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Xin Huang, Yangyu Guo, Yunhui Wu, Satoru Masubuchi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Zhongwei Zhang, Sebastian Volz, Tomoki Machida, Masahiro Nomura

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[Observation of phonon Poiseuille flow in isotopically purified graphite ribbons](https://mdr.nims.go.jp/datasets/bf141c90-3911-4b2c-9fbc-274dad05d5d0)

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Observation of phonon Poiseuille flow in isotopically purified graphite ribbonsArticle https://doi.org/10.1038/s41467-023-37380-5Observation of phonon Poiseuille flow inisotopically purified graphite ribbonsXin Huang1,6, Yangyu Guo1,6, Yunhui Wu 1, Satoru Masubuchi 1,KenjiWatanabe 2, Takashi Taniguchi 1,3, ZhongweiZhang1, SebastianVolz 1,4,Tomoki Machida 1 & Masahiro Nomura 1,5In recent times, the unique collective transport physics of phonon hydro-dynamics motivates theoreticians and experimentalists to explore it in micro-and nanoscale and at elevated temperatures. Graphitic materials have beenpredicted to facilitate hydrodynamic heat transport with their intrinsicallystrong normal scattering. However, owing to the experimental difficulties andvague theoretical understanding, the observation of phonon Poiseuille flow ingraphitic systems remains challenging. In this study, based on a microscaleexperimental platform and the pertinent occurrence criterion in anisotropicsolids, we demonstrate the existence of the phonon Poiseuille flow in a 5.5 μm-wide, suspended and isotopically purified graphite ribbon up to a temperatureof 90K.Our observation is well supported by our theoreticalmodel based on akinetic theory with fully first-principles inputs. Thus, this study paves the wayfor deeper insight into phonon hydrodynamics and cutting-edge heatmanipulating applications.The classical Fourier’s law well describes the diffusive phonon trans-port inmacroscale materials at high temperatures, where the frequentUmklapp phonon-phonon scatterings damp the heat flux. Cooling ordown-scaling of the systems invalidates the Fourier’s law and gives riseto non-Fourier heat transport behaviors1–4, such as coherent5–8,ballistic9–11, and hydrodynamic12–16 transport. In contrast to ballistic orcoherent phonon transport dictated by the boundary and interface,hydrodynamic transport is governed by intrinsically momentum-conserving normal phonon-phonon scattering. The frequent normalprocesses lead to exceptionally collective behaviors of phonons simi-lar to those of fluids, including second sound in transient-state14,15 andphonon Poiseuille flow in steady-state16,17. The theoretical predictionand experimental observation of phonon hydrodynamics in solids areof vital significance for both the fundamentals of lattice dynamics dueto its unusual physics and the potential applications in thermal man-agement due to its excellent transport properties.The second sound, named analogously to the first sound (pressurewave), denotes the temperature wave propagating in solid-statematerials18,19. The phonon Poiseuille flow is similar to that of viscousfluids under the pressure gradient in a pipe. The Poiseuille flow ofphonons results from the interplay between normal scattering and dif-fuse boundary scattering events in the structure with a finite width.Phonon momenta are transferred along the gradient of drift velocityfrom the sample center to the sides by normal processes and destroyedat the boundaries13,20,21, inducing a parabolic heat flux profile (Fig. 1a).The experimental detection of second sound in solids has a long historyand has been widely reported owing to its direct wavy feature. Thedrifting second sound was first observed unambiguously in solid He4crystals22, later in various other crystals23–28 with heat-pulse and light-scattering methods at low temperatures, whereas the driftless coun-terpart has been detected very recently in Ge even at room temperatureunder a rapidly varying temperature field19. However, owing to the dif-ficulty in observation and lack of direct evidence, there are limitedexperimental reports on the phonon Poiseuille flow16,29,30. Furthermore,this is also partially caused by the ambiguous criterion to confirm theevidence of phonon Poiseuille flow, as to be shown in the present study.Received: 18 January 2022Accepted: 10 March 2023Check for updates1Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan. 2Research Center for Functional Materials, National Institute for MaterialsScience, Tsukuba 305-0044, Japan. 3International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044,Japan. 4LIMMS, CNRS-IIS IRL 2820, The University of Tokyo, Tokyo 153-8505, Japan. 5Research Center for Advanced Science and Technology, The Universityof Tokyo, Tokyo 153-0041, Japan. 