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[Hiroya Sakurai](https://orcid.org/0000-0003-1964-6023), [Yoshihiko Takano](https://orcid.org/0000-0002-1541-6928)

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[Superconducting lanthanum nickel oxides with bilayered and trilayered crystal structures](https://mdr.nims.go.jp/datasets/da8d53f9-262a-4e5e-9edb-eb2b33a4489c)

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Superconducting Lanthanum Nickel Oxides with Bilayered and Trilayered Crystal StructuresSuperconducting Lanthanum Nickel OxidesHiroya Sakurai and Yoshihiko TakanoNational Institute for Materials ScienceFebruary 19, 2026AbstractIn 2023, superconductivity in La3Ni2O7 was discovered under high pressuresabove approximately 14 GPa. In addition to its high transition temperature(Tc ≃ 80 K), the structural resemblance to high-Tc cuprates has strongly stimu-lated research, soon followed by the discovery of superconductivity in La4Ni3O10.These compounds belong to the Ruddlesden–Popper phases, comprising double-and triple-layered NiO2 square lattices separated by LaO rock-salt slabs.Research on these systems has rapidly developed along three major direc-tions, as in other prominent families of superconductors such as the cupratesand iron arsenides: expanding the chemical variety of compounds, enhancingTc through elemental substitution, and elucidating the superconducting mech-anism. These challenges, being closely interconnected, continue to drive thefield. The clarification of the pairing mechanism encounters a particular diffi-culty, since the key experiments must be performed under high pressures. Thissituation highlights the significance of developing nickel oxides that exhibit su-perconductivity at much lower pressures, ideally at ambient pressure, whichwould in turn broaden the scope of chemical tuning and detailed physical char-acterization.In this context, it is timely and meaningful to summarize the present stateof knowledge. Here, we emphasize sample synthesis and characterization, whichare already well established and often decisive for progress in unconventionalsuperconductors, while providing a brief overview of the currently available elec-tronic properties.1 IntroductionIn 2023, La3Ni2O7 was discovered to exhibit superconductivity under pressuresabove approximately 14 GPa [1]. The transition temperature reaches nearly80 K. In addition to this relatively high Tc, the compound shares a structuralfeature with high-Tc cuprates, as revealed shortly thereafter, which triggeredintense research on this compound and related materials. Indeed, supercon-ductivity was soon reported in La4Ni3O10 above 33 GPa [2], and several new1arXiv:2603.17657v1  [cond-mat.supr-con]  18 Mar 2026https://arxiv.org/abs/2603.17657v1nickel oxides with similar structural features have since been synthesized to ex-plore superconductivity, despite the experimental challenges under high pressure[3, 4, 5, 6, 7].La3Ni2O7 and La4Ni3O10 crystallize in the Ruddlesden–Popper (RP) phase,with the general formula Lan+1NinO3n+1. Their crystal structure consists of al-ternating LaNiO3 perovskite slabs and LaO rock-salt slabs, as shown in Fig. 1.In the perovskite slabs, NiO6 octahedra are corner-sharing, forming n-layerNiO2 square lattices, where Ni ions occupy the sites equivalent to Cu in theCuO2 planes of cuprates. It should be noted that the Ni valence in the RPseries increases from divalent to trivalent as n increases. For example, La2NiO4(n = 1), which contains only divalent Ni ions, is a two-dimensional antiferro-magnetic insulator with a charge-transfer energy gap [8, 9], similar to the parentcompounds of cuprates. Upon partial substitution with trivalent Ni ions, thiscompound exhibits a charge and spin stripe state, which has been extensivelystudied for its possible relevance to stripe order in cuprates [10]. On the otherhand, LaNiO3 (n = ∞), which contains only trivalent Ni ions, remains metal-lic down to the lowest temperatures, though it can become insulating whenthe bandwidth is reduced by substituting La with smaller lanthanide ions [11].Furthermore, LaNiO3 can be regarded as the parent compound of infinite-layernickelate superconductors, which have also attracted much attention due totheir electronic states analogous to those of cuprates [12, 13]. The Ni valence inthe infinite-layer nickelates lies between monovalent and divalent, in contrast toLa3Ni2O7 and La4Ni3O10, which are situated on the opposite side of the phasediagram with respect to La2NiO4 [14].These diverse electronic states indicate that the superconductivity in La3Ni2O7and La4Ni3O10 is likely unconventional, rooted in their unique electronic struc-tures. The purpose of this review is to summarize the experimental results onthese nickel oxides available at present, with the hope of stimulating further re-search that will lead to the development of a wider variety of related materialsand a deeper understanding of their rich physical properties.2 Crystal Structure2.1 Ruddlesden–Popper phaseThe crystal structure of the Ruddlesden–Popper (RP) phase, An+1MnX3n+1,consists of AMX3 perovskite slabs and AX rock-salt slabs, as described abovefor Lan+1NinO3n+1. To highlight this, the chemical composition can also beexpressed as (AX)(AMX3)n. The ideal structure has tetragonal I4/mmm sym-metry for n ̸= ∞. Within each slab, the A and X atoms adopt a cubic close-packed arrangement, whereas the interface between the slabs deviates from closepacking, as shown in Fig. 1. In the perovskite slabs, the M ions form n layersof square lattices as previously mentioned.The ideal perovskite AMX3 (n = ∞) has cubic Pm3̄m symmetry, and itslattice constant corresponds to the distance between neighboring M atoms. For2Figure 1: Crystal structures of La3Ni2O7 (a) and La4Ni3O10 (b), with or-thorhombic Amam and monoclinic P21/a symmetries, respectively. Panel (c)shows a top view of the perovskite and rock-salt slabs. Green, red, and bluecircles represent La, Ni, and O atoms, respectively. Solid and dashed blacklines in panel c indicate the actual unit cell of La3Ni2O7 and a virtual unit cellassuming tetragonal I4/mmm symmetry, respectively.3Figure 2: (a) Average Ni–O–Ni bond angle in perovskite nickel oxides, and (b)average Ni–Ni distance together with√2(rR + rO), plotted as functions of theionic radius of the rare-earth ion for ninefold coordination estimated by Shannon[16].smaller A ions, the structure is stabilized by rotations of the AMX6 octahe-dra. More specifically, when the tolerance factor t = (rA + rX)/√2(rM + rX)(with rX the ionic radius of X) is larger than about 0.9, the ideal perovskitestructure is favored. For smaller t, the MX6 octahedra tilt to reduce the A-sitevolume [15]. Typically, the LaAlO3-type rhombohedral (R3̄c) structure appearsfor relatively large t, while the GdFeO3-type orthorhombic (Pbnm) structureoccurs for smaller t, in the range 0.7 < t < 0.9. In fact, LaNiO3 adopts theLaAlO3-type distortion, whereas RNiO3 (R = rare earths) perovskites withsmaller ionic radii show the GdFeO3-type structure for R = Pr–Dy, and furtherdistorted monoclinic (P21/n) structures for R = Dy–Lu [11]. In these nickelateperovskites, the Ni–O bond length is almost independent of the rare-earth ion[11], with NiO6 octahedra behaving nearly as rigid units. Only the average Ni–O–Ni bond angle varies systematically with the ionic radius of R, as illustratedin Fig. 2a.The rock-salt structure is realized for rA/rX = 0.41–0.73 [17], which corre-sponds to the condition that A atoms occupy the octahedral sites created byclose-packed X atoms. However, many rock-salt compounds, such as CsF andBaO, exhibit significantly larger values than 0.73, indicating that for large Aions, both A and X atoms contribute comparably to the close packing. Thisapplies to LaO rock-salt slabs, since rLa/rO = 0.87 > 0.73. The lattice constantof an ideal rock-salt slab can be estimated as aRS = 2(rA + rX), which matchesthe diagonal spacing of the square M lattice in the ideal RP structure. Thus, nomismatch arises between the rock-salt and perovskite slabs when√2(rA + rX)equals the M–M distance. Because the X atoms in the rock-salt slabs are sharedwith the perovskite slabs, the way to tilt the MX6 octahedra is severely restrictedin RP phases with n ̸= ∞. This likely explains why only a limited number of4RP phases with n = 2 or 3 are known. For example, with A = La and X = O,only La3Ni2O7, La4Ni3O10, and La4Co3O10 have been reported when contain-ing only a single transition-metal element, which is surprising given that LaMO3perovskites exist for all 3d transition metals. The variety of A ions for Ni ox-ides is also limited, likely becausue the mismatch between the perovskite androck-salt slabs expands for smaller A ions as suggested in Fig. 2b. For a singleA ion, no n = 2 Ni oxide was known except for R = La, and for n = 3, onlyR = Pr and Nd compounds were reported [18], which is in sharp contrast withthe variety of ANiO3 perovskite [11]. Unlike the case of perovskites, no simpleand widely applicable parameter like the tolerance factor has been establishedfor RP phases, presumably because it is difficult to describe the matching be-tween perovskite and rock-salt slabs, although several attempts have been made[19, 20, 21].2.2 La3Ni2O7 and La4Ni3O10For La3Ni2O7, three types of crystal structures at ambient pressure and roomtemperature have been widely accepted: tetragonal I4/mmm (#139) [22], or-thorhombic Fmmm (#69) [22, 23, 24, 25], and orthorhombic Amam (#63)[26, 27, 28, 29]. Other symmetries have also been suggested [30, 31], but theyare not generally accepted. The tetragonal symmetry is observed for oxygen-deficient La3Ni2O6.35, suggesting that it is stabilized when the oxygen contentis significantly below the stoichiometric value. By contrast, the origin of thetwo orthorhombic structures remains unclear, and in some samples both areobserved to coexist [32]. Care should be taken in structure determination, asgood refinement of X-ray diffraction (XRD) data with the Fmmm model doesnot