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M. Bejas, V. Zimmermann, D. Betto, T. D. Boyko, R. J. Green, T. Loew, N. B. Brookes, G. Cristiani, G. Logvenov, M. Minola, B. Keimer, [H. Yamase](https://orcid.org/0000-0003-0328-5657), A. Greco, M. Hepting

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[Plasmon dispersion in bilayer cuprate superconductors](https://mdr.nims.go.jp/datasets/f28622b6-c4ea-4366-9197-efdd7b126acd)

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Plasmon dispersion in bilayer cuprate superconductorsPHYSICAL REVIEW B 109, 144516 (2024)Plasmon dispersion in bilayer cuprate superconductorsM. Bejas ,1 V. Zimmermann ,2 D. Betto,3 T. D. Boyko ,4 R. J. Green ,5,6 T. Loew,2 N. B. Brookes ,3 G. Cristiani,2G. Logvenov,2 M. Minola ,2 B. Keimer ,2 H. Yamase ,7,* A. Greco ,1,† and M. Hepting 2,‡1Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física de Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina2Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany3European Synchrotron Radiation Facility, B.P. 220, 38043 Grenoble, France4Canadian Light Source, Saskatoon, Saskatchewan, Canada S7N 2V35Department of Physics & Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A26Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z17Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan(Received 2 November 2023; revised 27 March 2024; accepted 28 March 2024; published 16 April 2024)The essential building blocks of cuprate superconductors are two-dimensional CuO2 sheets interspersed withcharge reservoir layers. In bilayer cuprates, two closely spaced CuO2 sheets are separated by a larger distancefrom the subsequent pair in the next unit cell. In contrast to single-layer cuprates, prior theoretical work onbilayer systems has predicted two distinct acoustic plasmon bands for a given out-of-plane momentum transfer.Here we report random phase approximation (RPA) calculations for bilayer systems which corroborate theexistence of two distinct plasmon bands. We find that the intensity of the lower-energy band is negligibly smallin most parts of the Brillouin zone, whereas the higher-energy band carries significant spectral weight. Wealso present resonant inelastic x-ray scattering (RIXS) experiments at the O K-edge on the bilayer cuprateY0.85Ca0.15Ba2Cu3O7 (Ca-YBCO), which show only one dispersive plasmon branch, in agreement with theRPA calculations. In addition, the RPA results indicate that the dispersion of the higher-energy plasmon bandin Ca-YBCO is not strictly acoustic but exhibits a substantial energy gap of approximately 250 meV at thetwo-dimensional Brillouin zone center.DOI: 10.1103/PhysRevB.109.144516I. INTRODUCTIONCuprate high-temperature superconductors have garnered aprominent position in modern condensed matter research [1].In addition to superconductivity, a broad range of phenomena,including the pseudogap, spin and charge density wave orders,and the strange metal phase [2–4], occur when charge carriersare introduced into the CuO2 sheets, which are periodicallystacked along the z direction.The electrodynamics of such a layered configuration drawsclose parallels to that of the layered electron gas (LEG) model[5–7], which was predicted to host unconventional plasmonexcitations [8,9]. Specifically, the plasmon spectrum in a LEGsystem consists of one optical and several acoustic modes witha characteristic dispersion as a function of both the in-plane(q‖) and the out-of-plane momentum (qz). For qz = 0, theplasmon dispersion corresponds to the optical branch for all*yamase.hiroyuki@nims.go.jp†agreco@fceia.unr.edu.ar‡hepting@fkf.mpg.dePublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI. Openaccess publication funded by Max Planck Society.values of q‖. For a finite qz, the plasmon dispersion followsthe acoustic branch, where the lowest-energy mode is reachedwhen qz = π . This distinct three-dimensional (3D) depen-dence of the