Introduction

High-temperature superconductivity continues to be a challenging topic in correlated quantum matter since multiple electronic phases emerge in proximity to each other, masking the leading interaction for electron pairing. The newly discovered superconducting infinite-layer nickelates provide a new platform to study unconventional superconductivity1. A central question that has soon arisen for these systems is to what extent they are analogs of cuprate superconductors. Understanding the similarities and distinctions between these two families of materials may help bring to light new aspects of high-Tc superconductivity, and in particular, the pairing mechanism2. Recent experiments have revealed significant differences between these two classes of materials. For instance, cuprates are charge-transfer insulators, while the parent compounds of infinite-layer nickelates are likely Mott-Hubbard type3, or somewhere between the charge-transfer and Mott-Hubbard regime4 according to the Zaanen–Sawatzky–Allen (ZSA) scheme3,4,5,6,7; unlike the cuprates, the rare-earth spacing layers in infinite-layer nickelates hybridize with Ni 3d orbitals, leading to 5d metallic states at the Fermi level3,5. Despite these differences, infinite-layer nickelates share several general characteristics with cuprates, including a linear temperature dependence of resistivity for the normal state8, a dome-like shape for Tc as a function of doping in the phase diagram9,10,11,12,13,14, a sizable magnetic exchange interaction15,16,17,18, possible charge density wave instabilities17,18,19, and a possible d-wave superconducting gap20,21,22. These properties corroborate that the superconductivity in infinite-layer nickelates is unconventional.

An approach to tackling the pairing mechanism is to directly manipulate Tc with controllable knobs, and simultaneously examine the response of the bosonic excitations to these perturbations. By examining the coupling between low-energy excitations and external tuning parameters, one may identify the leading interaction channels accounting for the superconductivity. The use of different substrates has been proposed as a pathway for tuning Tc in thin films of infinite-layer nickelates1. However, it is highly challenging to obtain superconducting films on substrates other than SrTiO3 (STO). Nevertheless, recent experiments have shown an increase of Tc by 40% for the Pr0.8Sr0.2NiO2 films grown on (LaAlO3)0.3(Sr2TaAlO6)0.7 (LSAT) (with respect to STO)23 therefore calling for a connection with strain. Similarly, enhancing Tc through strain engineering has been observed in La2−xSrxCuO4 thin films grown on different substrates24,25,26, where the increase of Tc is attributed to strengthening the magnetic exchange interaction by compressive strain27. Although long-range magnetic ordering has not been found in infinite-layer nickelates to date, resonant inelastic x-ray scattering (RIXS) studies on Nd1−xSrxNiO2 have revealed propagating spin excitations, resembling the ones of the spin-1/2 antiferromagnet (AFM) on the square lattice, with a large spin exchange energy ~60 meV in the parent compound15. Presently, how the magnetic excitations couple to strain in nickelates is still unexamined due to the limited scattering volume precluding inelastic neutron scattering.

Here, using high-resolution Ni L3-edge RIXS, we explored the role of the substrate on the magnetic excitations in infinite-layer PrNiO2 thin films. We observe spin excitations regardless of the strain level. Moreover, the substrate tuning shows a marginal influence on the bandwidth of magnon dispersion, in contrast to the enhancement of Tc for superconducting films. These results suggest that the energy scale of spin fluctuations in parent compounds exerted with different strain values does not directly correlate with Tc, at odds with what has been reported in cuprates27,28,29. Considering that the magnetism of the parent compound is the starting point to reach superconductivity upon doping, our results provide important implications for elucidating the electron pairing mechanism in superconducting infinite-layer nickelates.

Results

Figure 1a shows a schematic of the scattering geometry of our RIXS measurements. It has been shown that the magnetic excitations in cuprates can be detected with RIXS by using either grazing-in geometry with σ (linear vertical) incident light polarization or grazing-out geometry with π (linear horizontal) polarized incident photons30. We adopted the former for the majority of the data in the present study, which allowed for the detection of both magnetic excitations and phonons. The x-ray absorption spectra (XAS) of the PrNiO2 films display a much stronger absorption peak in σ than π polarization at the Ni L3 edge (2p63d9−2p53d10 transition) (see Fig. 1c, d). This linear dichroism reflects the \(d_{x^{2}{\mbox{-}}y^{2}}\) symmetry of the 3d hole in PrNiO2. In the PrNiO2 film on LSAT, the strong signal at ~850 eV is associated with the La M4 edge (3d – 4f transition) in the LSAT substrate. A clear linear dichroism is observed at both the La M4 and Ni L3 absorption edges. A comparable linear dichroism at the Ni L3 absorption edge corroborates that the films on both STO and LSAT have equivalent sample quality.

