Phase Diagram of Nickelate Superconductors Calculated by Dynamical Vertex Approximation

Abstract

We review the electronic structure of nickelate superconductors with and without effects of electronic correlations. As a minimal model, we identify the one-band Hubbard model for the Ni 3 orbital plus a pocket around the A-momentum. The latter, however, merely acts as a decoupled electron reservoir. This reservoir makes a careful translation from nominal Sr-doping to the doping of the one-band Hubbard model mandatory. Our dynamical mean-field theory calculations, in part already supported by the experiment, indicate that the Γ pocket, Nd 4f orbitals, oxygen 2p, and the other Ni 3d orbitals are not relevant in the superconducting doping regime. The physics is completely different if topotactic hydrogen is present or the oxygen reduction is incomplete. Then, a two-band physics hosted by the Ni 3 and 3orbitals emerges. Based on our minimal modeling, we calculated the superconducting T_c vs. Sr-doping x phase diagram prior to the experiment using the dynamical vertex approximation. For such a notoriously difficult to determine quantity as T_c, the agreement with the experiment is astonishingly good. The prediction that T_c is enhanced with pressure or compressive strain has been confirmed experimentally as well. This supports that the one-band Hubbard model plus an electron reservoir is the appropriate minimal model.

1 Introduction

Twenty years ago, Anisimov, Bukhvalov, and Rice [1] suggested high-temperature (T_c) superconductivity in nickelates based on material calculations that showed apparent similarities to cuprates. Subsequent calculations [2–4] demonstrated the potential to further engineer the nickelate Fermi surface by heterostructuring. Two years ago, Li, Hwang et al. [5] discovered superconductivity in Sr-doped NdNiO₂ films grown on a SrTiO₃ substrate and protected by a SrTiO₃ capping layer. These novel Sr_xNd_1–xNiO₂ films are isostructural and formally isoelectric to the arguably simplest, but certainly not the best superconducting cuprate: infinite layer CaCuO₂ [6–9].

However, the devil is in the details, and here cuprates and nickelates differ. For revealing such material-specific differences, band-structure calculations based on density functional theory (DFT) are the method of choice. They serve as a starting point for understanding the electronic structure and, subsequently, the phase diagram of nickelate superconductors. Following the experimental discovery of nickelate superconductivity, and even before that, numerous DFT studies have been published [10–20]. Based on these DFT calculations, various models for the low-energy electronic structure for nickelates and the observed superconductivity have been proposed. Besides the cuprate-like Ni 3 band, DFT shows an A and a Γ pocket which originate from Nd 5d_xy and 5 bands, but with major Ni 3d admixture in the region of the pocket. The importance of the Ni 3 orbital has been suggested in some studies [21–24], and that of the Nd-4f orbitals in others [20, 25]. Furthermore, there is the question regarding the relevance of the oxygen 2p orbitals. For cuprates, these are, besides the Cu 3 orbitals, the most relevant. Indeed, cuprates are generally believed to be charge-transfer insulators [26]. This leads to the three-orbital Emery model [27] visualized in Figures 1A,C as the minimal model for cuprates. The much more frequently investigated Hubbard model [28–30] may, in the case of cuprates, only be considered as an effective Hamiltonian mimicking the physics of the Zhang–Rice singlet [31].

FIGURE 1

At first glance, nickelates appear to be much more complicated with more relevant orbitals than in the case of the cuprates. In this article, we review the electronic structure of nickelates in comparison to that of cuprates and the arguments for a simpler description of nickelate superconductors, namely, a Hubbard model for the Ni 3 band plus a largely decoupled reservoir corresponding to the A pocket. This A pocket is part of the Nd 5d_xy band which has, however, a major admixture of Ni 3d_xz/yz and O 2p_z states around the momentum A. This leaves us with Figures 1B,D as the arguably simplest model for nickelates [32, 33]. This (our) perspective is still controversially discussed in the literature. However, as we will point out below, a number of experimental observations already support this perspective against some of the early suggestions that other orbitals are relevant. Certainly, other perspectives will be taken in other articles of this series on “Advances in Superconducting Infinite-Layer and Related Nickelates.” The simple picture of a one-band Hubbard model, whose doping needs to be carefully calculated since part of the holes in Sr-doped Sr_xNd_1−xNdO₂ go to the A pocket, allowed us [33] to calculate T_c, see Figure 5, at a time when only the T_c for a single doping x = 20% was experimentally available. To this end, state-of-the-art dynamical vertex approximation (DΓA) [36–38], a Feynman diagrammatic extension of dynamical mean-field theory (DMFT) [39–42] has been used. For such a notoriously difficult to calculate physical quantity as T_c, the agreement of the single-orbital Hubbard model calculation with subsequent experiments [34, 35] is astonishingly good. This further supports the modeling by a single-orbital Hubbard model which, thus, should serve at the very least as a good approximation or a starting point.

