Paper

Ultrafast optical evidence of coexisting density waves in bilayer nickelate La₃Ni₂O₇

The discovery of high-temperature superconductivity in La3⁢Ni2⁢O7 at pressures above 14 GPa has spurred extensive research efforts. Yet, fundamental aspects of the superconducting phase, including the possibility of a filamentary character, are currently subjects of controversial debates. Conversely, a crystal structure with NiO6 octahedral bilayers stacked along the 𝑐-axis direction was consistently posited in initial studies on La3⁢Ni2⁢O7. Here, we reassess this structure in optical floating zone-grown La3⁢Ni2⁢O7 single crystals that show signs of filamentary superconductivity. Employing scanning transmission electron microscopy and single-crystal x-ray diffraction under high pressures, we observe multiple crystallographic phases in these crystals, with the majority phase exhibiting alternating monolayers and trilayers of NiO6 octahedra, signifying a profound deviation from the previously suggested bilayer structure. Using density functional theory, we disentangle the individual contributions of the monolayer and trilayer structural units to the electronic band structure of La3⁢Ni2⁢O7, providing a firm basis for advanced theoretical modeling and future evaluations of the potential of the monolayer-trilayer structure for hosting superconductivity.在高于 14 GPa 的压力下， La3⁢Ni2⁢O7 中高温超导性的发现激发了广泛的研究。然而，超导相的基本性质，包括其丝状特征的可能性，目前仍存在争议。相反，在对 La3⁢Ni2⁢O7 的早期研究中，人们一直假设其晶体结构为沿 𝑐 轴方向堆叠的 NiO6 八面体双层。本文中，我们重新评估了光学浮区法生长的 La3⁢Ni2⁢O7 单晶中的这种结构，这些单晶表现出丝状超导的迹象。利用扫描透射电子显微镜和高压下的单晶 X 射线衍射技术，我们观察到这些晶体中存在多种晶相，其中主要相由 NiO6 八面体的单层和三层交替构成，这表明其与先前提出的双层结构存在显著偏差。利用密度泛函理论，我们分离出单层和三层结构单元对 La3⁢Ni2⁢O7 电子能带结构的各个贡献，为高级理论建模和未来评估单层-三层结构承载超导性的潜力提供了坚实的基础。

本文对双层镍酸盐La₃Ni₂O₇单晶开展了最高16.51 GPa静水压下的系统电子拉曼散射研究，追踪自旋密度波（SDW）态随压力的演化。结果显示，SDW能隙能量和SDW转变温度均随压力增加而总体上升，而无量纲耦合比2Δ_SDW/(k_BT_SDW)始终保持在约7.5，表明SDW态具有稳健的强耦合特征；同时，SDW拉曼峰随压力逐渐展宽，说明长程SDW有序相干性逐步减弱。该工作揭示了一种不同寻常的压力演化行为：SDW能量尺度被增强，但SDW态变得越发不连贯，为双层镍酸盐中与高温超导相关的磁关联提供了光谱学约束。

该研究采用第一性原理路径积分分子动力学方法，全面纳入原子核量子多体效应，构建自由能面，在H3S和La3Ni2O7中发现了晶格量子无序（LQD）相。通过对比经典与量子相边界，LQD相在压力-温度相图中形成一个三角形区域，其左边界与超导穹顶左侧翼的转变温度精确对齐，且LQD相的最高温度Tcmax与超导Tc最大值重合，同位素效应亦被准确捕捉。这揭示穹顶左翼超导起源于低对称相向量子无序相的转变，超导态完全位于高对称相，从而推翻双相解释。研究确立了LQD相作为统一框架，不仅阐明了非常规超导电性中晶格量子多体效应的关键作用，还为预测更高Tc超导体及解释凝聚态反常现象开辟了新途径。

本研究针对高压下高转变温度的La₃Ni₂O₇块材与常压超导但Tc减半的薄膜之间的差异，发展了一种基于对称性的唯象方法，结合DFT+U计算和实验确定的Tc及结构对称性，分析超导能隙结构。研究发现，两者均呈现s±波配对对称性和两带超导，但主导的微观配对构型不同：加压块材中超导主要由Ni-dz²轨道的面外配对主导，而薄膜中则由Ni-dx²-y²轨道的面内配对主导。Tc的降低归因于薄膜中层间与层内跳跃比率减小，导致主导配对类型从面外转向面内。该结果揭示了对称性在非常规超导中的关键作用，所发展的方法有望推广至其他非常规超导体。

The family of nickelate superconductors have long been explored as analogs of the high temperature cuprates. Nonetheless, the recent discovery that certain stoichiometric nickelates superconduct up to high $T_c$ under pressure came as a surprise. The mechanisms underlying the superconducting state remain experimentally unclear. In addition to the practical challenges posed by working in a high pressure environment, typical samples exhibit anomalously weak diamagnetic responses, which have been conjectured to reflect inhomogeneous `filamentary’ superconducting states. We perform wide-field, high-pressure, optically detected magnetic resonance spectroscopy to image the local diamagnetic responses of as grown La$_3$Ni$_2$O$_7$ samples \emph{in situ}, using nitrogen vacancy quantum sensors embedded in the diamond anvil cell. These maps confirm significant inhomogeneity of the functional superconducting responses at the few micron scale. By spatially correlating the diamagnetic Meissner response with both the local tensorial stress environment, also imaged \emph{in situ}, and stoichiometric composition, we unravel the dominant mechanisms suppressing and enhancing superconductivity. Our wide-field technique simultaneously provides a broad view of sample behavior and excellent local sensitivity, enabling the rapid construction of multi-parameter phase diagrams from the local structure-function correlations observed at the sub-micron pixel scale.

