted widespread interest over the last decades, both from academic and application points of view. This can be ascribed to their wide range of functionalities that originates from the interplay between lattice, electronic, and magnetic degrees of freedom.1 Among all perovskites, rare-earth nickelates R 3+ Ni3+ O3 (R = Lu-La, Y) might be considered as a prototypical case because they posses almost all possible degrees of freedom present in these materials. Nickelates were intensively studied during the nineties 2, 3 and have regained interest in the last few years due to their great potential for engineering novel electronic and magnetic states.4,5,6,7,8,9,10,11

Except for R = La, all rare-earth nickelates undergo a metal–insulator phase transition (MIT) at a temperature T MI, accompanied by a symmetry lowering from Pbnm to P21/n yielding two different Ni sites.2, 3 In the insulating phase, a subsequent breathing distortion of the NiO6 cage octahedra is observed, producing a rock-salt-like pattern of large and small NiO6 groups. In the following, we will use the notation NiS and NiL to refer to the Ni cations belonging to the small and large NiO6 groups, respectively. At T N ≤ T MI, they undergo an antiferromagnetic (AFM) phase transition leading to a quadrupling of the magnetic unit cell (\(\overrightarrow{k}=(\frac{1}{2}\mathrm{,0,}\frac{1}{2})\) with respect to the Pbnm primitive 20-atom unit cell) and possible collinear or non collinear spin orderings.12,13,14,15,16 The electronic structure is also characterized by strong overlaps between O-2p and Ni-3d states inducing large covalent effects.2 As a consequence, external stimuli, such as temperature, or chemical or hydrostatic pressure, can modify the electronic bandwidth and influence the MIT.17,18,19,20 Nickelates have thus been used as a platform to search for novel electronic phases through strain engineering or confinement.10, 11, 21,22,23

Nevertheless, the description of the insulating phase of nickelates—and its origin—is still under debate. Originally, the MIT has been ascribed to ionic charge disproportionation effects of Ni cations from 2Ni3+ to \({{\rm Ni}_{\rm S}^{(3+\delta )+}}\) + \({{\rm Ni}_{\rm L}^{(3-\delta )+}}\),24,25,26 that result in the rock-salt-like pattern of large and small NiO6 groups in the low symmetry phase.25 This presence of the breathing distortion has been argued to be an alternative to Jahn–Teller distortions—Ni3+ cations are expected to be nominally Jahn–Teller active—, by removing electronic degeneracies between Ni sites and rendering a localized charge-ordered state opening a band gap.27 In other words, through such a distortion, the e g levels of large NiO6 groups are stabilized with respect to those of small NiO6 groups. This scenario was first suggested by several neutron diffraction studies,16, 24, 25, 28, 29 for which a charge ordering of Ni sites was derived from Ni–O bond lengths analysis based on the empirical valence bond model of Brown.30 X-ray absorption spectroscopy (XAS) measurements26 on several elements present also characteristics of a Ni valence splitting at the MIT, with a magnitude modulated by the rare-earth. Finally, the strongest evidence is provided by Mössbauer spectroscopy showing a clear-cut split of the Ni charges when entering the insulating phase.29

Most recently, an alternative covalent vision has emerged and appears to be somehow contradictory with this original scenario.31, 32 The starting point of this theory is the ability to transfer charges from the ligands to the Ni cations due to the strong covalent character of nickelates. It follows that Ni cations adopt a Ni2+ \(3{d}^{8}\underline{L}\) electronic configuration in the metallic phase—the notation \(\underline{L}\) defines a ligand hole that is created on the surrounding oxygens—rather than the expected Ni3+ 3d 7 configuration. In the insulating phase, this configuration transforms into a NiL 3d 8 and a NiS \(3{d}^{8}{\underline{L}}^{2}\) electronic configurations through a transfer of charge from the oxygens from the small NiO6 groups—the notation \({\underline{L}}^{2}\) labels two holes created on the oxygens—, which is compatible with the strong breathing mode observed in this phase.31 Consequently, this mechanism is rather labeled as a bond disproportionation effect and implies similar valence states for both Ni sites and de facto an absence of charge ordering in the insulating phase. However, these theoretical observations are at odds with the clear experimental evidences of charge disproportionation extracted through several techniques.26, 29

