nature nanotechnology Article https://doi.org/10.1038/s41565-023-01421-3 Electron cooling in graphene enhanced by plasmon–hydron resonance Received: 11 January 2023 Xiaoqing Yu 1, Alessandro Principi 2 , Klaas-Jan Tielrooij , Mischa Bonn 3,4 1 & Nikita Kavokine 1,5 Accepted: 15 May 2023 Published online: 22 June 2023 Evidence is accumulating for the crucial role of a solid’s free electrons in the Check for updates dynamics of solid–liquid interfaces. Liquids induce electronic polarization and drive electric currents as they flow; electronic excitations, in turn, participate in hydrodynamic friction. Yet, the underlying solid–liquid interactions have been lacking a direct experimental probe. Here we study the energy transfer across liquid–graphene interfaces using ultrafast spectroscopy. The graphene electrons are heated up quasi-instantaneously by a visible excitation pulse, and the time evolution of the electronic temperature is then monitored with a terahertz pulse. We observe that water accelerates the cooling of the graphene electrons, whereas other polar liquids leave the cooling dynamics largely unaffected. A quantum theory of solid–liquid heat transfer accounts for the water-specific cooling enhancement through a resonance between the graphene surface plasmon mode and the so-called hydrons—water charge fluctuations—particularly the water libration modes, which allows for efficient energy transfer. Our results provide direct experimental evidence of a solid–liquid interaction mediated by collective modes and support the theoretically proposed mechanism for quantum friction. They further reveal a particularly large thermal boundary conductance for the water–graphene interface and suggest strategies for enhancing the thermal conductivity in graphene-based nanostructures. Free electrons in graphene exhibit rather unique dynamics in the been shown to arise in the graphene electron–phonon interaction6, and terahertz frequency range, including a highly nonlinear response to plasmon–phonon coupling between graphene and a polar substrate has photoexcitation by terahertz pulses1,2. Graphene’s distinctive dynamic been demonstrated7–9. More recently, it has been theoretically proposed properties on picosecond time scales have found several applications that similar effects are at play when graphene has a liquid environment: in, for example, ultrafast photodetectors, modulators and receivers3–5. then, the interaction between the liquid’s charge fluctuations—dubbed The terahertz frequency range acquires particular importance at room hydrons—and graphene’s electronic excitations tunes the hydrodynamic temperature T, where it corresponds to the typical frequency of ther- friction at the carbon surface10,11. This ‘quantum friction’ mechanism mal fluctuations: kBT/ℏ ≈ 6 THz, where kB is Boltzmann’s constant and holds the potential for entirely new strategies for controlling liquid flows ℏ is Planck’s constant. One may therefore expect non-trivial couplings on the nanometre scale12,13; it is therefore of interest to experimentally between the graphene electrons and the thermal fluctuations of their probe the underlying electron–hydron interaction. environment. These couplings have been intensively studied in the In this article, we probe solid–liquid interactions by measuring case of a solid environment: for instance, non-adiabatic effects have energy transfer at the solid–liquid interface (Fig. 1a). Specifically, we Max Planck Institute for Polymer Research, Mainz, Germany. 2School of Physics and Astronomy, University of Manchester, Manchester, UK. 1 3 Catalan Institute of Nanoscience and Nanotechnology (ICN2), BIST and CSIC, Campus UAB, Bellaterra, Barcelona, Spain. 4Department of Applied Physics, TU Eindhoven, Eindhoven, Netherlands. 5Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA. e-mail: nikita.kavokine@mpip-mainz.mpg.de Nature Nanotechnology | Volume 18 | August 2023 | 898–904 898 Article https://doi.org/10.1038/s41565-023-01421-3 a b c Water Flowing water Electrons Water Hot electrons V Plasmon Phonons Phonons Electron cloud Graphene lattice Quantum friction Quantum ET Classical friction Classical ET Fig. 1 | Heat transfer and friction at the solid–liquid interface. a, Schematics of friction, but also directly to the electrons through quantum friction. c, Energy the system under study: the interface between water and a graphene sheet. The transfer (ET) processes at the solid–liquid interface. In the typically assumed picture emphasizes the electron cloud and its wave-like plasmon excitation. b, ‘classical’ pathway, hot electrons first transfer energy to the phonons, which Momentum transfer processes