Possible strain-induced enhancement of the superconducting onset transition temperature in infinite-layer nickelates

Abstract

The mechanism of unconventional superconductivity in correlated materials remains a great challenge in condensed matter physics. The recent discovery of superconductivity in infinite-layer nickelates, as an analog to high-T_c cuprates, has opened a new route to tackle this challenge. By growing 8 nm Pr_0.8Sr_0.2NiO₂ films on the (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 substrate, we successfully raise the superconducting onset transition temperature T_c in the widely studied SrTiO₃-substrated nickelates from 9 K into 15 K, which indicates compressive strain is an efficient protocol to further enhance superconductivity in infinite-layer nickelates. Additionally, the x-ray absorption spectroscopy, combined with the first-principles and many-body simulations, suggest a crucial role of the hybridization between Ni and O orbitals in the unconventional pairing. These results also suggest the increase of T_c be driven by the change of charge-transfer nature that would narrow the origin of general unconventional superconductivity in correlated materials to the covalence of transition metals and ligands.

Introduction

The discovery of the high-T_c superconducting cuprates is one of the greatest surprises in quantum materials. Due to the complexity stemming from multiple intertwined orders, the pairing mechanism of cuprates remains an enigma¹. Many theories have proposed that the unconventional superconductivity is likely associated with the strong correlation nature of the d-orbital electrons in transition-metal oxides (TMOs)², which can be characterized by the Mott insulating parent state at the filling of one hole per unit cell. This picture has motivated the study of superconductivity in several TMOs and, more recently, magic-angle twisted bilayer graphene³. However, the anticipated analogy materials, such as iridates and ruthenates, do not show high T_c as cuprates. Theoretically, it remains a question whether the single-band Hubbard model, as a prototype of correlated electron systems, indeed gives long-range ordered superconductivity^4,5.

The recent discovery of superconducting nickelates provides a new opportunity to shed light on the puzzle^{6,7,8,9,10,11}. Due to nickelates’ structural and chemical similarities with cuprates, a comparative study of these two materials may help establish the pairing mechanism and guide us to design materials with higher T_c^{12,13,14,15,16,17,18,19,20,21,22,23}. Experiments have shown infinite-layer nickelates exhibiting similar magnetic excitations with cuprates²⁴. However, nickelates show a much lower T_c. A possible reason is that the nickelates RNiO₂ (R = La, Pr, and Nd) are Mott-Hubbard insulators^18,22,25,26, while cuprates are charge-transfer insulators with more covalent-bond nature near the parent compound. The charge-transfer property of the latter is characterized by a prominent pre-peak in the oxygen K-edge x-ray absorption spectrum (XAS), usually referred to as the Zhang-Rice peak²⁷. Given that the charge-transfer gap was also found to anti-correlate (and thus are indirectly proportional) with T_c in various cuprate compounds^28,29, it is conceivable that the oxygen-orbital components of dopants may play a crucial role in pairing. A critical step to assess this is to manipulate the charge-transfer extent of nickelates and demonstrate the possibility of enhancing superconductivity^30,31,32.

Here, we report an enhancement of T_c by growing the Pr_0.8Sr_0.2NiO₂ film on a compressive substrate (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 (LSAT). Combining first-principles and many-body simulations, we reveal that the in-plane compressive strain strengthens the p − d hybridization between oxygen and nickel orbitals. Thus, a considerable number of doped carriers distribute into oxygen orbitals, increasing the pre-peak in the oxygen K-edge XAS. This comparison reflects the dominant role of the charge-transfer nature in unconventional superconductivity, including but not restricted to nickelates and cuprates. This comparative study of the impact of microscopic ingredients on T_c further motivates a systematic and efficient approach to designing superconductivity in TMOs.

