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David Barcons Ruiz, Niels C.H. Hesp, Hanan Herzig Sheinfux, Carlos Ramos Marimón, Curdin Martin Maissen, Alessandro Principi, Reza Asgari, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Marco Polini, Rainer Hillenbrand, Iacopo Torre, Frank H.L. Koppens

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[Experimental signatures of the transition from acoustic plasmon to electronic sound in graphene](https://mdr.nims.go.jp/datasets/fdf7f4c2-38f8-4dbb-b6a1-2b4d1cba3477)

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Experimental signatures of the transition from acoustic plasmon to electronic sound in graphenePHYS ICSExperimental signatures of the transition from acousticplasmon to electronic sound in grapheneDavid Barcons Ruiz1, Niels C.H. Hesp1, Hanan Herzig Sheinfux1, Carlos Ramos Marimón1,Curdin Martin Maissen2†, Alessandro Principi3, Reza Asgari4,5, Takashi Taniguchi6,Kenji Watanabe7, Marco Polini1,8,9, Rainer Hillenbrand2,10,11, Iacopo Torre1*,Frank H.L. Koppens1,12*Fermi liquids respond differently to perturbations depending on whether their frequency is higher (collisionlessregime) or lower (hydrodynamic regime) than the interparticle collision rate. This results in a different phasevelocity between the collisionless zero sound and the hydrodynamic first sound. We performed terahertz pho-tocurrent nanoscopy measurements on graphene devices, with a metallic gate close to the graphene layer, toprobe the dispersion of propagating acoustic plasmons, the counterpart of sound modes in electronic Fermiliquids. We report the observation of a change in the plasmon phase velocity when the excitation frequencyapproaches the electron-electron collision rate that is compatible with the transition between the zero andthe first sound mode.Copyright © 2023 TheAuthors, somerights reserved;exclusive licenseeAmerican Associationfor the Advancementof Science. No claim tooriginal U.S. GovernmentWorks. Distributedunder a CreativeCommons AttributionNonCommercialLicense 4.0 (CC BY-NC).INTRODUCTIONThe Fermi liquid paradigm (1, 2) is one of the cornerstones ofmodern condensed matter theory, providing an effective descrip-tion of the many-body systems whose elementary excitations areweakly interacting fermionic quasi-particles. The theory of Fermiliquids provides an understanding of why conduction electrons inmetals behave essentially as noninteracting particles.Fermi liquids can support collective modes in the form of longi-tudinal density oscillations that are analogous to sound in classicalfluids. Their propagation depends on whether the angular frequen-cy ω of the mode is higher or lower than the interparticle collisionrate (3) τ� 1coll. Liquid3He, a neutral Fermi liquid, was the first systemin which the transition (a change in the phase velocity and attenu-ation of the propagating mode) from the first sound mode(ω� τ� 1coll, i.e., in the hydrodynamic regime) to the zero soundmode (ω� τ� 1coll, i.e., in the collisionless regime) was observed (4).In electronic Fermi liquids with long-range Coulomb interac-tions, where the electron-electron (ee) scattering time τee playsthe role of τcoll, first and zero sound collapse into a plasmonmode (5). In such a mode, the smooth crossover from the collision-less to the hydrodynamic regime manifests in the dispersionrelation ω(q) only at subleading order in the wave vector q of themode (5), and it is therefore very challenging to observe.However, two-dimensional (2D) electron liquids allow for sufficientscreening of the long-range part of the Coulomb interaction by anearby metallic gate electrode, the first sound and zero sound reap-pear (5–7), and a transition between the two can be observed. In 2Delectron liquids with screened ee interactions, the zero sound modeis known as acoustic plasmon and has been extensively studied ex-perimentally in hexagonal boron nitride (hBN)–encapsulated gra-phene devices (8, 9). However, to the best of our knowledge, theelectronic first sound mode, the closest electronic analog of ordi-nary sound, has never been observed experimentally.In this work, we report a change in the phase velocity of acousticplasmons that is compatible with the transition between an acousticplasmon and the electronic first sound. We probe this transition atroom temperature (RT) using a terahertz (THz) source whoseangular frequency ω can be tuned around the ee collision rateτ� 1ee , with τee being 0.1 to 0.2 ps in doped graphene (10–12).Figure 1A and movies S1 and S2 show the evolution of the distribu-tion function of electrons during the propagation of a plasmonmode. While for ω� τ� 1ee the distribution function differs signifi-cantly from the equilibrium one, for ω� τ� 1ee ee, collisions havetime to smooth the distribution to a circle, leading to a quasi-equi-librium, fluid-like response (13). This results in a change in thephase velocity vp of the mode between the two regimes (seeFig. 