# Fileset [Bi2223_Plasmon_PRB.pdf](https://mdr.nims.go.jp/filesets/cecdc52e-e2a3-45d1-82d9-570b870cc238/download) ## Creator [S. Nakata](https://orcid.org/0000-0003-1675-9532), [M. Bejas](https://orcid.org/0000-0003-4254-0622), [J. Okamoto](https://orcid.org/0000-0003-0538-6603), [K. Yamamoto](https://orcid.org/0000-0002-4270-6410), [D. Shiga](https://orcid.org/0000-0003-4500-6214), [R. Takahashi](https://orcid.org/0000-0002-6099-5201), H. Y. Huang, [H. Kumigashira](https://orcid.org/0000-0003-4668-2695), H. Wadati, [J. Miyawaki](https://orcid.org/0000-0002-0602-907X), [S. Ishida](https://orcid.org/0000-0001-9445-8079), [H. Eisaki](https://orcid.org/0000-0002-8299-6416), [A. Fujimori](https://orcid.org/0000-0003-4876-0634), [A. Greco](https://orcid.org/0000-0001-5958-5080), [H. Yamase](https://orcid.org/0000-0003-0328-5657), D. J. Huang, [H. Suzuki](https://orcid.org/0000-0003-2973-0579) ## Rights ©2025 American Physical Society [In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Out-of-phase plasmon excitations in the trilayer cuprate

{Bi}_{2} {Sr}_{2} {Ca}_{2} {Cu}_{3} O_{10 + δ}

](https://mdr.nims.go.jp/datasets/8dceebb9-17d5-4314-8142-3972577ea7a9) ## Fulltext Out-of-phase Plasmon Excitations in the Trilayer Cuprate Bi2Sr2Ca2Cu3O10+δS. Nakata,1 M. Bejas,2 J. Okamoto,3 K. Yamamoto,4 D. Shiga,5 R. Takahashi,1 H.Y. Huang,6 H. Kumigashira,5 H. Wadati,1 J. Miyawaki,4 S. Ishida,7 H. Eisaki,7A. Fujimori,6, 8, 9 A. Greco,2 H. Yamase,10, ∗ D. J. Huang,6 and H. Suzuki11, 5, †1Department of Material Science, Graduate School of Science,University of Hyogo, Ako, Hyogo 678-1297, Japan2Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura and Instituto de F́ısica de Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina3National Synchrotron Radiation Research Center, Hsinchu 300092, Taiwan4NanoTerasu Center, National Institutes for Quantum Science and Technology, Sendai, 980-8572, Japan5Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan6National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan7National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan8Center for Quantum Science and Technology and Department of Physics,National Tsing Hua University, Hsinchu 30013, Taiwan9Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan10Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan11Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan(Dated: May 19, 2025)Within a homologous series of cuprate superconductors, variations in the stacking of CuO2 layersinfluence the collective charge dynamics through the long-range Coulomb interactions. We use OK-edge resonant inelastic x-ray scattering to reveal plasmon excitations in the optimally-dopedtrilayer Bi2Sr2Ca2Cu3O10+δ. The observed plasmon exhibits nearly qz-independent dispersion anda large excitation gap of approximately 300 meV. This mode is primarily ascribed to the ω− mode,where the charge density on the outer CuO2 sheets oscillates out of phase while the density in theinner sheet remains unaltered at qz = 0. The intensity of the acoustic ω3 mode is relatively weakand becomes vanishingly small near (qx, qy) = (0, 0). This result highlights a qualitative change inthe eigenmode of the dominant low-energy plasmon with the number of CuO2 layers.I. INTRODUCTIONHigh-temperature superconductivity in the cupratesemerges by introducing charge carriers into antiferromag-netic Mott insulators that contain the CuO2 sheets [1, 2].The proximity to the Mott insulating state commonly in-vokes theoretical description based on a single-band Hub-bard model that describes the itinerant Cu 3dx2−y2 holeson a square lattice and their on-site Coulomb repulsion.This approach describes the antiferromagnetic order inthe parent compounds and the d-wave superconducting(SC) pairing in the doped compounds, arising from thevirtual exchange of antiferromagnetic spin fluctuations[3].Despite the success of spin-fluctuation theories in cap-turing key features of the cuprate superconductivity, itdoes not fully explain its material dependence. One of themajor unanswered questions is the dependence of transi-tion temperature (Tc) on the number of CuO2 layers (n).Within a homologous series of cuprates, the Tc mono-tonically increases with n up to n = 3 and decreases forn ≥ 4 [4]. A theoretical proposal explaining this n de-pendence of Tc [5] suggests the importance of collectivecharge fluctuations arising from the long-range Coulombinteraction and its efficient screening by the SC pairs,thereby enhancing Tc. Thus, it is crucial to reveal then dependence of charge fluctuations in the momentumspace.Recent advances in the energy resolution of resonantinelastic x-ray scattering (RIXS) [6] have enabled themeasurement of elementary excitations relevant to super-conductivity, including paramagnons [7, 8], phonons [9],and plasmons [10–14]. The plasmon excitations highlightthe crucial role of the long-range Coulomb interaction inthe low-energy charge dynamics of the cuprates, which isneglected in the Hubbard-model descriptions that incor-porate only the local interactions.Plasmon dispersions identified in single-layer (n =1) compounds La2−xCexCuO4 [10, 11], La2−xSrxCuO4[12, 13], and Bi2Sr1.6La0.4CuO6+δ (Bi2201) [12] areall acoustic-like, with a small energy gap in the limit(qx, qy) → (0, 0). In contrast, the infinite-layer (n = ∞)Sr0.9La0.1CuO2 has a gapped plasmon branch [14]. Thisenergy gap arises from interlayer electron hopping [16],which brings the system closer to the three-dimensionalelectron gas with a gapped plasma frequency [17]. Inboth cases, the plasmons exhibit a large qz dispersionnear (qx, qy) = (0, 0) caused by an evolution from an opti-cal mode at qz = 0 to acoustic-like modes at qz ̸= 0. In bi-layer (n = 2) compounds, the Coulomb interaction acrossthe bilayers generates two plasmon branches at each qz[18], and the intra-bilayer electron hopping generates anenergy gap in one of these branches [19]. Experimentaldata of the bilayer Y0.85Ca0.15Ba2Cu3O7 indicate thatthe gapped branch carries the dominant spectral weight2 !"#!$%"&'()'(*'(+'),')-')./01"1!2#!#34&25#6782 9#:4#2;<= 2>1?@AB=5C75D75E7FG8)!H%8" #=3$FE$%FG&'&22222222 !( I ("%)"#FIG. 1. (a) Crystal structure of Bi2Sr2Ca2Cu3O10+δ (Bi2223)[15]. The inter-trilayer distance c′ is half of the crystallo-graphic lattice constant c = 37.16 Å. (b) Scattering geometryof the OK-edge resonant inelastic x-ray scattering (RIXS) ex-periment. The incident x-ray photons were σ-polarized, andthe polarization of the scattered photons was not analyzed.(c) O K-edge x-ray absorption spectrum (XAS) measured at24 K with the σ polarization. The arrow indicates the absorp-tion peak (527.9 eV) to the Zhang-Rice singlet (ZRS) states.[19]. However, the lack of experimental data on cupratecompounds with a finite n ≥ 3 hinders the understand-ing of the evolution of plasmon dispersions as a functionof n, and the role of plasmons in the superconductingmechanism of the cuprates.In this work, we present a RIXS investigation ofthe plasmon excitations in the optimally-doped trilayercuprate Bi2Sr2Ca2Cu3O10+δ (Bi2223), which shows thehighest Tc of 110 K among the Bi-based cuprates [20].The observed plasmon branch exhibits a nearly two-dimensional dispersion with a significant gap of approx-imately 300 meV at the two-dimensional Brillouin zonecenter. Theoretical calculations of the charge susceptibil-ity within the random phase approximation (RPA) pre-dict three plasmon branches. We show that the observeddispersion is mainly described by the ω− mode, whichrepresents the out-of-phase oscillation of the charges onthe outer CuO2 sheets at qz = 0 and exhibits a weakqz dispersion, strikingly different from the plasmons ob-served in single-layer cuprates.(c)(b)(a)FIG. 2. (a) Measurement paths in the three-dimensional qspace. (b) Representative OK-edge RIXS spectrum of Bi2223at q = (-0.05, 0, -2.0). The spectrum includes the elastic line(blue), phonon (orange), plasmon (green), bimagnon (red),and dd excitations (purple). (c) RIXS intensity map at 24K along the q = (H, 0, L) direction taken with a maximalscattering angle of 2θ = 150◦.II. RESULTS AND DISCUSSIONFigure 1(a) shows the crystal structure of Bi2223 andthe key parameters that determine the plasmon disper-sion. The CuO2 trilayers (light blue planes) are sep-arated by spacer layers, with an inter-trilayer distanceof c′ = 18.58 Å, which is half the crystallographic lat-tice constant c = 37.16 Å [21]. This large inter-trilayerdistance suppresses electron hopping across the trilay-ers. In contrast, the interlayer distance within a trilayer,d = 3.30 Å, allows a finite interlayer hopping tz. Thetz combined with the long-range Coulomb interaction Vyields the gapped dispersion of the plasmon branch.To identify collective charge fluctuations in the hole-doped Bi2223, we employed O K-edge RIXS. The scat-tering geometry of the RIXS experiment is illustrated inFig. 1(b). The incident x-ray photons were σ-polarized,and the outgoing photons with both σ and π polariza-tions were collected. Other experimental conditions aredetailed in the Supplemental Material [20].The resonant transition to the itinerant O 2p holeswas determined by an O K-edge x-ray absorption spec-trum (XAS) collected with σ-polarized photons. TheXAS lineshape shown in Fig. 1(c) reproduces previousresults for both Pb-doped and Pb-free Bi2223 [22]. ForRIXS measurements, we set the incident energy to 527.9eV (indicated by the arrow), which corresponds to thetransition to the Zhang-Rice singlet states formed by thehybridization between the Cu 3dx2−y2 and O 2p orbitals[23]. This condition enhances the RIXS cross section ofthe charge scattering from the itinerant holes in the hole-doped cuprates [12, 13].The investigated q paths in the reciprocal space aredepicted in Fig. 2(a). We first provide an overview30.10.05H = 00.010.005(f) (g) (h)(a)L = -2 L = -2.5(-0.06, 0, -L)2.75L = 3.02.52.252.0(e)q = (-H, 0, -2)H = 0.010.050.1(-H, 0, -2.5)(c)(b) (d)(-H, -H, -2) (-H, -H, -2.5)0.10.05H = 00.010.005H = 0.010.050.1FIG. 3. RIXS spectra at q =(H, 0, L) and (H, H, L), collected at fixed L values (a,b) L = -2 and (c,d) L = -2.5. (e) RIXSspectra collected along the L direction at q = (−0.06, 0, L). Red circles indicate the plasmon peak energies. (f)-(h) RIXSintensity maps corresponding