at the solid–liquid interface. A flowing liquid (the transfer energy to the liquid. An alternative ‘quantum’ pathway consists in the flow profile is shown by the thin blue arrows) may not only transfer momentum to electrons transferring energy directly to the liquid through Coulomb coupling. the crystal lattice (exciting phonon vibrations) through classical hydrodynamic use a femtosecond visible pulse to introduce a quasi-instantaneous We show that this condition is met upon optically exciting a graphene– temperature difference between the electrons of a graphene sample water interface, due, in particular, to graphene’s weak electron– and their environment. The cooling rate of the electronic system is fol- phonon coupling29,30. lowed in real-time using terahertz pulses. Such optical pump–terahertz probe spectroscopy is a well-established tool for probing electron Time-resolved electron cooling relaxation in two-dimensional materials14–19. In high-quality graphene, Our experimental set-up is schematically represented in Fig. 2a. A mon- it has been used to identify the interaction of hot carriers with optical olayer graphene sample grown by chemical vapour deposition (CVD) phonons17,18 and with substrate phonons as the main electron cooling was transferred onto a fused silica flow cell, filled with either nitrogen mechanisms20; it has also identified the role of Coulomb interactions gas or a liquid of our choice (Supplementary Information, section 1.1). in the interlayer thermal conductivity of graphene stacks16. Here, we The graphene chemical potential was in the range 100–180 meV as measure the electron relaxation time in the presence of different polar determined from Raman measurements (Supplementary Informa- liquids to probe the electron–hydron interaction, which we find to be tion, section 1.4). In a typical experiment, the graphene electrons were comparable to the electron–optical phonon interaction only when excited using an ~50 fs laser pulse with 800 nm central wavelength. the liquid is water. A complete theoretical analysis shows that this Then, the attenuation of an ~1 ps THz probe pulse (precisely, the modu- specificity of water is explained by the strong coupling of its terahertz lation of the peak electric field) was monitored as a function of the (libration) modes to the graphene surface plasmon, with the electron– pump–probe delay (Supplementary Information, section 1.2). After electron interactions in graphene playing a crucial role. absorption of the exciting pump pulse, the non-equilibrium electron distribution typically thermalizes over a sub-100 fs time scale through Solid–liquid heat transfer electron–electron scattering31: it can then be described as a Fermi–Dirac The energy transfer between a solid and a liquid is usually considered distribution at a given temperature. A hotter electron distribution to be mediated by molecular vibrations at the interface because most results in a lower terahertz photoconductivity because hotter electrons of a solid’s heat capacity is contained in its phonon modes21. Even if an are less efficient at screening charged impurities32,33. The pump–probe optical excitation of the solid’s electrons is used to create the tempera- measurement thus gives access to the electron temperature dynamics ture difference, the electrons are typically assumed to thermalize with after photoexcitation (Fig. 2b). phonons on a very short time scale, so that the solid’s phonons ulti- Regardless of the medium that the graphene is in contact with, the mately mediate the energy transfer to the liquid’s vibrational modes22,23. electronic temperature T(t) exhibits a relaxation that can be approxi- However, if the electrons were to transfer energy to the liquid faster than mated by an exponential function: ΔT(t) = T(t) − T0 = ΔT0e−t/τ. This allows to the phonons, the interfacial thermal conductivity would contain a us to extract the cooling times τ for the different liquids and different non-negligible contribution from near-field radiative heat transfer24,25 initial electronic temperatures (determined by the excitation laser flu- (Fig. 1c). Such an electronic or ‘quantum’ contribution to heat transfer ence), displayed in Fig. 2c. We observe that the cooling time is longer is in close analogy with the quantum contribution to hydrodynamic for an initially hotter electron distribution, in agreement with previous friction. Quantum hydrodynamic friction relies on momentum being reports18. Now, for all initial temperatures, we consistently observe transferred directly between the solid’s and the liquid’s charge fluctua- the same dependence of the cooling time on the sample’s