1B) from the collisionless value vc to the hydrodynamicvalue vh.The plasmon velocity difference has, at long wavelength, an in-tuitive physical interpretation thanks to an analogy with viscoelasticmaterials (14). Materials respond as solids (with an elastic shearforce) to shear deformations that are faster than a certain equilibra-tion time scale τcoll and as fluids (with a dissipative shear force) toshear deformations slower than τcoll. This time scale greatly varies(even by orders of magnitude) depending on the material and di-verges for ordered solids. Electrons are no exception to this behav-ior, τcoll = τee. In the collisionless regime, the elastic shear force,which is not present in the hydrodynamic regime, adds to the1 ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science andTechnology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain.2CIC nanoGUNE, 20018 Donostia-San Sebastián, Spain. 3Department of Physicsand Astronomy, The University of Manchester, M13 9PL Manchester, UK. 4Schoolof Physics, Institute for Research in Fundamental Sciences, IPM, Tehran 19395-5531, Iran. 5School of Physics, University of New South Wales, Kensington, NSW2052, Australia. 6International Center for Materials Nanoarchitectonics, National In-stitute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 7ResearchCenter for Functional Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba 305-0044, Japan. 8Dipartimento di Fisica, Università di Pisa,Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy. 9Istituto Italiano di Tecnologia, Gra-phene Labs, Via Morego 30, I-16163 Genova, Italy. 10CIC nanoGUNE BRTA and De-partment of Electricity and Electronics, UPV/EHU, 20018 Donostia-San Sebastián,Spain. 11IKERBASQUE, Basque Foundation for Science, Bilbao, Spain. 12ICREA-Insti-tucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain.*Corresponding author. Email: frank.koppens@icfo.eu (F.K.); iacopo.torre@icfo.eu(I.T.)†Present address: Bruker Switzerland AG, Industriestrasse 26, CH-8117 Fällanden,Switzerland.S C I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 1 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023mailto:frank.koppens@icfo.eumailto:iacopo.torre@icfo.euhttp://crossmark.crossref.org/dialog/?doi=10.1126%2Fsciadv.adi0415&domain=pdf&date_stamp=2023-09-29Coulomb and pressure forces (14) in sustaining the plasmon mode.This makes the plasmon mode stiffer, increasing its velocity. Toobserve this effect, it is, however, necessary to screen the otherwisedominant Coulomb force.The electronic first sound pertains to the hydrodynamic regime,which is characterized by τee being the shortest time scale of thesystem (15–17). Quantitatively, this happens when τee ≪ τ, ω−1,(qvF)−1, where τ is the momentum relaxation time, ω−1 is thetime over which the phase of the mode changes significantly, and(qvF)−1 (q being the wave vector of the mode) is the time it takesto an electron traveling at the Fermi velocity vF to cross a substantialfraction of a spatial oscillation of the mode (the correspondingranges of frequencies relevant for our experiment are depicted inFig. 1C). Because both τ� 1ee and τ−1 increase with temperature, typ-ically at different rates, the hydrodynamic regime can only be real-ized in high-mobility electronic systems for a limited window ofexperimental conditions (10, 18, 19). The hydrodynamic regimehas been demonstrated experimentally in encapsulated graphenesamples (10, 11, 20–23) or GaAs/AlGaAs quantum wells (19, 24).The transition between the zero sound and the first sound in 2Delectronic liquid was studied theoretically (5–7) using simplifiedmodels based on the semiclassical Boltzmann transport equationthat captures nonlocal effects (9, 25). The magnitude of the differ-ence between the plasmon velocity in the two regimes is controlledby the screening parameter (7)Λ ¼Ce2N�1thBN½nm�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijn½1012� cm� 2� jp ð1Þwhere C is the capacitance per unit area between the electron liquidand the metallic gate, e is the unit charge, and N is the electronicdensity of states at the chemical potential. The definition of Λused here differs slightly from the one used in (7) in that thedensity of states N appearing in Eq. 1 is the observed or renormal-ized one, not the bare one. The second relation holds for the specificcase of single-layer graphene with carrier density n and separatedFig. 