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[J. Schultz](https://orcid.org/0000-0001-5649-0983), [A. Lubk](https://orcid.org/0000-0003-2698-8806), F. Jerzembeck, [N. Kikugawa](https://orcid.org/0000-0003-3975-4478), M. Knupfer, [D. Wolf](https://orcid.org/0000-0001-5000-8578), [B. Büchner](https://orcid.org/0000-0002-3886-2680), [J. Fink](https://orcid.org/0000-0002-5286-1684)

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[Optical and acoustic plasmons in the layered material Sr2RuO4](https://mdr.nims.go.jp/datasets/956faabf-a543-46ec-9731-9204fecd9863)

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Optical and acoustic plasmons in the layered material Sr2RuO4Article https://doi.org/10.1038/s41467-025-58978-xOptical and acoustic plasmons in the layeredmaterial Sr2RuO4J. Schultz 1 , A. Lubk 1,2, F. Jerzembeck3, N. Kikugawa 4, M. Knupfer1,D. Wolf 1, B. Büchner 1,2 & J. Fink 1,2The perfect linear temperature dependence of the electrical resistivity in avariety of “strange” metals is a real puzzle in condensed matter physics. Forthese materials also other non-Fermi liquid properties are predicted ordetected. In particular we mention the results derived from holographic the-ories which conclude that plasmons should be overdamped due to a lowenergy continuum in the electronic susceptibility. These predictions weresupported by electron energy-loss spectroscopy in reflection on cuprates andruthenates. Here we use electron energy-loss spectroscopy in transmission tostudy collective charge excitations in the layermetal Sr2RuO4. Thismetal has atransition from a perfect Fermi liquid below T ≈ 30K into a “strange” metalphase above T ≈ 800K. In this compound we cover a complete range betweenin-phase and out-of-phase oscillations. Outside the classical range of electron-hole excitations, leading to a Landau damping, we observe well-defined plas-mons. The optical (acoustic) plasmondue to an in-phase (out-of-phase) chargeoscillation of neighbouring layers exhibits a quadratic (linear) positive dis-persion. Using amodel for the Coulomb interaction of the charges in a layeredsystem, it is possible to describe the range of optical plasmon excitations athigh energies in a mean-field random phase approximation without takingcorrelation effects into account. In contrast, resonant inelastic X-ray scatteringdata show at low energies an enhancement of the acoustic plasmon velocitydue to correlation effects. This difference can be explained by an energydependent effective mass which changes from ≈ 3.5 at low energy to 1 at highenergy near the optical plasmon energy. There are no signs of over-dampedplasmons predicted by holographic theories.“Strange” metals are at present one of the most interesting researchfields in solid state physics1. Due to the strong on-site interactionbetween their charge carriers, they show a deviation from a Fermi-liquid behavior, e.g., they do not show a quadratic but a linear tem-perature dependence of the electrical resistivity or linear in energyscattering rate in Angle-Resolved Photoemission Spectroscopy(ARPES)2,3. Moreover, there is no saturation at the Mott-Ioffe-Regellimit4. The unconventional and in some cases high-temperaturesuperconductivity detected in these materials is supposed to be rela-ted to their “strange” normal-state electronic structure. Doped cup-rates are prototypes of these “strange” metals. The non-Fermi liquidproperties could be explained by a continuum of excitations up to anultraviolet cutoff frequencyωc in the electronic susceptibility5, leadingto a phenomenological marginal Fermi liquid theory. Integrating overReceived: 2 February 2024Accepted: 7 April 2025Check for updates1Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstraße 20, 01069 Dresden, Germany. 2TU Dresden, Institute of Solid State andMaterials Physics, Haeckelstraße 3, 01069 Dresden, Germany. 3Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, D-01187Dresden, Germany. 4National Institute for Materials Science, Tsukuba 305-0003, Japan. e-mail: j.schultz@ifw-dresden.de; j.fink@ifw-dresden.deNature Communications |         (2025) 16:4287 11234567890():,;1234567890():,;http://orcid.org/0000-0001-5649-0983http://orcid.org/0000-0001-5649-0983http://orcid.org/0000-0001-5649-0983http://orcid.org/0000-0001-5649-0983http://orcid.org/0000-0001-5649-0983http://orcid.org/0000-0003-2698-8806http://orcid.org/0000-0003-2698-8806http://orcid.org/0000-0003-2698-8806http://orcid.org/0000-0003-2698-8806http://orcid.org/0000-0003-2698-8806http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0001-5000-8578http://orcid.org/0000-0001-5000-8578http://orcid.org/0000-0001-5000-8578http://orcid.org/0000-0001-5000-8578http://orcid.org/0000-0001-5000-8578http://orcid.org/0000-0002-3886-2680http://orcid.org/0000-0002-3886-2680http://orcid.org/0000-0002-3886-2680http://orcid.org/0000-0002-3886-2680http://orcid.org/0000-0002-3886-2680http://orcid.org/0000-0002-5286-1684http://orcid.org/0000-0002-5286-1684http://orcid.org/0000-0002-5286-1684http://orcid.org/0000-0002-5286-1684http://orcid.org/0000-0002-5286-1684http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58978-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58978-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58978-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58978-x&domain=pdfmailto:j.schultz@ifw-dresden.demailto:j.fink@ifw-dresden.dewww.nature.com/naturecommunicationsthis continuum yields a linear in energy imaginary part of the self-energy or scattering rate6. Electron Energy-Loss Spectroscopy (EELS) isa suitable experimental method to verify the existence of such a con-tinuum because it measures the imaginary part of the electronic sus-ceptibility = χðq,ωÞ� �. Here q is themomentum andω is the energy. Innearly-free electron metals, there exists a 2kF (with kF equal to theFermi wave vector) wide stripe of a continuum of particle-holeexcitations7 which starts at q ≈ω/vF, where vF is the Fermi velocity.In simple metals collective excitations (plasmons) exist belowthe critical wave vector qcrit ≈ ωP/vF which is determined by theplasmon energy ωP and vF. Above this momentum the plasmonsmerge into the continuum and therefore are (Landau) damped bya decay into particle-hole excitations. Usually, qcrit is between halfand one Å−1.Surprisingly, early transmission EELS (T-EELS) studies of thehighly correlated doped cuprates, using dedicated T-EELSspectrometers8–10 with high momentum resolution, showed thisbehaviorwith aweakdampingof plasmons below qcrit11–16. On the otherhand, T-EELS studies using transmission electron microscopes withweak momentum resolution17,18 showed no plasmon but a continuum.The difference can be easily explained by the poor momentum reso-lution in the TEMexperiments (Δq = 10 and 