Core-Level X-Ray Spectroscopy of Infinite-Layer Nickelate: LDA+DMFT Study

Motivated by recent core-level x-ray photoemission spectroscopy (XPS), x-ray absorption spectroscopy (XAS) and resonant inelastic x-ray scattering (RIXS) experiments for the newly-discovered superconducting infinite-layer nickelate, we investigate the core-level spectra of the parent compounds NdNiO$_2$ and LaNiO$_2$ using the combination of local density approximation and dynamical mean-field theory (LDA+DMFT). Adjusting a charge-transfer energy to match the experimental spectra, we determine the optimal model parameters and discuss the nature of the NdNiO$_2$ ground state. We find that self-doping from the Nd 5$d$ states in the vicinity of the Fermi energy prohibits opening of Mott-Hubbard gap in NdNiO$_2$. The present Ni $L_3$ XAS and RIXS calculation for LaNiO$_2$ cannot explain the difference to NdNiO$_2$ spectra.


I. INTRODUCTION
High-T c superconductivity of cuprates has been a focal point of 3d transition-metal oxide (TMO) physics over the past 30 years [1][2][3]; nevertheless, the underlying mechanism remains elusive. Superconductivity [4] reported recently in layered nickelate Nd 0.8 Sr 0.2 NiO 2 (T c = 9-15 K) with a similar crystal structure may provide new clues. The fundamental question is whether the electronic structure of NdNiO 2 (and LaNiO 2 ) is similar to that of high-T c cuprates. Naively, one might presume that Ni in the undoped systems is monovalent and, thus, hosts the d 9 (S = 1/2) ground state similar to cuprates. However, theoretical studies [5][6][7][8][9] suggest a self-doping from Nd (or La) 5d orbitals. Additionally, holes doped to a low-valence Ni 1+ compound may reside in Ni 3d orbitals, unlike in cuprates [2,3,10] or NiO with Ni 2+ [11], where they occupy the O 2p states.
Further information can be obtained with chargeconserving spectroscopies XAS and RIXS. The Ni L 3edge XAS and RIXS spectra are measured in both NdNiO 2 [9,13] and LaNiO 2 [9]. Interestingly, a side peak (852.0 eV) is observed in L 3 -XAS of LaNiO 2 , while it is absent in NdNiO 2 . A low-energy RIXS feature (E loss =0.6 eV) associated with the XAS side peak is observed in LaNiO 2 . The difference between the Ni L 3 XAS and RIXS spectra of NdNiO 2 and LaNiO 2 poses an open question.
Material-specific DMFT calculations for NdNiO 2 or LaNiO 2 were performed by several authors, leading to contradictory conclusions, which can be sorted into two groups: (i) Multiorbital (Hund's metal) physics is crucial [31][32][33][34], and (ii) (single-orbital) Mott-Hubbard physics is relevant with little influence of charge-transfer effects or with a small self-doping by Nd 5d electrons [35][36][37]. The differences, recently addressed blueby Karp, Hampel, and Millis [38], can be traced to the model parameters, which are not uniquely defined, such as the interaction strength, orbital basis, and, in particular, the double-counting correction. To settle the debate, an experimental input is needed to provide a benchmark for selecting the model parameters.