Fig. 1: X-ray absorption spectra (XAS) and high energy (dd) excitations of PrNiO2 films grown on STO and LSAT.
Fig. 1: X-ray absorption spectra (XAS) and high energy (dd) excitations of PrNiO2 films grown on STO and LSAT.
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a Crystal structure of PrNiO2 and scattering geometry of the RIXS experiments. The polarization of the incoming photon is fixed to σ or π, where σ and π represent, respectively, the polarization components, perpendicular and parallel to the scattering plane. The 2θ scattering angle is fixed at 150° (or 154°) to maximize the in-plane momentum transfer, which is tuned by rocking the sample. q// (q⊥) refers to the momentum transfer that is parallel (perpendicular) to the nickel-oxide plane, respectively. b The d-level splitting of Ni ion in the D4h crystal field. c, d The XAS of the PrNiO2 films grown on STO and LSAT substrate measured by σ and π polarization. All the XAS measurements were performed with the grazing-in geometry at an incident angle of 30 degrees. e, f The high energy dd excitations of the PrNiO2 films grown on STO and LSAT at a representative momentum, respectively. The blue dashed lines represent the peak positions of the dxy and dyz/xz orbital excitations.

Moving to the RIXS spectra, the high energy (dd) excitations provide valuable information regarding the local configurations of the 3d orbitals in Ni ions which is determined by the symmetry of the crystal field. In infinite-layer nickelates, the expected D4h crystal field leads to the splitting of the Ni 3d orbitals with the \(d_{x^{2}{\mbox{-}}y^{2}}\) at the highest energy, followed in sequence by the dxy, dyz/xz, and \(d_{3z^{2}{\mbox{-}}r^{2}}\), as illustrated in Fig. 1b. We show in Fig. 1e, f the dd excitations in the RIXS spectra of PrNiO2 on STO and LSAT, respectively. Both spectra exhibit four major features marked by the shaded areas. The spectral peaks in the energy-loss range of 1−4 eV correspond to the crystal field splitting as illustrated in Fig. 1b, and the peak at ~0.7 eV arises from the hybridization between Ni and Pr ions, which is similar to studies on ANiO2 (A= La, Nd)3,15,17,18,19,31. As denoted by the blue dashed lines, the peaks assigned to the transition to the dxy orbital are comparable for both samples regarding the peak center of mass positions as well as line shapes; the peak assigned to the transition to dyz/xz orbital moves toward lower energy for PrNiO2/LSAT compared with PrNiO2/STO. The studies on various cuprate families have revealed a power law relationship Exy ∝ a−4 between the Exy state energy referring to the x2 - y2 ground state and the in-plane lattice constant a32. This simple relation might also be applied to the case of infinite-layer nickelates and would yield an increase of 4% in Exy ~0.05 eV, which is hard to resolve in the broad RIXS spectra for dd excitations. This explains the nearly equal Exy state energies for the films on both substrates, as shown in Fig. 1e, f. The transition to the dyz/dxz orbital shifts toward lower energy for PrNiO2/LSAT (1% compressive strain) in comparison to PrNiO2/STO, in contrast to the observations made on cuprate films27. A possible picture to reconcile the inconsistency is that Ni-O-Ni bond angles deviate from 180◦, and this deviation is further enhanced under compressive strain, leading to the decrease of the absolute value of the x2 - y2 orbital energy. Qualitatively, the energy of xz/yz orbital (pdπ overlap) is less affected than that of x2 - y2 orbital (pdσ overlap) by compressive strain, thus the energy shift in Exz/yz is dominated by the x2 - y2 orbital energy. All in all, the discernible changes of dd excitations suggest that the electronic structures of PrNiO2 are effectively modified by the epitaxial strain variation which is confirmed by the reciprocal space maps (RSM) collected around the (103) reflections for the PrNiO2 films on the different substrates LSAT and STO (see the supplementary materials for details).