liquid envi- tion modes, coupled by Coulomb forces (Fig. 1b): the two processes ronment. In the presence of water (H2O) and heavy water (D2O), the are mediated by the same solid–liquid interaction. graphene electrons cool faster than they do intrinsically, in an inert To probe this interaction through a hydrodynamic friction meas- nitrogen atmosphere. Conversely, methanol and ethanol have almost urement, one needs to ensure that quantum friction dominates over no effect on the electron cooling time. Interestingly, we observe an the classical surface roughness contribution: this imposes stringent isotope effect in the electron cooling process: there is a difference in constraints on the sample’s surface state, in already technically difficult the cooling times in the presence of H2O and D2O that greatly exceeds experiments26–28. Similarly, in the case of energy transfer, the quantum experimental uncertainties. contribution needs to be comparable to the classical phonon-based We are thus led to hypothesize, as anticipated above, that the contribution to become measurable; however, this condition is easier liquid provides the electrons with a supplementary cooling pathway, to satisfy since it is insensitive to the sample’s surface roughness. which, in the case of water, has an efficiency comparable to the intrinsic Nature Nanotechnology | Volume 18 | August 2023 | 898–904 899 Article https://doi.org/10.1038/s41565-023-01421-3 a b c Te = 623 K 1.0 2.4 SiO2 cell Liquid N2 Te = 1,241 K H2O Graphene 0.8 2.2 D2O Cooling time (ps) Normalized ∆T ∆E(t) Methanol THz probe 0.6 Ethanol 2.0 Te = 1,023 K 0.4 1.8 Te = 770 K 0.2 1.6 Te = 623 K Optical pump 0 1.4 0 1 2 3 4 5 6 O 2O Et ol l no 2 N Pump–probe delay (ps) n 2 H D ha ha et M Fig. 2 | Measurement of picosecond hot electron relaxation in graphene. lines represent raw data and the full lines are exponential fits. c, Electron cooling a, Schematic of the experimental set-up. A graphene sample (Fermi level in time obtained through exponential fitting (see b) for the different liquids that the range 100−180 meV; Supplementary Information, section 1.4) is placed in have been placed in the flow cell and different initial electron temperatures, contact with a liquid inside a fused silica flow cell. An optical excitation pulse set by the excitation laser fluence. Faster cooling is observed in the presence quasi-instantaneously heats up the graphene electrons, and the electron of water and heavy water. Error bars represent 95% confidence intervals of the temperature dynamics are then monitored with a THz probe. b, Normalized exponential fits, and the centre point corresponds to the result of the least- electron temperature as a function of time after photoexcitation. The dotted squares fitting procedure. cooling pathway via phonons. We then interpret the faster cooling as definition of which is given in Supplementary Information, section a signature of ‘quantum’ electron–liquid energy transfer. We assess 2.3. For the liquids under consideration, it will be sufficient to use the the pertinence of this hypothesis by developing a complete theory of long-wavelength-limit expression of the surface response function: quantum energy transfer at the solid–liquid interface. ϵl (ω) − 1 gl (q → 0, ω) = , (2) Theoretical framework ϵl (ω) + 1 To tackle the interaction between a classical liquid and an electronic system whose behaviour is intrinsically quantum, we describe the liquid where εl(ω) is the liquid’s bulk dielectric function. For two-dimensional in a formally quantum way. Following ref. 10, we represent the liquid’s graphene, we show in Supplementary Information, section 2.3 that the charge density as a free fluctuating field with prescribed correlation surface response function can be expressed as functions. This naturally leads to a Fourier-space description of the solid–liquid interface in terms of its collective modes, rather than the e2 ge (q, ω) = − χ(q, ω), (3) usual molecular-scale interactions. Within this description, the quan- 2ϵ0 q tum solid–liquid energy transfer amounts to electron relaxation upon coupling to a bosonic bath, a problem that has been extensively studied where χ(q,ω) is graphene’s charge susceptibility. in condensed matter systems34. Interestingly, in the case of graphene, The result in equation (1) has been derived for two solids separated many of these studies are carried out within a single-particle Boltzmann by a vacuum gap in the framework of fluctuation-induced electromag- formalism, which may incorporate multiple screening effects only in an netic phenomena24,38,39; our non-equilibrium framework, however, is ad hoc fashion18,29,35. These