Here we performed density functional theory (DFT) calculations and a Wannier analysis on various representative nickelates in order to reconcile these two apparently conflicting visions of the insulating phase. Our simulations reveal the co-existence of strong ionic effects, with a clear-cut split of the electronic structure of the inequivalent Ni sites resembling a high-spin \({{\rm{Ni}}}_{{\rm{L}}}^{2+}\) and a low-spin \({{\rm{Ni}}}_{{\rm{S}}}^{4+}\), and of covalent features, with O-2p electrons shared with the charge-depleted NiS cations leaving virtually no charge ordering. Our results therefore propose a unified picture for the description of the insulating phase. By performing calculations on several species, we build a theoretical phase diagram in which all the key features of nickelates can be recognized, and most notably the correct AFM states and the crossover from a second to a first-order metal–insulator phase transition for R = Nd and Pr. Finally, we unveil a new pathway to control electronic and magnetic phases in perovskites by tuning the level of covalence.

Results

Structural properties

First, we performed full geometry relaxations considering 80-atom supercells of both possible Pbnm and P21/n structures with different magnetic orderings: ferromagnetic (FM) as well as complex E-type, S-type, and T-type AFM orderings 5 based on ↑↑↓↓ spin chains in the (ab)-plane with different stackings along the \(\overrightarrow{c}\) axis (see Fig. 1a–c). The results are presented for spins treated at the collinear level, although we have carefully checked that allowing the spin–orbit interaction and non collinear spin arrangements only yields metastable magnetic phases and does not change the main conclusions of the study (see Supplementary Material). We employed the PBEsol functional33 in combination with a U correction of 2 eV on Ni-3d states as implemented by Dudarev et al. 34 in order to account for electronic correlations. Several nickelates (R = Y, Dy, Tb, Gd, Eu, Sm, Nd, and Pr) were considered, covering the phase diagram as a function of rare-earth radius. All nickelates relax to a P21/n insulating ground state with complex AFM structures (S-type or T-type depending on the rare-earth) and band gaps compatible with experiments 35, 36 (see Table 1). All our Pbnm phases favor a metallic FM solution, while classical AFM orders—A, C and G type—yield solutions that are higher in energy by at least 180 meV per 80 atoms unit cell (see supplementary Material). We checked the reliability of our DFT + U calculations by changing the U correction to either 0 eV or 5 eV in SmNiO3. While the ground state is unchanged when no U-correction is applied—the gap is eventually decreased to 91 meV—, imposing U = 5 eV yields a P21/n FM and insulating solution that is much more stable than the considered complex AFM orderings (ΔE≃160 meV per 80-atom unit cell). This further supports our choice of a relatively small Hubbard correction for the Ni-3d electrons.

Fig. 1
figure 1

Sketch of the three complex colinear AFM orderings used in the calculations. a E-type AFM ordering. b S-type AFM ordering. c T-type AFM ordering. The A cations are not displayed for clarity. The dashed lines represent the size of the crystallographic unit cell

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Table 1 Key quantities of the different relaxed ground states
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Our optimized ground state structures are characterized by three main lattice distortions (the optimized lattice parameters are provided in the Supplementary Material). First, they feature two antiferrodistortive (AFD) modes that can be described, respectively, as a − a − c 0 and a 0 a 0 c + patterns using Glazer’s notation.37 These AFD modes are the main features of the phase with Pbnm symmetry. Second, we have a breathing of the O6 octahedra, B oc (see Fig. 2a–c). The breathing mode only appears in the P21/n symmetry and produces the rock-salt-like pattern of small and large NiO6 groups, automatically resulting in two different Ni sites (see Fig. 2c).