plasmon dispersion in a LEG sets it apart fromthe√q dispersion in a purely two-dimensional (2D) systemand the q2 dependence in an isotropic 3D metal.To calculate the details of the corresponding plasmon dis-persion in cuprates, several computational methods have beenemployed [10–17], including the random phase approxima-tion (RPA) [5,8,9,18,19] and a large-N theory for the layeredt − J model with long-range Coulomb interaction (t − J −V model) [20–24]. Yet, the majority of theoretical studiesconsidered cuprates with equidistantly stacked CuO2 sheets[Fig. 1(a)], whereas the cuprates with the highest supercon-ducting transition temperatures (Tc) are multilayer systems[1]. Specifically, in a bilayer system, two closely spaced CuO2planes are separated by a substantially larger distance from thenext set of planes in the subsequent unit cell [Fig. 1(b)]. Forsuch bilayer systems, the generalized RPA approach by Griffinand Pindor predicted that for a given qz the acoustic plasmonmodes separate into high- (ω+) and low-energy (ω−) bands[25]. This is markedly distinct from a system with equidis-tantly spaced planes [Fig. 1(a)], where one band comprisesall acoustic plasmon modes. Furthermore, the bilayer RPAapproach [25] indicated that, at a given q‖, the plasmon modesof the ω− band are essentially degenerate and do not dependon the value of qz. By contrast, the plasmon modes in the ω+2469-9950/2024/109(14)/144516(8) 144516-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-4254-0622https://orcid.org/0000-0001-5378-410Xhttps://orcid.org/0000-0002-7221-5778https://orcid.org/0000-0003-1849-1576https://orcid.org/0000-0002-1342-9530https://orcid.org/0000-0003-4084-0664https://orcid.org/0000-0001-5220-9023https://orcid.org/0000-0003-0328-5657https://orcid.org/0000-0001-5958-5080https://orcid.org/0000-0002-5824-8901https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.109.144516&domain=pdf&date_stamp=2024-04-16https://doi.org/10.1103/PhysRevB.109.144516https://creativecommons.org/licenses/by/4.0/M. BEJAS et al. PHYSICAL REVIEW B 109, 144516 (2024)CuBaYCaOt'ztztza bcdd(c)(b)(a)FIG. 1. (a) Sketch of a system with equally spaced conductingsheets (red shaded planes). Hopping between the sheets is denotedby tz. The black boxes represent unit cells with the lattice param-eters a, b, and c. The distance between the planes is denoted byd . (b) Sketch of a bilayer system that contains two closely spacedsheets within a unit cell. The intrabilayer hopping is denoted by tz andthe interbilayer hopping by t ′z. The distance between the two closelyspaced planes is denoted by d . (c) Schematic of the crystal structureof the bilayer cuprate Ca-YBCO. The red shaded planes are the CuO2sheets.band exhibit a qz-dependence that is qualitatively similar tothat of systems with equidistant planes.Similarly to the majority of theoretical studies, pre-ceding resonant inelastic x-ray scattering (RIXS) exper-iments investigating plasmons in cuprates focused oncompounds with equally spaced CuO2 sheets, including(La, Nd)2−xCexCuO4 [10,26,27], La2−xSrxCuO4 [23,28–30],and Bi2Sr0.16La0.4CuO6+δ (Bi-2201) [23]. RIXS is a partic-ularly versatile tool to study the plasmon phenomenologyin cuprates, because it provides high momentum resolutionin all three spatial directions along with polarization analy-sis, which facilitated the identification of acoustic plasmons[10,23,28]. In addition, recent RIXS experiments revealed thatthe nominally acoustic plasmon branch in cuprates deviatesfrom a pure acoustic dispersion due to an energy gap atthe 2D Brillouin zone (BZ) center [24] when charge carrierhopping (tz) between the CuO2 planes is present [20]. Despitethese advances, RIXS studies have yet to explore the plasmondispersion in multilayer cuprates, including bilayer systemssuch as YBa2Cu3O6+δ (YBCO) and Bi2Sr2CaCu2O8+δ (Bi-2212). Specifically, the impact of the relatively large interlayerhopping within a bilayer unit on