Results and discussion

Substrate-induced strain and enhanced T _c

As illustrated in Fig. 1a, b, we prepared Pr_0.8Sr_0.2NiO₂ films of equal thickness (8 nm) on two different substrates, SrTiO₃ and LSAT, using pulsed laser deposition (PLD) followed by a CaH₂-assisted chemical reduction method (see Supplementary Note 1 for details). Figure 1c, d shows the high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) images of the two superconducting Pr_0.8Sr_0.2NiO₂ films on the two substrates, respectively. The contrast difference of diffraction spots reveals the uniform growth of the films with an ordered structure of a coherent infinite layer, displaying no evidence of minority phases. The well-defined infinite-layer phase of the Pr_0.8Sr_0.2NiO₂ films is further disclosed in the annular bright-field (ABF) STEM images in Fig. 1e, f. The enlarged views of the areas inside the rectangles in Fig. 1e, f are shown in Fig.1g, h for better visibility of the lattice structures of the films and the substrates near the interface. The clear and sharp interfaces between the films and the substrates confirm the former epitaxially strained on the latter. Besides, the reciprocal space maps (RSM) around the (103) reflection demonstrate that the films are strained to the substrates (see Supplementary Note 2 for details). Based on both measurements, the in-plane lattice constants of the nickelate films match those of the substrates, i.e., a = 3.868 Å, yielding −1% compressive in-plane strain in the films grown on LSAT compared to SrTiO₃¹⁰.

Fig. 1: Cross-sectional scanning transmission electron microscopy (STEM) images of two types of superconducting Pr_0.8Sr_0.2NiO₂ films.

a, b The schematic illustrations of heterostructures of Pr_0.8Sr_0.2NiO₂ films on two different substrates SrTiO₃ and (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 (LSAT) respectively. c, d The high-angle annular dark field (HAADF) STEM images of films on SrTiO₃ and LSAT substrates, respectively. e, f The annular bright field (ABF) STEM images of film on SrTiO₃ and LSAT substrates, respectively. g, h The enlarged views of the areas inside the rectangles in (e, f), respectively.

With the nickelate films on distinct substrates, we first examine their transport properties using a Quantum Design Physical Property Measurement System (PPMS) with a standard four-probe configuration. Figure 2a shows the temperature-dependent resistivity of these two Pr_0.8Sr_0.2NiO₂ films on SrTiO₃ and LSAT substrates, respectively. Both films exhibit a sudden decrease in resistivity, characteristic of the superconducting transition (see Supplementary Note 3 for similar measurements on more samples). Here, we employ the onset temperature to quantify the superconductivity critical temperature T_c, resulting in ${T}_{c}^{(onset)}$ = 9 K and 15 K (highlighted with arrows) for samples grown on SrTiO₃ and LSAT, respectively. To further confirm the superconducting behavior, we investigate the transport properties with a magnetic field (up to 9.0-tesla) applied perpendicular to the ab-plane of the film. As shown in Fig. 2c, d, the field dependence exhibits similar behaviors: In the normal state, the films display negligible variation, whereas, below the superconducting transition, the conductance is suppressed significantly with an increasing magnetic field.

Fig. 2: Transport properties of superconducting Pr_0.8Sr_0.2NiO₂ films on two different substrates.

a Temperature-dependent resistivity of the Pr_0.8Sr_0.2NiO₂ on SrTiO₃ and (LaAlO₃)_0.3(Sr₂AlTaO₆)_0.7 (LSAT) substrates, respectively. b The phase diagram of Pr_1−xSr_xNiO₂ films adapted from Ref. ¹⁰ The T_c varies from 7 K to 12 K with thickness for Pr_0.8Sr_0.2NiO₂ films adapted from Ref. ⁹ (pink bar). The ${T}_{c}^{(onset)}$≈ 9 K (red diamond) and ${T}_{c}^{(onset)}$≈ 15 K (blue diamond) of our 8 nm-thick uncapped Pr_0.8Sr_0.2NiO₂/SrTiO₃ and Pr_0.8Sr_0.2NiO₂/LSAT. c, d Magnetic-field response of superconducting Pr_0.8Sr_0.2NiO₂ film on SrTiO₃ and LSAT substrates, at a varying magnetic field perpendicular to the (a, b) plane. The onset of superconducting transition at 9 K for the film on SrTiO₃ is smaller than that of the film at 15 K on LSAT substrate. Zero-resistance is obtained at 2.5 K and 2.0 K for the films on LSAT and SrTiO₃, respectively.