6These authors contributed equally: Xin Huang, Yangyu Guo. e-mail: nomura@iis.u-tokyo.ac.jpNature Communications |         (2023) 14:2044 11234567890():,;1234567890():,;http://orcid.org/0000-0003-0786-287Xhttp://orcid.org/0000-0003-0786-287Xhttp://orcid.org/0000-0003-0786-287Xhttp://orcid.org/0000-0003-0786-287Xhttp://orcid.org/0000-0003-0786-287Xhttp://orcid.org/0000-0001-7039-6694http://orcid.org/0000-0001-7039-6694http://orcid.org/0000-0001-7039-6694http://orcid.org/0000-0001-7039-6694http://orcid.org/0000-0001-7039-6694http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-2069-9480http://orcid.org/0000-0003-2069-9480http://orcid.org/0000-0003-2069-9480http://orcid.org/0000-0003-2069-9480http://orcid.org/0000-0003-2069-9480http://orcid.org/0000-0002-1938-7415http://orcid.org/0000-0002-1938-7415http://orcid.org/0000-0002-1938-7415http://orcid.org/0000-0002-1938-7415http://orcid.org/0000-0002-1938-7415http://orcid.org/0000-0003-3706-4836http://orcid.org/0000-0003-3706-4836http://orcid.org/0000-0003-3706-4836http://orcid.org/0000-0003-3706-4836http://orcid.org/0000-0003-3706-4836http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37380-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37380-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37380-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37380-5&domain=pdfmailto:nomura@iis.u-tokyo.ac.jpGraphitic materials, owing to their intensive normal scatteringdue to the strong anharmonicity, and the high density of states of thelow-lying flexural (or bending) phonon modes, are considered as thesuitable systems for demonstrating phonon hydrodynamics at ele-vated temperatures13,20,21. The second sound has been recentlyobserved in highly oriented pyrolytic graphite (HOPG) using transientthermalmeasurement techniques at recordinghigh temperatures14,15,31.Despite the numerous theoretical investigations of phonon Poiseuilleflow in graphitic materials17,20,32–34, the experimental observationremains challenging owing to its more stringent observation windowcondition compared to that of the second sound, as to be elucidated inthis work. It requires a special temperature range to realise the dom-inance of normal scattering and well-designed suspended micro-structures to establish the hydrodynamic phonon flow. The indicationof phonon Poiseuille flow was reported in a recent experimental workon bulk-scale natural graphite samples35. However, it still remainsinconclusive due to the pending theoretical explanation of theanomalous thickness-dependent trend and the ambiguous criterion.Additionally, the isotope-phonon scattering, as a momentum-destroying process, has been predicted to play an indispensable rolein suppressing the occurrence of phonon Poiseuille flow12,13. However,the impact of isotope content in graphitic samples on the phononhydrodynamic phenomena remains experimentally unexplored.In this work, we present an unambiguous experimental evidenceof phonon Poiseuille flow in graphitic materials. We design and fabri-cate submicroscale-suspended graphite ribbons and measure thethermal conductivity using a non-contact microsecond-scale time-domain thermoreflectance (μ-TDTR) technique. In addition, weinvestigate hydrodynamic phonon transport in both the natural andisotopically purified graphite samples in a wide temperature range of10−300 K. Supported by our first-principles-based theoretical model-ing, weuncover the impact of the anisotropic natureof graphite on thecriterion of phonon Poiseuille flow, and the appreciable influence ofisotope content on its occurrence.ResultsSamples and thermal conductivity measurementOur isotopically purified graphite crystals are synthesised using thehigh-pressure/high-temperature (HPHT) technique36,37, and the iso-topic abundance of 13C is measured to be 0.02% using a time-of-flightsecondary ion mass spectrometry (TOF-SIMS) (Supplementary Note 1and Supplementary Fig. 1a). The 13C isotope concentration in the nat-ural graphite crystal is 1.1%. A Raman spectroscopy is employed tocharacterise the crystallinity of the sample. As seen in SupplementaryFig. 1b, theRamanspectra show twomainpeaks for both crystals: theGpeak at ~ 1561 cm−1 and the 2D peak at ~2710 cm−1, representing the sp2bonding of carbon atoms and perfect crystallite of the samples38–40.The only difference between these two samples is the isotopic con-centration of 13C. Both samples are treated under the same conditionsin the entire fabrication and measurement process. Starting froma ~50× 150μm2 graphite flake, we first pattern and fabricate severalsuspended graphite ribbons connecting the graphite islands and thegold heat sinks from both sides (Fig. 1b). All the ribbons have the samelength of 30μm, and various widths from 1.3 to 5.5μm. Next, we applythe pump-probe technique for investigating the in-plane phonontransport through the graphite ribbons with a circular aluminumFig. 1 | Isotopically purified graphite ribbons and measurement method.a Illustration of phonon Poiseuille flow in graphite ribbon. In the hydrodynamicregime, the heat flux (represented by the red arrows) manifests a parabolic profilemaintained by the collective motion of phonons through a ribbon structure with afinite width. b SEM image of suspended graphite ribbons with various widths.c Schematic of the μ-TDTR measurement with the pump-probe method.Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 2transducer deposited on the central island, as depicted in Fig. 1c. Ourμ-TDTR setup ensures the precise measurement of the steady-statethermal conduction in the ribbons at the microsecond-scale (seedetails in Methods). The thickness of the ribbons is identical to that ofthe initial flake, and it is estimated to be ~85 nm from the cross-sectionview of the scanning electron microscope (SEM) image (Supplemen-tary Fig. 1a). Other sample conditions, including the impurity con-centration and surface roughness, are assumed to be consistent for allthe adjacent ribbons from the same flake to justify the width-dependent investigation in the following content.Figure 2 shows the in-plane thermal conductivity of the iso-topically purified (0.02% 13C) and natural (1.1% 13C) graphite ribbonswith the same designed width of 5.5 μm. We measure the thermalconductivity of both the ribbons from 300 K down to 10 K, whichsufficiently covers the temperature range of hydrodynamic windowcondition in graphite as reported in previous theoretical andexperimental studies14,17,35. With the decrease in temperature from300 K, as the Umklapp phonon scattering becomes weaker, thethermal conductivities of both samples follow an increasing trenduntil they reach their peaks. The peak value of natural sample ismeasured as ~ 1477 Wm−1K−1, which is lower than that of bulk HOPGwith the same isotope contents35 reported in a recent work owing tothe strong size effect from structure down-scaling41–43 (Supplemen-tary Fig. 2). While the thermal conductivity of isotopically purifiedsample is peaked at around 150 K as ~ 2635 Wm−1K−1 owing to theisotopic enrichment. The phonon Umklapp process is further wea-kened below 150 K, where appreciable difference emerges betweenthe thermal conductivities of the two samples with different 13Cconcentrations. The isotope scattering plays an important role in thisregime, resulting in an enormous enhancement of thermal con-ductivity of isotopically purified graphite ribbon compared to that ofthe natural one by 105% at 90 K, which also qualitatively affects theoccurrence of phonon Poiseuille flow as discussed in the followingcontent. The isotopic effect remains prominent until the tempera-ture goes down to 50 K, below which the thermal conductivities ofthe two samples become comparable again due to the gradualdominance of boundary scattering over isotope scattering. Thesetrends of the thermal conductivities are similar in natural and iso-topically purified graphite ribbons with the widths of 1.3μm and3.3μm, as shown in Supplementary Fig. 3. The effect of isotopicenrichment on the increase of thermal conductivity has also beenobserved in many other materials within the temperature range of128−380 K, such as GaN (15%)44, boron phosphide (17%)45, graphene(36%)46, diamond (50%)47, cubic boron nitride (90%)48, and SiNWs (150%)49.Phonon Poiseuille flowIn the hydrodynamic regime, the collective motion of thermal pho-nons due to the dominant intrinsic normal process demonstrates aPoiseuille flow of heat, as shown in Fig. 1a. In this regime, the prob-ability of phonons losing momentum (due to resistive scattering) isnotably reduced along their transport paths. In contrast, the phononmomentum is frequently destroyed in the ballistic regime due to thediffuse boundary-phonon scattering events. Therefore, a fasterincrease in thermal conductivity than the ballistic limit is considered asthe indicator of the phonon Poiseuille flow12,13,20. Quantitatively, asimple kinetic formula estimates the thermal conductivity as κ ~Cvl,with C, v, and l being the heat capacity, group velocity, and mean freepath (MFP), respectively. The effective momentum-destroying MFP inthe hydrodynamic transport can be obtained from the random walktheory as50,51: l ~W2/lN, withW as the sample width and lN as the MFP ofnormal process. Onemay feature the hydrodynamic thermal transportin principle from the temperature-dependence of l associated with thestrength of the normal process, as detected in some crystals30,52. TheMFP is instead limited by the sample width (l ~W) in the ballistic limit:thus, κ ~CvW. For most common three-dimensional (3D) materials, inthe hydrodynamic regime, meaning that at very low temperatures, theheat capacity follows the well-known Debye T3 law, and the groupvelocity is approximately the speed of sound as a constant. Therefore,as the temperature increases, with the enhancement of normal scat-tering (lN decreases), the thermal conductivity also boosts morerapidly than the ballistic limit (T3). As a result, the increase in thermalconductivity, with a temperature-dependent exponent larger than 3,has been adopted as a criterion to confirm the hydrodynamic phononflow in several 3D crystals, such as SrTiO3 (T~3.5)16, Bi (T3.5)52, and He4(T8)53, within the temperature ranges of 6 − 13 K, 1.5 − 2.4 K, and0.7 −0.9 K, respectively. For a two-dimensional (2D) system like gra-phene, the ballistic thermal conductance (or conductivity) followsT1.6841, and a faster increasing trend of thermal conductivity over T1.68 isused to indicate the phonon Poiseuille flow, and it has been obtainedby varying the width of graphene ribbon in a previous theoreticalstudy20.However, the situation becomes quite different for graphite,where single graphene layers are bonded through weak van der Waalsforce. The anisotropic nature of graphite makes the temperaturescaling of the thermal properties different from those of its 2D coun-terpart (graphene) and other isotropic 3D materials. The measuredheat capacity of graphite deviates from the Debye law and shows asmaller power