necessarily mean that the true structure is Fmmm [33].In the orthorhombic structures, the a and b axes correspond to the diagonaldirections of the c-plane unit cell of the I4/mmm structure, so their lengthsare approximately√2 times the a parameter of the tetragonal phase. In theFmmm structure, NiO6 octahedra alternately contract and expand along thea and b axes without tilting from the c axis, and the Ni–O–Ni bonds along cremain linear. In contrast, in the Amam structure, the Ni–O–Ni bonds arebent to about 168◦ due to tilting of the NiO6 octahedra. It is widely acceptedthat the superconducting samples have the orthorhombic Amam structure atambient pressure, as superconductivity is thought to emerge when this bendingis removed by a pressure-induced structural transition.The orthorhombic Amam phase of La3Ni2O7 becomes unstable at around 10GPa at room temperature, undergoing a transition to the orthorhombic Fmmmstructure near 15 GPa [1]. This structural transition has been supported byseveral XRD measurements [1, 34, 35] and DFT calculations [1], although theslopes of the phase boundary reported in these studies are not consistent witheach other [34, 35]. As mentioned above, distinguishing these structures by XRDpatterns is difficult, particularly near the phase boundary, as the transition isfirst-order and the phases may coexist. In general, DFT calculations do notaccount for effects arising at finite temperatures; thus, careful characterization5is required to accurately determine the structural transition.At low temperatures under high pressure, the Fmmm structure furthertransforms into the tetragonal I4/mmm structure, accompanied by a clearerchange in the XRD pattern [34], strongly suggesting that superconductivity oc-curs in the tetragonal phase, the ideal structure of the RP phase. Since thetetragonal I4/mmm structure is also stabilized at high temperatures aboveapproximately 450◦C under ambient pressure [31], the Fmmm phase, if it ex-ists, may be inserted between the I4/mmm phases or separates them. ForLa2PrNi2O7, the Amam structure transforms directly into the I4/mmm struc-ture even around room temperature, without passing through the Fmmm phase[36].Also for La4Ni3O10, three types of crystal structures are recognized at roomtemperature and ambient pressure: orthorhombic Fmmm (#69) [18], orthorhom-bic Bmeb (#64) [26, 27, 28, 29, 37, 38], and monoclinic P21/a (#14) [39, 37, 38].At higher temperatures above approximately 750◦C, the tetragonal I4/mmmstructure appears [31, 39]. The orthorhombic Bmeb structure can be metastable[37]; in fact, it is reported to occur at temperatures between approximately600◦C and 700◦C [39]. In the Fmmm structure, the bridging angle of Ni–O–Ni along the c axis is straight, as in La3Ni2O7, whereas in the orthorhombicBmeb and monoclinic P21/a structures, the angle bends from 180◦. After thediscovery of superconductivity in La4Ni3O10 [2], it was found that the super-conductivity also occurs in the I4/mmm structure [40, 41, 42]. Differently fromLa3Ni2O7, this ideal RP structure is preserved even at room temperature underpressures above approximately 33 GPa [40, 41, 42].The occurrence of the I4/mmm structures in La3Ni2O7 and La4Ni3O10 un-der pressure appears reasonable because the oxygen anions are likely to shrinkmore under pressure than the cations. Assuming the pressure dependence of rNiand rLa is negligible, the derivative of the tolerance factor with respect to theoxygen ionic radius, ∂t/∂rO = (rNi − rLa)/√2(rNi + rO)2, is negative becauserNi < rLa.2.3 Isovalent Substitution of La Ions – Chemical Pressure–As mentioned above, superconductivity in La3Ni2O7 and La4Ni3O10 occurswhen the compounds adopt the tetragonal I4/mmm symmetry. This is con-sistent with theoretical expectations that the hopping integral between two Niions along the c-axis, t⊥, is crucial for the emergence of superconductivity [43].In this context, one might consider that substitution of La ions with smallertetravalent ions could induce chemical pressure, removing the orthorhombic ormonoclinic distortion and enabling superconductivity at ambient pressure. How-ever, this simple idea cannot be applied. The orthorhombicity, b/a, increaseswhen smaller ions such as Pr or Nd substitute La, as shown in Table 1, re-flecting the reduction of the tolerance factor for smaller ions. This trend hasbeen confirmed in systematic experiments [44, 45]. Consequently, for smallerR ions, higher pressures are required to eliminate the distortion, as suggested6by theoretical calculations for La3Ni2O7 [46] and supported by experimentalobservations [44]. This illustrates that internal chemical pressure and externalapplied pressure can have different effects on the structural transition.There are two distinct La sites in La3Ni2O7 and La4Ni3O10: La(1) in theperovskite slabs, and La(2) in the rock-salt slabs. Substitution of La ions withsmaller ions may affect the transition pressure differently depending on whichsite is predominantly substituted. For example, if the La(2) sites were fully sub-stituted while the La(1) sites remained unchanged, the perovskite slabs wouldbe compressed in the ab plane relative to the pure La compounds, which couldreduce the transition pressure. From this perspective, it is noteworthy that asample of La2PrNi2O7 showed a lower transition pressure of 11 GPa comparedwith 14 GPa for La3Ni2O7 [36], despite the smaller average ionic radius of theLa/Pr sites relative to pure La, although the site occupancy of Pr ions is notreported. Since the site occupancy of La and substituted ions depends on syn-thesis conditions, such as cooling rate and oxygen partial pressure, the transitionpressure may be tunable by adjusting these conditions.2.4 Oxygen NonstoichiometryThe crystal structures of La3Ni2O7 and La4Ni3O10 are not well defined as men-tioned above, and their physical properties are sample-dependent, as shownlater. In such cases, oxygen content is often the primary cause of the sampledependence. Indeed, the oxygen content of Ni oxides can vary depending onsynthesis conditions. Here, oxygen defects are discussed from the viewpointof crystal structures, while their precise control will be addressed in a latersection. A deficiency of La and/or Ni ions may also be possible, consideringperovskite-related materials such as ReO3, La2/3TiO3 [64], and La1−ϵMn1−ϵO3[65]. However, no reports on cation deficiency in La3Ni2O7 or La4Ni3O10 areavailable.Oxygen deficiency readily occurs in the perovskite slabs. It is well knownthat perovskite oxides, AMO3, can accommodate oxygen loss. When 0.5 oxy-gen atoms per formula unit are systematically removed, brownmillerite-typestructures appear, as in Sr2Fe2O5 [66]. The crystal structure of the infinite-layer nickelate superconductor is also derived from perovskite Ni oxides [12].La3Ni2O6 and La4Ni3O8 can be obtained by nearly topotactic removal of oxy-gen atoms located between two Ni atoms along the c-axis (inner apical oxygen),although the rock-salt slabs between perovskite layers transform into fluorite-type structures in both compounds [67, 68, 69]. The critical oxygen contentrequired to preserve the rock-salt structure remains unclear. However, since thefluorite structure in La4Ni3O8 reverts to the rock-salt structure above 21 GPa[70], the enthalpy gain associated with the rock-salt to fluorite transformationis likely small. These facts suggest that oxygen deficiency predominantly occursin the perovskite slabs, specifically at the inner apical oxygen sites. Indeed,oxygen loss at these sites has been directly observed in La3Ni2O7 by neutrondiffraction [71] and multislice electron ptychography [72]. Upon oxygen loss, ad-jacent NiO5 pyramids face each other base-to-base. Since this structural motif7Table 1: Lattice parameters of La3Ni2O7 with orthorhombic Amam structure,La4Ni3O10 with monoclinic P21/a structure, and their La-substituted materi-als at room temperature and ambient pressure. The oxygen contents of somematerials are nominal. ∗: Fmmm was tentatively used to estimate the latticeparameters.Composition Space Group a (Å) b (Å) c (Å) β (◦) b/a Ref.La3Ni2O7.02 Amam 5.39283 5.44856 20.5185 – 1.01033 [26]La3Ni2O7.05 Amam 5.39710 5.45011 20.5074 – 1.00982 [26]La3Ni2O7 Amam 5.3981 5.4494 20.502 – 1.0095 [27]La3Ni2O7 Amam 5.392 5.447 20.517 – 1.010 [28]La3Ni2O7 Amam 5.3959 5.4495 20.5350 – 1.0099 [29]La3Ni2O7.01 Amam 5.4000 5.4384 20.455 – 1.0071 [47]La3Ni2O6.93 Amam 5.3920 5.4480 20.5311 – 1.0104 [48]La3Ni2O6.93 Amam 5.38996 5.44719 20.5305 – 1.01062 [35]La3Ni2O7.01 Amam 5.3903 5.4464 20.507 – 1.0104 [49]La3Ni2O7 Amam 5.392 5.447 20.517 – 1.010 [28]La3Ni2O6.92 Amam 5.4018 5.4557 20.537 – 1.0100 [50]La3Ni2O7 Amam 5.39885 5.45130 20.5163 – 1.00972 [51]La3Ni2O7 Amam 5.407 5.4176 20.490 – 1.002 [52]La2PrNi2O7.01 Amam 5.37318 5.45274 20.41213 – 1.01481 [36]La4Ni3O10 P21/a 5.4151 5.4714 14.2277 100.818 1.0104 [37]La4Ni3O10 P21/a 5.4160 5.4656 27.9750 90.179 1.0092 [31]La4Ni3O10 P21/a 5.4243 5.4748 28.0053 90.192 1.0093 [53]La4Ni3O10 P21/a 5.42335 5.47315 28.00414 90.1475 1.00918 [54]La4Ni3O10.03 P21/a 5.403 5.454 14.24 100.70 1.009 [55]La4Ni3O10 P21/a 5.4142 5.4647 14.2208 100.66 1.0093 [40]La4Ni3O10 P21/a 5.4164 5.4675 14.2279 100.752 1.0094 [42]La4Ni3O10 P21/a 5.4247 5.4596 14.2422 100.9663 1.0064 [56]La4Ni3O10 P21/a 5.4132 5.4627 14.2337 100.71 1.0091 [57]La4Ni3O10 P21/a 5.4131 5.4638 14.2461 101.26 1.0094 [57]La4Ni3O10 P21/a 5.4082 5.4533 14.2548 100.94 1.0083 [57]La4Ni3O10 P21/a 5.4184 5.4655 14.2278 100.841 1.0087 [58]La4Ni3O10 P21/a 5.4185 5.4678 14.2296 100.7245 1.0091 [59]La4Ni3O9.99 P21/a 5.4162 5.4642 27.984 90.256 1.0089 [60]Pr4Ni3O10 Fmmm∗ 5.370 5.462 27.528 – 1.017 [18]Pr4Ni3O10 Fmmm∗ 5.372 5.462 27.532 – 1.017 [61]Pr4Ni3O10.06 Fmmm∗ 5.369 5.464 27.523 – 1.018 [61]Pr4Ni3O10.1 Fmmm∗ 5.356 5.463 27.548 – 1.020 [61]Pr4Ni3O10 Fmmm 5.3714 5.4611 27.5271 – 1.0167 [62]Pr4Ni3O10 P21/a 5.3816 5.4711 14.0284 100.646 1.0166 [37]Pr4Ni3O10.1 P21/a 5.3705 5.4637 27.5728 90.303 1.0174 [63]Pr4Ni3O10.1 P21/a 5.37556 5.46462 27.5463 90.283 1.01657 [31]Pr4Ni3O10 P21/a 5.3826 