Abstract
The discovery of unconventional superconductivity in hole doped NdNiO_{2}, similar to CaCuO_{2}, has received enormous attention. However, different from CaCuO_{2}, RNiO_{2} (R = Nd, La) has itinerant electrons in the rareearth spacer layer. Previous studies show that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) and rareearthd orbitals is very weak and thus RNiO_{2} is still a promising analog of CaCuO_{2}. Here, we perform firstprinciples calculations to show that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons in RNiO_{2} is substantially stronger than previously thought. The dominant hybridization comes from an interstitials orbital rather than rareearthd orbitals, due to a large intercell hopping. Because of the hybridization, Ni local moment is screened by itinerant electrons and the critical U_{Ni} for longrange magnetic ordering is increased. Our work shows that the electronic structure of RNiO_{2} is distinct from CaCuO_{2}, implying that the observed superconductivity in infinitelayer nickelates does not emerge from a doped Mott insulator.
Introduction
Since the discovery of hightemperature superconductivity in cuprates^{1}, people have been attempting to search for superconductivity in other materials whose crystal and electronic structures are similar to those of cuprates^{2,3}. One of the obvious candidates is La_{2}NiO_{4} which is isostructural to La_{2}CuO_{4} and Ni is the nearest neighbor of Cu in the periodic table. However, superconductivity has not been observed in doped La_{2}NiO_{4}^{4}. This is in part due to the fact that in La_{2}NiO_{4}, two Nie_{g} orbitals are active at the Fermi level, while in La_{2}CuO_{4} only Cu\({d}_{{x}^{2}{y}^{2}}\) appears at the Fermi level. Based on this argument, a series of nickelates and nickelate heterostructures have been proposed with the aim of realizing a single orbital Fermi surface in nickelates. Those attempts started from infinitelayer nickelates^{2,5,6}, to LaNiO_{3}/LaAlO_{3} superlattices^{7,8,9,10}, to tricomponent nickelate heterostructures^{11,12} and to reduced Ruddlesden–Popper series^{13,14}. Eventually, superconductivity with a transition temperature of about 15 K has recently been discovered in hole doped infinitelayer nickelate NdNiO_{2}^{15}, injecting new vitality into the field of highT_{c} superconductivity^{16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33}.
However, there is an important difference between infinitelayer nickelate RNiO_{2} (R = Nd, La) and infinitelayer cuprate CaCuO_{2} in their electronic structures: in infinitelayer cuprates, only a single Cu\({d}_{{x}^{2}{y}^{2}}\) band crosses the Fermi level, while in infinitelayer nickelates, in addition to Ni\({d}_{{x}^{2}{y}^{2}}\) band, another conduction band also crosses the Fermi level^{6,21,22,23}. Firstprinciples calculations show that the other nonNi conduction band originates from rareearth spacer layers^{6,21,22,23}. Hepting et al.^{20} propose that itinerant electrons on rareearthd orbitals may hybridize with Ni\({d}_{{x}^{2}{y}^{2}}\) orbital, rendering RNiO_{2} an “oxideintermetallic” compound. But previous studies find that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) and rareearthd orbitals is very weak^{21,22,23,29}. Therefore other than the selfdoping effect^{27}, infinitelayer nickelates can still be considered as a promising analog of infinitelayer cuprates^{21,16}.
In this work, we combine density functional theory (DFT)^{34,35} and dynamical meanfield theory (DMFT)^{36,37} to show that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons in rareearth spacer layers is substantially stronger than previously thought. However, the largest source of hybridization comes from an interstitials orbital due to a large intercell hopping. The hybridization with rareearthd orbitals is weak, about one order of magnitude smaller. We also find that weaktomoderate correlation effects on Ni lead to a charge transfer from Ni\({d}_{{x}^{2}{y}^{2}}\) orbital to hybridization states, which provides more itinerant electrons to couple to Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. In the experimentally observed paramagnetic metallic state of RNiO_{2}, we explicitly demonstrate that the coupling between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons screens the Ni local moment, as in Kondo systems^{38,39,40}. Finally we find that the hybridization increases the critical U_{Ni} that is needed to induce longrange magnetic ordering.