Fig. 2
figure 2

Rare-earth nickelates ground state structural properties. a, b and c Schematic pictures of the three main lattice distortions appearing in the ground state of nickelates: a − a − c 0 AFD motion, a 0 a 0 c + AFD motion and breathing of the oxygen cage octahedra B oc. d, e Ni–O–Ni angle (in degrees) produced by the a − a − c 0 and a 0 a 0 c + AFD motions. f Oxygen cage octahedra volume expansion/contraction (in %) produced by the breathing mode

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As usual in perovskites, the magnitude of the metal–oxygen–metal bond angles associated with the O6 rotations is governed by steric effects (see Fig. 2d, e), and nickelates with low tolerance factors (i.e., smaller R cations)38 are more distorted. The alternating expansion/contraction pattern of the oxygen cage associated with the B oc breathing mode also appears to be modulated by the rare earth (see Fig. 2f), as smaller R cations yield larger distortions. Finally, we observe a Jahn–Teller distortion in the ground state that is one to two orders of magnitude smaller than the breathing mode or the two AFD motions, and it is also much smaller than the Jahn–Teller motion appearing in prototype compounds like LaMnO3 39 (amplitudes of distortions of our nickelates ground state structures are reported in the Supplementary Material). Hence, the relaxed structures indicate that there is no significant orbital order in these systems, although the 3d 7- \({{\rm{t}}}_{{\rm{2g}}}^{6}{{\rm{e}}}_{{\rm{g}}}^{1}\) electronic configuration of Ni3+ in the high temperature Pbnm phase is nominally Jahn–Teller active.27

Disproportionation signatures

The electronic structure of the optimized ground states is characterized by strong hybridizations between O-2p and Ni-3d levels, as inferred from the projected density of states (pDOS, see Fig. 3a for the representative case of SmNiO3). Comparing the pDOS corresponding to the 3d levels of the two different Ni sub-lattices reveals some small differences, likely reflecting weak disproportionation effects and a small charge ordering (see Fig. 3b). Although atomic charges are not uniquely defined in DFT calculations and there are no unambiguous ways to extract them,40, 41 sphere integrations around the Ni cations can provide some insight into the possible charge ordering, keeping in mind that they might not reflect the real ionization state40 (We emphasize that the number of electrons extracted from the sphere integrations is not reminiscent of the real site occupancy. We have performed calculations on BaNiO3—a non magnetic insulator adopting a P6 3 cm hexagonal structure and in which the Ni cations are formally in a 4+ oxidation state—and La2TiNiO6—an insulator adopting a double perovskite P2 1/n phase and in which Ni cations are formally in a 2+ oxidation state. Those calculations reveal that the total Ni site occupancy increases with increasing the oxidation state of Ni cations—QNi = 9.631 e in BaNiO3, QNi = 9.376 e in SmNiO3 (Pbnm phase) and QNi = 9.176 e in La2TiNiO6. At the same time, the extracted d occupancy on Ni sites remains contants for the three valence states—QNi,d = 8.257 e for BaNiO3, QNi,d = 8.276 e for SmNiO3 (Pbnm phase) and QNi,d = 8.272 e for La2TiNiO6). Figure 3c reports the occupancy of both Ni sites as a function of the rare earth. A weak and rather constant level of charge ordering is observed between the Ni sites, going from δ = 0.13 (YNiO3) to δ = 0.11 (PrNiO3), but the sign of δ is opposite to what is expected: the NiS cations, sitting at the center of the smallest O6 octahedra, appear to hold more electrons than the NiL cations, located in the largest oxygen cages. Since the breathing mode B oc enhances the crystal field splitting at the small NiO6 groups, the e g levels of NiS lie higher in energy than those of NiL 27, 42 and therefore NiS should have fewer electrons than NiL associated to it.

Fig. 3
figure 3

Key electronic properties of rare-earth nickelates ground state. a, b Total and projected spin-polarized density of states (arbitrary units). The Fermi level is located at 0 eV. The upper (lower) panels correspond to spin-up (down). c Total number of electrons ascribed to both Ni sites during the sphere integrations. d Average BECs of both Ni sites. e Computed magnetic moment (in μ B) of the NiL site. The magnetic moment of NiS is exactly zero in the calculation. f Charge disproportionation δ extracted from the empirical Brown valence bond model

