# Fileset

[Doping_dependence_of_plasmons_in_LSCO.pdf](https://mdr.nims.go.jp/filesets/334b5811-f762-40ff-8e8e-57f497ca205e/download)

## Creator

M. Hepting, T. D. Boyko, V. Zimmermann, M. Bejas, Y. E. Suyolcu, P. Puphal, R. J. Green, L. Zinni, J. Kim, D. Casa, M. H. Upton, D. Wong, C. Schulz, M. Bartkowiak, K. Habicht, E. Pomjakushina, G. Cristiani, G. Logvenov, M. Minola, [H. Yamase](https://orcid.org/0000-0003-0328-5657), A. Greco, B. Keimer

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Evolution of plasmon excitations across the phase diagram of the cuprate superconductor <math>  <mrow>    <msub>      <mi>La</mi>      <mrow>        <mn>2</mn>        <mo>−</mo>        <mi>x</mi>      </mrow>    </msub>    <msub>      <mi>Sr</mi>      <mi>x</mi>    </msub>    <msub>      <mi>CuO</mi>      <mn>4</mn>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/0b444b5f-3027-4ad3-a825-9e7aa0af2fbf)

## Fulltext

Evolution of plasmon excitations across the phase diagram of the cupratesuperconductor La2−xSrxCuO4M. Hepting,1, ∗ T. D. Boyko,2 V. Zimmermann,1 M. Bejas,3 Y. E. Suyolcu,1 P. Puphal,1 R. J. Green,4, 5L. Zinni,6 J. Kim,7 D. Casa,7 M. H. Upton,7 D. Wong,8 C. Schulz,8 M. Bartkowiak,8 K. Habicht,8, 9E. Pomjakushina,10 G. Cristiani,1 G. Logvenov,1 M. Minola,1 H. Yamase,11 A. Greco,3, † and B. Keimer11Max-Planck-Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany2Canadian Light Source, Saskatoon, Saskatchewan S7N 2V3, Canada3Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura and Instituto de F́ısica de Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina4Department of Physics & Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada5Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada6Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura (UNR),Avenida Pellegrini 250, 2000 Rosario, Argentina7Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA8Helmholtz-Zentrum Berlin für Materialien und Energie,Hahn-Meitner-Platz 1, D-14109 Berlin, Germany9Institute of Physics and Astronomy, University of Potsdam,Karl-Liebknecht-Straße 24/25, D-14476 Potsdam, Germany10Laboratory for Multiscale Materials Experiments (LMX),Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland11International Center of Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba 305-0047, Japan(Dated: February 22, 2023)We use resonant inelastic x-ray scattering (RIXS) at the O K- and Cu K-edges to investigatethe doping- and temperature dependence of low-energy plasmon excitations in La2−xSrxCuO4. Weobserve a monotonic increase of the energy scale of the plasmons with increasing doping x in theunderdoped regime, whereas a saturation occurs above optimal doping x & 0.16 and persists at leastup to x = 0.4. Furthermore, we find that the plasmon excitations show only a marginal temperaturedependence, and possible effects due to the superconducting transition and the onset of strange metalbehavior are either absent or below the detection limit of our experiment. Taking into accountthe strongly correlated character of the cuprates, we show that layered t-J-V model calculationsaccurately capture the increase of the plasmon energy in the underdoped regime. However, thecomputed plasmon energy continues to increase even for doping levels above x & 0.16, which isdistinct from the experimentally observed saturation, and reaches a broad maximum around x =0.55. We discuss whether possible lattice disorder in overdoped samples, a renormalization of theelectronic correlation strength at high dopings, or an increasing relevance of non-planar Cu and Oorbitals could be responsible for the discrepancy between experiment and theory for doping levelsabove x = 0.16.I. INTRODUCTIONIn spite of intense experimental and theoretical re-search efforts, superconducting cuprates have retained anenigmatic character for more than three decades. On theone hand, it is well known that their high-temperature su-perconductivity emerges when charge carriers are dopedinto the CuO2 planes, suppressing the long-range antifer-romagnetic order that is prevalent in the Mott insulatingparent compounds [1–4]. For instance, in the prototypi-cal cuprate La2−xSrxCuO4, the doping of p ≈ 0.07 holesper CuO2 unit is sufficient to stabilize superconductivity[5], and the highest transition temperature Tc is realizedfor p ≈ 0.16. On the other hand, a consensus on the mi-croscopic mechanism mediating the superconductivity in∗ hepting@fkf.mpg.de† agreco@fceia.unr.edu.arcuprates is still lacking. This elusive nature of supercon-ductivity in cuprates is mostly rooted in the fact that—incontrast to conventional superconductors—cuprates arestrongly correlated systems and their Cooper pairs donot condense from a uniform and well-understood metal-lic state. Instead, the superconducting dome as a func-tion of charge carrier doping is embedded in a variety ofother enigmatic phases that also exhibit a strongly cor-related character, and either promote or compete withsuperconductivity in a nontrivial fashion. Most promi-nently, these phases comprise the pseudogap, spin andcharge order, as well as the strange metal regime [1].Insights into the electronic structure and the dynam-ics of doped charge carriers in these diverse phasescan be gained by various spectroscopic techniques.For instance, x-ray absorption spectroscopy (XAS) andelectron-energy loss spectroscopy (EELS) at the O K-edge or Cu L-edge have revealed how doped holes dis-tribute within the CuO2 planes, that is, they residemailto:hepting@fkf.mpg.demailto:agreco@fceia.unr.edu.ar2within the hybridized Cu 3dx2−y2 and O 2px,y orbitals,with a dominant influence of the latter orbitals [6–9].This behavior of the doped holes is commonly translatedinto a low-energy effective model where the quasiparti-cles are Zhang-Rice singlets (ZRS) [10, 11], which corre-spond to plaquettes of holes and oxygen ions around a Cuion, taking on the same role as fully occupied or emptysites in an effective single-band Hubbard model [12]. Thesingle-band approach is also employed in the t−J model[13–16], which additionally discards all doubly occupiedstates. Nevertheless, an accurate description of specificproperties of cuprates, such as pd charge-transfer relatedeffects, requires the explicit consideration of all three pla-nar orbitals (Cu 3dx2−y2 and O 2px,y) [17–26], and insome cases also non-planar orbitals [27–41].A different technique for investigating the charge dy-namics of cuprates, both in the normal and supercon-ducting state, is infrared/optical spectroscopy [42–44].Yet, in contrast to EELS, optical spectroscopy is a probethat is essentially limited to the center of the Brillouinzone (BZ), due to the small momentum of optical pho-tons.Recently, resonant inelastic x-ray scattering (RIXS)[45] has emerged as a versatile tool to study electronicand magnetic excitations in cuprates, providing both mo-mentum resolution and a relatively high energy resolu-tion. Along these lines, soft x-ray RIXS at the Cu L3and O K-edge was employed to investigate collective spinexcitations [46–49], and hard x-ray RIXS at the Cu K-edge was used to probe electronic inter- and intrabandtransitions [50–55]. Moreover, collective charge excita-tions were detected in Cu L3 and O K-edge RIXS exper-iments, which were attributed to acoustic plasmons [56–59]. More recently, a high-resolution Cu L3-edge RIXSstudy revealed that these low-energy plasmon branchesare not strictly acoustic, but they exhibit an energy gapat the two-dimensional BZ center [60].Plasmons are a fundamental collective excitation ofthe charge carrier density in metallic systems, medi-ated by long-range Coulomb interactions. An isotropicthree-dimensional metal usually exhibits only one opti-cal plasmon branch, while systems with stacked conduct-ing planes that interact via poorly-screened interlayerCoulomb interactions exhibit a set of acoustic plasmonbranches that disperse almost linearly towards zero en-ergy at the BZ center [61–65]. However, in the presenceof single-electron tunneling between the planes, the latterbranches are not strictly acoustic, but exhibit a gap at theBZ center, which is proportional to the interlayer hoppingintegral tz [60, 66]. In doped cuprates, which are com-posed of stacked conducting CuO2 planes separated bydielectric spacer layers, early transmission EELS and op-tical spectroscopy experiments already detected the opti-cal plasmon, which manifests itself as a plasma edge in re-flectance spectra and a peak in the loss function around 1eV [67, 68]. Yet, direct evidence of the acoustic-like plas-mon branches was only provided later by high-resolutionRIXS with polarization analysis [56] and the capacity toindependently vary the in- and out-of-plane momentumtransfer to probe the distinct plasmon dispersion alongdifferent directions in the BZ [56, 58]. The plasmon dis-persion is accurately captured by a large-N theory of thelayered t-J-V model [58, 60, 66, 69, 70], which intrin-sically accounts for the strongly correlated character ofcuprates. Other methods to describe the plasmon disper-sion in cuprates include the random phase approximation(RPA) [71, 72], a combination of determinant quantumMonte Carlo (DQMC) and RPA in a layered Hubbardmodel [56], and an extended variational wave functionapproach [73].In a broader context, low-energy plasmons in cupratesattracted considerable attention already before their de-tection with RIXS [67, 68], as they were suggested toplay a role in the superconducting pairing [72], and theirevolution across the superconducting transition could re-flect possible savings of kinetic energy of the charge car-riers [74]. Moreover, low-energy plasmon-phonon modes[75, 76] were proposed to mediate superconductivity, orcontribute constructively to the high Tc of cuprates [77].In addition, the evolution of plasmon excitations acrossthe phase diagram of cuprates might encode critical infor-mation about the charge carrier dynamics in the normalstate. Specifically, a debate is ongoing about the natureof the charge carriers in the strange metal state [78, 79],which is characterized by a linear-in-temperature resis-tivity [80]. Several theories suggest that the standardLandau quasiparticle description for the charge carriersbreaks down in this regime [81–83], inhibiting the emer-gence of collective charge excitations. As a consequence,strange metals could hamper the propagation of plas-mons, while they rapidly decay into a quantum criticalcontinuum [84]. Indeed, the lack of well-defined plasmonpeaks in EELS data acquired in a reflection geometrywas interpreted as a sign of such an anomalous plasmondamping in overdoped cuprates, [85, 86], which is stilldiscussed controversially [87, 88].In this work, we use soft x-ray RIXS at the O K-edge and hard x-ray RIXS at the Cu K-edge to com-prehensively map the evolution of low-energy plasmonexcitations across the phase diagram of La2−xSrxCuO4.Whereas the doping dependence of plasmon excitationsin electron-doped cuprates was addressed in earlier stud-ies [56, 57], the evolution of the plasmon as a function ofhole-doping has not been systematically investigated byRIXS, to the best of our knowledge. We observe plas-mon excitations for several Sr substitution levels fromx = 0.05 to 0.4 and at various temperatures, includingthe strange metal regime around x = 0.2, where previ-ous works proposed the decay of the plasmon into anenergy- and momentum-independent continuum [84–86].We model the doping dependence of the plasmon energyin the framework of layered t-J-V model calculations,which capture the experimentally observed trends butshow a deviation for dopings above x & 0.16 (optimaldoping). We discuss the presence of disorder, a renor-malization of the electronic correlation strength in the30 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00120 . 0 0 . 1 0 . 2 0 . 3 0 . 402 04 0     x  =0 . 0 5  0 . 1 0  0 . 1 60 . 2 00 . 3 0  0 . 4 0  � ab (mΩ cm)T e m p e r a t u r e  ( K )( x  0 . 