The outline of this article is as follows: In Section 2,we first compare the electronic structure of nickelates to that of cuprates, starting from DFT but also discussing effects of electronic correlations as described, for example, by DMFT. Subsequently, we argue in Section 3, orbital-by-orbital, that the other orbitals besides the Ni 3 and the A pocket are, from our perspective, not relevant. This leaves us with the one-3-band Hubbard model plus an electron reservoir representing the A pocket of Figures 1B,D, which is discussed in Section 4, including the translation of Sr-doping to the filling in the Hubbard model and the reservoir. In Section 5, we discuss the effect of non-local correlations as described in DΓA and the calculated superconducting phase diagram. Section 6 shows that topotactic hydrogen, which is difficult to detect in the experiment, completely overhauls the electronic structure and the prevalence of superconductivity. Finally, Section 7 summarizes the article.

2 Electronic Structure: Nickelates vs. Cuprates

Let us start by looking into the electronic structure in more detail and start with the DFT results. On a technical note, the calculations presented have been performed using the wien2k [43, 44], VASP [45], and FPLO [46] program packages, with the PBE [47] version of the generalized gradient approximation (GGA). For further details, see the original work [33]. Figure 2 compares the bandstructure of the two simple materials: CaCuO₂ and NdNiO₂. Here, we restrict ourselves to only the Brillouin zone path along the most relevant momenta for these compounds: Γ (0, 0, 0), X (π, 0, 0), and A (π, π, π). In DFT, both the cuprate and nickelate parent compounds are metals with a prominent Cu or Ni 3 band crossing the Fermi energy. In other aspects, both materials differ (for a review cf [48]): In the case of cuprates, the oxygen bands are much closer to the Fermi energy. Hence, if electronic correlations split the DFT bands into two Hubbard bands as indicated in Figure 2 by the arrows and the spectral function in the left side panel, we get a charge-transfer insulator [26]. For this charge-transfer insulator, the oxygen 2p orbitals are the first orbitals below the Fermi level (E_F) and receive the holes that are induced by doping. The Cu 3 lower Hubbard band is below these oxygen orbitals, and the Cu 3 upper Hubbard band is above the Fermi level. Let us note that we here refer to oxygen 2p orbitals and Cu 3 orbitals even though the hybridization between both is very strong. Indeed, the two sets of orbitals strongly mix in the resulting effective DFT bands of Figure 2.

FIGURE 2

Because cuprates are charge-transfer insulators, the one-band Hubbard model can only be considered an effective Hamiltonian mimicking the Zhang–Rice singlet [31]. As already pointed out in the Introduction, more appropriate is the Emery model of Figure 1. The correlation-induced splitting into the Hubbard bands [40], as well as the Zhang–Rice singlet [49], can be described already by DMFT [39, 40, 42]. Two-dimensional spin-fluctuations and superconductivity, however, cannot be described. For describing such physics, non-local correlations beyond DMFT are needed.

For the nickelates, the oxygen bands are at a much lower energy. Hence, as indicated in the right side panel of Figure 2, the lower Ni 3 Hubbard band can be expected to be closer to the Fermi energy than the oxygen p orbitals [32, 33]. Consequently, undoped nickelates would be Mott–Hubbard insulators if it was not for two additional bands that cross E_F around the Γ- and A-momentum. These form electron pockets, as visualized in Figure 2 (bottom right), and self-dope the Ni 3 band away from half-filling. As the 3 is doped, it develops, even when the Coulomb interaction is large, a quasiparticle peak at the Fermi energy as displayed in the right side panel of Figure 2.

3 Irrelevance of Various Orbitals

Next, we turn to various orbitals that may appear relevant at first glance but turn out to be irrelevant for the low-energy physics when taking electronic correlations properly into account. To account for the latter, we use DFT + DMFT [50–54], which is the state of the art for calculating correlated materials.

