Abstract
The recently discovered nickelate superconductors appear, at first glance, to be even more complicated multiorbital systems than cuprates. To identify the simplest model describing the nickelates, we analyse the multiorbital system and find that it is instead the nickelates which can be described by a oneband Hubbard model, albeit with an additional electron reservoir and only around the superconducting regime. Our calculations of the critical temperature T_{C} are in good agreement with experiment, and show that optimal doping is slightly below 20% Srdoping. Even more promising than 3d nickelates are 4d palladates.
Introduction
Following the discovery of superconductivity in the cuprates^{1} and the seminal work by Anderson^{2}, the theoretical efforts to understand hightemperature superconductivity have been focusing to a large extent on a simple model: the oneband Hubbard model^{3,4,5}. However, superconducting cuprates need to be doped, and the doped holes go into the oxygen orbitals^{6,7,8}. This requires a more elaborate multiband model such as the threeorbital Emery model^{9}. If at all, the Hubbard model may mimic the physics of the ZhangRice singlet^{10} between the moment on the copper sites and the oxygen holes.
The recent discovery of superconductivity in Sr_{0.2}Nd_{0.8}NiO_{2} by Li et al.^{11} marked the beginning of a new, a nickel age of superconductivity, ensuing a plethora of experimental and theoretical work; see, among others, refs ^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. Similar as for the cuprates, the basic structural elements are NiO_{2} square lattice planes, and Ni has the same formal 3d^{9} electronic configuration.
But at second glance, there are noteworthy differences, see Fig. 1a. For the parent compound NdNiO_{2}, density functional theory (DFT) calculations show, besides the Ni \({d}_{{x}^{2}{y}^{2}}\) orbital, additional bands around the Fermi level E_{F}^{15,20,21,35} that are of predominant Nd5d character and overlap with the former. Note that the Nd5d bands in Fig. 1a extend below E_{F} despite their centre of gravity being considerably above E_{F}.
Such bands or electron pockets have the intriguing effect that even for the parent compound NdNiO_{2}, the Ni\({d}_{{x}^{2}{y}^{2}}\) orbital is hole doped, with the missing electrons in the Nd5d pockets. In other words, the parent compound NdNiO_{2} already behaves as the doped cuprates. First calculations^{16,19,27} for superconductivity in the nickelates that are valid at weak interaction strength hence started from a Fermi surface with both, the Nd5d pockets and the Ni\({d}_{{x}^{2}{y}^{2}}\) Fermi surface. Such a multiorbital nature of superconductivity has also been advocated in refs ^{18,29}.
There is additionally the Nd4f orbital which in DFT spuriously shows up just above the Fermi energy. But these 4f orbitals will be localised which can be mimicked by DFT + U or by putting them into the core, as has been done in Fig. 1a. Choi et al.^{32} suggest a ferromagnetic, i.e., antiKondo coupling of these Nd4f with the aforementioned Nd5d states. A further striking difference is that the oxygen band is much further away from the Fermi level than for the cuprates, see Fig. 1a. Vice versa, the other Ni3d orbitals are closer to the Fermi energy and slightly doped because of their hybridisation with the Nd5d orbitals.
This raises the following issues: which orbitals are depopulated upon Sr doping NdNiO_{2}; the validity of a singleorbital description in the superconducting doping range; the doping range in which Sr_{x}Nd_{1−x}NiO_{2} is actually superconducting; and the upper superconducting transition temperature T_{C}.
Addressing these points calls for a proper treatment of electronic correlations since e.g., the formation of Hubbard side bands has the potential to alter the orbitals that host the holes induced by Sr doping. In a materialspecific, ab initio way this can be done using DFT + dynamical meanfield theory (DMFT)^{36} and several groups have done such DFT + DMFT studies^{25,26,28,33,34}. Most of these focus on the undoped parent compound and arrive at a picture that the main players are the Ni \({d}_{{x}^{2}{y}^{2}}\) orbital plus A (and Γ) pocket.
However, if we further Srdope the system this picture must break down. If we have e.g., two holes in the Ni orbitals as for LaNiO_{2}H^{28}, it is clear that because of Hund’s rule coupling we need at least two Ni orbitals. Hence somewhere inbetween the undoped parent compound and the high Srdoping regime, there must be a crossover from a Ni\({d}_{{x}^{2}{y}^{2}}\)orbital picture plus hole pockets to a multiNiorbital picture. Whether this crossover happens before, within or after the superconducting Srdoping regime is one of the key questions for identifying the minimal model of superconductivity in nickelates.