We summarize the critical temperatures of various nickelate experiments by the gray area in Fig. 2b to compare with previous experiments. Considering the sensitivity to the thickness, we further mark the T_c varying from 7 K to 12 K for existing Pr_0.8Sr_0.2NiO₂ studies⁹. The ${T}_{c}^{(onset)}$ ≈ 9 K (denoted by the red diamond) of our 8 nm-thick uncapped Pr_0.8Sr_0.2NiO₂ film grown on SrTiO₃ is lower than ${T}_{c}^{(onset)}$ ≈ 12 K from the previous studies on the film with equal thickness. Interestingly, when substrated by LSAT, the same uncapped 8 nm-thick Pr_0.8Sr_0.2NiO₂ film exhibits a considerably higher ${T}_{c}^{(onset)}$ ≈ 15 K, beyond the record of any (capped and uncapped) STO-substrated samples for the same doping. The uncapped films on both substrates in our study show a more significant broadening of superconducting transition, which is probably due to the detailed topotactic reduction and common in various superconducting nickelate films^11,33. Nevertheless, the zero resistance develops at 2.0 K and 2.5 K in zero-field for the films on SrTiO₃ and LSAT, respectively. It is noticed that such an observation of enhanced T_c was also reported in Nd_1-xSr_xNiO₂ films on LSAT compared to those on SrTiO₃ for different doping levels³⁴. While we cannot completely rule out some other extrinsic effects such as crystalline disorder that might partially affect T_c, it seems natural to attribute this ${T}_{c}^{(onset)}$ enhancement in the comparative study to the crystal structure of Pr_0.8Sr_0.2NiO₂, which is compressively strained by the LSAT substrate. We were aware that an even higher ${T}_{c}^{(onset)}$ ≈ 17 K was reported in Pr_0.82Sr_0.18NiO₂/STO³⁵, which does not necessarily conflict with our attribution of the enhancement to strain since it corresponds to a different doping level that expects to result in a higher ${T}_{c}^{(onset)}$ than that of Pr_0.80Sr_0.20NiO₂/STO¹⁰. Besides, the capping of STO layers may help improve the sample quality.

Pre-edge peak in x-ray absorption spectra

To unveil the connection between the strain and the enhancement of T_c in Pr_0.8Sr_0.2NiO₂/LSAT, we further examine their electronic structure using XAS. The XAS at the Ni L_3,2 and Pr M_5,4 absorption edges have been collected for Pr_0.8Sr_0.2NiO₂ films to characterize the ionic valence states (see Supplementary Note 4 for further details). We select the O K-edge to reveal the nature of dopant carriers since its characteristic peaks have been regarded as a fingerprint in cuprates²⁷. Considering that the top layer of the films can be oxidized due to air exposure as well as that the substrates are insulators, we have employed the total fluorescence yield (TFY) mode to collect the XAS at the oxygen K absorption edge. Similar to other recent studies, we find that the K-edge XAS of Pr_0.8Sr_0.2NiO₂/SrTiO₃ does not exhibit any pre-edge peak^22,36, suggesting that the parent compound of superconducting nickelates (Pr_0.8Sr_0.2NiO₂/SrTiO₃) falls into the Mott-Hubbard insulator regime. This property is in sharp contrast to cuprates, where an extra pre-edge mobile carrier peak (MCP) arises with hole doping and carries intensity comparable with the doping concentration. We illustrate the difference in XAS features via the partial density of states (PDOS) distribution sketched in Fig. 3c. As charge-transfer insulators, cuprates have a relatively strong on-site Coulomb interaction U_d (primarily for the Cu 3d electrons) compared to the charge-transfer energy ∆_h in the hole language. Therefore, a doped hole primarily occupies the oxygen 2p orbital, which hybridizes with the Cu 3d orbital as the Zhang-Rice singlet band. XAS selectively probes the unoccupied PDOS on oxygen, resulting in the coexistence of the pre-edge MCP and the upper Hubbard band (UHB) peaks. Nevertheless, the ∆_h is slightly larger than the U_d in nickelates, leading to tiny dopant holes on oxygen orbitals and, accordingly, a main absorption peak with a shoulder instead of a pre-edge peak. It is important to note that the distinction between charge-transfer and Mott-Hubbard insulators occurs suddenly at ∆_h ~ U_d only in the atomic limit²⁵, while the presence of orbital hybridization leads to a crossover in the intensity of the pre-edge peak (see Supplementary Note 5 for detailed discussions).