Oxygen Orbitals

For nickelates, the oxygen 2p orbitals are approximately 3 eV lower in energy than in cuprates within DFT. Hence, some DFT + DMFT calculations did not include these from the beginning [32, 33], and those that did [22] also found the oxygen 2p orbitals at a lower energy than the lower Ni 3 Hubbard band. Hence, while there is still some hybridization and mixing between the O 2p states and the Ni 3 states, a projection onto a low-energy set of orbitals without oxygen appears possible.

Ni 3 and t_2g Orbitals

Instead of the oxygen 2p orbitals, the DFT calculation in Figure 2 and elsewhere [10, 11, 17, 48] show other Ni 3d orbitals closely below the Ni 3 band. In fact, these other 3d orbitals are somewhat closer to the Fermi level than in the case of cuprates. Electronic correlations can strongly modify the DFT band structure. In particular, the Hund’s exchange J tends to drive the system toward a more equal occupation of different orbitals, especially if there is more than one hole (more than one unpaired electron) in the Ni 3d orbitals. This is not only because a larger local spin is made possible but also because the inter-orbital Coulomb interaction U′ between two electrons in two different orbitals is smaller than the intra-orbital Coulomb interaction U = U′ + 2J for two electrons in the same orbital. This tendency is countered by the crystal field splitting (local DFT potentials) which puts the 3 orbital above the Ni 3 orbital and the other (t_2g) Ni 3d orbitals because of the absence of apical O atoms in NiO₄ squares.

Figure 3 shows the DFT + DMFT spectral function for Sr_xLa_1−xNiO₂ from 0 to 25% Sr-doping. In these calculations [33], all Ni 3d and all La 5d orbitals have been taken into account in a wien2wannier [56] projection supplemented by interactions calculated within the constrained random phase approximation (cRPA) [55] to be U′ = 3.10 eV (2.00 eV) and Hund’s exchange J = 0.65 eV (0.25 eV) for Ni (La). On a technical note, the DMFT self-consistency equations [39] have been solved here at room temperature (300 K) by continuous-time quantum Monte Carlo simulations in the hybridization expansions [57] using the w2dynamics implementation [58, 59] and the maximum entropy code of ana_cont [60] for analytic continuation.

FIGURE 3

Clearly, Figure 3 indicates that for up to 20% Sr-doping, the other Ni 3d orbitals besides the 3 orbital are not relevant for the low-energy physics of Sr_xLa_1−xNiO₂; they are fully occupied below the Fermi energy. With doping, these other Ni 3d orbitals, however, shift more and more upward in energy. At around 25% Sr-doping, they touch the Fermi energy and hence a multi-orbital Ni description becomes necessary at larger dopings. That is, between 20 and 30% Sr-doping, the physics of Sr_xLa_1−xNiO₂ turns from single to multi-orbital. In the case of Sr_xNd_1−xNiO₂, this turning point is at slightly larger doping [33]. Later, in Section 6, we will see that for the Ni 3d⁸ configuration, which in Section 6 is induced by topotactic hydrogen and here would be obtained for 100% Sr doping; the two holes in the Ni 3d orbitals form a spin-1 and occupy two orbitals: 3 and 3. In Figure 3,we see at 30% doping, the first steps into this direction. Importantly, within the superconducting doping regime which, noteworthy, is below 24% Sr-doping for Sr_xNd_1−xNiO₂ [34, 35] and 21% for Sr_xLa_1−xNiO₂ [61], a single 3 Ni-orbital is sufficient for the low-energy modeling. A one-band Hubbard model description based on DMFT calculations was also concluded in [32] for the undoped parent compound.

DFT + DMFT calculations by Lechermann [22, 23] stress, on the other hand, the relevance of the 3 orbital. Let us note that also in [22] the number of holes in the 3 orbital is considerably less than in that in the 3. However, for low Sr-doping, also a small quasiparticle peak develops for the 3 band [22, 23]. An important difference to [33, 55] is that the 5d Coulomb interaction has been taken into account in [33, 55] and that the Coulomb interaction of [22] is substantially larger. The 5d Coulomb interaction pushes the Γ pocket above the Fermi energy (see next paragraph). As much of the holes in the 3 orbital stem from the admixture of this orbital to the Γ pocket, this difference is very crucial for the occupation of the 3 orbital in some calculations [21, 23]. On the other hand, in GW + extended DMFT calculations by Petocchi et al. [24], the 3 orbital is pushed to the Fermi energy for large k_z instead, that is, around the R, Z, and A point. Except for this large k_z deviation, the Fermi surface, the effective mass of the Ni 3 orbital etc. of [24] are similar to our calculation [33, 62].