to panels (a)-(e).of the charge fluctuations in Bi2223 by employing thelargest scattering angle of 2θ = 150◦ to maximize themomentum transfer |q|. The in-plane component wasscanned by rotating the sample angle (blue circles). Sub-sequently, we investigate the three-dimensional disper-sions along straight lines in the q space by simultaneouslytuning the 2θ and sample angle θi. Specifically, we ex-plore the q = (H, 0, L) and (H,H,L) paths on the fixedL = −2 (green) and L = −2.5 (pink) planes, as well asthe q = (−0.06, 0, L) path in the range −3.0 < L < −2.0(red).A representative O K-edge RIXS spectrum at q =(−0.05, 0,−2.0) is presented in Fig. 2(b). In addition tothe elastic line at ω = 0 eV (blue), it contains multipleelementary excitations in the spin-conserving channel,including phonon (orange), plasmon (green), bimagnon(red), and the tail of the dd excitations (purple). In thesubsequent discussion, we will focus on the dispersion ofthe plasmon branch.A colormap of RIXS spectra at 24 K collected witha constant scattering angle of 2θ = 150◦ is shown inFig. 2(c). A pronounced plasmon dispersion is ob-served, originating from the two-dimensional zone cen-ter and extending up to approximately 1 eV. While asharp plasmon peak is well-defined for small |H| values,the linewidth steadily increases with larger |H|, resultingin significant broadening for |H| ≳ 0.2. This broaden-ing suggests the damping of the plasmon excitations intoincoherent electron-hole pairs, likely due to the intersec-tion of plasmons with the electron-hole continuum. ThisLandau damping of plasmons was not identified in pre-vious investigations [10–14], as the investigated in-planeq range was limited to |H| ≲ 0.2. Despite the damping,the charge continuum retains significant spectral weightin |H| ≳ 0.2. In addition to the plasmon excitations, weobserve additional intensity around |H| = 0.25 in the low-energy region (ω < 0.2 eV). This intensity is attributedto the charge density wave (CDW) and the concomitantphonon anomaly. The details of these CDW-related fea-tures will be published elsewhere.We now examine the three-dimensional dispersion re-lation in more detail. In Figs. 3 (a)-(d), we show theRIXS spectra at 24 K at q =(H, 0, L) and (H, H, L),collected at fixed L values L = -2 [(a,b)] and L = -2.5[(c,d)]. The plasmon peak positions are also shown as redcircles. The corresponding intensity colormaps are pre-sented in Figs. 3(f) and (g), together with the peak po-sitions. In contrast to the plasmons in single- [10, 12, 13]and infinite-layer [14] cuprates with significant qz dis-persion, the dispersion of the observed branch in Bi2223is nearly identical on the L = −2 and L = −2.5 planes,highlighting its two-dimensional character. Furthermore,it exhibits a large excitation gap of approximately 300meV at the two-dimensional Brillouin zone center.To directly validate the small qz dispersion, we presentRIXS spectra along the L direction at q = (−0.06, 0, L)in Fig. 3(e). The corresponding colormap is presented inFig. 3(h). At this in-plane momentum, the plasmon line-shape is almost independent of L, and the peak energyshows only a weak dispersion from 0.51 eV at L = −2 to0.53 eV at L = −3. This energy difference at these equiv-alent q vectors shows that the periodicity of the plas-mon intensity is determined by the inter-trilayer distancec′, instead of the crystallographic lattice constant c [10].This qz dispersion is significantly smaller than the plas-mons in single- [10–13] and infinite-layer [14] cuprates.The small qz dispersion is a characteristic feature of theω− branch in trilayer cuprates, as shown below.To quantitatively describe the observed plasmon dis-persion, we have computed the dynamical charge sus-ceptibility of Bi2223 within the RPA. The theoreticaltrilayer model and the explicit form of the long-rangeCoulomb interactions are detailed in the SupplementalMaterial [20]. Our theoretical treatment is analogous toRef. [18], but we adopt a tight-binding model with afinite intra-trilayer hopping tz, which yields gapped plas-mon branches at (qx, qy) = (0, 0). The Fermi surfaces ofthe model [inset of Fig. 4(a)] are in good agreement withangle-resolved photoemission data of optimally-dopedBi2223 [24, 25]. We do not include the electron hoppingbetween the outer CuO2 sheets within a trilayer, whichmight become relevant only in the overdoped region [26].Figures 4(a) and (b) show the intensity colormaps ofcomputed charge susceptibility [-Imχ(q, ω)] along the ex-perimental q paths on the L = −2 and −2.5 planes,respectively. The experimental plasmon peak positions4 !"#!$#!" %&'%()"!"*+%"+%) ,(-,""! "!." "!"$!%&'%()!+%",(/, %0%)#!$.!$.!""!$"1'2345%(26,"!." "!"$!%&'%()!+%",(7, %0%)# 8 9 "!.""!"$"!%&'%()!+%)!,"!.""!"$"!%&'%()!