2 5 )T c (K)S r  c o n t e n t  xFIG. 1. In-plane resistivity ρab of La2−xSrxCuO4 films withvarious hole-doping levels x. The x = 0.05 curve is scaled bya factor 0.25. The gray symbols in the inset correspond tothe superconducting transition temperatures Tc of the films.The orange symbol indicates the Tc of the La1.8Sr0.2CuO4single-crystal.overdoped regime, and an increasing relevance of non-planar Cu and O orbitals as possible origins for this dis-crepancy.II. RESULTSA. Electronic transportThin films of La2−xSrxCuO4 were synthesized byozone-assisted molecular beam epitaxy (MBE) (see Ap-pendix A for details). Our film with the lowest doping(x = 0.05) is non-superconducting and becomes insulat-ing at low temperatures [Fig. 1], whereas the subsequentdoping level x = 0.1 shows a clear superconducting tran-sition. This is consistent with previous La2−xSrxCuO4thin film studies [89–91] and bulk measurements, wherethe onset of the superconducting dome is situated aroundx ∼ 0.07 [5]. Note that the long-range antiferromag-netic (AFM) order, which is prevalent in the parent com-pound La2CuO4, vanishes already around x ∼ 0.016 [92].For higher dopings, spin and charge stripe ordered statesemerge and are most pronounced around x ∼ 0.125. Inthe bulk, a low-temperature-tetragonal (LTT) and low-temperature less orthorhombic (LTLO) structural phasetransition also occur in this doping regime [93]. In thinepitaxial La2−xSrxCuO4 thin films, however, these struc-tural transitions are suppressed [90].As expected, we detect the highest superconductingtransition of our film series for the optimal doping x= 0.16, that is, Tc = 41 K. The Tc decreases slightlyin the subsequent film with x = 0.2, where the electri-cal resistivity varies almost perfectly linear as a func-tion of temperature in the normal state at least up to300 K [Fig. 1]. Such T -linear behavior is commonly re-garded as a key manifestation of the strange metal phe-nomenology [79, 81, 82]. Previous transport studies onLa2−xSrxCuO4 found that the T -linear behavior is re-stored at low temperatures when the superconductingtransition is suppressed in strong magnetic fields, andthat the strange metal phase emanates in a fan-like shapefrom a putative quantum critical point at p∗ ∼ 0.19 [94].In addition to a series of thin films, we have synthesizeda single-crystal with a doping concentration close to x =0.2. The temperature dependence of the transport andthe Tc of the single-crystal are closely similar to the corre-sponding film, and will be discussed in more detail belowin the context of the Cu K-edge RIXS measurements.For the overdoped x = 0.3 film, we observe a decreaseof Tc to 26 K. The highly overdoped x = 0.4 film ex-hibits an onset of a superconducting transition around 5K. This behavior in the latter two samples differs fromreports on bulk La2−xSrxCuO4, where the superconduct-ing dome typically terminates around x ∼ 0.25 [92], andfrom prior thin film studies that reported only a slightlyextended dome [89]. In the case of our films, we attributethe substantially extended superconducting dome to thehighly oxidizing growth atmosphere [95] and the stabiliz-ing effect of the epitaxial strain of the substrate, whichcan possibly enhance the solubility limit of Sr [96] and al-leviate the formation of oxygen vacancies [97, 98]. A sim-ilar trend was recently observed in ozone-assisted MBEgrown La2−xCaxCuO4 films, where the superconductingtransition persisted for Ca-substitutions as high as x =0.5 [99].B. X-ray absorption at the O K-edgeThe XAS signal across the near-edge fine structureof the O K-edge contains hallmark fingerprints of theelectronic structure of the cuprates [7, 8, 11, 21, 31,33]. Figures 2a-f display the O K-edge XAS of theLa2−xSrxCuO4 films with dopings between x = 0.05 and0.4. The peak near 529.5 eV is particularly pronouncedfor low doping levels and corresponds to the upper Hub-bard band (UHB), which arises as the on-site part ofthe Coulomb repulsion U splits the Cu 3d band into afully occupied lower Hubbard band (LHB) and the emptyUHB. The Cu 3d band-derived features are visible in theO K-edge XAS due to the strong 2p-3d hybridizationbetween O and Cu. The peak in the XAS at lower ener-gies (∼ 528 eV) typically emerges in hole-doped cuprates[100], and is associated with the ZRS states [11]. With in-creasing hole density, spectral weight is transferred fromthe UHB peak to the hole-peak. Notably, we observe thatwith increasing Sr-substitution the intensity of the hole-peak increases continuously and the peak energy shiftsto lower energies, even up to x = 0.4. This trend is con-sistent with the notion of a continuous creation of newdoped hole states and that the ZRS stays intact up tohighest doping levels [11, 21].40.040.080.12H =a b c d e fhole-peakUHBg h i j k lx=0.05T=300 Kσ-pol.x=0.1 x=0.16x=0.16x=0.2x=0.2x=0.3 x=0.4x=0.4x=0.1x=0.05T=300 Kσ-pol.x=0.3XAS intensity (arb. units)530528526Photon Energy (eV)530528526Photon Energy (eV)530528526Photon Energy (eV)530528526Photon Energy (eV)530528526Photon Energy (eV)530528526Photon Energy (eV)RIXS intensity (arb. units)-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)plasmonbimagnonfit-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)-1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)FIG. 2. (a)-(f) O K-edge XAS of La2−xSrxCuO4 films with various hole-doping levels x. The spectra were collected at T=300 K with σ-polarized photons at an incident angle θ = 35◦. The peak associated with the doped holes and the upperHubbard band (UHB) peak are indicated. (g)-(l) O K-edge RIXS of the La2−xSrxCuO4 films, for various momenta along theH direction, while K = 0 and L∗ ≈ 0.37. Orange symbols correspond to H = 0.04, light red symbols to H = 0.08, and redsymbols to H = 0.12. The spectra were taken with σ-polarized photons at energies tuned to the hole-peak in the correspondingXAS. A high-energy background was subtracted from the spectra (see Appendix B). The solid gray lines are fits, which includethe dispersive plasmon (blue shaded peak) and the non-dispersive bimagnon (dashed black peak). The other contributions tothe fit are omitted for clarity (for details of the fitting procedure see Appendix B). Curves for different momenta are offset inthe vertical direction for clarity.C. RIXS at the O K-edgeIn order to gain deeper insights into the dynamicsof the doped holes and the ZRS state for each dopinglevel, we measured RIXS with the incident photon en-ergy tuned to the maximum of the hole-peak in the cor-responding XAS from Figs. 2a-f. In the following, wedenote the momentum transfer by (H,K,L) in recipro-cal lattice units (2π/a, 2π/b, 2π/c), where a, b, and c arethe lattice constants of La2−xSrxCuO4. However, thecrystallographic unit cell of La2−xSrxCuO4 contains twoCuO2 planes, whereas only the distance d = c/2 betweenadjacent CuO2 planes is relevant for the periodicity ofthe plasmon dispersion [56]. Hence, we utilize the indexL∗ = L/2 instead.The obtained RIXS spectra for in-plane momenta H= 0.04, 0.08, and 0.12 are displayed in Figs. 2g-l. Theout-of-plane momentum transfer of the spectra is on av-erage L∗ ∼ 0.37 (see Appendix A for details). A broadbackground contribution in the RIXS signal from fluores-cence and dd-excitations [48] is already subtracted fromthe spectra in Figs. 2g-l. The raw spectra without back-ground subtraction and details about the fitting proce-dure can be found in Appendix B.In Figs. 2g-l, the spectra are decomposed into severalcomponents, including the elastic line centered at zero-energy loss, a non-dispersive feature around 510 meV,and a dispersive peak that evolves between 200 and 500meV. Along the lines of previous O K-edge RIXS stud-ies, we assign the ∼510 meV peak (dashed black linein Figs. 2g-l) to a non-dispersive bimagnon excitation[48, 49, 58, 59]. Note that long-range AFM order is ab-sent for the investigated doping levels, hence, this exci-tation corresponds strictly speaking to a bi-paramagnon[46, 47] and not a bimagnon. The essentially unchangedenergy scale of our observed non-dispersive feature is con-50 . 0 0 . 1 0 . 2 0 . 3 0 . 40 . 00 . 20 . 40 . 60 . 81 . 01 . 2 H  =  0 . 0 4  H  =  0 . 0 8H  =  0 . 1 2z o n e  c e n t e r  [ 4 4 ]Plasmon energy (eV)D o p i n g  xH = 0 . 1 2z o n e  c e n t e rH = 0 . 0 8H = 0 . 0 4FIG. 3. Doping-dependence of the plasmon energy. The cir-cles are the plasmon energies extracted from fits of the OK-edge RIXS spectra of the La2−xSrxCuO4 films in Figs. 2g-l. The energies are grouped according to the different in-planemomentum transfer H (light to dark red). The gray trianglescorrespond the plasmon energy at the three-dimensional Bril-louin zone center H = K = L = 0 (optical plasmon), reportedfor La2−xSrxCuO4 single-crystals in Ref. 44. The dashed linesare computed plasmon energies obtained from layered t-J-Vmodel calculations. The labels next to the dashed arrows indi-cate the momenta for which the calculations were performed.sistent with the evolution of paramagnon excitations inCu L-edge RIXS experiments on La2−xSrxCuO4 filmswith doping levels between x=0.1 and 0.4 [47]. In thefollowing, we will focus on the dispersive peak (solid blueline in Figs. 2g-l), which we attribute to a low-energyplasmon excitation. Notably, the bimagnon and the plas-mon peak overlap strongly for momenta between H =0.04 and 0.12 [Figs. 2g-l], which precluded the identifica-tion of two distinct peaks in early RIXS experiments onLa2−xSrxCuO4 [49, 101]. Nevertheless, recent O K-edgeRIXS studies with improved energy resolution [58, 59]clearly discerned the two excitations and determined theplasmon energy as a function of momentum transfer H.Similarly, we extract the energy of the dispersive plas-mon peak from our fits, while keeping the energy ofthe bimagnon peak fixed. For La1.84Sr0.16CuO4, we ob-tain closely similar plasmon energies [Fig. 2i and Fig. 3]as reported in Ref. 58 for the same doping (see Ap-pendix C). Furthermore, our extracted plasmon energiesfor La1.9Sr0.1CuO4 [Fig. 2h and Fig. 3] are not far fromthose reported for La1.88Sr0.12CuO4 in Ref. 59 (see alsoAppendix C). Yet, to the best of our knowledge, the plas-mon energy for other doping levels of La2−xSrxCuO4 hasnot yet been reported. Thus, we closely inspect Figs. 2g-lin the following to extract a possible systematic trend inthe doping-dependence of the plasmon excitation.Figure 2g shows the RIXS spectra for the lightly dopedcase x = 0.05, where our fits reveal that the broad shoul-der next to the elastic line contains the bimagnon andthe plasmon excitation, similar to the x = 0.1 and 0.16cases [Figs. 2h,i]. However, the energy scale and the dis-persion of the plasmon for x = 0.05 are substantiallysmaller than for x = 0.1 and 0.16. This difference is alsovisible in Fig. 3, which plots the doping dependence ofthe plasmon energy for each momentum H, suggestinga monotonic increase of the energy as a function of x.An increase of the plasmon energy with increasing dop-ing concentration was also observed in RIXS experimentson the electron-doped cuprate La2−xCexCuO4 [56, 57].Note that this trend does not depend on the details ofour fitting procedure, such as the exact treatment of thebimagnon peak. For instance, a fit that captures boththe plasmon and the bimagnon feature by a single broadpeak function still yields an increase of the peak energywith increasing doping concentration (see Appendix B).For dopings beyond x = 0.16, we find that the plasmonenergy does not continue to grow, but instead tends tosaturate for x = 0.2, 0.3, and 0.4 [Figs. 2j-l and Fig. 3].The onset of a flattening of the plasmon energy increasein the overdoped regime was already noticed for electron-doped La2−xCexCuO4 and attributed to a variation inthe band dispersion and Fermi surface reconstruction[56], or strong correlations [57]. However, the investi-gated range for La2−xCexCuO4 was between x = 0.11and 0.18, whereas our present study on La2−xSrxCuO4extends over a much wider range from x = 0.05 to 0.4,which can allow for more robust conclusions.Besides the systematic evolution of the plasmon en-ergy [Fig. 3], we find that the integrated intensity of theplasmon peak changes only marginally, especially in theoverdoped regime [Figs. 2j-l]. On the other hand, a smalldecrease in the linewidth of the plasmon for high dopingsmight be discernible in Figs. 2j-l. Yet, because of thestrong overlap with the bimagnon, the statistical error ofthe integrated intensities and linewidths extracted fromthe fits are large and we refrain from further discussionsof these two quantities.Nonetheless, the RIXS data in Figs. 2g-l clearly demon-strate that plasmon excitations pervade the cupratephase diagram at least from x = 0.05 to 0.4 and for mo-menta as large as H = 0.12. In particular, the unaffectedemergence of plasmon quasiparticles in the overdopedregime raises the question, whether such an observationis compatible with previous proposals based on reflectionEELS experiments [85, 86, 88] that suggested the van-ishing of coherent plasmon excitations specifically in thestrange metal regime of cuprates. Hence, in the followingwe focus on the doping level x = 0.2, which is situated6XAS int. (arb. units)9.019.008.998.98Photon energy (keV)1.00.50.0Plasmon energy (eV)6.86.66.46.21.00.50.0Plasmon energy (eV)0.150.100.050.00ac dH=K=L*=6.50.150.130.110.070.090.030.05L*=L*=6.56.8AResistance (mΩ)B Cπ-pol.