dependence of T2.5 with an exponent between those of2D and 3D systems54. In a recent experimental report with naturalHOPG sample, an increase in the ratio of thermal conductivity over T2.5or over heat capacitywith increasing temperature has been adopted toindicate the phonon hydrodynamics35. In the following part, we willillustrate that the ballistic limit (or ballistic thermal conductance(Gballistic), equivalently) of graphite shows a different temperaturedependence from that of heat capacity, i.e., different from T2.5.Therefore, a faster increase of κ than Gballistic with increasing tem-perature is expected for the presence of phonon Poiseuille flow ingraphite. Thus, we will demonstrate unambiguously the evidence ofphonon Poiseuille flow in isotopically purified graphite ribbons withsufficiently large widths.The results of κ/T2.5 and κ/Gballistic as a function of temperature aregiven in Fig. 3a and b, respectively, for our isotopically purified gra-phite ribbons. The ballistic thermal conductance for graphite is cal-culated by the first-principles method (see details in Methods) asFig. 2 | Temperature-dependent in-plane thermal conductivity (κ) of iso-topically purified and natural graphite ribbons with a designed width of5.5μm. The dark and light blue dots represent the results of isotopically purified(0.02% 13C) and natural (1.1% 13C) graphite ribbons, respectively. Note that the actualwidth of the natural graphite ribbon is 0.8 μm wider than that of the isotopicallypurified one due to the deviation in fabrication, resulting in the minor flip ofthermal conductivities at very low temperatures. Inset: thermal conductivity from50 to 100K. Errorbars depict the standarddeviations of differentmeasurements onthe same ribbon.Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 3follow:Gballistic =XpZvðkÞ_ωðkÞ ∂feq∂Tdkð2πÞ3, ð1Þwherep represents phononpolarization, and v(k),k,ℏ,ω(k), and feq arethe group velocity, wave vector, reduced Planck’s constant, frequency,and Bose-Einstein equilibrium phonon distribution, respectively. Notethat we also calculate the ballistic thermal conductance based on anempirical atomic interaction potential. The quantitative value of theballistic thermal conductance will be influenced by the atomicinteraction potential. However, the qualitative temperature-scalingbehavior and the conclusion will be not much changed. More detaileddiscussions are given in Supplementary Note 2 and Supplemen-tary Fig. 4.As shown in Fig. 3a, κ/T2.5 increases with temperature from 10 to~40 K for all the isotopically purified graphite ribbons with differentwidths, including in the narrowest case of 1.3μm, where momentum-destroying boundary scattering is expected to remain appreciable. Wefurther examine the temperature dependence of κ/T2.5 in a 500nm-wide ribbon, where the heat transport should lie within the ballisticregime. However, an unexpected raise of κ/T2.5 as temperatureincreases is observed, as shown inSupplementary Fig. 5a. This could beexplained by the faster increase of the Gballistic than T2.5 in the sametemperature range, as illustrated in the inset of Supplementary Fig. 5a.In other words, a faster increase of κ than T2.5 or the heat capacity maynot indicate the occurrence of hydrodynamic phonon flow definitely.Thus, amore relevant criterion to demonstrate phonon Poiseuille flowin graphite wouldbe the temperature-dependent trend of κ/Gballistic, asshown in Fig. 3b. In the graphite ribbon with a width of 1.3μm,κ/Gballistic continuously decreases with increasing temperature, as asign of the transition from ballistic to diffusive transport. The trend issimilar in the case of 500 nm-wide ribbon, as indicated in Supple-mentary Fig. 5b. In these cases, the sample widths are too narrow forenough normal scatterings to occur. However, in the graphite ribbonwith a larger width of 3.3μm, κ/Gballistic starts to increase withincreasing temperature from 50 to 80 K, where the normal scatteringstarts to play an increasing role to cancel and prevail the effect ofmomentum-destroying phonon scatterings. When the width of thegraphite ribbon is sufficiently large as 5.5μm, the normal scatteringbecomes frequentwhile the resistive (Umklapp and isotope) scatteringis still scarce, and an apparent enhancement of κ/Gballistic is observedfrom 40 K. The temperature window of phonon Poiseuille flow isexpanded and lasts to an elevated upper limit of 90 K owing to thedominance of momentum-conserving normal scattering in the ribbonwith larger width. This super-ballistic scaling of thermal conductivitywith temperature is a clear evidence of phonon Poiseuille flow. Athigher temperatures (>100 K), κ/Gballistic shows a dramatic decreasewith increasing temperature in the graphite ribbons with all the widthsdue to the increasing rate of Umklapp scattering.To observe the phonon Poiseuille flow, the rate of normal scat-tering should be dominant over those of the resistive ones, such asUmklapp scattering and phonon-isotope scattering12,13. To this end, wealso investigate the isotope effect on the phonon Poiseuille flow bycomparing the results of isotopically purified (0.02% 13C) and natural(1.1% 13C) graphite ribbons.We first examine the occurrence of phononPoiseuille flow based on the temperature dependence of