the plasmon spectrum aswell as the anticipated coexistence of ω+ and ω− plasmonbands [25] have remained open questions, fundamental to acomprehensive description of their electrodynamics.In this work, we adapt an RPA framework for bilayersystems beyond that of Ref. [25], incorporating a cuprate-specific electronic band dispersion and intrabilayer hoppingtz. This enables us to evaluate the relative spectral intensitiesof plasmon bands in bilayer cuprates. Utilizing RIXS at theO K edge on the bilayer cuprate Y0.85Ca0.15Ba2Cu3O7 (Ca-YBCO) [Fig. 1(c)], we identify a single dispersive plasmonmode. This observation is compatible with our RPA analysis,which predicts that one of the two plasmon bands carriesalmost vanishing spectral weight. Furthermore, the experi-mentally observed plasmon dispersion is captured when usingcuprate-typical effective interaction parameters in the RPA,while an extrapolation of the plasmon dispersion to the 2DBZ center suggests the presence of a gap of the acoustic-likebranch of approximately 250 meV.II. RESULTSFor the RPA calculations, we consider a bilayer setting[Fig. 1(b)] under the presence of a long range Coulomb in-teraction. In order to extract fundamental characteristics ofbilayer systems without compromising generality, we startwith a generic model. The Hamiltonian is given by H =H0 + HI , with the single particle HamiltonianH0 =∑k,σ,α,βc†k,σ,α ĥαβ (k)ck,σ,β , (1)where ĥαβ are the elements of the matrixĥ(k) =(ε‖k ε⊥kε⊥ ∗k ε‖k). (2)Here ε‖ is the in-plane dispersion,ε‖k = −2t (cos kx + cos ky), (3)which contains the nearest neighbor hopping t , and ε⊥ is theout-of-plane dispersion,ε⊥k = −tzeikzd , (4)which contains only the intrabilayer hopping tz, while omit-ting the interbilayer hopping t ′z [Fig. 1(b)]. The electronannihilation operator is denoted by ck,σ,α , where σ is the spinindex and α, β = 1, 2 indicates each plane in the unit cell.The symbol “∗” means complex conjugate. In the following,the in-plane hopping t is considered as the energy unit.The Coulomb interacting Hamiltonian is given byHI =∑q,α,βnα (q)Vαβ (q)nβ (−q), (5)where nα (q) is the electron density at a given plane α, and thetwo forms of the Coulomb interactions areVαα (q) = Vc|q‖|(sinh |q‖|ccosh |q‖|c − cos qzc), (6)andVαβ (q) = Vc|q‖|(sinh |q‖|(c − d ) + e−iqzc sinh |q‖|dcosh |q‖|c − cos qzc)e−iqzd ,(7)with Vαα = Vββ and Vαβ = V ∗βα (α �= β) [25]. Note that, incontrast to a simplified treatment of the conducting sheetsin a 2D LEG picture in Ref. [25], our approach incorporatestight-binding dispersions [see Eqs. (3) and (4)], which allowfor the accommodation of bandstructure effects.Within RPA, the dressed charge susceptibility χ̂ (q, iωl ) isgiven byχ̂ (q, iωl ) = [I − χ̂ (0)(q, iωl )V̂ (q)]−1χ̂ (0)(q, iωl ), (8)where χ̂ (0) is the bare charge susceptibility. Here, χ̂ (0) and χ̂are 2 × 2 matrices. The dielectric function ε(q, iωl ) can becalculated as the determinant of the matrix [I − χ̂ (0)V̂ ]. In the144516-2PLASMON DISPERSION IN BILAYER CUPRATE … PHYSICAL REVIEW B 109, 144516 (2024)ω / t(q, 0) / π 0 2 4 6 8 100 0.1 0.2 0.3 0.410-410-310-210-1χ" c (q, ω)qz = 0ω+ω-(a)(q, 0) / π0 0.1 0.2 0.3 0.4qz = 0.2 πω+ω-(b)(q, 0) / π0 0.1 0.2 0.3 0.4qz = πω+ω-(c)qz / π0 0.2 0.4 0.6 0.8 1qx = qy = 0.1πω+ω-(d)FIG. 2. Computed intensity maps of the RPA charge response for a generic bilayer system. (a-c) Intensity maps in the q‖ − ω plane forqz = 0, 0.2π , and π , respectively. Red dashed lines correspond to the zeros of the dielectric function (ω+ and ω−). White dotted lines areguides to the eye marking the upper bound above which the spectral weight of the particle-hole continuum becomes insignificant. (d) Intensitymap in the qz − ω plane for qx = qy = 0.1π .above expressions, the bare susceptibility χ̂ (0) is defined asthe convolution of two Green’s functionsχ(0)αβ (q, iωl ) = 2TNs∑k,iνnG(0)αβ (k, iνn)G(0)βα (k + q, iνn + iωl ),(9a)withĜ(0)(k, iνn) =(iνn − (ε‖k − μ) −ε⊥k−ε⊥ ∗k