Impact of pressure and apical oxygen vacancies on superconductivity in La₃Ni₂O₇

Abstract

The bilayer nickelate La₃Ni₂O₇ under pressure has recently emerged as a promising system for high-T_c superconductivity. In this work, we investigate the fate of the superconducting properties in La₃Ni₂O₇ under pressure, focusing on the effects of structural deformation and apical oxygen vacancies. Employing a low-energy effective t-J_∥-J_⊥ model for the $3{d}_{{x}^{2}-{y}^{2}}$ orbitals within the slave-boson mean-field approach, we demonstrate that the pairing strength is significantly enhanced in the high-pressure tetragonal I4/mmm phase compared to the ambient pressure orthorhombic Amam phase. Furthermore, by simulating random configurations of apical oxygen vacancies, we show that oxygen vacancies suppress both pairing strength and superfluid density. These results underscore the critical role of pressure and oxygen stoichiometry in tuning the SC of La₃Ni₂O₇, providing key insights into optimizing its high-T_c behavior.

Introduction

Ruddlesden-Popper (RP) phase nickelates series La_n+1Ni_nO_3n+1 have been a subject of sustained and intense research due to their potential for realizing high-T_c superconductivity (SC)^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}. In a recent pivotal development, SC phenomena exhibiting a critical temperature T_c of approximately 80 K was discovered in La₃Ni₂O₇ (n = 2) under a pressure exceeding 14 GPa^{19,20,21,22,23,24}, attracting great interests^{24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87}. Furthermore, SC with a T_c ≈ 40 K was realized in La₃Ni₂O₇ thin films under ambient pressure (AP)^88,89, representing a crucial step forward toward practical applications and experimental investigations^90,91,92,93. Additionally, experimental evidence of SC with a T_c ≈ 20 K in another RP-phase material, La₄Ni₃O₁₀ (n = 3)^94,95,96, has also attracted significant attention^{33,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121}. A recent discovery of SC was found in La₅Ni₃O₁₁ under pressure with T_c ≈ 60 K¹²². These advancements underscore the important role of this nickelate series as a crucial platform for exploring high-T_c SC.

The phase diagram of bilayer La₃Ni₂O₇ has been explicitly explored across a wide range of temperatures and pressures^19,84, revealing a complex and rich phenomenology. In particular, the complex nature of the phase diagram and the potential for unconventional SC have attracted a subsequent surge of theoretical investigations aimed at revealing the underlying SC mechanism^{47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,86,87}. Despite these intensive theoretical efforts, the precise nature of the pairing mechanism and symmetry in La₃Ni₂O₇ remains an open question and a subject of ongoing debate. Two key factors, pressure and oxygen stoichiometry, play an important role and complicate the problem.

Experimental studies have definitively demonstrated that the emergence of SC in La₃Ni₂O₇ is strongly and sensitively influenced by the application of external pressure^{19,20,21,22,23,24,27,28,29,45}. At AP, the material adopts an orthorhombic Amam crystal structure. High-quality single crystals are metallic down to low temperatures, while oxygen deficiency can induce weakly insulating behavior. Under increasing pressure, La₃Ni₂O₇ undergoes sequential structural phase transitions^19,84. Initially, it transforms from the Amam phase to an orthorhombic Fmmm phase around 14 GPa, accompanied by an inter-bilayer Ni-O-Ni bond angle from approximately 168° toward 180°, which significantly enhances inter-bilayer coupling. Subsequently, at high pressure (HP), it transforms into a tetragonal I4/mmm phase. SC is presumed to emerge near the first structural phase transition into the Fmmm phase, with the superconducting transition temperature T_c subsequently decreasing as pressure increases^19,84.

The realization of SC is also highly sensitive to oxygen stoichiometry^{29,36,42,43,46,84,88,92}, emphasizing that sufficient oxygen content is crucial for stabilizing robust SC^49,87,123. Oxygen vacancies significantly alter the electronic structure and orbital interactions critical to SC. Experimental observations, including scanning transmission electron microscopy and X-ray diffraction (XRD)^42,84, have provided direct visualization of oxygen vacancies, particularly attributed to apical oxygen sites. Transport measurements have revealed that oxygen vacancies can reduce the superconducting volume fraction and, in some samples, induce a filamentary SC state⁸⁸. Additionally, ozone annealing has been employed to manipulate oxygen stoichiometry, demonstrating the importance of oxygen control in optimizing SC⁹².