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Let us now consider better defined—and experimentally measurable—quantities, such as Born effective charges (BECs) that measure the amount of charge displaced upon the movement of individual atoms. Figure 3d reports the average of the diagonal components of the tensor for the different nickelates (see the Supplementary Material for the full tensors). In the representative case of SmNiO3, we obtain \({Z}_{{\rm{N}}{i}_{{\rm{L}}}}\approx +2.5\), which is not far from the nominal oxidation state of 2+ that this Ni site is associated with in the complete-charge-disproportionation picture. However, we find a similar \({Z}_{{\rm{N}}{i}_{{\rm{S}}}}\approx +2.1\), which sharply deviates from the expectation value (4+) in the charge-disproportionation picture. As shown in Fig. 3d, we observe the same behavior across the various studied compounds; an approximately constant difference of Born charges, of about 0.4 electrons, is found across the whole series. Hence, from the point of view of the effective charges, the two Ni sites behave in a rather similar fashion, the disproportionation effects being weak and not complying with the usual picture.

However, our computed magnetic moments on both Ni sites appear to be in contradiction with the conclusion of the charge analysis. Indeed, as shown in Fig. 3e, we observe a large difference between NiL—with a moment estimated between 1.25 μ B and 1.15 μ B when going from R = Y to R = Pr—and NiS—for which the magnetic moment is null –, which suggests two very different electronic states.

Finally, Fig. 3f reports the charge disproportionation extracted from the empirical Brown bond valence model,30 which is just an alternative way to monitor the breathing mode magnitude. This model suggests a differentiation of the Ni valences in our optimized insulating phases, with a disproportionation δ modulated by the rare-earth, but the result of this phenomenological model is at odds with the charges—statics or dynamics—extracted directly from our simulations. However, this simple quantification yields values that are very similar to experiments.16, 24, 25, 28, 29 It is worth emphasizing that this latter model predicts the expected sign for the charge ordering.

Wannier analysis

The conclusion of the previous discussion is that, in the RNiO3 compounds, all Ni atoms seem to display a similar oxidation state. Yet, the presence of a significant breathing distortion and of a strong difference in the local magnetic moments clearly suggest two markedly different electronic states. Note that similar results have been reported in previous theoretical works using a variety of methods.23, 31, 43, 44 Here, to gain insights into the real site occupancies, we ran a maximally-localized Wannier function (MLWF) analysis of our first-principles results, which allowed us to resolve this pending issue.

We used the Wannier90 package 45,46,47 to determine the MLWFs that reproduce the occupied electronic manifold. More precisely, our purpose was to count how many occupied MLWFs are centered at the different Ni cations, and how many at the surrounding oxygen anions, and to characterize them. Further, we wanted to run our analysis without having to make any assumption on the precise character of the occupied Ni and O orbitals, which complicated the choice of the initial atomic orbitals onto which the Kohn–Sham states are projected in order to extract the initial gauge matrix for the localization procedure. Nevertheless, we found the following robust strategy to proceed. We considered the whole occupied manifold and sought to extract from it (i.e., to disentangle) a set of 2 × (144 + 10) MLWFs, where 2 × 144 = 2 × 3 × 48 is the total number of O-2p orbitals available in our 80-atoms supercell and 2 × 10 = 2 × 5 × 2 is the number of Ni-3d orbitals corresponding to two specific Ni atoms. (Note that we ran separate MLWFs optimizations for the spin-up and spin-down channels.) Hence, we used the following initial atomic orbitals: three generic p orbitals centered at each O anion and 5 generic d orbitals centered at two neighboring Ni cations; this couple of Ni cations were chosen to be first-nearest neighbors, so that we considered one NiL and one NiS. The basic qualitative results of this optimization were the same for all the nickelates considered, and thus the following discussion is not compound specific.