5.4717 27.583 90.284 1.0166 [53]Pr4Ni3O9.97 P21/a 5.372 5.458 14.02 100.81 1.016 [55]Nd4Ni3O10 Fmmm∗ 5.362 5.454 27.410 – 1.017 [18]Nd4Ni3O10.1 P21/a 5.36373 5.45221 27.4100 90.292 1.01650 [31]Nd4Ni3O10 P21/a 5.3719 5.46 27.4506 90.299 1.0164 [53]Nd4Ni3O9.93 P21/a 5.351 5.441 13.93 100.79 1.017 [55]La3PrNi3O10.1 P21/a 5.396 5.458 14.17 100.71 1.011 [55]La2Pr2Ni3O10.01 P21/a 5.382 5.457 14.12 100.71 1.014 [55]LaPr3Ni3O10 P21/a 5.382 5.466 14.07 100.78 1.016 [55]La3NdNi3O9.95 P21/a 5.401 5.466 14.19 100.68 1.012 [55]La2Nd2Ni3O9.99 P21/a 5.375 5.463 14.14 100.67 1.016 [55]LaNd3Ni3O9.98 P21/a 5.363 5.459 14.08 100.73 1.018 [55]Pr3NdNi3O9.95 P21/a 5.361 5.451 13.98 100.79 1.017 [55]Pr2Nd2Ni3O9.93 P21/a 5.358 5.449 13.96 100.78 1.017 [55]PrNd3Ni3O10.07 P21/a 5.356 5.447 13.95 100.80 1.017 [55]8Figure 3: Local structure of La2NiO4.25 [76]. Numbered circles represent Laatoms at their respective sites, while blue and pink atoms denote excess andoriginal oxygen atoms, respectively.is also found in other compounds such as YBa2Cu3O7−δ, this type of oxygendeficiency appears to be a common feature of RP phases with n ̸= 1.No detailed information has been reported regarding the incorporation of ex-cess oxygen in La3Ni2O7 and La4Ni3O10. However, detailed studies of La2NiO4[73, 74] strongly suggest that excess oxygen is located within the rock-salt slabs.The rock-salt slabs of La2CuO4 also accommodate excess oxygen [75], support-ing this assignment. Thus, if excess oxygen exists in La3Ni2O7 and La4Ni3O10,it is also most likely located in the rock-salt slabs. For La2NiO4+δ, excess oxy-gen atoms form commensurate ordered structures at δ = 1/4 (0.25), 1/6 (0.17),1/8 (0.13), and so on [74]. In La2NiO4.25, the introduction of excess oxygeninduces a strong distortion of the crystal structure from orthorhombic Bmabto monoclinic C2 [76]. In this phase, four of the eight La sites with Wyckoffposition 4c in the rock-salt slabs are located adjacent to an excess oxygen atom,as illustrated in Fig. 3. Because excess oxygen in the rock-salt slabs modifiesthe tilting pattern of NiO6 octahedra [77], it significantly alters the crystal fieldsat the Ni sites. Thus, excess oxygen in La3Ni2O7 and La4Ni3O10 can stronglyinfluence physical properties beyond the rigid-band picture.The kinetics of excess oxygen atoms is also important. RP phases of nickeloxides exhibit such high oxygen-ion conductivity that they are considered aspotential cathode materials for solid oxide fuel cells (SOFCs) [78]. Thus, even ifhomogeneous samples are obtained at high temperatures, they may undergo or-der–disorder transitions of excess oxygen atoms upon cooling, leading to phase9separation. Indeed, La2NiO4+δ exhibits multiple phases below room temper-ature [73, 10, 79, 80], indicating that oxygen content—and hence band fill-ing—cannot be controlled continuously and homogeneously at ambient temper-atures. Furthermore, the order–disorder transitions are slow [79, 80], potentiallyreducing reproducibility of experimental results unless special care is taken whenexcess oxygen is present in the rock-salt slabs. Notably, phase separation alsooccurs in single crystals.As mentioned above, La3Ni2O7 and La4Ni3O10 can accommodate both oxy-gen deficiency and excess oxygen. Thus, even nominally stoichiometric oroxygen-deficient samples may contain excess oxygen in the rock-salt slabs, be-cause oxygen atoms located between Ni ions along the c-axis can easily migrateto the rock-salt layers. This type of defect is widely recognized in solids and isknown as a Frenkel defect. In short, La3Ni2O7 and La4Ni3O10 have a strongtendency to form oxygen Frenkel defects. Interestingly, nearly stoichiometricLa3Ni2O7 exhibits two NQR (nuclear quadrupole resonance) peaks for La(2)sites, indicating two distinct local environments [81]. The smaller peak hasbeen attributed to La(2) sites adjacent to oxygen vacancies between Ni ionsalong the c-axis. From the relative intensities of the two peaks, the oxygen defi-ciency was estimated to be approximately 1/8, which coincides with the δ valuethat induces a commensurate superstructure in La2NiO4.17, despite the nearlystoichiometric oxygen content of the sample (7.01) [49]. This suggests that ap-proximately one-eighth of the oxygen atoms migrate to the rock-salt slabs asFrenkel defects. Notably, even single crystals contain Frenkel defects—likely ingreater abundance than powders—since they are typically synthesized at muchhigher temperatures. Because excess oxygen in the rock-salt slabs can inducephase separation, the small superconducting volume fractions discussed belowmay originate from phase separation caused by such Frenkel defects.2.5 Stacking Faults and Related MaterialsStacking faults have been studied more extensively and are often consideredmore significant than Frenkel defects. RP-phase compounds usually containstacking faults because of the structural similarity among phases with differ-ent n, which likely possess comparable formation enthalpies. High-resolutiontransmission electron microscopy clearly reveals stacking faults in La3Ni2O7and La4Ni3O10 [33, 82, 83, 84], although their quantitative evaluation remainsdifficult. These common defects were once regarded as highly relevant to thesuperconductivity of La3Ni2O7, leading to the proposal that superconductiv-ity might be filamentary, arising at the interfaces with La4Ni3O10 inclusions,present as stacking faults, rather than being a bulk property [85]. This viewwas supported by the small superconducting volume fraction estimated from ACsusceptibility [86]. However, this scenario is now less convincing, as large super-conducting volume fractions of ∼ 41% at 22.0 GPa were reported for La3Ni2O7[87], and La2PrNi2O7 also exhibits large values of approximately 57% at 20 GPaand 97% at 19 GPa [36].Interestingly, a new polymorph of La3Ni2O7 has been discovered [3, 4, 35],10Figure 4: Crystal structure of the newly discovered 1313 phase of lanthanumnickelate.which is related to stacking-fault–like structural motifs. The conventional La3Ni2O7is composed of alternating double perovskite slabs and single rock-salt slabs, asdescribed earlier. By contrast, the new La3Ni2O7 consists of alternating singleand triple perovskite slabs, replacing the double perovskite slabs of the con-ventional structure, as illustrated in Fig. 4. In short, it is a chimera of theconventional n = 1 and 3 phases, and it has been named the 1313 phase to dis-tinguish it from the conventional La3Ni2O7, now referred to as the 2222 phase.The space group of the 1313 phase is reported as orthorhombic Fmmm (#69)[3] or Cmmm (#65) [4, 35] at ambient pressure. In either case, the Ni–O–Nibridging angles along the c-axis in the n = 3 perovskite slabs deviate from 180◦.For the Fmmm structure, this angle increases gradually under pressure above∼6 GPa and reaches 180◦ via a structural transition at 12.3 GPa [3]. Super-conductivity then emerges above ∼8 GPa, with transition temperatures com-parable to those of the 2222 phase. For the Cmmm structure, no high-pressuremeasurements have yet been reported, and thus it remains uncertain whethersuperconductivity occurs. At ambient pressure, one sample shows semiconduct-ing behavior [35], while another displays metallic behavior with an anomaly inresistivity near 134 K, likely due to density-wave formation, as observed in the2222 phase [70]. Oxygen nonstoichiometry, Frenkel defects, stacking faults, andother sample-dependent factors are, of course, also expected to influence thephysical properties of the 1313 phase.The discovery of polymorphism in La3Ni2O7 broadens the landscape ofRP-related materials. For example, shortly thereafter, La2NiO4·La3Ni2O7 (=La5Ni3O11) was reported to adopt a chimera structure combining the n = 1and n = 2 phases [6]. This compound no longer follows the general RP chem-11ical formula An+1MnX3n+1. The 1212 phase crystallizes in the orthorhombicImmm (#71) structure at ambient pressure, where the Ni–O–Ni bridging anglealong the c-axis in the n = 2 perovskite slabs is 180◦. Its resistivity shows ametal–semiconductor crossover with temperature, but no superconductivity hasbeen observed in either resistivity or magnetic susceptibility measurements onas-grown single crystals [6]. In this review, La3Ni2O7 refers to the 2222 phaseunless otherwise specified.3 Synthesis & Characterization3.1 Powder SynthesisLa3Ni2O7 and La4Ni3O10 were synthesized intentionally for the first time in1981 [88] and 1979 [89], respectively, although these phases had previously beenrecognized as impurities in synthetic LaNiO3 or La2NiO4. Since then, manyattempts have been made to synthesize high-quality samples, especially afterthey were identified as promising SOFC cathode materials in 2006 [24, 78]. Formost syntheses, the Pechini method (a sol–gel process) has been employed, al-though the number of reports is too large to be fully summarized here. Inthe solution-based process, atoms are homogeneously dispersed in a gel whengelation is well controlled, drastically facilitating the approach to equilibrium.Although the details of sol–gel processes vary among reports, the representa-tive Pechini method [24] is briefly introduced here. A stoichiometric mixture ofLa(NO3)3 and Ni(NO3)2 is dissolved in water, to which excess citric acid andethylene glycol are added. The solution is then dried to form a gel, which issubsequently heated at 750◦C in air to remove organic components. The result-ing powder is pressed into pellets and fired at 1100◦C for La3Ni2O7 or 1050◦Cfor La4Ni3O10 for several days. Another frequently used method is the solid-state reaction of nitrates. Compared with simple oxides, nitrates are unstableand their powders are usually fine, which enhances the reaction rate. However,even with nitrates, obtaining homogeneous samples is difficult [88, 83]. Thesefindings suggest that direct solid-state reactions from La2O3 and NiO powdersdo not proceed smoothly to equilibrium. In fact, several reports explicitly statethat single-phase samples could not be obtained from such reactions [22, 