Our work provides the microscopic origin of a substantial hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons in RNiO_{2}, which leads to an electronic structure that is distinct from that of CaCuO_{2}. As a consequence of the hybridization, spins on Ni\({d}_{{x}^{2}{y}^{2}}\) orbital are affected by itinerant electrons and the physical property of RNiO_{2} is changed. This implies that the observed superconductivity in infinitelayer nickelates does not emerge from a doped Mott insulator as in cuprates.
The computational details of our DFT and DMFT calculations can be found in the Method section. For clarity, we study NdNiO_{2} as a representative of infinitelayer nickelates. The results of LaNiO_{2} are very similar (see Supplementary Note 1 and Note 2 in the Supplementary Information).
Results
Electronic structure and interstitials orbital
In Fig. 1a, b, we show the DFTcalculated band structure and Wannier function fitting of NdNiO_{2} and CaCuO_{2} in the nonspinpolarized state, respectively. We use altogether 17 Wannier projectors to fit the DFT band structure: 5 Ni/Cud orbitals, 5 Nd/Cad orbitals, 3 Op orbitals (for each O atom), and an interstitials orbital. The interstitials orbital is located at the position of the missing apical oxygen. The importance of interstitials orbitals has been noticed in the study of electrides and infinitelayer nickelates^{22,41,42}. Our Wannier fitting exactly reproduces not only the band structure of the entire transitionmetal and oxygen pd manifold, but also the band structure of unoccupied states about 5 eV above the Fermi level. In particular, the Ni/Cu\({d}_{{x}^{2}{y}^{2}}\) Wannier projector is highlighted by red dots in Fig. 1a, b. The details of the Wannier fitting can be found in Supplementary Note 3 in the Supplementary Information. For both compounds, Ni/Cu\({d}_{{x}^{2}{y}^{2}}\) band crosses the Fermi level. However, as we mentioned in the Introduction, in addition to Ni\({d}_{{x}^{2}{y}^{2}}\) band, another conduction band also crosses the Fermi level in NdNiO_{2}. Using Wannier analysis, we find that the nonNi conduction electron band is mainly composed of three orbitals: Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy}, and interstitials orbitals. The corresponding Wannier projectors are highlighted by dots in the panels of Fig. 1c–e. An isovalue surface of the three Wannier functions (Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy}, and interstitials orbitals) is explicitly shown in Fig. 1f–h. We note that interstitials orbital is more delocalized than Nd\({d}_{3{z}^{2}{r}^{2}}\) and Ndd_{xy} orbitals. Because all these three orbitals are located in the Nd spacer layer between adjacent NiO_{2} planes, if these three orbitals can hybridize with Ni\({d}_{{x}^{2}{y}^{2}}\) orbital, then they will create a threedimensional electronic structure, distinct from that of CaCuO_{2}^{20}.
Analysis of hybridization
However, from symmetry consideration, within the same cell the hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and any of those three orbitals (Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy}, and interstitials) is exactly equal to zero^{22}, which leads to the conclusion that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) and rareearthd orbitals is weak^{20,22,29}. While this conclusion is correct by itself, the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) and interstitials orbital has been omitted in previous studies^{20,21,22,23,27,29}. We find that due to a large intercell hopping, Ni\({d}_{{x}^{2}{y}^{2}}\) orbital hybridizes with interstitials orbital much more substantially than with rareearthd orbitals by about one order of magnitude.
The direct intercell hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) and any of the three orbitals (Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy} and interstitials) is negligibly small. The effective hopping is via Op orbitals. Figure 2 shows the intercell hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and the other three orbitals via one Op orbital. Among Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy} and interstitials orbitals, we find that the largest effective hopping (via one Op orbital) is the one with interstitials orbital (see Table 1). The effective hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) and Ndd_{xy}/\({d}_{3{z}^{2}{r}^{2}}\) orbitals is one order of magnitude smaller because Nd atom is located at the corner of the cell, which is further from the O atom than the interstitial site is. Furthermore, the energy difference between interstitials and Op orbitals is about 1 eV smaller than that between Ndd_{xy}/\({d}_{3{z}^{2}{r}^{2}}\) and Op orbitals (see Table 1). These two factors combined lead to the fact that Ni\({d}_{{x}^{2}{y}^{2}}\) has a significant coupling with interstitials orbital, substantially stronger than that with Ndd orbitals. This challenges the previous picture that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons in the Nd spacer layer is weak^{20,21,22,23,27,29}.