1. Electronic sound transition in graphene. (A) Fermi surface deformation associated with an acoustic plasmon propagating along the positive x direction in thecollisionless (top row) and hydrodynamic (bottom row) regime. The green shading represents the electron’s distribution function, and the black line represents the set ofpoints where the distribution function is 1/2. The dashed line marks the position of the equilibrium Fermi sphere. The oscillation represents the potential density per-turbation. The phase is given by qx − ωt. See also movies S1 and S2 for an animation of the Fermi surface deformation. (B) Qualitative sketch of the dispersion of acousticplasmons (thick red line) highlighting the change in phase velocity between the two regimes. The yellow and blue lines represent the phase velocity in the collisionlessand hydrodynamic regimes, respectively. The black dashed line marks the Fermi velocity. (C) Frequency scales involved in our experiment. The laser frequency ω can betuned to be larger or smaller than the ee collision rate τ� 1ee , while themomentum relaxation rate τ−1 is always the slowest mechanism. qvF is always smaller than ω becausethe plasmon phase velocity never reaches the Fermi velocity vF for the values of the screening parameter Λ in our experiment. τ� 1ee has been calculated for our devices’parameters (note S7), while τ−1 has been extracted from the high-frequency data (note S5). (D) Screening parameter Λ for single-layer graphene as a function of the hBNthickness thBN and carrier density. Red lines represent the data ranges for our devices once the air gap is taken into account (note S3). (E) Schematic view of the dual-gateddevice geometry and THz nanoscopy experiment. The inset highlights the relevant geometric dimensions.S C I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 2 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023from a nearby metallic gate by an hBN spacer of thickness thBN.Figure 1D shows the values of Λ that can be reached as a functionof the experimental parameters.The sought effect is negligible for Λ≈ 0 but becomes strong for Λ≈ 1. When the latter condition is reached, the hydrodynamicplasmon velocity becomes even smaller than vF. In the extremecase of very large screening (Λ ≫ 1), the collisionless plasmon ve-locity tends to vF, while the hydrodynamic plasmon velocity tends tothe 2D energy-wave (second sound) velocity vF=ffiffiffi2p(26). The con-vergence of these two modes to the same limiting velocity can beunderstood because they both approach charge-neutral oscillations.In the case of the second sound, this happens because of the chargecompensation between electrons and holes, while in the case ofacoustic plasmons, the same happens because of the compensationdue to induced image charges in the metallic gate.On the basis of the theoretical model presented in (5–7) andmaking an approximation that is well justified in single-layer gra-phene (27) (i.e., neglecting the first-order spin-symmetric Landauparameter Fs1 that controls the many-body renormalization of theDrude weight), it is possible to derive (note S4) a simple relationbetween the collisionless plasmon velocity vc and the hydrodynamicplasmon velocity vhvh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2c þ vcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2c � v2Fp2sð2Þwith vF as the (renormalized) Fermi velocity. From this formula, weimmediately see that the difference between the two velocities isnegligible if vc ≫ vF (corresponding to small values of Λ) andbecomes most important when vc ≈ vF. Even in this extreme case,the relative difference between the two velocities cannot exceed ∣vh− vc∣/vc ≲ 29%.RESULTSIn this work, we probe the transition between the collisionless andthe hydrodynamic regime of electrons in graphene by measuringthe phase velocity of acoustic plasmons at different angularfrequencies close to the expected value of τ� 1ee that we calculatedfor the specific structures of our experiment (see Fig. 1C and noteS7). To this aim, we fabricated two hBN-encapsulated single-layergraphene devices, dubbed device 1 and device 2, respectively, withdifferent gate-graphene separations. The transition effect is ob-served in device 1, while device 2 is a control device in which thiseffect is predicted to be negligible. The two devices share the samesplit-gate configuration depicted in Fig. 1E. They consist of hBN/graphene/hBN heterostructures on top of metallic palladiumgates. Each metallic gate is split into two halves whose voltagescan be controlled separately. This allows the creation of a sharp p-n junction in the sample, which enables the thermoelectric detec-tion of the plasmonic field (28).The only relevant difference between the two devices is the thick-ness of the bottom hBN spacer (thBN in Fig. 1E) that is chosen to beas small as possible (thBN = 2.0 nm, leading to a design value of Λ ≈0.5 at the carrier densities used in the experiment) in