effects turn out to be crucial for the solid– better suited to the solid–liquid system under consideration. We note liquid system under consideration: we have therefore developed an ab that equation (1) takes the form of a Landauer formula for the transport initio theory of solid–liquid heat transfer based on the non-equilibrium of bosonic quasiparticles—elementary excitations of the solid’s and the Keldysh formalism36, which has only very recently been considered for liquid’s charge fluctuations modes25. It involves the difference in the problems of interfacial heat transfer37. Our computation, detailed in Bose distribution functions between the solid and the liquid, and the Supplementary Information, section 2.2, is closely analogous to the one product of surface response functions plays the role of a transmission carried out for quantum friction in ref. 10. The theoretical framework coefficient for the quasiparticles. One may count either the energy or can formally apply to fully non-equilibrium situations and take interac- the momentum transported by the quasiparticles: the former corre- tions into account to arbitrary order. However, to obtain a closed-form sponds to near-field heat transfer, the latter to quantum friction. This result, assume that the liquid and the solid internally equilibrated at quasiparticle picture thus makes explicit the fundamental connection temperatures Tl and Te, respectively. Furthermore, we take electron– between the two processes. electron and electron–liquid Coulomb interactions into account at the random phase approximation (RPA) level. With these assumptions, we Plasmon–hydron resonance obtain the electron–liquid energy transfer rate as The graphene electrons may relax either through direct interaction +∞ with the liquid, or through emission of optical phonons. The latter 1 process has been well studied, both theoretically and experimen- 𝒬𝒬Q = ∫dq ∫ dω ℏω [ nB (ω, Te ) 2π3 0 tally18,29. Our non-equilibrium formalism applies in principle to any (1) Im [ge (q, ω)]Im [gl (q, ω)] electron–boson system: when applied to the electron–phonon system, −nB (ω, Tl ) ] , it recovers the result for the energy transfer rate 𝒬𝒬ph (from electrons |1 − ge (q, ω)gl (q, ω)|2 to phonons) obtained in ref. 18 (Supplementary Information, section 2.3.2). Then, within a three-temperature model, where the electrons, Here, nB(ω,T) = 1/(eℏω/T − 1) is the Bose distribution and the ge,l are surface liquid and phonons are assumed to be internally equilibrated at tem- response functions of the solid and the liquid, respectively. These are peratures Te, Tl and Tph, respectively, we may determine the evolution analogues of the dielectric function for semi-infinite media, the precise of the electron temperature according to Nature Nanotechnology | Volume 18 | August 2023 | 898–904 900 Article https://doi.org/10.1038/s41565-023-01421-3 Frequency (THz) a 10 20 30 40 50 b c 0.4 0.2 0.2 H2O Te = 623 K Te = 623 K Surface excitation spectrum D2O ne ne 0.3 Methanol 0.15 0.15 co co Ethanol Frequency (eV) Frequency (eV) rac rac Di Di 0.2 0.1 0.1 Plasmon–hydron resonance 0.1 0.05 0.05 Plasmon 0 0 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 –1 Frequency (eV) Wavevector (nm ) Wavevector (nm–1) d e f 0.8 1 Hindered rotation (libration) 0.8 Cooling rate (ps–1) Cooling rate (ps–1) 0.7 0.6 Coulomb interaction 0.4 Surface 0.6 plasmon Total rate (experiment) 0.2 Liquid contribution (theory) 0.5 0 Electron cloud O O l l O O l l no no no no 2 2 N N 2 2 2 2 H D H D ha ha ha ha et Et et Et M M Fig. 3 | Mechanism of electron–liquid heat transfer. a, Surface excitation liquids, for an initial electron temperature Te = 623 K. Error bars represent 95% spectra Im [gl (ω)] of the different liquids studied here obtained according to confidence intervals of the exponential fits to the temperature decay curves. equation (2) from the experimentally measured bulk dielectric permittivities. e, Theoretical prediction for the liquid contribution to the electron cooling rate, The arrows indicate the libration modes of H2O and D2O. b, Graphene surface reproducing the experimentally observed trend in terms of the nature of the excitation spectrum Im [ge (q, ω)], calculated at a chemical potential μ = 100 meV liquid. The symbol size in the vertical direction represents the variation in the and temperature Te = 623 K. The main feature is the collective plasmon mode. theoretical prediction when the graphene chemical potential spans the range c, Theoretical prediction for the graphene–water energy transfer rate resolved 100–180 meV. f, Schematic of the water-mediated electron cooling mechanism in frequency–wavevector space. The