26, 27].Samples prepared from stoichiometric mixtures of La2O3 and NiO gener-ally contain significant amounts of other RP phases [90]. The large intensity ofsecondary-phase peaks in the sample fired at 1200◦C suggests that high-qualitysamples cannot be obtained, even after repeated grinding and firing intendedto produce “single-phase” products. Nevertheless, the synthesis temperaturecannot be raised significantly, as only a narrow temperature window is avail-able for each compound. In the samples fired at 1000◦C, the raw materialsof La2O3 and NiO remained, suggesting that the reaction temperature is toolow. By contrast, La3Ni2O7 decomposes into La2NiO4 and NiO above 1366◦C,and La4Ni3O10 decomposes into La3Ni2O7 and NiO above 1277◦C at an oxygenpartial pressure of pO2= 1 bar [91].12The combination of ball milling and CIP (cold isostatic pressing) has been re-ported to be effective for synthesizing La3Ni2O7 and La4Ni3O10 samples suitablefor SOFC applications from La2O3 and NiO powders [92], although it remainsuncertain whether the quality is sufficient for microscopic measurements of theirelectronic properties. A La3Ni2O6.97 sample for SOFC studies, prepared eitherfrom ball-milled mixtures of La2O3 and NiO or from nitrates [91], exhibitedsignificant NQR (nuclear quadrupole resonance) signals arising from La4Ni3O10intergrowth [81].Recently, a new method was developed to synthesize homogeneous La3Ni2O7and La4Ni3O10 directly from La2O3 and NiO [2, 90]. In this approach, thesamples are preliminarily heated at 1300◦C (for La3Ni2O7) or 1200◦C (forLa4Ni3O10), and then reduced under flowing 10% H2/Ar gas to yield a mixtureof La2O3 and Ni metal. This method was originally developed for the synthesisof Y2Ba4Cu7O15−x to obtain more homogeneous samples than those producedby the polymerized-complex method (a type of solution process) [93]. As shownin Fig. 5a, the XRD peaks of Ni metal in a reduced sample are broad, indicat-ing that the Ni particles are very fine. Indeed, Ni atoms are dispersed withinindividual micron-sized grains, as observed in Fig. 5b–d. The reduced samplesare subsequently fired twice at 1300◦C for La3Ni2O7 or 1200◦C for La4Ni3O10,yielding single-phase products [2, 49, 81]. La3Ni2O7 synthesized by this methodexhibits no NQR signals from La4Ni3O10 [81]. By contrast, La4Ni3O10 samplesstill contain intergrowth of La3Ni2O7, as evidenced by NQR signals from thelatter [81]. Indeed, the resistivity of La4Ni3O10 under high pressure shows adrop below ∼80 K, probably due to the La3Ni2O7 intergrowth. To suppressthis, HIP (hot isostatic pressing) annealing at 1200◦C under high oxygen pres-sure (pO2= 400 atm) is effective, because La3Ni2O7 decomposes into La4Ni3O10and La2O3 under such conditions [91].3.2 Single CrystalSingle crystals of La3Ni2O7 and La4Ni3O10 are grown by the FZ (floating zone)and flux methods. These compounds exhibit incongruent melting at pO2= 1 bar[91] as mentioned above; Lan+1NinO3n+1 with larger n decomposes stepwise intophases with smaller n as temperature increases. The Ni valence for RP phaseswith larger n is higher, so at higher temperatures oxygen gas is released from theoxides due to the larger entropy of the gas phase. Fortunately, there exist oxygenpartial pressures that allow congruent melting, although their exact meltingpoints have not been reported. These pressures are approximately 14 bar forLa3Ni2O7, 16–30 bar for La4Ni3O10, and above 50 bar for LaNiO3 [37]. Thus,the FZ growth of single crystals is performed at 15 bar for La3Ni2O7 [94, 35, 1,3, 47, 52, 72, 95] and 20 bar for La4Ni3O10 [38, 40, 37, 41, 42]. No informationon the melt temperature, except 1650◦C for the growth of La4Ni3O10 [38], hasbeen reported, but high-power Xenon arc lamps were used. Since the crystalgrowth of Pr4Ni3O10 requires pO2> 100 bar [37], the synthesis of La-substitutedcrystals likely requires larger pO2 than that of the non-doped crystals.For crystal growth by the FZ method, the occurrence of stacking faults is13Figure 5: XRD pattern (a), SEM (scanning electron microscopy) image (b), andEDX (energy-dispersive X-ray spectroscopy) mapping of La (c) and Ni (d) inthe sample reduced by hydrogen gas after preliminary synthesis.almost inevitable, especially for La3Ni2O7, because the samples pass throughthe temperature window in which La4Ni3O10 is stable at high pO2 while be-ing cooled from the melts. Indeed, stacking faults were clearly observed byTEM [86], and some unidentified peaks were observed in 139La NMR signals[96]. This seems to be a major reason why the 1313 phase was obtained bythe FZ method [4, 3, 35], as supported by annealing powder samples at variouspO2 [51], although this is unlikely the only reason, considering that the 1212phase was synthesized by the flux method [6]. It is unclear whether the stack-ing faults can be eliminated by annealing the crystals at low oxygen pressures,such as pO2= 1 atm at 1300◦C (where La4Ni3O10 decomposes into La3Ni2O7and NiO, as mentioned above). By contrast, La4Ni3O10 crystals have stackingfaults of double perovskite slabs in some cases [38], although they may disap-pear by high pO2 and temperature annealing, such as HIP treatment. As-castLa4Ni3O10 crystals can have the orthorhombic Bmeb structure, which changesinto the monoclinic P21/a structure upon annealing, supporting that the Bmebstructure is metastable [37, 38].Unlike the FZ method, the flux method does not require very high tempera-tures or high pO2, although it carries the risk of inclusions in the grown crystalsand side reactions during flux removal. For successful synthesis, it is impor-tant to choose proper flux materials, although no systematic principle exists.Several groups have succeeded in this regard, and according to their reports,K2CO3 or a NaCl/KCl mixture works well as a flux at 1050◦C or below forLa3Ni2O7 [97, 98], whereas only K2CO3 flux has been reported for La4Ni3O10to date [59, 56, 58]. The crystals are almost square or rectangular plates withdimensions of ∼ 1002 µm2 for both compounds [97, 59].14Some single crystals grown by the flux methods sometimes have differentcrystallographic symmetries from orthorhombic Amam, Fmmm, Bmeb, andmonoclinic P21/a. La3Ni2O7 crystals made by the K2CO3 flux method adoptthe monoclinic P21/m structure [97], whereas those made by the NaCl/KCl fluxmethod adopt the orthorhombic Amam structure [98]. The as-cast crystals bythe latter method were electrically insulating, and after slow cooling from 500◦Cto 50◦C at 10–15 bar or 150 bar oxygen gas, orthorhombic Amam and tetrag-onal I4/mmm crystals showing metallic behavior were obtained. Although theoxygen content of the tetragonal crystals was estimated to be 6.96 by X-raystructural analysis, it most likely exceeds 7. La3Ni2O7 transforms from the or-thorhombic to the tetragonal structure upon annealing at extremely high pO2,as mentioned later. The underestimation is probably due to the assumption ofno excess oxygen atoms in the rock-salt slabs. On the other hand, tetragonalLa4Ni3O10 crystals with I4/mmm symmetry were obtained directly by the fluxmethod in flowing oxygen gas. Since monoclinic P21/a crystals were obtainedfrom flux growth in air, the tetragonal structure of La4Ni3O10 is likely causedby excess oxygen atoms in the rock-salt slabs, suggesting that the flux methodis effective for introducing excess oxygen atoms.The flux methods are also applicable to La3−xRxNi2O7 (R = Pr–Er) [97],corresponding to isovalent substitution of La3Ni2O7. Interestingly, Sr-dopedLa3Ni2O7 crystals were prepared under high pressure of 20 GPa at 1400◦C[99]. Although the mechanism of crystal growth is unknown, application ofhigh pressure seems reasonable considering that high Ni valence is stabilizedunder high pressures. The dimensions of the crystal were 0.059 × 0.047 × 0.031mm3.3.3 Estimation of Oxygen ContentThe physical properties of La3Ni2O7 and La4Ni3O10 are sensitive to their oxygencontents, as mentioned later. The oxygen contents need to be determined with aprecision on the order of 1/100. Since the formula weights of these compoundsare approximately 646 and 892, respectively, the sample weight used for thedetermination must be measured with a precision of roughly 0.02%, which iscomparable to the repeatability of typical electronic balances even for relativelylarge sample masses of about 1 g.Among common methods to estimate oxygen content or Ni valence, thermo-gravimetric (TG) analysis by hydrogen reduction or redox titration can generallyachieve the required precision. Concerning TG analysis, however, conventionalTG systems may not reach this accuracy because it is difficult to suppress fluctu-ations and thermal and time-dependent drifts. Such fluctuations and drifts mayoccur even if they appear to be removed by software provided with commercialsystems. Therefore, weighing for oxygen-content estimation may need to beperformed at room temperature to minimize errors due to air convection andchange in buoyancy, using a high-repeatability balance with a large amount ofpure sample, and in an inert atmosphere to prevent water absorption by La2O3in the reduced sample [2]. Water absorption, if it occurs, causes underestima-15tion of oxygen content, which could explain why many reported values are lowerthan stoichiometric values. On the other hand, redox titration methods suchas iodometry also seem unsuitable for these Ni oxides because oxygen bubblesare immediately generated when the sample is soaked in acid, indicating sidereactions unrelated to the redox indicator.Despite these challenges, reported oxygen contents are summarized here:6.92–7.15 for La3Ni2O7 [22, 24, 25, 26, 71, 100, 101, 102, 103, 50, 48, 2, 49] and9.51–10.26 for La4Ni3O10 [33, 24, 25, 26, 18, 100, 101, 102, 104, 2, 105]. Notethat most reports do not describe the care taken in the determination, althoughvalues are typically reported with 1/100-order precision. In some cases, thesamples used included secondary phases, meaning that the reported oxygencontents reflect only the average Ni valence of the sample. Even for single-phase samples, stacking