First experimental hints on the (ir) relevance of the 3 can be obtained from resonant inelastic X-ray scattering (RIXS) experiments [63, 64]. Higashi et al. [65] analyzed these RIXS data by comparison with DFT + DMFT and obtained good agreement with the experiment. They conclude that NdNiO₂ is slightly doped away from 3d⁹ because of a small self-doping from the Nd 5d band and that only the 3 Ni orbital (not the 3 orbital) is partially filled and that the Ni–O hybridization plays a less important role than for the cuprates.

Γ Pocket

A feature clearly present in DFT calculations for the nickelate parent compounds LaNiO₂ and NdNiO₂ is the Γ pocket, see Figure 2D. However, when the Coulomb interaction on the La or Nd sites is included, it is shifted upward in energy. Furthermore, Sr-doping depopulates the Ni 3 orbital and the A and Γ pocket, and, thus, also helps pushing the Γ pocket above the Fermi energy. Clearly in the DFT + DMFT k-resolved spectrum of Figure 3, the Γ pocket is above the Fermi energy. To some extent, the presence or absence of the Γ pocket also depends on the rare-earth cation. For NdNiO_2, we obtain a Γ pocket for the undoped compound [33], which only shifts above the Fermi energy with Sr-doping in the superconducting region, whereas for LaNiO_2, it is already above the Fermi level without Sr-doping. We can hence conclude that while there might be a Γ pocket without Sr-doping, DFT + DMFT results suggest that it is absent in the superconducting doping regime.

Briefly after the discovery of superconductivity in nickelates, it has also been suggested that the Nd 5d orbitals of the pockets couple to the Ni 3 spin giving rise to a Kondo effect [20, 66]. However, Table 1 shows that the hybridization between the relevant Ni 3 and the most important La or Nd 5d_xy and 5 vanishes by symmetry. Also, the full 5 Ni and 5 Nd band DMFT calculations in Figure 3 do not show a hybridization (gap) between A pocket and Ni bands. This suggests that the Γ and A pocket are decoupled from the 3 orbitals. There is no hybridization and hence no Kondo effect.

Nd 4f Orbitals

Finally, the importance of the Nd 4f orbitals has been suggested in the literature. Treating these 4f orbitals in DFT is not trivial because DFT puts them in the vicinity of the Fermi level. This neglects that electronic correlations split the Nd 4f into upper and lower Hubbard bands as they form a local spin. This effect is beyond DFT. One way to circumvent this difficulty is to put the Nd 4f orbitals in the core instead of having them as valence states close to the Fermi energy. This is denoted as “GGA open core” instead of standard “GGA” in Table 1. The localized Nd 4f spins might, in principle, be screened through a Kondo effect. However, the hybridization of the Nd 4f with the Ni 3 orbital at the Fermi energy is extremely small, see Table 2 and [67]. Hence, the Kondo temperature is zero for all practical purposes. In spin-polarized DFT + U, there is, instead, a local exchange interaction between the Nd 4f and the predominately Nd 5d Γ pocket [68]. However, as pointed out in the previous paragraph, the Γ pocket is shifted above the Fermi level in the superconducting Sr-doping regime. Hence in [33], we ruled out that the Nd 4f is relevant for superconductivity. This has been spectacularly confirmed experimentally by the discovery of superconductivity in nickelates without f electrons: Ca_xLa_1−xNiO₂ [69] and Sr_xLa_1−xNiO₂ [61] have a similar T_c.

4 One-Band Hubbard model Plus Reservoir

Altogether, this leaves us with Figures 1B,D as the arguably simplest model for nickelate superconductors, consisting of a strongly correlated Ni 3 band and an A pocket. This A pocket is derived from the Nd 5d_xy band which, however, crosses the Ni 3d orbitals and hybridizes strongly with the Ni t_2g orbitals so that at the bottom of the A pocket, that is, at the momentum A, it is made up primarily from Ni t_2g, whereas the Nd 5d_xy contribution is here at a lower energy. This makes the A pocket much more resistive to shifting up in energy than the Γ pocket.

