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Xin Huang, Satoru Masubuchi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Tomoki Machida, Masahiro Nomura

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[Super-Ballistic Width Dependence of Thermal Conductivity in Graphite Nanoribbons and Microribbons](https://mdr.nims.go.jp/datasets/d8a50f40-cd31-4ce4-8c5a-5502d6c54a1e)

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Super-Ballistic Width Dependence of Thermal Conductivity in Graphite Nanoribbons and MicroribbonsCitation: Huang, X.; Masubuchi, S.;Watanabe, K.; Taniguchi, T.; Machida,T.; Nomura, M. Super-Ballistic WidthDependence of Thermal Conductivityin Graphite Nanoribbons andMicroribbons. Nanomaterials 2023, 13,1854. https://doi.org/10.3390/nano13121854Academic Editor: Gyaneshwar P.SrivastavaReceived: 17 May 2023Revised: 9 June 2023Accepted: 10 June 2023Published: 13 June 2023Copyright: © 2023 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).nanomaterialsArticleSuper-Ballistic Width Dependence of Thermal Conductivity inGraphite Nanoribbons and MicroribbonsXin Huang 1,* , Satoru Masubuchi 1 , Kenji Watanabe 2 , Takashi Taniguchi 1,3 , Tomoki Machida 1and Masahiro Nomura 1,*1 Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan2 Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan3 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan* Correspondence: huangxin@iis.u-tokyo.ac.jp (X.H.); nomura@iis.u-tokyo.ac.jp (M.N.)Abstract: The super-ballistic temperature dependence of thermal conductivity, facilitated by collectivephonons, has been widely studied. It has been claimed to be unambiguous evidence for hydrodynamicphonon transport in solids. Alternatively, hydrodynamic thermal conduction is predicted to be asstrongly dependent on the width of the structure as is fluid flow, while its direct demonstrationremains an unexplored challenge. In this work, we experimentally measured thermal conductivity inseveral graphite ribbon structures with different widths, from 300 nm to 1.2 µm, and studied its widthdependence in a wide temperature range of 10–300 K. We observed enhanced width dependence ofthe thermal conductivity in the hydrodynamic window of 75 K compared to that in the ballistic limit,which provides indispensable evidence for phonon hydrodynamic transport from the perspective ofpeculiar width dependence. This will help to find the missing piece to complete the puzzle of phononhydrodynamics, and guide future attempts at efficient heat dissipation in advanced electronic devices.Keywords: phonon hydrodynamics; thermal conductivity; width dependence; graphite;nanoribbons; microribbons1. IntroductionThe effective manipulation of thermal phonons in nanoscale and microscale electronicdevices has been of ongoing interest to researchers in the semiconductor industry [1,2].Nowadays, with the increasing attention being given to environmental issues and the de-mand for energy resources, thermoelectrics has been developed as a promising energyharvesting technology involving direct conversion of thermal energy into electrical energy.For example, nanostructured thermoelectric generators (TEGs) based on Si materials havebeen designed for simple and economical power generation [3]. By matching or mismatch-ing of the vibration spectrum of the terminal interfaces, to conduct or block the heat flow, astable “on” or “off” state can be formed, thus making thermal logic operation possible forcomputing [4]. In addition, the development of lithography technique with a thermally con-trolled scanning tip has demonstrated its potential for high-resolution and large-throughputnanostructure patterning on the substrates [5].In addition, another important objective of thermal management is the efficient dis-sipation of unexpected heat, to prevent device overheating. In recent years, the ballisticbehavior of phonons has been exploited to achieve this goal, and many investigations havebeen carried out on different structures and materials [6–11]. With rare internal scatteringbetween other partners, ballistic phonons can travel for a relatively long distance (up to15 µm [12]), yet leave no temperature trace in the structures during their propagation.However, a ballistic property requires the dominance of long-wavelength phononmodes, which can only be populated at extremely low temperatures, as their mean freeNanomaterials 2023, 13, 1854. https://doi.org/10.3390/nano13121854 