faults may cause a distribution of Ni valence within thephase. The oxygen contents obtained by careful measurements are shown inTables 2 and 3. As seen there, the oxygen contents of La3Ni2O7 and La4Ni3O10after annealing at pO2= 1 atm are nearly stoichiometric.3.4 Excess OxygenExcess oxygen atoms can be introduced into the rock-salt slabs by annealingat high oxygen partial pressures, as shown in Tables 2 and 3. Interestingly,the introduction of excess oxygen into La3Ni2O7 causes phase separation [90].XRD patterns of La3Ni2O7.01, La3Ni2O7.12, and La3Ni2O7.17 are shown in Fig.6, clearly indicating the phase separation of the latter two samples. For example,the 020 and 200 peaks of La3Ni2O7.01 partially merge into single peaks in theother samples at 2θ ≃ 32.85◦, and the 0,0,10 peak at 2θ ≃ 44.12◦ for La3Ni2O7.01is partially shifted to 2θ ≃ 44.88◦, suggesting the coexistence of orthorhombicand tetragonal phases of La3Ni2O7. This indicates that the oxygen contentcannot be adjusted continuously and homogeneously above 7.01 around roomtemperature, similar to La2NiO4 as mentioned above. No phase separation ordecomposition upon excess oxygen introduction was observed for La4Ni3O10.The weight ratio of the orthorhombic and tetragonal phases of La3Ni2O7was estimated to be approximately 35% and 65%, respectively [90]. Thus, as-suming the composition of the orthorhombic phase is La3Ni2O7.01 based onthe nearly unchanged peak positions, the composition of the tetragonal phasecan be calculated to be La3Ni2O7.17 (= (LaO1+1/6)(LaNiO3)2). This estima-tion seems reasonable because the excess oxygen of δ = 0.17 exactly matchesone of the δ values that cause commensurate superstructures in La2NiO4+δ(=(LaO1+δ)(LaNiO3)) as mentioned in the previous section. The occurrence ofsuch superstructures may drive the phase separation, although no superstruc-ture has yet been observed for La3Ni2O7. Interestingly, La3Ni2O7.17 crystallizesin a tetragonal structure with higher symmetry than the original orthorhombicstructure, whereas La2NiO4.25 exhibits reduced symmetry, from orthorhombicBmeb in La2NiO4 to monoclinic C2.16Table2:OxygencontentsandNivalencesofLa3Ni 2O7annealedundervariousatmospheres.OxygencontentNivalenceAtmosphereTemperatureNote7.012.51flowingO2Slowlycooledfrom1200◦C7.002.51airSlowlycooledfrom600◦C7.122.62pO2=300atm(HIP)600◦ Cphaseseparation7.172.67pO2=400atm(HIP)1200◦Cphaseseparationanddecomposition6.962.46flowingH2200◦ C6.842.34flowingH2250◦ Cpossiblephaseseparation6.502.00flowingH2300◦ CTable3:OxygencontentsandNivalencesofLa4Ni 3O10annealedundervariousatmospheres.OxygencontentNivalenceAtmosphereTemperatureNote9.982.66flowingO2Slowlycooledfrom800◦CwithouttheLa 3Ni 2O7intergrowth9.942.63flowingArSlowlycooledfrom800◦CwithouttheLa 3Ni 2O7intergrowth9.992.66pO2=300atm(HIP)600◦ C10.042.69pO2=400atm(HIP)1200◦C9.882.59flowingH2200◦ C9.582.34flowingH2250◦ Cmultiphases8.951.97flowingH2300◦ C17Figure 6: XRD patterns of La3Ni2O7.01, La3Ni2O7.12, and La3Ni2O7.17. Indexesare given assuming orthorhombic Amam symmetry.3.5 Oxygen DeficiencyOxygen deficiency can be introduced mainly at the oxygen sites between two Nisites along the c axis (the inner apical oxygen sites), as mentioned in the pre-vious section. However, the amount of deficiency cannot be widely controlledsimply by varying pO2from 0 to 1 atm, as seen in Tables 2 and 3. This is insharp contrast with high-Tc cuprates [106], although Ni and Cu ions in theseoxides have valences higher than 2, the most common valence. To achieve signif-icant reduction of the oxygen content in La3Ni2O7 and La4Ni3O10, more activereduction methods are required, such as hydrogen gas or CaH2 treatment. Theformer is generally used to control oxygen content in La3Ni2O7 and La4Ni3O10,whereas the latter was used to synthesize La3Ni2O6 from La3Ni2O7 [68].Oxygen atoms in La3Ni2O7 and La4Ni3O10 are indeed partially removedby heating in flowing hydrogen gas at appropriate temperatures. However, theamount of removed oxygen is not well reproducible at a given temperature. Forexample, the oxygen contents of La3Ni2O7 samples were 6.35 and 6.84 whenreduced at 450◦C in 11% H2/Ar and 11% H2/N2, respectively [22], stronglysuggesting that the samples did not reach equilibrium. This is further supportedby the fact that oxygen content depends on the duration of reduction: reductionin 10% H2/Ar at 450◦C yields La3Ni2O6.38 after 12 hrs, but decomposition intoLa2O3 and Ni metal occurs after 60 hrs [71]. The oxygen contents reportedin Tables 2 and 3 were obtained using samples reduced in approximately 5%H2/Ar for 12 hrs.Temperature-dependent weight changes in thermogravimetric (TG) analysisby hydrogen reduction often exhibit step-like structures [22, 18, 50, 67, 48, 103,71], although the exact shapes of the curves vary significantly between reports.These differences are most likely due to excessive heating rates. Because of theheating rate, all characteristic temperatures, such as decomposition tempera-18tures, tend to be overestimated. For example, TG data suggest decomposition ofLa3Ni2O7 into La2O3 and Ni metal occurs above 600–900◦C [22, 50, 48, 103, 71],although it can occur even at 450◦C, as noted above [71]. La3Ni2O7 shows a sin-gle weight-loss step over approximately 100◦C around 500◦C, corresponding tooxygen contents of 6.35 [22], 6.38 [71], and 6.45 [70, 103]. Considering the phaseat this step is particularly robust against temperature changes, it is likely thatthe oxygen content at the step is exactly 6.5, corresponding to purely divalentNi ions in La3Ni2O6.5. By contrast, La4Ni3O10 exhibits two steps correspond-ing to La4Ni3O9 and La4Ni3O8 [67], although the step structure is sensitive toheating rate and hydrogen concentration.La3Ni2O6.84 and La4Ni3O9.58 listed in Tables 2 and 3 likely correspond to in-termediate states below the step temperatures. Some XRD peaks of La3Ni2O6.84are broader than those of orthorhombic La3Ni2O6.96 and tetragonal La3Ni2O6.50,suggesting phase separation. For La4Ni3O9.58, XRD patterns clearly showphase separation between the orthorhombic (or monoclinic) phase and tetrag-onal La4Ni3O9. These observations indicate that oxygen deficiency in thesecompounds, as well as excess oxygen, is discontinuous in a single phase whenprepared by simple reduction.3.6 Aliovalent Substitution of La Ions – Filling Control –Filling control is one of the most interesting and important challenges in su-perconductivity research. Aliovalent substitution of La ions, if feasible, wouldbe particularly valuable, considering that filling control via oxygen content isstrongly limited for the chemical reasons discussed above. Furthermore, randompotentials introduced to the conducting electrons by aliovalent substitution ofLa ions are expected to be much smaller than those caused by oxygen defects;as mentioned above, excess oxygen atoms in the rock-salt slabs can tilt NiO6octahedra irregularly, and oxygen deficiency at the inner apical oxygen sites canreduce the hopping integrals between two Ni atoms along the c axis.Unfortunately, no tetravalent ions can substitute La ions, so there is no op-portunity to reduce the Ni valence via La-site substitution. All tetravalent ionsare too small for the La sites. By contrast, divalent alkaline-earth ions canpartially substitute La ions in La3Ni2O7. For A = Ca, Sr, and Ba, their con-tents x in La3−xAxNi2O7 have been reported to reach x = 0.8 [107, 108], 0.2[107, 109, 99], and 0.075 [107], respectively. For Ca substitution, the electricalresistivity increases with increasing x [108], whereas Sr substitution reduces theresistivity at low pressures [107, 109]. For La2.8Sr0.2Ni2O7, however, the resistiv-ity under high pressures becomes larger above approximately 10 GPa comparedwith La3Ni2O7 [99]. No superconductivity has been observed for aliovalent sub-stitution of La ions in bulk samples. For thin films, superconductivity has beenobserved in La3−xSrxNi2O7 (0 ≤ x ≤ 0.21) [110].In general, careful examination is needed to ensure that a dopant is indeedincorporated into the target compound, especially when the dopant concentra-tion is small and/or the sample contains secondary phases. Even if the dopant19is incorporated, the dopant amount does not exactly correspond to carrier con-centration, which also depends on the content of other elements, such as oxygen.4 Electronic PropertiesStudies on the physical properties of La3Ni2O7 and La4Ni3O10 have been vig-orously pursued, despite experimental challenges in synthesizing high-qualitysamples and in conducting measurements under high pressure, particularly sincethe discovery of superconductivity in these compounds. Nevertheless, a com-plete consensus on their fundamental properties has not yet been reached, andresearch remains ongoing. Thus, hereafter, we briefly introduce representativeexperimental results and indicate directions for future studies.4.1 Electronic StructureFor La3Ni2O7, first-principles calculations indicate that three types of elec-tronic bands, denoted as the α, β, and γ bands, lie in the vicinity of theFermi level [111, 43]. This result is consistent with experimental observationsobtained by synchrotron-based and laser angle-resolved photoemission spec-troscopy (ARPES) [112]. The α and β Fermi surfaces have large Ni 3dx2−y2orbital weight, although their bonding-antibonding splitting occurs due to thehybridization with the 3dz2 orbitals. On the other hand, the γ bands originatemostly from the bonding orbitals of Ni 3dz2 orbitals.The α and β bands form cylindrical Fermi surfaces extended along the kzdirection, reflecting two-dimensional nature. The estimated doping levels areapproximately 0.2 electrons/Ni and 1.25 holes/Ni, respectively [112]. Accordingto band calculations without on-site Coulomb interactions [111, 43, 112], thesebands extend broadly from about −1 eV to 2.5 eV. Synchrotron ARPES revealedthat they are narrowed by renormalization factors of about ∼ 2 [112].The energy position of the γ band relative to the Fermi level remains unset-tled, and it is still under debate whether this band