iνn − (ε‖k − μ))−1(9b)the 2 × 2 bare electronic Green’s function. ωl (νn) is a bosonic(fermionic) Matsubara frequency. For the momentum q‖ (qz),we use the units of the in-plane (out-of-plane) lattice constanta (c). T (Ns) is the temperature (number of sites).We compute the 2 × 2 dressed RPA charge susceptibility,and execute the analytical continuation iωn = ω + i�, where� is in principle infinitesimally small. We then take the imag-inary part χ ′′αβ (q, ω) = −Imχαβ (q, ω + i�) of the chargesusceptibility. The full charge response χ ′′c (q, ω) isχ ′′c (q, ω) =∑α,βχ ′′αβ (q, ω). (10)We employ the following generic parameter set: c = 3aand d = c/3 for the structural parameters, tz/t = 0.1 for thehopping, and the temperature T = 0. The chemical potentialμ and the broadening � are set to μ/t = −1 and �/t = 0.1,respectively. The Coulomb repulsion strength is parameter-ized by Vc. For the purpose of illustration, we use the valueVc/t = 50, which allows us to emphasize the intensity of bothω+ and ω− modes.Figure 2 shows the computed intensity maps of the chargeresponse, i.e., the imaginary part of the charge susceptibility.Panels (a)–(c) present results within the q‖ − ω plane for qz =0, 0.2π , and π , respectively. Red dashed lines in each panelindicate the two zeros of the dielectric function, ω+ and ω−,which are situated above the particle-hole continuum (whitedotted lines). Note that a logarithmic scale was used for theintensities in the maps to accentuate the contrast between thespectral weights of ω+, ω−, and the particle-hole continuum.Interestingly, although both modes are zeros of the dielectricfunction, the spectral weight of the ω− mode is much lowerthan that of the ω+ mode. In particular, for qz = 0, the spectralweight of ω− is exactly zero [Fig. 2(a)] for all q‖ values.Conversely, ω+ corresponds to the conventional optical plas-mon branch, experiencing a decline in spectral intensity asq‖ decreases, reaching exactly zero intensity at q‖ = (0, 0).For qz = 0.2π , some subtle spectral weight emerges for ω−at large q‖ values [Fig. 2(b)]. For qz = π , the spectral weightof ω− becomes more discernible across the entire q‖ range[Fig. 2(c)]. In addition, we note that for finite qz values[Figs. 2(b) and 2(c)], an energy gap is present for the ω+ bandat q‖ = (0, 0), whereas ω− approaches zero energy. Hence, incontrast to the purely acoustic plasmon bands in Ref. [25], wefind that a gap opens for the ω+ band as a direct consequenceof the inclusion of a finite tz in our model.For a more detailed analysis of the evolution of the plasmonspectral weight along the out-of-plane direction, we plot theintensity map within the qz − ω plane for qx = qy = 0.1πin Fig. 2(d). Notably, the dispersion of ω− appears to bealmost independent of qz, whereas the behavior of ω+ witha maximum at qz = 0 and a minimum at qz = π is reminis-cent of that of plasmons in systems with equidistant sheets[10,21,23]. In addition, we find that the spectral weight ofω− decreases strongly with decreasing qz and is nearly in-visible for small qz. In comparison, in layered systems withequidistantly stacked planes, such as single-layer cuprates,the qz = π acoustic branch, corresponding to an out-of-phasecharge distribution, does not exhibit zero intensity [10,21,23].Conversely, a configuration comprising solely two isolatedplanes is characterized by the presence of only two plasmonbranches, corresponding to in-phase and out-of-phase chargedistributions [31], with the latter exhibiting zero spectral in-tensity. Systems with periodically stacked bilayers, such asbilayer cuprates, present a combination of elements fromboth previous cases. As a result, the ω− mode and generallyconfigurations with partial out-of-phase charge distributionsmanifest diminished but nonzero spectral intensities, and zerointensity at qz = 0.To assess the predictive power of our theoretical model,we perform RIXS experiments in resonance to