Fig. 3: Total fluorescence-yield x-ray absorption spectra (XAS) of two superconducting nickelate films.

a, b The XAS at the O K edge of Pr_0.8Sr_0.2NiO₂ thin films with the linear polarization perpendicular to the films’ c axis. The inset in (b) shows the fitted spectral features near the pre-edge peak (the regime inside of the dashed square). The mobile carrier peak (MCP) is absent in the film on SrTiO₃ substrate of (a). c Schematic density of states (DOS) of transition-metal oxides (TMOs) and their corresponding XAS features (i): a charge-transfer insulator, represented by cuprates, distributes doped holes primarily to oxygen orbitals, resulting in a prominent pre-edge peak in XAS; (ii): a Mott-Hubbard insulator, represented by nickelates on STO substrates, distributes doped holes primarily to transition-metal 3d orbitals, resulting in negligible pre-edge peaks; (iii): strained nickelates exhibit stronger hybridization between 3d and 2p orbitals, producing a pre-edge peak. UHB and LHB represent the upper and lower Hubbard bands, respectively.

Unlike the STO-substrated nickelates, the XAS of our Pr_0.8Sr_0.2NiO₂/LSAT sample exhibits an evident pre-peak to the O K main edge at 530.2 eV, reminiscent of the pre-peaks in cuprates²⁷. The energy position of 530.2 eV for the pre-peak is nearly equal to the observed hybridization peak in La₄Ni₃O₈, which belongs to another family of nickelate materials (R_n+1Ni_nO_2n+2)³⁷. Therefore, we can attribute this pre-edge peak in Pr_0.8Sr_0.2NiO₂/LSAT to the MCP. Since the charge-transfer nature appears to be one of the most significant differences between cuprates and nickelates, the emergence of the pre-edge XAS peak in LSAT-substrated nickelates provides insight for its enhanced T_c, consistent with previous results^28,29,38.

Numerical explanation of the electronic structure’s change

To reveal the impact of the strained lattice on nickelates’ electronic structure, we simulate the doped carriers and spectral evolution upon straining using density functional theory (DFT) and the extracted many-body model. We first calculate the electronic structure of the pristine and −1% strained PrNiO₂ using Perdew-Burke-Ernzerhof (PBE) functional (see Methods). As shown in Fig. 4, the simulated PDOS reflects that the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular (LUMO) orbitals primarily consist of Ni 3d_x2−y2 orbitals. As pointed out in previous studies²², the rare-earth band also resides at low energy, leading to a “self-doping” effect. To address the experimental features observed in the oxygen K-edge XAS, we focus on the oxygen components of the electronic wavefunction. By projecting the simulated Kohn-Sham orbitals to single-particle Wannier orbitals, we extracted two oxygen orbitals (2p_x and 2p_y) which hybridize with the active Ni 3d_x2−y2 orbital [see the inset of Fig. 4a]. We employ these three orbitals to construct a tight-binding model, the first four terms in Eq. (1) in the Methods section, where the hopping parameters and site energies are extracted from the Kohn-Sham Wannier orbitals. For the pristine PrNiO₂, we obtain t_pd = 1.36 eV, t_pp = 0.57 eV, and E_d - E_p = 3.90 eV, consistent with previous simulations²² for the −1% strained PrNiO₂, t_pd are increased into 1.41 eV, while the changes of other parameters are less than 2%.

Fig. 4: Theoretical analysis of strain effects on the electronic structures and x-ray absorption spectra (XAS) of Praseodymium Nickelates.

a Band structures of unstrained PrNiO₂ from density functional theory (DFT) and Wannier tight-binding model. Nickel’s 3d_x2−y2 and oxygen’s p_x and p_y orbitals are shown at the top, where gray, purple, and green spheres represent Pr, Ni, and O, respectively. The site energies of E_d and E_p, as well as the hopping parameters t_pd (hopping between Ni: 3d_x2−y2 and O: p_x, p_y) and t_pp (hopping between O: p_x and p_y) in the Wannier tight-binding model. These band parameters are employed in exact diagonalization simulations. b Exact diagonalization XAS with a charge-transfer energy ∆_h = 8 eV and a varying t_pd from 1.0 eV to 1.5 eV. The dashed and solid lines represent the spectra for 0 and 12.5 % hole-doping, respectively. The pre-peak area (illustrated by the shade) increases with an increasing t_pd. The pre-peak area (c) and the average hole number per p_x,_y orbital (d) as a function of ∆_h and t_pd are shown by the intensity plots. Both increase with an increasing t_pd and a decreasing ∆_h. The blue arrows indicate the change of t_pd from 1.36 to 1.41 eV when 1 % compressive epitaxial strain is applied.

