source:nature
Abstract
We provide a set of computational experiments based on ab initio calculations to elucidate whether a cuprate-like antiferromagnetic insulating state can be present in the phase diagram of the low-valence layered nickelate family (Rn+1NinO2n+2, R= rare-earth, n=1−∞) in proximity to half-filling. It is well established that at d9 filling the infinite-layer (n=∞) nickelate is metallic, in contrast to cuprates wherein an antiferromagnetic insulator is expected. We show that for the Ruddlesden-Popper (RP) reduced phases of the series (finite n) an antiferromagnetic insulating ground state can naturally be obtained instead at d9 filling, due to the spacer RO2 fluorite slabs present in their structure that block the c-axis dispersion. In the n=∞ nickelate, the same type of solution can be derived if the off-plane R-Ni coupling is suppressed. We show how this can be achieved if a structural element that cuts off the c-axis dispersion is introduced (i.e. vacuum in a monolayer of RNiO2, or a blocking layer in multilayers formed by (RNiO2)1/(RNaO2)1).
Introduction
Superconductivity in cuprate-like systems has been long sought for1. Among different approaches to achieve cuprate-analog materials, targeting nickelates has been an obvious strategy as nickel and copper are next to each other in the periodic table2. After a 30-year search, Li and coworkers reported superconductivity in Sr-doped infinite-layer NdNiO2 in 20193,4, and more recently in PrNiO25,6 and LaNiO27,8. These infinite-layer nickelates are formed by NiO2 planes (with a square-planar oxygen environment for the cation, like the CuO2 planes in cuprates), with Ni adopting the unusual Ni1+ valence, with nine d-electrons, (analogous to Cu2+)9,10. As of now, superconductivity has only been observed in thin films, prompting work on the role played by interfacial effects11,12. Importantly, RNiO2 (R= rare-earth) materials belong to a larger series represented by Rn+1NinO2n+2 (n=1−∞) in which each member contains n-NiO2 layers13 opening up the door to find a whole family of nickelate superconductors (see Fig. 1). Indeed, very recently the n= 5 Nd-based member of the series (Nd6Ni5O12) was found to be a superconductor as well14, and proposals exist for superconductivity to be realized in the n= 3 compound (R4Ni3O8) upon electron doping15, so that it falls within the superconducting dome in terms of d-filling in a general phase diagram applicable for the whole series.
In spite of the similarities between this nickelate family and the cuprates described above, a remarkable difference is that while parent cuprates are antiferromagnetic (AF) insulators9, parent infinite-layer nickelates are metallic and there is no signature of long-range magnetic order in LaNiO216,17,18 or NdNiO219. Based on these findings, magnetism was at first ruled out as being a key ingredient for superconductivity in RNiO2-based systems, even though many theories of superconductivity in cuprates rely on the existence of strong AF correlations20. Recent RIXS experiments in NdNiO2 point in this direction as they have revealed the presence of strong AF correlations with a superexchange ∼ 60 meV21. Also, an NMR study has found the presence of AF fluctuations and quasi-static AF order below 40 K in Sr-doped NdNiO222. Muon spectroscopy has shown that the ground state is formed by local moments coupled AF23. Along these lines, first-principles calculations do predict that the main correlations in RNiO2 are indeed AF, but less strong than in the cuprates10,24,25,26,27,28, and embedded in a metallic environment. While magnetism in these infinite-layer nickelates is still under intense study22,29,30,31,32,33, it is then important to understand: i) Why parent infinite-layer nickelates are metallic and do not show any signature of long-range magnetic order (their cuprate analog CaCuO2 is AF and insulating34). ii) Whether the phase diagram of these newly discovered superconducting nickelates can host an AF insulating phase close to half-filling by modifying parameters such as dimensionality. iii) If other (finite n) members of the layered nickelate family are more similar to cuprates at d9 filling in terms of their electronic structure.
