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

High-$T_{c}$ superconductivity of cuprates has been a focal point of 3$d$ transition-metal oxide (TMO) physics over the past 30 years Bednorz and Müller (1986); Imada et al. (1998); Dagotto (1994); nevertheless, the underlying mechanism remains elusive. Superconductivity Li et al. (2019) reported recently in layered nickelate Nd_0.8Sr_0.2NiO₂ ($T_{c}$ = 9–15 K) with a similar crystal structure may provide new clues. The fundamental question is whether the electronic structure of NdNiO₂ (and LaNiO₂) 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 Botana and Norman (2020); Krishna et al. (2020); Lee and Pickett (2004); Zhang et al. (2020); Hepting et al. (2020) suggest a self-doping from Nd (or La) 5$d$ orbitals. Additionally, holes doped to a low-valence Ni¹⁺ compound may reside in Ni 3$d$ orbitals, unlike in cuprates Zaanen et al. (1985); Imada et al. (1998); Dagotto (1994) or NiO with Ni²⁺ Kuneš et al. (2007a), where they occupy the O 2$p$ states.

The Ni 2$p_{3/2}$ core-level x-ray photoemission spectroscopy (XPS) Fu et al. (2019), x-ray absorption spectroscopy (XAS), and resonant inelastic x-ray scattering (RIXS) Hepting et al. (2020); Rossi et al. (2020) are employed to probe the electronic structure of infinite-layer nickelates. A shoulder observed in the main line of the Ni 2$p_{3/2}$ XPS spectra in NdNiO₂ Fu et al. (2019) is attributed to Ni-Ni charge-transfer (CT) response to the creation of the core hole, a process traditionally called nonlocal screening (NLS) van Veenendaal and Sawatzky (1993). Generally, NLS provides valuable information about the electronic structure of TMOs van Veenendaal (2006); Hariki et al. (2017); Taguchi and Panaccione (2016); Taguchi et al. (2008). For high-$T_{c}$ cuprates, the NLS in Cu 2$p_{3/2}$ XPS is extensively used to determine key parameters, such as the CT energy $\Delta_{dp}$, and more recently to analyze electronic reconstructions due to doping Taguchi et al. (2005a); Horio et al. (2018); Okada and Kotani (1995); van Veenendaal et al. (1994); Taguchi et al. (2005b).

Further information can be obtained with charge-conserving spectroscopies XAS and RIXS. The Ni $L_{3}$-edge XAS and RIXS spectra are measured in both NdNiO₂ Hepting et al. (2020); Rossi et al. (2020) and LaNiO₂  Hepting et al. (2020). Interestingly, a side peak (852.0 eV) is observed in $L_{3}$-XAS of LaNiO₂, while it is absent in NdNiO₂. A low-energy RIXS feature ($E_{\rm loss}$=0.6 eV) associated with the XAS side peak is observed in LaNiO₂. The difference between the Ni $L_{3}$ XAS and RIXS spectra of NdNiO₂ and LaNiO₂ poses an open question.

In this paper, we use the local-density approximation (LDA) + dynamical mean-field theory (DMFT) Metzner and Vollhardt (1989); Georges et al. (1996); Kotliar et al. (2006) to calculate XPS, XAS, and RIXS spectra Hariki et al. (2017, 2018, 2020); Ghiasi et al. (2019); Hariki et al. (2020); Kolorenč (2018) of undoped infinite-layer nickelates. By comparison with the available experimental data, we identify the most appropriate CT energy and use it for classification within the Zaanen-Sawatzky-Allen scheme Zaanen et al. (1985).

Material-specific DMFT calculations for NdNiO₂ or LaNiO₂ were performed by several authors, leading to contradictory conclusions, which can be sorted into two groups: (i) Multiorbital (Hund’s metal) physics is crucial Wang et al. (2020); Kang and Kotliar (2021); Petocchi et al. (2020); Lechermann (2020), and (ii) (single-orbital) Mott-Hubbard physics is relevant with little influence of charge-transfer effects or with a small self-doping by Nd 5$d$ electrons Karp et al. (2020a); Kitatani et al. (2020); Karp et al. (2020b). The differences, recently addressed blueby Karp, Hampel, and Millis Karp et al. (2021), 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.