https://www.mdpi.com/journal/nanomaterialshttps://doi.org/10.3390/nano13121854https://doi.org/10.3390/nano13121854https://creativecommons.org/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/nanomaterialshttps://www.mdpi.comhttps://orcid.org/0000-0002-6950-5651https://orcid.org/0000-0001-7039-6694https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-1938-7415https://doi.org/10.3390/nano13121854https://www.mdpi.com/journal/nanomaterialshttps://www.mdpi.com/article/10.3390/nano13121854?type=check_update&version=1Nanomaterials 2023, 13, 1854 2 of 11path is greatly reduced at higher temperatures, due to the frequent Umklapp or defectscattering. On a parallel track, rapid heat dissipation in solid-state materials driven bythe hydrodynamic properties of phonons has been frequently revisited and graduallyrealized by researchers, as another promising candidate for modern thermal managementat elevated temperatures [13–20].Phonon hydrodynamics, analogous to fluid dynamics (or hydrodynamics), is a fieldof research that focuses on the extraordinary collective behavior of momentum-conservedphonons in solids. The phonon Poiseuille flow [16,20] and the second sound [13,17,18] insteady and transient regimes are the two representative phenomena. Normal phonon–phonon scattering, which preserves the momentum-conserved nature, is an indispensablecondition for demonstrating phonon hydrodynamic conduction. Recent studies have shownthat normal scattering in graphite material is sufficiently large, due to its high Debye tem-perature, whereas Umklapp scattering is absent in small wavevector states and in thestrong anharmonic interactions of the interlayers [21,22]. It is recognized that graphitepossesses even more substantial hydrodynamic properties than its two-dimensional coun-terpart (i.e., graphene), owing to the additional channels for stronger three-phonon normalscattering participated by out-of-plane modes (i.e., bending acoustic modes (BA)) in thehydrodynamic temperature window [22].Thermal transport modeling in a hydrodynamic regime was first described by themacroscopic hydrodynamic equations derived from the phonon Boltzmann transport equa-tion (BTE) in common bulk crystals [23]. While the accurate prediction of thermal transportin nanostructures and microstructures requires a direct solution of the phonon BTE, thephonon collision term is very complex, making the solution of the phonon BTE very difficult.To simplify the problem, single-mode relaxation time approximation (SMA) has been used,to solve the phonon BTE in most cases where phonons are treated to be independent ofeach other during the scattering process [24,25]. Previous studies have shown that theSMA is a reasonable simplified model for obtaining thermal conductivity when the nor-mal process is weak or negligible in common crystals [26,27]. A peculiarity is that whatlies behind hydrodynamic phonon transport is the dominance of momentum-conservedcollective phonon transport over individual phonon transport, which undergoes strongmomentum-unconserved resistive scattering; therefore, Callaway’s dual relaxation approxi-mation allows the correct description of phonon transport in systems with significant normalprocesses, such as graphite and graphene [13,28], and has been used extensively to modelhydrodynamic thermal transport, by both semi-analytical [22] and numerical [28] methods.It is obvious that hydrodynamic phonon transport (or the dominance of momentum-conserved normal scattering) is highly dependent on the temperature, and this dependencein graphite has recently been studied extensively [20,22,28–30]. Moreover, similar to the vis-cous effect in fluid dynamics, in an ideal hydrodynamic phonon flow, the diffuse scatteringof phonons at the boundaries is the only resistive process: thus, the boundary of the structureplays a crucial role in hydrodynamic transport. Theoretical work has predicted a peculiarwidth dependence of the thermal conductivity in graphene [31] and graphite [22], separately,and has explained it by the phonon viscous damping effect in the phonon hydrodynamicregime; however, direct observation of the width dependence of thermal conductivity ingraphite or graphene remains experimentally challenging.For this paper, we carried out a pioneering experimental study of thermal conductionin free-standing graphite ribbon structures fabricated from the same graphite flake, withwidths ranging from nanoscale to microscale. Based on a µs time-domain thermoreflectance(µ-TDTR) technique, we studied the width dependence of thermal conductivity over a widetemperature range of 10–300 K, to explore the different thermal conduction characteristicsand their dependence on the structure width in ballistic, hydrodynamic and diffusiveregimes. We aimed to provide an experimental support and a deeper insight into thewidth dependence of phonon hydrodynamics, and