In this paper, we employ strong correlated analysis to investigate the effects of pressure and random apical oxygen vacancies on the superconducting properties of La₃Ni₂O₇. To gain insight into the pairing mechanism, we focus on a low-energy effective t-J_∥-J_⊥ model⁵⁴ for the $3{d}_{{x}^{2}-{y}^{2}}$ orbital. We consider two representative regimes: AP Amam phase and HP I₄/mmm phase at 30 GPa. At 30 GPa, any intralayer anisotropy between the x and y directions is typically minimal. Therefore, for simplicity and to effectively capture the near-tetragonal high-pressure environment, we adopt the I4/mmm phase with equivalent hopping integrals along these in-plane directions. Our numerical results reveal a superconducting state in the AP Amam phase, while the pairing strength is significantly weaker than that in the HP I₄/mmm phase. Furthermore, we simulate the influence of random oxygen vacancy configurations, demonstrating that apical oxygen vacancies act to suppress both pairing strength and superfluid density. These findings provide valuable theoretical insights into the impact of pressure and oxygen vacancies on the superconducting properties of La₃Ni₂O₇, offering guidance for future experimental and theoretical investigations.

Results

Effective bilayer single-orbital model

The electronic properties of La₃Ni₂O₇ are dominated by the two E_g orbitals⁴⁷ within Ni^2.5+ ions. Direct valence counting shows that the $3{d}_{{z}^{2}}$ orbital is nearly half-filled, while the $3{d}_{{x}^{2}-{y}^{2}}$ orbital is approximately quarter-filled. Besides, La₃Ni₂O₇ exhibits strong electronic correlations, as evidenced by optical measurements showing the suppression of electron kinetic energy²⁵ and band renormalization revealed from angle-resolved photoemission spectroscopy (ARPES) data²⁶, placing the system proximity to a Mott regime. Under strong Hubbard repulsion, $3{d}_{{z}^{2}}$-orbital electrons become effectively localized^26,47. SC primarily contributes from the $3{d}_{{x}^{2}-{y}^{2}}$ orbital, which shares similarities with overdoped cuprates. However, the in-plane superexchange interaction between the $3{d}_{{x}^{2}-{y}^{2}}$ electrons alone is not sufficient to account for high-T_c SC.

A critical aspect of the pairing mechanism in La₃Ni₂O₇ lies in the interplay between the $3{d}_{{z}^{2}}$ and $3{d}_{{x}^{2}-{y}^{2}}$ orbitals. The corresponding effective t-J-J_H model that captures this orbital entanglement is illustrated in Fig. 1. The strong interlayer antiferromagnetic (AFM) superexchange associated with the $3{d}_{{z}^{2}}$ spins is transmitted to the $3{d}_{{x}^{2}-{y}^{2}}$ orbital via Hund’s coupling⁵⁴. This mediates an effective interlayer spin exchange J_⊥ between $3{d}_{{x}^{2}-{y}^{2}}$ spins, which is essential for the emergence of SC in this system. The electronic properties of the $3{d}_{{x}^{2}-{y}^{2}}$ orbital under strong Hund’s coupling can be captured by a bilayer t-J_∥-J_⊥ model^54,55,57.

Fig. 1: Schematic illustration of the two-orbital bilayer t-J-J_H model.

The charge carriers primarily reside in the $3{d}_{{x}^{2}-{y}^{2}}$ orbitals, as indicated with wavy lines on the lattice, while the $3{d}_{{z}^{2}}$ orbitals are depicted as localized spins due to their single occupancy. These localized spins interact via interlayer spin exchange J_⊥, reflecting the strong correlations present in the system. The onsite E_g orbitals tend to form a spin-triplet configuration driven by Hund’s rule coupling J_H.

The minimal effective model consists of a bilayer t-J_∥-J_⊥ Hamiltonian can be expressed as follows:

$$H = {\sum}_{\langle i,j\rangle \alpha \sigma }-{t}_{\langle i,j\rangle }^{\parallel }\left({c}_{{x}^{2}i\alpha \sigma }^{{\dagger} }{c}_{{x}^{2}j\alpha \sigma }+{{\rm{h.c.}}}\right) - {t}_{\perp }{\sum}_{i\sigma }\left({c}_{{x}^{2}i1\sigma }^{{\dagger} }{c}_{{x}^{2}i2\sigma }+{{\rm{h.c.}}}\right)\\ + {\sum}_{\langle i,j\rangle \alpha }{J}_{\parallel }{{{\boldsymbol{S}}}}_{{x}^{2}i\alpha }\cdot {{{\boldsymbol{S}}}}_{{x}^{2}j\alpha }+{J}_{\perp }{\sum}_{i}{{{\boldsymbol{S}}}}_{{x}^{2}i1}\cdot {{{\boldsymbol{S}}}}_{{x}^{2}i2},$$