On the other hand, the A pocket does not interact with the Ni 3 band; that is, does not hybridize in Table 1. Hence, we can consider the A pocket as a mere hole reservoir which accommodates part of the holes induced by Sr-doping, whereas the other part goes into the correlated Ni 3 band which is responsible for superconductivity. Figure 4 shows the thus obtained Ni 3 occupation as a function of Sr-doping in the DFT + DMFT calculation with 5 Ni and 5 Nd(La) orbitals.

FIGURE 4

It is noted that NdNiO₂ shows for Sr-doping below about 10% more holes in the Ni 3 orbital and a weaker dependence on the Sr-doping since here the Γ pocket is still active, not only taking away electrons from Ni but also first absorbing some of the holes from the Sr-doping until it is completely depopulated (shifted above the Fermi energy) before superconductivity sets in.

In the subsequent one-band calculation, presented in the next paragraph, we employ the occupation from the Ni 3 orbital as calculated in this full DMFT calculation with 5 Ni and 5 Nd orbitals. This accounts not only for the electron pocket in the DMFT calculation but also for minor hybridization effects between the Ni 3 and 3 orbital, for example, along the Γ-X direction. In principle, this hybridization effect, which inter-mixes the orbital contribution to the bands, should not be taken into account in the one-band Hubbard model. This aims at modeling the effective 3 band which is crossing the Fermi level and which is predominantly Ni 3 (but also has admixtures from the other orbitals because of the hybridization). In the case of the Ni 3 orbital, this hybridization is very weak [33] (Supplemental Material) and can be neglected as a first approximation [62] (Supplemental Material). For other Ni 3d orbitals, this hybridization has a sizeable effect on their respective occupation. For example, the Ni 3 orbital which is strongly hybridizing with the Nd 5 orbital only has an occupation of 1.85 electrons per site in our multi-orbital calculation [62] (Supplemental material), whereas the effective Ni 3 orbital, including contributions from the hybridization, is fully occupied with two electrons per site as it is completely below the Fermi level.

The hopping parameters for the Ni-3 model from a one-band Wannier projection are shown in Table 3 and compared to that of the same orbital in a 10-band and 17-band Wannier projection. Here, denotes the hopping by R_i unit cells in the i direction. That is, t₀₀₀ is the on-site potential, t = − t₁₀₀; t' = − t₁₁₀, and t^″ = − t₂₀₀ are the nearest, next-nearest, and next-next-nearest neighbor hopping, respectively; and t_z = − t₀₀₁ is the hopping in the z-direction perpendicular to the NiO₂ planes. The hopping parameters are strikingly similar for LaNiO₂ and NdNiO₂ and the different Wannier projections.

Besides the doping from Figure 4 and the hopping for the one-band Wannier projection from Table 3, we only need the interaction parameter for doing realistic one-band Hubbard model calculations for nickelates. In cRPA for a single 3 orbital, one obtains U = 2.6 eV [15, 17] at zero frequency. But, the cRPA interaction has strong frequency dependence because of the screening of all the other Ni and Nd (La) orbitals close by. To mimic this frequency dependence, the static U parameter needs to be slightly increased. Expertise from many DFT + DMFT calculations for transition metal oxides shows that it typically needs to be about 0.5 eV larger, so that U = 3.2 eV = 8t is reasonable. Altogether, this defines a one-band Hubbard model for nickelates at various dopings. For the conductivity and other transport properties, the A pocket may be relevant as well, but superconductivity should arise from the correlated 3 band that is hardly coupled to the A pocket.

5 Non-Local Correlations and Superconducting Phase Diagram

DFT provides a first picture of the relevant orbitals, and DMFT adds to this effect of strong local correlations, such as the splitting into Hubbard bands, the formation of a quasiparticle peak, and correlation-induced orbital shifts, such as the upshift of the Γ pocket. However, at low temperatures, non-local correlations give rise to additional effects. Relevant factors are as follows: the emergence of strong spin fluctuations and their impact on the spectral function and superconductivity.