30Å−1) whichaverages overthe whole Brillouin zone (BZ) and therefore measures predominantlythe Landau continuum above qcrit.At variance with the early T-EELS measurements of collectiveexcitations in hole doped cuprates great attention attracted the pre-diction of over-damped plasmons in “strange” metals and a replace-ment of these excitations by a continuum19. The authors point out thata novel theoretical model of strongly interacting matter may benecessary. They propose, that such a model would be potentially alsorelated to high-Tc superconductivity. The over-damping of plasmons isexplained by holographic theories. Different from classical Landaudamping at higher momentum the latter predict strongly enhanceddamping also for long wavelength plasmons caused by quantum cri-tical fluctuations. Recently, this work was supported by similarcalculations20.Furthermore, there are several recent EELS experiments inreflection (R-EELS), supporting the theories which predict over-damped plasmons in “strange” or highly correlated metals. Only atvery small momenta a well-defined plasmon exists followed by atransition into a featureless momentum-independent constant-in-frequency continuum well below qcrit21–24. Moreover, there is a veryrecent ARPES study on doped cuprates in which these holographictheories are supported by an asymmetric line shape at higherenergies25.There are other differences between T-and R-EELS results fromhole-doped cuprates: at small q, i.e., long wave length, T-EELS datashow a positive dispersion which can be explained in RPA using anunrenormalized band structure11,12,14,26. On the other hand, R-EELS datashow a negative dispersion, which may indicate a more localizedelectron liquid. The hybridization of the d-bands with the s-band in thealkali metals or many-body effects, when moving from Na to Cs issupposed to turn the plasmondispersion frompositive to negative27,28.On the other hand, the different result between T-EELS and R-EELSpossibly can be explained by different response functions with respectto surface and bulk properties29,30.In this context, we mention that in various cuprates, well pro-nounced dispersive acoustic plasmons were detected by resonantinelastic X-ray scattering (RIXS)31–33. In all thesemeasurements, weaklydamped plasmons were detected for momentum ranging up to half ofthe size of the BZ.For understanding the differences between T- and R-EELS oncuprates and to understand the influence of correlation effects oncollective charge oscillation in general, we present here T-EELS data onthe relatedmetal Sr2RuO434,35. It is in someway intermediate between anormal Fermi liquid metal and a “strange” metal. Below ≈ 30K it is aperfect Fermi liquid which transforms at low temperatures Tc = 1.5 Kinto an unconventional superconductor36. It has other similarities tothe cuprates: it has a perovskite structure formed by transition metaloxides layers. The essentially 2D correlated electronic structure isformed by three bands and has a van Hove singularity close to theFermi level.Deviating from the cuprate high- Tc superconductors, it is a stoi-chiometric compound without crystallographic disorder due todopant ions. Furthermore, the temperature dependence of the trans-port properties are more complicated. Above T ≈ 30K there is acrossover region in which Sr2RuO4 exceeds at TMIR≈ 800K the Mott-Ioffe-Regel limit, i.e., it turns to a “bad metal”. These transport prop-erties are partially related to Hund’s rule coupling which causes strongcorrelation effects far from the insulating state37.Recently, we have studied the electronic structure of Sr2RuO4by an investigation of the optical plasmon excitations withmomenta parallel to the layers38 using a dedicated T-EELSspectrometer10. Also in this highly correlated material, a well-defined plasmon could be detected near 1.5 eV. The plasmon has apositive dispersion and decays into a continuum of particle-holeexcitations due to Landau damping, which could be explained inthe framework of the random phase approximation (RPA) usingan unrenormalized band structure.Most of the previous momentum dependent EELS studies onlayered materials were performed for a wave vector parallel to thelayers. The reason for this is that thin samples (T-EELS) or cleansurfaces (R-EELS) are easily prepared by a cleavage of the crystalsparallel to the layers. In the present work, by focused ion beammilling, we are able to prepare a thin electron transparent lamellain which the layers are perpendicular to the surface. Using suchsamples, we could map out a complete set of plasmon excitationswith momentum between parallel and perpendicular to the layersalmost in the entire BZ. In this way, it is possible to control the-oretical work on plasmon excitations in layered compounds,which is available since many decades39–42.At present, there is a strong discussion, whether spectroscopicresults on the damping and dispersion of plasmons in “strange”metalscan be explained on the basis of mean field theories such as RPA orwhether we need new theories to explain valence band EELS results(see also the recent EELS review43 which contrast the conflicting resultsof T-EELS andR-EELS). The present article strongly supports the resultsderived from T-EELS.The dynamic structure factor is determined by the Fouriertransformation of the charge density-density correlation function. Itcan be expressed44 by the dynamical susceptibility χ(q, ω)Sðq,ωÞ / = χðq,ωÞ� � / = � 1ϵðq,ωÞ� �: ð1ÞHere, ϵ(q, ω) is the complex dielectric function.The calculation of the Lindhard-Ehrenreich-Cohen susceptibilityof the many-body system of the charge carriers in solids is a challen-ging task. The susceptibility χ0 for a non-interacting one-band electronliquid is given by14,45χ0ðω,qÞ=ZBZMðq, kÞ 2FðkÞΔEðq, kÞðω+ iΓÞ2 � ΔE2ðq, kÞd3k: ð2ÞHere ΔE = Ek+q − Ek, Ek are the band energies of the electrons having amomentum k, M(q, k) is related to matrix elements, Γ is the lifetimebroadening of the particle-hole excitations, and F(k) is the Fermifunction.Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 2www.nature.com/naturecommunicationsWhile χ0 is the Lindhard-Ehrenreich-Cohen susceptibility forsingle-particle excitations related to an external field, stemming fromthe field of the scattering electron, χ is the susceptibility for the totalfield, including the induced one. Running a self-consistency cycle, weobtain in the mean field RPA the resultχRPAðq,ωÞ= χ0ðq,ωÞϵb � V ðqÞχ0ðq,ωÞ: ð3ÞHere, V(q) is the Fourier transformed Coulomb interactionbetween the charge carriers and ϵb is the background dielectric con-stant. The dielectric function can be calculated fromϵðq,ωÞ= ϵb � V ðqÞχ0ðq,ωÞ: ð4ÞIn this approximation and for small damping, there are in additionto the single-particle excitations collective excitations, termed plas-mons. The energy of the plasmon is determinedby the zerosof the realpart of the denominator of Eq. (3). The long wavelength energy of theplasmon in the RPA is given byωPð0Þ2 =4πNe2ϵbm*ð5Þwith N being the density of the charge carriers, m* the effective massand e the elementary charge.For small but finite momentum, up to q2 and within the RPA, thedispersion is given byωPðqÞ=ωPð0Þ+ARPAq2 + :::::;ARPA = ðA1 +A2Þ: ð6ÞA1 is related by the finite compressibility or the squared averagedFermi velocity of the electron liquid and is always positive. A2, which isalways negative, is proportional to the size of the effective mass26.For free-electron metals onlyA1 =1ωPð0Þ310v2F� �q ð7Þdetermines the optical plasmon dispersion. Here, the averagedsquared Fermi velocity along the q direction is defined by14v2F� �q =q_q∂Ek∂k�  2* +: ð8ÞFor metals where the band dispersion is strongly reduced by afinite effective mass enhancement m*/m0, the negative A2 may dom-inate the optical plasmon dispersion, leading in total to a negativedispersion26.In the following we discuss the structure factor V(q). In a homo-geneous electron systemV ðqÞ= 4πe2q2: ð9ÞFor a system, built up by 2D layers separated by the distance d we usethe Fetter model with the Coulomb potentialV ðqÞ=V ðqjj,q?Þ=4πe2q2qjjd2sinhðqjjdÞcoshðqjjdÞ � cosðq?dÞ, ð10Þwhereq∣∣ (q⊥) is themomentumparallel (perpendicular) to the layers40.Therefore, in a layered system, the plasmon dispersion is not onlydetermined by the compressibility and by the effective mass of theelectron liquid (see Eq. (6)) but also by the structure factor V(q),depending on q∣∣d and q⊥dωPðqÞ2 = s2q2 +ωPð0Þ2qjjd2sinhðqjjdÞcoshðqjjdÞ � cosðq?dÞð11Þwith s2 = 12 v2F� �q. In the low q approximation Eq. (11) yields again aquadratic plasmon dispersionωPðqÞ=ωPð0Þ 1 +12112d2 +s2ω0ð0Þ2" #q2jj !: ð12ÞFor a dielectric function that is consistent with the above dispersionrelation we refer to the Method section. For small q∣∣ and q⊥d =0, thestructure factor V(q∣∣, q⊥) is the same as in the homogeneous electrongas. In this case, the charge oscillations in the layers are in phase andthe plasmon dispersion is the same as in a homogeneous 3D electronsystem. For q⊥d =π the charge oscillations between neighboring layersare out of phase.This leads to an acoustic plasmon, with a rather small energy gapdue to the layer interaction46 in the long wavelength limit. The dis-persion of the acoustic plasmon for small q∣∣ is given byωpðqjjÞ=qjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 +ωpð0Þ2d24s: ð13ÞVery often, thefirst term in the square root is considerably smaller thanthe second. Then the phase velocity of the acoustic plasmon along thedirection of q∣∣ is given byvp =ωpð0Þd2: ð14ÞBesides the hydrodynamic Fetter model, the plasmon dispersion of alayered system was also derived by means of RPA, leading to similarresults47.ResultsT-EELSmeasurementswereperformedon single-crystalline Sr2RuO4 atroom temperature. Figure 1 shows the crystal structure of Sr2RuO4which consists of RuO2 layers stacked with SrO spacer layers along thec-axis direction. The thin films were characterized by in-situ electrondiffraction and the crystallographic axes were oriented with respect tothe (a, c)-plane (see Fig. 2a). The distance d = 6.36 Å between the RuO2layers is half the c-axis lattice constant. The thin lamellas for T-EELSwith a normal direction parallel to the b-axis were cut with a focusedion beam. In this way, EELS experiments were possible with themomentum parallel to the (a, c)-plane. We emphasize that thismomentum range is different from our previous EELS study onSr2RuO438 where we covered the momentum range in the (a, b)-plane.T-EELS was performed with a primary electron energy of 80 keVand with an energy and momentum resolution of 120meV and0.04Å−1, respectively. The momentum-resolved EELS data weresequentially recorded in the (q∣∣ = qa, q⊥) momentumplane as depictedin Fig. 2b. In Fig. 3 we show typical EELS intensities as a function of theenergy for various q∣∣ and q⊥ values. With increasing q∣∣ the plasmonenergy slightly increases, whereas for increasing q⊥ the plasmonenergies decrease. At low total momentum it is difficult to see a welldefinedplasmondue to thehigh intensity of the quasi-elastic peak. Thesame holds for high totalmomentumbecause the T-EELS cross sectionis decreasing with 1/q2 (see below). Except for the described cases, welldeveloped and dispersing plasmon excitations below 1.8 eV are visible.From the fit to the loss data with a Drude function, we obtain theenergy, the width, and the intensities (see Methods).The energy is plotted in Fig. 4 as a function of q∣∣ for various q⊥values together with theoretical curves calculated in the framework ofArticle https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 3www.nature.com/naturecommunicationsthe Fetter model (see Section II). We use the parametersωp(0) = 1.48 eV (from optical spectroscopy48), d = 6.36Å, andv2F� �100 = 4.91 (eV Å)2. The latter value for the three Ru 4d t2g bandscrossing the Fermi level was derived from a tight-binding (TB) bandstructure49 (see Methods). For all plasmon energies above the con-tinuum, within error bars, there is rather good agreement betweentheory and experiment. There is a continuous transition between theoptical plasmon for q⊥ =0 (purple data) at higher energy to theacoustic plasmon (q⊥ =0.4Å−1) close to q⊥ =π/d = 0.49Å−1 (red data) atlower energy. Due to our finite energy resolution we cannot follow theacoustic plasmon to zero energy. Furthermore, the momentum reso-lution of the instrument is limited by the finite width of the collectionaperture of our spectrometer. Since the signal is integrated over thelatter we observe a drop of the plasmon energy also for the opticalplasmon near q⊥ =0 (see purple data in Fig. 4).The difference between extrapolated RPA plasmon dispersionand experimental data for q∣∣≥ qcrit (see Fig. 4) can be explained byLandau damping. For these wave vectors, the plasmon decays intoover-damped plasmon excitations and into spectral weight which iscaused by the Lindhard continuum. This leads to a reduction of theenergy of the maximum in the total loss function (see Fig. 5). Thisreduction has been observed in our calculations presented in ref. 38. Asimilar behavior has been also observed for Al50. In Fig. 4 we also showthe continuum of the single-particle intra-band transitions χ0 (see thegray region) calculated using an unrenormalized TB band structure49(see Methods). The optical plasmon merges into the continuum nearq∣∣ ≈0.4Å−1. We assign this value to the critical momentum qcrit. Thisvalue is further corroborated by the rapid increase of the plasmonwidth at this wave vector (see Fig. 6).DiscussionOptical and acoustic plasmons in layered “strange” metalsThe