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[A. Chainani](https://orcid.org/0000-0002-5639-5393), M. Horio, C.-M. Cheng, D. Malterre, [K. Sheshadri](https://orcid.org/0000-0002-5264-5113), M. Kobayashi, K. Horiba, H. Kumigashira, [T. Mizokawa](https://orcid.org/0000-0002-7682-2348), M. Oura, [M. Taguchi](https://orcid.org/0000-0001-7047-4955), Y. Mori, A. Takahashi, T. Konno, T. Ohgi, H. Sato, [T. Adachi](https://orcid.org/0000-0002-9223-8483), Y. Koike, [T. Mochiku](https://orcid.org/0000-0003-2208-4279), [K. Hirata](https://orcid.org/0000-0001-8241-5100), S. Shin, [M. K. Wu](https://orcid.org/0000-0001-6681-0098), [A. Fujimori](https://orcid.org/0000-0003-4876-0634)

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[Oxygen on-site Coulomb energy in Pr1.3−xLa0.7CexCuO4 and Bi2Sr2CaCu2O8+δ and its relation with Heisenberg exchange](https://mdr.nims.go.jp/datasets/76ea432b-a264-49e5-b318-e6b438b3b928)

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Oxygen on-site Coulomb energy in ${\rm Pr}_{1.3-x}{\rm La}_{0.7}{\rm Ce}_x{\rm CuO}_{4}$ and ${\rm Bi}_2{\rm Sr}_2{\rm CaCu}_2{\rm O}_{8+\delta }$ and its relation with Heisenberg exchangePHYSICAL REVIEW B 107, 195152 (2023)Oxygen on-site Coulomb energy in Pr1.3−xLa0.7CexCuO4 and Bi2Sr2CaCu2O8+δand its relation with Heisenberg exchangeA. Chainani ,1,2 M. Horio,3,4 C.-M. Cheng,1 D. Malterre,5 K. Sheshadri ,6 M. Kobayashi,7 K. Horiba,8 H. Kumigashira,9T. Mizokawa ,10 M. Oura,2 M. Taguchi ,2,* Y. Mori,11 A. Takahashi,11 T. Konno,11 T. Ohgi,11 H. Sato,11 T. Adachi ,12Y. Koike,11 T. Mochiku,13 K. Hirata ,13 S. Shin,2,† M. K. Wu ,14 and A. Fujimori 3,15,11Condensed Matter Physics Group, National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan, Republic of China2RIKEN SPring-8 Centre, 1-1-1 Sayo-cho, Hyogo 679-5148, Japan3Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan4Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan5Institut Jean Lamour, Université de Lorraine, UMR 7198 CNRS, Bôite Postale 70239, 54506 Vandoeuvre lés Nancy, France6226, Bagalur, Bangalore North, Karnataka State 562149, India7Department of Electrical Engineering and Information Systems and Center for Spintronics Research Network,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan8National Institutes for Quantum and Radiological Science and Technology (QST), Sayo, Hyogo 679-5148, Japan9Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan10Department of Applied Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan11Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan12Department of Engineering and Applied Sciences, Sophia University, Tokyo 102-8554, Japan13National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan14Institute of Physics, Academia Sinica, Taipei 115201, Taiwan, Republic of China15Center for Quantum Technology, and Department of Physics, National Tsing Hua University,Hsinchu 30013, Taiwan, Republic of China(Received 25 February 2023; revised 6 May 2023; accepted 8 May 2023; published 30 May 2023)We study the electronic structure of electron-doped Pr1.3−xLa0.7CexCuO4 (PLCCO; Tc = 27 K, x = 0.1)and hole-doped Bi2Sr2CaCu2O8+δ (Bi2212; Tc = 90 K) cuprate superconductors using x-ray absorption spec-troscopy and resonant photoemission spectroscopy (Res-PES). From Res-PES across the O K-edge and CuL-edge, we identify the O 2p and Cu 3d partial density of states (PDOS) and their correlation satellites, whichoriginate in two-hole Auger final states. Using the Cini-Sawatzky method, analysis of the experimental O2p PDOS shows an oxygen on-site Coulomb energy for PLCCO to be Up = 3.3 ± 0.5 eV, and for Bi2212,Up = 5.6 ± 0.5 eV, while the copper on-site Coulomb correlation energy is Ud = 6.5 ± 0.5 eV for Bi2212.The expression for the Heisenberg exchange interaction J in terms of the electronic parameters Ud , Up,charge-transfer energy �, and Cu-O hopping tpd obtained from a simple Cu2O cluster model is used to carryout an optimization analysis consistent with J known from scattering experiments. The analysis also providesthe effective one-band on-site Coulomb correlation energy Ũ and the effective hopping t̃ . PLCCO and Bi2212are shown to exhibit very similar values of Ũ/t̃ ∼ 9–10, confirming the strongly correlated nature of the singletground state in the effective one-band model for both materials.DOI: 10.1103/PhysRevB.107.195152I. INTRODUCTIONSince its discovery more than 35 years ago [1], an under-standing of superconductivity in high-transition temperature(Tc) cuprate superconductors continues to attract researcherseven today. Extensive experimental and theoretical efforts tounderstand the cuprates have identified important aspects oftheir electronic structure, such as spin- and charge-ordering[2–12], a dx2−y2 -type superconducting gap [13,14], the role*Present address: Toshiba Nanoanalysis Corporation, Kawasaki212-8583, Japan.