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Electronic structure and correlations in planar trilayer nickelate Pr4Ni3O8

The recent discovery of superconductivity in hole-doped planar nickelates R1-xSrNiO2 (R=Pr,Nd) raises the foundational question of how the electronic structure and electronic correlations of these Ni1+ compounds compare to those of the Cu2+ cuprate superconductors. Here, we present an Angle-Resolved Photoemission Spectroscopy (ARPES) study of the trilayer nickelate Pr4Ni3O8, revealing an electronic structure and Fermi surface very similar to that of the hole-doped cuprates but with a few critical differences. Specifically, the main portions of the Fermi surface are extremely similar to that of the bilayer cuprates, with an additional piece that can accommodate additional hole doping. We find that the electronic correlations are about twice as strong in the nickelates and are almost k-independent, indicating that they originate from a local effect-likely the Mott interaction, whereas the cuprate interactions are somewhat less local. Nevertheless, the nickelates still demonstrate an approximately linear in energy and linear in temperature scattering rate. Understanding the similarities and differences between these two related families of strongly-correlated novel superconductors is an important challenge.


Introduction
As 3d transition metal oxides, various nickelate compounds have long been studied for correlated electron properties and as candidates for high-Tc superconductivity. In particular, special attention has been paid to two-dimensional Ni 1+ compounds that might mimic the Cu 2+ state of the cuprates that is known to support both the strange-metal physics and high Tc superconductivity. However, even with the same formal electron count between the Ni 1+ and Cu 2+ states, concerns have been raised that these two states would not be the same and so could not support the same exotic physics as the cuprates, with concerns arising due to reduced 3d-2p mixing and different on-site correlation energies [ 1 ]. Such differences could arise due to the varying energetics of the average transition metal d bands as a function of atomic number, as described in the Zaanen-Sawatzky-Allen classification schemes of correlation gaps as arising from Mott correlations that principally center on the transition metal (farther left in the periodic table) or charge-transfer physics (farther right) [ 2 ] that includes the oxygen ligands more explicitly and brings in the Zhang-Rice singlet that is a combination of the Cu and O orbitals [ 3 ].
Recently, the spotlight has been turned to square-planar nickelate materials following the breakthrough discovery of superconductivity in thin films of the planar infinite layer nickelates, Srdoped RNiO2 (R=Pr, Nd) [ 4,5 ], and more recently the 5-layer variant [ 6 ]. It is natural that proposals have been put forward that the superconducting nickelate should express exotic electronic correlations similar to those found in other unconventional superconductors, such as the doped charge-transfer cuprates [ 7,8,9,10,11 ]. However, the topotactic reduction process used in the synthesis of these nickelate thin films, which is required to remove the apical oxygen atoms and thus to create the square planar network, is known to degrade the surface. This has so far prohibited investigations necessary to make direct measurements of the electronic structure and test such conjectures because the most direct probes of the electronic structure and carrier dynamics (e.g., ARPES) require surface-sensitive techniques.
We focus on the planar trilayer nickelates Pr4Ni3O8 that can be grown in bulk, cleavable crystalline form suitable for ARPES [ 12 ]. In this material, the apical oxygen has been removed from the Ruddlesden-Popper (R-P) phase Pr4Ni3O10 structure by reduction, leading to square planar nickel assembled into triple NiO2 layers that alternate with fluorite-like R2O2 layers (Fig. 1A). In contrast to the R-P nickelates such as RNiO3, R2NiO4, and R4Ni3O10 that several previous studies have focused on [ 13,14,15,16,17,18 ], the planar nickelates-e.g., R1-xSrxNiO2, R4Ni3O8, are proximate to the d 9 electron count per transition metal ion that is common to the cuprates. Based on the electron counting of the 3d orbital states, the planar trilayer nickelates possess a 1/3-hole doping with average orbital occupancy d 8.67 , which nominally is on the heavily overdoped side of the high TC cuprate phase diagram (Fig. 1B). Furthermore, a recent study has revealed a large orbital polarization of dx 2 -y 2 character in the 3d conduction bands of R4Ni3O8--a key signature of the cuprate electronic structure [ 19 ]. Nevertheless, no experimental measurement of the electronic structure and carrier dynamics of these materials has yet been reported.