II. COMPUTATIONAL METHOD
The XPS, XAS and RIXS simulations start with a standard LDA+DMFT calculation [16,[25][26][27][28]39]. First, LDA bands for the experimental crystal structure of NdNiO 2 and LaNiO 2 [4,40] are calculated using the Wien2K package [41] [42] and projected onto Wannier basis spanning the Ni 3d, O 2p, and Nd (La) 5d orbitals [43,44]. The model is augmented with a local electron-electron interaction within the Ni 3d shell, parametrized by Coulomb's U =5.0 eV and Hund's J=1.0 eV [31,32,45]. The strong-coupling continuoustime quantum Monte Carlo impurity solver [46][47][48][49] is employed with the DMFT cycle to obtain the Ni 3d selfenergy Σ(iω n ), which is analytically continued [50] to real frequency after having reached the self-consistency . The calculations are performed at temperature T = 290 K. The XPS, XAS, and RIXS spectra are calculated from the Anderson impurity model augmented with the 2p core states and the real-frequency hybridization function discretized into 40-50 levels (per spin and orbital). To this end, we use the configuration-interaction solver; for details, see Refs. 16 and 29 for XPS and Refs. 27, 28, and 51 for XAS and RIXS simulation.
Determination of Ni 3d site energies in the model studied by DMFT involves subtracting the so-called doublecounting correction µ dc from the respective LDA values (ε LDA d ), a procedure accounting for the effect of the dd interaction present in the LDA description. It is clear that µ dc is of the order of Hartree energy U n d , but a generally accepted universal expression is not available [26,52,53]. While a similar uncertainty exists also for interaction parameters U and J, impact of their variation on physical properties is usually minor (see Supplemental Material [54] for NdNiO 2 -specific discussion). Variation of µ dc , on the other hand, may have a profound effect. Therefore we choose to adjust µ dc by comparison to the experimental data. Although µ dc is the parameter entering the calculation, in the discussion we use its linear function , which sets the scale for the energy necessary to transfer an electron from O 2p to Ni 3d orbital. Here, U dd = U − 4 9 J is the average interorbital interaction, and 9 is the Ni 3d occupation in the Ni + formal valence (similar to the definition of the charge-transfer energy in the cluster model [28,29,55]).  Figure. 1 shows the orbitally resolved spectral densities (projected density of states) of NdNiO 2 obtained by LDA and LDA+DMFT for ∆ dp = 4.9 eV, which we later identify as the optimal parameter choice. Both the LDA and LDA+DMFT yield a metallic state with the Ni x 2 −y 2 orbital character dominating around the Fermi level. This general picture is valid in the entire range of studied ∆ dp = 2.9-6.9 eV. In Fig. 2, we show the dependence of Ni x 2 − y 2 and 3z 2 − r 2 spectra on ∆ dp . Increasing ∆ dp corresponds to an upward shift of the bare Ni 3d site energies, which is indirectly reflected in the shift of the 3z 2 −r 2 band. The x 2 −y 2 peak at the Fermi level, rather than being shifted, exhibits an increased mass renormalization (reduced width). The amplitude of the x 2 − y 2 hybridization function around the Fermi level is reduced with increasing ∆ dp ; in particular, the sizable decrease just below the Fermi level (blue region) has an important implication for the XPS spectra as discussed later. The evolution of x 2 − y 2 and 3z 2 − r 2 occupancies in Fig. 4 shows that, up to ∆ dp ≈ 7 eV the 3z 2 − r 2 is completely filled (the deviation from 2.0 is due to hybridization with empty bands). The physics is , thus, effectively of a single-orbital Hubbard model, and the Ni ion takes a monovalent (Ni 1+ , d 9 ) character.

III. ELECTRONIC STRUCTURE
Different from cuprates, the stoichiometric parent compound is metallic. In order to analyze the role of Nd d bands, we study two modified models: (i) hybridization between NiO 2 planes and the Nd orbitals is switched off, and (ii) Nd orbitals are removed from the model. In the former case (i) self-doping of the NiO 2 planes from Nd orbitals is possible, while in the latter case (ii) the stoichiometry of the NiO 2 planes cannot change. The evolution of the x 2 −y 2 spectral density with ∆ dp for (i) and (ii) is shown in Fig. 3. Like the full model, the low-energy spectrum of model (i) remains metallic over the whole studied range of ∆ dp . Removing the Nd orbitals (ii) results in progressive mass renormalization with increasing ∆ dp and eventually opening of a gap above ∆ dp = 5.9 eV. This can be understood as a result of effective weakening of the Ni-O hybridization, i.e., a bandwidth-driven Mott transition. The NiO 2 layers in NdNiO 2 can, thus, be viewed as a strongly correlated system in the vicinity of Mott transition, where the insulating state is precluded by the presence of Nd 5d bands [56].