data presented in Fig. 3 demonstrate that T-EELS in layered sys-tems is capable of probing not only in-phase (optical) plasmons withmomentum parallel to the layers but also out-of phase (acoustic)collective charge oscillations in neighboring layers. A suitable samplepreparation is important for such studies. Thus, we show that T-EELScan compete with recent RIXS studies of acoustic plasmons ofcuprates31,32,46. This is an important extension of T-EEL spectroscopy.Furthermore, we emphasize that the present work shows that acousticplasmons exist also in non-cuprate “strange" metal layer systems.Well defined plasmons exist in the complete momentum rangewhich is not covered by the range of single-particle excitations calcu-lated in the mean-field RPA theory. Optical plasmons exist in ≈ 15% ofthe BZ. The rest is determined by a continuum of intra-band singleparticle excitations, which strongly dampen the plasmon excitation inexcellent agreement with previous studies38. There is no sign of areduction of the coherent plasmon range due to an over-dampingdiscussed in terms of holographic theories19,20.Fig. 1 | Schematic of the charge oscillations in the layered material. Crystalstructure of Sr2RuO4 (left hand site) and schematic representation of in-phase andout-of-phase two dimensional charge oscillations (right hand site).Fig. 2 | Electrondiffractionon the Sr2RuO4 crystal. a Indexed electron diffractionpattern (white dots) in the (q∣∣, q⊥) plane, including the reduced Brillouin zone(green) for equal layers along the c-axis. b Brown momentum range [see (a)] inwhich loss spectra are recorded for various q∣∣ and q⊥ values. The colors purple,blue, green, yellow, and red correspond to q⊥ equal 0, 0.1, 0.2 0.3, and 0.4Å−1,respectively. The diameter of the filter entrance aperture [indicated by the coloredcircles in (b)] defining the momentum range in one EEL spectrum (momentumresolution) corresponds to a momentum of 0.04Å−1.Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 4www.nature.com/naturecommunicationsNext we discuss the dispersion of the optical plasmon in themomentum range q∣∣ ≤ qcrit ≈0.4Å−1 (see Fig. 4 purple data). It can bewell fitted by Eqs. (1) and (12), relations which are derived by a mean-field theory in the Fetter-Apostol model40,47. We obtain a dispersioncoefficient Aexp = 2.1 ± 0.2 eV Å2. Calculating the dispersion coeffi-cient by Eq. (12) using v2F� �q =4:91 (eV Å)2 we obtain A = 2.8 eV Å2. Thesmall difference between experimental and calculated dispersioncoefficients can be explained e.g. by a 1.5Å thickness of the RuO2layers which reduces the nominal d to dred = 4.85Å. This reductionbrings the theoretical A value very close to the experimental one.Moreover, our calculations show that an enlargement of the halfwidth of Γ from about 0.1 to the experimental value of ≈ 1 eV reducesthe dispersion coefficient A by ≈0.2 eVÅ2. Thus the experimentaloptical plasmon dispersion can be well described by an unre-normalized band structure. Moreover, using a Fermi velocity which isreduced by an effective massm*/m0 = 3.5, we derive a correspondingFig. 3 | Electron energy-loss intensities for various q∣∣ and q⊥ values. The indi-cated numbers (1–6) and the colors correspond to themomentum values depictedin Fig. 2. Experimental spectra are depicted twice (normal scale + 30x/60x scaledup) to show both the zero loss peak and the plasmon peak. The black dashed linescorrespond to fits of a superposition of a Voigt profile (zero loss), a Drude function(plasmons), and a background (see Methods).Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 5www.nature.com/naturecommunicationsdispersion coefficient A = 0.8 eVÅ2 which is at variance with theexperimental data.In Fig. 5 we compare the T-EELS data of the optical plasmon dis-persion with those derived from theoretical calculations of the lossfunction [Eq. (1)]. For the susceptibility [Eq. (2)]weuse the tight bindingband structure from ref. 49. The dielectric function is derived using Eq.(4) together with the Coulomb potential of a layered electronic systemgiven in Eq. (10). A small Γ =0.1 eV is used to better visualize the smalldispersion. In Fig. 5wepresent the results for the susceptibility, the lossfunction, and the optical plasmon dispersion for an unrenormalizedband structure (m*/m0 = 1), a constant effective mass (m*/m0 = 3.5), andan energy dependent effectivemass ranging from3.5 at low energies to1 above ≈0.2 eV. The latter was taken from optical spectroscopy48together with an extrapolation to 1 at higher energies.For the unrenormalized band structure (m*/m0= 1) and for theone, which is renormalized at low energies only, the calculations pre-dict a classic plasmon dispersion, which merges at qcrit ≈0.4Å−1 intothe Lindhard continuum (see columns 2 and 3). In both cases (see row 1and 3, column 4) the calculated dispersion agrees well with theexperimental one. However, for the renormalized band structure,using the low-energy averagem*/m0= 3.5 from optical spectroscopy48,the calculated data deviates considerably from the experimental one.The continuum is strongly lowered in energy, preventing the mergingof the plasmon into a continuum and hence as a strong increase of thepeak width above qcrit. Moreover, the plasmon dispersion is reduced,in disagreement with the experiment. The effective mass dependenceof the plasmon dispersion was already discussed in our previouspaper38.Using the energy dependent effective mass from optical spec-troscopy below ω =0.2 eV plus an extrapolation to m*(ω)/m0 = 1 athigher energies, we see the expected strong renormalization and anintensity increase in the low energy/momentum range. However, thesusceptibility is similar to that calculated from an unrenormalizedband structure at higher energy/momentum. This yields a plasmondispersion, which is very close to an unrenormalized dispersion. Thusour present and the previous experimental data of the dispersion ofthe optical plasmon indicates that the long-wavelength dispersion canbe explained in the framework of a mean-field RPA theory using aneffective mass of one. Because an energy dependent effective mass isexpected also for the cuprates, the present result can potentially alsoexplain the unrenormalized plasmon dispersion detected in thecuprates14,26.The calculations clearly demonstrate that the low energy renor-malization of the susceptibility/optically conductivity does not trans-fer into the high-energy plasmon dispersion. A similar behavior waspredicted for the case of electron-phonon coupling51. This was alsodiscussed in a standard solid state text book52 where it was stated thatwell above the Debye energy phonons do not renormalize the bandstructure. On the other hand, the renormalized acoustic plasmondispersion observed by RIXS at low energies in