Results
We performed high resolution ARPES measurements on single crystals of trilayer nickelate Pr4Ni3O8, cleaved in-situ in the ultra-high vacuum environment of the ARPES spectrometer. To explore the electronic structure, we first present the Fermi surface measured at T=22K in Fig. 1C,D.
By using different photon energies (84eV and 55eV), we take advantage of the matrix element effects to highlight distinct parts of the Fermi surface. The Fermi surface maps clearly display an electron pocket centered at Γ (first zone) and Γ′ (second zone) (Fig. 1C), and a hole pocket or pockets centered around the zone corner (Fig. 1D). Pr4Ni3O8 has a much simpler Fermi surface than that of the related trilayer Ruddlesden-Popper nickelate La4Ni3O10, which has a significantly different electron count, d 7.33 [ 13 ], and an additional piece of Fermi surface near the Γ point not found in either the cuprates or the present compound. In Fig. 1E, we sketch the two main Fermi pockets from panels C&D in solid line, while including a third pocket in dashed line that is more difficult to resolve from those experimental panels. We compare these results with the Fermi surface of optimally doped bilayer cuprate Bi2Sr2CaCu2O8+x (Bi2212) (Fig.1F)  splitting of the hole band. Another possibility is that the actual band splitting is much weaker than predicted by DFT, and the actual splitting is less than our instrumental resolution.
The polarization dependence of our ARPES results (part of the matrix element effects discussed above) indicates that the states which cross the Fermi energy in Pr4Ni3O8 originate from predominantly dx 2 -y 2 symmetry orbitals -a fact that is confirmed by our DFT calculations (see Supplementary Text S1 and Fig. S1 for both the ARPES and DFT results). This finding is consistent with polarization-dependent X-ray absorption measurements reported on these materials [ 19 ]. The dominant dx 2 -y 2 orbital character of Pr4Ni3O8 is a further critical aspect of the electronic structure analogy between these compounds and the cuprates.  The strong electron correlation in Pr4Ni3O8, signified by the large mass enhancement, also manifests through the quasiparticle scattering rates, which are nominally proportional to the ARPES peak widths. More precisely, we can extract the imaginary part of the electron self-energy function if the spectral dispersion can be approximated by a linear function (see detailed explanation in Supplementary Text S5). In this case, the imaginary self-energy Σ″(ω) is given by [ 24,25,28 ]: where ΓMDC(ω) is the MDC width and vB is the bare Fermi velocity, which we obtained from the bare (unrenormalized) DFT band velocity. The orange circles in Fig. 3E show the extracted Σ″(ω) from the "nodal cut" (Γ-M cut or "cut 2", see Fig. S9 for comparison of the MDCs and fits) where the band dispersion is nearly linear. For all energies above ~ 15 meV, the extracted Σ″(ω) displays a linear energy dependence that deviates from the quadratic behavior of the conventional Fermi liquid model (red dashed curve) which we write in a generalized form [ 25 ] as: where T is the temperature, λ defines the general coupling scale, β is a scaling factor for kBT, S0 ¢¢ is an offset parameter that accounts for impurity or disorder scattering and wN is a normalization frequency that maintains the proper dimensionality of the self-energy to cuprates on both the imaginary self-energy and the mass enhancement indicates that the electronic correlation in Pr4Ni3O8 is of a similar exotic form but significantly stronger than that in cuprates.