†Deceased.of antiferromagnetic correlations [15–17], electron-phononcoupling [18], a temperature and momentum-dependent pseu-dogap [19,20], etc. The charge ordering favors localizationof carriers and competes with superconductivity of dopedcarriers in the CuO2 layers, thereby leading to novel transport,thermodynamic, and spectroscopic phenomena that suggestquantum critical behavior [21–25]. However, the origin for thehigh-Tc superconductivity in the cuprates still remains an openproblem [26].Several important models have emphasized the complexnature of the superconductivity and electronic structure of thecuprates. Starting with the one-band Hubbard model [27,28],theoretical models evolved along several different routes, suchas the resonating valence bond theory [29], the three-band2469-9950/2023/107(19)/195152(12) 195152-1 ©2023 American Physical Societyhttps://orcid.org/0000-0002-5639-5393https://orcid.org/0000-0002-5264-5113https://orcid.org/0000-0002-7682-2348https://orcid.org/0000-0001-7047-4955https://orcid.org/0000-0002-9223-8483https://orcid.org/0000-0001-8241-5100https://orcid.org/0000-0001-6681-0098https://orcid.org/0000-0003-4876-0634http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.107.195152&domain=pdf&date_stamp=2023-05-30https://doi.org/10.1103/PhysRevB.107.195152A. CHAINANI et al. PHYSICAL REVIEW B 107, 195152 (2023)Hubbard model [30,31], the t-J model [32], spin fluctuationtheory [33], marginal Fermi liquid theory [34], the pair densitywave model [35], electron-phonon coupling-induced pairingwhich goes beyond the BCS model [36], etc. Although the ori-gin of superconductivity in the cuprates remains a challenge, itis generally accepted that the quasi-two-dimensionality of theCuO2 layers and strong on-site Coulomb correlations providea suitable starting point for describing the electronic structureof the cuprates [27–35,37–42].Early studies using the Cini-Sawatzky method based on thetwo-hole Auger correlation satellite [43,44] showed that theO on-site Coulomb energy Up can be large (∼5–6 eV) andclose to the copper on-site Coulomb energy Ud (∼6–8 eV)in YBa2Cu3O7 (YBCO) [45,46], Bi2Sr2CaCu2O8+δ (Bi2212)[47], and La2−xAxCuO4 (A = Sr, Ba) [48–50]. Further,Up ∼ Ud is also known for several oxides across the 3d tran-sition metal (TM) series : titanium/vanadium oxides (SrTiO3,V2O3, VO2, V2O5) [51–53], LaMO3 (M = Mn-Ni) per-ovskites [54,55], and cuprates (including Cu2O and CuO)[45–48,56]. A theoretical study on rare-earth nickelates(RNiO3) with values of Ud (=7 eV) and Up (=5 eV) showedthe relation of a novel charge-order involving ligand holeswith the metal-insulator transition in RNiO3 [57]. Very re-cently, the relation of the intersite Heisenberg exchangeinteraction J with Ud and Up was recognized for the parentcuprates as well as hole-doped cuprates [58]. In particular, itwas shown that J could be used as a bridge to connect theelectronic parameters Ũ and t̃ of the widely used effectiveone-band Hubbard model with the parameters Ud , Up, �,and tpd known from the three-band Hubbard model, clustermodel calculations applied to core-level spectroscopy as wellas resonant inelastic x-ray scattering [58], and from ab initioelectronic structure calculations [59].Surprisingly, there is no experimental estimate of Up usingthe Cini-Sawatzky method in electron-doped cuprates whichpossess CuO2 planes without the apical oxygen site, i.e.,the cuprates crystallizing in the so-called T ′ structure. ForBi2212, the estimate of Ud and Up was made using knowncluster model parameters [56] to explain the Res-PES spectra[47]. While optimally doped Bi2212 (TC ∼ 90 K) has beenextremely well-studied using soft- and hard-x-ray photoe-mission [47,60–64], as well as low-energy angle-resolvedphotoemission spectroscopy (ARPES) studies of its banddispersions and Fermi surfaces [13,18–21,38], there is noestimate of Ud and Up using the experimental Cu 3d and O2p partial density of states (PDOS). Thus, we felt it impor-tant to experimentally quantify on-site Coulomb energies inan electron-doped system in comparison with a well-studiedhole-doped system. Further, recent studies on the T ′ struc-ture Pr1.3−xLa0.7CexCuO4 (PLCCO) showed the importanceof reduction annealing to achieve electron-doped supercon-ductivity [65–68]. From careful ARPES studies, it was shownthat the superconducting state was found to extend over awide electron doping range with an optimal TC ∼ 27 K [69].Interestingly, a sharp quasiparticle feature was observed onthe entire Fermi surface of optimally doped PLCCO withno signature of the antiferromagnetic (AF) pseudogap whichindicated a reduced AF correlation length [66]. However, thesuperconducting gap still showed a dx2−y2 symmetry like thewell-known results for the hole-doped Bi2212 [13] and forelectron-doped NCCO [14], and it suggests the importanceof spin-fluctuations as a viable source of pairing even forPLCCO [67].In this work, we have used the Cini-Sawatzky method toobtain Ud (= 6.5 ± 0.5 eV for Bi2212) and Up values (=5.6 ± 0.5 eV for Bi2212 and 3.3 ± 0.5 eV for PLCCO). How-ever, since the Pr 3d core level overlaps with the Cu 2p corelevel and also the Pr 4 f valence-band states overlap the Cu3d states, we could not separate out the Cu 3d states fromthe Pr 4 f states of PLCCO. Hence, we could not estimate Udfor PLCCO, but instead we use the Ud estimated