In Fig. 2, we show the low-energy loss RIXS spectra for PrNiO2/STO and PrNiO2/LSAT along high symmetry directions (h, 0) and (h, h) in momentum space. The spectra of PrNiO2/STO consist of three major features composed of an elastic peak at zero energy loss, a clear excitation at ~60 meV, and a broad peak in the range of 100–400 meV. The peak at ~60 meV represents the phonon excitation, reminiscent of the ~70 meV phonon mode that prevails in cuprates15,33. The broad peak is ascribed to the magnetic excitations as it disperses as a function of momentum, resembling what was observed in NdNiO2 and LaNiO215,17,18. In PrNiO2/LSAT, the elastic peak is more prominent, probably owing to the contribution from the La M4 edge in the substrate. Nevertheless, the excitations can still be reliably extracted by fitting the RIXS spectra to a combination of a Voigt function for the elastic peak, a Gaussian function for the phonon, a damped harmonic oscillator (DHO) to account for the magnetic excitation, and a smoothly varying background. The DHO function χ″(q, ω) is given by

$${\chi }^{{\prime\prime} }\left(q,\, \omega \right)=\frac{{\gamma }_{q}\omega }{{\left({\omega }^{2}-{\varepsilon }_{q}^{2}\right)}^{2}+4{\gamma }_{q}^{2}{\omega }^{2}}$$
(1)

where εq is the undamped mode energy, and γq is the damping factor30. As shown in Fig. 2, the fitting overall describes well the experimental spectra.

Fig. 2: Momentum resolved RIXS spectra along high symmetry directions.
Fig. 2: Momentum resolved RIXS spectra along high symmetry directions.
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a, b RIXS spectra of the PrNiO2 film grown on STO along (h, 0) and (h, h) directions. c, d RIXS spectra of the PrNiO2 film grown on LSAT along (h, 0) and (h, h) directions. The filled black circles represent the data and the solid red curves fit the data set, using a combination of an elastic scattering contribution (green), a Gaussian profile for the phonon peak (orange), a DHO function for the magnetic excitation (purple), and background (gray). All the measurements were taken at 40 K.

To better represent the magnon dispersion, we include in Fig. 3a, b the magnetic spectra map of the magnetic excitations after subtracting the elastic peak, phonon peak, and background from the raw data. As shown in Fig. 3a, a clear magnon dispersion can be visualized along both (h, 0) and (h, h) directions for PrNiO2/STO, while in the case of PrNiO2/LSAT (see Fig. 3b), the magnetic excitations appear to be weak and less dispersive. To directly characterize the response of magnetic excitations to strain-tuning, we show in Fig. 3c–f the RIXS spectra at the zone boundaries. Here π incident light polarization and grazing-out scattering geometry were used to enhance the magnon intensity. Again, the DHO function was used to fit the magnon peaks. The gray dashed lines represent the magnon energies from the fits, while the blue dashed lines denote the energies of peak maximum. The magnons move toward lower energies in PrNiO2/LSAT compared with PrNiO2/STO.

Fig. 3: Dispersion of the magnetic excitations in PrNiO2.
Fig. 3: Dispersion of the magnetic excitations in PrNiO2.
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a, b RIXS intensity map of the PrNiO2 films grown on SrTiO3 (a) and LSAT (b) along (h, h) and (h, 0) directions at 40 K, which were obtained by subtracting the elastic peak, phonon, and the background components for better visualization. The insets in (a, b) show the trajectory in momentum space of the RIXS measurements. c–f RIXS spectra of the PrNiO2 film grown on STO and LSAT at the zone boundaries measured by π polarization. The filled black circles represent the data and the solid red curves fit the data set, using a combination of an elastic scattering contribution (green), a Gaussian profile for the phonon peak (orange), a DHO function for the magnetic excitation (purple), and a smoothly varying background (gray). The gray dashed lines represent the mode energy (εq) of the magnetic excitations obtained by fitting with the DHO function and the blue dashed lines represent the peak energy of the magnon.

Figure 4a shows the momentum dependence of the fitted values of the magnetic excitation energy εq and the damping factor γq as defined in Eq. (1) for both samples. Considering the uncertainty for the fitting, the magnon dispersions are comparable in both cases, with an energy maximum close to (0.5, 0) along (h, 0) direction, and close to (0.25, 0.25) along (h, h) direction; they are similar to the magnon dispersions for the spin-1/2 Heisenberg AFM on the square lattice. The magnon bandwidth in PrNiO2/LSAT appears to be slightly reduced compared to that in PrNiO2/STO. The damping factors of the two samples are comparable and vary little in both directions. To model the energy scale of the spin excitations, we fit the extracted magnetic dispersion by resorting to linear-spin-wave theory34. The Hamiltonian is given by

$$H={J}_{1}{\sum }_{\left\langle {ij}\right\rangle }{S}_{i}\cdot {S}_{j}+{J}_{2}{\sum }_{\left\langle i{i}^{{\prime} }\right\rangle }{S}_{i}\cdot {S}_{{i}^{{\prime} }}$$
(2)