cuprates53 can beexplained in this framework of a renormalized band structure.In Fig. 6 we present the optical plasmon width as a function of q∣∣.We show data derived from Fig. 3 and from our previous EELSexperiments38. Within error bars there is a good agreement betweenthe two datasets. The plasmon width at zero momentum is smallerthan the plasmon energy, indicating a coherent collective chargeexcitation. The width below q∣∣ ≈0.4Å−1 is nearly constant. Nearq∣∣ = 0.4Å−1 the width increases, indicating the merging of the plasmondispersion into the single particle continuum at a qcrit ≈ 0.4Å−1 (Seealso Fig. 4). We compare the experimental results with those derivedfrom the calculated loss function. For the effective mass equal to 3.5there is no merging of the plasmon line into the continuum becausethe latter is strongly lowered in energy (see also Fig. 5). On the otherhand, the calculation for an unrenormalized effective mass using anoffset of 1 eV is in qualitative agreement with the experiment. Theoffset will be explained in detail below. The difference in qcrit of 0.1Å−1may be caused by the absence of spin-orbit interaction and matrixelements in the calculations.This indicates again that at the relative high plasmon energy, theeffective mass is close to one and supports the formation of resilientquasi-particles48 which were predicted by DFT+DMFT calculations54.Already in our previous paper38, we pointed out that the plasmonwidth below qcrit is nearly constant. From this we conclude that theplasmon width in our system is not related to electron-electroninteraction, since for the latter a quadratic increase of the width,starting at zero energy,wouldbeexpectedwith increasingmomentumtransfer55. Furthermore, the finite width at zero energy cannot beexplained in terms of a temperature dependent scattering rate, pro-portional to the imaginary part of the self-energy (ℑΣ). Recent ARPESexperiments on Sr2RuO456 report near 20K ℑΣ ≈0.01 eV and at roomtemperature ℑΣ ≈0.15 eV. These values are well below the observedwidth of 1 eV. Essentially, no temperature dependence of thewidth hasbeen detected in our previous experiments38. In addition, the finitewidth at zero energy is likely not caused by electron-phononinteraction57 nor by impurity scattering (different for super-conducting cuprates, Sr2RuO4 is a stoichiometric compound withoutdoping ions).A smaller width in comparison to the energy hints that the over-damping is not caused by quantum critical fluctuations which is inconflict with theoretical predictions19. In fact, it is caused by a decayinto interband transitions27,58–60, as it is the case for the majority ofmetallic systems investigatedbyT-EELS. The origin of these interbandsare back-folded bands from the second to the first BZ due to a finiteFig. 4 | Plasmon dispersion along the momentum q∣∣ parallel to the layers forseveral q⊥ perpendicular to the layers (squares) together with calculationswithin the framework of the Fettermodel (solid lines). The q⊥ values 0, 0.1, 0.2,0.3, and 0.4 Å−1 are marked by purple, blue, green, yellow, and red color symbols,respectively (see Fig. 3). We have added also the data from the optical plasmondispersion derived in our previous publication (dark purple diamonds)38. Thehorizontal error bars originate from the finite momentum resolution while thevertical ones are related to the finite spectral resolution of the EEL spectrometer(the fitting error is small in comparison). The regionmarked in gray corresponds tothe susceptibility χ0 calculated from a tight binding band structure (see below andMethods). The excitations in the continuum range are marked by open symbols.The colored solid (dashed) curves correspond to theoretical data consider-ing (neglecting) finite momentum resolution, i.e., integration within the EEL col-lection aperture.Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 6www.nature.com/naturecommunicationspseudo-potential. In recent RIXS data on p- and n-typed cuprates53, thedifference in acoustic plasmon width could be described in thisframework.In Fig. 7 we depict the plasmon intensities as a function of q∣∣ andq⊥ (seeMethods section for details of the evaluation). The decay of thetotal spectral weight of the plasmon resonances at large momentumtransfers is approximately proportional to q−2 as observed for con-ventional bulk plasmons and predicted by the longitudinal f-sum ruleand the theoretical dielectric function of the Fetter model (see Meth-ods). The decay observed for q⊥ ≠0 in the longwavelength limit is alsoconsistent with the theoretical dielectric response as well as a versionof the long wavelength sum rule implying that the full spectral weightat q = 0 is concentrated in the bulk plasmon q⊥ =0mode (seeMethodsand ref. 61 for the sum rules). The shift of the maximal integrated lossintensity towards smaller momentum transfer predicted by theorymay be attributed to shortcomings of the Fetter model dielectricfunction (see Methods), impact of the zero loss, and limited experi-mental resolution.In the following we discuss the acoustic plasmon data (see reddata and line in Fig. 4). Despite the gap at low energy due to a finiteenergy resolution, the dispersion extrapolates to zeroenergy typical ofan acoustic plasmon. The derived experimental plasmon velocity isvP ≈ 4.7 eV Å. From Eq. (13) we derive vP ≈ 4.8 eV Å in very goodagreement with the experimental value. This indicates that the firstterm in Eq. (13) due to the finite Fermi velocity is small compared to theterm which only depends on ωP(0) and d. Thus, for a given ωP(0) theacoustic plasmon dispersion only depends on d and is not influencedby a large Fermi velocity but hints to a reduced one due to correlationeffects (enhanced effective mass). Unfortunately, the finite energyresolution in the present T-EELS experiment does not allow a quanti-tative determination of the effective mass at low energies. However,low-energy RIXS data on cuprates indicate, that an enhanced effectiveFig. 5 | Calculated imaginary part of the susceptibility χ0(ω, q∣∣, q⊥ =0) (secondcolumn), loss function = � 1ϵðω,qjj ,q? =0Þn o(third column), and plasmon disper-sion (fourthcolumn) along themomentum (q∣∣,q⊥ =0). In the calculation variouseffective masses were used: first row m*/m0 = 1, second row m*/m0 = 3.5, and anenergy dependent effectivemassm*(ω)/m0 shown in the third row.The red labels inthe susceptibility in the first row are related to the origin of the three electronicbands in Sr2RuO4. In column four we compare the experimental optical plasmondispersion (shown in Fig. 4) with the calculated one for m*/m0 = 1 and 3.5, and aband structure which is renormalized only at low energies. The horizontal errorbars originate from the finite momentum resolution while the vertical ones arerelated to the finite spectral resolution of the EEL spectrometer.Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 7www.nature.com/naturecommunicationsmass is necessary to describe the plasmon velocity of the acousticplasmons53.Neglecting the Fermi velocity term, the linear acoustic plasmondispersion [see Eq. (13)] can be explained in the following way. Thephase difference π of the oscillations between neighboring layersreduces the plasmon energy from ωP(0) to zero. When adding amomentum q∣∣ the phase difference between neighboring layers isincreased toπ + q∣∣d and therefore, using a linear relation, the energy ofthe acoustic plasmon should increase by (ωp(0)dq∣∣)/πwhich is close toEq. (14).In summary, the long wavelength q∣∣ dispersion of the q⊥ depen-dent plasmons including the optical and the acoustic collective exci-tations and the decay of the optical plasmon by Landau damping canbe all explained in terms of a mean-field RPA model. It is possible tounderstand this interpretation of the present results in the followingway. Long wavelength charge excitations are not influenced by localinteractions such as on-site Coulomb and Hund’s exchange interac-tions. This behavior is different from ARPES studies, in which localproperties play an important role. In this context it is also important tonote that monopole (a single hole) excitations detected in ARPES aredifferently screened compared to dipole excitations recorded in EELS.We further emphasize that our present analysis of the acoustic plas-mon dispersion is also important for the understanding ofprevious31,32,46 and future RIXS studies on cuprates.PerspectivesThe present study has demonstrated that optical and acoustic plas-mons can be investigated by T-EELS in the complete BZ in layeredsystems. Therefore, with the advent of higher energy resolution,T-EELSwill be competitive at lower energies with RIXS, also taking intoaccount that momentum-resolved T-EELS provides a direct probe ofthe dynamic susceptibility. It will be possible to study in more detailthe different influence of correlation effects on optical and acousticplasmons, caused by an energy dependent effective mass. The latterwas predicted by a combined density functional/dynamical mean-fieldtheory (DFT + DMFT) calculation54 and experimentally detected byoptical spectroscopy48. Furthermore, in “strange” metals, it will bepossible to study low-energy and high-momentum charge excitationswhichwerepredicted in ref. 62 to dependon correlation effects. In thisway it will be possible to evaluate the spatial dependence of thedensity-density fluctuations in “strange” metals.MethodsDielectric response of the layered plasmon systemThe dielectric function corresponding to the Fetter model of a systemof coupled 2D layers reads40ϵðω,qjj,q?Þ= 1�2πNe2qjj=mω ω+ iΓð Þ � s2q2jjsinhðqjjdÞcoshðqjjdÞ � cosðq?dÞ: ð15ÞThis dielectric function is an approximation assuming a perturbationcharge that is confined to the 2D layers supporting the plasmons,which is violated by the electron beam resulting in deviations to theexperimental dielectric response. However, the intensities of theplasmon peaks can be derived from this dielectric function by calcu-lating the loss probability (T-EELS signal) using Eq. (1) and integrationalongω (see Fig. 7). A version of the longwavewavelength sum rule forthe dynamic susceptibility reads61limq!0=1ϵ q,ωð �( != � πωP2δ ω� ωP� �� δ ω+ωP� �� �: ð16ÞThus, in the q→0 limit, the dynamic susceptibility is determined by thelongitudinal plasmon.SamplesSr2RuO4 crystals were grown using the floating-zone method63. Thesuperconducting transition temperature of the sample was Tc = 1.5 K.Thin TEM lamellas of Sr2RuO4 with the normal pointing along theb-axis were prepared by Focused Ion Beam (FIB) using a Thermofisherinstrument. The target thickness of the lamellas was 80 nanometers.Low ion energy polishingwasused asfinal step to thinGa-ion damagedsurface layers.EELS measurementsThe momentum-resolved loss function was recorded at a non-monochromized Hitachi HF3300S at 60 kV acceleration voltage (pre-characterization) and a FEI Titan3 TEM equipped with a Wien mono-chromator and a Gatan Tridiem imaging filter (GIF) at 80 kV accel-eration voltage in a serial way (i.e., one EEL spectrum per fixedmomentum). The energy resolution is around 120meV (FWHMof zeroloss peak). At a camera length (i.e., effective distance between sampleand detector) of 1.15m, the GIF entrance aperture was used to selectthe different momentum values and covers a momentum range of0.04Å−1 (0.13mrad semi collection angle). The acquisition times arepresented in Table 1.Due to instabilities of the monochromator and the rather longcollection times required at large momentum transfers, the recordedspectra are subject to substantial noise as well as mutual randomfluctuations/offsets. In order to mitigate these effects, each spectrumFig. 6 | Plasmon width at half maximum in dependence on the momentumtransfer parallel to the layers at q⊥ =0, determined from a fit to the EELS data(see Methods). The horizontal error bars originate from the finite momentumresolution while the vertical ones are related to the residual error of the fit. Forcomparison the corresponding data from a Drude fit of our previous EELS mea-surements reported in ref. 38 was added as dark purple diamonds. The energy ofthe maximum of the excitations in the continuum range (above q∣∣ ≈0.4 Å−1) aremarked by open symbols. The experimental values are comparedwith data derivedfrom RPA calculations of the loss function for an unrenormalized (red line) and arenormalized (black line) band structure. For the direct comparison, the y-axiscorresponding to the calculated width is shifted by 1 eV to recognize decay intointerband transitions, not included in the calculations of the loss function. Fur-thermore, the width of the unrenormalized curve is evaluated for q∥ <0.55Å−1 onlydue to splitting of the spectral weight into a damped plasmon and a Lindhardcontinuum above qcrit (see Fig. 4 upper row, third column).Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 8www.nature.com/naturecommunicationswas separately aligned with the help of the quasi-elastic peak. Afteralignment of the zero loss position a superposition of the followingthree functions was fitted to the spectra. (i) an asymmetric Pseudo-Voigt-profile [V = c η 11 + ðω�ω0Þ2=ðλðωÞσÞ2+ 1� ηð Þ exp � lnð2Þ ω�ω0σ� �2n o