Our optimization renders 2 × 3 MLWFs centered at each oxygen anion (i.e., three spin-up MLWFs and three—very similar—spin-down MLWFs), suggesting that all oxygens in our nickelates are in a 2- oxidation state. The oxygen-centered MLWFs have a clear p character, as can be appreciated in Fig. 4c, d. We also obtained 2 × 3 t 2g -like MLWFs centered at each of the two considered Ni atoms, indicating that the t 2g states are fully occupied and there is no magnetic moment associated to them. Further, we obtained 2 e g-like spin-up MLWFs centered at the NiL site (see Fig. 4a, b), indicating that this cation is in a 2+ oxidation state and has a significant magnetic moment associated to it. Finally, as regards the other initial atomic orbitals centered at the chosen NiL (two spin-down d orbitals) and NiS (2 × 2 d orbitals) atoms, they did not lead to any MLWF centered at those sites. Instead, the maximal-localization procedure resulted in MLWFs centered at Ni and R cations in the vicinity of the considered NiL–NiS pair. Thus, in particular, it was impossible to localize any e g-like MLWFs at a NiS site, which strongly indicates that these Ni cations are close to a 4+ oxidation state. We note that a Ni close to a 4+ oxidation state was experimentally observed for some Ni compounds through 61Ni Mössbauer spectroscopy.48 These conclusions were ratified by considering larger clusters of Ni sites for the MLWFs optimization, as well as individual Ni’s and/or optimizations in which the oxygen bands were not included.

Fig. 4
figure 4

Orbital occupancies in rare-earth nickelates. a \({d}_{{x}^{2}-{y}^{2}}^{\uparrow }\) MLWF on two consecutive Ni sites. The NiL site bears a spin up. b \({d}_{{z}^{2}}^{\uparrow }\) MLWF on two consecutive Ni sites. The NiL site bears a spin up. c O-2p MLWFs oriented along the Ni–O bonds on the two consecutive Ni sites in the spin-up channel. d O-2p MLWFs oriented along the Ni–O bonds on the two consecutive Ni sites in the spin down channel. e Schematic picture of the electronic structure of rare-earth nickelates ground state

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Hence, the Wannier analysis yields a picture of strong charge disproportionation between the Ni sites, which is clearly at odds with the quantitatively similar behavior discussed in the section above. To resolve this apparent contradiction, we now inspect in more detail the oxygen-centered MLWFs.

Figure 4c, d report representative results of the O-2p-like MLWFs oriented along the Ni–O–Ni bonds for SmNiO3. As it is clearly visible in the figures for both spin channels, these MLWFs have their centers significantly shifted towards the NiS cations and away from the NiL sites. The shift—as quantified by the distance between the oxygen position and MLWF center—is ∼0.231 Å in average. Thus, while the NiS cations appear to be in a 4+ state when we count how many MLWFs are centered at them, they also receive a significant fraction of electrons coming from the surrounding oxygens, with which they are strongly hybridized. Therefore, this is the explanation why quantitative measures of the charge around the NiS ions render results that are similar to those of the NiL and suggest valence state much more reduced than the expected 4+. Across the series, we observe that O-2p centers get closer to the NiS cations for smaller rare earths, increasing the localization of the charge on NiS. Additionally, the O-2p MLWFs behave similarly irrespective of their spin polarization, so that their shifting does not result in any magnetic moment at the NiS sites.

We have run additional MLWF optimizations by initially projecting the Kohn-Sham states on all the “missing” e g states of the small NiS sites and by restricting the localization procedure to the 16 first unoccupied states (2 e g-like MLWFs × 8 NiS = 16 MLWFs per spin channel). We are able to obtain the unnocupied e g-like MLWFs on NiS (see Supplementary Material); these MLWFs exhibit a certain O-2p character, which gives further evidence for the hybridization between oxygen and NiS states.

The Wannier analysis therefore leads to the conclusion that a strong disproportionation occurs in the system, going toward clearly distinct \({{\rm{Ni}}}_{{\rm{S}}}^{4+}\) (low-spin, non-magnetic) and \({{\rm{Ni}}}_{{\rm{L}}}^{2+}\) (high-spin, magnetic) sites. Simultaneously, the O-2p MLWFs approach the \({{\rm{Ni}}}_{{\rm{S}}}^{4+}\) sites, ultimately yielding NiS and NiL that are nearly equivalent from the point of view of the charge (static and dynamic) in their vicinity.