In this paper, we show that, if we properly include electronic correlations by DMFT, up to a Srdoping x of about 30%, the holes only depopulate the Ni3\({d}_{{x}^{2}{y}^{2}}\) and Nd5d bands. Only for larger Srdopings, holes are doped into the other Ni3d orbitals, necessitating a multiorbital description. The hybridisation between the Ni3\({d}_{{x}^{2}{y}^{2}}\) and Nd5d orbitals as calculated from the DFTderived Wannier Hamiltonian is vanishing. We hence conclude that up to a Srdoping of around 30% marked as dark blue in Fig. 1b, a single Ni3\({d}_{{x}^{2}{y}^{2}}\) band description as in the oneband Hubbard model is possible. However, because of the Nd5d pocket(s), which acts like an electron reservoir and otherwise hardly interacts, only part of the Srdoping [lower xaxis of Fig. 1b] goes into the Ni3\({d}_{{x}^{2}{y}^{2}}\)band [upper xaxis], cf. Supplementary Note 2 for the functional dependence. We also take small Srdoping out of the blueshaded region since for such small dopings there is, besides the Nd5d Apocket, the Γpocket which interacts with the Nd4f moments ferromagentically^{32} and might result in additional correlation effects. At larger doping and when including the Ndinteraction in DMFT the Γpocket is shifted above E_{F}, see Fig. 2 below and e.g., refs ^{16,25,28,32}.
Figure 1b further shows the superconducting critical temperature T_{C} of the thus derived and doped Hubbard model, calculated by a method that is appropriate in the strong coupling regime: the dynamical vertex approximation (DΓA^{37}). The agreement between DΓA and experiment in Fig. 1b is reasonable given that the experimental T_{C} can be expected to be lower because of e.g., impurity scattering and the theoretical one is somewhat overestimated (The back coupling of the particle–particle channel to the particlehole and transversal particlehole channel is not included. While a (tiny) hopping t_{z} in the zdirection is necessary to overcome the MerminWagner theorem, somewhat larger t_{z}’s may suppress T_{C}^{38,39}).
Results and discussion
DDFT + DMFT multiorbital calculations
Let us now discuss these results in more detail. We start with a DFT calculation [cf. Supplementary Note 1] which puts the Nd4f orbitals just above E_{F}. But since their hybridisation with the Ni3\({d}_{{x}^{2}{y}^{2}}\) orbital is weak^{40}, \( {V}_{{x}^{2}{y}^{2},4f} =25\)meV, see Supplementary Note 2, they will localise and not make a Kondo effect. This localisation can be described e.g., by the spinsplitting in DFT+U, or by including the Nd4f states in the core. It leaves us with a well defined window with just five Nd5d and five Ni3d around the Fermi energy. For these remaining ten orbitals we do a Wannier function projection (see Supplementary Note 2) and subsequent DMFT calculation with constrained random phase approximation (cRPA) calculated interorbital interaction \(U^{\prime} =3.10\)eV (2.00 eV) and Hund’s exchange J = 0.65 eV (0.25 eV) for Ni (Nd)^{28}. Figure 2 presents the calculated DMFT spectral function for these ten bands. Let us first concentrate on Sr_{0.2}Nd_{0.8}NiO_{2} for which also the kresolved spectral function on the right hand side is shown. Clearly in DMFT there is a single, compared to the DFT strongly renormalized Ni3d band of \({d}_{{x}^{2}{y}^{2}}\) character crossing E_{F} = 0, see the zoom in Fig. 2e. Besides, there is also a pocket around the Apoint of predominately Nd5d_{xy} character, but the Γpocket is shifted above E_{F}, cf. Supplementary Notes 3 and 4 for other dopings.
Hence, we have two bands of predominately Ni3\({d}_{{x}^{2}{y}^{2}}\) and Nd5d_{xy} character. Their hybridisation is zero, see Supplementary Table III, which can be inferred already from the lack of any splitting around the DFT crossing points in Fig. 2d, e. There is some hybridisation of the Nd5d_{xy} with the other Nibands, which results in minor spectral weight of Nd5d_{xy} character in the region of the other Ni3d bands at −0.5 to −2.5 eV in Fig. 2a–c and viceversa of the Ni3\({d}_{{z}^{2}}\) orbital between 1 and 3 eV. This admixing is however so minor, that it can be described by properly admixed, effective orbitals that are away from the Fermi energy, without multiorbital physics.