 �with λ(ω) = 1∣ω<=0 and λ(ω) > 1∣ω>0] reflecting the quasi elastic peakincluding ultra-low-loss excitations such as phonons, (ii) a Drude-likefunction I ωð Þ=a ω2PωΓω2�ω2Pð Þ2 + ωΓð Þ2corresponding to the plasmonpeaks and(iii) a phenomenological background following a linear function.Finally, the spectral positions and half widths of the plasmons werederived from the fitted parametersωP and Γ of theDrude function. Theintensity (Fig. 7) corresponds to the area under the fitted Drude peak,which is obtained by integrating the latter over ω.Calculation of the average Fermi velocity, susceptibility, andloss functionFor the calculation of the Fermi velocity along the [100] direction, thebare particle susceptibility χ0(q∣∣,ω) and the loss function we used a TBband structure49 based on an LDA band calculation. χ0(q∣∣, ω) is calcu-lated from a multi-band version of Eq. (2), taking only intra-bandexcitationswith the samematrix element into account, thus neglectinginter-band excitations which may be caused by a strong spin-orbitcoupling, which leads to a k-dependence of the orbital character of thebands64. Mainly the XY and the YZ bands contribute to v2F� �100 and χ0.The dielectric function and the loss functionwere calculated from Eqs.(1), (2), and (4). ϵb and the matrix element were fixed by using theplasmon energy from optical spectroscopy. For the calculation weused a half width of Γ =0.1 eV.Data availabilityAll data supporting the findings are provided as figures in the article.Datafiles for allfigures are available at https://opara.zih.tu-dresden.de/handle/123456789/1382 and from the corresponding authors onrequest.References1. Zaanen, J. Planckiandissipation,minimal viscosity and the transportin cuprate strange metals. SciPost Phys. 6, 061 (2019).2. Cooper, R. A. et al. Anomalous criticality in the electrical resistivityof La2−xSrxCuO4. Sci. 323, 603 (2009).3. Valla, T. et al. Evidence for quantum critical behavior in theoptimally doped cuprate Bi2Sr2CaCu2O8+δ. Sci. 285, 2110(1999).4. N. E. Hussey, K. T. & Takagi, H. Universality of the Mott-Ioffe-Regellimit in metals. Philos. Mag. 84, 2847 (2004).5. Varma, C. M., Littlewood, P. B., Schmitt-Rink, S., Abrahams, E. &Ruckenstein, A. E. Phenomenology of the normal state of Cu-Ohigh-temperature superconductors. Phys. Rev. Lett. 63, 1996(1989).Table 1 | Acquisition times (in sec) for the used momentumtransfers (in Å−1)q⊥ 0.08 0.12 0.16 0.33 0.49 0.66q∣∣0 2 5 5 5 10 100.1 5 5 5 5 10 100.1 5 5 5 5 10 100.3 10 10 10 10 10 100.4 10 10 10 10 10 10.Fig. 7 | Integrated loss intensities under the fitted plasmon peaks as a functionof in-plane (q∣∣) and out-of-plane (q⊥) momentum transfer. The horizontal errorbars originate from the finite momentum resolution while the vertical ones arerelated to the residual error of the fit. The q⊥ values 0, 0.1, 0.2, 0.3, and 0.4 Å−1 aremarked by purple, blue, green, yellow, and red color symbols, respectively (seeFig. 3). The q∣∣ values 0.08, 0.12, 0.16, 0.33, 0.49, and 0.66Å−1 are marked by tipdown triangles, squares, tip top triangles, circles, diamonds, and pentagons,respectively. The dashed lines and open symbols indicate the single particle con-tinuum regime. The insets show the dependency predicted by the coupled 2Dplasmon model due to ref. 40. At q⊥ = q∣∣ =0 the curves diverge indicated by adashed line.Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 9https://opara.zih.tu-dresden.de/handle/123456789/1382https://opara.zih.tu-dresden.de/handle/123456789/1382www.nature.com/naturecommunications6. Norman,M. R. & Pépin, C. Theelectronic nature of high temperaturecuprate superconductors. Rep. Prog. in Phys. 66, 1547 (2003).7. Lindhard, J. Kgl. Danske Videnskab. Selskab, Mat-phs. Medd 28,8 (1954).8. Raether, H. Excitation Of Plasmons And Interband Transitions ByElectrons 1st edn, Vol. 198 (Springer Verlag, Berlin, 1980).9. Schnatterly, S. Inelastic electron scattering spectroscopy. SolidState Phy. 34, 275–358 (1979).10. Fink, J. Recent developments in energy-loss spectroscopy. Adv.Electron Phys. 75, 121–232 (1989).11. Nücker, N. et al. Plasmons and interband transitions inBi2 Sr2CaCu2O8. Phys. Rev. B 39, 12379 (1989).12. Wang, Y.-Y., Feng,G.&Ritter, A. L. Electron-energy-loss andoptical-transmittance investigation of Bi2 Sr2 CaCu2O8. Phys. Rev. B 42,420 (1990).13. Romberg, H. et al. Dielectric function of YBa2Cu3O7−δ between 50meV and 50 eV. Zeitschrift für Physik B Condensed Matter. 78,367 (1990).14. Nücker, N., Eckern, U., Fink, J. & Müller, P. Long-wavelength col-lective excitations of charge carriers in high-Tc superconductors.Phys. Rev. B 44, 7155 (1991).15. Knupfer, M., Roth, G., Fink, J., Karpinski, J. & Kaldis, E. Plasmondispersion and the dielectric function in YBa2Cu4O8 single crystals.Phys. C Supercond. 230, 121 (1994).16. Roth, F., Revcolevschi, A., Büchner, B., Knupfer, M. & Fink, J. Evi-dence for an orbital dependent Mott transition in the ladders of (La,Ca)xSr14−xCu24O41 derived by electron energy loss spectroscopy.Phys. Rev. B 101, 195132 (2020).17. Terauchi, M. et al. Electron-energy-loss spectroscopy of oxidesuperconductor Bi2Sr2CaCu2O8. Jpn. J. Appl. Phys. 34, L1524(1995).18. Terauchi, M., Tanaka, M., Tsuno, K. & Ishida, M. Development of ahigh energy resolution electron energy-loss spectroscopy micro-scope. J. Microsc. 194, 203–209 (1999).19. Romero-Bermúdez, A., Krikun, A., Schalm, K. & Zaanen, J. Anom-alous attenuation of plasmons in strange metals and holography.Phys. Rev. B 99, 235149 (2019).20. Van den Eede, S. T., van Stralen, T. J. N., Flipse, C. F. J. & Stoof, H. T.C. Plasmons in a layered strange metal using the gauge-gravityduality. Phys. Rev. B 109, 085119 (2024).21. Mitrano,M. et al. Anomalous densityfluctuations in a strangemetal.Proc. Natl. Acad. Sci. USA 115, 5392 (2018).22. Husain, A. A. et al. Crossover of charge fluctuations across thestrange metal phase diagram. Phys. Rev. X 9, 041062 (2019).23. Husain, A. A. et al. Pines’demonobserved as a 3D acoustic plasmonin Sr2RuO4. Nature 621, 66–70 (2023).24. Chen, J. et al. Consistency between reflectionmomentum-resolvedelectron energy loss spectroscopy and optical spectroscopy mea-surements of the long-wavelength density response ofBi2Sr2CaCu2O8+x. Phys. Rev. B 109, 045108 (2024).25. Mauri, E., Smit, S., Golden, M. S. & Stoof, H. T. C. Gauge-gravityduality comes to the laboratory: Evidence of momentum-dependent scaling exponents in the nodal electron self-energy ofcuprate strange metals. Phys. Rev. B 109, 155140 (2024).26. Grigoryan, V. G., Paasch, G. & Drechsler, S.-L. Determination of aneffective one-electron spectrum from the plasmon dispersion ofnearly optimally dopedBi2Sr2CaCu2O8. Phys. Rev. B60, 1340 (1999).27. vom Felde, A., Sprösser-Prou, J. & Fink, J. Valence-electron excita-tions in the alkali metals. Phys. Rev. B 40, 10181 (1989).28. Fleszar, A., Stumpf, R. & Eguiluz, A. G. One-electron excitations,correlation effects, and the plasmon in cesium metal. Phys. Rev. B55, 2068 (1997).29. Damascelli, A. et al. Fermi surface, surface states, and surfacereconstruction in Sr2RuO4. Phys. Rev. Lett. 85, 5194 (2000).30. Chandrasekaran, A. et al. On the engineering of higher-order vanHove singularities in two dimensions. Nat. Commun. 