for Bi2212.Next, using the estimated Ud and Up values, and known valuesof � and tpd , we obtain a set of parameter values for PLCCOand Bi2212 consistent with the experimental J known fromneutron or x-ray scattering using an optimization procedure[58]. The method also provides the effective one-band pa-rameters Ũ and t̃ consistent with the experimental J . Theresults show that Ũ/t̃ ∼ 9–10 for both PLCCO and Bi2212,and they confirm the strongly correlated nature of the effectiveone-band singlet state in spite of the significantly differentvalues of Up.II. EXPERIMENTWe have carried out x-ray absorption (XAS) and resonantphotoemission spectroscopy (Res-PES) across the O K-edgeof electron-doped Pr1.3−xLa0.7CexCuO4 (PLCCO, with x =0.1; Tc = 27 K) and hole-doped (Bi2212; Tc = 90 K) toestimate Up. For Bi2212, we also measured XAS and Res-PES across the Cu L-edge to estimate Ud . In addition, XASand Res-PES across the O K-edge was measured for PLCCOwith x = 0.0, which shows an antiferromagnetic metal groundstate, to check the doping dependence of the two-hole Augersatellite. The Bi2212 single-crystal samples were preparedby the traveling solvent floating zone method as reportedin the literature [70], and characterized for their supercon-ducting Tc = 90 K. Res-PES across the O K-edge and CuL-edge for Bi2212 was performed at BL17SU of SPring-8,Japan, with an energy resolution �E = 0.2 eV. Bi2212 waspeeled with scotch-tape in UHV and measured at T = 20 K.The Fermi level EF of gold was measured to calibrate theenergy scale. Low-energy off-resonant synchrotron valence-band PES measurements (hν = 22.0 and 53.0 eV) werecarried out at BL21 of Taiwan Light Source, NSRRC, Tai-wan. The energy resolution was set to �E = 15 meV and thesample temperature was T = 10 K. Single crystals of PLCCOwith x = 0.0 and 0.10 were synthesized by the traveling-solvent floating-zone method and were protect annealed for24 h at 800 ◦C [65]. The x = 0.1 composition showed a su-perconducting TC = 27 K. XAS and Res-PES across the OK-edge for PLCCO was performed at BL2A of Photon Fac-tory, Japan, with an energy resolution �E = 0.2 eV. The XASand Res-PES measurements were carried out at T = 200 K.Low-energy synchrotron PES with hν = 16.5 and 55.0 eVfor PLCCO was performed at BL9A HiSOR and BL28A ofPhoton Factory, Japan, respectively. The energy resolutionwas set to �E = 15 meV at HiSOR and at BL28A of PhotonFactory. The measurements were carried out at T = 9 K, andEF of gold was measured to calibrate the energy scale.195152-2OXYGEN ON-SITE COULOMB ENERGY IN … PHYSICAL REVIEW B 107, 195152 (2023)Intensity (arb. units)534532530528526Photon energy (eV)PLCCO x = 0.1 annealedO K-edge XASFIG. 1. The O K-edge (1s-2p) x-ray absorption spectrum ofPLCCO, x = 0.1.III. RESULTS AND DISCUSSIONSFigure 1 shows the O K-edge (1s-2p) XAS spectrum ofPLCCO, x = 0.1, measured at T = 200 K over the incidentphoton energy range of hν = 526–535 eV. It shows a smallpeak at ∼528.7 eV and a broad structure between 530 and534 eV, with a weak shoulder at ∼531 eV. The states above530 eV are attributed to the overlapping La, Ce, and Pr 5dstates hybridized with O 2p states [71], while the 528–530 eVstates are due to Cu 3d–O 2p hybridized states. The peak at528.7 eV is quite similar to the lowest energy peak featureseen in the O K-edge XAS of electron-doped NCCO, whichwas analyzed as the unoccupied upper Hubbard band associ-ated with Cu 3d states hybridizing with O px, py states, whilethe pz states are mixed into the tail of the ∼531 eV shoulder[71].Figure 2(a) shows the O 1s-2p Res-PES spectra of PLCCO,x = 0.1, obtained using incident photon energies labeled byvertical tick marks in Fig. 1. The main valence-band spectrashow three features consisting of a rounded peak at about1.5 eV binding energy (BE), a small sharp feature at around2.5 eV BE, and a broad feature spread over 2.5–7.5 eV BE.The rounded peak is attributed to the mainly Pr3+ occupied4 f 2 states, which have a strong cross-section at these hνvalues compared to Cu 3d states, which are also expectedover the same energies but hidden below the Pr 4 f states.The small sharp feature at 2.5 eV BE is due to the Ce3+occupied 4 f 1 states. This is confirmed by comparing the O1s-2p Res-PES spectra of PLCCO, x = 0.0, which do notcontain Ce, as discussed in the Appendix. The broad featureat 2.5–7.5 eV BE mainly consists of the O 2p states. Thevalence-band spectrum measured with hν = 55.0 eV is alsoshown in Fig. 2(a). It confirms the suppression of the Ce andPr 4 f states due to their low photoionization cross-sections atlow incident hν, and it also confirms the dominantly O 2pPDOS character of the broad feature spread over 2.5–7.5 eVBE.In this work, our main interest is to measure over higherbinding energies and check for the O KVV Auger satellitefeature, which originates from a two-hole final state and pro-vides a measure of Up. As can be seen in Fig. 2(a), a weakFIG. 