If we study the doping dependence in Fig. 2a–c, we see that with Srdoping all bands move upwards. More involved and beyond a rigidband picture, also the Ni3\({d}_{{x}^{2}{y}^{2}}\) band becomes less and less correlated. The effective mass enhancement or inverse quasiparticle weight changes from m^{*}/m = 1/Z = 4.4 for the undoped compound to m^{*}/m = 2.8 at 20% Srdoping to m^{*}/m = 2.5 at 30% Srdoping. The Hubbard bands gradually disappear.
At 30% Srdoping we are in the situation that the other Ni3d bands are now immediately below the Fermi energy. Hence around this doping it is no longer justified to employ a Ni\({d}_{{x}^{2}{y}^{2}}\)band plus Ndd_{xy}pocket around A picture to describe the lowenergy physics. The other Ni3d orbitals become relevant. Because of the weak hybdization, the Ndd_{xy}pocket only acts as an electron reservoir, which changes the doping of the Ni\({d}_{{x}^{2}{y}^{2}}\)band from the lower xaxis in Fig. 1b to the upper xaxis.
For this \({d}_{{x}^{2}{y}^{2}}\)band we have done a separate Wannier function projection which results in the hopping parameters t = 395 meV, \(t^{\prime} =95\) meV, t″ = 47 meV between first, second, and thirdnearest neighbors on a square lattice. We employed the onsite interaction U = 3.2 eV = 8t which is slightly larger than the cRPA zero frequency value [U = 2.6 eV^{16,20}] to mimic the frequency dependence, as well as beyond cRPA contributions^{41}. In the Supplementary Note 5 we present further calculations within a realistic range of U values. The hopping parameters in the zdirection are negligibly small t_{z} = 34 meV, leaving us with a—to a good approximation—twodimensional oneband Hubbard model.
T _{C} in DΓA
This twodimensional oneband Hubbard model with the properly translated doping according to Fig. 1b can now be solved using more sophisticated, numerically expensive methods such as the DΓA^{37}. DΓA starts from a local irreducible vertex, which depends on three frequencies and is calculated at DMFT convergence. From this, DΓA builds nonlocal vertices and selfenergies through ladder or parquet diagrams. The types of diagrams are similar as in the random phase approximation (RPA) or fluctuation exchange (FLEX), but the important difference is the starting vertex which is nonperturbative. As a consequence all local DMFT correlations are included as well nonlocal spin, charge or superconducting fluctuations. This way, among others, (quantum) critical exponents, pseudogaps and superconductivity in the oneband Hubbard model can be calculated^{37}. Applying ladder DΓA to cuprates yields quite reasonable T_{C}’s^{42}, giving us some confidence to now explore the largely unknown nickelates. As a matter of course effects beyond the Hubbard model, such as disorder and phonons or a strengthening of charge fluctuations^{43} by nonlocal interactions which all are also considered to be of some relevance for superconductivity^{44} are not included. Studying a (tetragonal) rotational symmetry breaking^{45} would require a full parquet DΓA or an eigenvalue analysis like we do for dwave superconductivity here.
Figure 3 shows the thus obtained DΓA Fermi surfaces at different dopings and two different temperatures. For the superconducting Sr_{0.2}Nd_{0.8}NiO_{2}, which corresponds to \(n_{d_{x^{2}y^{2}}}=0.822\) electrons per site in the Ni\(d_{x^{2}y^{2}}\)(Wannier)band, we have a well defined holelike Fermi surface, whereas for \(n_{d_{x^2y^2}}=0.9\) and, in particular for \({n}_{{d}_{{x}^{2}{y}^{2}}}=0.95\), we see the development of Fermi arcs induced by strong antiferromagnetic spin fluctuations. The A = (π, π, π)pocket is not visible in Fig. 3 because it is only included through the effective doping in the DΓA calculation. In any case it would be absent in the k_{z} = 0 plane and only be visible around k_{z} = π.
