Low Valence Nickelates: Launching the Nickel Age of Superconductivity

The discovery of superconductivity in thin films ($\sim$10 nm) of infinite-layer hole-doped NdNiO$_2$ has invigorated the field of high-temperature superconductivity research, reviving the debate over contrasting views that nickelates that are isostructural with cuprates are either (1) sisters of the high-temperature superconductors, or (2) that differences between nickel and copper at equal band filling should be the focus of attention. Each viewpoint has its merits, and each has its limitations, suggesting that such a simple picture must be superseded by a more holistic comparison of the two classes. Several recent studies have begun this generalization, raising a number of questions without suggesting any consensus. In this paper, we organize the findings of the electronic structures of $n$-layered NiO$_2$ materials ($n$= 1 to $\infty$) to outline (ir)regularities and to make comparisons with cuprates, with the hope that important directions of future research will emerge.

RNiO 2 materials are the n=∞ member of a larger series of layered nickelates with chemical formula (RO 2 ) − [R(NiO 2 ) n ] + (R=La, Pr, Nd; n = 2, 3, . . . , ∞) that possess n cuprate-like NiO 2 planes in a squareplanar coordination. Except for the n=∞ case, groups of n-NiO 2 layers are separated by R 2 O 2 blocking layers that severely limit coupling between adjacent units. These layered square-planar compounds are obtained via oxygen deintercalation from the corresponding parent perovskite RNiO 3 (n=∞) 2 and Ruddlesden-Popper R n+1 Ni n O 3n+1 (n =∞) phases 1 . As shown in Fig. 1, the (RO 2 ) − [(RNiO 2 ) n ] + series can be mapped onto the cuprate phase diagram in terms of the nickel 3d-electron count, with nominal fillings running from d 9 (n=∞) to d 8 (for n=1). That superconductivity arises in this series suggests that a new family of superconductors has been uncovered, currently with two members, n=∞ and n=5, 44 the only ones (so far) where an optimal Ni valence near d 8.8 has been attained.
Some overviews on experimental and theoretical findings in this family of materials have been recently published [45][46][47][48] . In this paper, we focus on the electronic structure of layered nickelates, confining ourselves to materials with the basic infinite-layer structure: n square planar NiO 2 layers each separated by an R 3+ ion with-out the apical oxygen ion(s) that are common in most cuprates and nickelates.
In parent RNiO 2 materials, Ni has the same formal 3d 9 electronic configuration as in cuprates. As mentioned above, superconductivity in RNiO 2 materials emerges via hole doping, with T c exhibiting a dome-like dependence 12,13,49 akin to cuprates, as shown in Fig. 1. However, in parent infinite-layer nickelates the resistivity shows a metallic T-dependence (but with a low temperature upturn) 7,8 and there is no signature of longrange magnetic order, even though the presence of strong antiferromagnetic (AFM) correlations has recently been reported 50 . This is in contrast to cuprates, where the parent phase is an AFM charge-transfer insulator.
Noteworthy differences from cuprates were already reflected in early electronic structure calculations as well 51,52 . For the parent material RNiO 2 , non-magnetic density functional theory (DFT) calculations show that besides the Ni-d x 2 −y 2 band, additional bands of R-5d character cross the Fermi level. The electronic structure of RNiO 2 is three-dimensional-like, with a large c-axis dispersion of both (occupied) Ni and (nearly empty) R-5d z 2 bands due to the close spacing of successive NiO 2 planes along the c-axis. The R-5d z 2 dispersion leads to the appearance of electron pockets at the Γ and A points of the Brillouin zone which display mainly R-5d z 2 and R-5d xy character, respectively, that self-dope the large hole-like Ni-d x 2 −y 2 Fermi surface. This self-doping effect (absent in the cuprates) introduces a substantial difference between nominal and actual filling of the Ni-d  bands, accounting for conduction and possibly also disrupting AFM order. The presence of the 5d electrons is consistent with experimental data, which reveal not only metallic behavior but also evidence for negative charge carriers as reflected in the negative Hall coefficient 8,12,13 . However, as the material becomes doped with Sr, the R-5d pockets become depopulated, the Hall coefficient changes sign 8 , and the electronic structure becomes more single-band, cuprate-like 39,53 .