Discussion

While the charge—static or dynamic—analysis yields very similar Ni sites occupancies, the Wannier analysis leads to a picture with a strong disproportionation of Ni oxidation states going toward \({{\rm{Ni}}}_{{\rm{S}}}^{4+}\) and \({{\rm{Ni}}}_{{\rm{L}}}^{2+}\) sites in the insulating phase. Our results thus appear to be compatible with the ionic disproportionation effects originally proposed to occur at the MIT. They are indeed reminiscent of this picture, with the observation of a subsequent breathing distortion modulated by the rare-earth (see Fig. 2f)—which results in the varying ionization state according to the phenomenological Brown valence bond model (see Fig. 3f)—and a charge disproportionation between Ni sites complying with numerous experimental works, and most notably with XAS and Mössbauer measurements.

They are also compatible with the model of Mizokawa et al.49 as well as with recent dynamical mean field theory studies and exact Hartree–Fock calculations,31, 32, 50,51,52,53 proposing a ligand-hole structure in rare-earth nickelates. Indeed, our Wannier analysis indicates that we have a 3d 8 electronic configuration for the NiL cations. More importantly, it is also compatible with the \(3{d}^{8}{\underline{L}}^{2}\) configuration proposed for the NiS sites.31 According to our MLWF analysis, such a situation would correspond to having all six oxygens around NiS sharing the 2p electrons that occupy orbitals along the Ni–O–Ni bonds. Further, given that our integrated and dynamical Born charges suggest that the NiS and NiL sites host a similar number of electrons, our results do in fact point to a situation in which each O6 cage shares approximately two electrons with the NiS at its center, exactly as in the ligand-hole picture. More generally, our results are consistent with the self-regulation response upon a change of the oxidation state of transition metals 40, 54—we note here that this mechanism explains that similar sphere integrated charges around transition metals might be evaluated irrespective of a change of the oxidation state of the elements.

As regards the magnetic moments at the different Ni sites, our results are also clear and compatible with both proposed pictures: NiL bears a magnetic moment greater than 1 μ B, as it corresponds to having Ni2+ in a high-spin configuration—we note here that while the magnetic moment associated with NiL should be around 2 μ B in a purely ionic vision, screening effects from oxygens reduce the magnetic moment beared by NiL. Then, NiS has no magnetic moment associated with it, as it would correspond to a nominal Ni4+ low-spin configuration. The latter result is partly a consequence of the fact that the electrons shared by NiS and its surrounding oxygens are spin paired, which can be interpreted as a ligand-hole screening.

Our first-principles calculations, based on the DFT framework, shed light on the nature of the ground state of rare-earth nickelates and agree with the different visions proposed to occur at the MIT. The electronic structure is summarized in Fig. 4e and is based on having (i) a strong charge disproportionation between NiL and NiS sites accompanied by a breathing mode and (ii) nearly two electrons from surrounding oxygens “shared” with the depleted NiS site, leaving the impression of two similar Ni2+ sites in the system. This electronic structure therefore yields a NiL site with a dominant ionic character and a NiS site with a dominant covalent character, as it was already suggested by a structural analysis.55

Although the electronic structure is similar between all nickelates, the rare-earth cations do have an impact on the level of covalence of the system through their induced lattice distortions. An example is their influence on the magnetic moment of the NiL cations. Note that our computed NiL magnetic moments are always far from the nominal value of 2 μ B, as a consequence of the hybridization between Ni-3d and O-2p states. This can be seen in Fig. 4c, d where slightly larger overlaps between O-2p orbitals over the NiL site arise in the spin-up channel (this particular NiL site has a net spin up). From Fig. 3e, the level of covalence clearly increases with the tolerance factor (a calculation assuming FM order yields similar conclusions, see Supplementary Material). This is further confirmed by the increase of the average BECs on both NiS and NiL, following the interpretation of such effects in other perovskite oxides.56 We also observe increasing O-2p overlaps with NiL sites going from R = Y to Pr. Therefore, the level of covalence increases with the tolerance factor of the nickelate, i.e. increasing with the rare-earth ionic radius.