main contribution originates from a inferred from the combination of theoretical and experimental results. The resonance between the graphene plasmon mode and the water libration mode. cooling proceeds through the Coulomb interaction between the graphene d, Experimentally measured electron cooling rate in the presence of the various plasmon mode and the hindered molecular rotations (librations) in water. dTe (t) the theory reproduces the isotope effect, that is, the slightly slower C(Te ) = −𝒬𝒬Q (Te , Tl ) − 𝒬𝒬ph (Te , Tph ), (4) cooling observed with D2O as compared with H2O. dt We may now exploit the theory to gain insight into the micro- scopic mechanism of the liquid-mediated cooling process. In equa- where C(Te) is the graphene electronic heat capacity at temperature tion (1), the difference of Bose distributions decreases exponentially Te. We focus in the following on the liquid contribution to the electron at frequencies above Te/ℏ ≈ 100 meV. At frequencies below 100 meV, cooling rate, defined as 1/τ = 𝒬𝒬Q (Te , Tl )/(C(Te ) × (Te − Tl )), which may the graphene spectrum is dominated by a plasmon mode that cor- be compared with the experimental results. The quantitative evalua- responds to the collective oscillation of electrons in the plane of the tion of τ requires the surface response functions of graphene and of graphene layer40 (Fig. 3b). In this same frequency range, water and the various liquids. We compute the graphene surface response func- heavy water have a high spectral density due to their libration modes tion according to equation (3) by numerical integration40 at the chem- that correspond to hindered molecular rotations41 (Fig. 3a). As a result, ical potential determined for our samples by Raman spectroscopy the energy transfer rate resolved in frequency–momentum space (Supplementary Information, section 1.4). For the liquids, we use the (the integrand in equation (1), plotted in Fig. 3c) has its main contri- expression in equation (2), with the bulk dielectric function determined bution from the spectral region where the two modes overlap. We by infrared absorption spectroscopy (Fig. 3a and Supplementary Infor- conclude that the particularly efficient electron–water cooling is due mation, section 1.3). to a resonance between the graphene plasmon mode and the water Our theoretical prediction for the contribution of the various libration modes. This conclusion is further supported by the isotope liquids to the electron cooling rate is shown in Fig. 3e. Quantitatively, effect. Indeed, the libration of the heavier D2O is at slightly lower we obtain cooling rates of the order of 1 ps−1, in excellent agreement frequency than that of the lighter H2O, and a higher-frequency mode with the experimentally observed range (Fig. 3d): our theory indicates makes a larger contribution to the cooling rate due to the factor ℏω that the quantum electron–liquid cooling is a sufficiently efficient pro- in equation (1). In the Landauer picture, the quasiparticle transport cess to compete with the intrinsic phonon contribution, estimated at rates are almost the same for the graphene–H2O and graphene–D2O around 0.6 ps−1 from the cooling rate in the absence of liquid. Moreover, systems, but in the case of H2O each quasiparticle carries more energy. our theory reproduces the experimentally observed trend in cooling Overall, our experiments evidence a direct interaction between the rates, with a significant liquid contribution arising only for water and graphene plasmon and water librations, as shown schematically in heavy water; the dependence of the cooling rate on initial electron Fig. 3f. We note that plasmons have been shown to play a role in temperature is also well reproduced (Supplementary Fig. 7). Finally, the energy transfer between two graphene sheets42; however, a Nature Nanotechnology | Volume 18 | August 2023 | 898–904 901 Article https://doi.org/10.1038/s41565-023-01421-3 a b c 1 10 0.2 60 Energy transfer rate (meV Å2 s–1) Te = 623 K –1 q0 = 0.15 nm 0.15 First order Cooling rate (ps–1) Frequency (eV) Full theory 40 q0 = 0.15 nm–1 No e–e interactions Bare Im (ge(q0,ω)) 10 0 Full theory 0.1 Renormalized First order 20 0.05 0 0.1 0.2 Frequency (eV) –1 10 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 O O l l no no Frequency (eV) 2 2 H D –1 Wavevector (nm ) ha ha et Et M Fig. 4 | Strong plasmon–hydron coupling. a, Theoretical prediction for the renormalized by the presence of water according to equation (5). The white graphene electron cooling rate in contact with different liquids, within different dashed lines are guides to the eye showing the strongly coupled plasmon–hydron treatments