(1)

where ${c}_{{x}^{2}i\alpha \sigma }^{{\dagger} }$ is the electron creation operator for $3{d}_{{x}^{2}-{y}^{2}}$ orbital at the lattice site i; α = 1, 2 is the layer index for the NiO₂ plane and σ = ↑, ↓ is the spin index; ${{{\boldsymbol{S}}}}_{{x}^{2}i\alpha }=\frac{1}{2}{c}_{{x}^{2}i\alpha }^{{\dagger} }[{{\boldsymbol{\sigma }}}]{c}_{{x}^{2}i\alpha }$ is the spin operator for $3{d}_{{x}^{2}-{y}^{2}}$ electron, with Pauli matrices σ = (σ_x, σ_y, σ_z). The summation ${\sum}_{\langle i,j\rangle }$ takes over all the nearest-neighboring (NN) pair 〈i, j〉 within the NiO₂ plane. In the following study, t is set as the energy unit. The super-exchange interaction is effectively approximately as ${J}_{\parallel }=4{t}_{\parallel }^{2}/U$, where U is the Hubbard interaction in the $3{d}_{{x}^{2}-{y}^{2}}$ orbital. We consider the whole picture, where the hole density is given by δ_h = 1 − 2x. Here, x represents the electron filling for the ${d}_{{x}^{2}-{y}^{2}}$-orbital, with x = 0.25 indicating quarter-filling.

This effective $3{d}_{{x}^{2}-{y}^{2}}$ orbital t-J_∥-J_⊥ model could potentially exhibit high-T_c SC with interlayer s-wave pairing^{54,57,67,69,71}. The important role of Hund’s coupling in stabilizing SC has also been numerically confirmed^73,74. The strong interlayer AFM exchange, crucial for SC, has been experimentally revealed by spin excitation studies, including resonant inelastic X-ray scattering^83,93 and inelastic neutron scattering⁸⁵. These studies indicate a strong interlayer value for SJ_⊥ ≈ 60–70 meV, while the intralayer one is much weaker. We further predict element substitution could enhance the SC based on this pairing mechanism⁶⁸, a trend that is qualitatively consistent with recent experiments on La₂ReNi₂O₇ (Re:rare Earth)^24,44.

Pressure and SC

Pressure induces changes in the lattice structure, consequently modifying the underlying electronic properties. Figure 2a, b illustrate the intra-layer and inter-layer hopping parameters, respectively, denoted as ${t}_{\langle i,j\rangle }^{\parallel }$. In the AP phase, spatial inversion symmetry ensures that inter-layer order parameters are identical for symmetry-equivalent positions. Density functional theory (DFT) calculations⁸² provide the AP and HP hopping parameters as shown in Table 1. In the strong coupling limit, the spin superexchange interaction J scales proportionally to t². Consequently, J is expected to become stronger under pressure due to the increased hopping parameters.

Fig. 2: Schematic diagram for the hopping process and order parameters.

a Schematic diagram for the intra-layer nearest-neighboring hopping process at ambient pressure, and the two types of intra-layer order parameters(Δ_1,2, χ_1,2). b Schematic diagram for the inter-layer order parameters (Δ_⊥, χ_⊥).

Table 1 Table of the hopping parameters at AP and HP

We apply the slave-boson mean-field (SBMF) approach^124,125 on the bilayer t-J_∥-J_⊥ model to investigate the SC properties under pressure. For a typical Hubbard interaction value of U = 4 eV, the simulated order parameters at AP 0 GPa and HP 30 GPa are calculated and summarized in Table 2. The filling dependence of the pairing strengths for AP (0 GPa) and HP (30 GPa) conditions is depicted in Fig. 3a. Evidently, the inter-layer pairing amplitude Δ_⊥ increases with increasing filling levels. With increasing pressure, the superexchange interaction is enhanced, leading to an increasing pairing amplitude Δ_⊥, from 2.40 × 10⁻³ to 1.38 × 10⁻² eV. Due to the larger inter-layer superexchange, the inter-layer pairing amplitude Δ_⊥ is significantly stronger than the intra-layer one Δ_∥.

Table 2 Table of the AP and HP order parameters (OPs) as well as the physical electron filling x

Fig. 3: Filling-dependent superconducting strength.

a Inter-layer pairing gap Δ_⊥ versus filling level x. b Superconducting T_c versus filling level x. The position of ambient pressure (AP) and high pressure (HP) physical filling is indicated by black dashed line (x = 0.25) and red dashed line (x = 0.3) respectively.