To further confirm that the hybridization is substantial, we downfold the full band structure to a noninteracting model that is based on the above four orbitals (Ni\({d}_{{x}^{2}{y}^{2}}\), Nd\({d}_{3{z}^{2}{r}^{2}}\), Ndd_{xy}, and interstitials orbitals). Equation (1) shows the Wannierbased Hamiltonian 〈0∣H_{0}∣a_{1}〉 = H_{0}(a_{1}) in the matrix form (not the usual Hamiltonian 〈0∣H_{0}∣0〉 = H_{0}(0), H_{0}(0) is shown in Supplementary Note 3 in the Supplementary Information). The important information is in the first row. The largest hopping is the one between neighboring Ni\({d}_{{x}^{2}{y}^{2}}\) orbitals (this is due to the σ bond between Ni\({d}_{{x}^{2}{y}^{2}}\) and Op_{x}/p_{y} orbitals). However, the hopping between Ni\({d}_{{x}^{2}y2}\) and interstitials orbitals is even comparable to the largest hopping. By contrast, the hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) and Ndd_{xy}/\({d}_{3{z}^{2}{r}^{2}}\) orbitals is about one order of magnitude smaller than the hopping between Ni\({d}_{{x}^{2}{y}^{2}}\) and interstitials orbitals, which is consistent with the preceding analysis.
Charge transfer and screening of Ni local moment
Since infinitelayer nickelates are correlated materials, next we study correlation effects arising from Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. We focus on whether the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons in the rareearth spacer layer may affect the correlated properties of NdNiO_{2}, such as magnetism.
We use the above four orbitals (see Eq. (1)) to build an interacting model:
where \(mm^{\prime}\) labels different orbitals, i labels Ni sites and σ labels spins. \({\hat{n}}_{i\sigma }\) is the occupancy operator of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital at site i with spin σ and the onsite Coulomb repulsion is only applied on the Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. H_{0}(k) is the Fourier transform of the Wannierbased Hamiltonian H_{0}(R)^{9} and \({\hat{V}}_{{\mathrm{dc}}}\) is the double counting potential. That we do not explicitly include Op states in the model is justified by noting that in NdNiO_{2} Op states have much lower energy than Nid states, which is different from perovskite rareearth nickelates and chargetransfertype cuprates^{20,19}. In the model Eq. (2), the Ni\({d}_{{x}^{2}{y}^{2}}\) orbital is the correlated state while the other three orbitals (interstitials and Nd\({d}_{3{z}^{2}{r}^{2}}\)/d_{xy}) are noninteracting, referred to as hybridization states.
We perform dynamical meanfield theory calculations on Eq. (2). We first study paramagnetic state (paramagnetism is imposed in the calculations). Figure 3a–c shows the spectral function with an increasing U_{Ni} on Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. At U_{Ni} = 0 eV, the system is metallic with all the four orbitals crossing the Fermi level (the main contribution comes from Ni\({d}_{{x}^{2}{y}^{2}}\)). As U_{Ni} increases to 3 eV, a quasiparticle peak is evident with the other three orbitals still crossing the Fermi level. We find a critical U_{Ni} of about 7 eV, where the quasiparticle peak becomes completely suppressed and a Mott gap emerges. As U_{Ni} further increases to 9 eV (not shown in Fig. 3), a clear Mott gap of about 1 eV is opened.