device 1 andlarger (thBN = 11.8 nm, corresponding to Λ ≈ 0.08) in device 2. Thismeans that the Coulomb interaction should be strongly screened indevice 1 (quantified by higher values of Λ as seen in Fig. 1D), whereacoustic plasmons are expected to propagate with low velocity (withvp/vF reaching values as low as 1.5). This yields a sizable change in vpbetween the two regimes. On the contrary, in device 2, the Coulombinteraction is less screened (see Fig. 1D) and vp/vF never goes below2.5. This means that, in device 2, vp is almost the same in the tworegimes, and no significant transition effect is expected.We performed THz photocurrent nanoscopy (8) measurementsat RT (T = 295 K) in a commercial scanning near-field optical mi-croscope (SNOM). We used a methanol gas laser to measure at fourdifferent frequencies ( f = 1.84, 2.52, 3.11, 4.25 THz). For each laserfrequency, we scanned the tip repeatedly along the white dashedlines indicated in Fig. 2 (A and B), for a set of gate voltages V1,while the other gate is kept at a voltage V2 chosen to maximizethe photocurrent signal (see Materials andMethods). The dominat-ing plasmon launching mechanism differs between the two devicesdue to the very different vertical confinement of the plasmon [see(9) and note S1]. In device 1, the sharp metallic edge at the junctionlaunches a plane wave propagating perpendicular to it. By scanningFig. 2. Photocurrent nanoscopy of acoustic graphene plasmons. (A) Optical micrographs of device 1, indicating the electrodes used for collecting the photocurrentsignal and the gate electrodes. Thewhite dashed linemarks where datasets in (C) were acquired. The green dashed lines delimitate the area covered by graphene, and thegreen arrows the location of the junction. Scale bars, 5 μm. (B) Same as (A) for device 2, here, the white dashed line marks where datasets in (D) were acquired. (C) Near-field photocurrent signal acquired in device 1 along the line shown in (A) at four different frequencies. The carrier density is fixed at n ≈ 1012 cm−2. In this configuration(perpendicular to the junction), only λ-fringes appear. Data are shifted vertically for more clarity. (D) Same as in (C) for device 2, along the line marked in (B). In thisconfiguration (parallel to the junction), only λ/2-fringes appear. The color code is the same as in (C).S C I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 3 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023the tip perpendicular to the junction, we measure λ-fringes (i.e.,fringes with a periodicity equal to the plasmon wavelength). Indevice 2, the tip launches a circular plasmonic wave. By scanningthe tip parallel to the junction, we detect λ/2-fringes (i.e., fringeswith a periodicity equal to half the plasmon wavelength) due tothe standing wave originating from the plasmons reflected at thegraphene edge and traveling back to the tip (29). Instead, we donot observe plasmonic oscillations in device 1 when scanning par-allel to the junction, nor in device 2 when scanning perpendicular tothe junction. In both cases, the SNOM tip serves as a local probe torectify the plasmonic field, generating heat, which is then convertedinto photocurrent at the p-n junction via thermoelectric effect (28).Figure 2 (C and D) displays the real part of the photocurrentsignal (recorded at the first harmonic of the tip frequency) acquiredfor both devices at the four studied frequencies, at a carrier density n≈ 1012 cm−2. Both devices, and particularly device 1, display veryhigh electronic quality with plasmon lifetimes of about 0.5 to 1 ps(note S5). This is evident from the quality of the data in Fig. 2 (C andD), which show up to 7 clearly visible oscillation fringes. The pres-ence of a good number of fringes is pivotal for the reliable extractionof the plasmon wavelength. Because of the different scanning direc-tions, in device 1, there is an additional decay of the signal along thescanning direction as the tip moves away from the junction (28).Conversely, in device 2, the tip-junction distance is kept constant,but there is a geometric decay due to the cylindrical plasmonic waveradiating away from the tip (notes S1 and S2). The high signal-to-noise ratio typical of our technique and the high mobility of ourdevices allows us to accurately extract the plasmon wavelength λp= 2π/qp for carrier densities above 0.5 × 1012 cm−2 for device 1and above 0.3 × 1012 cm−2 for device 2 (see note S2 for the fulldata sets and the fitting procedure details).From these measured values of λp, we extract the plasmon phasevelocity as a function of the gate voltage (Fig. 3, A and B), measuredwith respect to the charge neutrality point, determined by two-probe transconductance measurements, VG = V1 − V1,CNP, fordevice 1 and 2, respectively. We