2. (a) The Res-PES spectra across the O K-edge (1s-2p) ofPLCCO, x = 0.1, measured at photon energies marked with verticalbars in Fig. 2. The spectra are normalized at 8 eV BE. The off-resonance valence-band spectrum measured with hν = 55.0 eV isalso shown. (b) The difference spectra obtained for higher energieswith respect to the hν = 526.2 eV spectrum.feature seen at ∼11 eV BE shows a small increase in inten-sity on increasing the incident hν from 526.2 to 527.2 eV.For higher hν > 527.2 eV, the feature gets strongly enhancedand shifts to higher BEs tracking the increase in hν [reddashed line in Fig. 2(a)]. This behavior is a signature of theAuger two-hole satellite. To characterize the evolution of thesatellite, in Fig. 2(b), we have plotted the difference spectrawith respect to hν = 526.2 eV for all higher hν. The differ-ence spectra show a small intensity increase of the satellitefeature at ∼11 eV BE for hν = 527.2 eV (see Fig. 3 for anexpanded y-scale figure). On increasing hν, it shows a system-atic increase in intensity with an energy shift and a coupledsuppression of the main O 2p valence-band intensity. Theenergy shift is seen with a small increase in intensity up tohν = 529.7 eV, but a small increase of the main valence-bandintensity is also observed at hν = 529.7 eV. The La and Ce5p states are observed in Fig. 2(a) as weak bumps between∼15 and 18 eV BE, while the Pr 5p states are between ∼20and 23 eV and overlap with the O 2s states at ∼23 eV. A verysimilar behavior was observed in the O K-edge XAS and O1s-2p Res-PES spectra of PLCCO, x = 0.0 (detailed in theAppendix), indicating a very similar O KVV Auger two-holesatellite.To estimate Up using the Cini-Sawatzky method, we plotthe PLCCO, x = 0.1 valence-band spectra with hν = 16.5and 55.0 eV, as shown in Figs. 3(a) and 3(b), respectively.At these energies, the overall valence-band spectrum is dom-inated by O 2p states but it can be seen that the spectrumwith hν = 55.0 eV is slightly broader than at hν = 16.5 eV.From a numerical self-convolution of the one-hole valence-band spectra, we obtained the two-hole spectra, also shownin Figs. 3(a) and 3(b). Comparing the two-hole spectra with195152-3A. CHAINANI et al. PHYSICAL REVIEW B 107, 195152 (2023)FIG. 3. (a) The valence-band spectrum of PLCCO for x = 0.1measured with hν = 16.5 eV. From a numerical self-convolution ofthe one-hole valence-band spectrum, we obtained the two-hole spec-tra. Comparing the two-hole spectrum with the difference spectraobtained at hν = 527.2 eV (which shows the correlation satellitefeature for x = 0.1), we could estimate Up = 3.3 ± 0.5 eV for hν =16.5 eV. (b) Similarly, we could estimate Up = 3.0 ± 0.5 eV forhν = 55 eV. The correlation satellite feature for x = 0.0 also lies atthe same energy as for x = 0.1.the difference spectra of x = 0.0 and 0.1 obtained at hν =527.2 eV, which is the lowest energy that shows the two-holecorrelation satellite feature, we estimate Up = 3.3 ± 0.5 eVfor hν = 16.5 eV and Up = 3.0 ± 0.5 eV for hν = 55.0 eV.Thus, the estimated Up from the analyses using hν = 16.5 and55.0 eV for x = 0.1 are quite close to each other. Interestingly,as seen in Fig. 3, since the two-hole correlation satellite fea-ture for x = 0.0 is observed at the same energy as for x = 0.1,it suggests that the strength of Up does not depend on theelectron doping content.In Fig. 4, we plot the O K-edge (1s-2p) XAS spectrumof Bi2212 measured at T = 20 K over the incident photonenergy range of hν = 526–536 eV. The spectra are quite sim-ilar to early reports of the XAS of Bi2212 [47]. It shows asmall peak at ∼529.3 eV and a shoulder at ∼531.35 eV, whichextends as a broad feature up to nearly 535 eV. The shouldermarks the onset of the upper Hubbard band associated withCu 3d states bonding to O px, py states, while states aboveare attributed to the Bi, Sr, and Ca states hybridized with O2p states. It is well-known that the peak at 529.3 eV shows anintensity proportional to the doped hole states [72,73]. SimilarFIG. 4. The O K-edge (1s-2p) x-ray absorption spectrum ofBi2212. The photon energies labeled (a)–(f) were used to measurethe Res-PES spectra across the O K-edge as discussed in Fig. 5.behavior was also seen in O K-edge XAS spectra of hole-doped La2−xSrxCuO4 [74]. At photon energies labeled (a)–(f),we then carried out O 1s-2p Res-PES spectra of Bi2212 tocheck for the two-hole Auger correlation satellite.As shown in Fig. 5, the O 1s-2p Res-PES spectra ofBi2212, measured over a wide BE range of 30 eV, exhibitmany shallow core levels, which are due to Bi 5d , Ca 3p, O 2s,and Sr 4p as labeled in Fig. 8. The shallow core features be-tween 17 and 30 eV BE allow us to consistently calibrate theon-resonance spectra in spite of the relatively weak intensitiesof the main valence-band spectra between EF and about 7 eVBE. Importantly, we see that the peak feature at 12.8 eV BEsystematically increases in intensity on increasing the incidenthν from 527.7 to 529.3 eV (a)–(c). At hν = 530.6 eV, theintensity reduces, reflecting the dip in the XAS spectrum andthen increases again for hν = 531.3–534.3 eV. From hν =529.3 to 534.3 eV, the feature systematically shifts to higherBEs tracking the increase in hν, confirming its Auger two-holesatellite character.To estimate Up for Bi2212, we measured the valence-bandspectrum with hν = 53.0 eV, as plotted in Fig. 6. The spec-trum shows the dominantly O 2p states hybridized with Cu 3dstates, centered at about 3.5 eV BE, and very weak intensityFIG. 