The presence of hybridization states means that there could be charge transfer between correlated Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and interstitials/Ndd orbitals. We calculate the occupancy of each Wannier function N_{α} and study correlationdriven charge transfer in NdNiO_{2}. Figure 3d shows N_{α} of each hybridization state and Ni\({d}_{{x}^{2}{y}^{2}}\) orbital as well as the total occupancy of hybridization states as a function of U_{Ni}. We first note that at U_{Ni} = 0, the total occupancy of hybridization states is 0.14, which is significant. As U_{Ni} becomes larger, the total occupancy of hybridization states first increases and then decreases. This is because when U_{Ni} is small, the system is still metallic with all the hybridization states crossing the Fermi level, while the upper Hubbard band of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital is just formed and pushed to higher energy. This leads to charge transfer from Ni\({d}_{{x}^{2}{y}^{2}}\) orbital to hybridization states, providing more itinerant electrons to couple to Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. However, when U_{Ni} is large, hybridization states are also pushed above the Fermi level, which causes electron to transfer back to Ni\({d}_{{x}^{2}{y}^{2}}\) orbital (in the lower Hubbard band). In the strong U_{Ni} limit where the Mott gap opens, itinerant electrons in the Nd spacer layer disappear. Figure 3d also shows that for all U_{Ni} considered, the occupancy on interstitials orbital is always the largest among the three hybridization states, confirming the importance of the interstitials orbital in infinitelayer nickelates. We note that because we calculate the occupancy at finite temperatures, even when the gap is opened, the occupancy of hybridization states does not exactly become zero.
Because of the hybridization, we study possible screening of Ni local magnetic moment by itinerant electrons. We calculate local spin susceptibility of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital:
where S_{z}(τ) is the local spin operator for Ni\({d}_{{x}^{2}{y}^{2}}\) orbital, at the imaginary time τ. g denotes the electron spin gyromagnetic factor and β = 1/(k_{B}T) is the inverse temperature. Figure 3e shows \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) for two representative values of U_{Ni}. The blue symbols are \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) for U_{Ni} = 7 eV when the system becomes insulating. The local spin susceptibility nicely fits to a Curie–Weiss behavior, as is shown by the black dashed line in Fig. 3e. \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) has a strong enhancement at low temperatures. However, for U_{Ni} = 2 eV when the system is metallic, we find a completely different \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\). The local spin susceptibility has very weak dependence on temperatures (see Fig. 3f for the zoomin). In particular, at low temperatures (T < 250 K), \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) reaches a plateau. We note that the weak temperature dependence of \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) is consistent with the experimental result of LaNiO_{2} paramagnetic susceptibility^{5}, in particular our simple model calculations qualitatively reproduce the lowtemperature plateau feature that is observed in experiment^{5}.
To explicitly understand how the hybridization between itinerant electrons and Ni\({d}_{{x}^{2}{y}^{2}}\) orbital affects local spin susceptibility, we perform a thoughtexperiment: we manually “turn off” hybridization, i.e., for each R, we set \(\langle s {H}_{0}({\bf{R}}) {d}_{{x}^{2}{y}^{2}}\rangle =\langle {d}_{xy} {H}_{0}({\bf{R}}) {d}_{{x}^{2}{y}^{2}}\rangle =\langle {d}_{3{z}^{2}{r}^{2}} {H}_{0}({\bf{R}}) {d}_{{x}^{2}{y}^{2}}\rangle =0\). Then we recalculate \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) using the modified Hamiltonian with U_{Ni} = 2 eV. The chemical potential is adjusted so that the total occupancy remains unchanged in the modified Hamiltonian. The two local spin susceptibilities are compared in Fig. 3f. With hybridization, \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) saturates at low temperatures, implying that μ_{eff} decreases or even vanishes with lowering temperatures. However, without hybridization, \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) shows an evident enhancement at low temperatures and a Curie–Weiss behavior is restored (black dashed line). This shows that in paramagnetic metallic NdNiO_{2}, the hybridization between itinerant electrons and Ni\({d}_{{x}^{2}{y}^{2}}\) orbital is substantial and as a consequence, it screens the Ni local magnetic moment, as in Kondo systems^{38,39,40}. Such a screening mechanism may be used to explain the lowtemperature upturn in the resistivity of NdNiO_{2} observed in experiment^{27,15}. We note that while we only fix the total occupancy by adjusting the chemical potential, the occupancy of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital is almost the same in the original and modified models. In Fig. 3f, “with hybridization”, Ni\({d}_{{x}^{2}{y}^{2}}\) occupancy is 0.84 and “without hybridization”, Ni\({d}_{{x}^{2}{y}^{2}}\) occupancy is 0.83. This indicates that the screening of Ni moment is mainly due to the hybridization effects, while the change of Ni\({d}_{{x}^{2}{y}^{2}}\) occupancy (0.01e per Ni) plays a secondary role.