future directions in which to exploit thepeculiar physics of collective phonons in thermal applications.Nanomaterials 2023, 13, 1854 3 of 112. Materials and MethodsIn addition to 12C, which occupies ∼98.9% of naturally occurring isotopes in carbonmaterials, 13C, as a secondary stable carbon isotope, accounts for the remaining ∼1.1%: itseffect in suppressing thermal transport, due to sufficient momentum-unconserved isotope–phonon scattering, has been intensively claimed in regard to graphene [32], graphite [20],and diamond [33]. In order to avoid the additional isotopic effect on our investigation ofhydrodynamic thermal transport, we used a world-class graphite crystal with an enriched12C content of 99.98% in the current work.The raman spectroscopy experiment measured that the thermal conductivity of free-standing graphene exceeds 2500 W·m−1·K−1 [34] at room temperature, while the thermalconductivity of graphene supported by copper is dramatically reduced to 370 W·m−1·K−1.The presence of a supporting substrate results in significant phonon–substrate interactions,which tend to affect the study of intrinsic thermal properties in the sample: hence, in orderto avoid the considerable heat loss to the supported substrate [35], and to ensure that theheat was only dissipated through the structures of interest, the fabrication of suspendedgraphite ribbons was indispensable for the precise investigation of phonon hydrodynamicsin this work.To this end, we first mechanically exfoliated 150 nm thick flakes from the mothergraphite flake, and bonded them to a 2.5-µm-thick SiO2 sacrificial layer on an Si substrate.In order to study the steady-state hydrodynamic heat transport in the graphite ribbons,the desired ribbon length is expected to be at least 5 times larger than the ribbon width,to ensure the formation of phonon hydrodynamic behavior and the observation of thehydrodynamic flow of phonons, as predicted by a recent work [28]: thus, the typical size ofa flake required to pattern the ribbon structures is ∼100 × 100 µm2. A symmetrical airbridgeconfiguration was designed, with a ribbon length of 30 µm, connecting the transducerand the heat sinks from both sides, as shown in Figure 1: this allowed us to obtain thethermal properties in the in-plane direction, which was of interest to the current work, andwhich allowed the heated structure to be sufficiently cooled down before the next heatingpulse, for accurate measurement. The thermal properties of many other organic [36] andinorganic [11] nanomaterials and micromaterials have been investigated using a similarairbridge method.Pump laserGraphiteHeat dissipation AluminumProbe laserFigure 1. Thermal characterization of graphite nanoribbons and microribbons, using the pump–probemethod.Nanomaterials 2023, 13, 1854 4 of 11To demonstrate the width dependence of thermal conductivity, and to maintain theprecision of the widths in this work, we used an JEOL JBX-6300FS electron-beam lithography(EBL) system, with an overlay accuracy of 9 nm or less, to pattern the designed structures onthe graphite flake. A 6 × 6 µm2 island was patterned in the center of the ribbons on the sameflake, to load the transducer for thermal measurement in the following process. After theEBL patterning of the structures, we combined an electron beam physical vapor deposition(EBPVD) and a standard metal lift-off process, to deposit and form 100-nm-thick Al airbridgestructures as transfer masks. The Al ribbon patterns were then transferred to the underlyinggraphite flake, using a reactive ion etching (RIE) method to etch the remaining graphite byO2 plasma. After the RIE process, we applied the Al etchant with mixed acids, to remove theAl masks and release the desired graphite ribbon structures on the SiO2 substrate.Unlike the suspension process of samples fabricated on a semiconductor-on-insulatorwafer—where vapor-phase hydrofluoric (VHF) acid enters through slits opened on thesemiconductor device layer, and removes specific parts of SiO2 underneath, to suspend thestructure (as demonstrated in previous works [37,38])—the suspension of graphite ribbonstructures exfoliated and transferred onto SiO2 substrate is more challenging and complex.Metal VHF stoppers are required, to protect the graphite ribbons from falling down to thesubstrate or peeling off during the SiO2 removal, because the etching rate of SiO2 under themetal is lower than that under the graphite. To this end, we combined the laser lithographyand EBPVD, to fabricate two 250 × 400 µm2 Au heat sinks, which were used as metal VHFstoppers, to cover the graphite ribbons from both sides. A 70-nm-thick round Al pad wasthen deposited