κ/Gballistic in1.3μm-wide graphite ribbons (Fig. 4a). As illustrated in Fig. 4d,κ/Gballistic of both samples show a decreasing trend as the temperatureincreases due to the predominant diffuse phonon-boundary scatteringin relatively narrow ribbons. A steeper decrease is found in the naturalgraphite sample due to the additional effect of the momentum-destroying isotope scattering of phonons. When the ribbon width issufficiently large, namely the 3.3μm case (Fig. 4b), κ/Gballistic shows anon-monotonous trend in the isotopically purified sample, while itcontinuously decreases in the natural one with increasing tempera-ture, as shown in Fig. 4e. It infers that the phonon Poiseuille flow isdeteriorated by the resistive phonon-isotope scattering in naturalgraphite sample. Further enlargement of the ribbon width to 5.5μm(Fig. 4c) makes the difference of κ/Gballistic between isotopically pur-ified and natural samplesmore pronounced, as depicted in Fig. 4f. Thisis caused by the larger space for sufficient momentum-conservingnormal scattering to occur in the purified sample. On the opposite,widening of the ribbon also makes more momentum-destroyingphonon-isotope scattering to occur in the natural counterpart.The aforementioned tendencies of experimental data are gen-erally consistent with our theoretical modeling results in Fig. 4g–ibased on a direct solution of phonon Boltzmann transport equation(BTE) with full first-principles inputs (see details in Methods). There issome difference between the absolute values of κ/Gballistic in experi-mental and theoretical results, as we consider infinite thickness in theBTE modeling. However, a good agreement is found in terms of therelative trends of isotopically purified and natural abundance graphiteribbons and the temperatures where the minimum and maximumemerge. A direct solution of phonon BTE for hydrodynamic heattransport in graphite ribbon with finite length, width and thickness is achallenging task, as it requires a numerical solution in both 3D reci-procal space and 3D real space. To the authors’ best knowledge, it isFig. 3 | The criterion and evidence of phonon Poiseuille flow in isotopicallypurified graphite ribbons. a The usual criterion, namely the ratio of thermalconductivity (κ) over T2.5 as a function of temperature (T) for graphite ribbons withvarious widths (W). b The present criterion, namely the ratio of thermalconductivity over Gballistic as a function of temperature. Error bars depict thestandard deviations of different measurements on the same ribbon. The thermalconductivities of the isotopically purified graphite ribbons are shown here.Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 4only reported that the Monte Carlo solution of phonon BTE with abinitio full scattering term for such situation from one group in veryrecent studies15,55. However, due to huge computational cost, relativelycoarse grids in both reciprocal and real spaces have been adopted.Apparently, there is still some space to further improve the accuracy ofthe numerical solution and its agreement with experimental result15.On the other hand, as the thickness effect on basal-plane heat trans-port in graphite remains an open question35,55, the present modelingand experimental study are mainly focused on the effects of finitelength and width. Our semi-quantitative theoretical modeling gen-erally provides a good guide for the observation of phonon Poiseuilleflow in finite-sized isotopically purified graphite ribbons.DiscussionThe steady-state phonon hydrodynamic phenomenon, i.e., phononPoiseuille flow, appears only when a strict condition is satisfied. Itrequires normal scattering to be more sufficient than boundary scat-tering, which is further more substantial than other resistive phononscattering events, such as Umklapp and isotope scatterings. Innanoscale structures, the frequent interaction between phonons andthe structure edges brings heat conduction to the ballistic regime. Inmuch larger and isotopically-impure samples, the resistive scatteringdeteriorates the hydrodynamic phonon flow. Thus, the Poiseuille flowof phonons can be well-established in the graphite sample with apurified isotope concentration and a width in between the MFPs ofnormal and resistive scatterings56 (lN≪W, lRlN≫W2). A recent theore-tical work predicted that the width window condition of phononhydrodynamics is approximately 2 − 20μm in the temperature rangeof 50−90 K in graphite ribbons with 0.1% isotope content17. Ourobservation of phonon Poiseuille flow in 3.3μm- and 5.5μm-wide iso-topically purified graphite ribbons, therefore, confirms this theoreticalprediction. We also provide a detailed quantitative demonstration ofwhy the hydrodynamic window condition is satisfied only in the iso-topically purified graphite ribbon in Supplementary Note 2 and Sup-plementary Fig. 6. On the other hand, it explicitly demonstrates amorestringent condition for the observation of phonon Poiseuille flow thanthat of the second sound, which has been observed instead in naturalgraphite recently14,15,31 (detailed explanation in Supplementary Note 3).We have shown that the temperature dependence of κ/Gballistic (orequivalently κ/κballistic) is a more relevant criterion to confirm thephonon Poiseuille flow. In