In this paper, using first-principles calculations, we try to address these questions. We argue that the metallic non-cuprate-like behavior of parent infinite-layer nickelates can be attributed to the strong Ni-R-Ni off-plane hopping. This gives rise to extremely dispersive R-d bands, strongly hybridized with the Ni-d occupied bands. These R-d bands cross the Fermi level leading to a deviation from half-filling in the Ni-dx2−y2 band via a self doping mechanism. We show that if structural modifications are used to provide a means to suppress the Ni-Ni off-plane hopping, an AF insulating state can be obtained. As such, we reveal that in the purely two-dimensional limit, RNiO2 materials are indeed AF insulators. This very same phase can be realized naturally in the finite-n members of the layered nickelate family (with a fluorite blocking slab between NiO2 planes), including the recently found n=5 layered nickelate superconductor. As such, our calculations show that these finite-n nickelates can be AF insulators at d9 filling without any structural changes. All in all, our results illustrate that a crucial part of the phase diagram of the superconducting cuprates (the parent insulating antiferromagnetic phase) can indeed be present for the whole series of layered nickelates. In the case of the n = ∞ compound it is simply hindered by strong off-plane band dispersions but a dimensionality reduction can bring it to light.
Results
Higher-order layered nickelates
In order to unravel the nature of the ground state of the whole layered-nickelate family when approaching the d9 nominal filling, we will start by analyzing the electronic structure of the reduced RP compounds as n is varied. As mentioned above, the n=5 Nd-based layered nickelate has been recently found to be superconducting, increasing the interest in these higher-n layered nickelates.
We have used the I4/mmm structure of the n=3 compound35,36 as a model for constructing the structures of the n=4−6 counterparts37, since the latter have not been experimentally resolved yet. The structure of the n=3−6 materials (reduced from Ruddlesden-Popper phases38,39) can be visually inspected in Fig. 1. The structure is formed by a series of n-NiO2 planes where the Ni coordination is a square plane of oxygens. These n planes are sandwiched by RO2 fluorite blocking layers that confine the Ni-d states along the c-direction conferring the Ni-dz2 bands a localized, molecular-like character40. The structural parameters (nearest-neighbor distances and lattice parameters) obtained after relaxation for the La-based n=4−6 nickelates are summarized in Table 1 compared to the experimentally resolved ones for the n=3 material41.
Once the relaxations are carried out, we have artificially moved the chemical potential in all compounds (using the virtual crystal approximation) so that a nominal d9 filling is achieved42, in order to explore what magnetic and electronic phase is the ground state in this situation. It turns out that a C-type checkerboard AF state is the ground state for all systems within GGA and GGA+U. The GGA checkerboard AF solutions at d9 filling are metallic, though a U is needed on the Ni-d states to open up a gap. A summary of the derived AF band structures within GGA+U for the n=3−6 compounds is presented in Fig. 2. In all cases, we show the band structures for the aforementioned AF checkerboard C-type ground state, where the S=1/2 (nominally Ni1+) cations couple AF due to the hole in the dx2−y2 orbital. The derived Ni magnetic moments are ∼ 0.8-0.9 μB once a U is added (with some slight variation in inner, outer, and mid-layers, as shown in Table 2) and hence consistent with this Ni1+ (S=1/2) picture. In all cases, a gap appears in the spectrum but its value gets reduced as n increases, that is, as one moves towards the infinite-layer (three-dimensional) limit37,43. This gap opens up between states that are predominantly dz2 in character at the top of the valence band and dx2−y2 at the bottom of the conduction band, as expected (see Supplemental Material). In this manner, the electronic structure resembles that of parent cuprates, where the ground state at d9 filling is also an AF insulator. Importantly, our VCA calculations qualitatively reproduce previously reported electronic structure calculations for the n=3 case, where this material was explicitly electron-doped with a 4+ cation: Ce, Zr, or Th15.
These results give rise to the following questions: why are the infinite-layer counterparts metallic at the same nominal d-filling and what happens to the AF correlations that should exist due to the partly filled dx2−y2 band? In the following, we will try to answer this question carefully with the aid of electronic structure calculations through computational experiments in the infinite-layer materials.