actually crosses the Fermilevel to form a Fermi surface. As shown in Fig. 7, the top of the γ band liesclose to the Fermi level. Its energy is shifted downward by the inclusion ofon-site Coulomb interactions, whereas it is shifted upward by the application ofexternal pressure [112]. These opposing effects make a definitive determinationof the γ-band position difficult. In addition, the γ band is relatively flat andexhibits a strong tendency toward renormalization due to electron correlations.The calculated bare bandwidth is approximately 1 eV, and the renormalizationfactor has been estimated to be in the range of 5–8 [112], which is significantlylarger than those for the α and β bands. Such large renormalization factorssuggest that electron correlations are substantially stronger in the γ band thanin the α and β bands.The electronic band structure near the Fermi level can be well described bya four-orbital tight-binding model consisting of dx2−y2 -like and dz2-like Wannierorbitals derived from two Ni atoms stacked along the c axis [43], as illustrated in20Fig. 7. In this model, tetragonal I4/mmm symmetry is assumed, which leads toa notation of high-symmetry points in the Brillouin zone that differs from that ofthe orthorhombic Amam structure. In the kz plane, the orthorhombic Brillouinzone can be viewed as the tetragonal one folded along the lines connecting themidpoints of its edges (indicated by broken lines in Fig. 7). This folding providesa geometric representation of the symmetry lowering from I4/mmm to Amam.Several features of the band structure can be qualitatively understood byconsidering a cluster consisting of two NiO6 octahedra stacked along the c axis.Because the NiO6 octahedron is slightly elongated along the c axis, the Ni dz2orbital lies lower in energy by ∆E than the dx2 − y2 orbital. For these twoorbitals, a total of 1.5 electrons are expected to be distributed. Consequently,the dz2 orbital is half-filled, while the remaining 0.5 electrons occupy the dx2−y2orbital. Furthermore, within the cluster, the two dz2 orbitals pointing towardthe inner apical oxygen give rise to a large bonding–antibonding energy splittingthrough a sizable interlayer hopping integral, t⊥. As a result, the center ofgravity of the γ band is located at a lower energy than those of the α and βbands and is (nearly) occupied, as shown in Fig. 7. This simple picture providesa reasonable description of the band structure along the Γ–X direction, wherehybridization between the dz2 and dx2−y2 orbitals is forbidden by symmetry.By contrast, along the Γ–N direction, hybridization between these orbitals liftsthe degeneracy of the α and β bands, leading to the formation of distinct αand β Fermi surfaces at the Fermi level. Although this cluster-based picture isnecessarily simplified, it highlights the crucial roles of ∆E and t⊥ in shapingthe low-energy electronic structure.The electronic structure of La4Ni3O10 [2] can be understood within a similarcluster-based model. Because its crystal structure is trilayered, three Ni atomsstacked along the c axis must be considered. Consequently, the coupling ofthree Ni dz2 orbitals gives rise to a nonbonding orbital in addition to bondingand antibonding orbitals. As a result, the electronic bands exhibit a slightlyincreased complexity, as illustrated in Fig. 7. Nevertheless, the overall electronicstructure remains very similar to that of La3Ni2O7. Specifically, a relatively flatband is located around and just below the Fermi level, while more dispersivebands extend around and above it.Based on these band structures, pronounced nesting features are expected[43, 113, 114, 115, 116], which is consistent with the observation of density-waveformations discussed in the following section.4.2 Density WaveAt ambient pressure and zero field, La3Ni2O7 exhibits metallic conductivitybelow approximately 700 K , as shown in Fig. 8a [117]. Around 550 K, atransition-like behavior from a good to a poor metallic state is observed, ac-companied by changes in magnetic susceptibility (Fig. 8b). The origin of thishigh-temperature transition remains unclear, although it may be related to thestructural transition from the I4/mmm to the Amam space group reportednear 700 K in some samples [31].21Figure 7: Electronic band structures of La3Ni2O7 calculated using a four-orbitaltight-binding model (a) [43], and of La4Ni3O10 using a six-orbital tight-bindingmodel (b) [2] Both calculations assume tetragonal I4/mmm symmetry. Panels(c) and (d) show the Fermi surfaces at kz = 0 for La3Ni2O7 and La4Ni3O10,respectively. These panels were kindly provided by Prof. Sakakibara at TottoriUniversity. The broken lines and the notations in panels (c) and (d) were addedby the authors.22Figure 8: Electronic resistivity (a) and magnetic susceptibility (b) of La3Ni2O7and La4Ni3O10 as functions of temperature [117].23At lower temperatures, two additional transitions are observed at approxi-mately 153 K and 110 K [47]. Below 153 K, the magnetic susceptibility per-pendicular to the c-axis decreases slightly, accompanied by minor changes inresistivity. Below 110 K, the resistivity shows a modest increase without pro-nounced anisotropy in the magnetic response. Specific-heat anomalies are sub-tle due to the relatively high temperatures, but a small peak near 153 K isdiscernible. These low-temperature transitions are not universally observedin all samples, and the transition temperatures exhibit sample dependence.In some cases, the resistivity exhibits semiconducting behavior below 300 Kor 140 K [26, 33, 107, 84, 50, 32], particularly in oxygen-deficient samples[49, 109, 118]. Samples showing semiconducting behavior tend to exhibit up-turns in the temperature dependence of the magnetic susceptibility at low tem-peratures [26, 107, 100, 32], likely due to oxygen vacancies that generate randompotentials and localize magnetic moments on surrounding Ni sites. Consistently,the upturns were more pronounced in samples slightly reduced by hydrogen gas.The 153 K and 110 K transitions are generally attributed to density wave for-mation. Although it was initially interpreted as charge density wave (CDW) for-mation [100], current evidence favors a spin density wave (SDW) scenario. NQRmeasurements on high-quality La3Ni2O7 samples suggest a magnetic structurein which spins align ferromagnetically along the a-axis and antiferromagneti-cally along the b-axis within NiO2 planes, while spins in adjacent planes alongthe c-axis are antiferromagnetically coupled [81]. This model is almost con-sistent with muon spin rotation (µSR) [50], resonant inelastic X-ray scattering[119], and neutron scattering experiments [120]. Also in La2PrNi2O7, magneticorder is observed below 161 K, as indicated by µSR and neutron diffraction[120, 121]. In some samples, CDW–SDW coexistence or CDW has been sug-gested [122, 123], implying near-degenerate electronic states whose manifesta-tion depends on crystallographic defects or sample-specific details.Temperature-dependent measurements of the La nuclear spin-lattice relax-ation rate divided by temperature (1/T1T ) exhibit divergent behavior near150 K [81], consistent with a second-order density wave transition. Typically,both CDW and SDW transitions are of second-order with the gaps openinggradually below the transition temperatures as order parameters. However, op-tical conductivity studies indicate that the associated energy gap remains nearlyconstant below the transition [124], suggesting a sharp, first-order-like behav-ior. These discrepancies suggest that the low-temperature electronic state ofLa3Ni2O7 is an open field for further investigation.In La4Ni3O10, metallic behavior persists below 700 K down to 150 K (Fig.8a) [117]. A small anomaly near 550 K may arise from minor inclusions ofLa3Ni2O7 or stacking faults. Magnetic susceptibility gradually decreases above400 K, resembling the behavior of La3Ni2O7 between 300 K and 550 K. Consider-ing that La4Ni3O10 undergoes a structural transition to I4/mmm at 873–1000 K[31, 39] —approximately 200 K higher than La3Ni2O7— a high-temperaturemetal–metal transition-like behavior similar to that in La3Ni2O7 may occurabove 700 K.At low temperatures, La4Ni3O10 also shows density wave formations. Semi-24conducting behavior appears when large amount of oxygen deficiency exists asin the case of La3Ni2O7 [57] although the sample dependence is not so promi-nent. For crystals with P21/a symmetry, a transition occurs at 138.6 K, whileBmeb crystals transition at 147.5 K [37]. These transitions involve a temporaryincrease in resistivity below the transition temperatures [26, 33, 18, 37, 4, 100,57, 55, 56, 125] with anomalies in lattice-related properties such as lattice con-stants and thermal expansion coefficients [37, 54, 53], suggesting CDW state. Onthe other hand, magnetic measurements reveal anisotropic responses to appliedfields [37, 14], suggesting SDW formations. In fact, complex density wave stateof both spin and charge degrees of freedom has been proposed [125, 126, 60].Angle-resolved photoemission spectroscopy indicates a temperature-dependentgap, being consistent with second-order transition of the density wave forma-tion [115]. No experimental results show a first-order-like signatures. Substi-tuting La with Pr or Nd does not significantly shift the transition temperatures[18, 37, 55, 53, 127].4.3 Phase diagramThe electronic states are determined by thermodynamic variables such as tem-perature (T ), pressure (P ), and magnetic field (H), and by the band filling. Thefillings in La3Ni2O7 and La4Ni3O10 are directly related to oxygen stoichiometry.Thus, the electronic states of La3Ni2O7 and La4Ni3O10 can be considered func-tions of these four variables. In the following, we describe the electronic statesof these compounds under representative combinations of these parameters. Inmany experimental studies, not all four variables are precisely controlled; inparticular, oxygen content is often imperfectly defined. Consequently, some ofthe phase diagrams discussed here should be