Reconciling the ionic and covalent pictures in rare-earth nickelates

The properties of AMO3 perovskite oxides, where M is a 3d transition metal, depend strongly on the level of covalency between the metal d and oxygen p orbitals. With their complex spin orders and metal-insulator transition, rare-earth nickelates verge between dominantly ionic and covalent characters. Accordingly, the nature of their ground state is highly debated. Here, we reconcile the ionic and covalent visions of the insulating state of nickelates. Through first-principles calculations, we show that it is reminiscent of the ionic charge disproportionation picture (with strictly low-spin 4+ and high-spin 2+ Ni sites) while exhibiting strong covalence effects with oxygen electrons shifted toward the depleted Ni cations, mimicking a configuration with identical Ni sites. Our results further hint at strategies to control electronic and magnetic phases of transition metal oxide perovskites.

Transition metal oxides with an AMO 3 perovskite structure have attracted widespread interest over the last decades, both from academic and industrial 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+ Ni 3+ O 3 (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 few last 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 P bnm to P 2 1 /n [2,3]. In this P 2 1 /n phase, a Ni-site splitting is observed; this is usually associated with the appearance of charge disproportionation [12][13][14] from 2Ni 3+ to Ni (3+δ)+ + Ni (3−δ)+ and/or a breathing distortion of O 6 octahedra that leads to a rock-salt-like pattern of small and large NiO 6 groups [13]. At T N ≤ T MI , nickelates undergo an antiferromagnetic (AFM) phase transition yielding a quadrupling of the magnetic unit cell ( k=( 1 2 , 0, 1 2 ) with respect to the P bnm primitive cell) and possible collinear or non collinear spin orderings [15][16][17][18].
The electronic structure is also characterized by strong overlaps between O-2p and Ni-3d states leading to 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 [19][20][21][22]. Efforts have thus been devoted to search for novel electronic phases in nickelates, mainly using strain engineering or confinement [10,11,[23][24][25].
In spite of all these research efforts, the structural, electronic and magnetic properties of the bulk ground state are still under debate. This can be ascribed to the scarcity of systematic bulk studies, from both the experimental and theoretical sides. On one hand, bulk nickelates are hard to synthesize and mainly thin films have been studied [3]. On the other hand, no theoretical systematic studies have been performed due to the difficulty of reproducing the RNiO 3 ground state using density functional theory (DFT). In the context of DFT-based calculations, the choice of the Hubbard U correction for Ni-3d levels remains ambiguous; indeed, a great diversity of values, ranging from very weak to quite strong corrections, have been proposed and argued for in different works [5,24,[26][27][28][29]. Moreover, the identified ground state is usually ferromagnetic [24,26,30], in contrast to the established antiferromagnetic ordering. Here we performed a systematic study of various representative nickelates using the standard DFT+U formalism. We find that a small Coulombian correction on Ni-3d states is appropriate to reproduce the key ground state properties of these compounds. We then use this theory to discuss the electronic ground state of the nickelates, revealing the co-existence of ionic (Ni electronic states featuring a complete and strict charge disproportionation) and covalent (oxygen-p electrons shared with the charge-depleted Ni cations) features, and providing an unified picture of these materials that is easy to reconcile with existing (and apparently conflicting) proposals in the literature. Finally, we unveil a new pathway to control electronic and magnetic phases in perovskites by tuning the level of covalency.