We start by describing the basic electronic structure of infinite-layer nickelates and how we can understand the differences between the n=∞ case and the n≠∞ oxygen-reduced RP compounds described in the previous section.
Figure 3 shows the GGA band structure and density of states (DOS) of bulk LaNiO2 in its AF C-type (in-plane AF checkerboard) ground state. The metallic character can be seen to be brought about, as it has been analyzed in previous works29,31,44,45, by the presence of strongly dispersive bands (in particular along the Γ-Z direction) of mixed La-d and Ni-dz2 character. In the non-magnetic state, La-d bands also cross the Fermi level, as pointed out before10,25,46. Introducing a large U, either in the La-d or the Ni-d bands, does not lead to a band splitting that would open up a gap in this system29. As such, the R-d bands lead to a so-called self-doping effect, i.e. the depletion in occupation of the Ni-dx2−y2 band, away from the nominal half-filling that would correspond to a Ni1+ cation (see Supplemental Material for the dx2−y2 fat bands). Note that the presence of a flat Ni-dz2 band pinned at the Fermi level in connection to metallicity in the C-type AF state of infinite-layer nickelates has been highlighted in previous DFT work associated to instabilities that could limit AF order at low temperature。
Now the question arises: is there a way one can get an insulating AF phase from here? To answer this question, we start by taking the system towards the purely two-dimensional limit. For that sake, we have constructed a RNiO2 monolayer from the constituting NiO2 planes, with the standard square planar environment for the Ni cations (see Fig. 4). The additional rare-earth cations are placed alternatively on top and below the NiO2 planes to ensure the stoichiometry and provide the highest possible symmetry. If the rare-earth cations are all placed on a single plane above or below the NiO2 plane, the resulting total energy is much higher.
Figure 4 shows the band structure of this single-layer LaNiO2 system at the GGA level (without introducing correlations via DFT+U) in an AF checkerboard ground state. The La-d bands do not cross the Fermi level and the only Ni-d band that remains unoccupied is a single minority-spin dx2−y2 band per Ni (see Supplemental Material for the dx2−y2 fat bands). The La-d bands cannot hybridize strongly with the Ni-d ones anymore (once the periodicity along c is lost) and as a consequence there is no largely dispersive band crossing the Fermi level, as it occurs in the bulk. As such, the system is no longer self-doped and the nominally Ni1+ purely ionic description works nicely, like in parent cuprates and reduced RP nickelates. The half-filled dx2−y2 band yields a spin-1/2 scenario with a strong in-plane AF coupling, that appears even at the GGA level (U= 0 limit). This solution is similar to that obtained for cuprates (such as isostructural CaCuO2) that are AF insulators at the GGA level as well10. The magnetic moments within GGA are also somewhat larger (∼ 0.9 μB) than those appearing in bulk LaNiO2 (∼ 0.6 μB) that get reduced by the self-doping effect. If an on-site Coulomb repulsion is introduced, the gap becomes widened as U increases by pushing the unoccupied Ni-dx2−y2 bands higher in energy, and the magnetic moments approach 1.0 μB per Ni. Even though we have presented our results for La, the same type of solution can be derived for the NdNiO2 counterpart (see Supplemental Material).