Besides the presence of R-5d electrons, infinite-layer nickelates have some other relevant differences from the cuprates, particularly their much larger charge-transfer energy between the metal 3d and oxygen 2p states. In cuprates, the charge-transfer energy ε 3d -ε 2p is as small as 1-2 eV 54 , indicative of a large p-d hybridization, and enabling Zhang-Rice singlet formation. In RNiO 2 , the charge-transfer energy is much larger, ∼4.4 eV, as obtained from the on-site energies derived from a Wannier analysis 39 . This is consistent with the lack of a prepeak in x-ray absorption data at the oxygen K-edge 18 . Because of this increase in charge transfer energy, the nickelate is more Mott-like, whereas the cuprate is more charge-transfer-like, in the scheme of Zaanen, Sawatzky and Allen 35,38 . Moreover, the doped holes tend to be on the Ni sites, as opposed to cuprates where they tend to reside on the oxygen sites. This in turn brings up the issue of the nature of the doped holes on the Ni sites. That is, do they behave as effective d 8 dopants, and if so, is d 8 high-spin or low-spin? If the former, then these materials would fall in the category of Hund's metals 40,[55][56][57][58] , and thus would deviate substantially from cuprates. This im-portant matter has yet to be resolved, though ab initio calculations point towards a low-spin picture due to the large crystal-field splitting of the e g states in a square planar environment 53 .
Because of their lower degree of p − d hybridization, the superexchange in RNiO 2 , as determined by resonant inelastic x-ray scattering experiments 50 , is about half that of the cuprates. Still, its value (J=64 meV) confirms the existence of significant AFM correlations 30,50 . Long-range AFM order has however not been reported, with NMR data suggesting the ground state is paramagnetic 59 , and susceptibility data interpreted as spin-glass behavior 60 . Neél type order is consistently obtained in DFT studies 24,27,29,51 , as in d 9 insulating cuprates. The predicted AFM ground state in DFT+U calculations 61 is characterized by the involvement of both d x 2 −y 2 and d z 2 Ni bands 62 . This state is peculiar in that it displays a flat-band one-dimensional-like van Hove singularity of d z 2 character pinned at the Fermi level. These flat-band instabilities should inhibit but not eliminate incipient AFM tendencies 62 .
Discussing the origin of superconductivity in RNiO 2 , as in the cuprates, is a controversial topic. But certainly the reduced T c of the nickelate compared to the cuprates is consistent with the reduced value of the superexchange, and the larger charge-transfer energy. t-J model and RPA calculations building from tight-binding parameters derived from DFT calculations show that the dominant pairing instability is in the d x 2 −y 2 channel, as in cuprates 63 . Indeed, single-particle tunneling measurements on the superconducting infinite-layer nickelate have revealed a V-shape feature indicative of a d-wave gap 64 . On a broader level, several theoretical papers have speculated that the superconductivity is instead an interfacial effect of the infinite-layer film with the SrTiO 3 substrate 65-68 , though recently superconductivity has been observed when other substrates are used 69 .
In this context, it should be noted that superconductivity has not been observed in bulk samples yet; since the precursor is cubic, there is no set orientation for the c-axis, meaning the bulk is far less ordered than the film 20,21 .
B. The superconducting n = 5 material Recently, a second superconducting member has been found in the (RO 2 ) − [(RNiO 2 ) n ] + family: the n=5 layered nickelate Nd 6 Ni 5 O 12 , also synthesized in thin-film form 44 . As schematically shown in Fig. 1, this material has a nominal valence near that of the optimally-doped infinite-layer material (that is, Ni 1.2+ : d 8.8 nominal filling) and so, in contrast to its infinite-layer counterpart, it is superconducting without the need for chemical doping. While RNiO 2 displays NiO 2 layers separated by R ions, this quintuple-layer material (with five NiO 2 layers per formula unit) has an additional fluorite R 2 O 2 slab separating successive five-layer units. Further, each successive five-layer group is displaced by half a lattice constant along the a and b directions (i.e., the body centered translation of the I4/mmm space group). These additional structural features effectively decouple the fivelayer blocks, so the c-axis dispersion of this material is much weaker than its infinite-layer counterpart. Despite these significant structural differences, T c is similar to that of the doped infinite-layer materials (with the onset of the superconducting transition taking place at ∼ 15 K), reducing the chances that yet to be synthesized low valence nickelates will have substantially higher transition temperatures.
