Antiferromagnetic Ground State, Charge Density Waves and Oxygen Vacancies Induced Metal-Insulator Transition in Pressurized La₃Ni₂O₇

Introduction.—Over the past decades, exploring and understanding superconductors with high superconducting transition temperatures ($\textit{T}_{\textit{c}}$) have been the pivotal focuses in condensed matter physics [1, 2]. Recently, the Ruddlesden–Popper nickelate La₃Ni₂O₇ was found to exhibit a $\textit{T}_{\textit{c}}$ of nearly 80 K under pressure [3]. Subsequent experiments consistently confirmed this observation, as evidenced by both zero resistance [4, 5, 6, 7, 8] and diamagnetic susceptibility [8]. The electronic configuration with Ni^2.5+ in La₃Ni₂O₇ deviates from the 3$\textit{d}^{\textit{9}}$ configuration of transition metal ions in infinite-layer nickelates and cuprates [3, 9, 10]. Despite diverse views in extensive theoretical studies, broad consensuses have emerged regarding the crucial roles played by interlayer antiferromagnetic coupling, hybridization of $\textit{e}_{\textit{g}}$ orbitals, Hubbard interaction, Hund coupling, etc., for the onset of superconductivity [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 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].

Apart from superconductivity, the system undergoes a potential charge density wave (CDW) transition around 110-140 K under low pressures, as evidenced by nuclear magnetic resonance and anomalous kinks of resistivity, magnetization, specific heat and optical conductivity [49, 50, 51, 52, 53, 5, 54, 4, 7]. On the other hand, despite some La₃Ni₂O₇ samples being identified as metals under ambient pressure, they undergo an unexpected metal-insulator (MI) transition under 2-4 GPa [3]. In contrast, the calculations based on the non-magnetic (NM) state reveal a robust metallic character under both low and high pressures [3, 55, 34, 56]. Therefore, an integrated and reasonable explanation for these experimental results remains elusive so far.

In this Letter, we perform comprehensive calculations based on the density functional theory (DFT) to examine structural, electronic, and phonon properties of La₃Ni₂O₇ under pressures. We find that La₃Ni₂O₇ possesses an antiferromagnetic ground state under both low and high pressures, with the strong Fermi surface (FS) nesting of electronic flat band that gives rise to the phonon softening at high symmetric points and electronic instabilities. We elucidate that CDW orders are originated from Peierls instability and trigger MI transitions. It is also found that the competition between the half distortion and full distortion phases with apical oxygen vacancies induces an MI transition under pressures. These calculated results are in good agreement with recent experimental findings. In addition, the electron-phonon coupling (EPC) is uncovered insufficient to be responsible for superconductivity, suggesting that the Cooper pairing mechanism is unconventional in pressurized La₃Ni₂O₇, and is probably from antiferromagnetic fluctuations.

Superconductivity not from BCS.—The crystal structure of La₃Ni₂O₇ changes from the Cmcm to the Fmmm space group at about 15 GPa [FiG. 1(a)] [3], coinciding with the onset of superconductivity. Based on Bardeen-Cooper-Schrieffer (BCS) scenario, we calculate the superconducting properties of La₃Ni₂O₇ as listed in Table 1. Despite a considerable electronic density of states at the Fermi level N($\textit{E}_{\textit{F}}$), we find that the EPC constant $\lambda$ remains below 0.25 for both Cmcm and Fmmm phases, and both N($\textit{E}_{\textit{F}}$) and $\lambda$ decrease with increasing pressure. By calculations we reveal that the low $\lambda$ results in zero $\textit{T}_{\textit{c}}$ under 0-50 GPa, indicating that the EPC alone cannot account for experimental $\textit{T}_{\textit{c, exp.}}$ [3]. This shows that the superconducting pairing mechanism in pressurized La₃Ni₂O₇ is unconventional.