For including such non-local correlations, diagrammatic extensions of DMFT, such as the dynamical vertex approximation (DΓA) [36–38, 70], have been proven extremely powerful. Such calculations are possible down to the temperatures of the superconducting phase transition in the correlated regime and for very large lattices so that the long-range correlations close to a phase transition can be properly described. Even (quantum) critical exponents can be calculated [71–74]. DΓA has proven reliable compared with the numerically exact calculations where these are possible [75] and, in particular, provide for a more accurate determination of T_c [76] since the full local frequency dependence of the two-particle vertex is included. Such local frequency dependence can affect even the non-local pairing through spin fluctuations. In, for example, RPA, this frequency dependence and the suppression of the pairing vertex for small frequencies can only be improperly mimicked by (quite arbitrarily) adjusting the static U.

This simple one-band Hubbard model in DΓA has been the basis for calculating the phase diagram T_c vs. Sr-doping in Figure 5 [33]. At the time of the calculation, only a single experimental T_c at 20% Sr-doping was available [5]. The physical origin of the superconductivity in these calculations is strong spin fluctuations which form the pairing glue for high-temperature superconductivity. Charge fluctuations are much weaker; the electron–phonon coupling has not been considered and is also too weak for transition metal oxides to yield high-temperature superconductivity. The theoretical T_c in Figure 5 at 20% doping was from the very beginning slightly larger than in the experiment. Most likely, this is because in the ladder DΓA [36, 37] calculation of T_c, the spin fluctuations are first calculated and then enter the superconducting particle–particle channel [76]. This neglects the feedback effect of these particle–particle fluctuations on the self-energy and the spin fluctuations, which may, in turn, suppress the tendency toward superconductivity somewhat. Such effects would be only included in a more complete parquet DΓA calculation [77–79]. Also, the ignored weak three-dimensional dispersion will suppress Tc. Let us note that antiferromagnetic spin fluctuations have recently been observed experimentally [64, 80].

FIGURE 5

Given the slight overestimation of T_c from the very beginning, the agreement with the subsequently obtained experimental T_c vs. Sr-doping x phase diagram [34, 35] in Figure 5 is astonishingly good. We further see that the superconducting doping regime also concurs with the doping regime where a one-band Hubbard model description is possible for Sr_xNd_1−xNiO₂, as concluded from a full DFT + DMFT calculation for 5 Ni plus 5 Nd bands. This regime is marked dark blue in Figure 5 and, as already noted, extends to somewhat larger dopings [33] than for Sr_xLa_1−xNiO₂ shown in Figure 3. Concomitant with this is the fact that the experimental superconducting doping range for Sr_xLa_1−xNiO₂ extends to a larger x than for Sr_xNd_1−xNiO₂. For dopings larger than the dark blue regime in Figure 5, two Ni 3d bands need to be included. As we will show in the next section, this completely changes the physics and is not favorable for superconductivity. For dopings smaller than the dark blue regime in Figure 5, on the other hand, the Γ pocket may become relevant for Sr_xNd_1−xNiO₂, as well as its exchange coupling to the 4f moments.

Our theoretical calculations also reveal ways to enhance T_c. Particularly promising is to enhance the hopping parameter t. This enhances T_c because (i) t sets the energy scale of the problem, and a larger t means a larger T_c if U/t, t′/t, and t^″/t and doping are kept fixed. Furthermore, the ratio U/t = 8 for nickelates is not yet optimal. Indeed, (ii) a somewhat smaller ratio U/t would imply a larger T_c at fixed t [33]. Since the interaction U is local, it typically varies much more slowly when, for example, applying compressive strain or pressure and can be assumed to be constant as a first approximation (for secondary effects, see [81, 82]). Thus, compressive strain or pressure enhance (i) t and reduce (ii) U/t. Both effects enhance T_c. This prediction made in [33] has been confirmed experimentally: applying a pressure of 12 GPa increases T_c from 18K to 31 K in Sr_0.18Pr_0.82NiO₂ [83]. This is so far the record T_c for nickelates, and there are yet no signs for saturation or maximum, indicating that even higher T_c’s are possible at higher pressures.

Alternatives to enhance t are 1) to substitute the SrTiO₃ substrate by a substrate with smaller in-plane lattice constants since the nickelate film in-plane axis parameters will be locked to that of the substrate. Furthermore, one can 2) replace 3d Ni by 4d Pd, that is, try to synthesize Nd (La)PdO₂ [11]. Since the Pd 4d orbitals are more extended than the 3d Ni orbitals, this should enhance t as well.