Whereas the Ni3\({d}_{{x}^{2}{y}^{2}}\)band is strongly correlated and has a Fermi surface that is prone to highT_{C} superconductivity, the Apocket is weakly correlated and hardly hybridises with the former. Nonetheless for some physical quantities, different from superconductivity, it will play a role. For example, the Apocket will give an electronlike (negative) contribution to the Hall coefficient. Experimentally, the Hall coefficient^{11} is large and electronlike (negative) for NdNiO_{2}, whereas it is smaller and even changes its sign from negative (electronlike) to positive (holelike) below 50 K for Sr_{0.2}Nd_{0.8}NiO_{2}. This doping dependence at low temperatures is strikingly different to the cuprates and can be naturally explained from the balance between a 3\({d}_{{x}^{2}{y}^{2}}\) holecontribution (like for cuprate) and an additional electronlike Apocket contribution. In Fig. 3, we observed a pseudogap behaviour in the underdoped region. This should reduce the hole contribution somewhat as in underdoped cupates^{46}. We expect hence that the electronlike Apocket contribution wins in the low doping regime, but the 3\({d}_{{x}^{2}{y}^{2}}\) holelike contribution might well dominate for larger dopings, qualtitatively explaining the sign change of the Hall coefficient.
Moreover, in Fig. 3 we further see that for T = 92 K = 0.02t, we have a strong scattering at the antinodal point k = (π, 0), whereas at the lower temperature T = 46 K = 0.01t, we have a more well defined band throughout the Brillouin zone. This indicates that the 3\({d}_{{x}^{2}{y}^{2}}\) hole contribution is suppressed at higher temperatures, whereas the uncorrelated Apocket contribution essentially remains the same. This possibly explains why the Hall coefficient changes sign^{11} when increasing temperature for Sr_{0.2}Nd_{0.8}NiO_{2}.
At the temperatures of Fig. 3, Sr_{x}Nd_{1−x}NiO_{2} is not yet superconducting. But we can determine T_{C} from the divergence of the superconducting susceptibility χ, or alternatively the leading superconducting eigenvalue λ_{SC}. These are related through, in matrix notation, χ = χ_{0}/[1 − Γ_{pp}χ_{0}]. Here χ_{0} is the bare superconducting susceptibility and Γ_{pp} the irreducible vertex in the particle–particle channel calculated by DΓA; λ_{SC} is the leading eigenvalue of Γ_{pp}χ_{0}. If λ_{SC} approaches 1, the superconducting susceptibility is diverging.
In Fig. 4 we plot this λ_{SC} vs. temperature, and see that it approaches 1 at e.g., T = 36 K = 0.008t for \({n}_{{d}_{{x}^{2}{y}^{2}}}=0.85\). But outside a narrow doping regime between \({n}_{{d}_{{x}^{2}{y}^{2}}}=0.9\) and 0.8 it does not approach 1. There is no superconductivity. Let us also note that the phase transition is toward dwave superconductivity which can be inferred from the leading eigenvector corresponding to λ_{SC}.
Altogether, this leads to the superconducting dome of Fig. 1b. Most noteworthy Sr_{0.2}Nd_{0.8}NiO_{2} which was found to be superconducting in experiment^{11} is close to optimal doping \({n}_{{d}_{{x}^{2}{y}^{2}}}=0.85\) or Sr_{0.16}Nd_{0.84}NiO_{2}. Our results call for a more thorough investigation of superconductivity in nickelates around this doping, which is quite challenging experimentally^{12,13,14}.
Besides a slight increase of T_{C} by optimising the doping, and further room of improvement by adjusting \(t^{\prime}\) and t″, our results especially show that a larger bandwidth and a somewhat smaller interactiontobandwidth ratio may substantially enhance T_{C}, see Supplementary Fig. 7, qualitatively similar as in fluctuation exchange (FLEX) calculations^{16}. One way to achieve this is compressive strain, which enlarges the bandwidth while hardly affecting the interaction. Compressive strain can be realised by e.g., growing thin nickelate films on a LaAlO_{3} substrate with or without SrTiO_{3} capping layer, or by Cadoping instead of Srdoping for the bulk or thick films. Another route is to substitute 3d Ni by 4d elements, e.g., in Nd(La)PdO_{2} which has a similar Coulomb interaction and larger bandwidth^{17,47}.
Note added: In an independent DFT+DMFT study Leonov et al.^{48} also observe the shift of the Γpocket above E_{F} with doping for LaNiO_{2} and the occupation of further Ni 3d orbitals besides the 3\({d}_{{x}^{2}{y}^{2}}\) for large (e.g., 40%) doping.