at the center of the graphite bridge by another EBPVD process, as a transducerfor optical/thermal signal detection. Finally, we applied VHF acid, to remove the SiO2 layer,in order to suspend the patterned ribbon structures on the substrate.3. ResultsTo study the thermal conduction in the graphite ribbons, we used the pump–probemethod, in order to characterize the thermal properties of our graphite ribbons in a TDTRsetup on the microsecond scale: as a non-contact measurement technique, this significantlyminimized the difficulties of sample preparation (no electrical heaters and sensors), andallowed rapid measurement on a real-time scale. This approach has been widely usedto study phonon transport, and to evaluate the thermal properties in various graphiticmaterials with high thermal conductivity, such as graphite [17,18,39] and diamond [40,41].During the measurement, our samples were placed in a cryostat with a high vacuumlevel (<10−5 Pa), to avoid unexpected heat loss to the ambient environment. Inside, ametal heating plate and a liquid helium flow system were used, to ensure the preciseadjustment of the sample temperature. We first induced excitation in the Al transducer,using a pump laser (λ = 642 nm) with a 10 µs repetitive pulse at a rate of 1 kHz. Meanwhile,a probe laser (λ = 785 nm) continuously detected the optical response of the transducer.The thermoreflectance measurements were based on the temperature dependence of thematerial’s reflectivity, where the reflectivity of the metal transducer followed the sameresponses as the temperature of the transducer [42], as ∆R/R = Cth∆T, where Cth was thethermoreflectance coefficient. During the pump laser heating, the temperature-dependentreflectivity changed dramatically, as shown by the increase in ∆R/R (the normalizedthermoreflectance signal) in Figure 2a.The conductive Al transducer and the underlying graphite structure reached thermalequilibrium in less than 1 µs, due to the low thermal boundary resistance, as investigated inour previous work [20]: this allowed the heat to be conducted through the graphite structureson a much longer timescale, where the phonon contribution to the thermal conductivitywas dominant. In semiconductors, such as the graphene and graphite mentioned in thecurrent work, or insulators (e.g., organic materials), electrons carry a relatively small oreven negligible amount of the heat [30,43,44], as determined by the Wiedemann–Franz law(κe = LσT, where L is the Lorenz number), using the experimentally measured electronconductivity (σ).Nanomaterials 2023, 13, 1854 5 of 11w3 �mPumpPumpProbe(a)(c)(b)Figure 2. (a) Normalized thermoreflectance signal; (b) Scanning electron microscope (SEM) image of asuspended graphite ribbon with a width of 600 nm; (c) Experimentally measured thermal conductivity(dots) compared to calculated results from Ref. [25] (line), as a function of the graphite ribbon width at300 K.After removal of the pump pulse, the reflectivity change shows exponential decay(exp(−t/τ)), as heat is dissipated through the graphite ribbon structures, where the decaytime τ is a crucial parameter for characterizing the thermal property. At each temperaturepoint, we performed 3–4 measurements for each ribbon structure, to collect sufficient decaytime data to extract the thermal conductivity, as indicated by the error bars of the thermalconductivity results in the figures.We then carried out finite element method modeling, to simulate the heat dissipationthrough the structures, and to reproduce the decay times measured in our experiments. Toextract the thermal conductivity of our samples, we swept a range of thermal conductivities,and their corresponding decay times were also obtained, by fitting the exponential decaycurves from the simulations. The in-plane thermal conductivity values of the measuredgraphite ribbons were thus extracted, by interpolating a linear function between the simu-lated thermal conductivity and the measured decay time. A detailed method for extractingthe thermal conductivity can be found in our previous works [37,38].To investigate the width dependence of the thermal transport in our graphite samples,we fabricated and measured the thermal conductivity of graphite ribbons with differentwidths of 300 nm, 600 nm, and 1.2 µm, as shown in Figure 2b. We used an ultra-high-resolution scanning electron microscope (SEM) (Hitachi SU9000) to characterize the dimen-sions of our structures. Although it was not possible to accurately measure the roughnessby SEM, our high-resolution images show that the roughness of the edges was no morethan 50 nm, as detailed in Figure S1 in the Supplementary Materials.Due