IV. COMPARISON TO EXPERIMENTAL X-RAY SPECTROSCOPIES
A. Ni 2p 3/2 XPS Next, we investigate the impact of the variation of ∆ dp on the core-level spectra. Figure 5 shows the calculated Ni 2p 3/2 XPS spectra of NdNiO 2 together with the experimental data [12]. The Ni 2p 3/2 XPS spectrum consists of two components: the main-line (852-857 eV) and the CT satellite (861 eV) [16,55]. The core hole created by x rays represents an attractive potential, which induces CT from surrounding atoms to the empty 3d orbital on Unlike A, the peak B is absent in the cluster-model spectra [14,29] and, thus, can be ascribed to NLS [12]. The sensitivity of the relative intensity of A and B to ∆ dp can be used to locate its value to the interval 4.9-5.9 eV. The observed behavior of the NLS feature B reflects the amplitude of the hybridization function just below the Fermi level [16], the shaded area in Fig. 2(c). The NLS (B) is known to dominate over the local screening (A) in cuprates, as shown in Fig. 5 for Cu 2p 3/2 XPS in La 2 CuO 4 [19]. For small ∆ dp = 2.9 eV, a typical value for high-T c cuprates [10,14,19,23,57], the spectra of NdNiO 2 resemble that of La 2 CuO 4 . Thus our analysis shows that ∆ dp in NdNiO 2 is by 2-3 eV larger than in cuprates. The relative size ∆ dp and the Hubbard U would place NdNiO 2 somewhere between the Mott-Hubbard (∆ dp > U ) and CT (∆ dp < U ) systems in the Zaanen-Sawatzky-Allen classification of TMOs [10,37,58,59]. The calculated occupations for doped Nd 0.775 Sr 0.225 NiO 2 , shown in Fig. 4 and in Supplementary Material [54], reveal that for optimal ∆ dp doped holes are almost equally shared by Ni, Nd and O sites. This is a remarkable difference to monovalent cuprates or divalent NiO. In these systems of strong charge-transfer character, the doped holes reside predominantly in O 2p orbitals, irrespective of a substantial 3d spectral weight just below the Fermi level [60]. Moreover, for the optimal ∆ dp values inferred above, the doped holes in NdNiO 2 do not enter the Ni 3z 2 − r 2 orbitals (Fig. 4). The single-band Hubbard description is thus valid for not only the parent NdNiO 2 but also the superconducting one Nd 0.8 Sr 0.2 NiO 2 , as suggested by Refs. [35][36][37].
Proximity to NiO 2 layers to a Mott state (precluded by self-doping from Nd) suggests that a superexchange interaction still plays a role despite the metallic state. Using the optimal ∆ dp we arrive [54] at the nearest Ni-  5. (a) Ni 2p 3/2 XPS spectra and (b) Ni 2p 3/2 XAS spectra of NdNiO2 calculated by the LDA+DMFT method for different ∆ dp values. The experimental data [9,12] are shown together. For comparison, experimental Cu 2p 3/2 XPS data of La2CuO4 are shown (gray) [19]. The spectral broadening is taken into account using a Lorentzian 300 meV (HWHM) and a Gaussian 250 meV (HWHM) for XAS and a Lorentzian 500 meV and a Gaussian 400 meV for XPS. The XPS spectra with different broadening widths can be found in Supplemental Material [54].
Ni anti-ferromagnetic exchange in the range 40-60 meV. Given the oversimplification of representing spin response of a metal in terms of local moments interactions, this value is consistent with 69 meV inferred from the RIXS experiment on a related compound La 4 Ni 3 O 8 [61] The calculated LaNiO 2 spectra in Fig. 6(a) show similar behavior to NdNiO 2 .