Unconventional charge and spin density-wave states are commonly observed in bilayer nickelates, drawing considerable attention due to their proximity to high-$T_c$ superconductivity in various phase diagrams. However, the nature and mechanisms of charge and spin density-waves (DWs) in nickelates remain poorly understood. Numerous experiments have reported that the charge-density-wave (CDW) transition temperature $T_{ cdw}$ and the spin-density-wave (SDW) transition temperature $T_{sdw}$ are closely related but distinct. However, in contrast to these experiments, previous mean-field-type analyses have yielded only a simple SDW phase. To resolve this key problem, this paper demonstrates that sizable CDW instabilities emerge in proportion to the SDW instability in La$3$Ni$2$O$7$.This behavior is driven by the paramagnon-interference (PMI) mechanism, which captures important electron correlations beyond mean-field theory. Therefore, (i) experimental CDW + SDW coexisting state is naturally explained. In addition, (ii) the CDW + SDW fluctuations cooperatively drive high-$T_c$ superconductivity. Notably, the predicted $s$-wave SC state is robust against the inner apical O vacancies. Furthermore, (iii) the CDW instability is highly sensitive to the size of the $d_{z^2}$-orbital hole pocket, allowing for the realization of CDW quantum criticality through carrier-doping and pressure application. We find that the coexistence of charge and spin fluctuations is essential in bilayer nickelates, with both playing a cooperative role in mediating high-$T_c$ superconductivity.

Unconventional charge- and spin-density-wave states are commonly observed in bilayer nickelates, drawing considerable attention due to their proximity to high-transition temperature ( $${T}_{{\rm{c}}}$$) superconductivity. However, the nature and origin of these density waves remain poorly understood. Experiments show that the charge-density-wave and spin-density-wave transition temperatures are closely related but distinct, while mean-field-type analyses typically have yielded only a simple spin-density-wave phase. To resolve this key problem, this paper demonstrates that sizeable charge-density-wave instabilities emerge in proportion to spin-density-wave instabilities in La3Ni2O7 due to the paramagnon-interference mechanism, which captures electron correlations beyond mean-field theories. Therefore, (i) the experimental charge- and spin-density-wave coexisting state is naturally explained, and (ii) charge- and spin-density-wave fluctuations cooperatively drive high- $${T}_{{\rm{c}}}$$superconductivity. Furthermore, the predicted s-wave superconducting state is robust against the inner-apical oxygen vacancies. We find that the coexistence of charge- and spin-fluctuations is essential in bilayer nickelates, with both playing a cooperative role in mediating high- $${T}_{{\rm{c}}}$$superconductivity.

The discovery of pressure-induced superconductivity in two- and three-layer Ruddlesden-Popper nickelates has generated significant interest in these materials as a platform for unconventional superconductivity. While their ground state exhibits magnetism, a direct determination of their magnetic structure remains elusive. Understanding this aspect is crucial, as magnetism may play a role in the pairing mechanism of superconductivity in these compounds. We resolve the magnetic structures of the bilayer (2222) polymorphs of La3Ni2O7 and La2PrNi2O7 using neutron powder diffraction (NPD) and muon-spin rotation/relaxation (muSR). Magnetic neutron scattering appears below approximately 150 K in both compounds and is observed at the (qx, 1/2, 0) position, with qx = 0 and 1/2 for La3Ni2O7 and qx = 0 for La2PrNi2O7. Within a single layer, alternating low (0.05 - 0.075 muB) and high (0.66 muB) magnetic moment stripes form. These layers stack antiferromagnetically along the c-direction to form bilayers. The presence of two propagation vectors (qx = 0 and 1/2) in undoped La3Ni2O7 suggests the coexistence of two magnetic stacking polymorphs within a single crystallographic phase. The muSR spectra further confirm these magnetic structures. Our findings provide a detailed understanding of the magnetic ground state in bilayer nickelates, offering insights into possible precursor states that may influence the emergence of superconductivity in these materials.

The recent discovery of superconductivity in La3Ni2O7−δ under high pressure with a transition temperature around 80 K (ref. 1) has sparked extensive experimental2–6 and theoretical efforts7–12. Several key questions regarding the pairing mechanism remain to be answered, such as the most relevant atomic orbitals and the role of atomic deficiencies. Here we develop a new, energy-filtered, multislice electron ptychography technique, assisted by electron energy-loss spectroscopy, to address these critical issues. Oxygen vacancies are directly visualized and are found to primarily occupy the inner apical sites, which have been proposed to be crucial to superconductivity13,14. We precisely determine the nanoscale stoichiometry and its correlation to the oxygen K-edge spectra, which reveals a significant inhomogeneity in the oxygen content and electronic structure within the sample. The spectroscopic results also reveal that stoichiometric La3Ni2O7 has strong charge-transfer characteristics, with holes that are self-doped from Ni sites into O sites. The ligand holes mainly reside on the inner apical O and the planar O, whereas the density on the outer apical O is negligible. As the concentration of O vacancies increases, ligand holes on both sites are simultaneously annihilated. These observations will assist in further development and understanding of superconducting nickelate materials. Our imaging technique for quantifying atomic deficiencies can also be widely applied in materials science and condensed-matter physics.