Next, we turn to the DΓA spectra, more precisely Fermi surfaces, in Figure 6. Here, beyond quasiparticle renormalizations of DMFT, non-local spin fluctuations can further impact the spectrum. Only the spectral function of the Hubbard model is shown, describing the 3 band. Please keep in mind that on top of the Fermi surface in Figure 6, there is also a weakly correlated A pocket. As one can see in Figure 6, antiferromagnetic spin-fluctuations lead to a pseudogap at the antinodal momenta (± π, 0) (0, ± π) if the filling of the 3 band is close to half-filling. Indeed is the filling for the undoped parent compound NdNiO₂ where the A- and Γ pocket have taken 5% of the electrons away from the Ni 3 band. An Sr-doping of 20% is in between and , see Figure 4. Comparing these theoretical predictions with the experimental Fermi surface, even the k-integrated spectrum is very much sought after. However, here, we face the difficulty that the superconducting samples require a SrTiO₃ capping layer or otherwise may oxidize out of vacuum. This hinders photoemission spectroscopy (PES) experiments as these are extremely surface-sensitive. Hitherto, PES is only available without the capping layer for Sr_xPr_1−xNiO₂ [84]. These show a surprisingly low spectral density at the Fermi energy despite the metallic behavior of the doped system, raising the question of how similar these films are to the superconducting films.

FIGURE 6

6 Topotactic Hydrogen: Turning the Electronic Structure Upside Down

The fact that it took 20 years from the theoretical prediction of superconductivity in rare-earth nickelates to the experimental realization already suggests that the synthesis is far from trivial. This is because nickel has to be in the unusually low oxidation state Ni⁺¹. The recipe of success for nickelate superconductors is a two-step process [85]: First, doped perovskite films Sr (Ca)_xNd (La,Pr)_1−xNiO₃ films are deposited on a SrTiO₃ substrate by pulsed laser deposition. Already, this first step is far from trivial, not least because the doped material has to be deposited with homogenous Sr (Ca) concentration. Second, Sr(Ca)_xNd(La,Pr)_1−xNiO₃ needs to be reduced to Sr(Ca)_xNd(La,Pr)_1−xNiO₂. To this end, the reducing agent CaH₂ is used. Here, the problem is that this reduction might be incomplete with excess oxygen remaining or that hydrogen from CaH₂ is topotactically intercalated in the Sr (Ca)_xNd (La,Pr)_1−xNiO₂ structure. A particular difficulty is that the light hydrogen is experimentally hard to detect, for example, it evades conventional X-ray structural detection.

In [55], we studied the possibility to intercalate hydrogen, that is, to synthesize unintendedly Sr_xNd(La)_1−xNiO₂H instead of Sr_xNd(La)_1−xNiO₂. For the reduction of, for example, SrVO₃ with CaH_2, it is well-established that SrVO₂H may be obtained as the end product [86]. Both possible end products are visualized in Figure 7. The extra H takes away one more electron from the Ni sites. Hence, we have two holes on the Ni sites which in a local picture are distributed to two orbitals and form a spin-1, due to Hund’s exchange.

FIGURE 7

The first question is how susceptible the material is to bind topotactic H. To answer this question, one can calculate the binding energy E (ABNiO₂) + 1/2 E (H₂)- E (ABNiO₂H) in DFT [55, 87]. The result is shown in Figure 8, which clearly shows that early transition metal oxides are prone to intercalate hydrogen, whereas for cuprates, the infinite layer compound without H is more stable. Nickelates are in-between. For the undoped compounds NdNiO₂, and even a bit more for LaNiO₂, it is favorable to intercalate H. However, for the Sr-doped nickelates, the energy balance is inverted. Here, it is unfavorable to bind hydrogen.

FIGURE 8

Let us emphasize that this is only the enthalpy balance. In the actual synthesis also the reaction kinetics matter, and the entropy which is large for the H₂ gas. Nonetheless, this shows that undoped nickelates are very susceptible to topotactic H. This possibly means that, experimentally, not a complete H-coverage as in ABNiO₂H of Figure 8 is realized, but some hydrogen may remain in the nickelates because of an incomplete reduction with CaH₂. Indeed, hydrogen remainders have later been detected experimentally by nuclear magnetic resonance (NMR) spectroscopy, and they have even been used to analyze the antiferromagnetic spin fluctuations [80].