+%)!,"#"$FIG. 4. Dynamical charge susceptibility computed within theRPA. The intensity maps are generated along the experimen-tal q paths. The notations of three plasmon branches (ω+,ω−, and ω3) follow those in Ref. [18]. The experimental plas-mon peak positions are overlaid as red open circles. The insetof panel (a) depicts the Fermi surfaces of the employed tri-layer model [20].are also overlaid as red circles. Reflecting the presenceof the three CuO2 sheets, three plasmon branches appearwithin the RPA. The ω+ mode has strong qz-dependence,resulting in different in-plane dispersions on the L = −2and −2.5 planes. As its dominant spectral weight lies inthe high-energy region (≳ 1.2 eV) that falls within thetail of the dd excitations (see Fig. 2), it is not resolvedin the present RIXS measurements. In the low-energyregion (≲ 1.2 eV), the dominant spectral weight lies inthe ω− branch, although the ω3 mode also has sizablespectral weight, particularly at L = −2.5. Due to thefinite tz and the long-range Coulomb interaction V , thedispersion of the ω− mode becomes gapped at the two-dimensional zone center, providing an excellent descrip-tion of the experimental plasmon dispersion. Note thatthe ω3 mode remains gapless at q = (0, 0,−2.5), but itsspectral weight near q = (0, 0,−2.5) becomes vanishinglysmall. Concomitantly, the bottom of the ω+ mode startsto carry sizable spectral weight near q = (0, 0,−2.5), re-producing the experimental peak positions.The intensity map along the q = (−0.06, 0, L) path,shown in Fig. 4(c), reveals the distinct qz dependenceof the three plasmon branches. The ω+ mode exhibits astrong L dependence with its minimum at L = −3, simi-lar to the plasmons in single- and infinite-layer cuprates.On the other hand, the energies of the ω− and ω3 modesdepend on L only weakly, and the slight increase of theω− mode from L = −2 to L = −3 agrees with the exper-imental data.Here, we mention the linewidths of the plasmon excita-tions. The plasmon linewidth of the RIXS data is quitebroad in the small in-plane (qx, qy) region presented inFig. 3. On the other hand, the calculated RPA chargesusceptibility predicts the three sharp plasmon branches.This is because the RPA does not incorporate spectralbroadening due to electron correlations, such as short-range antiferromagnetic correlations [27]. In the single-layer case, this effect is discussed in the strong-coupling t-J-V model [28]. It can be simulated phenomenologicallyby invoking a large broadening parameter comparable tothe peak widths of the observed plasmons [20].Our RIXS data and theoretical analysis have revealedthat intra-trilayer hopping in Bi2223 is responsible forthe finite energy gap in the ω− branch. Note thatthis mechanism differs from the energy gap generatedby inter-(multi)layer hopping [14], which generates thethree-dimensionality of the electronic band structures.For instance, the kz-dependent modulation of the Fermisurfaces in the overdoped La2−xSrxCuO4 (x = 0.22) withan interlayer distance of 6.61 Å shows that the magnitudeof the interlayer hopping is 7 % of the nearest-neighborhopping [29]. In contrast, the weak modulation observedin Tl2Ba2CuO6+δ, which has a long interlayer distanceof 11.60 Å, suggests that interlayer hopping is less than1.5 % of the nearest-neighbor hopping [29]. In Bi2223,the longer inter-trilayer distance, given by c′−2d = 11.98Å, likely leads to a negligibly small inter-trilayer electronhopping.Now, we discuss the n dependence of the plasmonbranches in the absence of inter-multilayer hopping. Inthe single-layer case (n = 1), only the ω+ mode exists,whose dispersion is acoustic-like, except for the opticalbranches appearing at integer L values [17]. In the bi-layer case (n = 2), both ω+ and ω− modes are present,and the intra-bilayer hopping generates an energy gap inone of them at (qx, qy) = (0, 0). Ref. [19] concluded thatthe ω+ mode carries the dominant spectral weight and ex-plains the observed dispersion in Y0.85Ca0.15Ba2Cu3O7,although recent analyses suggested that the observed dis-persion can be explained by the ω− mode [30, 31]. In thetrilayer case (n = 3), the three modes (ω+, ω−, and ω3)emerge, and we have shown that the ω3 and ω− modesexhibit a weak qz dependence. The observed plasmonin Bi2223 is primarily ascribed to the ω− branch withthe dominant spectral weight, supplemented by contri-butions from the ω3 mode at L = −2 and from the ω3and ω+ modes at L = −2.5.These results highlight the qualitative change in theeigenmode of the dominant charge fluctuations as a func-tion of n in cuprates. For simplicity, we describe theeigenmodes of the ω+, ω−, and ω3 modes at qz = 0. Theω+ mode, present in any n, is the in-phase charge oscil-lation among the multilayers. The ω− mode, present inn ≥ 2, is an out-of-phase charge oscillations within themultilayers. In the trilayer case, the charge density on theouter two sheets oscillates out of phase, while the den-sity on the inner sheet remains unaltered. The ω3 modeappears in n ≥ 3. In the trilayer case, the charge den-sity on the outer and inner sheets oscillates out of phase,with the larger amplitude in the inner sheet. While theplasmons in the single- and infinite-layer cuprates [10–14]originate from the ω+ mode, our results on the trilayerBi2223 provide the first identification of the ω− plasmonmode in cuprates. Our results also call for more com-prehensive data in bilayer cuprates [19], for the preciseassignment of the eigenmode.5A natural question is how the evolution of plasmonswith n