15, 9521(2024).31. Hepting,M. et al. Three-dimensional collective chargeexcitations inelectron-doped copper oxide superconductors. Nature 563,374 (2018).32. Nag, A. et al. Detection of acoustic plasmons in hole-dopedlanthanum and bismuth cuprate superconductors using reso-nant inelastic X-Ray scattering. Phys. Rev. Lett. 125, 257002(2020).33. Singh, A. et al. Acoustic plasmons and conducting carriers in hole-doped cuprate superconductors. Phys. Rev. B 105, 235105(2022).34. Maeno, Y. et al. Superconductivity in a layered perovskite withoutcopper. Nature 372, 532 (1994).35. Bergemann, C., Mackenzie, A. P., Julian, S. R., Forsythe, D. &Ohmichi, E. Quasi-two-dimensional Fermi liquid properties of theunconventional superconductor Sr2RuO4. Adv. Phys. 52,639 (2003).36. Mackenzie, A. P. et al. Extremely strong dependence of super-conductivity on disorder in Sr2RuO4. Phys. Rev. Lett. 80, 161(1998).37. de’ Medici, L., Mravlje, J. & Georges, A. Janus-faced influence ofHund’s rule coupling in strongly correlated materials. Phys. Rev.Lett. 107, 256401 (2011).38. Knupfer, M., Jerzembeck, F., Kikugawa, N., Roth, F. & Fink, J. Pro-pagating charge carrier plasmons in Sr2RuO4. Phys. Rev. B 106,L241103 (2022).39. Grecu, D. Plasma frequency of the electron gas in layered struc-tures. Phys. Rev. B 8, 1958 (1973).40. Fetter, A. L. Electrodynamics of a layered electron gas. II. Periodicarray. Ann. Phys. 88, 1 (1974).41. Das Sarma, S. &Quinn, J. J. Collective excitations in semiconductorsuperlattices. Phys. Rev. B 25, 7603 (1982).42. Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layeredhigh-Tc cuprates. Phys. Rev. B 94, 075139 (2016).43. Abbamonte, P. & Fink, J. Collective charge excitations studied byelectron energy-loss spectroscopy. Ann. Rev.Condensed MatterPhys. 16, 465 (2025).44. Platzman, P. M. & Wolff, P. A.Waves And Interactions In Solid StatePlasmas Vol. 304 (Academic Press, New York, 1973).45. Ehrenreich, H. & Cohen, M. H. Self-consistent field approach to themany-electron problem. Phys. Rev. 115, 786 (1959).46. Hepting, M. et al. Gapped collective charge excitations and inter-layer hopping in cuprate superconductors. Phys. Rev. Lett. 129,047001 (2022).47. Apostol, M. Plasma frequency of the electron gas in layered struc-tures. Z. für Phys. B Condens. Matter 22, 13 (1975).48. Stricker, D. et al. Optical response of Sr2RuO4 reveals universalfermi-liquid scaling andquasiparticles beyond Landau theory. Phys.Rev. Lett. 113, 087404 (2014).49. Liebsch, A. & Lichtenstein, A. Photoemission quasiparticle spectraof Sr2RuO4. Phys. Rev. Lett. 84, 1591 (2000).50. Batson, P. & Silcox, J. Experimental energy-loss function, Im[− 1/ε(q,ω)], for aluminum. Phys. Rev. B 27, 5224 (1983).51. Engelsberg, S. & Schrieffer, J. R. Coupled electron-phonon system.Phys. Rev. 131, 993 (1963).52. Ashcroft, N. W. & Mermin, N. D. Solid State Physics, Vol. 848(Saunders College, 1976).53. Nag, A. et al. Impact of electron correlations on two-particle chargeresponse in electron- and hole-doped cuprates. Phys. Rev. Res. 6,043184 (2024).54. Deng, X. et al. Howbadmetals turn good: Spectroscopic signaturesof resilient quasiparticles. Phys. Rev. Lett. 110, 086401 (2013).Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 10www.nature.com/naturecommunications55. DuBois, D. F. & Kivelson, M. G. Electron correlational effects onplasmon damping and ultraviolet absorption in metals. PhysicalReview 186, 409 (1969).56. Hunter, A. et al. Fate of quasiparticles at high temperature in thecorrelated metal Sr2RuO4. Phys. Rev. Lett. 131, 236502 (2023).57. Sturm, K. Electron energy loss in simple metals and semi-conductors. Adv. Phys. 31, 1 (1982).58. Paasch, G. Influence of interband transitions on plasmons in thealkali metals: pseudopotential calculation. Phys. stat. sol. 38,K123 (1970).59. Gibbons, P. C. &Schnatterly, S. E. Comment on the line shapeof theplasma resonance in simple metals. Phys. Rev. B 15, 2420 (1977).60. Sturm, K. & Gusarov, A. Dynamical correlations in the electron gas.Phys. Rev. B 62, 16474 (2000).61. Mahan, G. D.Many-Particle Physics, Third Edition Vol. 788 (Plenum,New York, 2000).62. Khaliullin,G.&Horsch, P. Theoryof thedensityfluctuation spectrumof strongly correlated electrons. Phys. Rev. B 54, R9600 (1996).63. Bobowski, J. S. et al. Improved single-crystal growth of Sr2RuO4.Condens. Matter 4, 6 (2019).64. Tamai, A. et al. High-resolution photoemission on Sr2RuO4 revealscorrelation-enhanced effective spin-orbit coupling and dominantlylocal self-energies. Phys. Rev. X 9, 021048 (2019).AcknowledgementsJ.F. thanks P. Abbamonte, R. von Baltz, S.-L. Drechsler, and A. Greco forhelpful discussions. This work is supported by a KAKENHI Grants-in-Aidsfor Scientific Research (Grants No. 18K04715, No. 21H01033, and No.22K19093), and Core-to-Core Program (No. JPJSCCA20170002) fromthe Japan Society for the Promotion of Science (JSPS) and by a JST-MiraiProgram (Grant No. JPMJMI18A3). J.S. received funding from the HOR-IZON EUROPE framework program for research and innovation undergrant agreement n. 101094299. A.L. and M.K. acknowledge fundingfrom the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation)-project-id 461150024. B.B. received funding from theWürzburg-Dresden Cluster of Excellence on Complexity and Topologyin Quantum Matter - ct.qmat (EXC 2147, project-id 390858490).Author contributionsJ.F., M.K., A.L., and B.B. conceived the experiment, J.S. and D.W. per-formed the EELS experiment. J.S. and J.F. analyzed the data. F.J. andN.K.prepared and characterized the samples. J.F., J.S., and A.L. calculatedthe susceptibility and the loss function. J.F., J.S., and A.L. wrote themanuscript with input from all authors.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-58978-x.Correspondence and requests for materials should be addressed toJ. Schultz or J. Fink.Peer review information Nature Communications thanks Nicolas Gau-quelin, and the other, anonymous, reviewers for their contribution to thepeer review of this work. A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article's Creative Commons licence, unlessindicated otherwise in a credit line to the material. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-58978-xNature Communications |         (2025) 16:4287 11https://doi.org/10.1038/s41467-025-58978-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Optical and acoustic plasmons in the layered material Sr2RuO4 Results Discussion Optical and acoustic plasmons in layered “strange” metals Perspectives Methods Dielectric response of the layered plasmon system Samples EELS measurements Calculation of the average Fermi velocity, susceptibility, and loss function Data availability References Acknowledgements Author contributions Funding Competing interests Additional information