where Si is the spin-1/2 operator on site i, and \(\left\langle {ij}\right\rangle\) (\(\left\langle i{i}^{{\prime} }\right\rangle\)) denotes the nearest neighbors (next-nearest neighbors). The best-fit to the spectra yields J1 (J2) = 66.5 (−7.5) meV for PrNiO2/STO, and J1 (J2) = 64 (−5.5) meV for PrNiO2/LSAT, similar to the findings for NdNiO2 on STO15. The magnon dispersions near the zone center exhibit a noticeable deviation from the fits to the linear-spin-wave dispersion (the solid lines). This is caused by the substantial overlapping spectra peaks of the magnon, phonon, and elastic signals near the zone center, leading to a relatively larger error in determining the magnon peak positions. All in all, the spin exchange coupling for PrNiO2/LSAT is nearly equal to that for PrNiO2/STO, suggesting that the in-plane compressive strain of ~1% has a marginal influence on the superexchange coupling J. Figure 4b presents an additional data set obtained using π (linear horizontal) polarized incident photons, where the one for PrNiO2/STO is from the reference and obtained on our sample35. A similar analysis using linear-spin-wave theory yields J1 (J2) = 70.0(−8.0) meV for PrNiO2/STO, and J1 (J2) = 65 (−6.5) meV for PrNiO2/LSAT, suggesting the magnon bandwidth is slightly reduced in response to the in-plane compressive strain of ~1% enforced by LSAT referring to STO. This is consistent with the result of the measurements using σ polarized incident photons.

Fig. 4: Dispersion of the magnetic excitations in PrNiO2 and comparison to the model calculations.
Fig. 4: Dispersion of the magnetic excitations in PrNiO2 and comparison to the model calculations.
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a, b The mode energy (εq) and damping factor (γq) in PrNiO2 grown on STO and LSAT obtained from fitting the RIXS spectra of PrNiO2 grown on STO (LSAT) measured by σ polarization (a) and π polarization (b); the data set for the PrNiO2 grown on STO in (b) is from ref. 35. The solid lines represent the best-fit for the model of the spin-1/2 Heisenberg antiferromagnet on the square lattice using linear-spin-wave theory. J1 and J2 represent the nearest-neighbor (NN) and next-nearest-neighbor (NNN), respectively. Error bars are estimated from the standard deviation obtained by the least-squares fitting algorithm and multiple times of fittings. c Illustration of possible structural distortions in the PrNiO2 films. d Schematic plot of the bond-stretching phonon modes that may suppress the superexchange coupling strength.

Discussion

In the RIXS studies on La2CuO4 thin films, the Coulomb and magnetic exchange interactions are strengthened by the compressive strain imposed by the substrates, which may account for the doubling of Tc in the doped films27. Empirically, the superexchange J in an insulator is expected to scale with inter-ion distance a by J ~ a−10 36. This would approximately lead to a ~10% enhancement of J for 1% compressive strain. First-principles calculations on this system predict that the magnon bandwidth increases by 7.8% for a −1% strain23, corresponding to ~15 meV. From our results, the energy scale of the spin excitations, which is determined by the superexchange interaction between the planar nearest-neighbor Ni spins, seems to be slightly reduced when a ~1% compressive strain is applied. Admittedly, the small increase in the magnon bandwidth could be obscured considering the uncertainty in the fitting of the magnon dispersions to a linear-spin-wave model. However, as shown in Fig. 4a, b, the spin excitations in PrNiO2/LSAT are systematically lower rather than higher in energy compared with those in PrNiO2/SrTiO3, demonstrating a small but nonetheless decrease of superexchange J under compressive strain. Besides, as shown in Fig. 3c–f, the RIXS spectra with well-defined magnon peaks at the zone boundaries show that the energies of spin excitations are of the same order of magnitude and, likewise, a small decrease in PrNiO2/LSAT compared with PrNiO2/SrTiO3.