In order to illustrate the lack of La-d - Ni-d hybridization in this case, we have highlighted in the band structure of Fig. 4 the Ni-dz2 character for both Ni atoms in the structure. These dz2 bands become completely occupied and are very flat (in contrast to the bulk plot of Fig. 3) and no Ni-dz2 character can be seen in the unoccupied part of the spectrum. The DOS plot helps locate the different characters of other bands shown. La-d bands start appearing above 1 eV (only one spin channel is analyzed for clarity). The O-p bands start to emerge below -3 eV in this case, further away from the Fermi level than in the cuprates。
In order to study the band(s) that bring about metallicity in the bulk in more detail, we have constructed a hypothetical system, like the one shown in Fig. 5. This is a multilayer that alternates one LaNiO2 and one LaNaO2 unit cell stacked along the (001) direction. This system provides continuity along the c-axis for the rare-earth La-d-La-d direct hopping. However, in this structure the hopping paths for the transition metal Ni-d bands are largely confined into the ab plane. Their off-plane hopping is largely blocked by the LaNaO2 layer, since the Na atoms do not provide continuity for a direct off-plane Ni-dz2 hopping. Also, the La-d-Ni-d hybridization is largely hampered by the NaO2 blocking layer. Note that LaNaO2 exists but it is not isostructural to LaNiO248. Here, for the purpose of comparison though, we assume the same crystal structures taking the in-plane lattice parameter of LaNiO2 as a basis, and then all the internal atomic positions and the off-plane lattice parameter have been relaxed. It is not relevant for our discussion whether this system is experimentally achievable or not, it just provides another way of suppressing the c-axis dispersion to confirm the reasoning exposed above. We note that the equivalent Li-based system (LiLa2NiO4) has been shown to be dynamically stable49.
The band structure of the multilayer is shown in Fig. 5. Similar to the LaNiO2 monolayer, the system is an AF insulator. The only subtle difference is that a very small U (∼ 1.5 eV) is needed for a gap to be opened up (an even smaller U is required in the case of NdNiO2, see Supplemental Material). This gapped AF solution occurs even though there is a continuity of the rare-earth sublattice along the off-plane direction. This confirms that the key factor in order to open up a gap is for the geometry to provide a way to suppress the R-Ni off-plane hopping that leads to hybridized bands with a large dispersion, crossing the Fermi level. When the Ni-dz2 bands (highlighted in the band structure shown) have no mechanism to hybridize along the c-direction, these bands remain very flat and fully occupied. Then, the R-d bands do not disperse below the Fermi level and the system is an analog of the reduced RP phases and parent cuprates at d9 filling: an AF insulator, with no self-doping. As such, at the electronic structure level, this computational experiment of a multilayered system works similar to the single-layer LaNiO2. All in all, we can conclude that reducing dimensionality is a way to reach a parent phase that is AF and insulating in parent RNiO2, like that in the cuprates.
It is important to explain this result in detail. If one looks at the band character of the band crossing the Fermi level in bulk LaNiO224 or NdNiO225, it is mostly R-d in character. If this were the whole story, in this multilayered system, such a band would still have room to be largely dispersive. What our calculations show is that its large dispersion requires a strong hybridization with the Ni-dz2 band. When this hybridization is cut by, e.g. introducing NaO2 planes in the structure, then this band is no longer living close to the Fermi level and the electronic structure becomes cuprate-like, AF correlations become stronger (larger moments closer to the nominal ionic S=1/2 value), and the ground state becomes AF and insulating even at the GGA level (or with a very small U in the case of the multilayer).
Discussion
We have shown that an AF insulating ground state can be obtained in Rn+1NinO2n+2 layered nickelates at d9 filling: in a natural manner in the reduced RP phases (n=3−6), or introducing a structural element to cut the c-axis dispersion in the n=∞ compounds.
In the infinite-layer nickelates (RNiO2) we have shown that the origin of the celeb self-doping effect is not simply in the rare-earth d bands – the existence of the electron pockets with R-d parentage requires a strong hybridization with the Ni-dz2 bands. This hybridization leads to the formation of wide bands that arise due to the large hopping taking place along the c-axis. This provides further evidence of the active role of the Ni-dz2 orbitals in the electronic structure of infinite-layer nickelates29,50,51,52. Furthermore, our calculations reveal how the hybridizing rare-earth bands seemingly mask the underlying two-band (Ni-dx2−y2 + Ni-dz2) Mott insulating state described in Ref.53.