Correlation strength and phase diagram
We estimate the correlation strength for NdNiO_{2} by calculating its phase diagram. We allow spin polarization in the DMFT calculations and study both ferromagnetic and checkerboard antiferromagnetic states. We find that ferromagnetic ordering cannot be stabilized up to U_{Ni} = 9 eV. Checkerboard antiferromagnetic state can emerge when U_{Ni} exceeds 2.5 eV. The phase diagram is shown in Fig. 4a in which M_{d} is the local magnetic moment on each Ni atom. M_{d} is zero until U_{Ni} ≃ 2.5 eV and then increases with U_{Ni} and finally saturates to 1 μ_{B}/Ni which corresponds to a \(S=\frac{1}{2}\) state. We note that the critical value of U_{Ni} is modeldependent. If we include Op states and semicore states, the critical value of U_{Ni} will be substantially larger^{43}. The robust result here is that with U_{Ni} increasing, antiferromagnetic ordering occurs before the metalinsulator transition. In the antiferromagnetic state, the critical U_{Ni} for the metalinsulator transition is about 6 eV, slightly smaller than that in the paramagnetic phase. The spectral function of antiferromagnetic metallic and insulating states is shown in Fig. 4b and c, respectively. Experimentally longrange magnetic orderings are not observed in NdNiO_{2}^{44}. The calculated phase diagram means that NdNiO_{2} can only be in a paramagnetic metallic state (instead of a paramagnetic insulating state), in which the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) and itinerant electrons screens the Ni local magnetic moment. We note that using our model Eq. (2), the calculated phase boundary indicates that Ni correlation strength is moderate in NdNiO_{2} with U_{Ni}/t_{dd} < 7 (t_{dd} is the effective hopping between the nearestneighbor Ni\({d}_{{x}^{2}{y}^{2}}\) due to the σ_{pd} bond). This contrasts with the parent compounds of superconducting cuprates which are antiferromagnetic insulators and are described by an effective singleorbital Hubbard model with a larger correlation strength (U_{Ni}/t_{dd} = 8–20)^{45,46,47,48}. Finally, we perform a selfconsistent check on the hybridization. When the system is metallic, the hybridization between itinerant electrons and Ni\({d}_{{x}^{2}{y}^{2}}\) orbital screens the spin on Ni site and reduces the local spin susceptibility \({\chi }_{{\rm{loc}}}^{\omega = 0}(T)\) in the paramagnetic phase. This implies that once we allow antiferromagnetic ordering, a smaller critical U_{Ni} may be needed to induce magnetism. To test that, we recalculate the phase diagram using the modified Hamiltonian with the hybridization manually “turned off”. The chemical potential is adjusted in the modified model so that the total occupancy remains unchanged. Figure 4d shows that without the hybridization, the Ni magnetic moment increases and the antiferromagnetic phase is expanded with the critical U_{Ni} reduced to 1.8 eV (U_{Ni}/t_{dd} ≃ 5). This shows that the coupling to the conducting electrons affects Ni spins and changes the magnetic property of NdNiO_{2}^{40}.