We then involve the strong Coulomb interactions, doping, and x-ray core-hole interactions on top of the DFT-extracted model to simulate the XAS spectra and analyze correlation effects. Based on the estimations from previous studies, we set the Hubbard-type Coulomb interactions on the Ni 3d and O 2p orbitals as U_d = 6 eV and U_p = 2 eV, respectively. We also set the core-hole attractive interaction U_c = 3 eV to match the experimental separation between the pre-edge and main absorption peaks. The XAS spectral intensity is simulated using Eq. (2) in the Methods section. As shown in Fig. 4b, we manipulate the strength of t_pd and fix other parameters, mimicking the primary impact on the electronic structure induced by the strain effect. Considering possible ambiguities of band parameters extracted from minimal Wannier-folded orbitals, we examine the spectral properties for a wide range of t_pd values, from 1.0 to 1.5 eV. The simulated XAS results suggest that increased t_pd, induced by the compressive strain effect, enhances the pre-edge peak, consistent with our experimental observations in Fig. 3a, b. As illustrated in Fig. 3c, although the strain does not switch the relative size of U_d to ∆_h in the atomic representation, the increased t_pd mixes oxygen and nickel PDOS at the top of the lower Hubbard band (LHB). Therefore, a larger portion of the doped holes resides in oxygen, resulting in the rise of the pre-edge XAS peak.

To further demonstrate the aforementioned mechanism, we calculate the average hole concentrations ‹n_p› on the oxygen 2p_x/y orbitals. By comparing the undoped and doped XAS results, the pre-edge peak reflects most dopant carrier concentrations in oxygen. Therefore, we evaluate the pre-peak intensity and ‹n_p› for a wide range of charge-transfer energy ∆_h and t_pd of the three-band Hubbard model. As shown in Fig. 4c, d, the parameter dependencies of these two quantities are consistent in the entire ∆_h − t_pd parameter space. We also highlight the DFT-extracted model parameters for the pristine and strained nickelates to guide the eye. In this context, the pre-edge peak intensity (Fig. 4c) rises by 2.8%, and the oxygen-hole concentration (Fig. 4d) increases by 3.1%. This increase is smooth in the DFT-extracted model, in contrast to the sharp transition in the atomic model (see Supplementary Note 5), but the correlation between the spectral and carrier properties is the same. Due to the simplicity of the downfolded three-band, our simulation has neglected contributions from other oxygen orbitals in nickelates and substrates, underestimating the main-peak intensity. Despite the mismatch of this main peak, the rise of the pre-edge peak reflects that the nickelates in LSAT substrate distribute more dopant carriers into oxygen orbitals.

The fact that Pr_0.8Sr_0.2NiO₂/LSAT displays a much higher T_c compared with Pr_0.8Sr_0.2NiO₂/STO suggests the strain enabled via substrate design is an efficient route to control, and especially enhance, the superconductivity in nickelates. More importantly, the comparison between strained and unstrained nickelates, as well as nickelates and cuprates, provides significant insights into the long-standing puzzle – the pairing mechanism of unconventional superconductivity in correlated materials. Our experimental evidence for the concurrent rise of T_c and the emergence of the pre-edge peak in oxygen K-edge XAS reflects the intrinsic connection between the TM-oxygen hybridization in the pairing mechanism. Narrowing the dominant factor into the hybridization, one can further interpret its microscopic origin. A possible explanation is the enhanced spin fluctuations due to these Ni-O hybridizations, similar to the recently observed strained-enhanced magnons in monolayer cuprates³⁹. According to our first-principles extracted three-band models, we simulate the dynamical spin structure factors of the pristine and strained PrNiO₂. The top of magnon excitations rises from 178 meV into 192 meV with the −1% compressive strain (see Supplementary Note 6 for further details). Large-momentum spin fluctuations have been popular candidates for the d-wave pairing glue of unconventional superconductivity. Moreover, such a 7.8% increase of spin excitations out of the 1% strain indicates the outsized nonlinear impact of the lattice structure. Thus, an alternative but non-exclusive explanation for the 60% enhancements lies in the possible contribution of electron-phonon to unconventional superconductivity^{40,41,42,43,44,45}. As a key difference from nickelates and other TMOs²⁶, the unique charge-transfer insulating parent compounds of cuprates cause doped holes primarily to reside on oxygen, whose vibrations constitute most phonon modes. The present experimental and simulation results suggest that the strain-enhanced hybridization between O and Ni orbitals also leads to the similar charge transfer of doped carriers, which may further help pairing.