Our analysis reveals key characteristics of the SC state, identifying it as having a s-wave pairing symmetry. For the bilayer La₃Ni₂O₇, the intralayer (x-y plane) and interlayer (z direction) pairing amplitudes are distinct due to the anisotropy between the x/y and z dimensions. There is no crystalline symmetry operation within the D_4h point group that transforms an intralayer bond into an interlayer one. Therefore, the intralayer and interlayer pairing amplitudes together reveal an overall A_1g pairing symmetry. Moreover, Δ_⊥ induces a superconducting gap that changes sign between the bonding and antibonding Fermi surfaces that arise from the bilayer crystal structure.

Figure 3b presents the numerically simulated transition temperature T_c for different pressures. The enhancement of Δ_⊥ indicates a corresponding rise in the transition temperature T_c. At the physical filling level of 0.25, T_c exhibits a dramatic increase from 15 K at 0 GPa to 87 K at 30 GPa. Our calculations also give a ratio of Δ_⊥/T_c ≈ 1.6, consistent with the predictions of BCS theory. Furthermore, our calculations yield a ratio of Δ_⊥/T_c ≈ 1.6, which is consistent with the predictions by BCS theory.

Furthermore, to evaluate the impact of symmetry changes on T_c within our model, we performed calculations at the physical electron filling of x = 0.25, eliminating the anisotropy by setting ${t}_{1}^{\parallel }$ and ${t}_{2}^{\parallel }$ to their averaged value $\sqrt{{t}_{1}^{\parallel }{t}_{2}^{\parallel }}$. The results show that the pairing strength Δ_⊥ changes only slightly from  −2.40 × 10⁻³ to  −2.59 × 10⁻³ eV, and the corresponding T_c increases modestly from approximately 15–16.1 K. These findings indicate that symmetry changes alone are unlikely to account for the pronounced variation in superconducting T_c across the structural phase transition.

Most experimental data have suggested that there is no bulk SC in the AP phase^{19,20,21,22,23,24}. The suppression of SC at ambient pressure can be attributed to two main factors: First, ARPES experimental²⁶ and DFT calculations⁴⁷ both reveal that in the AP phase, the $3{d}_{{z}^{2}}$ orbital is exactly below the Fermi surface and will not contribute to the electronic property. The electron filling of the $3{d}_{{x}^{2}-{y}^{2}}$ orbital is well-approximated by quarter-filling, which is significantly overdoped and itinerant, reducing its electronic correlations and, consequently, the pairing strength. Second, at the AP Amam phase, the inter-layer hopping of the $3{d}_{{z}^{2}}$ orbital is reduced due to the deviation of the Ni-O-Ni angle φ from 180°. This leads to a significant reduction in the inter-layer superexchange interaction J_⊥, further diminishing the pairing strength. Experimental, robust SC of La₃Ni₂O₇ is found in the phase with φ ≈ 180°^91,93, emphasizing the important role of the inter-layer coupling.

DFT calculations⁵³ also show that further increasing pressure decreases the ratio of the inter-layer NN hopping for the $3{d}_{{z}^{2}}$ orbital to the intra-layer NN hopping for the $3{d}_{{x}^{2}-{y}^{2}}$ orbital. This change directly affects the AFM superexchange interaction J, which scales as 4t²/U in the strong-coupling limit. As a result, the ratio of inter-layer to intra-layer exchange couplings, J_⊥/J_∥, decreases under pressure. Further studies indicate that the superconducting T_c depends on the ratio J_⊥/J_∥: as J_⊥/J_∥ increases, T_c rises sharply, and conversely, it drops quickly as J_⊥/J_∥ decreases⁵⁴. This suggests that the observed reduction in T_c at higher pressures can be attributed to the reduction of J_⊥/J_∥.

To further illustrate this phenomenon, we performed additional tests employing the effective single-orbital t-J_∥-J_⊥ model. As shown in Fig. 4, we fixed the hopping amplitude t, the inter-layer exchange coupling J_⊥, and the filling level while systematically increasing intra-layer coupling J_∥. The results show a rapid decrease in T_c, revealing the competition between interlayer and intralayer pairing channels. These findings strongly suggest that the suppression of T_c under higher pressure⁸⁴ is driven by the delicate balance between interlayer and intralayer exchange couplings. Upon further compression, the hybridization V between orbitals might become increasingly significant, potentially leading to a suppression of T_c^61,73.

Fig. 4: T_c versus J_∥.

The diagram shows the relationship between superconducting T_c and the intralayer superexchange J_∥, with fixed J_⊥ = 0.8t and x = 0.3.