In terms of its electronic structure 70 , the n=5 material is intermediate between cuprate-like and n=∞-like behavior. From DFT calculations, the charge-transfer energy of Nd 6 Ni 5 O 12 is ∼ 4.0 eV. This reduced energy compared to the undoped infinite-layer material means that the Ni-3d states are not as close in energy to the Nd-5d states, consistent with the presence of a pre-peak in the oxygen K-edge (similar to what happens with Sr-doped NdNiO 2 53 ). As a consequence, the electron pockets arising from the Nd-5d states are significantly smaller than those in the infinite-layer material (see Fig. 1). This reduced pocket size along with the large hole-like contribution from the Ni-3d states is consistent with experiment in that the Hall coefficient remains positive at all temperatures, with a semiconductor-like temperature dependence reminiscent of under-and optimally-doped layered cuprates. Aside from the appearance of these small Ndderived pockets at the zone corners, the Fermi surface of Nd 6 Ni 5 O 12 is analogous to that of multilayer cuprates with one electron-like and four hole-like d x 2 −y 2 Fermi surface sheets. Importantly, the Fermi surface of the quintuple-layer nickelate is much more two-dimensionallike compared to the infinite-layer nickelate material, as the presence of the fluorite blocking slab reduces the caxis dispersion, as mentioned above.
C. The n=3 material, the next superconducting member of the series?
The materials discussed above can be put into the context of earlier studies of bulk reduced RP phases with n=2, 3 NiO 2 layers 5,71-73 , separated by fluorite R 2 O 2 blocking slabs that enforce quasi-2D electronic and magnetic behavior.
The n=3 member of the series, R 4 Ni 3 O 8 (with Ni 1.33+ : d 8.67 filling), has been studied extensively over the past decade (both single crystal and polycrystralline samples) 71 . Since the charge-transfer energy decreases with decreasing n 70 , the n=3 class is more cuprate-like than its n=∞ and n=5 counterparts. Both La and Pr materials are rather similar regarding their high-energy physics, with a large orbital polarization of the Ni-e g states, so that the d 8 admixture is low spin 72,74 (but see Ref. [75]). The primary difference is that La 4 Ni 3 O 8 exhibits long-range diagonal stripe order 73,76 (similar to that seen in 1/3 hole-doped La 2 NiO 4 ), whereas its Pr counterpart appears to have short-range order instead 77 . This results in the La material being insulating 78 in its low-temperature charge-ordered phase 79 , whereas Pr 4 Ni 3 O 8 remains metallic at all temperatures 72 , with an intriguing linear T behavior in its resistivity for intermediate temperatures (similar to that of cuprates at a comparable hole doping). Nd samples have also been studied 80 , but the degree of insulating/metallicity behavior seems to be sample dependent.
The difference between La and Pr trilayer materials could be due to the reduced volume associated with Pr (one of the motivations for the authors of Ref. [8] to study Sr-doped NdNiO 2 rather than Sr-doped LaNiO 2 ). The Ni spin state and metal versus insulator character have indeed been calculated to be sensitive to modest pressure 74 . Another factor is possible mixed valency of Pr as observed in cuprates (though Pr-M edge data on Pr 4 Ni 3 O 8 did not indicate mixed valent behavior 72 ). Because of its decreased charge-transfer energy relative to n = 5, the rare-earth derived pockets no longer occur 81 . This lack of R-5d involvement is confirmed by the Hall coefficient that stays positive at all temperatures 44 , (it remains to be understood why the thermopower in the case of La 4 Ni 3 O 8 is always negative 78 , also seen in the metallic phase). In addition, these trilayer nickelates show a reduced charge-transfer energy (∼ 3.5 eV as obtained from a Wannier analysis 70 ) that, along with the larger effective doping level, is consistent with the strong oxygen K edge pre-peak seen in x-ray absorption data 72 . Oxygen K edge RIXS data indicate a significant contribution of oxygen 2p states to the doped holes 82 . As the effective hole doping level is 1/3, these materials are outside the range where superconductivity would be expected (see Fig. 1). Reaching the desired doping range for superconductivity might be possible via electron doping. This could be achieved by replacing the rare earth with a 4+ ion (such as Ce or Th) 83 , intercalating with lithium, or gating the material with an ionic liquid.
If superconductivity were to occur, one might hope for a higher T c as has indeed been predicted via t − J model calculations 84 . Recent RIXS measurements 77 , though, find a superexchange value for n=3 nearly the same as that reported for the infinite-layer material. This suggests the possibility that T c in the whole nickelate family may be confined to relatively low temperatures compared to the cuprates. The similar value of the superexchange for n=∞ and n=3 is somewhat of a puzzle. Though their t pd hoppings are very similar, the difference in the charge-transfer energy should have resulted in a larger superexchange for n=3. The fact that it is not larger is one of the intriguing questions to be resolved in these low valence layered nickelates.