σ-pol.plasmon.orbital exc.fitT=17 K, π-pol.x=0.2 (crystal)6.756.76.66.46.5H=0.09 K=0bfe H=0.09 K=0H,H (2π/a) L* (2π/d)-2.0 -1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)RIXS intensity (arb. units)-2.0 -1.5 -1.0 -0.5 0.0 0.5Energy loss (eV)6420250200150100500Temperature (K)FIG. 4. Cu K-edge RIXS of the La1.8Sr0.2CuO4 single-crystal. (a) Electrical transport of the crystal. (b) XAS acrossthe Cu K-edge measured with π- (solid line) and σ-polarized(dashed line) incident photons. (c) Cu K-edge RIXS spec-tra for various in-plane momenta along the diagonal direc-tion (H = K), while the out-of-plane momentum is fixed toL∗ = 6.5. The fit (gray line) to the experimental data (filledsymbols) includes the plasmon peak (blue shaded peak pro-file), an orbital excitation (dashed black line), and other con-tributions (not shown here) that are described in Appendix B.Curves for different momenta are offset in the vertical direc-tion for clarity. (d) RIXS spectra for momenta along the L∗direction, while H = 0.09 and K = 0 are fixed. (e,f) Plasmonenergy (red squares) extracted from the fits (e) along the diag-onal direction and (f) along the L∗ direction. The solid blacklines are the corresponding plasmon dispersions computed inthe t-J-V model.within the strange metal region in the La2−xSrxCuO4phase diagram [90, 94].D. RIXS at the Cu K-edgeTo corroborate the existence of plasmon excitations inthe strange metal regime and to obtain complementaryinformation, we use RIXS at a different absorption edge,that is, the Cu K-edge. An advantage of the high energyof the hard x-ray photons tuned to the Cu K-edge is thewide range of reciprocal space that becomes accessible forRIXS. However, due to the increased penetration depthof the photons, we carry out the RIXS measurementson a bulk single-crystal instead of the La1.8Sr0.2CuO4thin film that was used for the O K-edge experimentabove. Our single-crystal exhibits a Tc of approximately38 K and a similar T -linear transport behavior [Fig. 4a] asthe La1.8Sr0.2CuO4 thin film [Fig. 1], indicating that thesample is excellently suitable for complementary RIXSmeasurements.Figure 4b shows the XAS spectra of theLa1.8Sr0.2CuO4 single-crystal across the Cu K-edgemeasured with π- and σ-polarized incident photons,respectively. The main absorption line for the formerpolarization is centered at 8993.7 eV (peak A), whilethe latter polarization yields pronounced peaks around8998.1 eV (peak B) and 9004.9 eV (peak C). The shapesof our XAS spectra are in close agreement with those ofoverdoped La1.7Sr0.3CuO4 in Ref. 55. A relative shiftof the absolute energy scale of approximately 1 eV inRef. 55 is likely due to a different energy calibration.In order to identify the resonance energy where theemergence of the plasmon excitation is expected in RIXS,we briefly recall previous assignments of the Cu K-edgeXAS features of cuprates. Note that low-energy plas-mons were not observed in previous Cu K-edge RIXSexperiments on hole-doped cuprates, whereas recent CuK-edge RIXS studies on electron-doped cuprates re-vealed a low-energy excitation that is compatible withthe typical plasmon dispersion [102]. Yet, the XASof electron-doped cuprates includes several distinctionsfrom the hole-doped counterparts [51]. In hole-dopedLa2−xSrxCuO4, the XAS peak A was attributed to well-screened intermediate states, and previous RIXS stud-ies detected charge-transfer (CT) excitations around 4 eVenergy loss in resonance to this energy [50, 53, 55]. Forresonant energies in the region of peak B and C, which areassociated with poorly screened states, molecular-orbitalexcitations around 7.5 eV energy loss were observed withRIXS [50, 53, 55]. However, both types of high-energy ex-citations are clearly not associated with low-energy plas-mon modes. Instead, the plasmon excitation must be lo-cated in the CT gap, where indeed spectral weight below3 eV energy loss was reported, which showed a depen-dence on the in-plane momentum transfer [53–55]. Thisbroad spectral weight, whose intensity resonates at theenergy of peak A, was initially attributed to an inter-band excitation from a low-lying band below the Fermienergy to the ZRS band [54]. In contrast, an analysis ofthe spectral intensity specifically around 1 eV in Ref. 55suggested an intraband excitation as the origin, although7this assignment is incompatible with theoretical calcula-tions [103]. Nonetheless, Ref. 55 pointed out that for in-cident energies around peak A, the RIXS intensity insidethe CT gap contains an appreciable contribution fromanother type of excitation, possibly unresolved due to aninsufficient energy resolution of ∆E ∼ 350 meV in theRIXS experiment [55].Hence, we focus on incident energies in the regionaround peak A in our RIXS experiment, and use a muchimproved energy resolution of ∆E ∼ 80 meV. To reducethe strong elastic scattering at the Cu K-edge, we em-ploy π-polarized photons and keep the scattering angleclose to 90◦ for all measurements. This condition is real-ized for out-of-plane momenta revolving around L∗ = 6.5and small in-plane momenta. Further details about thescattering geometry are given in Appendix A.Figure 4c shows our CuK-edge RIXS spectra for differ-ent in-plane momenta along the diagonal direction withH = K, while the out-of-plane momentum is fixed toL∗ = 6.5. The RIXS spectra were taken with an inci-dent photon energy of 8993 eV, which is slightly belowthe maximum of the XAS peak A [Fig. 4b]. Below 2 eVenergy loss in the spectra, we resolve two prominent in-elastic features (for details about the fitting procedure,see Appendix B). The feature centered around 1.5 eV en-ergy loss (dashed black line in Fig. 4c) shows only littleor no dispersion as a function of momentum transfer,whereas the feature at lower energies (solid blue line inFig. 4c) disperses strongly for varied momentum transfer.Both features resonate within an incident energy intervalof more than 1 eV above and below the maximum of XASpeak A. Upon increasing the incident energy, we find thatthe inelastic feature that appears at 1.5 eV for an inci-dent energy of 8993 eV moves to higher energy losses.In contrast, we observed that the low-energy feature ex-hibits a Raman-like character and remains essentially atthe same energy loss for different incident energies (notshown here).Furthermore, the latter feature shows a distinct out-of-plane dispersion [Fig. 4d], while the 1.5 eV feature re-mains at a constant energy loss within the experimentalerror. These behaviors indicate that the 1.5 eV featurelikely corresponds to an inter- or intraorbital excitation,as proposed for the features detected in Refs. 53–55. Incontrast, the behaviors of our dispersive feature at lowerenergies are compatible with a plasmon excitation, inanalogy to the characteristic in- and out-of-plane disper-sion of the plasmon mode recently identified in an O K-edge RIXS study on optimally doped La1.84Sr0.16CuO4[58], as well as that of electron-doped La2−xCexCuO4in an Cu L-edge RIXS study [56]. Moreover, the energyrange of the dispersion of the low-energy feature in the CuK-edge RIXS spectra is comparable to the range of theplasmon energy of the x = 0.2 film in the O K-edge RIXSspectra in Fig. 2j and Fig. 3. However, we note that theO K-edge RIXS spectra are acquired along the axial Hdirection for L∗ = 0.37, whereas the Cu K-edge spectrain Fig. 4c are along the diagonal direction (H = K) forL∗ = 6.5, which prevents a direct comparison. We ruleout an assignment of the low-energy mode to a bimagnonexcitation, as magnetic excitations in layered cupratesexhibit a two-dimensional character [56], which would beat odds with the out-of-plane momentum dependence ofour low-energy mode in Fig. 4d. Note that previous CuK-edge RIXS studies detected a bimagnon excitation in(underdoped) La2−xSrxCuO4, with a maximum in theintensity at the in-plane momentum (π, 0) and vanishingintensity at (0, 0) [104]. On the other hand, it was sug-gested that the bimagnon intensity in O K-edge RIXS ismaximal around (0, 0), as the two types of RIXS mea-surements are sensitive to different parts of the bimagnoncontinuum [48]. The observation of a bi-(para)magnon inour O K-edge RIXS data [Figs. 2g-l] and its absence (orvery low intensity) in our Cu K-edge spectra [Figs. 4c,d]is therefore consistent with this notion.For a definitive assignment of the dispersive modein Figs. 4c,d, we turn to layered t-J-V model calcula-tions (for details see Appendix D). Specifically, we de-termine the plasmon energy as the maximum of thecomputed imaginary part of the charge susceptibilityχ′′(q, ω) [58, 60, 66, 69, 70] to model the dispersion ob-served in Fig. 4c along the (H,H) direction for fixedL∗ = 6.5, as well as the dispersion in Fig. 4d alongthe out-of-plane direction for fixed H = 0.09 and K= 0. For an unbiased comparison between the experi-mental and modeled dispersion [Figs. 4e,f], we computeχ′′(q, ω) of La1.8Sr0.2CuO4 using the same parametersas in Ref. 60 for La1.84Sr0.16CuO4, and only adjust thedifferent hole-doping level x = 0.2 in the calculation. Re-markably, Figs. 4e,f reveal that both the in- and out-of-plane dispersion extracted from our experimental data es-sentially match with the computed dispersion, althoughthe model appears to overestimate the plasmon energy.This small but systematic discrepancy will be discussedin more detail below. Nonetheless, the semiquantitativeagreement between the experimental and the computeddispersion corroborates our identification of the plasmonmode in the Cu K-edge RIXS spectra, which in turnsubstantiates the emergence of plasmon quasiparticles inLa2−xSrxCuO4 in the T -linear resistivity regime.E. Temperature dependence of the plasmonexcitationIn addition to the dependence of the plasmon exci-tation on the doping level x at a fixed temperature[Fig. 2], we investigate the temperature dependence ofthe plasmon for specific doping levels in the following.Figure 5a displays the Cu K-edge RIXS spectra of theLa1.8Sr0.2CuO4 crystal at T = 17, 35, and 100 K. Withinthe experimental error, the plasmon peak remains un-changed between 17 and 35 K, corresponding to mea-surements below and above the superconducting transi-tion (Tc ∼ 32 K), respectively. Even an increase of thetemperature to 100 K affects the plasmon excitation only81 . 0 0 . 5 0 . 0 - 0 . 5 1 . 0 0 . 5 0 . 0 - 0 . 5 1 . 0 0 . 5 0 . 0 - 0 . 51 7  K3 5  K1 0 0  KRIXS intensity (arb. units)E n e r g y  l o s s  ( e V )π- p o l .C u  K - e d g e a1 5  K1 0 0  KE n e r g y  l o s s  ( e V )π- p o l .x  =  0 . 2O  K - e d g e bx  =  0 . 1 61 5  K1 0 0  KE n e r g y  l o s s  ( e V )π- p o l .O  K - e d g e cx  =  0 . 3FIG. 5. Temperature dependence of the plasmon excitation.(a) Cu K-edge RIXS spectra of the La1.8Sr0.2CuO4 crystal atT = 17, 35, and 100 K, respectively. The momentum transferis H = K = 0.13 and L∗ = 6.5. (b,c) O K-edge RIXS spectraof the (b) La1.84Sr0.16CuO4 and (c) La1.7Sr0.3CuO4 film at T= 15 and 100 K, respectively. The momentum transfer is H= 0.1, K = 0, and L∗ = 0.5.marginally, that is, a subtle increase in the linewidth ofthe peak, likely due to thermal broadening of the collec-tive excitation. To confirm the minor temperature depen-dence of the plasmon, we also perform O K-edge RIXSmeasurements on La2−xSrxCuO4 films with x = 0.16 and0.3 [Figs. 5b,c], which exhibit a Tc of 41 and 26 K, respec-tively. Similarly to Fig. 5a, the temperature dependenceof the plasmon peak in the latter two samples is smallbetween 15 and 100 K and almost indiscernible from thenoise in the data, although a general trend of a linewidthbroadening at higher temperatures appears evident.The observed absence of strong renormalization effectsof the plasmon energy and linewidth across Tc is consis-tent with detailed ellipsometry experiments of the opticalplasmon branch in Bi-based cuprates in Ref. 74. Theseexperiments indicated that the temperature evolution ofthe optical plasmon branch with an energy of approx-imately 1 eV exhibits at most subtle anomalies at thesuperconducting transition. The detection of such subtleanomalies in the acoustic-like branches in Fig. 5 wouldrequire RIXS measurements with much higher statisticsand denser temperature sampling intervals around Tc.Moreover, to detect significant renormalization effectsof the acoustic-like plasmons across Tc, it may be nec-essary to conduct the RIXS measurements in immediatevicinity to the in-plane BZ center (H = K ≈ 0). Specif-ically, at the in-plane BZ center the energy of acoustic-like plasmons becomes minimal, and can be comparableto the energy scale of the superconducting gap 2∆, giventhat the interlayer-hopping tz of the material [60] is rel-atively small. Such proximity or a crossover of the twoenergy scales could affect the plasmon properties sub-stantially. While measurements in the present study wereperformed at momenta as small as H = 0.04, the intenseelastic line in the RIXS spectra progressively obscures theplasmon peak when approaching the in-plane BZ center.Thus, our present RIXS measurements do not providesufficient information to ascertain or exclude a relation-ship between plasmons and superconductivity. Neverthe-less, future RIXS investigations leveraging an improvedenergy resolution could yield new insights into this sub-ject.F. Theoretical modeling of the