Oxygen vacancies and SC

The inter-layer apical oxygen, positioned between the two Ni-O layers, plays a critical role for the SC of La₃Ni₂O₇^{19,49,84,88,92,123}. Experimental direct visualization studies suggest that the apical oxygen is particularly prone to vacancy formation^42,84. The schematic crystal structure, presented in Fig. 5, illustrates the system with apical oxygen vacancies. The removal of apical oxygen atoms induces substantial local structural distortions, reducing the NiO₆ octahedra into NiO₅ pyramids. These distortions significantly alter the symmetry of the local crystal structure, resulting in anisotropic lattice constant changes: a marked contraction along the c-axis, accompanied by a slight expansion of the in-plane lattice constants (a and b). These structural modifications are expected to further influence both intralayer and interlayer electron hopping, as well as orbital hybridization.

Fig. 5: Structural comparison between pristine La₃Ni₂O₇ and La₃Ni₂O_7−δ, projected along the [110] axis.

The left diagram displays the regular lattice structure of La₃Ni₂O₇, composed of alternating layers of NiO₂ and LaO planes. The right diagram illustrates La₃Ni₂O_7−δ with an oxygen vacancy at the inner apical position, marked by a black dashed circle.

To investigate the effects of apical oxygen vacancies, we performed real-space SBMF calculations on a finite-size bilayer lattice. For a given vacancy concentration, δ, a corresponding number of apical oxygen sites was chosen randomly. The presence of an apical oxygen vacancy disrupts the Ni-O-Ni bond path crucial for interlayer electronic processes. This is incorporated into our model by locally modifying the Hamiltonian: specifically, the interlayer hopping (t_⊥) and interlayer superexchange (J_⊥) terms associated with any Ni-Ni bond path interrupted by a vacant apical oxygen site are set to zero. To maintain generality and clarity, we normalize the energy scales by setting t_∥ = 1, J_∥ = 0.4, and J_⊥ = 0.8.

We first compute the effects of apical oxygen vacancies on the GS SC pairing. Interlayer pairing strength Δ_⊥ remains as the dominate pairing channel. The averaged Δ_⊥ is depicted in Fig. 6, which is significantly reduced as the concentration of apical oxygen vacancies increases. Figure 7a–c further displays the pairing order parameters for various random configurations at oxygen vacancy concentrations of δ = 5, 10, 15%. It is evident that, due to the breaking of Ni-O-Ni bonds, Δ_⊥ vanishes locally at defect sites. Additionally, the results highlight how Δ_⊥ is progressively suppressed in the vicinity of these vacancies, with the suppression intensifying as δ rises. A similar pattern of suppression is also observed for the in-plane pairing Δ_∥, as shown in Fig. 7d–f.

Fig. 6: Dependence of pairing order parameters and superconducting T_c on apical-oxygen vacancy concentrations δ.

a Dependence of pairing order parameters perpendicular to the Ni-O plane Δ_⊥ on δ. b Dependence of superconducting T_c on δ. Note that for each δ, we randomly computed results for 100 samples. The solid line represents the distribution of mean values, while the error bars are the standard deviation of the deviation of each sample’s data from the mean.

Fig. 7: Spatial distributions of Δ_⊥, Δ_∥, and the particle number density n versus apical-oxygen vacancy δ.

The top row a,b,c shows the distribution of Δ_⊥, the middle row d,e,f presents the distribution of Δ_∥, and the bottom row g,h,i illustrates the spatial distribution of n. The columns correspond to different apical-oxygen vacancy concentrations: the first column a,c,g for δ = 5%, the second column b,e,h for δ = 10%, and the third column c,f,i for δ = 15%. Color bars indicate the magnitude of each quantity, highlighting variations caused by oxygen vacancy in superconducting properties and charge distribution. Note that for each δ, we randomly computed results for 100 samples, and the presented findings here correspond to one randomly selected configuration.

The suppression of SC near oxygen vacancies is closely related to the reduction of electronic density of states (DOS). At the vacancy site, the disruption of the interlayer Ni-O-Ni bond reduces the electron hopping channels, thereby increasing the on-site energy of electrons at this location. Consequently, the local electron DOS around the apical oxygen vacancy decreases, as directly illustrated in Fig. 7g–i. According to BCS theory, the superconducting gap Δ and critical temperature T_c scales as $\propto \exp (\frac{1}{{N}_{F}{V}_{eff}})$, where N_F is the DOS near the Fermi level, and V_eff is the effective interaction strength. Therefore, the reduction in electronic DOS near the vacancies results in a weakening of the pairing strength. Additionally, as detailed in the Supplementary Note 1 Fig. S1, the distribution of the hopping order parameter χ at various vacancy concentrations exhibits a trend similar to that of Δ, further reinforcing this observation.