Structural properties
First, we performed full geometry relaxations considering 80-atom supercells of both possible P bnm and P 2 1 /n structures with different magnetic orderings: ferromagnetic (FM) as well as complex E-, S-, and T-type AFM orderings [5] based on ↑↑↓↓ spin chains in the (ab)-plane with different stackings along the c axis (see Figures 1.a, b and c). We employed the PBEsol functional [31] in combination with a U correction [32]   of rare-earth radius. All nickelates relax to a P 2 1 /n insulating ground state with complex antiferromagnetic structures (S-or T-type depending on the rare-earth) and band gaps compatible with experiments [33,34] (see Table I). All our P bnm phases favor a metallic FM solution [35]. We checked the reliability of our DFT+U calculations by changing the U correction to either 0 eV or 5 eV in SmNiO 3 . While the ground state is unchanged when no U-correction is applied [36], imposing U = 5 eV yields a P 2 1 /n ferromagnetic and insulating solution that is much more stable than the considered complex AFM orderings (∆E 160 meV per 80-atom unit cell). This further supports our choice of a relatively small Hubbard correction for the Ni-3d electrons.
Our optimized ground state structures are characterized by three main lattice distortions.
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    Figure 2.c). In the following we will use the notation Ni S and Ni L to refer to the Ni cations belonging to the small and large NiO 6 groups, respectively.
As usual in perovskites, the magnitude of the metal-oxygen-metal bond angles associated with the O 6 rotations is governed by steric effects (see Figures 2.d and 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 also appears to be modulated by the rare earth (see Figure 2.f), 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. Hence, the relaxed structures indicate that there is no significant orbital order in these systems, although the 3d 7 t 6 2g e 1 g electronic configuration of Ni 3+ in the high temperature P bnm phase is nominally Jahn-Teller active [28].

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 Figure 3.a for the representative case of SmNiO 3 ). 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 Figure 3.b).
Although atomic charges are not uniquely defined in DFT calculations [39], sphere integrations around the Ni cations can provide some insight into the possible charge ordering. Ni S cations, sitting at the center of the smallest O 6 octahedra, appear to hold more electrons than the Ni L cations, located in the largest oxygen cages. Since the breathing mode B oc enhances the crystal field splitting at the small NiO 6 groups, the e g levels of Ni S lie higher in energy than those of Ni L [28,40] and therefore Ni S should have fewer electrons than Ni L associated to it.
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 3.d 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 SmNiO 3 , we obtain Z Ni L ≈ +2.5, which is not far from the nominal oxidation state of 2+ that this Ni site is associated with in the complete-chargedisproportionation picture. However, we find a similar Z Ni S ≈ +2.1, which sharply deviates from the expectation value (4+) in the charge-disproportionation picture. As shown in 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 Figure 3.e, we observe a large difference between Ni L -with a moment larger than 1 µ B -and Ni S -for which the magnetic moment is null -, which suggests two very different electronic states.

Wannier analysis
The conclusion of the previous discussion is that, in the RNiO 3 compounds, all Ni atoms seem to display a similar oxidation state. Yet, the presence of a significant breathing distortion, and of the drastic 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 [25,26,30], but in our opinion a convincing explanation for this apparent contradiction is still missing. Here we ran a Wannier function (WF) analysis of our first-principles results, which allowed us to resolve this pending issue.
We used the Wannier90 package [41][42][43] to determine the maximally-localized WFs that reproduce the occupied electronic manifold. More precisely, our purpose was to count how many occupied WFs 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 seed functions that are needed for an efficient maximal-localization calculation. 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) WF functions, where 2×144 = 2×3×48 is the total number of O-2p orbitals available in our 80-atom 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 WF optimizations for the spin-up and spin-down channels.) Hence, for our initial WF seeds, we used 3 generic p orbitals centered at each of the O anions in our cell, 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 Ni L and one Ni S . 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 WFs centered at each oxygen anion (i.e., 3 spin-up WFs and 3 -very similar -spin-down WFs), suggesting that all oxygens in our nickelates are in a 2− oxidation state. The oxygen-centered WFs have a clear p character, as can be appreciated in Figures 4.c and d. We also obtained 2×3 t 2g -like WFs 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 WFs centered at the Ni L site (see Figures 4.a and b), indicating that this cation is in a 2+ oxydation state and has a significant magnetic moment associated to it. Finally, as regards the other seed functions centered at the chosen Ni L (2 spin-down d orbitals) and Ni S (2×2 d orbitals) atoms, they did not lead to any WF centered at those sites. Instead, the maximallocalization procedure resulted in WFs centered at Ni and R cations in the vicinity of the considered Ni L -Ni S pair. Thus, in particular, it was impossible to localize any e g -like WFs at a Ni S site, which strongly indicates that these Ni cations are in a 4+ oxidation state. These conclusions were ratified by considering larger clusters of Ni sites for the WF optimization, as well as individual Ni's and/or optimizations in which the oxygen bands were not included.
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 need to inspect in more detail the obtained oxygen-centered WFs. The Wannier analysis therefore leads to the conclusion that a full disproportionation occurs in the system, with clearly distinct Ni 4+ S (low-spin, non-magnetic) and Ni 2+ L (highspin, magnetic) sites. Simultaneously, the O-2p WFs approach the Ni 4+ S sites, ultimately yielding Ni S and Ni L that are nearly equivalent from the point of view of the charge (static and dynamics) around them.