doping dependenceAfter our confirmation of the emergence of the plasmonexcitation for x= 0.2 by CuK-edge RIXS [Fig. 4] and ourfinding of a marginal temperature dependence [Fig. 5],we return to the O K-edge RIXS data from Fig. 3 for adetailed analysis of the doping-dependence in the frame-work of the t-J-V model [58, 60, 66, 69, 70]. Specifically,Fig. 3 compares the experimentally extracted plasmonenergies (symbols) with the computed energies (dashedlines), which correspond to the maximum of the imagi-nary part of the charge susceptibility χ′′(q, ω) calculatedas a function of the doping x. The data are groupedaccording to the momenta H = 0.04, 0.08, and 0.12, re-spectively, while K = 0 and L∗ = 0.37. Similarly to ourcalculation in Figs. 4e,f, we employ the model parame-ters of La1.84Sr0.16CuO4 derived in Ref. 60 (for detailssee Appendix D), and only adjust the doping level. Forcompleteness, we also present the computed optical plas-mon branch, and superimpose the optical plasmon en-ergy of La2−xSrxCuO4 crystals (gray triangles) reportedin optical reflectivity experiments [44]. Hence, these datapoints correspond to the plasmon energy at the three-dimensional BZ center (H = K = L = 0), and werederived from the position of the peak in the loss function[44].Notably, the computed plasmon energies in Fig. 3 agreewell with the experimentally observed energies for lowdopings until approximately x ≈ 0.16. This indicatesthat t-J-V model calculations not only capture the be-havior of plasmons in cuprates at a fixed doping level[58, 60, 66, 69, 70], but also describe their doping de-pendence without additional tuning of the parameters,at least for small H and small dopings. The observedincrease of the plasmon energy with increasing chargecarrier concentration x is intuitively expected for a sys-tem with some correspondence to a simple free electronmodel, where the plasmon energy scales with the chargecarrier density. Yet, according to the basic plasmon the-ory [105], the plasmon energy should decrease with in-creasing hole doping, as it is proportional to√n/m∗,where n is the electron density and m∗ the effectiveelectron mass. Moreover, in RPA calculations (see Ap-pendix E for details) the plasmon energy follows the bandfilling effect, i.e., the plasmon energy decreases with in-creasing hole-doping [Fig. 6]. Specifically, the plasmonenergy computed in RPA is maximal for x = 0, while itdecreases with increasing x, and eventually vanishes forx = 1, where the band becomes completely empty. Ob-viously, our observed increase of the plasmon energy inhole-doped La2−xSrxCuO4 until x ≈ 0.16 is incompat-9ible with the notion of the basic plasmon theory [105]and the RPA results. A different theoretical calcula-tion for La2−xSrxCuO4 [76], which includes Hartree andexchange-correlation contributions, considered a mixedplasmon-phonon mode at very small in-plane momenta(H . 0.005). Whereas the plasmon-like branch of themixed mode is essentially doping-independent betweenx = 0.156 and 0.297, the phonon-like branch increases inenergy. However, the present RIXS data were acquired atmuch larger momenta which implies that such a couplingto phonons is not relevant for our plasmon dispersion inFig. 3. In addition, Ref. 76 considered the case of an un-coupled plasmon, which shows an increase in energy fordopings between x = 0.156 and x = 0.297 for momentaH . 0.005, whereas it is almost doping-independent forslightly larger momenta. For a direct comparison to thepresent RIXS data, it will be interesting to carry outfuture calculations in analogy to Ref. 76, but at largermomenta which coincide with the RIXS experiment andalso for dopings below x = 0.156.In the present study, we employ the t-J-V model toexplore the entire doping range of La2−xSrxCuO4 at mo-menta that directly correspond to the RIXS experiment.The t-J-V model takes into account strong correlationswhich are known to be responsible for the insulating stateat half-filling (x = 0) [4]. Accordingly, Figs. 3 and 6 il-lustrate that the plasmon energy computed in the t-J-Vmodel tends towards zero for x = 0, which is consistentwith the notion that insulators cannot exhibit plasmonexcitations composed of conduction electrons (or holes).Note, however, that the plasmon energy at x = 0 is ex-pected to be non-zero also in the present calculations,because a finite exchange interaction J induces a finitebandwidth even at x = 0, if antiferromagnetic order isnot considered [66]. Nonetheless, the present theory isapplicable at least down to x = 0.05 [Figs. 3 and 6],which is the lowest hole-doping for which plasmons weredetected in the experiment [Fig. 2].Interestingly, above optimal doping and especially inthe highly overdoped regime, the plasmon energies de-termined by RIXS tend to saturate around a doping thatis lower than the prediction of the t-J-V calculation, ex-cept for the momentum H = 0.04 [Fig. 3]. The opticalplasmon energy determined from optical reflectivity ex-periments [44] even decreases between x = 0.2 and 0.34,although we note that the decrease instead of a satu-ration might be due to possible sample quality issues forthe highest doping in Ref. 44. In general, the calculationssuggest that the saturation trend is due to a broad peakstructure of the plasmon energy as a function of doping[Fig. 6], which emerges naturally when considering thatthe plasmon energy tends towards zero for x = 0 and 1.In particular, in the limit x = 1, both the t-J-V modelcalculations and the RPA results exhibit a strong de-crease of the plasmon energy. This suggests that the t-J-V model recovers the band filling effect when approachingthe empty-band limit at x = 1. However, Fig. 3 indicatesthat the t-J-V model calculation predicts a saturation ofzone centerH=0.04H=0.08H=0.12Plasmon energy (eV)Doping xRPAt-J-V0.0 0.2 0.4 0.6 0.8 1.00.00.40.81.2FIG. 6. Doping-dependence of the plasmon energy computedin the layered t-J-V model (dashed lines) and the RPA (dot-ted lines). The parameters used in the RPA calculations arethe same as those used for the t-J-V model calculation, exceptfor Vc/t = 14, which was adjusted to match the experimen-tal values of the optical plasma frequency (ωoptpl ∼ 0.85) atx = 0.16 [43, 44].the plasmon energy at a doping level higher than thatobserved experimentally.III. DISCUSSION AND CONCLUSIONOur RIXS measurements at the O K-edge revealedthat from very low dopings up to x ≈ 0.2 the plasmon en-ergy increases with increasing hole-doping [Fig. 3]. Theincrease in this doping regime is accurately captured byt-J-V model calculations. The RPA calculations yieldan opposite trend [Fig. 6], namely, a decrease of theplasmon energy with increasing hole-doping. This dis-crepancy points to the critical importance of an explicitinclusion of strong electron correlations into the model-ing of plasmon excitations in cuprates, which is naturallyimplemented in the framework of the t-J-V calculations.However, our study reveals that the t-J-V model predictsa monotonic increase of the plasmon energy at least upto x ≈ 0.55, whereas the experimental data indicatesthat the plasmon energy increase tends to saturate be-yond x ≈ 0.2 [Fig. 3]. In the following, we discuss threepossible origins of this quantitative difference.First, we address potential experimental shortcomingsand note that the doping dependence of the plasmon en-ergy probed by RIXS is reminiscent of that of the opticalplasmon energy observed by optics experiments [44, 99].This similarity rules out that the saturation is due to er-rors or inaccuracies of our RIXS measurements or dataanalysis. However, we cannot fully rule out that struc-tural defects or chemical disorder of the dopant ions inour samples play a role in the saturation. In fact, the de-tails of the distribution of the dopant ions [106, 107] aswell as the emergence of oxygen vacancies [98] and the ef-fects of out-of-plane substitutional disorder [108–111] are10currently under debate for cuprates. Along these lines,we note that the room-temperature resistivity of the x= 0.4 film lies above that of the x = 0.3 film [Fig. 1],whereas the resistivities decrease sequentially from x =0.05 to 0.3. This suggests that the electronic propertiesare affected by structural defects and/or disorder at leastin our x = 0.4 film. Yet, we stress that a Raman spec-troscopy study reported that the formation of oxygenvacancies can be greatly alleviated in our ozone-assistedMBE grown La2−xSrxCuO4 films for dopings below x =0.35 [97]. Moreover, the onset of the plasmon energy sat-uration is observed already between x = 0.16 and 0.2,whereas the effects of disorder typically emerge only inthe overdoped and highly overdoped regime [98, 108–111]. These considerations suggest that while disordermight have some effect on the propagation of the col-lective plasmon excitation for our overdoped samples, itcannot fully account for the observed plasmon energysaturation.Second, there might be an overestimation of correla-tions in the t-J-V model, which pushes the crossover froma strongly correlated to a moderately or weakly corre-lated regime to higher dopings. In more detail, the broadmaximum of the plasmon energy between 0.4 . x . 0.6in the t-J-V calculation in Fig. 6 can be regarded as acrossover between strong and weak correlation physics.At high dopings, electrons are more dilute and the lo-cal constraint in the t-J-V model that prohibits doubleoccupancy of electrons as well as the nearest-neighborexchange interactions are less effective. This is the rea-son why the t-J-V and RPA calculations show the sametrend in the limit of x = 1 in Fig. 6. If the presentt-J-V calculations overestimate the correlation effect forhigh dopings in cuprates, a weaker correlation is expectedto yield the broad peak at lower doping, which wouldbe in better agreement with the experimental data inFig. 3. Whether the experimental plasmon energy alsoexhibits a crossover behavior as predicted by the theoryand hence starts to decrease for some doping level abovex = 0.4 is an interesting topic for future experimentsif La2−xSrxCuO4 films with such high dopings becomeavailable.Third, the activation of additional orbital degrees offreedom at large doping might play a significant role inthe doping dependence. Such an activation is not cap-tured by the one-band t-J-V model, which is an effec-tive model for the three-band Hubbard model, where theCu 3dx2−y2 orbitals are hybridized with the 2px and 2pyorbitals of the in-plane oxygen ions, forming the ZRSband [12]. Yet, there are theoretical indications thatother planar [17–20, 22–26] and non-planar orbitals [27–33, 35, 36, 38, 41] should be taken into account in com-prehensive models of cuprates. In the overdoped regime,specifically the Cu 3d3z2−r2 , interstitial 4s, and the 2pzorbital of the apical oxygen were proposed to become rel-evant [30, 34, 37, 39, 40]. For instance, a six-band Hub-bard model, which includes the Cu 3d3z2−r2 and O 2pzand 2p−z orbitals, was employed to compute the dopingdependence of the optical conductivity of La2−xSrxCuO4[34]. The obtained doping dependence is consistent withthe experimentally observed continuous increase in op-tical spectral weight below 1.5 eV with increasing hole-doping [44, 99]. Yet, the detailed dependence of the op-tical plasmon energy in the overdoped regime [Fig. 3] isdifficult to discern in Ref. 44 and might be affected bydisorder in the overdoped single-crystalline samples. Onthe other hand, a recent ellipsometry study on highly oxi-dized La2−xCaxCuO4 films [99] carefully decomposed theoptical spectral weight below 1.5 eV into a Drude peakand contributions from intra-ZRS transitions, revealingthat the former component as well as the (bare) plasmafrequency saturate above x ≈ 0.2. By contrast, the sec-ond component associated with the intra-ZRS transitionsand the Cu 3d3z2−r2 and O 2pz orbitals increases contin-uously up to the highest measured doping. The sum ofboth components is again compatible with the spectralweight evolution calculated in Ref. 34. These distinctbehaviors of the components in the optical spectra arestrikingly reminiscent of our RIXS and XAS data. In par-ticular, it seems likely that the peak in our RIXS spectrapredominantly reflects the dynamics of the Drude chargecarriers in the planar orbitals, whereas holes that areadded to the system beyond x ≈ 0.2 might be mostly as-sociated with non-planar orbitals and do not contributeto the collective plasmon excitation probed by RIXS. Onthe other hand, the continuous increase of the XAS spec-tral weight of the hole-peak up to x= 0.4 [Fig. 2a-f] mightreflect the total number of doped holes, both in the pla-nar and non-planar orbitals. In this context, note thatour XAS includes contributions from both planar andnon-planar orbitals due to the employed measurementgeometry (see Appendix A for details). Hence, futureXAS and RIXS experiments in a