affects the SC properties in cuprates. Wenote that the lineshape of the RIXS spectrum at q =(−0.1, 0,−2) at 300 K (> Tc) remains almost identicalto that at 25 K [20]. This indicates that the reconstruc-tion in the dynamical charge susceptibility in the mid-infrared region across Tc, as theoretically proposed [5],was not detected at this q. In Bi2223, the bottom en-ergy of the plasmons, ω ∼ 300 meV, is significantly largerthan 2∆, where ∆ is the amplitude of its d-wave SC orderparameter in the three Fermi surfaces [24, 25, 32]. Thisis not the case in single-layer cuprates with acoustic-likeplasmon dispersion. Future investigations of the detailedtemperature dependence of plasmon lineshape across Tc,both in single- and multilayer cuprates, will shed light onthe role of plasmons in the superconducting mechanism.III. CONCLUSIONIn conclusion, we have conducted an O K-edge RIXSstudy of plasmon excitations in the optimally doped tri-layer cuprate Bi2223. We have identified nearly two-dimensional plasmon excitations, which exhibit a largeexcitation gap of approximately 300 meV at (qx, qy) =(0, 0) generated by the long-range Coulomb interactionsand the intra-trilayer hopping. We attribute this modeprimarily to the ω− mode, in contrast to the ω+ modeidentified in single- and infinite-layer cuprates. These re-sults suggest a qualitative change in the eigenmode ofthe dominant plasmon with the number of CuO2 layersin cuprate superconductors.ACKNOWLEDGMENTSWe thank T. Tohyama for enlightening discussions.The project was supported by Grants-in-Aid for Scien-tific Research from JSPS (KAKENHI) (No. JP19H05823,JP22K13994). M.B. is indebted to MANA Short-TermInvitation Program and warm hospitality in NIMS; hewas also partially supported by JSPS KAKENHI (GrantNo. JP20H01856). A.F. was supported by JSPS KAK-ENHI (Grant No. JP22K03535), NSTC Taiwan (GrantNo. NSTC 113-2112-M-007-033), and the Yushan FellowProgram and the Center for Quantum Science and Tech-nology within the framework of the Higher EducationSprout Project under the MOE of Taiwan. H.Y. was sup-ported by JSPS KAKENHI (Grant No. JP20H01856) andthe World Premier International Research Center Initia-tive (WPI), MEXT, Japan. Part of the theoretical resultswas obtained by using the facilities of the CCT-RosarioComputational Center, a member of the High Perfor-mance Computing National System (SNCAD, MincyT-Argentina).DATA AVAILABILITYThe data that support the findings of this article areopenly available.∗ yamase.hiroyuki@nims.go.jp† hakuto.suzuki@tohoku.ac.jp[1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, andJ. Zaanen, From quantum matter to high-temperaturesuperconductivity in copper oxides, Nature 518, 179(2015).[2] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott in-sulator: Physics of high-temperature superconductivity,Rev. Mod. Phys. 78, 17 (2006).[3] D. J. Scalapino, A common thread: The pairing inter-action for unconventional superconductors, Rev. Mod.Phys. 84, 1383 (2012).[4] A. Iyo, Y. Tanaka, H. Kito, Y. Kodama, P. M. Shi-rage, D. D. Shivagan, H. Matsuhata, K. Tokiwa, andT. Watanabe, Tc vs n relationship for multilayered high-tc superconductors, J. Phys. Soc. Jpn. 76, 094711 (2007).[5] A. J. Leggett, Cuprate superconductivity: Dependenceof Tc on the c-axis layering structure, Phys. Rev. Lett.83, 392 (1999).[6] F. M. F. de Groot, M. W. Haverkort, H. Elnaggar,A. Juhin, K.-J. Zhou, and P. Glatzel, Resonant inelas-tic x-ray scattering, Nat. Rev. Methods Primers 4, 45(2024).[7] M. Le Tacon, G. Ghiringhelli, J. Chaloupka, M. M. Sala,V. 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).[8] Y. Y. Peng, G. Dellea, M. Minola, M. Conni, A. Amorese,D. Di Castro, G. M. De Luca, K. Kummer, M. Sal-luzzo, X. Sun, X. J. Zhou, G. Balestrino, M. Le Tacon,B. Keimer, L. Braicovich, N. B. Brookes, and G. Ghir-inghelli, Influence of apical oxygen on the extent of in-plane exchange interaction in cuprate superconductors,Nat. Phys. 13, 1201 (2017).[9] T. P. Devereaux, A. M. Shvaika, K. Wu, K. Wohlfeld,C. J. Jia, Y. Wang, B. Moritz, L. Chaix, W.-S. Lee, Z.-X.Shen, G. Ghiringhelli, and L. Braicovich, Directly charac-terizing the relative strength and momentum dependenceof electron-phonon coupling using resonant inelastic x-ray scattering, Phys. Rev. X 6, 041019 (2016).[10] M. Hepting, L. Chaix, E. W. Huang, R. Fumagalli, Y. Y.Peng, B. Moritz, K. Kummer, N. B. Brookes, W. C. Lee,M. Hashimoto, T. Sarkar, J. F. He, C. R. Rotundu, Y. S.Lee, R. L. Greene, L. Braicovich, G. Ghiringhelli, Z. X.Shen, T. P. Devereaux, and W. S. Lee, Three-dimensionalcollective charge excitations in electron-doped copper ox-ide superconductors, Nature 563, 374 (2018).[11] J. Lin, J. Yuan, K. Jin, Z. Yin, G. Li, K.-J. Zhou,X. Lu, M. Dantz, T. Schmitt, H. Ding, H. Guo, M. P. M.Dean, and X. Liu, Doping evolution of the charge ex-citations and electron correlations in electron-doped su-mailto:yamase.hiroyuki@nims.go.jpmailto:hakuto.suzuki@tohoku.ac.jphttp://dx.doi.org/10.1038/nature14165http://dx.doi.org/10.1038/nature14165https://doi.org/10.1103/RevModPhys.78.17https://doi.org/10.1103/RevModPhys.84.1383https://doi.org/10.1103/RevModPhys.84.1383https://doi.org/10.1103/PhysRevLett.83.392https://doi.org/10.1103/PhysRevLett.83.392https://doi.org/10.1038/s43586-024-00322-6https://doi.org/10.1038/s43586-024-00322-6http://dx.doi.org/10.1038/nphys2041https://doi.org/10.1038/nphys4248https://doi.org/10.1103/PhysRevX.6.041019https://doi.org/10.1038/s41586-018-0648-36perconducting la2−xcexcuo4, npj Quantum Mater. 