to the low atomic mass of carbon, and the strong intraplanar sp2 bonding ofcarbon atoms, heat in graphite conducts more efficiently in the direction parallel to thebasal plane: this results in much higher thermal conductivity than that of other commonthree-dimensional solid-state materials. Experimentally measured in-plane thermal conduc-tivity of bulk-like graphite ranges from ∼1370 W·m−1·K−1 to ∼1950 W·m−1·K−1 at roomtemperature [30,45,46]. As shown in Figure 2c, our measured value of thermal conductivityin a 1.2-µm-wide graphite ribbon was 1111 W·m−1·K−1. As the ribbon width was furtherreduced, the thermal conductivity followed a continuous decreasing trend, dropping to733 W·m−1·K−1 in a 300-nm-wide graphite ribbon, where the diffuse boundary scatteringof phonons became stronger, due to the reduction of the structure size. The strong depen-dence of the thermal conductivity on the characteristic size (i.e., width) of the pure graphiteNanomaterials 2023, 13, 1854 6 of 11sample was also well-demonstrated by Fugallo et al., in their calculations with the exactsolution of the Boltzmann transport equation for phonons [25]. The rough edges of thestructures may also explain the lower experimentally measured thermal conductivities inthis work, compared to the calculated results; however, a preferable agreement was stillfound, in terms of the qualitative trend of the width dependence.By contrast, heat conduction in the direction perpendicular to the basal planes waslargely limited by the weak interplanar van der Waals coupling between the adjacentgraphene layers. The thermal conductivity along the out-of-plane direction of pyrolyticgraphite is ∼6 W·m−1·K−1 at room temperature [45], which is ∼300 times lower than thatalong the in-plane direction. Meanwhile, previous works have observed a transition ofthermal conductivity from graphene to graphite within a few layers of increasing graphenethickness [47,48]. The thickness of our graphite ribbon was 150 nm (∼450 graphene monolay-ers): thus, we assumed a negligible effect of the multilayer graphite on the in-plane thermaltransport, due to the weak van der Waals interaction along the out-of-plane direction.As shown above in Figure 2c, the thermal conductivity decreased monotonically withthe narrowing of the ribbon width at 300 K; however, a previous work predicted that thewidth dependence of thermal conductivity varies according to temperature in differentheat transport regimes of graphite [22]. In contrast to diffusive heat transport at highertemperatures, the phonon flux is rapidly damped by momentum-unconserved Umklappprocesses. At lower temperatures, with the absence (or negligibility) of Umklapp scattering,it enters the hydrodynamic regime, where the thermal phonons tend to conserve theirmomenta, and behave as a whole, in a collective motion, under the frequent momentum-conserved normal processes. At sufficiently low temperatures, the phonons are mainlypopulated in the long wavelength states. Individual phonons rarely experience internalscattering, but are largely limited by the boundaries of the structures.To this end, we performed temperature-dependent thermal conductivity measurementsof graphite ribbons from 300 K down to 10 K, using the µ-TDTR setup, which covered theabove-mentioned phonon transport regimes [22,30,39]. As shown in Figure 3, with thetemperature drop from 300 K, the Umklapp phonon–phonon scattering was gradually hin-dered, resulting in an increase in thermal conductivity in the 1.2-µm-wide graphite sample.However, the thermal conductivity enhancement was less pronounced in the 300-nm-wideand 600-nm-wide cases, indicating that diffuse boundary scattering is comparatively strongin such narrow structures, even at high temperatures. A similar invariance of the ther-mal conductivity with decreasing temperature has also been observed in Si thin film withnanocone structures, due to the sufficient boundary scattering of phonons [37]. At tempera-tures below 150 K, the thermal conductivities of the three samples generally followed thesame trend, with temperature further decreasing; however, a significant difference in thethermal conductivity of the three samples was shown throughout the temperature range. Itis noteworthy that a stronger width dependence of the thermal conductivity was observedin the intermediate temperature range (i.e., ∼30–80 K) in the log–log plot.Many recent works have demonstrated the different temperature-dependent behaviorof thermal conductivity compared to the ballistic limit in hydrodynamic phonon transportregimes in graphite [20,22,29]. On the other hand, another important aspect in provinghydrodynamic phonon