DISCUSSION
Our results thus appear to be compatible with the disproportionation effects originally proposed to occur at the MIT [12][13][14]. While for a long time the nickelates were believed to possess an orthorhombic P bnm symmetry in the insulating phase [2], it is now established that they adopt a monoclinic P 2 1 /n phase at low temperatures [12,13]. This phase exhibits a breathing distortion whose magnitude decreases with increasing the tolerance factor of the perovskite [13], concomitantly accompanied by a charge disproportionation δ between the two Ni sites, leading to a Ni the Ni S at its center, exactly as in the ligand-hole picture.
As regards the magnetic moments at the different Ni sites, our results are also clear and compatible with both proposed pictures: Ni L bears a magnetic moment approaching 2 µ B , as it corresponds to having Ni 2+ in a high-spin configuration. Then, Ni S has no magnetic moment associated to it, as it would correspond to a nominal Ni 4+ low-spin configuration.
The latter result is partly a consequence of the fact that the oxygen-Ni S shared electrons are spin paired, which can be interpreted as a ligand-hole screening.
Our first-principles calculations therefore reconcile the two visions proposed to occur in the ground state of rare-earth nickelates. The electronic structure is summarized in  [48]. We also observe increasing O-2p overlaps with Ni L sites going from R=Y to Pr.
The level of covalency seems to correlate with the stability of the magnetic ordering and the insulating phase (see Table I). On one hand, with increasing covalency, the energy dif- Finally, the rare-earth atom is known to play 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 P 2 1 /n structure with a ferromagnetic 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 Figures 5.a and 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.
In contrast, for the monoclinic phase of NdNiO 3 and PrNiO 3 , the FM spin configuration is found to be metallic; further, the stability of the low-symmetry strucure with respect to the orthorhombic one drastically decreases. The complex antiferromagnetic ordering therefore appears to be a necessary condition for the MIT to occur in these two compounds.
Considering the case of SmNiO 3 under 8% of compression, we unveil a similar behaviour to bulk PrNiO 3 and NdNiO 3 , since the level of covalency increases. These results suggest that it is possible to control of the electronic and magnetic structures of these compounds by tuning the level of covalency in the system. We also emphasize that these observations again support our choice of a small Hubbard U correction on Ni-3d sites and demonstrate that DFT+U methods can capture the key physical properties of correlated systems.
In summary, we have used first-principles methods to investigate the ground state electronic structure of rare-earth nickelates. Our DFT simulations using a small Hubbard correction on Ni-3d states reproduce all features reported from experiments (insulating character, disproportionation effects, covalency, complex antiferromagnetic structures, structural trends). 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 being low-spin 4+ and high-spin 2+. At the same time, our simulations reveal a shift of the oxygen-p orbitals toward the depleted Ni cations, so that, ultimately, from the point of view of the integrated charge, the two Ni sites appear to be nearly identical. These findings are clearly reminiscent of the various pictures proposed in the literature to explain the ground state of these compounds, which can thus be reconciled according to our results. Finally, we unveil that a control of the level of covalency between oxygens and transition metal ions provides an alternative pathway to tune the electronic and magnetic phases in late transition-metal oxide perovskites.