D. The n=2 material
The n=2 member of the series, La 3 Ni 2 O 6 , has been synthesized and studied as well 5,85 . In terms of nominal filling, it lies further away from optimal d-filling, being nominally Ni 1.5+ : d 8.5 . Experimentally, it is a semiconductor with no trace of a transition occurring at any temperature, although NMR data suggest that the AFM correlations are similar to those of the n=3 material. Electronic structure studies 79 have predicted its ground state to have a charge-ordered pattern with Ni 2+ cations in a low-spin state and the Ni + : d 9 cations forming a S=1/2 checkerboard pattern. This charge-ordering between S=1/2 Ni + : d 9 and non-magnetic Ni 2+ : d 8 cations is somewhat similar to the situation in the n=3 material 79 . Calculations suggest that it is quite general in these layered nickelates that the Ni 2+ cations in this square-planar environment are non-magnetic. This has been shown by ab initio calculations to be the case also with the Ni 2+ dopants in the RNiO 2 materials 86 .

E. The n=1 case
The long-known R 2 NiO 4 materials, with the n=1 formula as above, contain Ni ions with octahedral coordination. We instead consider Ba 2 NiO 2 (AgSe) 2 (BNOAS) 87 , as it represents the extreme opposite of the n=∞ member, not only in regards to its d 8 valence, but also because its square planar coordination with long Ni-O bond is thought to promote 'high-spin' (magnetic) behavior, that is, one hole in d x 2 −y 2 and one hole in d z 2 . Unlike the other n cases, the charge balanced formula is (BaAg 2 Se 2 ) 0 (BaNiO 2 ) 0 ; both blocking and active layers are formally neutral. BNOAS is insulating, distinguished by a magnetic susceptibiliy that is constant, thus nonmagnetic, above and below a peak at T * ∼130 K. This increase from and subsequent decrease to its high-T value reflects some kind of magnetic reconstruction at T * that was initially discussed in terms of canting of high-spin moments. That interpretation does not account for the constant susceptibility above and below the peak.
Valence counting indicates Ni 2+ : d 8 , so a half-filled e g manifold. Conventional expectations are either (i) both 3d holes are in the d x 2 −y 2 orbital -a magnetically dead singlet that cannot account for the behavior around T * , or (ii) a Hund's rule S=1 triplet, which would show a Curie-Weiss susceptibility above the ordering temperature, but that is not seen in experiment. Correlated DFT calculations 88 predict an unusual Ni d 8 singlet: a singly occupied d z 2 orbital anti-aligned with a d x 2 −y 2 spin. This 'off-diagonal singlet' consists of two fully spin-polarized 3d orbitals singlet-coupled, giving rise to a 'non-magnetic' ion, however one having an internal orbital texture. Such tendencies were earlier noted 51 in LaNiO 2 , and related Ni spin states were observed to be sensitive to modest pressure in the n=2 and n=3 classes 74 . Attempts are underway 89,90 to understand this "magnetic transition in a non-magnetic insulator".

III. OUTLOOK
While this new nickelate family seems to be emerging as its own class of superconductors, its connections to cuprates -crystal and electronic structures, formal d count in the superconducting region, AFM correlations -retain a focus on similarities between the two classes. Apart from the obvious structural analogy, the cupratemotivated prediction of optimal d 8.8 filling has been realized in two nickelate materials, one achieved through chemical doping, the other layering dimensionality. In this context, the (so far) little studied n=6 and n=4 members of the series 70 may provide some prospect for superconductivity. Oxygen-reduced samples of these materials are so far lacking (though the n=4 member of the RP series has been epitaxially grown 91 ), and even if they are synthesized, they might require additional chemical tuning to achieve superconductivity. They share a similar electronic structure to the n = 5 material, but with slightly different nominal filling of the 3d bands 70 . Calculations show that as n decreases from n=∞ to n=3, the cuprate-like character increases, with the charge-transfer energy decreasing along with the self-doping effect from the rare earth 5d states. In contrast, the particular n=1 member discussed above seems distinct from other nickelates, and provides a different set of questions in the context of quantum materials 89,90 .