5, 4(2020).[12] 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 usingresonant inelastic x-ray scattering, Phys. Rev. Lett. 125,257002 (2020).[13] A. Singh, H. Y. Huang, C. Lane, J. H. Li, J. Okamoto,S. Komiya, R. S. Markiewicz, A. Bansil, T. K. Lee, A. Fu-jimori, C. T. Chen, and D. J. Huang, Acoustic plasmonsand conducting carriers in hole-doped cuprate supercon-ductors, Phys. Rev. B 105, 235105 (2022).[14] 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 charge ex-citations and interlayer hopping in cuprate superconduc-tors, Phys. Rev. Lett. 129, 047001 (2022).[15] K. Momma and F. Izumi, VESTA3 for three-dimensionalvisualization of crystal, volumetric and morphology data,J. Appl. Cryst. 44, 1272 (2011).[16] A. Greco, H. Yamase, and M. Bejas, Plasmon excitationsin layered high-Tc cuprates, Phys. Rev. B 94, 075139(2016).[17] A. L. Fetter and J. D. Walecka, Quantum Theory ofMany-Particle Systems (McGraw-Hill, Boston, 1971).[18] A. Griffin and A. J. Pindor, Plasmon dispersion rela-tions and the induced electron interaction in oxide super-conductors: Numerical results, Phys. Rev. B 39, 11503(1989).[19] 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,A. Greco, and M. Hepting, Plasmon dispersion in bi-layer cuprate superconductors, Phys. Rev. B 109, 144516(2024).[20] See supplemental material at [url inserted by publisher]for sample characterization, experimental conditions forthe rixs measurements, an additional rixs spectrum at300 k, and the details of the theoretical model and chargesusceptibility calculations, which includes refs. [33, 34].[21] T. Fujii, T. Watanabe, and A. Matsuda, Single-crystalgrowth of bi2sr2ca2cu3o10+δ (bi-2223) by tsfz method, J.Cryst. Growth 223, 175 (2001).[22] M. Karppinen, S. Lee, J. M. Lee, J. Poulsen, T. No-mura, S. Tajima, J. M. Chen, R. S. Liu, and H. Ya-mauchi, Hole doping in pb-free and pb-substituted(Bi,Pb)2sr2ca2cu3o10+δ superconductors, Phys. Rev. B68, 054502 (2003).[23] F. C. Zhang and T. M. Rice, Effective hamiltonian forthe superconducting cu oxides, Phys. Rev. B 37, 3759(1988).[24] S. Ideta, K. Takashima, M. Hashimoto, T. Yoshida,A. Fujimori, H. Anzai, T. Fujita, Y. Nakashima, A. Ino,M. Arita, H. Namatame, M. Taniguchi, K. Ono, M. Kub-ota, D. H. Lu, Z.-X. Shen, K. M. Kojima, and S. Uchida,Enhanced superconducting gaps in the trilayer high-temperature bi2sr2ca2cu3o10+δ cuprate superconductor,Phys. Rev. Lett. 104, 227001 (2010).[25] S. Kunisada, S. Adachi, S. Sakai, N. Sasaki,M. Nakayama, S. Akebi, K. Kuroda, T. Sasagawa,T. Watanabe, S. Shin, and T. Kondo, Observation ofbogoliubov band hybridization in the optimally dopedtrilayer bi2sr2ca2cu3o10+δ, Phys. Rev. Lett. 119, 217001(2017).[26] X. Luo, H. Chen, Y. Li, Q. Gao, C. Yin, H. Yan, T. Miao,H. Luo, Y. Shu, Y. Chen, C. Lin, S. Zhang, Z. Wang,F. Zhang, F. Yang, Q. Peng, G. Liu, L. Zhao, Z. Xu,T. Xiang, and X. J. Zhou, Electronic origin of high su-perconducting critical temperature in trilayer cuprates,Nat. Phys. 19, 1841 (2023).[27] P. Prelovšek and P. Horsch, Electron-energy loss spectraand plasmon resonance in cuprates, Phys. Rev. B 60,R3735 (1999).[28] A. Nag, L. Zinni, J. Choi, J. Li, S. Tu, A. C. Walters,S. Agrestini, S. M. Hayden, M. Bejas, Z. Lin, H. Yamase,K. Jin, M. Garćıa-Fernández, J. Fink, A. Greco, and K.-J. Zhou, Impact of electron correlations on two-particlecharge response in electron- and hole-doped cuprates,Phys. Rev. Res. 6, 043184 (2024).[29] M. Horio, K. Hauser, Y. Sassa, Z. Mingazheva, D. Sut-ter, K. Kramer, A. Cook, E. Nocerino, O. K. Forslund,O. Tjernberg, M. Kobayashi, A. Chikina, N. B. M.Schröter, J. A. Krieger, T. Schmitt, V. N. Stro-cov, S. Pyon, T. Takayama, H. Takagi, O. J. Lip-scombe, S. M. Hayden, M. Ishikado, H. Eisaki, T. Ne-upert, M. Månsson, C. E. Matt, and J. Chang, Three-dimensional fermi surface of overdoped la-based cuprates,Phys. Rev. Lett. 121, 077004 (2018).[30] H. Yamase, Theory of charge dynamics in bilayer electronsystem with long-range coulomb interaction, Phys. Rev.B 111, 085138 (2025).[31] N. Sellati and L. Benfatto, Ghost josephson plasmon inbilayer superconductors, arXiv:2412.14927.[32] S. Ideta, S. Johnston, T. Yoshida, K. Tanaka, M. Mori,H. Anzai, A. Ino, M. Arita, H. Namatame, M. Taniguchi,S. Ishida, K. Takashima, K. M. Kojima, T. P. Dev-ereaux, S. Uchida, and A. Fujimori, Hybridization ofbogoliubov quasiparticles between adjacent cuo2 layersin the triple-layer cuprate bi2sr2ca2cu3o10+δ studied byangle-resolved photoemission spectroscopy, Phys. Rev.Lett. 127, 217004 (2021).[33] A. Singh, H. Y. Huang, Y. Y. Chu, C. Y. Hua, S. W. Lin,H. S. Fung, H. W. Shiu, J. Chang, J. H. Li, J. Okamoto,C. C. Chiu, C. H. Chang, W. B. Wu, S. Y. Perng, S. C.Chung, K. Y. Kao, S. C. Yeh, H. Y. Chao, J. H. Chen,D. J. Huang, and C. T. Chen, Development of the SoftX-ray AGM–AGS RIXS beamline at the Taiwan PhotonSource, J. Synchrotron Rad. 28, 977 (2021).