measurement geome-try [31] that is exclusively sensitive to charge carriers inthe non-planar orbitals are highly desirable for confirm-ing the above scenario and gaining further insights intothe mobility of holes in the highly overdoped regime ofcuprates.Besides the observed saturation of the plasmon en-ergy, our results provide fresh insights into the debateon whether plasmons can exist at all doping levels inthe cuprate phase diagram, or if specific phases precludetheir emergence. In particular, previous reflection EELSexperiments on Bi2.1Sr1.9CaCu2O8+x [85, 86] concludedthat well-defined plasmon excitations are replaced bya featureless, temperature- and momentum-independentcontinuum in the strange metal regime, although the au-thors specified in a recent comment [88] that they iden-tified the spectral feature as a plasmon excitation formomenta H ≤ 0.12. For larger momenta, it was sug-gested that the decay of the plasmon into a continuumis unusually strong [85, 86, 88], and that the putativequantum critical nature of the continuum can be cap-tured by holographic theories [84]. Yet, our combinedO K- and Cu K-edge RIXS results on La2−xSrxCuO4clearly indicate that dispersive plasmon excitations ex-11ist for dopings between x = 0.05 and 0.4, including thestrange metal regime at x ∼ 0.2 for momenta at least upto H = 0.15. Moreover, between x = 0.05 and 0.2, theexcitation energies are theoretically well described by thet-J-V model without any fine-tuning of the parameters,which is a long-established model for cuprates. However,the linewidths of the plasmon excitations in Figs. 4c,dare comparable to the respective plasmon energies forall measured momenta, indicating that the mode is ap-preciably damped. In general, several mechanisms canbe responsible for a damping of plasmons in the long-wavelength limit around the BZ center, including scat-tering with phonons and Umklapp scattering [112], whichcan provide the momentum necessary to couple the plas-mon to the electron-hole continuum. Furthermore, thedamping around the BZ center depends on the detailedbalance between the correlation strength and the size ofthe plasmon gap due to the interlayer hopping tz [113].At large momenta, a pronounced plasmon decay is ex-pected even in a conventional Lindhard model picturedue to Landau damping when the plasmon traverses theenergy scale of intra- and interband transitions [87]. Adefinite distinction between the presence of these mecha-nisms and a quantum critical decay [84] appears unfeasi-ble based on the present RIXS data, whereas a compre-hensive study of the plasmon linewidth evolution for bothsmall and large momenta might give critical insights. Inparticular, while previous reflection EELS studies werefocused on relatively large momenta [85, 86], future CuK-edge RIXS experiments are anticipated to provide de-tailed information about the entire BZ.In a broader context, we point out that a cautioususe of the terminology for the plasmon phenomenologyin cuprates is important. For instance, Ref. 65 refers tothe ensemble of branches between the optical plasmonand the lowest-energy (quasi-)acoustic branch in termsof a plasmon continuum. Such a continuum is not di-rectly related to the decay of a nominally sharp plasmonexcitation into a continuum due to various classical orquantum critical damping mechanisms [84, 87, 112, 113].In fact, the degree of the continuum-like character ofthe plasmon branch ensemble is determined by the to-tal number n of CuO2 planes of a system, as the numberof (quasi-)acoustic branches is n − 1 [67]. Notably, fora given in-plane momentum, the signal probed by thereflection EELS technique is integrated over all (or atleast a broad range of) out-of-plane momenta [65]. Incontrast, the momentum resolution of RIXS is relativelyhigh in all directions of reciprocal space [114], which im-plies that especially for thin-film samples with a smallnumber of CuO2 planes the out-of-plane integration canbe close to the limit of covering only a single branch.Whether this intrinsic difference between the two experi-mental techniques is responsible for the distinct spectralfeatures observed below 1 eV energy loss remains to betested in future experiments. In particular, future CuK-edge RIXS studies on Bi2.1Sr1.9CaCu2O8+x and otherhole-doped cuprates are highly desirable for establish-ing the universality of plasmon excitations in the strangemetal regime of the cuprates and clarifying the nature ofthe plasmon decay.ACKNOWLEDGMENTSWe thank A.V. Boris, G. Kim, and C. Falter for fruit-ful discussions. A.G. acknowledges the Max-Planck-Institute for Solid State Research in Stuttgart for hos-pitality and financial support. H.Y. was supported byJSPS KAKENHI Grant No. JP20H01856, Japan. Partof the research described in this paper was performed atthe Canadian Light Source, a national research facilityof the University of Saskatchewan, which is supported bythe Canada Foundation for Innovation (CFI), the Natu-ral Sciences and Engineering Research Council (NSERC),the National Research Council (NRC), the Canadian In-stitutes of Health Research (CIHR), the Government ofSaskatchewan, and the University of Saskatchewan. Thisresearch used resources of the Advanced Photon Source,a U.S. Department of Energy (DOE) Office of Scienceuser facility operated for the DOE Office of Science byArgonne National Laboratory under Contract No. DE-AC02-06CH11357.Appendix A: Experimental detailsA series of La2−xSrxCuO4 films with a thickness of 15unit cells was grown by ozone-assisted molecular beamepitaxy (MBE) on (001) oriented LaSrAlO4 (LSAO) sub-strates. The nominal Sr substitution levels were x = 0.05,0.1, 0.16, 0.2, 0.3, and 0.4. The hole-doping concentra-tion in La2−xSrxCuO4 is directly proportional to the Sr-substitution level (p = x). The Van-der-Pauw methodwas used to determine the in-plane electrical resistivity.The Tc values for x = 0.1, 0.16, 0.2, 0.3, and 0.4 films are35 K, 41 K, 32 K, 26 K, and below 2 K, respectively. Thex = 0.05 film did not show an onset of a superconductingtransition. In addition to the films, a single-crystal witha stoichiometry close to La1.8Sr0.2CuO4 was grown bythe optical floating zone technique. A standard CrystalSystems CSC FZ 1000 equipped with 300 W lamps wasused and the crystal was grown at a rate of 1 mm/h inmixed Ar with 20% O2 atmosphere at 3 bar. The refine-ment of single-crystal x-ray diffraction (XRD) data ontwo pieces broken off from the crystal indicated a Sr con-tent of x = 0.196±0.03 and 0.17±0.03, respectively. En-ergy dispersive x-ray spectroscopy (EDS) on the crystalsurface yielded x = 0.21 when normalized to a nominallyideal Cu stoichiometry. The Tc of the crystal was about38 K.The doping dependence of the plasmon excitation inthe La2−xSrxCuO4 films was measured using O K-edgeRIXS at the REIXS beamline of the Canadian LightSource (CLS). The RIXS spectra were collected at 300K using a Rowland circle spectrometer with a combined12energy resolution ∆E ∼ 190 meV and linearly polarizedphotons (σ-polarization). The films were mounted suchthat the c- and the a/b-axes were lying in the scatter-ing plane. The crystal structure of La2−xSrxCuO4 is or-thorhombic, but as the difference between the a and baxes is small, and the films are twinned in the ab-plane,we do not distinguish between a and b in the following.The scattering angle 2θ was kept fixed at 90◦, while theangle θ between the sample surface and the incident x-rays was varied to probe different momentum transfers.The XAS data were taken with σ-polarized photons atan incident angle θ = 35◦ in partial fluorescence yieldusing a silicon drift detector, collecting only the O Kαemission line.We denote the momentum transfer by (H,K,L) in re-ciprocal lattice units (2π/a, 2π/b, 2π/c), where a, b, and care the lattice constants of La2−xSrxCuO4. Note that thecrystallographic unit cell of La2−xSrxCuO4 contains twoCuO2 planes, whereas only the distance d = c/2 betweenadjacent CuO2 planes is relevant for the periodicity ofthe plasmon dispersion [56]. Hence, the index L∗ = L/2is used in the following. Since the scattering angle 2θ wasfixed for the measurements at the REIXS beamline, eachvariation of the in-plane momentum transfer also leads toa (minor) change of the out-of-plane momentum. Specif-ically, a change of H from 0.04 to 0.12 in the O K-edgeRIXS measurements involves a concomitant change of L∗from 0.393 to 0.346. However, since the plasmon disper-sion as a function of L∗ is relatively shallow between ∼0.3and 0.4 [66], we neglect the change of L∗ in the follow-ing and employ the averaged value of L∗ = 0.37 for theanalysis of the O K-edge RIXS data.Additional O K-edge RIXS measurements on selectedLa2−xSrxCuO4 films were conducted at the U41-PEAXISbeamline [115, 116] of BESSY II at the Helmholtz-Zentrum Berlin (HZB). The spectra were collected atvarious temperatures between 15 and 100 K, with a com-bined energy resolution ∆E ∼ 90 meV. Typically, σ-polarized photons are used for soft x-ray measurementsof plasmon excitations in cuprates [56–58, 60], as thecharge signal is enhanced over the magnetic signal. How-ever, since only π-polarization was available for our RIXSexperiment at U41-PEAXIS, the temperature-series wastaken at momenta that correspond to incident angles θclose to normal incidence. Accordingly, the projectionof the polarization vector of the electric field of the π-polarized photons onto the CuO2 planes was maximized,enhancing the charge signal as much as possible. Thescattering angle 2θ at U41-PEAXIS can be varied acrossa broad range. For our measurement, 2θ was approx-imately 137◦. The corresponding in-plane momentumtransfer was H = 0.1 and the out-of-plane momentumtransfer L∗ = 0.5.The Cu K-edge RIXS measurements on theLa1.8Sr0.2CuO4 single-crystal were carried out atthe MERIX spectrometer at sector 27 at the AdvancedPhoton Source (APS). RIXS spectra were collectedbetween 17 and 100 K. The incident photons weremonochromatized by a four-bounce monochromatorwith asymmetrically cut Si (4, 0, 0) crystals. A sphericaldiced Ge (3, 3, 7) analyzer was used and the flight pathbetween the analyzer and sample/detector was approx-imately 1 meter. The combined energy resolution was∆E ∼ 80 meV. The incident angle θ and the scatteringangle 2θ could be varied independently from each other.To minimize the strong elastic scattering signal in theRIXS spectra, π-polarized photons and a scatteringangle 2θ close to 90◦ was chosen. Specifically, for allmeasured H, K, and L values of momentum-dependenceof the plasmon excitation, the angle 2θ remained closeto 90◦. The scattering geometry of the RIXS experimentwas such that the c-axis and the diagonal between the a-and b-axis of La1.8Sr0.2CuO4 were lying in the scatteringplane. The XAS data were collected in the fluorescenceyield mode with π- and σ-polarized photons, respec-tively, at a shallow incident angle θ (close to grazingincidence). Hence, the polarization vector of the electricfield of the incoming photons lies within the a/b planeof the crystallographic unit cell for σ-linearly polarizedphotons, whereas it is mostly parallel to the c-axis forπ-polarized photons, but with a small projection on thea/b plane due to a small θ.Appendix B: RIXS raw data and fitsFigure 7a shows all components of a representative fitof the OK-edge RIXS spectra of the La2−xSrxCuO4 filmsof Fig. 2 of the main text. The displayed RIXS spectrumwas taken on the x = 0.16 film at momentum H = 0.08,K = 0, and L∗ = 0.37. The elastic peak was modeled bya Gaussian and the other contributions in the spectra byanti-symmetrized Lorentzians [56, 58], convoluted withthe energy resolution of ∆E = 190 meV via Gaussianconvolution. The anti-symmetrized Lorentzian profilesensure zero intensity at zero energy loss (prior to convolu-tion) for the inelastic features. The inelastic features areassigned to a phonon, a plasmon, and a non-dispersivebimagnon. In the fits, the energy of the bimagnon peakswas kept fixed, while all other fit parameters were var-ied. Furthermore, the spectra contain a broad and in-tense background (bkg) feature, which peaks beyond 2eV energy loss. This feature consists of fluorescence anddd-excitations [48]. In the fits, we included the tail of thefeature below 2.1 eV energy loss. For clarity, the signaloriginating from this feature was subtracted in the RIXSspectra shown in Fig. 2 of the main text. A collectionof the raw data of the O K-edge RIXS spectra of theLa2−xSrxCuO4 films is shown in Fig. 8.The observed trend of an increase of the peak energywith increasing doping concentration up to x = 0.16 doesnot depend on the details of our fitting procedure, such asthe exact treatment of the bimagnon peak. For instance,a fit that captures both the plasmon and the bimagnonfeature by a single broad peak function yields similarresults to Fig. 3 of the main text. This is illustrated in13FIG. 