5. The Res-PES spectra measured across the O K-edge(1s-2p) of Bi2212 at photon energies labeled (a)–(f) in Fig. 4. Thespectra are normalized to the incident photon flux.195152-4OXYGEN ON-SITE COULOMB ENERGY IN … PHYSICAL REVIEW B 107, 195152 (2023)FIG. 6. The off-resonance valence-band spectrum of Bi2212measured with hν = 53 eV, which represents the dominantly O 2pPDOS hybridized with Cu 3d states. The numerical self-convolutionof the valence-band spectrum is compared with the on-resonancespectrum obtained with hν = 529.3 eV in order to estimate Up.with a step at the EF . We have also measured the valenceband with hν = 22.0 eV (see Fig. 9, inset), but it is known thatthe Bi2212 spectrum shows relatively high intensity featuresdue to the Bi-O derived O 2p states between 4 and 8 eVBE [60]. Since we are interested in knowing the Up for theCuO2-plane oxygen sites, we used the hν = 53.0 eV spectrumto estimate Up. The numerical self-convolution of the one-holehν = 53.0 eV valence-band spectrum is plotted together withthe on-resonance spectrum obtained with the hν = 529.3 eVspectrum, as shown in Fig. 6. The energy separation of themain peaks between these two spectra provides an estimateof Up = 5.6 ± 0.5 eV. Thus, the estimated Up = 5.6 ± 0.5 eVfor Bi2212 is larger than the value of Up = 3.3 ± 0.5 eV forPLCCO, and it indicates that Up values can vary significantlyfor different families of cuprates. While the origin of thisdifference in Up between PLCCO and Bi2212 is not clear,it is generally considered that the on-site Coulomb energyin a solid is strongly reduced from the atomic values dueto solid-state screening. Considering the differences in thecrystal structure of PLCCO and Bi2212, the smaller Up forPLCCO may be attributed to the generally smaller � (equiv-alently, the smaller charge-transfer gap) of the electron-dopedcuprates compared to the hole-doped ones.Next, we do the same exercise of estimating on-siteCoulomb energy but for the Cu site, Ud , in Bi2212. Fig-ure 7 shows the Cu L-edge XAS spectrum, which exhibitsa typical single peak feature for the L3 and L2 edges. Thisis consistent with early work on Bi2212 [61,73], which alsoreported polarization-dependent studies to characterize theCu 3d states. It was shown that the single peak featurewas dominated by the 3dx2−y2 states, but also included about15% 3dz2−r2 contribution [61,73]. At photon energies labeled(a)–(g) marked in Fig. 7, we measured the Cu 2p-3d Res-PES spectra of BI2212 to check for the Cu two-hole Augercorrelation satellite. Figure 8 shows the valence-band spectrameasured over a wide energy range of 30 eV BE includingthe shallow core levels of Bi 5d , Ca 3p, O 2s, and Sr 4p.The shallow core-level positions help us to confirm the energycalibration. The spectral changes consist of a suppression orFIG. 7. The Cu L-edge (2p-3d) x-ray absorption spectrum ofBi2212. The photon energies labeled (a)–(g) were used to measurethe Res-PES spectra across the Cu L-edge, as discussed in Fig. 8.antiresonance behavior of the main valence band, coupled toa large increase of the feature at about 12.5 eV BE. This peakshows a tenfold increase in intensity on changing hν from930.6 to 933.4 eV corresponding to a resonant enhancement.Please note that the spectrum obtained with hν = 933.4 eV isdivided by a factor of 10. The spectrum with hν = 933.4 eVis very similar to the early study by Brookes et al., whichshowed a strong resonant enhancement of the ∼12.5 eV satel-lite feature [63]. The authors further identified the feature at∼12.5 eV as the atomic like 1G-state, the very weak feature at∼16 eV as the 1S-state and the weak feature at ∼10 eV as the3F -state, as shown on an expanded scale in Fig. 9.On increasing hν further from 933.4 to 940.0 eV, thefeature at 12.5 eV BE systematically moves to higher BE,tracking the increase in hν, and this indicates that the featureis the Cu L3VV two-hole Auger satellite, consistent with earlyreports [47]. To estimate Ud , we then measured the valenceband of Bi2212 with hν = 22.0 eV and compared it withthe off-resonance spectrum obtained with hν = 927.9 eV, asshown inset of Fig. 9. The hν = 22.0 eV spectrum repre-sents the valence-band spectrum dominated by O 2p PDOS,which are hybridized with Bi and Cu valence-band states. InFIG. 