interpreted as projections of afour-dimensional electronic phase space.4.3.1 P–T phase diagramThe evolution of electronic and structural properties of La3Ni2O7 and La4Ni3O10under pressure has been a central subject here. Of particular interest is how thestructural transition from I4/mmm to Amam or P21/a, as well as the changefrom a good to a poor metal observed at ambient conditions, evolves at elevatedpressures. Unfortunately, however, available P–T phase diagrams extend onlybelow 300 K, leaving the high-temperature regime unexplored.For La3Ni2O7, as noted earlier, the sequential transitions at 153 K and 110 Khave been observed at ambient pressure although they are sample-dependentand are not always clearly resolved. Nevertheless, a P–T phase diagram suchas that shown in Figure 9 has been broadly accepted [1, 128, 94, 86, 48, 96,129, 130, 131]. The higher-temperature transition of density wave formationshifts upward with increasing pressure, whereas the lower-temperature transi-tion shifts downward. Both transitions vanish near the pressure at which theAmam structure transforms into the Fmmm phase. Although the structuralphase boundary between Amam and Fmmm symmetries is not fully under-25Figure 9: Schematic P–T phase diagram of La3Ni2O7. SDW , DW , ST ,SM , and SC denote the spin-density wave, density wave, structural transition,strange metal, and superconducting phases, respectively.stood, it is generally assumed to be nearly vertical in the pressure axis [34, 35].The first-order nature of the transition at the boundary implies the presenceof a two-phase coexistence region, making the precise determination of phaseboundaries challenging as in many cases. The Amam structure becomes un-stable above approximately 10 GPa, and so the Fmmm phase likely startsto appear around 10 GPa. Although subtle superconductivity emerges in thisregime, clear one appear above approximately 15 GPa, just above the struc-tural transition pressure. The superconducting transition temperature, Tc, istypically about 80 K there and decreases gradually with further compression.While density-wave states and superconductivity often compete in correlatedsystems such as organic conductors, in La3Ni2O7 these two states are separateddiscontinuously by the first-order structural transition, leaving their relationshipunresolved.A strange metallic state has been reported just above Tc, in which resistiv-ity exhibits a linear temperature dependence [1, 94, 48, 129]. This has beeninterpreted as evidence for non-Fermi-liquid behavior and as a hallmark of un-conventional superconductivity, as speculation from the high-Tc curate case.Although the superconductivity in La3Ni2O7 is most likely unconventional in-deed, key signatures of the strange metal observed in cuprates—such as abruptchanges in resistivity derivatives or Hall coefficients outside the strange metalregime—have not yet been demonstrated in La3Ni2O7. Moreover, this strangemetallic phase is not adjacent to a Mott insulating state as in the cuprates.Whether the observed metallic state is genuinely anomalous and whether it isdirectly related to the unconventional nature of superconductivity remain open26questions. Some studies propose the presence of an additional density-wavephase in the strange-metal region [130].In a sample of La2PrNi2O7, the structural transition sometimes occurs di-rectly from Amam to I4/mmm near 11 GPa at room temperature [36]. Thistransition pressure is slightly lower than that of La3Ni2O7, although the or-thorhombicity increases owing to the samller ion of Pr as mentioned above.Even more intriguing is the appearance of superconducting-like transitions atsignificantly lower pressures, around 8 GPa, well below the nominal structuraltransition. While this may be attributed to local inhomogeneities that triggerstructural changes at lower pressures, it raises the possibility that tetragonalsymmetry may not be a strict prerequisite for superconductivity. As will bediscussed below, similar behavior is reported in La4Ni3O10 and Pr4Ni3O10.La4Ni3O10 undergoes a pressure-induced structural transition from P21/ato I4/mmm around 15 GPa [40, 41]. A report claims that there is a Bmebphase between the P21/a and I4/mmm phase [132]. Beyond this transition,superconductivity emerges with a maximum Tc of 36 K [105]. Immediately afterthe discovery of the superconductivity in La4Ni3O10 [2], the P–T phase diagramwas quickly constructed [40, 41, 132, 57, 133]. Two kinds of P–T phase diagramhave been reported. In the one, superconductivity appears below the structuraltransition pressure [40, 41], whereas in the other, it emerges just above thetransition pressure [132, 57, 133]. In either case, unlike La3Ni2O7, the highestTc is not found just above the transition pressure. The phase diagrams are moresimple than that of La3Ni2O7: there is a single transition of the density waveformation with transition temperature decreasing from around 140 K at ambientpressure with increasing pressure, although in a sample, magnetic structurechange was observed at 90 K [60]. Neither strange-metal behavior nor additionaldensity wave phase at higher pressures has been observed in any P–T region.Pr4Ni3O10 exhibits even more distinctive behavior [127]. The structuraltransition to I4/mmm occurs at approximately 35 GPa, yet superconductiv-ity emerges already at around 25 GPa. Meanwhile, density-wave order persistsup to pressures exceeding 40 GPa. Consequently, in the 25–40 GPa window,the system first undergoes a density-wave transition upon cooling, followed atlower temperatures by a superconducting transition. Although experimentaluncertainties related to hydrostaticity at such high pressures necessitate fur-ther confirmation, this coexistence regime, if intrinsic, provides critical insightsinto the interplay between structure and competing electronic states in layerednickelates.4.3.2 P–δ phase diagramPrecise determination of oxygen content in La3Ni2O7 and La4Ni3O10 remainsscarce, yet several studies have examined the dependence of superconductivityon oxygen stoichiometry. In La3Ni2O7, a reduction in oxygen content from thestoichiometric value of seven results in the immediate loss of metallic conductiv-ity [49]. The system transforms into an insulating state, most likely driven byAnderson localization. As noted earlier, the missing oxygen resides within the27perovskite block, and the resulting random potential strongly affects the NiO2planes. In contrast, excess oxygen exerts a different influence. When the oxygencontent exceeds stoichiometry, the superconducting transition temperature Tchas been reported to remain as high as that of the stoichiometric sample belowapproximately 40 GPa, whereas it is markedly suppressed at higher pressurescompared with the stoichiometric composition. However, the interpretation iscomplicated by the fact that the powder samples used for these measurementswere mixtures of orthorhombic La3Ni2O7.01 and tetragonal La3Ni2O7.17. Atpressures below 40 GPa, the observed superconductivity may originate fromLa3Ni2O7.01. Above 40 GPa, a distinct superconducting transition with lowerTc clearly emerges, which is associated with the oxygen-rich tetragonal phase.Supporting this interpretation, single crystals annealed under an oxygen partialpressure of 150 bar, yielding tetragonal La3Ni2O7+δ, did not exhibit supercon-ductivity up to 68.2 GPa [98]. These results suggest that oxygen excess shiftsthe superconducting onset to higher pressures and simultaneously reduces Tc.The situation is markedly different in La4Ni3O10[105]. For an oxygen-rich composition (La4Ni3O10.04), superconductivity emerges at approximately20 GPa, with sharp transitions observed beyond 30 GPa. The transition tem-perature increases to Tc ≈ 36 K at 48 GPa and then decreases gradually underfurther compression. In contrast, in slightly oxygen-deficient La4Ni3O9.99, su-perconductivity first appears near 30 GPa, but clear bulk transitions are onlyobserved above 70 GPa. In this case, Tc reaches a maximum of 22 K at 79.2 GPa.Thus, reducing the oxygen content shifts the onset of superconductivity tohigher pressures and lowers the maximum Tc, in contrast to the trend observedin La3Ni2O7. It should be noted, however, that the oxygen-deficient phase wassynthesized by HIP annealing at 600◦C, which is lower than the temperatureused to eliminate the La3Ni2O7 stacking faults. This raises the possibility ofresidual stacking disorder, which could influence the superconducting properties.4.3.3 Other phase diagramsCombinations of P , T , H, and δ can, of course, give rise to phase diagramsbeyond the P–T or P–δ types. Thus far, however, only H–T phase diagramshave been reported, primarily to estimate the upper critical fields of supercon-ductivity in La3Ni2O7, La4Ni3O10, and related compounds. These results willbe discussed in the final section.5 Thin FilmEpitaxial thin films of La3Ni2O7 have been fabricated by pulsed laser depo-sition (PLD) [134, 135, 136, 137], molecular beam epitaxy (MBE) [110, 138],and GOALL epitaxy methods [139], providing an important platform to ex-plore superconductivity under ambient pressure. When grown on substrateswith smaller in-plane lattice constants than that of La3Ni2O7, the films expe-rience compressive strain within the ab plane. For convenience, a pseudote-28tragonal lattice parameter of 3.833 Å, defined as half the diagonal length ofthe ab plane in the orthorhombic unit cell, is adopted as a reference value.On SLAO(001) [SrLaAlO4(001)], whose lattice constant is approximately 2%smaller than that of La3Ni2O7, superconducting transitions with Tc = 26–42 Kat ambient pressure have been observed [134]. By contrast, films grown onNGO(001) [NdGdO3(001)], which has a lattice constant about 0.7% larger,remain metallic but nonsuperconducting, while those deposited on STO(001)[SrTiO3(001)], with a lattice constant 1.9% larger, exhibit semiconducting be-havior [135].Beyond these three substrates, thin films have also been fabricated on LAST(001)[(LaAlO3)0.3(Sr2TaAlO6)0.7(001)], with a relative lattice