7. Representative fits of the RIXS spectra. (a) Fit ofthe O K-edge RIXS spectrum of the x = 0.16 film taken atmomentum H = 0.08, K = 0, and L∗ = 0.37. The individualcontributions of the fit are described in the text. (b) Fit ofthe Cu K-edge RIXS spectrum of the x = 0.2 single-crystaltaken at momentum momentum H = 0.09, K = 0.09, and L∗= 6.5.Fig. 9, which shows the energy of the plasmon peak (bluesquares, two-peak fit) extracted from a fit that includesseparate plasmon and bimagnon peaks (see also Fig. 3 ofthe main text), together with the energy of a single peak(light blue circles, single-peak fit) that comprises boththe plasmon and the bimagnon feature.The components of a representative fit of the Cu K-edge RIXS spectra of the La1.8Sr0.2CuO4 single-crystalof Fig. 4 of the main text is given in Fig. 7b. The dis-played RIXS spectrum was taken at momentum H = K= 0.09, and L∗ = 6.5. The elastic peak was modeled bya Gaussian and the other contributions in the spectra byanti-symmetrized Lorentzians, convoluted with the en-ergy resolution of ∆E = 80 meV via Gaussian convolu-tion. The inelastic features are assigned to a phonon,a plasmon, a non-dispersive orbital excitation around1.5 eV energy loss (orbital exc. 1), and an additionalorbital excitation at higher energy losses (orbital exc. 2)[54]. A collection of the raw data of the Cu K-edge RIXSspectra is shown in Fig. 10.Appendix C: Comparison to literatureFigure 11 shows the plasmon (bimagnon) energies oftwo representative La2−xSrxCuO4 film (x = 0.1 and0.16) as a function of the momentum transfer H. Inaddition, the plasmon (bimagnon) energies taken at sim-ilar momenta and dopings in Refs. [58, 59] are superim-posed. The plasmon (bimagnon) energies determined inthe present work are similar to those from Refs. [58, 59].Appendix D: Theoretical detailsFor the theoretical description of plasmon excitationsin cuprates, we employ the layered t-J-V model [66, 69],which incorporates the three-dimensional character of theplasmons [56] via the long-range (interlayer) Coulombinteraction V :H = −∑i,j,σtij c̃†iσ c̃jσ +∑〈i,j〉Jij(~Si · ~Sj −14ninj)+12∑i,jVijninj (D1)where the sites i and j run over a three-dimensional lat-tice. The hopping tij takes a value t (t′) between the first(second) nearest-neighbors on the square lattice. Thehopping integral between layers is scaled by tz. 〈i, j〉 de-notes a nearest-neighbor pair of sites. The exchange in-teraction Jij = J is considered only inside the plane; theexchange term between the planes (J⊥) is much smallerthan J [117]. c̃†iσ (c̃iσ) is the creation (annihilation) op-erators of electrons with spin σ(=↑, ↓) in the Fock spacewithout any double occupancy. ni =∑σ c̃†iσ c̃iσ is theelectron density operator and ~Si is the spin operator. Vijis the long-range Coulomb interaction on the lattice andis given in momentum space by [118]:V (q) =VcA(qx, qy)− cos qz, (D2)where Vc = e2d(2ε⊥a2)−1 andA(qx, qy) = α(2− cos qx − cos qy) + 1 . (D3)Here α = ε̃(a/d)2 , where ε̃ = ε‖/ε⊥, and ε‖ and ε⊥ arethe dielectric constants parallel and perpendicular to theplanes, respectively; e is the electric charge of electrons;a is the lattice spacing in the planes and the in-planemomentum q‖ = (qx, qy) is given in units of 1/a; similarlyd is the distance between the planes and the out-of-planemomentum qz is given in units of 1/d.14FIG. 8. (a)-(f) Raw RIXS spectra (filled symbols) of the La2−xSrxCuO4 films with various hole-doping levels x taken at the OK-edge, together with fits (gray lines). Curves for different momenta are offset in the vertical direction for clarity.FIG. 9. Results for alternative fits of the RIXS spectra ofthe La2−xSrxCuO4 films. (a) Fit results for the x = 0.05film. The blue squares (two-peak fit) indicate the plasmonenergy extracted from a fit that includes separate plasmonand bimagnon peaks, in analogy to Fig. 7a. The light bluecircles (single-peak fit) indicate the energy of a single peakthat comprises both the plasmon and the bimagnon feature(fit not shown here). (b) Fit results for the x = 0.16 film.To compute the plasmon excitations we use the large-N formalism given in Ref. 66. The charge-charge corre-lation function can be calculated asχ(q, iωn) = N(x2)2D11(q, iωn) . (D4)Here D11 is the element (1, 1) of the 6× 6 bosonic prop-agator[Dab(q, iωn)]−1 = [D(0)ab (q, iωn)]−1 −Πab(q, iωn) ,(D5)FIG. 10. Raw RIXS spectra (filled symbols) of theLa1.8Sr0.2CuO4 single-crystal taken at the Cu K-edge, to-gether with fits (gray lines). Curves for different momenta areoffset in the vertical direction for clarity. (a) RIXS spectra forvarious in-plane momenta along the (H,H) direction, whilethe out-of-plane momentum is fixed to L∗ = 6.5. (b) RIXSspectra for momenta along the L∗ direction, while H = 0.09and K = 0 are fixed.where ωn are the Matsubara frequencies and a,b run from1 to 6. The factor N in Eq. (D4) comes from the sum overthe N fermionic channels after the extension of the spinindex from 2 to N . The corresponding expressions for15FIG. 11. Comparison of the fit results of the RIXS spectrawith literature. (a) Blue squares (dark gray triangles) arethe plasmon (bimagnon) energies extracted from the O K-edge RIXS spectra of the La2−xSrxCuO4 film with x = 0.16.Blue circles (light gray triangles) are the plasmon (bimagnon)energies of a La2−xSrxCuO4 single-crystal with x = 0.16 fromRef. 58. (b) Fit results of the La2−xSrxCuO4 film with x = 0.1and the results of Ref. 59 on a La2−xSrxCuO4 single-crystalwith x = 0.12.the bare bosonic propagator D(0)ab (q, iωn) and the bosonicself-energy Πab(q, iωn) are given in Ref. 66.The large-N formalism successfully captured the ex-perimentally observed low-energy plasmon dispersionsin various cuprates for specific doping levels in priorworks [58, 60, 69, 70]. As also used in Ref. 60 to fitand reproduce the experimental plasmon dispersion inLa1.84Sr0.16CuO4, we use the parameters t/2 = 0.35 eV,t′/t = −0.2, tz/t = 0.01, J/t = 0.3, Vc/t = 31, andα = 3.5 for La2−xSrxCuO4 in the present work. Thenumber of planes is Nz = 30 and temperature T = 0.Appendix E: Summary of the RPA formalismIn the RPA the charge correlation function is given by[105]χRPA(q, iωn) =χ(0)(q, iωn)1− V (q)χ(0)(q, iωn), (E1)where χ(0)(q, iωn) is the Lindhard function which ac-counts for the particle-hole continuum. The electron dis-persion Ek isEk = E‖k + E⊥k , (E2)whereE‖k = −2t(cos kx + cos ky)− 4t′ cos kx cos ky − µ ,(E3)E⊥k = −2tz(cos kx − cos ky)2 cos kz .(E4)[1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida,and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature518, 179 (2015).[2] D. J. Scalapino, A common thread: The pairing inter-action for unconventional superconductors, Rev. Mod.Phys. 84, 1383 (2012).[3] N. P. Armitage, P. Fournier, and R. L. Greene, Progressand perspectives on electron-doped cuprates, Rev. Mod.Phys. 82, 2421 (2010).[4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mottinsulator: Physics of high-temperature superconductiv-ity, Rev. Mod. Phys. 78, 17 (2006).[5] H. Takagi, B. Batlogg, H. L. Kao, J. Kwo, R. J. Cava,J. J. Krajewski, and W. F. Peck, Systematic evolutionof temperature-dependent resistivity in La2−xSrxCuO4,Phys. Rev. Lett. 69, 2975 (1992).[6] N. Nücker, J. Fink, J. C. Fuggle, P. J. Durham, andW. M. Temmerman, Evidence for holes on oxygensites in the high-Tc superconductors La2−xSrxCuO4 andYBa2Cu3O7−y, Phys. Rev. B 37, 5158 (1988).[7] C. T. Chen, F. Sette, Y. Ma, M. S. Hybertsen, E. B.Stechel, W. M. C. Foulkes, M. Schulter, S.-W. Cheong,A. S. Cooper, L. W. Rupp, B. Batlogg, Y. L. Soo,Z. H. Ming, A. Krol, and Y. H. Kao, Electronic statesin La2−xSrxCuO4+δ probed by soft-x-ray absorption,Phys. Rev. Lett. 66, 104 (1991).[8] E. Pellegrin, N. Nücker, J. Fink, S. L. Molodtsov,A. Gutiérrez, E. Navas, O. Strebel, Z. Hu, M. Domke,G. Kaindl, S. Uchida, Y. Nakamura, J. Markl,M. Klauda, G. Saemann-Ischenko, A. Krol, J. L. Peng,Z. Y. Li, and R. L. Greene, Orbital character of statesat the Fermi level in La2−xSrxCuO4 and R2−xCexCuO4(R=Nd,Sm), Phys. Rev. B 47, 3354 (1993).[9] M. Haruta, Y. Fujiyoshi, T. Nemoto, A. Ishizuka,K. Ishizuka, and H. Kurata, Atomic-resolution two-dimensional mapping of holes in the cuprate super-conductor La2−xSrxCuO4±δ, Phys. Rev. B 97, 205139(2018).[10] L. H. Tjeng, B. Sinkovic, N. B. Brookes, J. B. Goed-koop, R. Hesper, E. Pellegrin, F. M. F. de Groot, S. Al-tieri, S. L. Hulbert, E. Shekel, and G. A. Sawatzky, Spin-Resolved Photoemission on Anti-Ferromagnets: DirectObservation of Zhang-Rice Singlets in CuO, Phys. Rev.Lett. 78, 1126 (1997).[11] N. B. Brookes, G. Ghiringhelli, A.-M. Charvet, A. Fu-jimori, T. Kakeshita, H. Eisaki, S. Uchida, and T. Mi-zokawa, Stability of the Zhang-Rice Singlet with Dopingin Lanthanum Strontium Copper Oxide Across the Su-perconducting Dome and Above, Phys. Rev. Lett. 115,027002 (2015).[12] F. C. Zhang and T. M. Rice, Effective hamiltonian forthe superconducting cu oxides, Phys. Rev. B 37, 3759(1988).[13] E. Dagotto, Correlated electrons in high-temperaturesuperconductors, Rev. Mod. Phys. 66, 763 (1994).[14] M. Ogata and H. Fukuyama, The t – J model for theoxide high- T c superconductors, Rep. Prog. Phys. 71,http://dx.doi.org/10.1038/nature14165http://dx.doi.org/10.1038/nature14165https://doi.org/10.1103/RevModPhys.84.1383https://doi.org/10.1103/RevModPhys.84.1383https://doi.org/10.1103/RevModPhys.82.2421https://doi.org/10.1103/RevModPhys.82.2421https://doi.org/10.1103/RevModPhys.78.17https://doi.org/10.1103/PhysRevLett.69.2975https://doi.org/10.1103/PhysRevB.37.5158https://doi.org/10.1103/PhysRevLett.66.104https://doi.org/10.1103/PhysRevB.47.3354https://doi.org/10.1103/PhysRevB.97.205139https://doi.org/10.1103/PhysRevB.97.205139https://doi.org/10.1103/PhysRevLett.78.1126https://doi.org/10.1103/PhysRevLett.78.1126https://doi.org/10.1103/PhysRevLett.115.027002https://doi.org/10.1103/PhysRevLett.115.027002https://doi.org/10.1103/PhysRevB.37.3759https://doi.org/10.1103/PhysRevB.37.3759https://doi.org/10.1103/RevModPhys.66.763https://doi.org/10.1088/0034-4885/71/3/03650116036501 (2008).[15] M. S. Hybertsen, E. B. Stechel, M. Schluter, and D. R.Jennison, Renormalization from density-functional the-ory to strong-coupling models for electronic states incu-o materials, Phys. Rev. B 41, 11068 (1990).[16] A. A. Aligia, M. E. Simón, and C. D. Batista, System-atic derivation of a generalized t-j model, Phys. Rev. B49, 13061 (1994).[17] V. J. Emery, Theory of high-Tc superconductivity inoxides, Phys. Rev. Lett. 58, 2794 (1987).[18] C. M. Varma, Non-fermi-liquid states and pairing insta-bility of a general model of copper oxide metals, Phys.Rev. B 55, 14554 (1997).[19] C. Weber, K. Haule, and G. Kotliar, Optical weights andwaterfalls in doped charge-transfer insulators: A localdensity approximation and dynamical mean-field the-ory study of La2−xSrxCuO4, Phys. Rev. B 78, 134519(2008).[20] C. Weber, C. Yee, K. Haule, and G. Kotliar, EPL 100,37001 (2012).[21] C.-C. Chen, M. Sentef, Y. F. Kung, C. J. Jia,R. Thomale, B. Moritz, A. P. Kampf, and T. P. De-vereaux, Doping evolution of the oxygen K-edge x-rayabsorption spectra of cuprate superconductors using athree-orbital Hubbard model, Phys. Rev. B 87, 165144(2013).[22] W. Chen and O. P. Sushkov, Implications of resonantinelastic x-ray scattering data for theoretical models ofcuprates, Phys. Rev. B 88, 184501 (2013).[23] H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, Thedynamics of a doped hole in a cuprate is not controlledby spin fluctuations, Nat. Phys. 10, 951 (2014).[24] H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, Dif-ferences between the insulating limit quasiparticles ofone-band and three-band cuprate models, J. Phys. Con-dens. Matter 28, 105603 (2016).[25] C. P. J. Adolphs, S. Moser, G. A. Sawatzky, andM. Berciu, Non-Zhang-Rice Singlet Character of theFirst Ionization State of T-CuO, Phys. Rev. Lett. 116,087002 (2016).[26] Y. F. Kung, C.-C. Chen, Y. Wang, E. W. Huang, E. A.Nowadnick, B. Moritz, R. T. Scalettar, S. Johnston,and T. P. Devereaux, Characterizing the three-orbitalhubbard model with determinant quantum monte carlo,Phys. Rev. B 93, 155166 (2016).[27] A. Bianconi, M. De Santis, A. Di Cicco, A. M.Flank, A. Fontaine, P. Lagarde, H. Katayama-Yoshida,A. Kotani, and A. Marcelli, Symmetry of the 3d9 ligandhole induced by doping in YBa2Cu3O7−δ, Phys. Rev. B38, 7196 (1988).[28] M. Grilli, C. Castellani, and C. Di Castro, Renormal-ized band structure of cuo2 layers in superconductingcompounds: A mean-field approach, Phys. Rev. B 42,6233 (1990).[29] H. Romberg, N. Nücker, M. Alexander, J. Fink,D. Hahn, T. Zetterer, H. H. Otto, and K. F. Renk,Density and symmetry of unoccupied electronic statesof Tl2Ba2CaCu2O8, Phys. Rev. B 41, 2609 (1990).[30] L. F. Feiner, M. Grilli, and C. Di Castro, Apical oxygenions and the electronic structure of the high-tc cuprates,Phys. Rev. B 45, 10647 (1992).