Discussion
Our minimal model Eq. (2) is different from the standard Hubbard model (singleorbital, twodimensional square lattice, and half filling) due to the presence of hybridization. It is also different from a standard periodic Anderson model in that (1) the correlated orbital is a 3dorbital with a strong dispersion instead of a 4f or 5f orbital whose dispersion is usually neglected^{20,49,50}; (2) the hybridization of Ni\({d}_{{x}^{2}{y}^{2}}\) with the three noninteracting orbitals is all intercell rather than onsite and anisotropic with different types of symmetries, which may influence the symmetry of the superconducting order parameter in the ground state^{51}. Figure 5 explicitly shows the symmetry of hybridization. The dominant hybridization of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital, the one with interstitials orbital, has \({d}_{{x}^{2}{y}^{2}}\) symmetry. Second, the hybridization of Ni\({d}_{{x}^{2}{y}^{2}}\) with Ndd_{xy} and Nd\({d}_{3{z}^{2}{r}^{2}}\) orbitals has \({g}_{xy({x}^{2}{y}^{2})}\) and \({d}_{{x}^{2}{y}^{2}}\) symmetries, respectively^{52}.
dwave superconducting states can be stabilized in the doped singleorbital Hubbard model from sophisticated manybody calculations^{53,54,55,56}. However, the hybridization between correlated Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons fundamentally changes the electronic structure of a singleorbital Hubbard model, in particular when the system is metallic. This probably creates a condition unfavorable for superconductivity^{51}, implying that new mechanisms such as interface charge transfer, strain engineering, etc. are needed to fully explain the phenomena observed in infinitelayer nickelates^{15}.
Before we conclude, we briefly discuss other models for RNiO_{2} (R = La, Nd). In literature, some models focus on lowenergy physics and include only states that are close to the Fermi level; others include more states which reproduce the electronic band structure within a large energy window around the Fermi level. Kitatani et al.^{57} propose that RNiO_{2} can be described by the oneband Hubbard model (Ni\({d}_{{x}^{2}{y}^{2}}\) orbital) with an additional electron reservoir, which is used to directly estimate the superconducting transition temperature. Hepting et al.^{20} construct a twoorbital model using Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and a R\({d}_{3{z}^{2}{r}^{2}}\)like orbital. Such a model is used to study hybridization effects between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and rareearth Rd orbitals. Zhang et al.^{28}, Werner et al.^{32}, and Hu et al.^{33} study a different type of twoorbital models which consist of two Nid orbitals. Hu et al.^{33} include Ni\({d}_{{x}^{2}{y}^{2}}\) and Nid_{xy} orbitals, while Zhang et al. and Werner et al.^{32,28} include Ni\({d}_{{x}^{2}{y}^{2}}\) and Ni\({d}_{3{z}^{2}{r}^{2}}\) orbitals. This type of twoorbital model aims to study the possibility of highspin S = 1 doublon when the system is hole doped. Wu et al.^{21} and Nomura et al.^{22} study threeorbital models. Wu et al.^{21} include Nid_{xy}, Rd_{xy}, and R\({d}_{3{z}^{2}{r}^{2}}\) orbitals. This model is further used to calculate the spin susceptibility and to estimate the superconducting transition temperature. Nomura et al.^{22} compare two choices of orbitals: one is Nid_{xy} orbital, R\({d}_{3{z}^{2}{r}^{2}}\) orbital, and interstitials; and the other is Nid_{xy}orbital, R\({d}_{3{z}^{2}{r}^{2}}\) orbital, and Rd_{xy}. The model is used to study the screening effects on the Hubbard U of Ni\({d}_{{x}^{2}{y}^{2}}\) orbital. Gao et al.^{23} construct a general fourorbital model B_{1g}@1a⨁A_{1g}@1b which consists of two Nid orbitals and two Rd orbitals. The model is used to study the topological property of the Fermi surface. Jiang et al.^{29} use a tightbinding model that consists of five Nid orbitals and five Rd orbitals to comprehensively study the hybridization effects between Nid and Rd orbitals; Jiang et al. also highlight the importance of Ndf orbitals in the electronic structure of NdNiO_{2}. Botana et al.^{16}, Lechermann^{26}, and Karp et al.^{58} consider more orbitals (including Ndd, Nid, and Op states) in the modeling of NdNiO_{2} with the interaction applied to Nid orbitals and make a comparison to infinitelayer cuprates. Botana et al.^{16} extract longerrange hopping parameters and the e_{g} energy splitting. Lechermann^{26} studies hybridization and doping effects. Karp et al.^{58} calculate the phase diagram and estimates the magnetic transition temperature.