Now that we have established that remainders of hydrogen can be expected for nickelates at low doping, the question is how this affects the electronic structure. The local picture of Figure 7 already suggested a very different electronic configuration. This is further corroborated by DFT + DMFT calculations for LaNiO₂H presented in Figure 9. Here, the DFT band structure shows a metallic behavior with two orbitals, Ni 3 and 3, crossing the Fermi level. There are no rare-earth electron pockets any longer. Thus, we have an undoped Ni 3d⁸ configuration without Sr-doping. If electronic correlations are included in DMFT, the DFT bands split into two sets of Hubbard bands. Above the Fermi level, one can identify the upper 3 and 3 Hubbard band below the flat f bands in Figure 9, with some broadening because of the electronic correlations. The lower Hubbard bands intertwine with the Ni t_2g orbitals below the Fermi energy.

FIGURE 9

Even if we dope LaNiO₂H, this electronic structure is not particularly promising for superconductivity. First, it is not two-dimensional because of the 3 orbitals, which makes the system more three-dimensional. More specifically, there is a considerable hopping process from Ni 3via H to the Ni 3 on the vertically adjacent layer, as evidenced in Figure 9 by the DFT dispersion of this band in the Γ-Z direction; the other 3 band is (as expected) flat in this direction. Second, the tendency to form local magnetic moments of spin-1 counteracts the formation of Cooper pairs from two spin-1/2s. Hence, altogether, we expect topotactic H to prevent high-temperature superconductivity.

7 Conclusion

In this article, we have discussed the physics of nickelate superconductors from the perspective of a one-band Hubbard model for the Ni 3 band plus an A pocket. Because of symmetry, this A pocket does not hybridize with the 3 band and merely acts as a decoupled electron reservoir. Hence, once the filling of the 3 band is calculated as a function of Sr- or Ca-doping in Sr(Ca)_xNd(La,Pr)_1−xNiO₂, we can, for many aspects, concentrate on the physics of the thus doped Hubbard model. This includes antiferromagnetic spin fluctuations and the onset of superconductivity. Other physical properties, such as transport and the Hall conductivity, depend, as a matter of course, also on the A pocket. This is in stark contrast to the cuprates, where the oxygen p orbitals are much closer to the Fermi level so that we have a charge-transfer insulator that needs to be modeled by the more complex Emery model.

The one-band Hubbard model picture for nickelates was put forward early on for nickelates [11, 19, 32, 33], and its proper doping, including correlation effects, has been calculated in [33]. This picture has been confirmed by many experimental observations so far. The Nd 4f states are, from the theoretical perspective, irrelevant because they form a local spin and barely hybridize with the 3 band. This has been confirmed experimentally by the observation of superconductivity in Sr (Ca)_xLa_1−xNiO₂. The minor importance of the other Ni 3d orbitals, in particular the 3 orbital, is indicated through the careful analysis [65] of RIXS data [63, 64]. Not confirmed experimentally is hitherto the prediction that the Γ pocket is shifted above the Fermi level in the superconducting doping regime.

Strong evidence for the one-band Hubbard model picture is the prediction of the superconducting phase diagram [33], confirmed experimentally in [34, 35]. A further prediction was the increase of T_c with pressure or compressive strain [33], which was subsequently found in the experiment with a record T_c = 31 K for nickelates under pressure [83]. The strength of antiferromagnetic spin-fluctuations as obtained in RIXS [64] also roughly agrees with that of the calculation [33]. Altogether, this gives us quite some confidence in the one-band Hubbard model scenario, which even allowed for a rough calculation of T_c. Notwithstanding, further theoretical calculations, in particular including non-local correlations also in a realistic multi-orbital setting [37, 70, 88], are eligible. On the experimental side more detailed, for example, k-resolved information is desirable as are further close comparisons between the experiment and theory.

A good analysis of the quality of the samples is also mandatory, especially against the background that superconducting nickelates have been extremely difficult to synthesize. Incomplete oxygen reduction and topotactic hydrogen [55, 87] are theoretically expected to be present because this is energetically favored, at least for low Sr-doping. This leads to two holes in two orbitals forming a high-spin state and a three-dimensional electronic structure, thus obstructing the intrinsic physics of superconducting nickelates.

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