[31] C. T. Chen, L. H. Tjeng, J. Kwo, H. L. Kao, P. Rudolf,F. Sette, and R. M. Fleming, Out-of-plane orbital char-acters of intrinsic and doped holes in La2−xSrxCuO4,Phys. Rev. Lett. 68, 2543 (1992).[32] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen,and O. K. Andersen, Band-Structure Trend in Hole-Doped Cuprates and Correlation with Tcmax, Phys.Rev. Lett. 87, 047003 (2001).[33] X. Wang, L. de’ Medici, and A. J. Millis, Theory of oxy-gen k edge x-ray absorption spectra of cuprates, Phys.Rev. B 81, 094522 (2010).[34] C. Weber, K. Haule, and G. Kotliar, Apical oxygens andcorrelation strength in electron- and hole-doped copperoxides, Phys. Rev. B 82, 125107 (2010).[35] H. Sakakibara, H. Usui, K. Kuroki, R. Arita, andH. Aoki, Two-Orbital Model Explains the Higher Tran-sition Temperature of the Single-Layer Hg-Cuprate Su-perconductor Compared to That of the La-Cuprate Su-perconductor, Phys. Rev. Lett. 105, 057003 (2010).[36] L. Hozoi, L. Siurakshina, P. Fulde, and J. Van DenBrink, Ab Initio determination of Cu 3d orbital ener-gies in layered copper oxides, Sci. Rep. 1, 1 (2011).[37] Y. Sakurai, M. Itou, B. Barbiellini, P. E. Mijnarends,R. S. Markiewicz, S. Kaprzyk, J.-M. Gillet, S. Waki-moto, M. Fujita, S. Basak, Y. J. Wang, W. Al-Sawai,H. Lin, A. Bansil, and K. Yamada, Imaging Doped Holesin a Cuprate Superconductor with High-ResolutionCompton Scattering, Science 332, 698 (2011).[38] H. Sakakibara, H. Usui, K. Kuroki, R. Arita, andH. Aoki, Origin of the material dependence of Tc in thesingle-layered cuprates, Phys. Rev. B 85, 064501 (2012).[39] C. E. Matt, D. Sutter, A. M. Cook, Y. Sassa,M. Månsson, O. Tjernberg, L. Das, M. Horio, D. De-straz, C. G. Fatuzzo, K. Hauser, M. Shi, M. Kobayashi,V. N. Strocov, T. Schmitt, P. Dudin, M. Hoesch,S. Pyon, T. Takayama, H. Takagi, O. J. Lipscombe,S. M. Hayden, T. Kurosawa, N. Momono, M. Oda,T. Neupert, and J. Chang, Direct observation of orbitalhybridisation in a cuprate superconductor, Nat. Com-mun. 9, 1 (2018).[40] K. P. Kramer, M. Horio, S. S. Tsirkin, Y. Sassa,K. Hauser, C. E. Matt, D. Sutter, A. Chikina, N. B. M.Schröter, J. A. Krieger, T. Schmitt, V. N. Strocov,N. C. Plumb, M. Shi, S. Pyon, T. Takayama, H. Takagi,T. Adachi, T. Ohgi, T. Kawamata, Y. Koike, T. Kondo,O. J. Lipscombe, S. M. Hayden, M. Ishikado, H. Eisaki,T. Neupert, and J. Chang, Band structure of over-doped cuprate superconductors: Density functional the-ory matching experiments, Phys. Rev. B 99, 224509(2019).[41] M. Jiang, M. Moeller, M. Berciu, and G. A. Sawatzky,Relevance of Cu−−3d multiplet structure in models ofhigh-Tc cuprates, Phys. Rev. B 101, 035151 (2020).[42] D. N. Basov and T. Timusk, Electrodynamics of high-Tcsuperconductors, Rev. Mod. Phys. 77, 721 (2005).[43] M. Suzuki, Hall coefficients and optical properties ofLa2−xSrxCuO4 single-crystal thin films, Phys. Rev. B39, 2312 (1989).[44] S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, andS. Tajima, Optical spectra of La2−xSrxCuO4: Effect ofcarrier doping on the electronic structure of the CuO2plane, Phys. Rev. B 43, 7942 (1991).[45] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux,J. P. Hill, and J. van den Brink, Resonant inelastic x-rayscattering studies of elementary excitations, Rev. Mod.Phys. 83, 705 (2011).[46] M. Le Tacon, G. Ghiringhelli, J. Chaloupka, M. M. Sala,https://doi.org/10.1088/0034-4885/71/3/036501https://doi.org/10.1103/PhysRevB.41.11068https://doi.org/10.1103/PhysRevB.49.13061https://doi.org/10.1103/PhysRevB.49.13061https://doi.org/10.1103/PhysRevLett.58.2794https://doi.org/10.1103/PhysRevB.55.14554https://doi.org/10.1103/PhysRevB.55.14554https://doi.org/10.1103/PhysRevB.78.134519https://doi.org/10.1103/PhysRevB.78.134519https://doi.org/10.1209/0295-5075/100/37001https://doi.org/10.1209/0295-5075/100/37001https://doi.org/10.1103/PhysRevB.87.165144https://doi.org/10.1103/PhysRevB.87.165144https://doi.org/10.1103/PhysRevB.88.184501https://doi.org/10.1038/nphys3130https://iopscience.iop.org/article/10.1088/0953-8984/28/10/105603https://iopscience.iop.org/article/10.1088/0953-8984/28/10/105603https://doi.org/10.1103/PhysRevLett.116.087002https://doi.org/10.1103/PhysRevLett.116.087002https://doi.org/10.1103/PhysRevB.93.155166https://doi.org/10.1103/PhysRevB.38.7196https://doi.org/10.1103/PhysRevB.38.7196https://doi.org/10.1103/PhysRevB.42.6233https://doi.org/10.1103/PhysRevB.42.6233https://doi.org/10.1103/PhysRevB.41.2609https://doi.org/10.1103/PhysRevB.45.10647https://doi.org/10.1103/PhysRevLett.68.2543https://doi.org/10.1103/PhysRevLett.87.047003https://doi.org/10.1103/PhysRevLett.87.047003https://doi.org/10.1103/PhysRevB.81.094522https://doi.org/10.1103/PhysRevB.81.094522https://doi.org/10.1103/PhysRevB.82.125107https://doi.org/10.1103/PhysRevLett.105.057003https://doi.org/10.1038/srep00065https://doi.org/10.1126/science.1199391https://doi.org/10.1103/PhysRevB.85.064501https://doi.org/10.1038/s41467-018-03266-0https://doi.org/10.1038/s41467-018-03266-0https://doi.org/10.1103/PhysRevB.99.224509https://doi.org/10.1103/PhysRevB.99.224509https://doi.org/10.1103/PhysRevB.101.035151https://doi.org/10.1103/RevModPhys.77.721https://doi.org/10.1103/PhysRevB.39.2312https://doi.org/10.1103/PhysRevB.39.2312https://doi.org/10.1103/PhysRevB.43.7942https://doi.org/10.1103/RevModPhys.83.705https://doi.org/10.1103/RevModPhys.83.70517V. Hinkov, M. W. Haverkort, M. Minola, M. Bakr, K. J.Zhou, S. Blanco-Canosa, C. Monney, Y. T. Song, G. L.Sun, C. T. Lin, G. M. De Luca, M. Salluzzo, G. Khal-iullin, T. Schmitt, L. Braicovich, and B. Keimer, In-tense paramagnon excitations in a large family of high-temperature superconductors, Nat. Phys. 7, 725 (2011).[47] M. P. M. Dean, G. Dellea, R. S. Springell, F. Yakhou-Harris, K. Kummer, N. B. Brookes, X. Liu, Y.-J. Sun,J. Strle, T. Schmitt, L. Braicovich, G. Ghiringhelli,I. Bozovic, and J. P. Hill, Persistence of magnetic ex-citations in La2−xSrxCuO4 from the undoped insulatorto the heavily overdoped non-superconducting metal,Nat. Mater. 12, 1019 (2013).[48] V. Bisogni, L. Simonelli, L. J. P. Ament, F. Forte,M. Moretti Sala, M. Minola, S. Huotari, J. van denBrink, G. Ghiringhelli, N. B. Brookes, andL. Braicovich, Bimagnon studies in cuprates withresonant inelastic x-ray scattering at the O K edge.I. Assessment on La2CuO4 and comparison with theexcitation at Cu L3 and Cu K edges, Phys. Rev. B 85,214527 (2012).[49] V. Bisogni, M. Moretti Sala, A. Bendounan, N. B.Brookes, G. Ghiringhelli, and L. Braicovich, Bimagnonstudies in cuprates with resonant inelastic x-ray scatter-ing at the O K edge. II. Doping effect in La2−xSrxCuO4,Phys. Rev. B 85, 214528 (2012).[50] Y. J. Kim, J. P. Hill, C. A. Burns, S. Wakimoto, R. J.Birgeneau, D. Casa, T. Gog, and C. T. Venkataraman,Resonant Inelastic X-Ray Scattering Study of ChargeExcitations in La2CuO4, Phys. Rev. Lett. 89, 177003(2002).[51] K. Ishii, K. Tsutsui, Y. Endoh, T. Tohyama,S. Maekawa, M. Hoesch, K. Kuzushita, M. Tsubota,T. Inami, J. Mizuki, Y. Murakami, and K. Yamada,Momentum Dependence of Charge Excitations in theElectron-Doped Superconductor Nd1.85Ce0.15CuO4: AResonant Inelastic X-Ray Scattering Study, Phys. Rev.Lett. 94, 207003 (2005).[52] K. Ishii, K. Tsutsui, Y. Endoh, T. Tohyama,K. Kuzushita, T. Inami, K. Ohwada, S. Maekawa,T. Masui, S. Tajima, Y. Murakami, and J. Mizuki, MottGap Excitations in Twin-Free YBa2Cu3O7−δ (Tc =93 K) Studied by Resonant Inelastic X-Ray Scattering,Phys. Rev. Lett. 94, 187002 (2005).[53] S. Wakimoto, Y.-J. Kim, H. Kim, H. Zhang, T. Gog,and R. J. Birgeneau, Resonant inelastic x-ray scatteringstudy of overdoped La2−xSrxCuO4, Phys. Rev. B 72,224508 (2005).[54] D. S. Ellis, J. Kim, H. Zhang, J. P. Hill, G. Gu,S. Komiya, Y. Ando, D. Casa, T. Gog, and Y.-J. Kim,Electronic structure of doped lanthanum cuprates stud-ied with resonant inelastic x-ray scattering, Phys. Rev.B 83, 075120 (2011).[55] S. Wakimoto, K. Ishii, H. Kimura, K. Ikeuchi,M. Yoshida, T. Adachi, D. Casa, M. Fujita, Y. Fuku-naga, T. Gog, Y. Koike, J. Mizuki, and K. Yamada,Resonant inelastic x-ray scattering study of intrabandcharge excitations in hole-doped high-Tc cuprates, Phys.Rev. B 87, 104511 (2013).[56] M. Hepting, L. Chaix, E. Huang, R. Fumagalli,Y. Peng, B. Moritz, K. Kummer, N. Brookes, W. Lee,M. Hashimoto, et al., Three-dimensional collectivecharge excitations in electron-doped copper oxide su-perconductors, Nature 563, 374 (2018).[57] J. Q. Lin, Jie Yuan, Kui Jin, Z. P. Yin, Gang Li, Ke-JinZhou, Xingye Lu, M. Dantz, Thorsten Schmitt, H. Ding,Haizhong Guo, M. P. M. Dean, and X. Liu, Doping evo-lution of the charge excitations and electron correlationsin electron-doped superconducting La2−xCexCuO4, npjQuantum Mater. 5, 4 (2020).[58] A. Nag, M. Zhu, M. Bejas, J. Li, H. C. Robarts, H. Ya-mase, A. N. Petsch, D. Song, H. Eisaki, A. C. Walters,M. Garćıa-Fernández, A. Greco, S. M. Hayden, and K.-J. Zhou, Detection of Acoustic Plasmons in Hole-DopedLanthanum and Bismuth Cuprate Superconductors Us-ing Resonant Inelastic X-Ray Scattering, Phys. Rev.Lett. 125, 257002 (2020).[59] A. Singh, H. Y. Huang, C. Lane, J. H. Li, J. Okamoto,S. Komiya, R. S. Markiewicz, A. Bansil, T. K. Lee,A. Fujimori, C. T. Chen, and D. J. Huang, Acousticplasmons and conducting carriers in hole-doped cupratesuperconductors, Phys. Rev. B 105, 235105 (2022).[60] M. Hepting, M. Bejas, A. Nag, H. Yamase, N. Coppola,D. Betto, C. Falter, M. Garcia-Fernandez, S. Agrestini,K.-J. Zhou, M. Minola, C. Sacco, L. Maritato, P. Or-giani, H. I. Wei, K. M. Shen, D. G. Schlom, A. Galdi,A. Greco, and B. Keimer, Gapped Collective ChargeExcitations and Interlayer Hopping in Cuprate Super-conductors, Phys. Rev. Lett. 129, 047001 (2022).[61] D. Grecu, Plasma frequency of the electron gas in lay-ered structures, Phys. Rev. B 8, 1958 (1973).[62] A. L. Fetter, Electrodynamics of a layered electron gas.ii. periodic array, Ann. Phys. 88, 1 (1974).[63] D. Grecu, Self-consistent field approximation for theplasma frequencies of an electron gas in a layered thinfilm, J. Phys. C Solid State 8, 2627 (1975).[64] F. Gabriele, C. Castellani, and L. Benfatto, Generalizedplasma waves in layered superconductors: A unified ap-proach, Phys. Rev. Research 4, 023112 (2022).[65] C. Boyd, L. Yeo, and P. W. Phillips, Probing the bulkplasmon continuum of layered materials through elec-tron energy loss spectroscopy in a reflection geometry,Phys. Rev. B 106, 155152 (2022).[66] A. Greco, H. Yamase, and M. Bejas, Plasmon exci-tations in layered high-Tc cuprates, Phys. Rev. B 94,075139 (2016).[67] I. Bozovic, Plasmons in cuprate superconductors, Phys.Rev. B 42, 1969 (1990).[68] J. Fink, M. Knupfer, S. Atzkern, and M. Golden,Electronic correlations in solids, studied using electronenergy-loss spectroscopy, J. Electron Spectrosc. 117-118, 287 (2001).[69] A. Greco, H. Yamase, and M. Bejas, Origin of high-energy charge excitations observed by resonant inelasticx-ray scattering in cuprate superconductors, Commun.Phys. 2, 3 (2019).[70] A. Greco, H. Yamase, and M. Bejas, Close inspection ofplasmon excitations in cuprate superconductors, Phys.Rev. B 102, 024509 (2020).[71] R. S. Markiewicz and A. Bansil, Dispersion anomaliesinduced by the low-energy plasmon in the cuprates,Phys. Rev. B 75, 020508(R) (2007).[72] V. Z. Kresin and H. Morawitz, Layer plasmons and high-Tc superconductivity, Phys. Rev. B 37, 7854 (1988).[73] M. Fidrysiak and J. Spa lek, Unified theory of spin andcharge excitations in high-Tc cuprate superconductors:A quantitative comparison with experiment and inter-pretation, Phys. Rev. B 104, L020510 (2021).http://dx.doi.org/10.1038/nphys2041https://doi.org/10.1038/nmat3723https://doi.org/10.1103/PhysRevB.85.214527https://doi.org/10.1103/PhysRevB.85.214527https://doi.org/10.1103/PhysRevB.85.214528https://doi.org/10.1103/PhysRevLett.89.177003https://doi.org/10.1103/PhysRevLett.89.177003https://doi.org/10.1103/PhysRevLett.94.207003https://doi.org/10.1103/PhysRevLett.94.207003https://doi.org/10.1103/PhysRevLett.94.187002https://doi.org/10.1103/PhysRevB.72.224508https://doi.org/10.1103/PhysRevB.72.224508https://doi.org/10.1103/PhysRevB.83.075120https://doi.org/10.1103/PhysRevB.83.075120https://doi.org/10.1103/PhysRevB.87.104511https://doi.org/10.1103/PhysRevB.87.104511https://doi.org/10.1038/s41586-018-0648-3https://doi.org/10.1038/s41535-019-0205-9https://doi.org/10.1038/s41535-019-0205-9https://doi.org/10.1103/PhysRevLett.125.257002https://doi.org/10.1103/PhysRevLett.125.257002https://doi.org/10.1103/PhysRevB.105.235105https://doi.org/10.1103/PhysRevLett.129.047001https://link.aps.org/doi/10.1103/PhysRevB.8.1958http://www.sciencedirect.com/science/article/pii/0003491674903972https://doi.org/10.1088/0022-3719/8/16/014https://doi.org/10.1103/PhysRevResearch.4.023112https://doi.org/10.1103/PhysRevB.106.155152https://doi.org/10.1103/PhysRevB.94.075139https://doi.org/10.1103/PhysRevB.94.075139https://doi.org/10.1103/PhysRevB.42.1969https://doi.org/10.1103/PhysRevB.42.1969https://doi.org/https://doi.org/10.1016/S0368-2048(01)00254-7https://doi.org/https://doi.org/10.1016/S0368-2048(01)00254-7https://doi.org/10.1038/s42005-018-0099-zhttps://doi.org/10.1038/s42005-018-0099-zhttps://doi.org/10.1103/PhysRevB.102.024509https://doi.org/10.1103/PhysRevB.102.024509https://link.aps.org/doi/10.1103/PhysRevB.37.7854https://doi.org/10.1103/PhysRevB.104.L02051018[74] J. Levallois, M. K. Tran, D. Pouliot, C. N. Presura,L. H. Greene, J. N. Eckstein, J. Uccelli, E. Gian-nini, G. D. Gu, A. J. Leggett, and D. van der Marel,Temperature-Dependent Ellipsometry Measurements ofPartial Coulomb Energy in Superconducting Cuprates,Phys. Rev. X 6, 031027 (2016).