the O K edge,known to provide significant plasmon intensity for hole-dopedcuprates [23,28,29]. Specifically, we carry out RIXS measure-ments on a detwinned Ca-YBCO single-crystal (p = 0.21)[32] with a superconducting transition temperature Tc ≈ 75K.144516-3M. BEJAS et al. PHYSICAL REVIEW B 109, 144516 (2024)In addition to this overdoped variant of YBCO, we also con-duct O K-edge RIXS measurements on an optimally dopedYBCO film (without Ca substitution), which yield qualita-tively comparable results (see Appendix A).Figure 1(c) shows the crystallographic unit cell of Ca-YBCO, with the lattice constants a = 3.89 Å, b = 3.88 Å,and c = 11.68 Å. The spacing between the two CuO2 planes(intrabilayer spacing) is d = 3.36 Å, and the distance from thecenter of the bilayer to the center of the next pair of planesalong the c-axis direction (interbilayer spacing) is 11.68 Å(i.e., equivalent to the c-axis lattice constant). The RIXSspectra were collected with high energy resolution (
E ≈27 meV) at T = 20K at the ID32 beamline of the ESRF [33].A similar scattering geometry as in Ref. [23] was employed,with the b and c axes of Ca-YBCO lying in the scatteringplane and incident photons linearly polarized perpendicularto the scattering plane (σ polarization). The scattering anglewas varied by continuous rotation of the RIXS spectrometerarm, which in combination with a rotation of the sampleenabled for the variation of the in-plane (q‖) and out-of-planemomentum transfer (qz) independently of each other.Figure 3(a) shows the XAS signal of Ca-YBCO across thenear-edge fine structure of the O K-edge. A prominent peakfeature emerges around 528.5 eV, similarly to the hole-peakreported in the O K-edge XAS of various other doped cuprates[34–38]. The hole-peak in cuprates is associated with theZhang-Rice singlet states, corresponding to plaquettes of hy-bridized states between copper and oxygen ions. In previousRIXS experiments on hole-doped cuprates with equidistantCuO2 sheets, dispersive plasmon excitations were detectedfor incident photon energies coinciding with the energy of thehole-peak in the O K-edge XAS [23,28,29].Figure 3(b) presents the RIXS spectra for various momen-tum transfers, acquired with an incident photon energy of528.5 eV. The momentum transfer is denoted by (H, K, L) inreciprocal lattice units (2π/a, 2π/b, 2π/c). Throughout theexperiment, the out-of-plane momentum is fixed to L = 0.9,while the in-plane momentum transfer is varied along the Kdirection, although we note that a qualitatively similar plas-mon dispersion can be expected for momentum transfer alongthe H direction of Ca-YBCO. We fit the RIXS spectra by thesum of the elastic line at zero-energy loss and several dampedharmonic oscillator functions for the inelastic features. Inaddition, intense peaks from fluorescence and dd-excitationsemerge beyond 2 eV energy loss, whose tails extend down to1 eV and below, and are captured by an exponential functionin our fits. Further details of the fitting procedure are given inAppendix B.Notably, the RIXS spectra exhibit a peak that dispersesapproximately from 350 to 600 meV when the momentumcomponent K increases from 0.02 to 0.1 [Fig. 3(b)]. Sucha rapid dispersion within a small variation of the in-planemomentum is reminiscent of that of plasmons in othercuprates [10,23,24,26–28]. In contrast, magnetic excitiationsin doped cuprates, such as bi-paramagnons, were found tobe (almost) nondispersive in O K-edge RIXS [23,28,39,40],and the bandwidth of the dispersion of paramagnons in CuL-edge RIXS is typically less than 300 meV. Moreover, theenergy scale of the dispersive peak in Fig. 3(b) is situated farabove that of phonons in cuprates, which is typically below(a)(b)K=0.10.08(0,K,0.9)T=20 K σ-pol.0.060.040.02RIXS intensity (arb. units)1.5 1.0 0.5 0.0Energy loss (eV)datafitplasmonXAS intensity(arb. units)534533532531530529528527526Photon energy (eV)FIG. 3. (a) XAS of Ca-YBCO across