[34] J. Miyawaki, K. Fujii, T. Imazono, K. Horiba, Y. Oht-subo, N. Inami, T. Nakatani, K. Inaba, A. Agui,H. Kimura, and M. Takahasi, Design of ultrahigh energyresolution rixs beamline at nanoterasu, J. Phys.: Conf.Ser. 2380, 012030 (2022).https://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://doi.org/10.1107/S0021889811038970https://doi.org/10.1103/PhysRevB.94.075139https://doi.org/10.1103/PhysRevB.94.075139https://doi.org/10.1103/PhysRevB.39.11503https://doi.org/10.1103/PhysRevB.39.11503https://doi.org/10.1103/PhysRevB.109.144516https://doi.org/10.1103/PhysRevB.109.144516https://doi.org/10.1103/PhysRevB.68.054502https://doi.org/10.1103/PhysRevB.68.054502https://doi.org/10.1103/PhysRevB.37.3759https://doi.org/10.1103/PhysRevB.37.3759https://doi.org/10.1103/PhysRevLett.104.227001https://doi.org/10.1103/PhysRevLett.119.217001https://doi.org/10.1103/PhysRevLett.119.217001https://doi.org/10.1038/s41567-023-02206-0https://doi.org/10.1103/PhysRevB.60.R3735https://doi.org/10.1103/PhysRevB.60.R3735https://doi.org/10.1103/PhysRevResearch.6.043184https://doi.org/10.1103/PhysRevLett.121.077004https://doi.org/10.1103/PhysRevB.111.085138https://doi.org/10.1103/PhysRevB.111.085138https://arxiv.org/abs/2412.14927https://doi.org/10.1103/PhysRevLett.127.217004https://doi.org/10.1103/PhysRevLett.127.217004https://doi.org/10.1107/S1600577521002897https://doi.org/10.1088/1742-6596/2380/1/012030https://doi.org/10.1088/1742-6596/2380/1/012030Supplemental Material forOut-of-phase Plasmon Excitations in the Trilayer Cuprate Bi2Sr2Ca2Cu3O10+δS. Nakata,1 M. Bejas,2 J. Okamoto,3 K. Yamamoto,4 D. Shiga,5 R. Takahashi,1 H.Y. Huang,3 H. Kumigashira,5 H. Wadati,1 J. Miyawaki,4 S. Ishida,6 H. Eisaki,6A. Fujimori,3, 7, 8 A. Greco,2 H. Yamase,9 D. J. Huang,3 and H. Suzuki10, 51Department of Material Science, Graduate School of Science,University of Hyogo, Ako, Hyogo 678-1297, Japan2Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura and Instituto de F́ısica de Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina3National Synchrotron Radiation Research Center, Hsinchu 300092, Taiwan4NanoTerasu Center, National Institutes for Quantum Science and Technology, Sendai, 980-8572, Japan5Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan6National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan7Center for Quantum Science and Technology and Department of Physics,National Tsing Hua University, Hsinchu 30013, Taiwan8Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan9Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan10Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, JapanI. Characterization of Bi2Sr2Ca2Cu3O10+δ single crystalsThe orientation of optimally-doped Bi2Sr2Ca2Cu3O10+δ (Bi2223) single crystal measured in the RIXS experimentwas determined from the Laue diffraction pattern [Fig. S1(a)]. The straight line corresponds to charge modulation inthe BiO planes along the q = (H,H,L) direction. The temperature dependence of magnetic susceptibility is shownin Fig. S1(b). The data were collected with an applied magnetic field of 3 Oe along the crystallographic c axis byfield cooling (FC) and zero-field cooling (ZFC) methods. The superconducting transition temperature (Tc = 110 K)was defined as the onset temperature of the Meissner diamagnetic signal. !"# #"$#%&'()*+,&*+-(./&01".2(+*34!5#!!#!##6#7#8#9):;)0&*20)./<4 !"#$$#!=.>.?)./14@AB@AC+?%-D2E&*+-(/&4FIG. S1. Laue diffraction pattern for a Bi2223 crystal. (b) Magnetization curves for the optimally-doped Bi2223 (Tc = 110K) measured with the field cooling (FC) and zero-field cooling (ZFC) methods. A magnetic field of 3 Oe was applied along thecrystallographic c axis.2II. Experimental conditions for RIXS measurementsO K-edge RIXS spectra at 24 K were collected using the AGS-AGM spectrometer [1] at Beamline 41A of TaiwanPhoton Source. The continuous rotation of the detector arm [1] permits q-space scans with fixed H or L values.The total energy resolution was set to 22 meV, as estimated from the full-width at half maximum (FWHM) of thenonresonant spectrum from carbon tape. In addition, a RIXS spectrum at 300 K in a wide energy region was collectedusing the 2D-RIXS spectrometer [2] at BL02U of NanoTerasu. Before installation in the measurement chamber, Bi2223single crystals were cleaved ex situ to prepare a fresh surface parallel to the CuO2 planes. In the main text, we usethe pseudotetragonal notation with lattice constants a = b = 3.85 Å and c = 37.16 Å [3]. The momentum transferq = (H,K,L) is expressed in the reciprocal lattice units (2π/a, 2π/b, 2π/c).III. RIXS spectrum at 300 K in a wide energy region !"#$%&'()*+,-./*01 .2./3'4&5.'5.3%1..%".6.