were able to extract the plasmonwavelength with a sufficient degree of accuracy at the four laser fre-quencies. Measuring at lower laser frequencies was not possiblebecause λp becomes too large compared with device dimensions,propagation, and cooling length, making the extraction of the wave-length not reliable enough. From the measured plasmon velocity,we find (see note S3) a smaller capacitance than the one expectedfrom the thickness of the exfoliated hBN flakes in both devices,yielding typical values of Λ of ≈0.2 and ≈0.04 in devices 1 and 2,respectively. We attribute this discrepancy to air gaps between themetallic gate and the hBN flake (see note S3). This effect can be de-scribed as an effective hBN thickness larger than the nominal one.We used this more realistic quantity to locate our devices in the pa-rameter space plotted in Fig. 1D.DISCUSSIONThe most notable observation is that the two devices show a differ-ent frequency dependence of the plasmon velocity. In contrast todevice 2 (Fig. 3B), the data of device 1 (Fig. 3A) show a clear fre-quency dependence, with the plasmon velocity slowing down by≈5% from the highest frequency to the lowest. The effect is empha-sized in the inset that shows the relative variation with respect to thehighest frequency. We will show that our findings are compatiblewith the transition from the collisionless to the hydrodynamicregime. To this aim, we compare Eq. 2 with our data by approximat-ing the collisionless velocity vc with our velocity data at the highestavailable frequency, 4.25 THz, and the Fermi velocity with its valuecalculated at the carrier density of n = 1012 cm−2, vF ≃ 1.1 × 106 m/s(note S3). For more clarity, instead of applying Eq. 2 directly, we fitthe experimental data with a simplified one-parameter (α) modelderived from the collisionless expression by removing quantum ca-pacitance and many-body renormalization effects (note S4)Fig. 3. Frequency dependence of the velocity of acoustic graphene plasmons. (A) Measured plasmon phase velocity as a function of the gate voltagewith respect tothe Dirac point ∣VG∣ and carrier density for device 1 for the frequencies indicated in the legend. The solid green line is a one-parameter fit vc(VG) of the data at 4.25 THz. Thegreen dashed line is the corresponding hydrodynamic velocity obtained by applying Eq. 2 to vc(VG). The inset shows the relative variation of the plasmon velocity withrespect to the fit, Δ = vp(VG, ω)/vc(VG) − 1, as a function of the gate voltage. The blue-shaded region indicates the area below the Fermi velocity. (B) Same as in (A) for thecontrol device 2. (C) Plasmon phase velocity as a function of the frequency for device 1 (top) and device 2 (bottom) for a VG corresponding to a carrier density of n ≈ 1012cm−2. The data points and error bars have been obtained by fitting the dispersions in (A) and (B) to the functional form explained in the text. The solid red lines follow theexpected plasmon velocity for our devices’ parameters (see table S1) using the model in (7). The dashed and dotted red lines correspond to a value of τee (0.16 ps fordevice 1 and 0.17 ps for device 2) reduced by a factor of 2 and 5, respectively. The difference between the three lines is not visible for device 2.S C I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 4 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023vcðVGÞ ¼ vF ðαffiffiffiffiffiffiffiffiffiffijVG jpþ 1Þð2αffiffiffiffiffiffiffiffiffiffijVG jpþ 1Þ� 1=2(green solid linein Fig. 3A). The parameter α, related to the capacitance quantifiesthe impact of gating on the plasmonic dispersion.We then apply Eq.2 to the fitted curve to obtain our theoretical estimate of the hydro-dynamic velocity (green dashed line in Fig. 3A). The theoretical linematches well with the data extracted at the lowest available frequen-cy of 1.84 THz. As expected, repeating the same procedure with thecontrol device, we find only a very small velocity shift that is com-patible with experimental errors, as shown in Fig. 3B. In Fig. 3C, wedisplay the plasmon velocity for a fixed carrier density n ≈ 1012cm−2 as a function of the laser frequency. To reduce the uncertaintyassociated with the use of a single experimental point in voltage, thedata points and their error bars have been obtained by fitting theexperimental plasmon dispersions in Fig. 3 (A and B) to a function-al expression, which allows fitting the dispersion in both the hydro-dynamic and collisionless regime: vp(VG) = a∣VG∣1/4(1 + b∣VG∣). Therationale behind this expression is that for small screening and ne-glecting all quantum and many-body effect, the plasmon velocity isproportional to ∣VG∣1/4, while all the effects beyond this approxima-tion can be captured in a small voltage interval by a linear