mismatch of +0.9%,and on LAO(001) [LaAlO3(001)], with −1.2% [134, 135, 138, 136]. Films ofsubstituted systems such as La3−xSrxNi2O7 (x = 0–0.45) and La3−xPrxNi2O7(x = 0.15, 1) have also been synthesized [110, 139, 137]. Nevertheless, su-perconductivity at ambient pressure has been realized only in films grown onSLAO(001). It is worth noting that films grown on SLAO(100) often contain re-gions of La2NiO4, La4Ni3O10, or amorphous-like domains, reflecting significantstructural imperfections [134, 135].Several additional factors are crucial for achieving superconductivity in thinfilms. One key parameter is film thickness: superconductivity has been observedonly in ultrathin films, just several unit cells thick along the c axis, most likelybecause sufficient substrate-induced strain is required. Oxygen stoichiometry isanother decisive factor. As-grown films exhibit nonmetallic behavior due to oxy-gen deficiency, and post-annealing in ozone is essential for restoring metallicityand superconductivity [134]. In La2PrNi2O7 thin films, systematic annealingstudies have shown that excessively high ozone concentrations decompose thefilms into perovskite phases, whereas insufficient annealing temperatures fail toinduce superconductivity [137]. Whether the absence of superconductivity afterlow-temperature annealing arises from excess oxygen incorporation or from in-complete oxygen uptake remains unresolved: at lower temperatures, ozone actsas a stronger oxidizing agent, while oxygen diffusion in oxides becomes slower.Another important aspect is temporal stability [134]. Films that initially showzero resistivity after annealing tend to degrade over time, with resistivity aboveTc gradually increasing, whereas re-annealing in ozone can restore supercon-ductivity. This strongly suggests progressive oxygen loss. Surface terminationlikely plays a role in stabilizing oxygen. In many superconducting films, a singleunit cell of SrTiO3 is deposited on the surface of La3Ni2O7, which may helpsuppress oxygen escape.The compressive strain imposed by SLAO(001) corresponds to an effectivepressure of 10–20 GPa relative to bulk La3Ni2O7. When plotted against thea-axis lattice constant, the superconducting transition temperatures of bulkand thin-film samples show a common trend: superconductivity emerges be-low a ≈ 3.79 Å and saturates at Tc ≈ 60–80 K for a ≲ 3.75 Å[134]. Nosuch universal relation has been identified with respect to the c-axis parameter.Detailed structural analyses reveal that decreasing substrate lattice constantsdrive the La3Ni2O7 layers from orthorhombic symmetry toward a more tetrag-29onal structure, as expected from the compressive strain [135]. Importantly,even films grown on the other substrates, such as LAO(001), NGO(001), andSTO(001) exhibit superconductivity once external pressure is applied, consis-tent with the P–T phase diagram of bulk La3Ni2O7 [136]. These diagrams alsodisplay a density-wave transition at low pressures, with its transition tempera-ture suppressed under pressure. Minor discrepancies remain, such as differencesin the semiconducting regimes between bulk and thin films. By contrast, theonset pressure for superconductivity does not systematically vary with the lat-tice mismatch among substrates: although SLAO(001) provides −2% strain,NGO(001) +0.6%, and STO(001) +1.9%, the pressure required to induce su-perconductivity in each case remains similar [136]. This observation suggeststhat substrate-induced strain alone cannot account for superconductivity in thinfilms. Other factors—such as oxygen stoichiometry or charge transfer across thefilm–substrate interface due to band alignment effects—may play equally signif-icant roles.6 Superconducting properties and parametersAt last, we would like to summarize the superconducting properties of La3Ni2O7and La4Ni3O10. In bulk form, both compounds become superconducting onlyunder very high pressures of several tens of GPa. Thus, the measurementsto determine the superconducting parameters have been carried out using dia-mond anvil cells or multi-anvil cell, which imposes the measurements on variousrestrictions. Consequently, precise measurements comparable to those at am-bient pressure remain challenging. In addition, the high-quality sample wellcharacterized is not always easy to obtain as mentioned above. By contrast, inthin-film form, superconductivity has been observed at ambient pressure. How-ever, such films are typically inhomogeneous, and the difficulty of determiningsample mass in thin films prevents reliable evaluation of extensive quantities.As a result, the superconducting properties of both La3Ni2O7 and La4Ni3O10have been investigated primarily through electrical resistivity measurements.Magnetization measurements exist only in a limited number of cases, and noreports of specific heat measurements are available.La3Ni2O7 exhibits superconducting transitions even under extremely highmagnetic fields, indicating that it is an extreme type-II superconductor. Pene-tration depth has been estimated for La2.85Pr0.15Ni2O7 film, to yield ∼4.5 µmat 1.8 K [139], far exceeding the coherence lengths listed in Table 4. The uppercritical field Hc2 has been measured only up to 14 T, and higher values havebeen extrapolated using the empirical Ginzburg–Landau relation,Hc2(T ) = Hc2(0)1 − t21 + t2, t = T/Tc.Although the accuracy of such extrapolations is uncertain, it is generally ac-cepted that Hc2(0) is extremely large. To date, no direct evidence has beenreported that Hc2(0) exceeds the Pauli limit. Thin-film studies at ambient30pressure show Tc values smaller than the maximum bulk values, yet coherencelengths of 1–2 nm, consistent with those inferred from bulk samples (Table 4).Anisotropy of Hc2 has also been reported, though the difference between fieldsapplied parallel and perpendicular to the c-axis is at most a factor of two. Sub-stitution at the La site (e.g., by Pr or Sm) yields similarly high estimates ofHc2.The highest reported onset-Tc for La3Ni2O7 is 86 K [129], while partialsubstitution yields up to 91 K in La2SmNi2O7 [97], though the uncertainty inonset values is not small. Critical current densities Jc have also been evaluated:bulk samples under 16.6 GPa show Jc = 0.85 kA/cm2 at 1.5 K [94], while thinfilms yield Jc = 0.32 kA/cm2 at 0.15 K (La3Ni2O7) [134], Jc = 10.4 kA/cm2at 1.4 K (La2PrNi2O7) [137], and Jc = 1.4 kA/cm2 at 2 K (La2.91Sr0.09Ni2O7)[110]. These results suggest ample room for improvement, considering possiblesample cracking under high pressure and the limited crystalline quality of films.La4Ni3O10 has also been considered to be an extreme type-II superconductoralthough neither lower critical field nor penetration depth have been measured.Its upper critical field Hc2 has been determined directly over a wide range ofapplied fields [140], yielding reliable results. The coherence length is estimatedto be 3.3 nm. Anisotropy between in-plane and out-of-plane fields amounts to afactor of ∼1.5 down to 0.8Tc, but diminishes at lower temperatures, convergingto identical values at 2 K. This behavior provides an important informationon assessing the reliability of extrapolated Hc2 values in La3Ni2O7. SimilarHc2(0) and coherence lengths have been reported in related compounds, includ-ing Pr4Ni3O10 (Table 4). The highest onset Tc values reported to date are 36 Kfor La4Ni3O10 [105] and 40.5 K for Pr4Ni3O10 [127].Finally, we turn to the issue of superconducting symmetry in nickel oxides,a subject of broad interest across the research community. Thus far, owing tothe experimental challenges outlined above, no decisive data have been obtainedfor bulk samples. In thin films of La-substituted La3Ni2O7, ARPES and STM(scanning tunneling microscopy) measurements suggest nodeless gap structures[141, 142], but the precise symmetry of the superconducting order parameterremains unresolved. Clarifying this issue is essential not only for establishingthe nature of the superconducting state, but also for uncovering the underlyingpairing mechanism.AcknowledgmentWe are grateful to our collaborators, in particular Prof. Kazuhiko Kuroki(University of Osaka), Prof. Hirofumi Sakakibara (Tottori University), Prof.Masayuki Ochi (University of Osaka), and Prof. Hidekazu Mukuda (Universityof Osaka), for their invaluable support and insightful discussions. The SEM andEDX images shown in Fig. 5 were obtained by Mr. Kazuki Yamane, and thecrystal structure illustrations in the figures were prepared using VESTA [143].This work was partly supported by the World Premier International ResearchCenter Initiative (WPI), MEXT, Japan, and by JSPS KAKENHI (Grant Nos.31Table4:Superconductingparameters.Tc,Hc2(0),Hmax,P,Criterion,ξ,andthicknessrepresentsuperconductingtransitionat0T,uppercriticalfield,themaximummagneticfieldappliedforthemeasurements,Pressure,thecriterionofestimationofTcrelatedtothemagnitudeoftheresistivitydrop,coherencelength,andthicknessofthefilmestimatedfromtheuppercriticalfield,respectively.Thecoherencelengthisshownonlywhenitisshowninthereference.(*:thisvaluewasestimatedthistime.Inthepaper,ξisestimatedtobe4.83nm.)CompositionformTc(K)Hc2(0)(T)Hmax(T)P(GPa)Criterion(%)ξ(nm)Directionthickness(nm)Ref.La3Ni 2O7bulk80158720.190c//H[98]727322.6731861418.9901.33*c//H[1]6912729.1659743.56786.68.514.590[48]66138925.1901.57[94]719720.51.84688326.61.99La2SmNi 2O7bulk85292723.7901.1[97]La3Ni 2O7film38132140901.8c//H5[134]97ab//HLa 2.85Pr 0.15Ni 2O7film34119140902.2c//H4.9[139]681.7ab//HLa2PrNi 2O7film48142140901.7c//H4[137]116ab//HLa2.91Sr 0.09Ni 2O7film3883.790901.98c//H5.21[110]110.3ab//HLa4Ni 3O10bulk18.524.43450.2503.3c//H[140]3.3ab//H233575390c//H[40]254463c//HPr 4Ni 3O10bulk3353775.0902.5c//H[127]32JP24K01333 and JP25K00959).References[1] Sun H, Huo M, Hu X, Li J, Liu Z, Han Y, Tang L, Mao Z, Yang P, WangB, Cheng J, Yao D X, Zhang G M and Wang M 2023 Nature 621 493–498.[2] Sakakibara H, Ochi M, Nagata H, Ueki Y, Sakurai H, Matsumoto R,Terashima K, Hirose K, Ohta H, Kato M, Takano Y and Kuroki K 2024Phys. 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