The dependence of the T_c on the concentration of apical oxygen vacancies δ, calculated using the finite-temperature SBMF framework, is presented in Fig. 6b. The variation in T_c consistently follows the behavior of Δ_⊥, underscoring the critical role that apical oxygen vacancies play in stabilizing the SC of La₃Ni₂O₇ under pressure.

In addition, we explored the impact of different oxygen vacancy concentrations on the superfluid density. As shown in Fig. 8, the superfluid density decreases rapidly with the increase in apical oxygen vacancies. This phenomenon arises from two primary factors. On the one hand, the apical oxygen vacancies reduce the superconducting pairing strength and simultaneously decrease the diamagnetic current density. On the other hand, the presence of oxygen vacancies introduces randomness, which significantly enhances the paramagnetic current density. Consequently, the decrease in diamagnetic current density coupled with the increase in paramagnetic current density inevitably leads to a reduction in the superfluid density. This result is consistent with experimental measurements, which show a significantly reduced diamagnetic fraction in samples with a higher concentration of apical oxygen vacancies²⁹.

Fig. 8: Dependence of superfluid density ρ_s on apical-oxygen vacancy concentrations δ.

Note that for each δ, we randomly computed results for 100 samples. The solid line represents the distribution of mean values, while the error bars are the standard deviation of the deviation of each sample’s data from the mean.

A reduction in superfluid density ρ_s is expected to cause an increase in the magnetic penetration depth (λ), since ρ_s ∝ λ⁻², consistently with weakened SC. This would also likely manifest as a weakened Meissner response in samples with higher vacancy concentrations.

Discussion

In summary, we have employed the SBMF method to investigate the effects of pressure and apical oxygen vacancies on the SC of La₃Ni₂O₇. As pressure increases, the system undergoes a structural phase transition from the Amam phase at ambient pressure to the I₄/mmm phase at HP. Our calculations show a significant enhancement in the superconducting transition temperature, which is qualitatively consistent with experimental observations.

We further investigated the influence of apical oxygen vacancies on SC by simulating the problem within the randomness in real space. The introduction of apical oxygen vacancies significantly weakens SC, a suppression that is also evident in the calculated pairing order parameters and density distributions. These results align with experimental observations of extremely weak diamagnetic signals in many samples, particularly those with high concentrations of oxygen vacancies.

Recent experiments on bulk La₃Ni₂O₇ at AP show signatures of high-T_c SC at 80 K, albeit with a low volume fraction (~0.2%)⁴⁵. Conversely, resistivity data reveal a broad resistance drop near 20 K, followed by a plateau at lower temperatures⁴⁵, hinting at a bulk-like SC phase at AP. Interestingly, this lower temperature is close to the theoretical predictions of weak SC around 15 K in this work (e.g., Fig. 3b). The  ~20 K resistivity feature observed experimentally aligns well with our theoretical prediction of a weak, intrinsic bulk SC phase with T_c ≈ 15 K for the homogeneous Amam phase. Thus, the experimental data might encompass an underlying, weaker bulk SC tendency captured by our homogeneous model.

Overall, our findings provide a theoretical explanation for the experimentally observed pressure dependence of SC, including the enhancement of SC under moderate pressure and its suppression by apical oxygen vacancies. These insights deepen our understanding of the delicate interplay between structural factors and SC in this bilayer nickelate system. Moreover, the perspective under AP phase points to a potential realization of AP SC in the bulk material.

Our effective ${d}_{{x}^{2}-{y}^{2}}$ orbital model for La₃Ni₂O₇ can be considered in light of its potential charge-transfer nature and the key role of oxygen 2p orbitals, as suggested by experiments⁴². In this charge-transfer scenario, Ni 3d⁸ configurations can form a Zhang-Rice singlet (ZRS)¹²⁶ by combining a Ni E_g hole with a ligand 2p hole. Crucially, these ZRS states differ based on the Ni orbital involved: ${d}_{{x}^{2}-{y}^{2}}$ orbitals form planar ZRS with O 2p_x,y orbitals (similar to cuprates), which are less affected by apical oxygens. In contrast, ${d}_{{z}^{2}}$ orbitals form out-of-plane ZRS that depend strongly on the apical oxygen 2p_z orbitals.

By mapping these ZRS to effective Ni d-holes, we can understand the impact of vacancies. An apical oxygen vacancy disrupts the local ${d}_{{z}^{2}}$-ZRS by breaking the Ni-O_apical-Ni bond, thereby significantly reducing the interlayer AFM coupling. This aligns directly with our model’s findings, where vacancies suppress SC by weakening this critical interlayer interaction. Therefore, the ZRS-based picture provides a justification for reducing the complex system to an effective E_g-orbital model, and subsequently to our ${d}_{{x}^{2}-{y}^{2}}$-focused approach. We acknowledge, however, the limitations of this effective model. A fully quantitative understanding, particularly concerning p-d effects near vacancies, will require more comprehensive multi-orbital models including oxygen 2p states, which is an important direction for future research.