most 3D materials close to isotropic struc-tures, the ballistic thermal conductance and heat capacity follow thesame temperature power law at low temperatures due to the lineardispersion relation of acoustic phonons20,57,58. The different tempera-ture scalings between Gballistic and heat capacity (T2.5) of graphite,shown in the inset of Supplementary Fig. 5a, is mainly attributed to theanisotropic nature and special phonon dispersion of graphite. Thehydrodynamic phonon transport ismainly contributed by the bendingacoustic (BA) modes in graphite17,33, the group velocity of whichincreases with frequency due to the quadratic dispersion curve13,59. Asexpressed in equation (1), Gballistic is determined by both the heatcapacity term and the group velocity term (v(k)). Therefore, with anincrease in temperature in the hydrodynamic window and morepopulated higher-frequency BA phonons, Gballistic of graphite boostsFig. 4 | Isotope effect on phonon Poiseuille flow in graphite ribbons. a–c SEMimages of suspended isotopically purified graphite ribbons with the widths of1.3μm, 3.3μm and 5.5μm, respectively. d–f Experimentally measured and (g–i)calculated thermal conductivity (κ) over ballistic thermal conductance (Gballistic) asa function of temperature corresponding to the three ribbons in (a–c). The dark(light) green, red (pink), and dark (light) blue dots represent the experimental dataof isotopically purified (natural) graphite ribbons. Error bars depict the standarddeviations of differentmeasurements on the same ribbon. The empty dots with thecorresponding colors denote the modeling results by BTE with first-principlesinputs. Note that the actual widths of the natural graphite ribbons are 1.6μm,3.7μmand 6.3μm, respectively, due to the deviation in fabrication, resulting in theminor flip of κ/Gballistic at very low temperatures.Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 5faster than the heat capacity (T2.5). As a result, the increase of κ/T2.5 withtemperature from few Kelvins to ~25 K in the recent experimentalreport35 is most probably contributed by the behavior of the ballisticthermal conductance.In addition, as a reference, we examine the isotope effect on theobservation of phonon Poiseuille flow based on the present criterion(κ/Gballistic) in silicon samples with natural and purified 28Si isotopeconcentrations60 (Supplementary Fig. 7). As temperature increases,κ/Gballistic of both natural and purified silicon samples drops mono-tonically, indicating the absence of phonon Poiseuille flow even in theisotopically purified silicon sample. This is explained by the well-known insufficient normal scattering in silicon to satisfy the hydro-dynamic window condition.According to the different temperature-dependent behaviors ofκ/Gballistic in our isotopically purified graphite ribbons with variouswidths, we observe the transition from the ballistic to the hydro-dynamic thermal transport when the ribbon width increases from1.3 μm to 5.5 μm. Besides, another important aspect to evidence thephonon hydrodynamic flow is the super-ballistic width dependence ofthermal conductivity, as clearly indicated by the effective mean freepath l ~W2/lN. The suspended microstructure system built up in thiswork provides a good platform for further experiments to investigatethe extraordinary super-linear width dependence of thermal conduc-tion in the hydrodynamic regime or the phonon Knudsen minimumphenomenon17,32,33,61.Finally, wewould like to note that the phonon Poiseuille flow couldbe modeled by macroscopic phonon hydrodynamic equations2,56 in amuchmore efficient way. It resembles the description of Poiseuille flowof classical fluids by the Navier-Stokes equation. The classical Guyer-Krumhansl (G-K) phonon hydrodynamic equation56 assumes gray pho-non properties and usually works well for traditional 3D isotropiccrystals at very low temperatures. However, the complex nonlinearfrequency-dependent phonon properties are important for phononhydrodynamics in graphitic materials where the original G-K equationmight be not able to be directly applied. Recently, a generalized G-Kheat transport equation (so-called kinetic-collective model, KCM62) hasbeen derived from phonon BTE taking into account the arbitrary scat-tering term and the nonlinear phonon properties63. In principle, theKCM, together with appropriate macroscopic boundary conditions,couldbe analternative formodeling thephononPoiseuilleflow infinite-sizedgraphite ribbon,which is however anontrivial task andbeyond thescope of the present study. On the other hand, the hydrodynamicmodel (or KCM) covers the heat transport in both kinetic62,64 andcollective65 limits. Apparently the debate about the definition of thehydrodynamic regime14,19 when heat transport could be described bythe KCM is still open. Nevertheless, the present work is mainly focusedon the collective limit, i.e. when normal scattering dominates. Thus wecould not provide definite remarks about this debate in the currentstage, and leave it for a future exploration.In summary, we have developed an integrated experimentalplatform to investigate the steady-state phonon hydrodynamics insuspended graphite submicron structures. In light of a more relevantcriterion due to the anisotropic nature of