Antiferromagnetic ground state.—We consider the interlayer nearest-neighbor (NN) coupling ($\textit{J}_{\textit{1}}$), intralayer NN coupling ($\textit{J}_{\textit{2}}$), and interlayer next NN coupling ($\textit{J}_{\textit{3}}$) among Ni atoms [FiG. 1(a)]. Combining all positive and negative values of $\textit{J}_{\textit{1}}$, $\textit{J}_{\textit{2}}$ and $\textit{J}_{\textit{3}}$, we calculate ferromagnetic (FM) and seven antiferromagnetic (AFM1-7) configurations, as depicted in FIG. S1 of supplemental materials (SM) [57]. As seen in FIG. 1(b), the energies of eight magnetic configurations form degenerate pairs, originating from $J_{3}\approx$ 0. Energies of all magnetic states are lower than that of NM state, suggesting magnetic tendencies favor the stability of system. The magnetic ground state is AFM1 (AFM4), which shows intralayer ferromagnetism and interlayer antiferromagnetism [FiG. 1(a)]. Notably, experimental results including muon spin relaxation ($\mu$SR), resonant inelastic X-ray scattering and magnetic susceptibility also support the AFM ground state [58, 59, 60, 52]. The AFM ground state suggests that the Cooper pairing in this nickelate may be mediated by AFM fluctuations.

Fermi surface nesting.—Under the AFM1 configuration, the band structures of the Cmcm phase at 0 GPa and the Fmmm phase at 30 GPa are depicted in FIG. 1(c) and (f), respectively. Similar to the NM scenario, their electronic states near $\textit{E}_{\textit{F}}$ are predominantly contributed by the $\textit{d}_{x^{2}-y^{2}}$ and $\textit{d}_{z^{2}}$ orbitals. However, a remarkably flat band emerges near $\textit{E}_{\textit{F}}$, with its flatness significantly increasing as pressure rises to 30 GPa, suggesting an enhanced correlated electron effect. More intriguingly, the FS formed by this flat band exhibits a perfect two-dimensional (2D) square-cylinder characteristic in FIG. 1(d) and (g). The band characteristics near $\textit{E}_{\textit{F}}$ in La₃Ni₂O₇ resemble the results of the Hubbard model on a 2D square lattice [61]. The strong nesting features of this square FS evoke potential electronic instabilities. The calculated FS nesting function $\xi(q)$ in FIG. 1(d) [62, 57], shows strong peaks along $\Gamma$-Z^′, which aligns with the nesting vector of the FS. Results at 30 GPa in FIG. 1(g) is similar, albeit with a different BZ and high symmetry points.

Stable CDW phases.—To explore its structural stability, the phonon spectra at 0 and 30 GPa are calculated in FIG. 2(a) and (g). For the Cmcm phase, two phonon branches exhibit imaginary frequencies along the $\Gamma$-Z^′ path, with four imaginary modes at $\Gamma$ and Z^′ points denoted as $\Gamma 1$, $\Gamma 2$, Z1, and Z2, respectively, each leading to distinct structural distortions. Following careful structural relaxation, stable phases are obtained named as CDW-$\Gamma 1$ (space group: $\textit{P2}_{1}\textit{/m}$), CDW-$\Gamma 2$ (Amm2), CDW-Z1 (Pnma) and CDW-Z2 (Pmmn), respectively. Detailed lattice information for these phases can be found in Table S3. These phases share a similarity in the in-plane expansion (fat) and contraction (skim) of oxygen octahedra alternating in each layer, with difference lying in the arrangement and orientation of the octahedra [FIG. 2(b) and S5]. For the CDW-Z1 phase corresponding to the lowest imaginary mode Z1 as an example, the phonon spectrum in FIG. 2(c) confirms its stability. By comparing the harmonic phonon energy of the NM state of pristine phase with the AFM1 state of the CDW-Z1 phase, we estimate the transition temperature of CDW phase ($\textit{T}_{\textit{CDW}}$) to be $\approx$120 K [FIG. 2(d)], consistent with experimental results [49, 51, 52, 53, 5, 54, 4, 7, 50].

For the Fmmm phase, only phonon branches near the M^′ point show imaginary frequencies, with a softer effect compared to the Cmcm phase at 0 GPa [FiG. 2(g)]. Two imaginary modes at the M^′ point are labeled as M1 and M2, corresponding to stable distorted structures named CDW-M1 (Cmmm) and CDW-M2 (Cmcm), respectively. CDW-M1 exhibits similar octahedral distortions to CDW-Z1 but includes an additional horizontal mirror symmetry of the Ni-O bilayer [FiG. 2(h)]. The estimated $\textit{T}_{\textit{CDW}}$ of CDW-M1 at 30 GPa is around 40 K, much lower than that at 0 GPa. Other CDW phases at various pressures also exhibit stable phonon spectra, as shown in FIG. S5 and S6.