8. The Res-PES spectra measured across the Cu L-edge(2p-3d) of Bi2212 at photon energies labeled (a)–(g) in Fig. 7. Thespectra are normalized to the incident photon flux, and in addition,for photon energies (c)–(e), the spectra were scaled by a factor tofacilitate a comparative evolution of the L3VV feature.195152-5A. CHAINANI et al. PHYSICAL REVIEW B 107, 195152 (2023)FIG. 9. The numerical self-convolution of the Cu 3d PDOSis compared with the on-resonance spectrum obtained with hν =933.4 eV in order to estimate Ud . Inset: The Cu 3d PDOS wasobtained as the difference between the valence-band spectrum ofBi2212 (hν = 22.0 eV) and the off-resonance spectrum [hν =927.9 eV (Fig. 8)].particular, it was shown that the features between ∼4 and 8 eVBE are dominated by the Bi-O hybridized states, which getsuppressed even with hν = 53.0 eV (see Fig. 6). Hence weused the hν = 53.0 eV spectrum to estimate Up for the O 2pstates associated with the CuO2 planes. On the other hand,since the Cu 3d cross section dominates at hν = 927.9, thehν = 927.9 eV spectrum is considered to have an enhancedcontribution of Cu 3d states, albeit hybridized with O 2pstates. To separate out the dominantly Cu 3d character PDOS,we normalized the spectra in the inset of Fig. 9 at 5.5 eV BEand obtained a difference spectrum, which is also plotted inthe same inset.We then carried out a numerical self-convolution of the dif-ference spectrum and compared it with the on-resonance hν =933.4 eV spectrum, which showed the Cu L3VV two-holeAuger satellite (Fig. 9, main panel). Although the numericalself-convolution shows weak features at BEs of 8, 10, and13 eV, we have checked that they are artifacts that arise fromthe structures between 4 and 6.5 eV BE in the differencespectrum (inset, Fig. 9) associated with the Bi-O states lyingat 4–8 eV BE. Hence, we used the main peak of the numericalself-convolution at 6 eV BE to get an estimate of averageUd in Bi2212. The energy separation between the main peakof the numerical self-convolution and the main peak of theCu L3VV two-hole Auger satellite provides an estimate ofUd = 6.5 ± 0.5 eV for Bi2212. The error bar of ±0.5 eV wasestimated by shifting the Auger spectrum by ±0.5 eV, whichleads to a width in fair agreement with the main peak of theself-convoluted two-hole spectrum. Using the same method, avalue of Ud = 6.5 ± 0.5 eV was also estimated recently forthe three-layer cuprate superconductor HgBa2Ca2Cu3O8+δ[75], which shows the highest Tc = 130 K at ambient pressure[76]. Having obtained estimates of Ud and Up, we appliedthem to determine the Heisenberg exchange J and the relationbetween the effective one-band and three-band Hubbard mod-els for PLCCO and Bi2212. But before that, we discuss belowthe very early work by deBoer et al. [77], which clarifiedthe difference between the Ud deduced from Auger spectracompared to the Hubbard Ud .For an atom M in a solid, the Ud obtained from the two-hole Auger satellite is the energy cost for the “reaction”2(M+) → M + (M2+). Then, the value of Ud (Auger) is thedifference between the first ionization energy (I1) and thesecond ionization energy (I2), i.e., Ud (Auger) = I2 − I1.However, the Hubbard Ud is the energy cost for the “reaction”2M → (M−) + (M+), i.e., Hubbard Ud = I1 − A, and it cor-responds to the difference between the first ionization energyI1 and the electron affinity A. While both the values representthe energy difference between one less electron and one moreelectron compared to a reference state, the reference states Mand M+ are obviously not the same. But the difference in theestimated values of Ud (Auger) and Hubbard Ud is expectedto be small due to solid-state screening effects [77]. It is notedthat for the value of Hubbard Ud for Cu, most of the literatureuses values between 6 and 8 eV [45–50,56,78–80], while weobtain Ud (Auger) = 6.5 ± 0.5 eV, confirming that they arenot very different.In a recent study [58], we developed an optimizationprocedure to estimate effective one-band Hubbard modelparameters Ũ and t̃ using reported three-band parameterstpd ,�,Ud , and Up from theoretical studies [78] as wellas cluster model calculations [79,80]. In this procedure,the Heisenberg exchange J calculated using a downfoldingmethod [81] for a Cu2O cluster model employing the three-band Hamiltonian in the hole picture is given byJ = 4t4pd�2[1Ud+ 1� + Up/2]. (1)This agrees with the expression obtained by fourth-order per-turbation theory [82–85], but in the approximation of intersiteCoulomb interaction Upd = 0 (which is typically smaller thanUp and Ud [42]) and the oxygen-oxygen hopping tpp = 0(since we used a Cu2O cluster). If we now write J = 4t̃2/Ũ ,then we can identifyt̃ = t2pd�,1Ũ= 1Ud+ 1� + Up/2. (2)As explained in Ref. [58], this expression for J does notlead to a satisfactory agreement with J = 121 meV(Pr2CuO4)reported from neutron scattering [86] and J = 161 meV(Bi2212) from x-ray scattering measurements [86,87], if wedirectly use reported values of three-band parameters. Wealso checked it by using the J obtained for a sample ofBi2212 doped into the antiferromagnetic regime, by scalingthe 2-magnon Raman scattering result [88] of J = 124 meVto an effective neutron scattering result of J = 132 meV,using La2CuO4 as a reference case, but we could not obtaina satisfactory agreement. Then, using an optimization pro-cedure, we first find values that provide a good agreementwith J known from the scattering experiments [86–88]. Itwas found that the energy cost was minimal for the secondoptimization procedure (described in Ref. [58]), in which wemodify the parameters (t̄pd , �̄; columns 3 and 4 in Table I) andobtain optimal values (tpd and �; columns 5 and 6 in TableI). The optimal values are sufficiently close to values of