of interactions. The cooling rate is strongly overestimated if no mode. Inset: bare and renormalized graphene spectra at fixed wavevector electron–electron interactions are taken into account (blue symbols), and q0 = 0.15 nm−1. c, Comparison between the spectrally resolved energy transfer underestimated if the electron–liquid interactions are considered only to first rates obtained to first order and to arbitrary order in the solid–liquid interaction. order (orange symbols). b, Graphene surface excitation spectrum Im [ge (q, ω)], Higher-order effects enhance the energy transfer rate at low frequencies. calculated at a chemical potential μ = 180 meV and temperature Te = 623 K, plasmon–hydron interaction does not appear to have been suggested fact the higher-order effect that is mainly responsible for the enhance- as a possible electron relaxation mechanism. ment of the electron cooling rate. As shown in Fig. 4c, taking into account solid–liquid interactions to arbitrary order mainly enhances Interactions and strong coupling the contribution of low frequencies to the energy transfer. The combination of theory and experiment allows us to identify the key physical ingredients that are required to account for energy transfer at Conclusions the water–graphene interface. First, our results reveal that electron– We have carried out ultrafast measurements of electron relaxation in electron interactions are crucial because they produce the plasmon graphene, revealing signatures of direct energy transfer between the mode that is instrumental to the energy transfer mechanism. Indeed, graphene electrons and the surrounding liquid. These results speak applying our theory to non-interacting graphene would result in a to the importance of electronic degrees of freedom in the dynamics strongly overestimated liquid contribution to the cooling rate (Fig. 4a). of solid–liquid interfaces, particularly interfaces between water and This precludes single-particle Boltzmann approaches—such as those carbon-based materials. Despite conventional theories and simulations that have been used for the electron–phonon interaction in gra- that describe the interface in terms of atomic-scale Lennard–Jones phene18,29—from accurately describing the water–graphene interaction. potentials22,23, or with electronic degrees of freedom in the Born– Furthermore, detailed examination of our theoretical result Oppenheimer approximation43,44, here we demonstrate experimen- reveals that the efficiency of the electron–water cooling is enhanced tally that the dynamics of the water–graphene interface need to be by the formation of a strongly coupled plasmon–hydron mode. Indeed, considered at the level of collective modes in the terahertz frequency the result in equation (1) involves bare surface response functions, range. In particular, our semiquantitative theoretical analysis attributes without any renormalization due to the presence of the other medium. the observed cooling dynamics to the strong coupling between the However, the denominator ∣1 − ge gl∣2 accounts for solid–liquid interac- graphene plasmon and water libration modes. tions to arbitrary order (at the RPA level) and contains the signature The experimental observation of such a collective mode inter- of any potential strong coupling effects. We find that these effects are action supports the proposed mechanism for quantum friction at indeed important: removing the denominator in equation (1) (that is, the water–carbon interface, which is precisely based on momentum treating the electron–liquid interactions only to first order) results transfer between collective modes10. The near-quantitative agreement in underestimation of the liquid-mediated cooling rate by about 30% between the experiment and theory obtained for energy transfer sug- (Fig. 4a). To gain physical insight into the nature of these higher-order gests that a similar agreement should be achieved for momentum effects, we can compute the graphene surface response function renor- transfer. The water–graphene quantum friction force is small if the malized by the presence of water, which is given by (Supplementary graphene electrons are at rest, but becomes important if they are Information, section 2.3) driven at a high velocity by a phonon wind or an applied voltage12. The quantum-friction-based driving of water flows by graphene elec- ge (q, ω) tronic currents appears as a promising avenue in light of our findings. gẽ (q, ω) = . (5) 1 − ge (q, ω)gl (q, ω) The electric circuit configuration would furthermore allow for noise thermometry45,46 to be used as a supplementary probe of the electron The renormalized surface excitation