flow is the super-ballistic width dependence of thermal conductiv-ity [16,31], as indicated by the effective mean free path (Λe f f ∼ W2/ΛN), from the randomwalk theory [49,50], where W and ΛN are the sample width and the phonon mean free pathof the normal process, respectively. Thus, in a hydrodynamic regime, thermal conductivity(κ ∼ Λe f f ) is proportional to Wα (α > 1), due to significant normal scattering (ΛN < W). Bycontrast, in a ballistic regime, thermal conductivity is linearly related to the width of thestructure, so that κ ∼ Wα (α = 1). Therefore, the superlinear width dependence of thermalconductivity is also considered to be robust evidence for the phonon Poiseuille flow.Nanomaterials 2023, 13, 1854 7 of 111 0 3 0 5 0 1 0 0 3 0 01 0 1 0 011 01 0 01 0 0 0Thermal conductivity, � (W⋅m-1 ⋅K-1 )T e m p e r a t u r e  ( K )  W  =   0 . 3  µm   0 . 6  µm   1 . 2  µm  Figure 3. Thermal conductivity as a function of temperature for graphite ribbons of different widths (W).As shown in Figure 4a, we plotted our experimentally measured thermal conductivityas a function of graphite ribbon width at several typical temperatures covering the entirephonon transport regimes of ballistic, hydrodynamic and diffusive. In principle, thethermal conductivity solely increases with the widening of the graphite ribbon structureover the entire temperature range; however, the rate of increase (or slope) varies for differentthermal transport regimes. Here, to illustrate the width dependence, we fitted the datapoints with linear functions at each temperature point, as shown by the solid lines. Thermalconductivity was thus related to the width of the graphite ribbon, using a simple function:κ ∼ Wα; therefore, we defined the exponent α as the coefficient that indicated how stronglythe thermal conductivity depended on the width at different temperatures.(a) (b)Figure 4. (a) Thermal conductivity as a function of graphite ribbon width at different temperatures;the lines show the linear fit to the measured data; (b) Coefficient of the width dependence, α, as afunction of temperature.As shown in Figure 4b, we compared the values of the width dependence coefficient(α) at different temperatures in the temperature range from 10 to 300 K. At 10 K, thermalNanomaterials 2023, 13, 1854 8 of 11conduction was typically in the ballistic regime, where ballistic phonons are mainly re-stricted by the size of the structures, and thermal conductivity is linearly proportional towidth (i.e., κ ∼ W); therefore, α is ideally equal to 1. However, in the actual experiment,due to other unavoidable defect phonon scattering mechanisms, we obtained a coefficientof ∼0.54 as a base value for the ballistic case.As the temperature increased, the scattering processes between the phonons weregradually excited. When the momentum-conserved normal phonon–phonon processeswere sufficient, it led to the peculiar collective transport of phonons in the hydrodynamicregime. At higher temperatures, normal scattering became more significant, the phononmean free path was further reduced, and the thermal conductivity showed stronger widthdependence, compared to that in the ballistic transport regime (i.e., κ ∼ W2/ΛN), as shownby the increase of α following the elevation of the temperature in Figure 4b. At 75 K,the Umklapp scattering was still neglectable, the normal scattering dominated over othermomentum-unconserved scatterings, and we observed the strongest width dependence,indicating the prominent hydrodynamic behavior of the phonons. After reaching its peak,the coefficient α dramatically decreased with further increase in temperature, indicating asmaller width dependence when Umklapp scattering was predominant, and hydrodynamicflow was ultimately destroyed in the diffusive regime at high temperatures.4. DiscussionFluid dynamics deals with the motion of the fluid as a continuum under force, andthe interaction between the fluid and the boundaries [51]. A commonly discussed case inmodern hydrodynamics is the steady-state flow of an incompressible fluid in a cylindricaltube, where the transfer rate of the molecular Poiseuille flow varies as the fourth power of theradius of the tube [52]. Similarly, investigation of the transfer rate of the phonon Poiseuilleflow has established that it follows the same power law of the radius as that of the molecularflow in the circular tube in previous works [16,23]. Therefore, for a phonon Poiseuille flowin a three-dimensional rectangular ribbon structure, as was the case in the present work, therate of heat flow scales as the cube of the width. Correspondingly, the thermal conductivityshould follow the square of the width