Conclusion
In summary, we use firstprinciples calculations to study the electronic structure of the parent superconducting material RNiO_{2} (R = Nd, La). We find that the hybridization between Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and itinerant electrons is substantially stronger than previously thought. The dominant hybridization comes from an interstitials orbital due to a large intercell hopping, while the hybridization with rareearthd orbitals is one order of magnitude weaker. Weaktomoderate correlation effects on Ni cause electrons to transfer from Ni\({d}_{{x}^{2}{y}^{2}}\) orbital to the hybridization states, which provides more itinerant electrons in the rareearth spacer layer to couple to correlated Nid orbital. Further increasing correlation strength leads to a reverse charge transfer, antiferromagnetism on Ni sites, and eventually a metalinsulator transition. In the experimentally observed paramagnetic metallic state of RNiO_{2}, we find that the strong coupling between Ni\({d}_{{x}^{2}{y}^{2}}\) and itinerant electrons screens the Ni local moment, as in Kondo systems. We also find that the hybridization increases the critical U_{Ni} that is needed to induce longrange magnetic ordering. Our work shows that the electronic structure of RNiO_{2} is fundamentally different from that of CaCuO_{2}, which implies that the observed superconductivity in infinitelayer nickelates does not emerge from a doped Mott insulator as in cuprates.
Methods
We perform firstprinciples calculations using density functional theory (DFT)^{34,35}, maximally localized Wannier functions (MLWF) to construct the noninteracting tightbinding models^{59}, and dynamical mean field theory (DMFT)^{36,37} to solve the interacting models.
DFT calculations
The DFT method is implemented in the Vienna ab initio simulation package (VASP) code^{60} with the projector augmented wave (PAW) method^{61}. The Perdew–Burke–Ernzerhof (PBE)^{62} functional is used as the exchangecorrelation functional in DFT calculations. The Nd4f orbitals are treated as core states in the pseudopotential. We use an energy cutoff of 600 eV and sample the Brillouin zone by using Γcentered kmesh of 16 × 16 × 16. The crystal structure is fully relaxed with an energy convergence criterion of 10^{−6} eV, force convergence criterion of 0.01 eV/Å, and strain convergence of 0.1 kBar. The DFToptimized crystal structures are in excellent agreement with the experimental structures, as shown in our Supplementary Note 1. To describe the checkerboard antiferromagnetic ordering, we expand the cell to a \(\sqrt{2}\times \sqrt{2}\times 1\) supercell. The corresponding Brillouin zone is sampled by using a Γcentered kmesh of 12 × 12 × 16.
MLWF calculations
We use maximally localized Wannier functions^{59}, as implemented in Wannier90 code^{63} to fit the DFTcalculated band structure and build an ab initio tightbinding model which includes onsite energies and hopping parameters for each Wannier function. We use two sets of Wannier functions to do the fitting. One set uses 17 Wannier functions to exactly reproduce the band structure of entire transitionmetal and oxygen pd manifold as well as the unoccupied states that are a few eV above the Fermi level. The other set uses 4 Wannier functions to reproduce the band structure close to the Fermi level. The second tightbinding Hamiltonian is used to study correlation effects when onsite interactions are included on Ni\({d}_{{x}^{2}{y}^{2}}\) orbital.