the O K edge measuredwith σ -polarized photons. The arrow indicates the incident photonenergy (528.5 eV) used for the RIXS experiment. (b) RIXS spectraof Ca-YBCO measured at various momenta along the K direction,while H and L were fixed to 0 and 0.9, respectively. The fit (blackline) to the experimental data (blue symbols) includes the plasmonpeak (orange peak profile) and other contributions (not shown here)that are described in Appendix B. Curves for different momenta areoffset in the vertical direction for clarity.100 meV. Hence, we tentatively assign the observed featureto a plasmon.At first sight, our observation of a single dispersive plas-mon mode in bilayer Ca-YBCO appears to conflict with theemergence of ω+ and ω− bands on equal footing [25], butmight be compatible with our RPA model predicting a van-ishingly small spectral weight of the ω− band. Thus, forfurther scrutiny, we next apply our bilayer RPA approach tofit the plasmon dispersion observed in the RIXS experimenton Ca-YBCO. To this end, we include the next-nearest neigh-bor hopping t ′ in the in-plane dispersion ε‖k = −2t (cos kx +cos ky) − 4t ′ cos kx cos ky, such that t ′/t = −0.3 with t =0.35eV, as discussed in Ref. [41] for YBCO. Furthermore,we choose Vc/t = 25 and tz/t = 0.06, along with the out-of-plane dispersion ε⊥k = −tz(cos(kx ) − cos(ky))2eikzd , whichwas proposed for YBCO in Ref. [41]. In cuprates, interbilayer144516-4PLASMON DISPERSION IN BILAYER CUPRATE … PHYSICAL REVIEW B 109, 144516 (2024)plasmon energy [eV]K [2π / a]0 0.2 0.4 0.6 0.80 0.02 0.04 0.06 0.08 0.10 0.1210-410-310-2>10-1χ" c (q , ω)qz = 0.2 πω+ω-FIG. 4. Computed intensity maps of the RPA charge responseusing renormalized effective parameters optimized for the bilayercuprate Ca-YBCO. The dispersion is along the K direction for qz =0.2π . The superimposed red symbols are the plasmon energies forCa-YBCO extracted from fits to the RIXS data [see Fig. 3(b)].hopping [Fig. 1(b)] is typically very small (t ′z   tz) [41], andis therefore not considered in the following. The charge carrierdoping is set to δ = 0.21, in accord with the Ca-YBCO sampleof the experiment. Since YBCO is a strongly correlated sys-tem, correlations are expected to affect the bare parameters.As discussed in Ref. [20], taking into account these renor-malization effects is important for an accurate description ofthe plasmon dispersions reported in RIXS experiments oncuprates [22–24,29], prompting only the renormalization ofthe bare parameters t, t ′, and tz by a factor δ. Note that theslope and the bandwidth of the computed plasmon dispersionare generally not dictated by a single parameter alone, butresult from a combination of the hopping parameters and theparameter Vc.Figure 4 shows the computed intensity map along theK direction for qz = 0.2π together with the experimentallyextracted plasmon energies (red symbols) at L = 0.9 (qz =1.8π ). We assume that this out-of-plane momentum employedin the experiment mirrors the essential features of the plasmondispersion at qz = 0.2π in the first BZ. For completeness, wealso present an intensity map computed for bare Ca-YBCOparameters without renormalization in Appendix C, yieldingqualitatively similar results. The dispersion of the ω+ mode(upper red dashed line) in Fig. 4 agrees remarkably well withthe RIXS experiment, corroborating our previous assignmentof the dispersive mode to a plasmon excitation. At the 2D BZcenter, the computed ω+ dispersion exhibits an energy gap ofapproximately 250 meV. In contrast, the computed dispersionof the ω− mode (bottom red dashed line) is essentially gapless,and its marginal spectral weight is almost indiscernible inthe color map in spite of a logarithmic intensity scale. Thissuggests that at least for L = 0.9 the ω− mode is below thedetection limit of RIXS experiments, whereas the ω+ modecan be prominently observed.III. DISCUSSION AND CONCLUSIONThe results of our study have several