graphite, we observed pro-minent Poiseuille flow of phonons in an isotopically purified graphiteribbon with a width of 5.5 μm up to 90 K, which is much elevatedcompared to the temperature range of previous observations in othersolid-state materials16,30,52,53. The phonon Poiseuille flow is prone to bedestroyed by the resistive phonon-isotope scattering in graphite rib-bons with natural abundance of carbon isotope. Our joint theoreticaland experimental study on phonon hydrodynamics in graphiticmaterials thus deepens the understanding of the collective physics ofphonons in anisotropic solids. The experimental platform will alsoopen innovative possibilities for tuning and manipulation of phononhydrodynamics, as well as its application in thermal management ofthe modern micro- and nanoelectronics.MethodsSample preparationSample fabrication begins with mechanical exfoliation of graphite toobtain graphite flakes, which are then transferred onto a SiO2 (2.4μm)/Si substrate right after O2 plasma surface treatment. The typical size ofa flake is ~50× 150μm2, and the flake thickness is approximately 85 nm(as shown in Supplementary Fig. 1a), measured by SEM. Next, we applyelectron-beam lithography (EBL) for patterning ribbon structures witha fixed length of 40 μm but varying widths. Furthermore, a 6 × 6μm2graphite island is patterned in the center of the ribbons on the sameflake.We also use an electron-beamphysical vapor deposition (EBPVD)to deposit 100 nm-thick aluminum on top of the graphite ribbons asmasks for O2 plasma etching. After exposing samples in a reactive ionetching chamber with an O2 plasma source, we remove the rest of thegraphite around the ribbon structures and release the aluminummasks to acquire the desired graphite ribbons. Moreover, we use laserlithography and another EBPVD to fabricate two 250× 400μm2 goldpads used as hydrofluoric acid (HF) stoppers, to cover all the ribbonsby 10μm-long from both sides. Following this, 70 nm-thick circularaluminum transducers with radius of 2.5μm are deposited on thecentral graphite island for TDTR measurement. HF vapor etching isused to remove SiO2 from the entire surface, with only a portion ofSiO2 remaining underneath the gold pads to support the pads. Beingattached and clamped by the gold pads from both sides, the ribbonsare finally suspended for investigating phonon hydrodynamic trans-port, with a length of 30 μm staying completely out-of-contact.Thermal characterisationWe employ a well-developed μ-TDTR setup to measure the thermalconductivity of our samples66. We place our samples in a vacuumcryostatwith apressure below10−5 Pa to avoid the convective heat loss.At 200 K, radiation heat loss from graphite ribbons is estimated as~1 nW, whereas the input power from the pump beam is ~ 200 nW. Attemperatures below 150 K, the loss through radiation is negligible67,68.A liquid helium flow system enables to cool down the cryostat to 4 K. Acryogenic temperature controller fromOxford Instruments adjusts thetemperature with a precision of 0.1 mK.In the μ-TDTR setup, two lasers are focused onto the center of thealuminum transducer located on the graphite island: one pulsed pumpbeam with a wavelength of 642 nm and one continuous probe beamwith a wavelength of 785 nm. The pump beam with a 2μs pulse dura-tion at 1 kHz repetition rate induces excitation in the transducer. Theprobe beam measures the reflectivity change from the base valuegenerated by the pump beam, and a photoreceiver monitors itsreflection with a maximum bandwidth of 200 MHz to detect theresponse every 5 ns. An oscilloscope captures the probe signal aver-aging 104 measurements into a thermal decay as a function of time, asshown in Supplementary Fig. 8. The fitting parameter, τ, is an indicatorof the inherent thermal properties of the measured sample. We mea-sure each ribbon with one single-width three to five times and calcu-lated its standard deviation as depicted by the error bars in figurescontaining the experimental data.To extract the thermal conductivity of the graphite ribbons, webuild up a numerical model identical to the actual experimental oneusing the finite element method (in COMSOL Multiphysics). Inthemodel, thematerial parameters ofmetals (gold and aluminum) areassumed as their bulk values due to neligible size effects, whereas thespecific heat of graphite is taken from the reported benchmark data69.The anisotropic nature of heat conduction in graphite is taken intoaccount by setting the bulk out-of-plane thermal conductivity from theliterature70 and leaving the in-plane thermal conductivity as the onlyfitting parameter. Thermal boundary conductance between metalsand graphite is also regarded as a crucial parameter in the model tocorrectly reproduce the experimental measurement, as furtherexplained in details in Supplementary Note 4 and SupplementaryArticle https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 6Fig. 9. By injecting a 2 μs heat flux pulse with a Gaussian spatialdistribution onto the aluminum transducer, we simulate the heatdissipation through the graphite ribbon structures to reproducethe decay times in the experiments by sweeping the values of in-plane thermal conductivity. The in-plane thermal conductivity ofgraphite ribbon is obtained when an optimal