Notably, the momenta with imaginary phonons in FIG. 2(a) and (g) correspond to those exhibiting strong peaks of $\xi(q)$ in FIG. 1(d) and (g), confirming that CDW originates from Peierls instability related to FS nesting. The imaginary phonons in the AFM1 state contrasts with the stable phonons in the NM state [55], implying the instabilities are related to antiferromagnetism.

The distortion of Ni-O bond length is less than 0.1 Å, posing challenges in probing predicted CDW structures in experiments. Simulated XRD spectra show that most CDW phases exhibit nearly identical peak positions and intensities to pristine phases at both 0 and 30 GPa, except for CDW-$\Gamma$1 with a distinct Bravais lattice [FiG. S9 and S10]. To aid in identifying CDW phases in experiments, we simulate diffraction patterns that can be measured using techniques such as XRD and transmission electron microscopy. Diffraction patterns of the CDW-Z1 phase show emergent spots with odd Miller indices K compared with pristine-Cmcm phases in FIG. 2(e) and (f). Similar patterns are observed in the CDW-M1 phase at 30 GPa in FIG. 2(l) and other CDW phases in FIG. S11 and S12.

We further analyze the magnetic properties of CDW phases. By comparing the energy of different magnetic states under pressure [FiG. S1], we find a gradual increase in the AFM $\textit{J}_{\textit{1}}$, alongside a decrease in the FM $\textit{J}_{\textit{2}}$ with increasing pressure, except for $\textit{J}_{\textit{3}}\approx$ 0 as seen in Fig. 3(d). Furthermore, $|\textit{J}_{\textit{1}}|$ consistently exceeds $\textit{J}_{\textit{2}}$, indicating strong interlayer AFM coupling. Employing Monte Carlo simulation based on the Heisenberg-type Hamiltonian: $\textit{H =--}\sum_{\textit{i}}^{\textit{1,2,3}}\textit{J}_{\textit{i}}\cdot\textbf{s}^{2}\textit{+E}_{\textit{0}}$, where $\textit{E}_{\textit{0}}$ is the energy independent of the spin configurations, we determine that the Neel temperature $\textit{T}_{\textit{N}}$ remains above 100 K under 0-50 GPa, surpassing the calculated $\textit{T}_{\textit{CDW}}$ in FIG. 2(g) and (j), suggesting that the presence of robust magnetic coupling is capable of inducing the CDW transition.

Unlike pristine phases with metallic behavior in FIG. 1(c) and (f), both CDW-Z1 and CDW-M1 phases exhibit band gaps of about 0.5 eV in FIG. 3(a) and (b). Similar gaps are observed in other CDW phases [FIG. S5 and S6]. The orbitals of Ni atoms in the expanded and contracted oxygen octahedra (named as Ni_fat and Ni_skim) are split, with the $\textit{d}_{x^{2}-y^{2}}$ orbital shifted downward and upward, respectively. It means that the distinct octahedra caused by CDW transitions leads to this MI transition.

All CDW phases have lower enthalpies compared to pristine phases under pressure in FIG. 3(c), indicating their energetic stability. Specifically, CDW-Z1 and CDW-M1 originating from the lowest imaginary phonon modes have the lowest energy under low and high pressure, respectively.

Role of Apical Oxygen Vacancies.—Experimental evidence suggests the existence of oxygen vacancies with a nonstoichiometry $\delta$ of La₃Ni₂O_7-δ as high as 0.34 [63]. Our calculations reveal that the apical oxygen vacancy has the lowest energy compared with vacancies at other sites, consistent with previous reports [63, 64]. After careful structural optimization for La₃Ni₂O_6.75 with apical oxygen vacancies, two unexpectedly stable structures emerge as shown in FIG. 4(a). One structure exhibits structural distortion only near the defect, while the other shows distortion in all NiO bilayers, labeled as half distortion (HD) and full distortion (FD) phases, respectively. Comparing their enthalpies (H) in FIG. 4(b), we observe a phase transition from the HD to the FD phase at 2 GPa. The H can be divided into two terms, H=E+PV, where E, P, and V represent energy, pressure, and volume, respectively. As seen in FIG. S13, although the HD phase consistently displays lower $E$ at 0-10 GPa, the smaller V of the FD phase makes it thermodynamically more stable under high pressure.