It is still challenging to examine if the magnetic or phonon-assisted mechanism, or their combined effect, may account for the more than 60% enhancement of T_c in Pr_0.8Sr_0.2NiO₂/LSAT. The quantitative assessment of their contributions may identify the general pairing mechanism in correlated materials and motivates future experiments. For example, resonant inelastic x-ray scattering has characterized both the magnon and phonon dispersions in STO-substrated Nd_1–xSr_xNiO₂²⁴. Comparative RIXS studies of nickelates on these two substrates may provide more insight into individual contributions from these excitations. Thus, our results, together with the further investigations of pairing mechanisms, should stimulate systematic engineering and design of high-T_c superconductivity.

Methods

Sample preparation and experimental characterization

The perovskite precursor Pr_0.8Sr_0.2NiO₃ films were prepared by using pulsed laser deposition (PLD). The corresponding infinite-layer phase was acquired by the soft-chemistry reduction method. The superconducting transition temperature was confirmed by transport measurements using a Quantum Design Physical Property Measurement System (PPMS) with a standard four-probe configuration. Samples for the cross-sectional scanning transmission electron microscopy (STEM) were prepared by focused ion beam (FIB, Helios 600i) techniques. The high-angle annular dark field (HAADF) and annular bright field (ABF) STEM images were acquired on the ARM-200F (JEOL, Japan) operated at 200 kV with a CEOS Cs corrector (CEOS GmbH, Germany). The x-ray absorption spectroscopy (XAS) measurements were performed at beamline 29-ID IEX at the Advanced Photon Source, Argonne National Laboratory. The beamline uses an electromagnetic undulator providing a soft x-ray from 250 eV to 3000 eV. The spectra were normalized by the incident x-ray intensity (I0) using a drain current from a gold mesh upstream of the sample. All spectra were collected at 30 K.

Density functional theory calculations

The electronic band structure of PrNiO₂ is calculated using the Quantum Espresso package⁴⁶ with Perdew-Burke-Ernzerhof⁴⁷ exchange-correlation functional, where the pseudopotential is based on the projected augmented wave (PAW) method⁴⁸. We adopt the Monkhorst-Pack sampling⁴⁹ with a Γ-centered k-mesh of 13 × 13 × 15 and a plane-wave cutoff energy of 40 rydberg is used to expand the wave function. The convergence criteria of structure relaxation and self-consistent field are set to 10⁻⁴ rydberg/bohr and 10⁻⁷ rydberg, respectively. For the −1% strained structure, we then change the lengths of the a- and b-axis directly while the c-axis remains unchanged.

To extract the site energies and hopping parameters, we employ the Wannier90 package⁵⁰ to fit our DFT results and construct the tight-binding models, where a total number of 16 Wannier functions: five Pr’s d orbitals, five Ni’s d orbitals, and six O’s p orbitals, are considered. In our calculations, the disentanglement procedure is employed with an energy window from −10 to +0.8 eV.