due to the hydrodynamic flow of phonons; however,the W2 trend could only occur by considering no Umklapp scattering, such as that influid dynamics. In a recent work, a width dependence of W1.17 was obtained at 100 K ina suspended graphene ribbon, using the Monte Carlo method [31], while it dropped toW0.17 at 300 K, due to the Umklapp scattering being stronger than the hydrodynamic effectat sufficiently high temperatures. Therefore, a superlinear width dependence in thermalconductivity clearly indicates hydrodynamic phonon transport.However, although we did not observe the superlinear (α > 1) width dependence of thethermal conductivity, due to the additional momentum-unconserved scattering of phononsthat was induced during the sample preparation, we could see that the width-dependentcoefficient α within the hydrodynamic temperature window (i.e., 45–75 K) was significantlymore pronounced than that outside the window, both in the ballistic (10 K) and diffusive(300 K) regimes, as shown in Figure 4b. The width dependence of the thermal conductivityhere showed an enhanced thermal conduction compared to that in the ballistic transportregime: this demonstrated the emergence of the hydrodynamic behavior of phonons in ourcarbon isotope-enriched graphite nanoribbons and microribbons. Moreover, superlinearwidth dependence represents an even stronger hydrodynamic phonon flow, and is moredifficult to observe, due to the additional size effect from the length of the structure, asrecently realized in a theoretical study of phonon hydrodynamic transport [28]. Further ex-perimental attempts that focus on the ultimate carbon isotope purification and prolongationof the ribbon structure may be beneficial for the optimization of the phonon hydrodynamictransport, and for the observation of superlinear width dependence in graphite.In this work, to explore the well-predicted prominent width dependence of hydrody-namic phonons, and to further confirm the existence of hydrodynamic thermal transport,we fabricated free-supporting nanoscale and microscale ribbons on isotopically enrichedNanomaterials 2023, 13, 1854 9 of 11graphite flake. Super-ballistic width dependence of the thermal conductivity was observedin our experimental results at 75 K, showing that potential hydrodynamic behavior mayoccur. This provides an adequate complement to a recent work on the observation ofphonon hydrodynamic flow [20] from a different point of view, by considering the widthdependence of thermal conductivity, and a new direction in which to make phonon hydro-dynamics practical in nanoscale and microscale heat management. Further experimentsare needed, to investigate the extraordinary superlinear width dependence of thermalconduction in the hydrodynamic regime, based on the suspended microstructure systemestablished in this work.Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/nano13121854/s1, Figure S1: Characterization of the roughnessand width of the structures.Author Contributions: Conceptualization, X.H. and M.N.; resources, S.M., K.W., T.T. and T.M.; method-ology, investigation and writing—original draft preparation, X.H.; writing—review and editing, X.H. andM.N.; supervision, M.N. All authors have read and agreed to the published version of the manuscript.Funding: This research was funded by Grant-in-Aid for JSPS Fellows (Grant No. 21J12652), JSPSKAKENHI (Grant Nos. 19H01820, 20H00127, 21H05232 and 21H04635) and CREST JST (Grant Nos.JPMJCR20B4 and JPMJCR19Q3). K.W. and T.T. acknowledge support from the JSPS KAKENHI (GrantNos. 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan.Data Availability Statement: The data that support the findings of this study are available from thecorresponding authors upon reasonable request.Acknowledgments: The authors thank Kazuhiko Hirakawa and Yue Tian for providing chem-icals and technical support in experiments, and Yangyu Guo, Yunhui Wu and Sebastian Volzfor discussions.Conflicts of Interest: The authors declare no conflict of interest.References1. Chen, G. Non-Fourier phonon heat conduction at the microscale and nanoscale. Nat. Rev. Phys. 2021, 3, 555–569. [CrossRef]2. Nomura, M.; Shiomi, J.; Shiga, T.; Anufriev, R. Thermal phonon engineering by tailored nanostructures. Jpn. J. Appl. Phys. 2018,57, 080101. [CrossRef]3. Yanagisawa, R.; Tsujii, N.; Mori, T.; Ruther, P.; Paul, O.; Nomura, M. 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MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.http://dx.doi.org/10.1038/nmat2753http://dx.doi.org/10.1103/PhysRevB.83.235428http://dx.doi.org/10.1070/PU1968v011n02ABEH003815 Introduction Materials and Methods Results Discussion References