This level of covalence seems to correlate with the stability of the magnetic ordering and the insulating phase (see Table 1 and Fig. 5c right scale). On one hand, with increasing covalence, the energy difference between the AFM (S or T) and FM solutions increases strongly. On the other hand, the insulating gap decreases. In order to further corroborate these trends, we performed additional calculations on SmNiO3 by applying a hydrostatic pressure of ±8% on the ground state volume and relaxed the atomic positions for each magnetic ordering (see methods). Under compression, we find a sizable enhancement of the covalent character as \({\mu }_{{\rm{N}}{i}_{{\rm{L}}}}\) decreases to 1.103 μ B—this is accompanied by a strong reduction of the breathing mode with a variation of the NiO6 octahedra volume of roughly 3.25% and by a reduction of the two a − a − c 0 and a 0 a 0 c + oxygen cage rotations. The stability of the S-type antiferromagnetic ordering with respect to the FM solution is therefore doubled (ΔE = −293 meV per 80-atom cell). Eventually, the band gap is decreased by 0.11 eV with respect to the ground state value. Under expansion, we observe a weakening of the covalent character of the system with \({\mu }_{{\rm{N}}{i}_{{\rm{L}}}}\) increasing to 1.308 μ B—this is accompanied by a strengthening of the breathing mode reaching 17% of NiO6 volume variation and by an increase of the a − a − c 0 and a 0 a 0 c + oxygen cage rotations; simultaneously, the stability of the complex AFM ordering is roughly reduced by a factor of 2 (ΔE = −67 meV per 80-atom cell).

Fig. 5
figure 5

Metal–insulator phase transition a Energy difference in meV per formula unit between the relaxed monoclinic and orthorhombic phases for a FM solution. b Band gap (in meV) of the FM monoclinic phase. c Left scale: phase diagram of rare-earth nickelates established on the basis of our first-principles calculations. Blue-filled circles and red-filled squares correspond to second order (T MI ≠ T N) and first order (T MI = T N) metal–insulator phase transitions, respectively. Right scale: energy difference in meV per formula unit between the monoclinic FM and AFM solutions for R = Y to Sm

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Finally, the rare-earth atom is known to play a key role in the nature of the MIT. Experimentally it is observed that T MI is different from the magnetic-ordering transition temperature T N for all nickelates except for those in which the rare earth is Pr or Nd. Interestingly, our calculations reflect this differentiated behavior. As already mentioned, we obtain an insulating solution for the AFM monoclinic ground state of all considered nickelates. Then, when we consider the P21/n structure with a FM spin arrangement, we also obtain an insulating phase for all R cations ranging from Y to Sm; the corresponding band gaps range from 77 to 41 meV and we observe a relatively large energy gain with respect to the orthorhombic phase (see Fig. 5a, b). Thus, the breathing distortion and disproportionation effects seem sufficient to open the band gap in these compounds, irrespective of the spin arrangement. As a consequence, our results indicate that these nickelates can potentially present an insulating, spin-disordered phase, as they indeed do experimentally for R = Lu to Sm. In contrast, for the monoclinic phase of NdNiO3 and PrNiO3, the FM spin configuration is found to be metallic; further, the stability of the low-symmetry structure with respect to the orthorhombic one drastically decreases. The complex AFM ordering therefore appears to be a necessary condition for the MIT to occur in these two compounds.

Considering the case of SmNiO3 under 8% of compression, we unveil a similar behavior to bulk PrNiO3 and NdNiO3, since the level of covalence increases. These results suggest that it is possible to control the electronic and magnetic structures of these compounds by tuning the level of covalence in the system.