[75] C. Falter and M. Klenner, Nonadiabatic and nonlo-cal electron-phonon interaction and phonon-plasmonmixing in the high-temperature superconductors, Phys.Rev. B 50, 9426 (1994).[76] T. Bauer and C. Falter, Impact of dynamical screeningon the phonon dynamics of metallic La2CuO4, Phys.Rev. B 80, 094525 (2009).[77] A. Bill, H. Morawitz, and V. Z. Kresin, Electronic collec-tive modes and superconductivity in layered conductors,Phys. Rev. B 68, 144519 (2003).[78] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink,E. Abrahams, and A. E. Ruckenstein, Phenomenologyof the normal state of Cu-O high-temperature supercon-ductors, Phys. Rev. Lett. 63, 1996 (1989).[79] P. W. Phillips, N. E. Hussey, and P. Abbamonte,Stranger than metals, Science 377, eabh4273 (2022).[80] C. M. Varma, Colloquium: Linear in temperature resis-tivity and associated mysteries including high temper-ature superconductivity, Rev. Mod. Phys. 92, 031001(2020).[81] J. Zaanen, Planckian dissipation, minimal viscosity andthe transport in cuprate strange metals, SciPost Phys.6, 061 (2019).[82] S. A. Hartnoll and A. P. Mackenzie, Colloquium:Planckian dissipation in metals, Rev. Mod. Phys. 94,041002 (2022).[83] D. Chowdhury, A. Georges, O. Parcollet, andS. Sachdev, Sachdev-ye-kitaev models and beyond:Window into non-fermi liquids, Rev. Mod. Phys. 94,035004 (2022).[84] A. Romero-Bermúdez, A. Krikun, K. Schalm, and J. Za-anen, Anomalous attenuation of plasmons in strangemetals and holography, Phys. Rev. B 99, 235149 (2019).[85] M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak,S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch,R. Zhong, G. D. Gu, and P. Abbamonte, Anomalousdensity fluctuations in a strange metal, P. Natl. Acad.Sci. U.S.A. 115, 5392 (2018).[86] A. A. Husain, M. Mitrano, M. S. Rak, S. Rubeck,B. Uchoa, K. March, C. Dwyer, J. Schneeloch,R. Zhong, G. D. Gu, and P. Abbamonte, Crossover ofCharge Fluctuations across the Strange Metal Phase Di-agram, Phys. Rev. X 9, 041062 (2019).[87] J. Fink, (2021), arXiv:2103.10268 [cond-mat.supr-con].[88] A. Husain, M. Mitrano, M. S. Rak, S. Rubeck, B. Uchoa,K. March, C. Dwyer, J. Schneeloch, R. Zhong, G. D. Gu,and P. Abbamonte, (2021), arXiv:2106.03301 [cond-mat.str-el].[89] H. Sato, A. Tsukada, M. Naito, and A. Matsuda,La2−xSrxCuOy epitaxial thin films (x = 0 to 2): Struc-ture, strain, and superconductivity, Phys. Rev. B 61,12447 (2000).[90] Y. Ando, S. Komiya, K. Segawa, S. Ono, and Y. Kurita,Electronic Phase Diagram of High-Tc Cuprate Super-conductors from a Mapping of the In-Plane ResistivityCurvature, Phys. Rev. Lett. 93, 267001 (2004).[91] I. Božović, X. He, J. Wu, and A. T. Bollinger, Depen-dence of the critical temperature in overdoped copperoxides on superfluid density, Nature 536, 309 (2016).[92] H. Takagi, T. Ido, S. Ishibashi, M. Uota, S. Uchida, andY. Tokura, Superconductor-to-nonsuperconductor tran-sition in (La1−xSrx)2CuO4 as investigated by transportand magnetic measurements, Phys. Rev. B 40, 2254(1989).[93] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. En-doh, Magnetic, transport, and optical properties ofmonolayer copper oxides, Rev. Mod. Phys. 70, 897(1998).[94] R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lip-scombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike,M. Nohara, H. Takagi, C. Proust, and N. E. Hussey,Anomalous Criticality in the Electrical Resistivity ofLa2−xSrxCuO4, Science 323, 603 (2009).[95] L. Sederholm, S. D. Conradson, T. H. Geballe, C.-Q. Jin, A. Gauzzi, E. Gilioli, M. Karppinen, andG. Baldinozzi, Extremely overdoped superconductingcuprates via high pressure oxygenation methods, Con-dens. Matter 6, 50 (2021).[96] I. Tanaka, J. Yamamoto, Y. Mori, H. Tanabe, M. Khan,and H. Kojima, Determination of the solubility ofalkaline-earth metals (Ca, Sr, Ba) into La2CuO4 by theSCFZ method, Physica C Supercond. 225, 185 (1994).[97] G. Kim, G. Christiani, G. Logvenov, S. Choi, H.-H.Kim, M. Minola, and B. Keimer, Selective formation ofapical oxygen vacancies in La2−xSrxCuO4, Phys. Rev.Materials 1, 054801 (2017).[98] D. Wang, J.-Q. Xu, H.-J. Zhang, and Q.-H. Wang,Anisotropic Scattering Caused by Apical Oxygen Va-cancies in Thin Films of Overdoped High-TemperatureCuprate Superconductors, Phys. Rev. Lett. 128, 137001(2022).[99] G. Kim, K. S. Rabinovich, A. V. Boris, A. N. Yaresko,Y. E. Suyolcu, Y.-M. Wu, P. A. van Aken, G. Chris-tiani, G. Logvenov, and B. Keimer, Optical con-ductivity and superconductivity in highly overdopedLa2−xCaxCuO4 thin films, Proc. Natl. Acad. Sci. U.S.A.118, e2106170118 (2021).[100] P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A.Sawatzky, and D. L. Feng, Spatially modulated ’Mot-tness’ in La2−xBaxCuO4, Nat. Phys. 1, 155 (2005).[101] K. Ishii, T. Tohyama, S. Asano, K. Sato, M. Fujita,S. Wakimoto, K. Tustsui, S. Sota, J. Miyawaki, H. Niwa,Y. Harada, J. Pelliciari, Y. Huang, T. Schmitt, Y. Ya-mamoto, and J. Mizuki, Observation of momentum-dependent charge excitations in hole-doped cuprates us-ing resonant inelastic x-ray scattering at the oxygen Kedge, Physical Review B 96, 115148 (2017).[102] K. Ishii, M. Kurooka, Y. Shimizu, M. Fujita,K. Yamada, and J. Mizuki, Charge Excitations inNd2−xCexCuO4 Observed with Resonant Inelastic X-ray Scattering: Comparison of Cu K-edge with Cu L3-edge, J. Phys. Soc. Japan 88, 075001 (2019).[103] C. J. Jia, C.-C. Chen, A. P. Sorini, B. Moritz, and T. P.Devereaux, Uncovering selective excitations using theresonant profile of indirect inelastic x-ray scattering incorrelated materials: observing two-magnon scatteringand relation to the dynamical structure factor, New J.Phys. 14, 113038 (2012).[104] D. S. Ellis, J. Kim, J. P. Hill, S. Wakimoto, R. J. Birge-neau, Y. Shvyd’ko, D. Casa, T. Gog, K. Ishii, K. Ikeuchi,A. Paramekanti, and Y.-J. Kim, Magnetic nature of the500 meV peak in La2−xSrxCuO4 observed with resonanthttps://doi.org/10.1103/PhysRevX.6.031027https://doi.org/10.1103/PhysRevB.50.9426https://doi.org/10.1103/PhysRevB.50.9426https://doi.org/10.1103/PhysRevB.80.094525https://doi.org/10.1103/PhysRevB.80.094525https://link.aps.org/doi/10.1103/PhysRevB.68.144519https://doi.org/10.1103/PhysRevLett.63.1996https://www.science.org/doi/10.1126/science.abh4273https://doi.org/10.1103/RevModPhys.92.031001https://doi.org/10.1103/RevModPhys.92.031001https://doi.org/10.21468/SciPostPhys.6.5.061https://doi.org/10.21468/SciPostPhys.6.5.061https://doi.org/10.1103/RevModPhys.94.041002https://doi.org/10.1103/RevModPhys.94.041002https://doi.org/10.1103/RevModPhys.94.035004https://doi.org/10.1103/RevModPhys.94.035004https://doi.org/10.1103/PhysRevB.99.235149https://doi.org/10.1073/pnas.1721495115https://doi.org/10.1073/pnas.1721495115https://link.aps.org/doi/10.1103/PhysRevX.9.041062https://arxiv.org/abs/2103.10268https://arxiv.org/abs/2106.03301https://arxiv.org/abs/2106.03301https://doi.org/10.1103/PhysRevB.61.12447https://doi.org/10.1103/PhysRevB.61.12447https://doi.org/10.1103/PhysRevLett.93.267001https://doi.org/10.1038/nature19061https://doi.org/10.1103/PhysRevB.40.2254https://doi.org/10.1103/PhysRevB.40.2254https://doi.org/10.1103/RevModPhys.70.897https://doi.org/10.1103/RevModPhys.70.897https://doi.org/10.1126/science.1165015https://www.mdpi.com/2410-3896/6/4/50https://www.mdpi.com/2410-3896/6/4/50https://doi.org/10.1016/0921-4534(94)90344-1https://doi.org/10.1103/PhysRevMaterials.1.054801https://doi.org/10.1103/PhysRevMaterials.1.054801https://doi.org/10.1103/PhysRevLett.128.137001https://doi.org/10.1103/PhysRevLett.128.137001https://pnas.org/doi/full/10.1073/pnas.2106170118https://pnas.org/doi/full/10.1073/pnas.2106170118http://www.nature.com/articles/nphys178https://doi.org/10.1103/PhysRevB.96.115148https://journals.jps.jp/doi/10.7566/JPSJ.88.075001https://doi.org/10.1088/1367-2630/14/11/113038https://doi.org/10.1088/1367-2630/14/11/11303819inelastic x-ray scattering at the Cu K-edge, Phys. Rev.B 81, 085124 (2010).[105] G. D. Mahan, Many-Particle Physics, 2nd ed. (Plenum,New York, 1990).[106] Y. E. Suyolcu, G. Christiani, P. A. van Aken, andG. Logvenov, Design of Complex Oxide Interfaces byOxide Molecular Beam Epitaxy, J. Supercond. Nov.Magn. 33, 107 (2020).[107] Y. E. Suyolcu, G. Christiani, P. T. Gemperline, S. R.Provence, A. Bussmann-Holder, R. B. Comes, P. A. vanAken, and G. Logvenov, Engineering ordered arrange-ments of oxygen vacancies at the surface of supercon-ducting La2CuO4 thin films, J. Vac. Sci. Technol. A 40,013214 (2022).[108] A. Narduzzo, G. Albert, M. M. J. French, N. Mangko-rntong, M. Nohara, H. Takagi, and N. E. Hussey, Vi-olation of the isotropic mean free path approximationfor overdoped La2−xSrxCuO4, Phys. Rev. B 77, 220502(2008).[109] N. R. Lee-Hone, V. Mishra, D. M. Broun, and P. J.Hirschfeld, Optical conductivity of overdoped cupratesuperconductors: Application to La2−xSrxCuO4, Phys.Rev. B 98, 054506 (2018).[110] F. Mahmood, X. He, I. Božović, and N. P. Armitage,Locating the Missing Superconducting Electrons in theOverdoped Cuprates La2−xSrxCuO4, Phys. Rev. Lett.122, 027003 (2019).[111] H. U. Özdemir, V. Mishra, N. R. Lee-Hone, X. Kong,T. Berlijn, D. M. Broun, and P. J. Hirschfeld, Effectof realistic out-of-plane dopant potentials on the super-fluid density of overdoped cuprates, Phys. Rev. B 106,184510 (2022).[112] A. vom Felde, J. Sprösser-Prou, and J. Fink, Valence-electron excitations in the alkali metals, Phys. Rev. B40, 10181 (1989).[113] L. Zinni, M. Bejas, H. Yamase, and A. Greco, Low-energy plasmon excitations in infinite-layer nickelates,Phys. Rev. B 107, 014503 (2023).[114] K.-J. Zhou, A. Walters, M. Garcia-Fernandez, T. Rice,M. Hand, A. Nag, J. Li, S. Agrestini, P. Garland,H. Wang, S. Alcock, I. Nistea, B. Nutter, N. Rubies,G. Knap, M. Gaughran, F. Yuan, P. Chang, J. Em-mins, and G. Howell, I21: an advanced high-resolutionresonant inelastic x-ray scattering beamline at diamondlight source, J. Synchrotron Rad. 29, 563 (2022).[115] K. Lieutenant, T. Hofmann, C. Schulz, M. V. Yablon-skikh, K. Habicht, and E. F. Aziz, Design concept ofthe high-resolution end-station PEAXIS at BESSY II:Wide-Q-range RIXS and XPS measurements on solids,solutions, and interfaces, J. Electron Spectros. Relat.Phenomena 210, 54 (2016).[116] C. Schulz, K. Lieutenant, J. Xiao, T. Hofmann,D. Wong, and K. Habicht, Characterization of the softX-ray spectrometer PEAXIS at BESSY II, J. Syn-chrotron Radiat. 27, 238 (2020).[117] T. Thio, T. R. Thurston, N. W. Preyer, P. J. Picone,M. A. Kastner, H. P. Jenssen, D. R. Gabbe, C. Y. Chen,R. J. Birgeneau, and A. Aharony, Antisymmetric ex-change and its influence on the magnetic structure andconductivity of La2CuO4, Phys. Rev. B 38, 905 (1988).[118] F. Becca, M. Tarquini, M. Grilli, and C. Di Castro,Charge-density waves and superconductivity as an al-ternative to phase separation in the infinite-U Hubbard-Holstein model, Phys. Rev. B 54, 12443 (1996).https://doi.org/10.1103/PhysRevB.81.085124https://doi.org/10.1103/PhysRevB.81.085124https://doi.org/10.1007/s10948-019-05285-4https://doi.org/10.1007/s10948-019-05285-4https://doi.org/10.1116/6.0001473https://doi.org/10.1116/6.0001473https://doi.org/10.1103/PhysRevB.77.220502https://doi.org/10.1103/PhysRevB.77.220502https://doi.org/10.1103/PhysRevB.98.054506https://doi.org/10.1103/PhysRevB.98.054506https://doi.org/10.1103/PhysRevLett.122.027003https://doi.org/10.1103/PhysRevLett.122.027003https://doi.org/10.1103/PhysRevB.106.184510https://doi.org/10.1103/PhysRevB.106.184510https://doi.org/10.1103/PhysRevB.40.10181https://doi.org/10.1103/PhysRevB.40.10181https://doi.org/10.1103/PhysRevB.107.014503https://doi.org/10.1107/S1600577522000601https://doi.org/10.1016/j.elspec.2015.08.009https://doi.org/10.1016/j.elspec.2015.08.009https://doi.org/10.1107/S1600577519014887https://doi.org/10.1107/S1600577519014887https://doi.org/10.1103/PhysRevB.38.905https://doi.org/10.1103/PhysRevB.54.12443 Evolution of plasmon excitations across the phase diagram of the cuprate superconductor La2-xSrxCuO4 Abstract Introduction Results Electronic transport X-ray absorption at the O K-edge RIXS at the O K-edge RIXS at the Cu K-edge Temperature dependence of the plasmon excitation Theoretical modeling of the doping dependence Discussion and Conclusion Acknowledgments Experimental details RIXS raw data and fits Comparison to literature Theoretical details Summary of the RPA formalism References