Exact diagonalization simulation for XAS spectra

We employ the three-band Hubbard model extracted from the DFT Wannier orbitals as a minimal description of the charge-transfer physics in PSNO. The Hamiltonian reads as:

$$Hv = - \mathop{\sum}\limits_{\langle i,j \rangle \alpha \sigma}{t}_{pd}^{ij}({d}_{i \sigma }^{+}{p}_{j \alpha \sigma } + h.c.)\\ - \mathop{\sum} \limits_{\langle j,j^{\prime} \rangle \sigma }{t}_{pp}^{jj^{\prime}}({p}_{jx\sigma }^{+}{p}_{j^{\prime}y\sigma }+h.c.)+\mathop{\sum} \limits_{i\sigma }{E}_{d}{n}_{i\sigma }^{d}+\mathop{\sum} \limits_{j\alpha \sigma }{E}_{p}{n}_{j\alpha \sigma }^{p}\\ + \mathop{\sum} \limits_{i}{U}_{d}{n}_{i\uparrow }^{d}{n}_{i\downarrow }^{d}+\mathop{\sum} \limits_{j\alpha }{U}_{p}{n}_{j\alpha \uparrow }^{p}{n}_{j\alpha \downarrow }^{p},$$

(1)

Here, ${d}_{i\sigma }^{+}({p}_{j\alpha \sigma }^{+})$ creates an electron with spin σ at a Ni site i (O site j), and ${n}_{i\sigma }^{d}({n}_{j\sigma }^{p})$ is the Ni (O) electron number. The flavor α = {x, y} labels the 2p_x,_y orbitals. The t_pd and t_pp determine the nearest-neighbor hopping amplitudes, and E_d (E_p) is the electronic site energy. These single-particle parameters are extracted from the Wannier orbitals of the DFT simulation. The last two terms are the on-site Hubbard interactions, whose strengths are set as ad hoc parameters U_d = 6 eV, U_p = 2 eV, comparable with previous studies^22,26. The charge-transfer energy in the hole representation is defined as ∆_h = E_d − E_p + U_d − U_p. At the pristine PSNO, the DFT simulation gives ∆_h = 8 eV, |t_pd | = 1.36 eV, |t_pp | = 0.57 eV, while in the −1% strained PSNO, the extracted parameters of |t_pd| and |t_pp| are = 1.41 eV and 0.58 eV. Beyond the single-particle Kohn-Sham band parameters, we perform many-body simulations for the XAS spectrum of the three-orbital Hubbard model on an 8-site cluster for two different doping levels: 0% (half-filling) and 12.5% (underdoped).

The many-body ground state $|G\rangle$ of the three-band valence-electron Hamiltonian Eq. (1) is solved by the exact diagonalization (ED). With this ground state, the zero-temperature XAS spectrum is calculated using

$${I}_{XAS}(\omega ,T)=-\frac{1}{\pi }{{\mbox{Im}}}\langle G|{D}^{+}\frac{1}{\omega +{E}_{G}-H+i\delta }D|G\rangle ,$$

(2)

where E_G is the ground-state energy and δ is the broadening corresponding to the inverse lifetime of the intermediate state. For the convenience of extracting the integrated peak intensity, we employed a small broadening δ = 0.1 eV in the simulation, where realistic values should be much larger. Different from the three-band valence-electron Hamiltonian in Eq. (1), the intermediate-state Hamiltonian H in Eq. (2) describes an x-ray excited state and is defined in an extended Hilbert space consisting of both the valence electrons and a single core-level band resonant to the x-ray edge. Thus, the dipole operators D define the electron transition between the oxygen 1 s (core-level) and the 2p orbitals (valence level) selected by the x-ray K-edge edge

$$D=\mathop{\sum} \limits_{\alpha \sigma }{p}_{j\alpha \sigma }^{+}{c}_{j\sigma },$$

(3)

We ignore the matrix elements since we do not compare absolute intensities among different edges. In addition to the valence-band tight-binding model H_v, the intermediate state in Eq. (2) contains an extra core hole and its attractive interaction

$${H}_{c}=\mathop{\sum} \limits_{i\sigma }{E}_{edge}(1-{n}_{i\sigma }^{c})-{U}_{c}\mathop{\sum} \limits_{i\sigma \sigma ^{\prime}}{n}_{i\sigma }^{p}(1-{n}_{i\sigma ^{\prime}}^{c}),$$

(4)

where E_edge is the edge energy E_edge = 530 eV and U_c = 3 eV the core-hole potential.