Methods

SBMF theory

In the slave-boson mean field theory, the $3{d}_{{x}^{2}-{y}^{2}}$-orbital electron operator is expressed as ${c}_{{{\boldsymbol{i}}}\alpha \sigma }^{{\dagger} }={f}_{{{\boldsymbol{i}}}\alpha \sigma }^{{\dagger} }{b}_{{{\boldsymbol{i}}}\alpha }$, where ${f}_{{{\boldsymbol{i}}}\alpha \sigma }^{{\dagger} }/{b}_{{{\boldsymbol{i}}}\alpha }$ is the spinon/holon creation/annihilation operator, with the local constraint $\sum\limits_{\sigma }{f}_{{{\boldsymbol{i}}}\alpha \sigma }^{{\dagger} }{f}_{{{\boldsymbol{i}}}\alpha \sigma }+{b}_{{{\boldsymbol{i}}}\alpha }^{{\dagger} }{b}_{{{\boldsymbol{i}}}\alpha }=1$. Introduce the intra-layer hoppings and pairings ones,

$${\chi }_{{{\boldsymbol{i}}}{{\boldsymbol{j}}}}^{(\alpha )} ={f}_{{{\boldsymbol{j}}}\alpha \uparrow }^{{\dagger} }{f}_{{{\boldsymbol{i}}}\alpha \uparrow }+{f}_{{{\boldsymbol{j}}}\alpha \downarrow }^{{\dagger} }{f}_{{{\boldsymbol{i}}}\alpha \downarrow },\\ {\Delta }_{{{\boldsymbol{i}}}{{\boldsymbol{j}}}}^{(\alpha )} ={f}_{{{\boldsymbol{j}}}\alpha \downarrow }{f}_{{{\boldsymbol{i}}}\alpha \uparrow }-{f}_{{{\boldsymbol{j}}}\alpha \uparrow }{f}_{{{\boldsymbol{i}}}\alpha \downarrow }$$

(2)

and the interlayer hoppings and pairings ones,

$${\chi }_{{{\boldsymbol{i}}},\perp } ={f}_{{{\boldsymbol{i}}}2\uparrow }^{{\dagger} }{f}_{{{\boldsymbol{i}}}1\uparrow }+{f}_{{{\boldsymbol{i}}}2\downarrow }^{{\dagger} }{f}_{{{\boldsymbol{i}}}1\downarrow },\\ {\Delta }_{{{\boldsymbol{i}}},\perp } = {f}_{{{\boldsymbol{i}}}2\downarrow }{f}_{{{\boldsymbol{i}}}1\uparrow }-{f}_{{{\boldsymbol{i}}}2\uparrow }{f}_{{{\boldsymbol{i}}}1\downarrow },$$

(3)

The holon will condense at low temperature and holon operator can be replaced by its condensation density below the condensation temperature, ${b}_{i\alpha } \sim {b}_{i\alpha }^{{\dagger} } \sim \sqrt{{\delta }_{h}}=\sqrt{1-2x}$, where x ~ 0.3 represents the electron filling of the ${d}_{{x}^{2}-{y}^{2}}$-orbital under pressure-induced self-doping effects⁶⁰.

The spin superexchange can be decoupled in the hopping and pairing channel, i.e., for the inter-layer one,

$${J}_{\perp }{{{\boldsymbol{S}}}}_{{x}^{2}i1}\cdot {{{\boldsymbol{S}}}}_{{x}^{2}i2}= -\left[{\chi }_{i\perp }\left({f}_{i1\uparrow }^{{\dagger} }{f}_{i2\uparrow }+{f}_{i1\downarrow }^{{\dagger} }{f}_{i2\downarrow }\right)+{{\rm{h.c.}}}-\frac{8| {\chi }_{i\perp }{| }^{2}}{3{J}_{\perp }}\right]\\ -\left[{\Delta }_{i\perp }\left({f}_{i1\uparrow }^{{\dagger} }{f}_{i2\downarrow }^{{\dagger} }-{f}_{i1\downarrow }^{{\dagger} }{f}_{i2\uparrow }^{{\dagger} }\right)+{{\rm{h.c.}}}-\frac{8| {\Delta }_{i\perp }{| }^{2}}{3{J}_{\perp }}\right],$$

(4)