Our first-principles results can be summarized by the computed phase diagram pictured in Fig. 5c. We have extrapolated the metal–insulator transition temperature (left scale of Fig. 5c) from our simulations by following the formula established in reference:57

$${k}_{B}{T}_{{\rm{MI}}}=\Delta -A\frac{\cos (\phi )}{{d}_{{\rm{Ni}}-{\rm{O}}}^{\mathrm{7/2}}}$$
(1)

where Δ is the charge-transfer energy (assumed to be constant along the series), ϕ is the average tilting angle of all NiO6 octahedra, d Ni−O is the average Ni–O bond lengths on the two Ni sites and A is a proportionality constant related to the bare bandwidth. By imposing our computed T MI to match those of the end members YNiO3 and PrNiO3, we fit a charge-transfer energy Δ of 0.893 eV, in very good agreement with experiments and Hartree–Fock methods,23, 58 and A = 8.741 eV Å7/2, again very close to recent reports.23 Remarkably, as is clear from Fig. 5c, when we evaluate Eq. 1 for the other nickelates considered here, we obtain a behavior for T MI that resembles very closely the experimental one.2, 3 On the right scale of Fig. 5c, we report the energy difference between the FM and the complex AFM orderings for R = Y to Sm. This quantity can be expected to correlate with the Néel temperature for these compound and, indeed we do observe that it exhibits a trend very close to experiments.2, 3 Our computed phase diagram further supports our choice of a small Hubbard U correction on Ni-3d sites and demonstrate that our DFT + U calculations capture all the key physical properties of rare-earth nickelates. More generally, this constitutes a good example of the efficiency and reliability of DFT to study strongly correlated systems.

In summary, we have used first-principles methods to investigate the ground state electronic structure and phase diagram of rare-earth nickelates. Our DFT simulations using a moderate Hubbard correction on Ni-3d states reproduce all features reported from experiments (insulating character, structural trends, complex AFM structures, disproportionation effects, covalence, apparent absence of charge ordering). In particular, we show that the insulating phase is characterized by a clear-cut split of the electronic states of the two Ni sites, which can be strictly described as resembling low-spin 4+ and high-spin 2+. At the same time, our simulations reveal a shift of the oxygen-2p orbitals toward the depleted NiS cations, so that, ultimately, from the point of view of the static and dynamical charge, the two Ni sites appear to be nearly identical. These findings are clearly related to the various pictures proposed in the literature to explain the ground state of these compounds, and they further highlight the efficiency of DFT to study the electronic properties of complex oxides. Performing a systematic study of various materials, we have been able to build a theoretical phase diagram of the rare-earth nickelates, underlining the subtle interplay between the level of covalence and the nature of the metal–insulator phase transition as well as the stability of the complex AFM ordering. These latter results hint at strategies to control electronic and magnetic phases of late transition-metal oxide perovskites.

Methods

First-principles calculations were performed with the Vienna ab-initio simulation package package.59, 60 The results are presented for spin arrangements treated only at the collinear level, although the role of spin–orbit interaction and non collinear spin arrangements have been carefully checked (see Supplementary Material). We used a 3 × 6 × 2 Γ-centered K-point mesh. The cut-off was set to 500 eV. We used projector augmented wave pseudopotentials 61 with the following electron configurations: 4 s23d8 (Ni), 2 s22p4 (O), 4 s24p65 s24d1 (Y), 4 s24p65 s2 (Pr, Nd, Sm, the extra 4f electrons are frozen in the pseudopotentials), 4p65 s2 (Gd, Eu, Tb, Dy, the extra 4f electrons are frozen in the pseudopotentials). We did not treat explicitly the f electrons as they order at very low temperature62 and they were included in the pseudopotential. Full geometry relaxations were treated until forces were lower than 0.001 eV/Å and energy was converged to 1 × 10−7 eV. BECs were computed using density functional perturbation theory.63 Symmetry adapted modes allowing the extraction of lattice distortion amplitudes were performed using the Bilbao crystallographic server.64, 65 The results presented in Fig. 2d–f have been obtained by freezing single lattice distortion in a cubic reference that has the same lattice parameters for all reported compounds. This allows to extract quantities that are strain independent for all nickelates. Wannier functions reported in Fig. 4a, b are plotted for isosurfaces equal to 2. Wannier functions reported in Fig. 4c, d are plotted for isosurfaces equal to 9. Parameters for the Brown valence bond model analysis can be found in reference.16 The calculations of SmNiO3 under hydrostatic pressure have been carried by considering the ground state unit cell lattice parameters with a homogeneous contraction or expansion of them (±2%). We then relaxed atomic positions by keeping lattice parameters fixed while considering the different magnetic orderings.