A possible explanation for this difference in comparison with La2CuO4 is that there may exist a structural distortion of the Ni-O plane, which would modify the bond angle of Ni-O-Ni under compressive strain, and hence the superexchange J. As illustrated in Fig. 4b, such structural distortion is commonly seen in the precursor phase RNiO3 (R is a rare-earth element such as La, Pr, Nd, and Sm)37,38. After the topotactic transformation, the structural distortion may exist in the infinite-layer phase39. Similarly, structural distortion with modified bond angles has been proposed to explain the stripe-like charge ordering in La4Ni3O8 and La3Ni2O640. Alternatively, the lattice fluctuation induced by the electron-phonon interaction suppresses the effective superexchange J, as recently proposed for parent compounds of cuprates41. In this scenario, bond-stretching phonons dynamically drag oxygen away from its equilibrium position, as illustrated in Fig. 4c; accordingly, the superexchange strength would be suppressed as the electron-phonon coupling increases under compressive strain. However, such a picture has been demonstrated only for one-dimensional (1D) systems, and whether it can be generalized to 2D systems remains unclear. In any case, this reduced magnetic exchange coupling due to 1% compressive strain in infinite-layer nickelates differs from that in the La2CuO4 thin films mentioned above27.

For unconventional superconductivity, spin fluctuations are the main candidate as the driving force for condensing electrons into pairs42. In this scenario, spin interactions primarily set the energy scale for superconductivity, which occurs in close proximity to an antiferromagnetic phase. In the large U limit, Tc is expected to scale with the superexchange J at the mean-field level for studies on cuprate superconductors28,43. This has also been used to estimate Tc for the potential LaNiO3/LaMO3 superconductors, where M = Al, Ga, and Ti44. In this context, an enhancement of ~27 meV in magnon bandwidth would be expected to scale with an increase of Tc by ~40%, which, however, contradicts our experimental observations, at odds with what has been recently reported for cuprates27,28,29. Admittedly, it is possible that besides the electronic modification via strain, other effects such as crystalline quality, interface reconstruction, and partial relaxation in the film may be at play as well8,45. This warrants further study to disentangle individual contributions to the enhancement of Tc as well as to the magnetic excitations45. We notice that freestanding superconducting infinite-layer nickelate membranes have recently been reported46,47, which offer opportunities for achieving even larger strain variation without the complexity from the sample quality or interface between film and substrates. Our study will motivate further exploration of superconductivity in infinite-layer nickelates by monitoring low-energy excitations versus strain variation using more different substrates33,48,49,50,51,52,53,54,55,56,57,58.

Methods

Sample preparation

Thin films of the precursor phase PrNiO3 with a thickness of ~7 nm were prepared by using pulsed laser deposition (PLD) on (001)-oriented SrTiO3 and LSAT substrates with a 248-nm KrF excimer laser23,59. The infinite-layer phase PrNiO2 was obtained by a soft-chemistry reduction process using CaH2 powder. Substrates were pre-annealed at 900 °C with an oxygen partial pressure of 1 × 10−5 Torr. During growth, the substrate was kept at 600 °C under an oxygen partial pressure of 150 mTorr. After deposition, the films were cooled to room temperature at a rate of 5 °C per minute in the same oxygen partial pressure. For CaH2 topotactic reduction, the as-grown nickelate films were sealed with 0.1 g CaH2 powder and annealed at a temperature of ~290 °C for ~3 h. After reduction, the PrNiO2 films were loaded back to the PLD chamber and capped with an amorphous SrTiO3 layer of 10 nm at room temperature to protect the films. After the reduced films were capped by SrTiO3, XRD measurements were carried out to confirm the high quality of the infinite-layer phase (see Supplementary Fig. S1).

XAS and RIXS measurements

The X-ray absorption spectroscopy (XAS) measurements at the Ni L-edge were performed at the resonant elastic inelastic X-ray scattering (10-ID2) of the Canadian Light Source, equipped with a 4-circle diffractometer in a 10−10 mbar ultrahigh-vacuum chamber. The photon flux is about 5 × 1011 photons per second and energy resolution reaches ΔE/E ~ 2 × 10−4. The incoming photons are fully polarized with two configurations linear vertical (σ) and horizontal (π).

High-resolution resonant inelastic X-ray scattering (RIXS) measurements were mainly performed at the SIX 2-ID beamline of NSLS-II using σ incident light polarization. The energy resolution was set to ∆E = 34 meV (full-width-at-half-maximum) at the Ni L3 edge60. Additional RIXS spectra (Figs. 3c–f, 4b) were taken with π incident light polarization at the I21 beamline at the Diamond Light Source, where the energy resolution was set to 39 and 54 meV for measurements on PrNiO2/STO and PrNiO2/LSAT, respectively61. The SIX (I21) spectrometer was positioned at the largest scattering angle of 150° (154°) to maximize the in-plane momentum transfer, and the sample temperature was set to 40 (16) K. All RIXS spectra are normalized to the area of the dd excitations (400–4000 meV).