t̄pd ,�̄ using the three-band model or cluster model calculationsreported in the literature. Next, we use our measurements ofUd and Up, and optimal values tpd ,�, to estimate the one-bandparameters Ũ and t̃ . The results are summarized in TableI. The results show that Ũ/t̃ ∼ 9–10 for both PLCCO and195152-6OXYGEN ON-SITE COULOMB ENERGY IN … PHYSICAL REVIEW B 107, 195152 (2023)TABLE I. Electronic parameters (U d , U p, t pd , �) for cuprates from the three-band Hubbard model/cluster model calculations. The tablealso shows an optimized parameter set of (tpd and �). J is the nearest-neighbor Heisenberg exchange deduced from scattering experiments.See the text for details.Optimized setCompound U d U p t pd � tpd � J (ref.no) Ũ t̃ Ũ/t̃(ref. no) ±0.5 eV ±0.5 eV ±1.0 eV ±0.2 eV eV eV meV eV eVPr2CuO4(80) 8.0 4.1 1.1 3.0 1.0 3.2 121 (86) 3.16 0.31 10.19PLCCO(80) with 6.5 3.3 1.1 3.0 0.96 3.0 121 (86) 2.7 0.29 9.31experimental Ud , UpBi2212(78) 8.5 4.1 1.13 3.2 1.1 3.5 161 (87) 3.34 0.37 9.03Bi2212(79) 7.7 6.0 1.5 3.5 1.2 3.7 161 (87) 3.59 0.38 9.44Bi2212(79) with 6.5 5.6 1.5 3.5 1.16 3.5 161 (87) 3.2 0.36 8.9experimental Ud , UpBi2212(78) 8.5 4.1 1.13 3.2 1.07 3.4 132 (88) 3.33 0.33 10.09Bi2212(79) 7.7 6.0 1.5 3.5 1.13 3.7 132 (88) 3.58 0.34 10.53Bi2212(79) with 6.5 5.6 1.5 3.5 1.10 3.7 132 (88) 3.24 0.33 9.81experimental Ud , UpBi2212 using experimental Ud and Up, and they confirm thestrongly correlated nature of the effective one-band singletstate [58]. It is very interesting to note that the estimatedone-band parameters Ũ and t̃ show small differences forPLCCO and Bi2212, although the Up values are significantlydifferent for them. If one looks at the small differences be-tween PLCCO and Bi2212 more closely, one can see thatthe smaller tpd for PLCCO (due to its longer in-plane latticeparameter) is responsible for the smaller J and t̃ , in spiteof the smaller � and Up. On the other hand, the smaller �and Up do play a major role in reducing Ũ in PLCCO. Incontrast, the larger tpd for Bi2212 (due to its shorter in-planelattice parameter) is responsible for the larger t̃ . Although alarger � and Up result in a relative increase in Ũ for Bi2212,the net result is ∼10% larger J for Bi2212 compared toPLCCO, based on the scaled Raman scattering estimate forBi2212.More interestingly, the obtained values of t̃ = 0.29 eV (forPLCCO) and t̃ = 0.33–0.36 eV (for Bi2212) are quite closeto the values of the primary or nearest-neighbor (NN) hoppingt = 0.26 eV (for PLCCO) and 0.36 eV (for Bi2212) estimatedfrom fitting the ARPES Fermi surfaces of PLCCO [68] andBi2212 [89]. It is noted that the tight-binding fits for PLCCOand Bi2212 also employed a second NN hopping t ′ (= 0.24tand 0.3t , respectively) and for Bi2212 an additional out-of-plane hopping t⊥ (= 0.3t), which are relatively small. Similarresults have been reported for La2CuO4 and Sr2CuO2Cl2 forwhich the neutron scattering results could be explained by aneffective extended one-band model. For La2CuO4, a dominantNN hopping t = 0.33 eV implied an effective U/t = 8.8 withU = 2.9 eV, but in addition to the NN exchange J = 138 meV,it was important to include a ring exchange term with Jc =38 meV, and the second NN and third NN exchange J ′ = J ′′ =2 meV [90]. For Sr2CuO2Cl2, the authors used a t-t ′-t ′′-Jmodel and obtained t = 0.35 eV, t ′ = 0.12 eV, t ′′ = 0.08 eV,and with J = 0.14 eV [91,92], it implied an effective U/t = 10with U = 3.5 eV. All these cases suggest that the NN hoppingt and U can be considered to be t̃ and Ũ of the effectiveone-band model.Thus, in spite of the differences in PLCCO and Bi2212, t̃plays an important role in determining the value of J and alsoresults in a very similar value of Ũ/t̃ ∼ 9–10. Several studieshave emphasized J as being one of the most important param-eters to achieve high-temperature superconductivity exhibitedby the family of cuprates [26,86,87,93–101]. It is clear fromEq. (1) that Up, Ud , �, and tpd all play an important role indetermining the Heisenberg exchange J . Finally, using Eq. (2)and writing J = 4t̃2/Ũ in the effective one-band Hubbardmodel form provides a bridge to understand the connectionbetween the effective one-band and three-band Hubbard mod-els of the cuprates [102,103]. While they have often beenconsidered as distinct models, in essence, as the present resultsshow, they are truly equivalent.IV. CONCLUSIONSIn conclusion, the Cini-Sawatzky method was employedto obtain the experimental values of Ud (= 6.5 ± 0.5 eV) forBi2212 and Up for Bi2212 (= 5.6 ± 0.5 eV) and PLCCO(= 3.3 ± 0.5 eV). This indicates that the Up values can varysignificantly in different families of cuprates. Using the esti-mated Ud and Up values, and known values of � and tpd , wecould obtain a set of optimal parameter values for PLCCOand Bi2212 consistent with the experimental J known fromneutron, x-ray, and Raman scattering. We also obtained theeffective one-band parameters Ũ and t̃ for the experimentalJ . The results show that Ũ/t̃ ∼ 9–10 for both PLCCO andBi2212, and they confirm the strongly correlated nature of theeffective one-band singlet state.ACKNOWLEDGMENTSThe