B. Ni 2p 3/2 XAS and RIXS
As expected for Ni 1+ systems with a d 9 configuration, the experimental Ni 2p 3/2 XAS of NdNiO 2 shows a sharp peak corresponding to the electron excitation from the 2p 3/2 to an empty x 2 − y 2 orbital [ Fig. 5(b)]. The XAS main peak is accompanied by a broad tail attributed to the hybridization with metallic bands. The theoretical results in Fig. 5(b) reproduce the experimental data reasonably well; however, the weak dependence on ∆ dp does not allow to draw conclusions about its value.
The RIXS spectra, on the other hand, exhibit fine changes with the ∆ dp values, see Fig. 7. The spectra at all ∆ dp values contain a strong Raman-like (RL) feature (at constant E loss irrespective of the incident photon energies E in ) at E loss ∼1 eV and a fluorescence-like (FL) feature (E loss linearly increases with E in ). The RL feature arises from t 2g → x 2 − y 2 excitation, and its width (in E loss ) reflects a rapid decay of this local "exciton". With increasing ∆ dp , the RL feature shifts to lower energies, due upward shift to the t 2g bands similar to 3z 2 − r 2 shown in Fig. 2(b), while the x 2 − y 2 peaks remain pinned in the vicinity of the Fermi level. The main variation of the RIXS spectra with increasing ∆ dp concerns the behavior of the FL part, the onset of which is pushed to higher E loss . For ∆ dp =4.9 eV, deduced from the XPS data, the FL feature sets in below the RL feature at around E loss ∼0.6 eV. The coexisting RL and FL features above well capture the experimental data by Hepting et al. [9] and Rossi et al. [13]. Artificial suppression of hybridization to Nd 5d states [ Fig. 7(e)] leads to a reduced intensity of the FL feature and only a moderate modification of the low-energy spectra supporting the conclusion about the electron-reservoir role of Nd 5d states. Finally we discuss XAS and RIXS spectra in LaNiO 2 (the experimental XPS data are not available at the moment). The experimental XAS spectra of LaNiO 2 [9] are clearly distinct from NdNiO 2 . A side peak at 852.0 eV is attributed to Ni-La hybridization effect by Hepting et al [9] based on a simplified impurity model simulation. The LDA+DMFT calculations (including Ni-La hybridization) do not support this conclusion as they do The Ni L3 RIXS spectra of NdNiO2 calculated for (a) ∆ dp = 3.9 eV, (b) ∆ dp = 4.9 eV, (c) ∆ dp = 5.9 eV, and (d) ∆ dp = 6.9 eV. (e) the Ni L3 RIXS spectra calculated for the model without the hybridization between Nd 5d and NiO2 plane (∆ dp =4.9 eV). The spectral broadening is considered using a Gaussian of 100 meV (HWHM).
not match the experimental XAS spectra. While large ∆ dp gives rise to a high-energy XAS shoulder (Fig. 6), it does not improve the agreement of the RIXS spectra, shown in Fig. 8. We have to conclude that the present LDA+DMFT description of LaNiO 2 does not match the experiment for any choice of ∆ dp . We propose that the problem lies on the experimental side; i.e., the measured spectra do not represent a perfect LaNiO 2 crystal. We argue by the success of the present method for a broad spectrum of transitionmetal oxides [16] including NdNiO 2 as well as the absence of an obvious source of difference between NdNiO 2 and LaNiO 2 . On the experimental side, we point out recent studies [62,63] reporting superconductivity in Srdoped LaNiO 2 , suggesting that NdNiO 2 and LaNiO 2 are not that different after all. Spectroscopic experiments on these new LaNiO 2 samples are needed to resolve the present discrepancy.

V. CONCLUSIONS
We have presented a comprehensive analysis of Ni 2p 3/2 core-level XPS, XAS, and RIXS in infinite-layer nickelates (NdNiO 2 and LaNiO 2 ) with the LDA+DMFT approach. Comparison to the experimental spectra allowed us to determine the CT parameter (doublecounting correction) and make the following conclusions about the electronic structure. Undoped NdNiO 2 is nearly monovalent (Ni 1+ , d 9 ) with a small self-doping from the Nd 5d band. Only the Ni x 2 − y 2 orbitals are partially filled and multiorbital physics does not play an important role for the stoichiometric as well as slightly hole-doped compound. Unlike in cuprates, the Ni-O hybridization does not play an important role in connection with doping -doped holes reside predominantly on the Ni sites. The physics of NdNiO 2 described effectively by a single-band Hubbard model [35][36][37] is consistent with the available core-level spectroscopies. While the present calculations provide a good description of the experimental core-level spectra of NdNiO 2 , we cannot explain the qualitative difference between the reported NdNiO 2 and LaNiO 2 XAS and RIXS spectra.