factor (1+ b∣VG∣).We note that (5–7) also predict a change in the plasmondamping rate Γ ≈ ω Re{qp}/Im{qp} at the crossover between the hy-drodynamic and collisionless regime. However, while the quality ofour results allows a reliable extraction of Re{qp}, the extraction ofIm{qp} has a larger uncertainty. On top of that, the plasmondecay due to Im{qp} needs to be disentangled (in the case ofdevice 1) from decay due to the varying tip-junction distance. Asa result, we do not aim to observe the transition in the dampingrate measurements (note S2).Motivated by the good matching between the expected hydrody-namic velocity and the experimental data at 1.84 THz, we aim tomake a thorough comparison between our experiment and the the-oretical model in (7). This model needs, as input, the value of Λ, τ,τee, vF, and of the zeroth-order Landau parameter (1) [which isrelated to the compressibility correction Fs0 (9)] and allows the cal-culation of the plasmon wave vector at every frequency. Our bestestimates of these parameters, either measured or calculated usingthe theory presented in (30, 31), are summarized in table S1, withdetails on how they are obtained given in notes S3 and S5 to S7. Thecalculated plasmon velocity according to these parameters is shownas a solid line in Fig. 3C for the two devices. While the trend iscorrect, the model predicts that only a smaller shift should be ob-served in our experimental range. We attribute this discrepancymainly to an overestimate of τee in our many-body calculation(τee = 0.16 ps for device 1 and τee = 0.17 ps for device 2) thatpushes the central frequency of the transition ftr = (2πτee)−1 toaround 1 THz, below our experimental points. To support thisnotion, we show in Fig. 3C the theoretical line calculated with thesame parameters but with τee reduced by a factor of 2, which makesftr fall inside our experimental range (dashed line), and factor of 5,which makes ftr above the highest measured frequency (dotted line).This shows that a better agreement is reached by assuming that thereal value of τee is reduced by around a factor of 2 with respect to thecalculated one. This discrepancy could motivate future theoreticalinvestigations.Further mismatch between the theoretical prediction and the ex-periment can be attributed to the estimate of other parameters or tomechanisms that are not captured by the simplified theoreticalmodel in (5–7). In particular, each angular harmonic of the one-particle distribution function relaxes, in principle, according to adifferent scattering rate (32, 33). In addition, considering effectsbeyond the energy-independent relaxation time approximationmay be important for a quantitative description of the transition.The dispersive behavior of the dielectric environment could alsointroduce a frequency dependence in our experiment. However, thehBN permittivity change in our frequency range is too small toexplain the effect (note S8), and the palladium gate electrode hasperfect mirror response in the same range (its plasma frequencybeing close to 10 eV) (34).In conclusion, we have experimentally demonstrated a shift inthe phase velocity of acoustic graphene plasmons in a graphenesample with a very thin (2 nm) hBN spacer when the frequency istuned from 4.25 to 1.84 THz. The same effect was not observed in adevice with a thicker hBN spacer. The magnitude of the observedshift and the frequency at which the shift is happening are in qual-itative agreement with the theoretical expectation for the collision-less to hydrodynamic transition.Two main ingredients have allowed us to observe the shift in thephase velocity. First, we have produced high-mobility graphenedevices in which the fastest scattering event is ee collisions.Second, we have incorporated a metallic gate electrode in veryclose proximity to the graphene sheet to ensure sufficient screeningof the long-range Coulomb interaction, achieving record-low valuesof acoustic graphene plasmon velocity. Our results can stimulatefurther experimental investigation on the dynamical aspects of thehydrodynamic regime of electronic transport.The ee collision rate strongly depends on temperature (10). Per-forming experiments in a variable-temperature cryogenic near-fieldmicroscope (35) would permit studying the evolution of the hydro-dynamic regime as a function of temperature. Moreover, the hydro-dynamic regime could be studied by using THz graphene plasmoncavities coupled with a continuously tunable THz source in the fewTHz range. Last, interesting nonlinear plasmonic effects are predict-ed to happen in the hydrodynamic regime due to the nonlinearitiesof the Navier-Stokes equations in graphene (36, 37).A recently published paper (38) reports the observation of hy-drodynamic plasmons and energy waves in graphene