This situation (a highly dispersive band crossing the Fermi level) does not occur in the parent cuprates. There are various reasons for this. In the isostructural infinite-layer cuprate (CaCuO2) the Ca-d bands are very far above the Fermi level to be able to produce any effects in the electronic structure. In other cuprates containing atoms with available unoccupied d bands (such as the La bands in La2CuO4 or the Y bands in the YBCO family), there are important structural differences. The case of La2CuO4 allows for a direct comparison with the layered nickelate family. Such a structure54 consists of CuO2 planes but separated by a spacing La2O2 blocking layer (the same can be said about YBCO). These structural elements provide a barrier for the off-plane hoppings (in a similar fashion to the fluorite slab in finite-n members of the layered nickelate family, and the NdNaO2 layer in the hypothetical multilayered system we described for the infinite-layer nickelate) ultimately allowing for an AF insulating phase to appear naturally in the d9 limit.
To summarize, we have shown that a parent insulating AF phase can take place in the phase diagram of layered nickelates (Rn+1NinO2n+2) close to half-filling. Our work shows yet another piece of evidence of the similarities between layered nickelates and cuprates, which were not apparent at first but can be elucidated digging deeper into the properties of these layered nickel-oxide materials.
Methods
Our electronic structure calculations in layered nickelates were performed within density functional theory55 using the all-electron, full potential code WIEN2K56,57 based on the augmented plane wave plus local orbital (APW+lo) basis set58 and the Vienna ab initio Simulation Package (VASP)59. We present the results obtained for R=La in all materials to avoid problems associated with the 4f electrons of Pr or Nd but no large changes can be anticipated from the R substitution in the electronic structure, as shown for the RNiO2 materials in a number of works10,31,44,60,61. We did indeed perform calculations in RNiO2 for a different rare-earth (Nd) and only minor changes not affecting the main conclusions of the paper arose, which are explained in the Supplemental Material. The generalized gradient approximation in the Perdew-Burke-Ernzerhof (GGA-PBE) scheme62 was used as the exchange-correlation functional. To deal with possible strong correlation effects we use GGA+U63 including an on-site repulsion U (U = 5 eV) and Hund’s coupling J (J= 0.7 eV) for the Ni 3d states.
For LaNiO2, when thin films are discussed, a sufficient vacuum of 20 Å was introduced, enough to guarantee the lack of interaction between two planes (periodic boundary conditions are utilized along the three spatial directions). Both in thin films and multilayers, atomic positions were fully relaxed within GGA-PBE using the in-plane lattice parameter of LaNiO2. In both cases, relaxations were done in a checkerboard antiferromagnetic state in a 2–√×2–√ supercell of the tetragonal (P4/mmm) cell. A C-type antiferromagnetic state has been chosen in the bulk as this is the ground state of the system as soon as a small U is included31. The muffin-tin radii used were: 2.50 a.u. for Nd and La, 2.00 a.u. for Ni and 1.72 a.u. for O. An RmtKmax = 7.0 was chosen for all calculations.
For the La-based n=4−6 compounds (La5Ni4O10 (n=4), La6Ni5O12 (n=5), and La7Ni6O14 (n=6)), whose structures have not been experimentally resolved yet, we conducted structural relaxations using VASP within GGA-PBE, optimizing both lattice parameters and internal coordinates in a 2–√×2–√ supercell of the tetragonal (I4/mmm) cell in a checkerboard (C-type) antiferromagnetic state. As above, a C-type antiferromagnetic state has been chosen as this is the ground state of the finite n materials as soon as a small U is included to account for correlations in the Ni 3d states. In the VASP calculations, an energy cutoff of 650 eV and a k-mesh of 6×6×1 were adopted with a force convergence criterion of 5 meV/Å.
With the VASP-optimized structures for the n=4−6 nickelates, we then performed further electronic structure calculations using the WIEN2K code. In these calculations, muffin-tin radii of 2.50 a.u., 2.00 a.u., and 1.72 a.u. were used for La, Ni, and O, respectively. We used RmtKmax = 7.0 and a k-grid of 21×21×21 for all materials in the irreducible Brillouin zone. To modify the chemical potential in the finite-n compounds, the virtual crystal approximation (VCA) was used in both the La-site and Ni-site (to achieve a nominal d9 filling for all materials), both yielding consistent results.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.