Score Card and Discussion
Our work provides the first direct observation of the electronic structure in the planar trilayer nickelate, where the Fermi pockets we found are similar to those found in the cuprates, consistent with its d orbital occupancy with dominant dx 2 -y 2 character. Importantly, the electron correlations found in Pr4Ni3O8 are of a very similar exotic type as in cuprates but are somewhat stronger, as reflected by both the larger mass enhancement and linear slope in Σ″(ω).
With this as well as with other experimental data we can put together a type of scorecard of the major physical attributes of the two families of compounds, as shown in Table 1 ) near EF will likely make these states the "drivers" of the superconductivity while the lower density of states from the zone diagonal will likely make them the "passengers". We argue that it is therefore likely that a gap of dx2-y2 symmetry, which derives most of its energy gain from pairing at the (p,0) points and which is generally supported by antiferromagnetic spin fluctuations of the type shown on row 2, should also likely be favored in the nickelate superconductors.
Previous works in cuprates have found that the correlation effects in the strange metal phase play a key role for enhancing Cooper pairing [ 35,38 ]. Likewise, a recent study on ferropnictides also shows a direct connection between the slope of the scattering rate and TC [ 33 ]. In this context, our result raises the important question of whether such stronger strange metal correlations could support high temperature superconductivity in Pr4Ni3O8 [ 35,38 ]. According to the recent DFT and X-ray absorption studies [ 8,19,20 ], the charge transfer energy between O-2p and Ni-3d in the planar trilayer nickelate is much smaller than that of the infinite layer nickelate superconductor, which places the planar trilayer nickelate closer to the cuprates. The hybridization of p-d states in Pr4Ni3O8 gives a substantial super-exchange coupling J of ~69 meV, as recently revealed by resonant inelastic Xray spectroscopy [ 39 ]. Based on the t-J model, the predicted TC of the electron-doped planar trilayer nickelate [ 20 ] would be significantly higher than the maximum TC (15K) of the infinite-layer nickelate [ 4 ] and potentially on par with optimally doped cuprates. The strong electron correlation we find in Pr4Ni3O8 places these studies on a firmer experimental footing.
In summary, the massive strange-metal correlations in Pr4Ni3O8 provide a new platform to explore strange metal physics and its connection to high temperature superconductivity. More generally, this distinct character to the 'overdoped' regime of planar nickelates may serve to bridge from our well-established descriptions of cuprates to our nascent understanding of the infinite layer nickelate superconductors. shown in the paper are integrated intensity over E ± 6 meV.

Materials and
Single-crystal growth and transport measurement. Pr4Ni3O8 single crystals (1-2 mm 2 × 0.5 mm) were obtained by reducing specimens cleaved from boules of Pr4Ni3O10 (flowing 4% H2/Ar gas, 350 °C, five days). High pressure single-crystal growth of Pr4Ni3O10 was performed in an opticalimage floating zone furnace (HKZ-1, SciDre GmbH) with 140 bar O2 [ 12 ]. The crystal structure was identified using X-ray diffraction [ 19 ]. Resistivity of Pr4Ni3O8 single crystals (Fig. 3F) was measured on a Quantum Design PPMS in the temperature range of 2-300 K using a conventional four-probe method with contacts made with silver paint.

DFT calculation.
The first principle calculations of La4Ni3O8 and Pr4Ni3O8 (result shown in Supplementary Materials The Pr4Ni3O8 DFT band structure at the Fermi level is dominated by the Pr 4f states, intersecting the Ni 3d bands (see Supplementary Materials Fig. S5). This is inconsistent with our experimental observations. If one includes the effects of electron correlation and applies different Coulomb repulsion values U to different orbital states, one can purposely move the 4f states away from Ef, and achieve a 3d band structure with similar Fermi surface and bandwidth to that of La4Ni3O8 [22,23]. However, in this work, we need a DFT bare band structure (no electron correlation) as a reference to extract the many-body information from the ARPES results, and the s, p, and d bands of Pr4Ni3O8and La4Ni3O8 are extremely similar. Thus, we choose to use the bare band structure of La4Ni3O8 in this work.    Table 1. Scorecard comparing the main physics topics between 2+ valence cuprates and 1+ valence nickelates. For the widest generality we have mixed the "best" or cleanest results from infinite layer and trilayer nickelates. The similarities between the two columns are overall quite dramatic, though with some numerical differences that may or not be important.