using on-chip THz spectroscopy. We believe that the findings of this comple-mentary work strengthen the validity and importance of our results.MATERIALS AND METHODSDevice fabricationWe start with the fabrication of the metallic gates. We use standardelectron beam lithography (EBL) at 30 kV to define two rectanglesseparated by 200- to 300-nm gaps on a 270-nm-thick polymethylmethacrylate layer. After developing, we perform a plasmadescum at low power to remove resist leftovers. We deposit 2 nmof Ti and 15 nm of Pd, both by electron beam evaporation. Last,to remove the spikes at the edges of the gates, we anneal thesamples at 300°C in Ar/H2 for 3 hours. We check with atomicforce microscopy (AFM) and choose only the gates withoutspikes. We find that the gap is in the order of 100 to 200 nm.We mechanically exfoliate hBN and graphene flakes on SiO2/Sichips and carefully choose the desired hBN flakes for our devices.To assemble the hBN/graphene/hBN heterostructure, we useSC I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 5 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023polycarbonate stamps and drop the heterostructure onto the pre-patterned metallic gates at 160°C (39). Last, we define the edge con-tacts (40) and shape the graphene channel with EBL and reactiveion etching.Before our near-field measurements, we clean the surface of thesamples with an AFM tip in contact mode (41), applying forcesbetween 30 and 60 nN. In fig. S1, we show an AFM image of thesurface of device 2 after the AFM cleaning.THz photocurrent nanoscopy measurementsAs the laser source, we used two far-infrared gas lasers: FIRL-100(Edinburgh Instruments Ltd.) and SIFIR-50 (Coherent Inc.). Bothlasers output the same THz lines at very similar powers.As the nearfield microscope, we used a neaSNOM (neaspecGmbH). Because we perform photocurrent measurements, weremoved the interferometer to maximize the power at the tip. Thephotocurrent signal (typically in the order of few nA) is read outthrough a photocurrent amplifier (DHPCA-100 from FEMTOMes-stechnik GmbH), working at gains between 104 and 106 V/A, de-pending on the device resistance and laser power. The amplifieroutput is fed to the neaSNOM lock-in input, such that the collectedsignal is demodulated at the harmonics of the tip frequency. Weused an Au-coated AFM tip with 250-nm radius at the apex and aforce constant of 3 N/m, model LRCH250 (Team Nanotec GmbH).The photocurrent signal is demodulated at either the first or secondharmonic of the tip frequency (~75 KHz), and the first harmonic ofthe mechanical phase is subtracted (9, 28). The typical tapping am-plitude is 80 to 120 nm.First, we locate a clean line perpendicular to the junction fordevice 1 and parallel to it but close enough to maximize thesignal for device 2. Whether we want to measure on the electronor hole doping regimes, we choose a different gate voltage for theother gate electrode to maximize the photocurrent. We scan alongthe same line for a range of gate voltages. In fig. S2, we display theraw measurements acquired for device 1 (left) and device 2 (right),where in the horizontal axis we scan the tip, and in the vertical axiswe step the gate voltage.We check for position and carrier density drifts between scansthat may alter the data. We always scan across the p-n junction(device 1) or across the graphene edge (device 2). This, togetherwith comparing forward and backward traces (which are recordedsequentially), allows us to discard sample drifts that could lead to anapparent change in the fringe spacing. To check for carrier densitydrifts, i.e., a drift of the charge neutrality point VCNP, we verify thatthe gate voltage at which the photocurrent signal changes its signremains the same. Moreover, we do not expect this to happen insamples with a local gate. This samples are much less affected bydrift and hysteresis than encapsulated samples directly on top ofSiO2 due to the lack of dielectric-dielectric interfaces that maytrap charges for long times.Supplementary MaterialsThis PDF file includes:Materials and MethodsNotes S1 to S8Figs. S1 to S11Table S1Legends for movies S1 and S2ReferencesOther Supplementary Material for thismanuscript includes the following:Movies S1 and S2REFERENCES AND NOTES1. D. Pines, P. 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Solid State Commun. 