implications. First,they underscore the distinct nature of charge excitations inbilayer systems, as already pointed out in earlier works inves-tigating the zeros of the dielectric function [25,31]. However,while our RPA calculations corroborate that the emergence ofω+ and ω− bands is a general characteristic of bilayer systems,we have revealed that the spectral weight of the ω− band isnegligibly small throughout most parts of the BZ. This alsorationalizes why only one dispersive plasmon band (ω+) isobserved in our RIXS data on bilayer Ca-YBCO.Another insight from our study is the remarkably largeplasmon gap of about 250 meV at the center of the BZ. Thisvalue of the gap in bilayer Ca-YBCO is possibly attributableto the large intrabilayer hopping tz/t = 0.06, which is consis-tent with the magnitude of the hopping discussed in Ref. [42].Despite our model’s ability to quantitatively and quali-tatively describe the observed plasmon mode in the bilayercuprate Ca-YBCO, there remain several open areas for fu-ture research. For instance, although the predicted spectralweight of the ω− band is diminishing, it might be detectablewith RIXS for momenta close to qz = π and for large q‖values, where its maximum intensity is expected accordingto Figs. 2(c) and 2(d). A telltale signature to discern be-tween the ω+ and ω− bands in this part of the BZ mightbe their dispersive versus nondispersive character in a RIXSscan along the qz direction. Furthermore, since cuprates arestrongly correlated systems, implementing the comprehensivet − J − V model calculation for the bilayer lattice includingthe intra- and interlayer hoppings should be a considerationfor future work.ACKNOWLEDGMENTSWe thank C. Falter for fruitful discussions and A.P. Schny-der for critical reading of the manuscript. A.G. acknowledgesthe Max Planck Institute for Solid State Research in Stuttgartfor hospitality and financial support. H.Y. was supported byJSPS KAKENHI Grant No. JP20H01856 and World PremierInternational Research Center Initiative (WPI), MEXT, Japan.Parts of the results presented in this work were obtained byusing the facilities of the CCT-Rosario Computational Cen-ter, member of the High Performance Computing NationalSystem (SNCAD, MincyT-Argentina). Part of the researchdescribed in this paper was performed at the Canadian LightSource, a national research facility of the University ofSaskatchewan, which is supported by the Canada Foundationfor Innovation (CFI), the Natural Sciences and EngineeringResearch Council (NSERC), the National Research Council(NRC), the Canadian Institutes of Health Research (CIHR),the Government of Saskatchewan, and the University ofSaskatchewan.APPENDIX A: COMPLEMENTARY RIXSMEASUREMENTSIn addition to the RIXS experiment on the Ca-YBCO sin-gle crystal, we investigated plasmon excitations in an YBCOthin film. The epitaxial film with a thickness of approximately50 nm was grown by pulsed laser deposition (PLD) on a(001) oriented SrTiO3 substrate. The lattice parameters werea, b = 3.87 Å and c = 11.72 Å. After the growth, the YBCOfilm was annealed in oxygen atmosphere in order to achievefull oxygenation, corresponding to a hole-doping of p = 0.19.The measured Tc was 83 K.The O K edge RIXS measurements were performed at theREIXS beamline of the Canadian Light Source (CLS). The144516-5M. BEJAS et al. PHYSICAL REVIEW B 109, 144516 (2024)(a)(b)YBCOYBCOT=300 K σ-pol.XAS intensity(arb. units)534533532531530529528527526Photon energy (eV)RIXS intensity (arb. units)1.5 1.0 0.5 0.0Energy loss (eV)(0.04, 0, 0.69)(0.08, 0, 0.66)(0.12, 0, 0.60)(0.16, 0, 0.51)FIG. 5. (a) XAS of the YBCO film across the O K edge. Thearrow indicates the incident photon energy (527.32 eV) used for theRIXS experiment. (b) RIXS spectra of the YBCO film measured atvarious momenta along the H direction between H = 0.04 and 0.16.Curves for different momenta are offset in the vertical direction forclarity.XAS data in Fig. 5(a) were taken with σ -polarized photons atan incident angle θ = 35◦ in partial fluorescence yield usinga silicon drift detector, collecting only the O Kα emissionline. The RIXS spectra in Fig. 5(b) were collected at 300 Kusing a Rowland circle spectrometer with a combined energyresolution 