DMFT calculations
We use DMFT method to calculate the 4orbital interacting model, which includes a correlated Ni\({d}_{{x}^{2}{y}^{2}}\) orbital and three noninteracting orbitals (interstitials, Ndd_{xy}, and Nd\({d}_{3{z}^{2}{r}^{2}}\)). We also crosscheck the results using a 17orbital interacting model which includes five Nid, five Ndd, six Op, and one interstitials orbital (the results of the 17orbital model are shown in Supplementary Note 4 of the Supplementary Information). DMFT maps the interacting lattice Hamiltonian onto an auxiliary impurity problem which is solved using the continuoustime quantum Monte Carlo algorithm based on hybridization expansion^{64,65}. The impurity solver is developed by K. Haule^{66}. For each DMFT iteration, a total of 1 billion Monte Carlo samples are collected to converge the impurity Green function and selfenergy. We set the temperature to be 116 K. We check all the key results at a lower temperature of 58 K and no significant difference is found. The interaction strength U_{Ni} is treated as a parameter. We calculate both paramagnetic and magnetically ordered states. For magnetically ordered states, we consider ferromagnetic ordering and checkerboard antiferromagnetic ordering. For checkerboard antiferromagnetic ordering calculation, we double the cell, and the noninteracting Hamiltonian is 8 × 8. We introduce formally two effective impurity models and use the symmetry that electrons at one impurity site are equivalent to the electrons on the other with opposite spins. The DMFT selfconsistent condition involves the selfenergies of both spins.
To obtain the spectral functions, the imaginary axis selfenergy is continued to the real axis using the maximum entropy method^{67}. Then the real axis local Green function is calculated using the Dyson equation, and the spectral function is obtained in the following equation:
where m is the label of a Wannier function. 1 is an identity matrix, H_{0}(k) is the Fourier transform of the Wannierbased Hamiltonian H_{0}(R). Σ(ω) is the selfenergy, understood as a diagonal matrix only with nonzero entries on the correlated orbitals. μ is the chemical potential. V_{dc} is the fully localized limit (FLL) double counting potential, which is defined as^{68}:
where N_{d} is the d occupancy of a correlated site. Here the Hund’s J term vanishes because we have a single correlated orbital Ni\({d}_{{x}^{2}{y}^{2}}\) in the model. A 40 × 40 × 40 kpoint mesh is used to converge the spectral function. We note that double counting correction affects the energy separation between Ni\({d}_{{x}^{2}{y}^{2}}\) and Ndd/interstitials orbitals. However, because the charge transfer is small (around 0.1e per Ni), the effects from the double counting correction are weak in the 4orbital model, compared with those in the p–d model in which double counting correction becomes much more important^{69}. That is because Op states are included in the p–d model. The double counting correction affects the p–d energy separation and thus the charge transfer between metald and oxygenp orbitals, which can be as large as 1e per metal atom for late transitionmetal oxides such as rareearth nickelates^{69}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The electronic structure calculations were performed using the proprietary code VASP^{60}, the opensource code Wannier90^{63}, and the opensource impurity solver implemented by Kristjan Haule at Rutgers University (http://hauleweb.rutgers.edu/tutorials/). Both Wannier90 and Haule’s impurity solver are freely distributed on academic use under the Massachusetts Institute of Technology (MIT) License.
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Acknowledgements
We thank Danfeng Li, JeanMarc Triscone, Andrew Millis, Dong Luan and Qiang Zhang for useful discussions. H.C. is supported by the National Natural Science Foundation of China under project number 11774236 and NYU University Research Challenge Fund. Y.G. and J.H. are supported by the Ministry of Science and Technology of China 973 program (Grant No. 2015CB921300, No. 2017YFA0303100, and No. 2017YFA0302900), National Science Foundation of China (Grant No. NSFC11334012), the Strategic Priority Research Program of CAS (Grant No. XDB07000000), and Highperformance Computing Platform of Peking University. Computational resources are provided by Highperformance Computing Platform of Peking University, NYU Highperformance computing at New York, Abu Dhabi, and Shanghai campuses.
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H.C. conceived the project. Y.G. and H.C. performed the calculations. S.Z. and X.W. analyzed data. J.H. participated in the discussion. H.C. and Y.G. wrote the manuscript. All the authors commented on the paper.
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Gu, Y., Zhu, S., Wang, X. et al. A substantial hybridization between correlated Nid orbital and itinerant electrons in infinitelayer nickelates. Commun Phys 3, 84 (2020). https://doi.org/10.1038/s420050200347x
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DOI: https://doi.org/10.1038/s420050200347x
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