Second note added: Given that our phase diagram Fig. 1b has been a prediction with only a single experimental data point given, the most recently experimentally determined phase diagram^{49,50} turned out to be in very good agreement. In particular if one considers, as noted already above, that the theoretical calculation should overestimate T_{C}, while the exerimentally observed T_{C} is likely suppressed by extrinsic contributions such as disorder etc. Supplementary Note 6 shows a comparison, which corroborates the modelling and theoretical understanding achieved in the present paper.
Third note added: In recent single particle tunnelling measurements^{51}, the authors observed the dominant dwave and the additional swave component of the superconducting gap function, which is apparently consistent with our scenario: the dominant dwave single orbital system with additional Fermi pockets.
Method
Density functional theory
We mainly employ the WIEN2K program package^{52} using the PBE version of the generalised gradient approximation (GGA), a 13 × 13 × 15 momentum grid, and R_{MT}K_{max }= 7.0 with a muffintin radius R_{MT} = 2.50, 1.95 and 1.68 a.u. for Nd, Ni and O, respectively. But we also double checked against VASP^{53}, which was also used for structural relaxation (a = b = 3.86 Å, c = 3.24 Å)^{40}, and FPLO^{54}, which was used for Fig. 1a and Supplementary Fig. 1. The Nd4f orbitals are treated as open core states if not stated otherwise.
Wannier function projection
The WIEN2K bandstructure around the Fermi energy is projected onto maximally localised Wannier functions^{55} using WIEN2WANNIER^{56}. For the DMFT we employ a projection onto five Ni3d and five Nd5d bands; for the parameterisation of the oneband Hubbard model we project onto the Ni3\({d}_{{x}^{2}{y}^{2}}\) orbital. For calculating the hybridisation with the Nd4f, we further Wannier projected onto 17 bands, with the Nd4f states now treated as valence bands in GGA.
Dynamical meanfield theory
We supplement the five Ni3d plus five Nd5d orbital Wannier Hamiltonian by a cRPA calculated Coulomb repulsions^{28} \(U^{\prime} =3.10\)eV (2.00 eV) and Hund’s exchange J = 0.65 eV (0.25 eV) for Ni (Nd or La). The resulting Kanamori Hamiltonian is solved in DMFT^{36} at room temperature (300 K) using continuoustime quantum Monte Carlo simulations in the hybridisation expansion^{57} implemented in W2DYNAMICS^{58}. The maximum entropy method^{59} is employed for the analytic continuation of the spectra.
Dynamical vertex approximation
For calculating the superconducting T_{C}, we employ the dynamical vertex approximation (DΓA), for a review see ref. ^{37}. We first calculate the particle–particle vertex with spinfluctuations in the particlehole and transversalparticle hole channel, and then the leading eigenvalues in the particle–particle channel, as done before in ref. ^{42}. This is like the first iteration for the particle–particle channel in a more complete parquet DΓA^{37}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes used are DFT/Wien2k (http://susi.theochem.tuwien.ac.at), DMFT/w2dynamics (https://github.com/w2dynamics) and ladder DΓA (https://github.com/ladderDGA). Additional scripts and modifications of the last are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank A. Hariki, J. Kaufmann, F. Lechermann and J. M. Tomczak for helpful discussions; and U. Nitzsche for technical assistance. M.K. is supported by the RIKEN Special Postdoctoral Researchers Program; L.S. and K.H. by the Austrian Science Fund (FWF) through projects P 30997 and P 32044M; L.S. by the China Postdoctoral Science Foundation (Grant No. 2019M662122); L.S. and Z.Z. by the National Key R&D Program of China (2017YFA0303602), 3315 Program of Ningbo, and the National Nature Science Foundation of China (11774360, 11904373); O.J. by the Leibniz Association through the Leibniz Competition. R.A. by GrantinAid for Scientific Research (No. 16H06345, 19H05825) from MEXT, Japan. Calculations have been done on the Vienna Scientific Clusters (VSC).
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All authors contributed with concepts, ideas and discussions. M.K. performed the DΓA calculations, L.S. the DFT and DMFT calculations, except for the FPLO calculations done by O.J. Coordination and writing was mainly done by K.H. with contributions from all authors.
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Kitatani, M., Si, L., Janson, O. et al. Nickelate superconductors—a renaissance of the oneband Hubbard model. npj Quantum Mater. 5, 59 (2020). https://doi.org/10.1038/s4153502000260y
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DOI: https://doi.org/10.1038/s4153502000260y
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