fitting was foundbetween the decay times obtained by simulation and by experiment(Supplementary Fig. 8).Theoretical modelingWe model the hydrodynamic heat transport through finite-sizegraphite ribbons by directly solving the phonon Boltzmanntransport equation (BTE) with fully first-principles inputs usingthe methodology developed in our recent work33. The requiredinput of phonon properties (including phonon dispersion rela-tions and scattering rates) are calculated in the open-sourcepackage SHENGBTE71, with the harmonic and anharmonic (third-order) force constants obtained by the first-principles calcula-tions implemented in the package QUANTUM ESPRESSO72. Toaccurately describe the inter-layer interaction in graphite, weadopt the most advanced van der Waals non-local density func-tional theory, with all the numerical details and validation of thefirst-principle calculations provided in our previous work33. Thefirst-principles phonon properties are also used in the calculationof the ballistic thermal conductance in equation (1). In the directnumerical solution of phonon BTE, we model the graphite ribbonwith the same length, width, and carbon isotope concentration asthose in the experimental measurement. Fully diffuse scatteringof phonons at the transverse boundaries of the graphite ribbonare considered. This is reasonable as the edges of the ribbon aregenerally roughened by etching process. Along the cross-planedirection of graphite ribbon, we assume infinite thickness andsolve the phonon BTE in 2D real space and 3D reciprocal space33.Such treatment is strictly valid when the surfaces in the thicknessdirection are fully specular, i.e., ideally smooth, which is anacceptable case when the sample is obtained via perfect exfolia-tion process as in multilayer graphene ribbon73,74. Since theabsolute thermal conductivity by the present BTE modeling islarger than the experimental one, we infer that there should beunknown surface imperfections in the thickness direction.Nevertheless, the present BTE modeling captures the dominanteffects from finite length and width, considering that thenumerical solution of BTE in both 3D real and reciprocal spaces iscomputationally too expensive for hydrodynamic heat transport.Also, our recent numerical methodology considering only finitelength and width33 remains to be further developed, whichrequires appreciable amount of future work and effort.Data availabilityThe data that support the findings of this study are available from thecorresponding authors upon reasonable request.References1. Wilson, R. &Cahill, D. G. Anisotropic failure of fourier theory in time-domain thermoreflectance experiments. Nat. Commun. 5,5075 (2014).2. Guo, Y. & Wang, M. Phonon hydrodynamics and its applications innanoscale heat transport. Phys. 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Dimensional crossover of thermal transport in few-layer graphene. Nat. Mater. 9, 555–558 (2010).AcknowledgementsX.H. acknowledges the support from the Grant-in-Aid for JSPS Fellows(Grant Number 21J12652). Y.G. acknowledges the support from theGrant-in-Aid for JSPS Fellows (Grant Number 19F19353). S.M. acknowl-edges the support from JSPS KAKENHI (Grant Number 19H01820). K.W.acknowledges the support from JSPS KAKENHI (Grant Number21H05233). T.M. acknowledges the support from JSPS KAKENHI (GrantNumbers 20H00127, 21H05232, 21H05234) and CREST JST (GrantNumber JPMJCR20B4). M.N. acknowledges the support from JSPSKAKENHI (Grant Number 21H04635) and CREST JST (Grant NumberJPMJCR19Q3).Author contributionsX.H., Y.G. and M.N. conceived this study. X.H. and Y.W. designed andconducted the fabrication. X.H. performed the TDTR and Raman spectrameasurements, analyzed the results, and wrote themanuscript. S.M. andT.M. provided the graphite samples, mechanical exfoliation techniqueand Raman spectroscopy technique. S.M. contributed to designing thefabrication and Raman spectra measurements. K.W. and T.T. synthesizedthe isotopically purified graphite crystals. Y.G. developed the theoreticalmodel and contributed to writing the manuscript. Z.Z. and S.V. con-tributed to interpreting the results. M.N. supervised this work. All authorscontributed to discussing the results and manuscript revision.Competing interestsThe authors declare no competing interests.Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 8Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-37380-5.Correspondence and requests for materials should be addressed toMasahiro Nomura.Peer review information Nature Communications thanks the otheranonymous reviewer(s) for their contribution to the peer review of thiswork. Peer review reports are 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 license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-37380-5Nature Communications |         (2023) 14:2044 9https://doi.org/10.1038/s41467-023-37380-5http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Observation of phonon Poiseuille flow in isotopically purified graphite ribbons Results Samples and thermal conductivity measurement Phonon Poiseuille flow Discussion Methods Sample preparation Thermal characterisation Theoretical modeling Data availability References Acknowledgements Author contributions Competing interests Additional information