spectrum Im [gẽ (q, ω)] is relaxation mechanisms. plotted in Fig. 4b for a chemical potential μ = 180 meV. We observe that Our results provide yet another example of the water–carbon the graphene plasmon now splits into two modes, which are both a interface outperforming other solid–liquid systems47. Indeed, the elec- mixture of the the bare plasmon and water librations. These are in fact tronic contribution to the graphene–water thermal boundary conduct- analogous to the coupled plasmon–phonon modes that have been ance is as high as λ = 0.25 MW m−2 K−1, exceeding the value obtained with predicted7 and measured8,9 for graphene on a polar substrate. It can the other investigated liquids by at least a factor of 2. This even exceeds be seen in the inset of Fig. 4b that coupling to the water modes also the thermal boundary conductance obtained for the graphene–hBN increases the spectral density at low frequencies (below the plasmon interface, at which particularly fast ‘super-Planckian’ energy transfer peak) compared with the bare graphene response function. This is in was observed20,35. Our investigation thus suggests that the density of Nature Nanotechnology | Volume 18 | August 2023 | 898–904 902 Article https://doi.org/10.1038/s41565-023-01421-3 modes in the terahertz frequency range is a key determinant for the 21. Phillpot, S. R. & McGaughey, A. J. Introduction to thermal thermal conductivity of graphene-containing composite materials. transport. Mater. Today 8, 18–20 (2005). 22. Gutierrez-Varela, O., Merabia, S. & Santamaria, R. 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Nature Nanotechnology | Volume 18 | August 2023 | 898–904 903 Article https://doi.org/10.1038/s41565-023-01421-3 Publisher’s note Springer Nature remains neutral with regard to article are included in the article’s Creative Commons license, unless jurisdictional claims in published maps and institutional affiliations. indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended Open Access This article is licensed under a Creative Commons use is not permitted by statutory regulation or exceeds the permitted Attribution 4.0 International License, which permits use, sharing, use, you will need to obtain permission directly from the copyright adaptation, distribution and reproduction in any medium or format, holder. To view a copy of this license, visit http://creativecommons. as long as you give appropriate credit to the original author(s) and the org/licenses/by/4.0/. source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this © The Author(s) 2023 Nature Nanotechnology | Volume 18 | August 2023 | 898–904 904 Article https://doi.org/10.1038/s41565-023-01421-3 Data availability Author contributions The experimental data supporting the findings M.B., K.-J.T. and N.K. conceived the project. X.Y. carried out the are available on Zenodo: https://doi.org/10.5281/ experiments and analysed the data. N.K. developed the theoretical zenodo.7738429 model and wrote the paper. All authors discussed the results and commented on the manuscript. Acknowledgements We acknowledge financial support from the MaxWater Funding initiative of the Max Planck Society. We thank X. Jia and Open access funding provided by Max Planck Society. H. Wang for carrying out preliminary experiments, M. Grechko and D.-W. Scholdei for assisting with the Fourier transform Competing interests infrared measurements, and M.-J. van Zadel and F. Gericke The authors declare no competing interests. for constructing the sample holder. X.Y. is grateful for support from the China Scholarship Council. K.J.T. acknowledges Additional information funding from the European Union’s Horizon 2020 research Supplementary information The online version contains supplementary and innovation programme under grant agreement number material available at https://doi.org/10.1038/s41565-023-01421-3. 804349 (ERC StG CUHL), RYC fellowship number RYC-2017-22330 and IAE project PID2019-111673GB-I00. A.P. acknowledges Correspondence and requests for materials should be addressed to support from the European Commission under the EU Nikita Kavokine. Horizon 2020 MSCA-RISE-2019 programme (project 873028 HYDROTRONICS) and from the Leverhulme Trust under Peer review information Nature Nanotechnology thanks the anonymous grant RPG-2019-363. N.K. acknowledges support from a reviewers for their contribution to the peer review of this work. Humboldt fellowship. The Flatiron Institute is a division of the Simons Foundation. We thank L. Reading-Ikkanda Reprints and permissions information is available at (Simons Foundation) for help with figure preparation. www.nature.com/reprints. Nature Nanotechnology