In the mean-field analysis, hoppings χ, pairings Δ and Lagrange multipliers λ_iα are replaced by their site-independent mean-field values,

$${\chi }_{ij}^{(\alpha )} = \frac{3}{8}{J}_{\parallel }\langle {f}_{j\alpha \uparrow }^{{\dagger} }{f}_{i\alpha \uparrow }+{f}_{j\alpha \downarrow }^{{\dagger} }{f}_{i\alpha \downarrow }\rangle \equiv {\chi }_{j-i}^{(\alpha )},\\ {\Delta }_{ij}^{(\alpha )} = \frac{3}{8}{J}_{\parallel }\langle {f}_{j\alpha \downarrow }{f}_{i\alpha \uparrow }-{f}_{j\alpha \uparrow }{f}_{i\alpha \downarrow }\rangle \equiv {\Delta }_{j-i}^{(\alpha )},\\ {\chi }_{i\perp } =\frac{3}{8}{J}_{\perp }\langle {f}_{i2\uparrow }^{{\dagger} }{f}_{i1\uparrow }+{f}_{i2\downarrow }^{{\dagger} }{f}_{i1\downarrow }\rangle \equiv {\chi }_{\perp },\\ {\Delta }_{i\perp } =\frac{3}{8}{J}_{\perp }\langle {f}_{i2\downarrow }{f}_{i1\uparrow }-{f}_{i2\uparrow }{f}_{i1\downarrow }\rangle \equiv {\Delta }_{\perp }.$$

(5)

The mean-field Hamiltonian for the spinon part is given by,

$${H}_{f,{{\rm{MF}}}} = {\sum}_{{{\boldsymbol{k}}}\alpha \sigma }{\varepsilon }_{{{\boldsymbol{k}}},\alpha }{f}_{{{\boldsymbol{k}}}\alpha \sigma }^{{\dagger} }{f}_{{{\boldsymbol{k}}}\alpha \sigma }- {\sum}_{{{\boldsymbol{k}}}}\left[\left(\frac{3}{8}{J}_{\perp }{\chi }_{z}+{t}_{\perp }{\delta }_{h}\right)\left({f}_{{{\boldsymbol{k}}}1\uparrow }^{{\dagger} }{f}_{{{\boldsymbol{k}}}2\uparrow }+{f}_{-{{\boldsymbol{k}}}1\downarrow }^{{\dagger} }{f}_{-{{\boldsymbol{k}}}2\downarrow }\right)+{{\rm{h.c.}}}\right]\\ +{\sum}_{{{\boldsymbol{k}}}\alpha }\left({F}_{{{\boldsymbol{k}}},\alpha }{f}_{{{\boldsymbol{k}}}\alpha \uparrow }^{{\dagger} }{f}_{-{{\boldsymbol{k}}}\alpha \downarrow }^{{\dagger} }+{{\rm{h.c.}}}\right)-\frac{3}{8}{J}_{\parallel }{\sum}_{{{\boldsymbol{k}}}}{\Delta }_{z}\left[\left({f}_{{{\boldsymbol{k}}}1\uparrow }^{{\dagger} }{f}_{-{{\boldsymbol{k}}}2\downarrow }^{{\dagger} }-{f}_{-{{\boldsymbol{k}}}1\downarrow }^{{\dagger} }{f}_{{{\boldsymbol{k}}}2\uparrow }^{{\dagger} }\right)+{{\rm{h.c.}}}\right]\\ -\frac{3}{8}{J}_{\parallel }N{\sum}_{\alpha }\left(| {\chi }_{x}^{(\alpha )}{| }^{2}+| {\chi }_{y}^{(\alpha )}{| }^{2}+| {\Delta }_{x}^{(\alpha )}{| }^{2}+| {\Delta }_{y}^{(\alpha )}{| }^{2}\right)+\frac{3}{8}{J}_{\perp }N\left(| {\chi }_{\perp }{| }^{2}+| {\Delta }_{\perp }{| }^{2}\right)$$

(6)

where the intralayer kinectic energy and pairing are,

$${\varepsilon }_{{{\boldsymbol{k}}},\alpha } =-\frac{3}{8}{J}_{\parallel }\left[{\chi }_{x}^{(\alpha )}{e}^{-i{k}_{x}}+{\chi }_{y}^{(\alpha )}{e}^{-i{k}_{y}}+{{\rm{h.c.}}}\right] -2t{\delta }_{h}\left[\cos {k}_{x}+\cos {k}_{y}\right]-{\mu }_{f},\\ {F}_{{{\boldsymbol{k}}},\alpha } =-\frac{3}{4}{J}_{\parallel }\left[{\Delta }_{x}^{(\alpha )}\cos {k}_{x}+{\Delta }_{y}^{(\alpha )}\cos {k}_{y}\right],$$

(7)

and μ_f is the chemical potential of the spinon field.