$''.6FIG. S2. RIXS spectrum at q = (−0.1, 0,−2) at 300 K in a wide energy region (red). The corresponding spectrum at 25 K(black) is also shown for comparison.Figure S2 presents the RIXS spectrum at q = (−0.1, 0,−2) in a wider energy region at 300 K. The correspondingspectrum at 25 K is also shown for comparison. We identify the charge-transfer excitation at ∼ 5.5 eV. We also notethat the lineshapes of the plasmon excitation are almost identical between 25 K and 300 K. This indicates that thesuperconducting transition at Tc = 110 K does not have an appreciable feedback effect on the plasmon excitations.IV. Trilayer model and RPA calculations of charge susceptibilityThe bare Green’s function for the trilayer model is expressed asG−10 (k, iνn) = iνn − εk −ε⊥k eikzd 0−ε⊥k e−ikzd iνn − εk −ε⊥k eikzd0 −ε⊥k e−ikzd iνn − εk , (1)where εk = −2t(cos kx + cos ky) − 4t′ cos kx cos ky − 2t′′(cos 2kx + cos 2ky) − µ; ε⊥k = −tz(cos kx − cos ky)2 − tz0; kxand ky are in units of 1/a; kz is in units of 1/c′; a is the lattice constant along the x and y directions; c′ is half of thelattice constant c since the unit cell contains two slabs of trilayers; the three layers are placed with equal distance din the trilayer slab; see Fig. 1. Definingε1 = εk +√2 ε⊥k , (2)ε2 = εk , (3)ε3 = εk −√2 ε⊥k , (4)3we obtainG0(k, iνn) =1D(iνn − εk)2 − (ε⊥k )2 (iνn − εk)ε⊥k eikzd (ε⊥k )2ei2kzd(iνn − εk)ε⊥k e−ikzd (iνn − εk)2 (iνn − εk)ε⊥k eikzd(ε⊥k )2e−i2kzd (iνn − εk)ε⊥k e−ikzd (iνn − εk)2 − (ε⊥k )2 , (5)with D = (iνn − ε1)(iνn − ε2)(iνn − ε3).The non-interacting susceptibility matrix χ0(q, iωn) is given byχ0,ij(q, iωn) =1Nx,yNz∑k,iνnG0,ij(k, iνn) G0,ji(k+ q, iνn + iωn) , (6)where Nx,y (Nz) is the number of points in the k∥ (kz) summation. Note that the kz dependence only appearsthrough the phases in the off-diagonal elements and the energies do not depend on kz. Therefore we can extract thephase factor by introducing a quantity dij such that dij = 0 for i = j, dij = d for (i, j) = (1, 2), (2, 3), dij = 2d for(i, j) = (1, 3), and dji = −dij . Consequently, Eq.(6) is written asχ0,ij(q, iωn) =1Nx,yNz∑k∥,kz,iνnG̃0,ij(k, iνn) G̃0,ji(k+ q, iνn + iωn)eikzdije−i(kz+qz)dij , (7)=e−iqzdijNx,y∑k∥,iνnG̃0,ij(k, iνn) G̃0,ji(k+ q, iνn + iωn) . (8)Here G̃0,ij(k, iνn) is equal to G0,ij(k, iνn) without the phase factor. Then the matrix χ0 for the stack of trilayersis equal to the χ0 for only three layers with open boundary conditions, with the addition of phase factors e±iqzd ande±i2qzd in the off-diagonal elements.Following the long-range Coulomb interaction obtained in Ref. [4], we employVq =V1 V2 V3V ∗2 V1 V2V ∗3 V ∗2 V1 , (9)withV1 =Vcq∥asinh(q∥c′)cosh(q∥c′)− cos(qzc′), (10)V2 =Vcq∥asinh[q∥(c′ − d)] + e−iqzc′sinh(q∥d)cosh(q∥c′)− cos(qzc′)eiqzd , (11)V3 =Vcq∥asinh[q∥(c′ − 2d)] + e−iqzc′sinh(q∥2d)cosh(q∥c′)− cos(qzc′)eiqz2d , (12)and q∥ =√q2x + q2y. The RPA susceptibility is then computed asχ = (1− χ0Vq)−1 χ0 . (13)Figure 4 in the main text presents −Imχtot = −Im∑i,j χij , where we use a doping δ = 0.18, temperature T = 0,t′ = −0.26, t′′ = 0.13, tz = 0.07, tz0 = 0.1, iωn = ω + iΓ, Γ = 0.1, Vc = 280; the energy unit is t = 0.1 eV; the latticeparameters are a = 3.85 Å, c′ = 18.58 Å, and d = 0.86a = 0.178c′.V. Charge susceptibility with larger broadeningThe broadening of plasmons is controlled by Γ in the present theory. In principle, Γ should be infinitesimally small,but we may invoke a finite value, e.g., Γ = 0.1, to mimic intrinsic broadening due to incoherent effects from electroniccorrelations [5, 6]. Phenomenologically, the choice of Γ can be arbitrary. When we respect the broadening closer tothe experimental linewidths [see the green curve in Fig. 2(b)], we may choose a larger Γ = 1.0. In this case, we obtain4 !"#!$#!" %&'%()"!"*+%"+%) ,(-,""!"$"!." "!"$!%&'%()!+%",(/, %0%)#!$.!$.!""!$"1'2345%(26,"!." "!"$!%&'%()!+%",(7, %0%)#"!.""!"$"!%&'%()!+%)!,"!.""!"$"!%&'%()!+%)!,FIG. S3. Dynamical charge susceptibility computed for a large broadening parameter Γ = 1.0.Fig. S3, which should be compared with Fig. 4 of the main text, where a smaller Γ = 0.1 is employed. The ω3 andω− branches are combined into one broader branch, which well reproduces the experimental plasmon lineshapes.[1] A. Singh, H. Y. Huang, Y. Y. Chu, C. Y. Hua, S. W. Lin, H. S. Fung, H. W. Shiu, J. Chang, J. H. Li, J. Okamoto, C. C.Chiu, C. H. Chang, W. B. Wu, S. Y. Perng, S. C. Chung, K. Y. Kao, S. C. Yeh, H. Y. Chao, J. H. Chen, D. J. Huang, andC. T. Chen, Development of the Soft X-ray AGM–AGS RIXS beamline at the Taiwan Photon Source, J. Synchrotron Rad.28, 977 (2021).[2] J. Miyawaki, K. Fujii, T. Imazono, K. Horiba, Y. Ohtsubo, N. Inami, T. Nakatani, K. Inaba, A. Agui, H. Kimura, andM. Takahasi, Design of ultrahigh energy resolution rixs beamline at nanoterasu, J. Phys.: Conf. Ser. 2380, 012030 (2022).[3] T. Fujii, T. Watanabe, and A. Matsuda, Single-crystal growth of bi2sr2ca2cu3o10+δ (bi-2223) by tsfz method, J. Cryst.Growth 223, 175 (2001).[4] A. Griffin and A. J. Pindor, Plasmon dispersion relations and the induced electron interaction in oxide superconductors:Numerical results, Phys. Rev. B 39, 11503 (1989).[5] P. Prelovšek and P. Horsch, Electron-energy loss spectra and plasmon resonance in cuprates, Phys. Rev. B 60, R3735 (1999).[6] A. Nag, L. Zinni, J. Choi, J. Li, S. Tu, A. C. Walters, S. Agrestini, S. M. Hayden, M. Bejas, Z. Lin, H. Yamase, K. Jin,M. Garćıa-Fernández, J. Fink, A. Greco, and K.-J. Zhou, Impact of electron correlations on two-particle charge response inelectron- and hole-doped cuprates, Phys. Rev. Res. 6, 043184 (2024).