E ∼ 190 meV and linearly polarized photons (σpolarization). The c and the a/b axes of the twinned film werelying in the scattering plane. The scattering angle was keptfixed at 90◦, while the angle θ between the sample surface andthe incident x rays was varied for momentum dependent mea-surements. In this scattering configuration, a variation of thein-plane momentum transfer q‖ also leads to a (small) changeof the out-of-plane momentum qz. This variation is differentfrom the experiment on Ca-YBCO in the main text, wherea variation of the in-plane (q‖) and out-of-plane momentumtransfer (qz) independently of each other allowed us to fix theout-of-plane momentum to L = 0.9.Figure 5(b) shows the RIXS spectra of the YBCO filmmeasured at various momenta along the H direction, normal-(0, 0.02, 0.9)(0, 0.04, 0.9)(0, 0.06, 0.9)Ca-YBCO(a)(b)(c)FIG. 6. Representative fits of the RIXS data. (a) Fit of the RIXSspectrum for momentum (0, 0.02, 0.9), with a model using a Gaus-sian for the elastic peak (dashed black line) and anti-symmetrizedLorentzians for the other contributions, convoluted with the energyresolution of 27 meV via Gaussian convolution. The individual con-tributions are described in the text. [(b) and (c)] Fit of the spectrumfor momentum (0, 0.04, 0.9) and (0, 0.06, 0.9), respectively.plasmon energy [eV]K [2π / a]0 0.2 0.4 0.6 0.80 0.02 0.04 0.06 0.08 0.10 0.1210-310-210-1χ " c (q, ω)qz = 0.2 πω+ω-FIG. 7. Computed intensity map in analogy to Fig. 4, but for bareparameters of Ca-YBCO.144516-6PLASMON DISPERSION IN BILAYER CUPRATE … PHYSICAL REVIEW B 109, 144516 (2024)ized to the incident flux. The spectra were acquired with anincident photon energy of 527.32 eV, corresponding to theenergy of the pre-peak in the XAS data [Fig. 5(a)]. The changeof H from 0.04 to 0.16 involves a concomitant change of Lfrom 0.69 to 0.51. Hence, the measured plasmon dispersionin the YBCO film is not directly comparable to that of theCa-YBCO crystal in Fig. 3(b) of the main text, where theout-of-plane momentum was fixed to L = 0.9. Nevertheless,the dispersive character and the energy scale of the dispersionof the plasmon in both samples are similar. Clear signatures ofthe ω− modes are not observed in Fig. 5(b), even for momentaclose to qz = π . This absence is consistent with expectationsaccording to our calculation for Ca-YBCO (Fig. 4), whereVc/t = 25 was utilized, resulting in a diminishing intensityfor the ω− mode, even for momenta near qz = π . However,it is possible that a weak ω− mode signal may be overshad-owed by the strong background signal in the O K-edge RIXSspectra shown in Fig. 5(b). Consequently, this underscoresthe necessity for future high-resolution RIXS experiments,particularly focused on the vicinity of qz = π , to uncover thesubtle indications of the ω− mode.APPENDIX B: RIXS RAW DATA AND FITSFigure 6 displays representative fits of RIXS spectra ofthe Ca-YBCO single crystal measured at the ID32 beamlineof the ESRF. The components fitted to the spectrum are theelastic peak modeled by a Gaussian, and the other contribu-tions are modeled by anti-symmetrized Lorentzians [10,23],convoluted with the energy resolution of 27 meV via Gaussianconvolution. The anti-symmetrized Lorentzian profiles ensurezero intensity at zero energy loss (prior to convolution) for theinelastic features. In detail, the inelastic features are assignedto two phonons, a plasmon, and dd excitations. We note thatin our fits the linewidth of the lowest-energy inelastic feature(denoted as phonon 1) is likely underestimated, due to overlapwith the highly intense elastic line. 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