The XPS, XAS and RIXS simulations start with a standard LDA+DMFT calculation Georges et al. (1996); Kotliar et al. (2006); Kuneš et al. (2009); Hariki et al. (2017, 2018, 2020). First, LDA bands for the experimental crystal structure of NdNiO₂ and LaNiO₂ Hayward et al. (1999); Li et al. (2019) are calculated using the Wien2K package Blaha et al. ¹¹1The Nd 4$f$ states in NdNiO₂ are treated as partially-filled core states. and projected onto Wannier basis spanning the Ni 3$d$, O 2$p$, and Nd (La) 5$d$ orbitals Kuneš et al. (2010); Mostofi et al. (2014). 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 Kang and Kotliar (2021); Wang et al. (2020); Ryee et al. (2020). The strong-coupling continuous-time quantum Monte Carlo impurity solver Werner et al. (2006); Boehnke et al. (2011); Hafermann et al. (2012); Hariki et al. (2015) is employed with the DMFT cycle to obtain the Ni $3d$ self-energy $\Sigma(i\omega_{n})$, which is analytically continued Jarrell and Gubernatis (1996) 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 2$p$ 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. Hariki et al., 2017; Ghiasi et al., 2019 for XPS and Refs. Hariki et al., 2018, 2020; Winder et al., 2020 for XAS and RIXS simulation.

Determination of Ni $3d$ site energies in the model studied by DMFT involves subtracting the so-called double-counting correction $\mu_{\rm{dc}}$ from the respective LDA values ($\varepsilon_{d}^{\rm LDA}$), a procedure accounting for the effect of the $dd$ interaction present in the LDA description. It is clear that $\mu_{\rm{dc}}$ is of the order of Hartree energy $Un_{d}$, but a generally accepted universal expression is not available Kotliar et al. (2006); Karolak et al. (2010); Haule (2015). 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 SM for NdNiO₂-specific discussion). Variation of $\mu_{\rm{dc}}$, on the other hand, may have a profound effect. Therefore we choose to adjust $\mu_{\rm dc}$ by comparison to the experimental data. Although $\mu_{\rm dc}$ is the parameter entering the calculation, in the discussion we use its linear function $\Delta_{dp}=(\varepsilon_{d}^{\rm LDA}-\mu_{\rm dc})+9U_{dd}-\varepsilon_{p}^{\rm LDA}$, which sets the scale for the energy necessary to transfer an electron from O $2p$ to Ni $3d$ orbital. Here, $U_{dd}=U-\tfrac{4}{9}J$ 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 de Groot and Kotani (2014); Ghiasi et al. (2019); Hariki et al. (2020)).

Figure. 1 shows the orbitally resolved spectral densities (projected density of states) of NdNiO₂ obtained by LDA and LDA+DMFT for $\Delta_{dp}=4.9$ 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 $\Delta_{dp}=2.9$–$6.9$ eV. In Fig. 2, we show the dependence of Ni $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ spectra on $\Delta_{dp}$. Increasing $\Delta_{dp}$ 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 $\Delta_{dp}$; 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 $\Delta_{dp}\approx 7$ 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.

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 NiO₂ 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 NiO₂ planes from Nd orbitals is possible, while in the latter case (ii) the stoichiometry of the NiO₂ planes cannot change. The evolution of the $x^{2}-y^{2}$ spectral density with $\Delta_{dp}$ 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 $\Delta_{dp}$. Removing the Nd orbitals (ii) results in progressive mass renormalization with increasing $\Delta_{dp}$ and eventually opening of a gap above $\Delta_{dp}=5.9$ eV. This can be understood as a result of effective weakening of the Ni-O hybridization, i.e., a bandwidth-driven Mott transition. The NiO₂ layers in NdNiO₂ 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 Hirayama et al. (2020).

We have presented a comprehensive analysis of Ni 2$p_{3/2}$ core-level XPS, XAS, and RIXS in infinite-layer nickelates (NdNiO₂ and LaNiO₂) with the LDA+DMFT approach. Comparison to the experimental spectra allowed us to determine the CT parameter (double-counting correction) and make the following conclusions about the electronic structure. Undoped NdNiO₂ is nearly monovalent (Ni${}^{1+},d^{9}$) with a small self-doping from the Nd 5$d$ 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 NdNiO₂ described effectively by a single-band Hubbard model Karp et al. (2020a); Kitatani et al. (2020); Karp et al. (2020b) is consistent with the available core-level spectroscopies. While the present calculations provide a good description of the experimental core-level spectra of NdNiO₂, we cannot explain the qualitative difference between the reported NdNiO₂ and LaNiO₂ XAS and RIXS spectra.