141, 262–266 (2007).AcknowledgmentsFunding: This work was supported by the Government of Spain through CEX2019-000910-S(MCIN/AEI/10.13039/501100011033) and European Union NextGenerationEU/PRTR (D.B.R., I.T.,N.C.H.H., H.H.S., C.R.M., and F.H.L.K.); the Fundació Cellex (D.B.R., I.T., N.C.H.H., H.H.S., C.R.M., andF.H.L.K.), the Fundació Mir-Puig (D.B.R., I.T., N.C.H.H., H.H.S., C.R.M., and F.H.L.K.); the Generalitatde Catalunya through CERCA (D.B.R., I.T., N.C.H.H., H.H.S., C.R.M., and F.H.L.K.); the Secretariad’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat deCatalunya and the European Social Fund (L’FSE inverteix en el teu futur)-FEDER (D.B.R.); theEuropean Union’s Horizon 2020 programme under the Marie Skłodowska-Curie grantagreement Ref. 843830 (H.H.S.); the European Union’s Horizon 2020 research and innovationprogramme under the Marie Skłodowska-Curie grant agreement ref. 665884 (N.C.H.H.); theSpanish Ministry of Science, Innovation, and Universities (MCIU) and State Research Agency(AEI) via the Juan de la Cierva fellowship ref. FJC2018-037098-I (I.T.); the ERC TOPONANOP(726001) (F.H.L.K.); the Government of Spain (PID2019-106875GB-I00, PHOTOTBG PCI2021-122020-2A) (F.H.L.K.); the Generalitat de Catalunya (AGAUR, SGR 1656) (F.H.L.K.); the EuropeanUnion’s Horizon 2020 under grant agreement no. 881603 (Graphene flagship Core 3) and ERC-POC PDC2022-138844-100 (F.H.L.K. and M.P.); the European Union’s Horizon 2020 under grantagreement no. 820378 (Quantum flagship) (F.H.L.K.); the The University of Pisa under the “PRA -Progetti di Ricerca di Ateneo” (Institutional Research Grants) (project no. PRA_2020-2021_92)“Quantum Computing, Technologies and Applications” (M.P.); the MUR–Italian Minister ofUniversity and Research under the “Research projects of relevant national interest - PRIN 2020”(project no. 2020JLZ52N), titled “Light-matter interactions and the collective behavior ofquantum 2D materials (q-LIMA)” (M.P.); the Swiss National Science Foundation (grant no.172218) (C.M.M.); the European Union’s Horizon 2020 research and innovation programmeunder the Marie Skłodowska-Curie grant agreement nos. 873028 (A.P.) and 843830 (H.H.S.); theEuropean Commission EU Horizon 2020 MSCA-RISE-2019 programme (project 873028HYDROTRONICS) and Leverhulme Trust grant RPG-2019-363 (A.P.); the Leverhulme Trust underthe grant agreement no. RPG-2019-363 (A.P.); JSPS KAKENHI (grant nos. 19H05790, 20H00354,and 21H05233; T.T. and K.W.); and the Spanish Ministry of Science and Innovation under theMaría de Maeztu Units of Excellence Program (CEX2020-001038-M) and the Projects RTI2018-094830-B-100 and PID2021-123949OB-I00. (R.H.). Author contributions: Conceptualization:I.T., D.B.R., and F.H.L.K. Sample fabrication: D.B.R. and N.C.H.H. Experiment: D.B.R., N.C.H.H.,H.H.S., and C.M.M. hBN crystals: K.W. and T.T. Data analysis: D.B.R., I.T., and C.R.M. Theoreticalcalculations: R.A., A.P., and I.T. Supervision: I.T., F.H.L.K., M.P., and R.H. Writing—original draft: I.T.,D.B.R., and F.H.L.K. Writing—review and editing: All authors. Competing interests: R.H. is thecofounder of Neaspec GmbH, a company producing scattering-type scanning near-field opticalmicroscope systems, such as the one used in this study. All other authors declare that they haveno competing interests. Data and materials availability: All data needed to evaluate theconclusions in the paper are present in the paper and/or the Supplementary Materials. All rawdata generated during the current study are available at 10.5281/zenodo.8298069.Submitted 30 March 2023Accepted 30 August 2023Published 29 September 202310.1126/sciadv.adi0415S C I ENCE ADVANCES | R E S EARCH ART I C L EBarcons Ruiz et al., Sci. Adv. 9, eadi0415 (2023) 29 September 2023 7 of 7Downloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023http://dx.doi.org/10.5281/zenodo.8298069Use of this article is subject to the Terms of serviceScience Advances (ISSN 2375-2548) is published by the American Association for the Advancement of Science. 1200 New York AvenueNW, Washington, DC 20005. The title Science Advances is a registered trademark of AAAS. Copyright © 2023 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claimto original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).Experimental signatures of the transition from acoustic plasmon to electronicsound in grapheneDavid Barcons Ruiz, Niels C.H. Hesp, Hanan Herzig Sheinfux, Carlos Ramos Marimón, Curdin Martin Maissen,Alessandro Principi, Reza Asgari, Takashi Taniguchi, Kenji Watanabe, Marco Polini, Rainer Hillenbrand, Iacopo Torre, andFrank H.L. KoppensSci. Adv. 9 (39), eadi0415.  DOI: 10.1126/sciadv.adi0415View the article onlinehttps://www.science.org/doi/10.1126/sciadv.adi0415Permissionshttps://www.science.org/help/reprints-and-permissionsDownloaded from https://www.science.org at National Institute for Materials Science on October 05, 2023https://www.science.org/content/page/terms-service INTRODUCTION RESULTS DISCUSSION MATERIALS AND METHODS Device fabrication THz photocurrent nanoscopy measurements Supplementary Materials This PDF file includes: Other Supplementary Material for this manuscript includes the following: REFERENCES AND NOTES Acknowledgments