Further band calculations reveal the HD phase is a metal similar to electron doped La₃Ni₂O₇ in FIG. 1(c). Conversely, the FD phase exhibits a gap of about 0.5 eV and a flat valence band. This indicates an MI transition accompanying the phase transition of the two vacancy structures, which is absent in the pure crystal phase. The FD phase of La₃Ni₂O_6.75 in FIG. 4(a) exhibits expansion and contraction for all oxygen octahedra, similar to the CDW phase in FIG. 2(h). This unexpected similarity is intriguing. The lower H of the FD phase above 2 GPa indicates harmony between oxygen vacancies and CDW under high pressure. On the contrary, the HD phase becomes more stable below 2 GPa, suggesting oxygen vacancies suppress structural distortions of CDW at this stage.

Phase Diagram and Discussions.— We summarize the main findings and related experimental data in a phase diagram shown in FIG. 5. It can be seen that the calculated $\textit{T}_{\textit{CDW, cal.}}$ of the CDW-Z1 phase gradually decreases with increasing pressure, reaching zero at 7.5 GPa, nicely aligning with experimental results [4, 52, 7]. Our calculations clearly reveal that the temperature-induced MI transition stems from the structural distortion of CDW [5, 6, 52, 65]. Beyond 27 GPa, the CDW-M1 phase emerges with $\textit{T}_{\textit{CDW, cal.}}>$ 30 K. Recent experiments demonstrate a pronounced insulating behavior under high pressure below 50 K compressed in the diamond anvil cell and above $\textit{T}_{\textit{c}}$ compressed in the cubic anvil cell [5]. These findings may be related to our calculated CDW phases under high pressure, suggesting a potential coexistence or competition between CDW and superconductivity.

We found that the nonstoichiometric La₃Ni₂O_6.75 undergoes an MI transition at 2 GPa, in good consistent with the pressure of MI transition in experiments ($\textit{P}_{\textit{M-I, exp.}}$) as seen in FIG. 5 [3]. The HD and FD phases identified in this work gain the insights for further theoretical and experimental analyses of the role of oxygen vacancies.

The $\mu$SR experiments reveal a bulk AFM transition near 150 K for La₃Ni₂O_7-δ [58, 59]. Additionally, a continuous increase in magnetic susceptibility suggests a potential AFM order below 350 K for La₃Ni₂O_7.05 [52]. Our calculated $\textit{T}_{\textit{N}}$ for the CDW-Z1, pristine-Cmcm and HD phases are 190, 393 and 95 K at 0 GPa, respectively [Table S2], showing that such variation in $\textit{T}_{\textit{N, cal.}}$ among different phases may explain the inconsistency in reported experimental $\textit{T}_{\textit{N}}$.

Summary.—In this work, we report by performing comprehensive DFT calculations that the La₃Ni₂O₇ possesses an antiferromagnetic ground state, with the strong FS nesting of the electronic flat band, which can drive the CDW transitions and trigger the MI transitions. In addition, we reveal that the competition between two different phases with apical oxygen vacancies can lead to a pressure-induced MI transition. These results are in good agreement with the experimental observations. The EPC in La₃Ni₂O₇ is found insufficient to be responsible for superconductivity, suggesting that the Cooper pairing mechanism is unconventional, and may be from the AFM fluctuations. A complex phase diagram consistent with experiments are given. Our calculations explain well the physical origins behind a few experimental results, and give intensive comprehension for the physical properties such as superconductivity, CDW orders and roles of apical oxygen vacancies in La₃Ni₂O₇, in which some results remain to be examined experimentally.

Supplementary Materials for  

Antiferromagnetic Ground State, Charge Density Waves and Oxygen Vacancies Induced Metal-Insulator Transition in Pressurized La₃Ni₂O₇

Yi et al.