synchrotron radiation experiments were performed atBL17SU, SPring-8, Japan with the approval of RIKEN (Pro-posal No. 20140019); BL 2A and BL 28 Photon Factory,Japan (2014G177, 2012G075, 2012S2-001); BL9A HiSOR,Japan; BL 21A Taiwan Light Source, NSRRC, Taiwan. Wethank H. Anzai, M. Arita, K. Ono, H. Suzuki, K. Koshiishi,and D. Ootsuki for valuable technical support. This work195152-7A. CHAINANI et al. PHYSICAL REVIEW B 107, 195152 (2023)was supported by JSPS KAKENHI (Grants No. JP19K03741,No. JP22K03535, and No. JP19H01841) and by the “Programfor Promoting Researches on the Supercomputer Fugaku”(Basic Science for Emergence and Functionality in QuantumMatter, JPMXP1020200104) from MEXT. A.C. thanks theNational Science and Technology Council (NSTC) of theRepublic of China, Taiwan for financially supporting this re-search under Contract No. NSTC 111-2112-M-213-031. A.F.acknowledges the support from the Yushan Fellow Programunder the Ministry of Education of Taiwan.APPENDIX: XAS AND RES-PES OF PLCCO, x = 0.0Figure 10 shows the O K-edge (1s-2p) XAS spectrum ofPLCCO, x = 0.0, measured at T = 200 K over the incidentphoton energy range of hν = 526–535 eV. It shows a smallpeak at ∼528.7 eV, a weak shoulder at ∼530.5 eV, and abroad structure at 532–535 eV. The high-energy states above532 eV are attributed to the La and Pr 5d states hybridizedwith O 2p states [71], while the 528–530 eV states are due toCu 3d–O 2p hybridized states. The peak at 528.7 eV is alsoquite similar to the lowest energy peak feature seen in the OK-edge XAS of electron-doped NCCO, which was analyzedas the unoccupied upper Hubbard band associated with Cu3d states hybridizing with O px, py states, while the shoulderat ∼530.5 eV is due pz states [71]. Comparing the x = 0.0and 0.1 spectra as shown in Fig. 10, the small peak associatedwith the upper Hubbard band at 528.7 eV shows relativelylower intensity in x = 0.1 compared to x = 0.0. This confirmsthe higher electron doping content in x = 0.1 with respect tox = 0.0.The O 1s-2p Res-PES spectra of PLCCO, x = 0.0 shownin Fig. 11(a) are quite similar to that of x = 0.1 shown inFig. 2(a). There are small differences, e.g., the small Ce3+peak at around 2.5 eV BE is missing in x = 0.0 and themainly Pr3+ occupied 4 f 2 states at 1.5 eV BE are sharper withslightly higher intensity. The Res-PES spectra also show thetwo-hole Auger satellite feature at ∼11 eV BE, which shiftsto higher BEs tracking the increase in hν [red dashed linein Fig. 11(a)]. The resonance behavior of the satellite wasIntensity (arb. units)534532530528526Photon energy (eV)PLCCO O K-edge XASFIG. 10. Comparison of the O K-edge (1s-2p) x-ray absorptionspectra of PLCCO, x = 0.0 and 0.1.FIG. 11. (a) The Res-PES spectra across the O K-edge (1s-2p)of PLCCO, x = 0.0, measured at photon energies marked with ver-tical bars in Fig. 1. The spectra are normalized at 8 eV BE. Theoff-resonance valence-band spectrum measured with hν = 55 eV forx = 0.1 is also shown for comparison. (b) The difference spec-tra for higher energies obtained with respect to the hν = 526.2 eVspectrum.confirmed by plotting the difference spectra with respect tohν = 526.2 eV, as shown in Fig. 11(b). The satellite startsgetting enhanced at hν = 526.2 eV, and its energy positionand spectral shape are very similar to the satellite observedfor x = 0.1, as shown in Fig. 3. For higher hν, the differ-ence spectra show an increase of the satellite intensity andshift to higher BE, coupled with a suppression of the mainvalence-band states until hν = 528.7 eV. This is followedby a suppression of the satellite coupled with a recovery ofthe main valence-band states at hν = 529.7 eV. The La 5pstates are observed in Fig. 11(a) as weak features between∼15 and 18 eV BE, while the Pr 5p states occur between∼20 and 23 eV and overlap with the O 2s states at ∼23 eV.The valence-band spectrum measured with hν = 55.0 eV forx = 0.1 is also shown in Fig. 11(a). It shows that the broad O2p states spread over 2.5–7.5 eV BE for x = 0.0 with higherhν are quite similar to the O 2p states for x = 0.1. It is notedthat although we did not measure the low-energy hν = 16.5or 55.0 eV valence-band spectra to estimate Up for x = 0.0,the BE shifts of the La 3d , Pr 3d , and O 1s core-level peakswere measured by x-ray photoemission spectroscopy [68].The results indicated a chemical potential shift of <0.3 eVfrom x = 0.0 to 0.1. Since the O 2p feature between 2.5and 7.5 eV BE for x = 0.1 matches closely with the O 2pfeature for the x = 0.0 spectra measured with higher hν, itindicates that for x = 0.0, the shift in the O 2p PDOS in thevalence band is also <0.3 eV. Accordingly, the change in Upfor x = 0.0 is considered to be within the error bar (±0.5 eV)of the Up estimated for x = 0.1.195152-8OXYGEN ON-SITE COULOMB ENERGY IN … PHYSICAL REVIEW B 107, 195152 (2023)[1] J. G. Bednorz and K. A. Mueller, Possible high Tc super-conductivity in the Ba-La-Cu-O system, Z. Phys. B 64, 189(1986).[2] J. Zaanen and O. Gunnarsson, Charged magnetic domainlines and the magnetism of high-Tc oxides, Phys. Rev. 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