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Wenhao Huang, Tathagata Paul, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Mickael L. Perrin, Michel Calame

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[Electronic Poiseuille flow in hexagonal boron nitride encapsulated graphene field effect transistors](https://mdr.nims.go.jp/datasets/59387634-9f26-4172-b3cf-e34c1d1d1761)

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Electronic Poiseuille flow in hexagonal boron nitride encapsulated graphene field effect transistorsPHYSICAL REVIEW RESEARCH 5, 023075 (2023)Electronic Poiseuille flow in hexagonal boron nitride encapsulated graphenefield effect transistorsWenhao Huang,1,2,* Tathagata Paul,1,*,† Kenji Watanabe,3 Takashi Taniguchi,4 Mickael L. Perrin,1,5,‡ and Michel Calame1,2,6,§1Empa, Swiss Federal Laboratories for Materials Science and Technology, Transport at Nanoscale Interfaces Laboratory,Überlandstrasse 129, CH-8600 Dübendorf, Switzerland2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan4International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba, Ibaraki 305-0044, Japan5Department of Information Technology and Electrical Engineering, ETH Zürich, 8092 Zürich, Switzerland6Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland(Received 1 November 2022; revised 10 March 2023; accepted 29 March 2023; published 1 May 2023)Electron-electron interactions in graphene are sufficiently strong to induce a correlated and momentum-conserving flow such that charge carriers behave similarly to the Hagen-Poiseuille flow of a classical fluid. In thecurrent work, we investigate the electronic signatures of such a viscous charge flow in high-mobility graphenefield effect transistors (FETs). In two complementary measurement schemes, we monitor differential resistanceof graphene for different channel widths and for different effective electron temperatures. By combining bothapproaches, the presence of viscous effects is verified in a temperature range starting from 178 K and extendingup to room temperature. Our experimental findings are supported by finite element calculations of the graphenechannel, which also provide design guidelines for device geometries that exhibit increased viscous effects. Thepresence of viscous effects near room temperature opens up avenues for functional hydrodynamic devices suchas geometric rectifiers like a Tesla valve and charge amplifiers based on the electronic Venturi effect.DOI: 10.1103/PhysRevResearch.5.023075I. INTRODUCTIONThe dynamics governing the flow of charges in conductorsis heavily dependent on the ratios of the different physicallength scales present in the system. These length scales dictatehow the charge carriers interact with each other, with thesystem boundaries, and with defects or impurities. Micro-scopically, the flow of charge carriers is resisted by scatteringfrom defects and lattice vibrations (phonons), characterizedby the length scale ldiff , which is the average distance traveledby the charge carriers between two such momentum-relaxingscattering events. The charge carriers also interact with eachother in a momentum-conserving fashion, with a length scalelee. The third relevant length scale is the channel width w(assuming channel length l � w), dictating how often chargecarriers interact with the boundary of the system.In most conductors, diffusive scattering prevails (ldiff <lee, w) and charge transport is Ohmic. Charge carriers*These authors contributed equally to this work.†tathagata.paul@empa.ch‡Mickael.Perrin@empa.ch§michel.calame@empa.chPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.traveling across the channel suffer many momentum relaxingcollisions leading to a constant drift velocity along the direc-tion of the applied electric field [1]. Alternatively, transportis ballistic, when the channel dimensions are the smallestlength scale in the system. Under these circumstances, chargecarriers travel collision-less from the source to the drain ter-minal, dissipating energy only at the contacts [2]. However,a third, and relatively unexplored transport regime emerges,when interparticle scattering dominates (lee < ldiff ,w). Thesestrong interparticle interactions lead to a correlated, viscouscharge flow, similar to the flow of fluids and governed by thetheory of hydrodynamics [3–10]. A direct consequence of thisbehavior is the nonuniformity of the carrier velocity in thechannel, which follows a Poiseuille-like flow profile, similarto fluids [Fig. 1(a)]. In the context of electrical transport,this translates to a (nonuniform) position-dependent currentdensity, being largest in the center of the channel and decreas-ing on approaching the channel edges, following a parabolicprofile.Fulfilling the criteria of charge hydrodynamics requireshigh-mobility devices with reduced scattering from defectsand phonons (large ldiff ) and strong interparticle interactions(small lee). Graphene fulfills both these requirements. First,the electron-electron interaction strength is enhanced dueto the 2D confinement effects and weaker mutual screen-ing of charges in graphene when compare to metals (smalllee) [11,12]. Second, graphene has a weak electron-phononcoupling strength [13–15] and advancements in fabrication2643-1564/2023/5(2)/023075(11) 023075-1 Published by the American Physical Societyhttp://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.5.023075&domain=pdf&date_stamp=2023-05-01https://doi.org/10.1103/PhysRevResearch.5.023075https://creativecommons.org/licenses/by/4.0/WENHAO HUANG et al. PHYSICAL REVIEW RESEARCH 5, 023075 (2023)-4 -2 0 2 4 604008001200   40 K 178 K 337 Kn (1011) (cm-2)-4 -2 0 2 4 60.30.60.91.21.5   40 K 178 K 337 K  n (1011) (cm-2)(a)(b)(c)XYVXFIG. 1. Device structure and electrical performance.(a) Schematic depiction of the graphene heterostructure usedin the current work. The Poiseuille flow profile is depicted abovethe top h-BN. The arrows and the curve represent the particlevelocity and the velocity profile expected for Poiseuille flow,respectively. (b) Transfer characteristics of a typical GrFET deviceat different values of T . (c) Diffusive scattering length scale(ldiff ) of GrFET device extracted from the transfer characteristicsin (b).techniques make it possible to limit scattering from extrinsicdisorders, in particular by encapsulation in hexagonal boronnitride (h-BN). h-BN also serves as an ideal substrate forgraphene, due to its atomically flat surface and low latticemismatch (∼1.7%) [16] [Fig. 1(a)]. These lead to grapheneheterostructures having outstanding electronic properties andlarge ldiff .In recent years, hydrodynamic transport of charge has at-tracted significant theoretical attention [5,17–21], and severalexperimental signatures have been reported [6–8,22–25]. Ofparticular interest is the direct observation of the parabolicflow profile. Recent investigations utilizing direct imaging ofHall field [26] and local magnetic field [23,27] have hinted atthe presence of viscous effects in charge transport in grapheneand layered Weyl semimetals. Additionally, unique transportproperties like formation of vortices, negative local resistanceand superballisticity have also been linked to a hydrodynamiccharge flow [6,24,28]. Apart from the Coulomb mediatedelectron-electron interactions [5,17–20], recent reports alsoindicate the possibility of phonon mediated electronic inter-actions generating a correlated charge flow at higher carrierdensities [27,28]. Though electrical signatures of Poiseuilleflow in graphene have been reported [6,24], a comparativestudy of the transport signatures from multiple types of ex-periments and a determination of the temperature range forthe viscous effects is lacking.In the current work, we perform electrical transportmeasurements in h-BN encapsulated graphene field effecttransistors (GrFETs) to investigate the presence of electronicPoiseuille flow. For this purpose, two separate measurementschemes are used: (1) current bias measurements (Sec. III)and (2) width-dependent measurements (Sec. IV). For the cur-rent bias measurements, we look into the scaling of channelresistance as a function of DC bias current in a sample ofconstant width [schematic in Fig. 2(a)]. The width-dependentmeasurements investigate the flow of charge carriers in chan-nels of different widths [schematic in Fig. 3(a)]. We observesignatures of correlated charge flow for temperatures startingfrom ∼178 K and persisting upto 300 K. Further, finite el-ement calculations are performed to verify the experimentalobservations (Sec. V). Our results demonstrate the presenceof strong electronic correlations near room temperature open-ing up possibilities of utilizing the viscous electronic flow ingraphene for functional devices.II. ELECTRICAL PERFORMANCE OF GrFETSThe quality of the fabricated GrFETs was assessed bymeasuring the conductance of the channel as a function of gatevoltage, as called a transfer characteristic at different tempera-tures [Fig. 1(b)]. The devices were fabricated using a standarddry transfer technique [29] and measurements performed ina four-probe geometry to remove the contact resistance (seeMethods). h-BN encapsulation allows fabrication of edge con-tacts, which are known to outperform conventional surfacecontacts [30]. Figure 1(b) demonstrates a typical graphenetransfer characteristic, with a maximum of resistance nearthe Dirac point and a sharp decrease with increasing numberdensity (n). This corresponds to the transition from insulatingbehavior, caused by the formation of charge puddles nearthe neutrality point, to metallic behavior [31,32]. The highquality of our FETs is demonstrated by a large carrier mo-bility (μ � 2 × 105 cm2/(Vs)) (Fig. S1 in Ref. [33]) and alarge łdiff � 1 μm [Fig. 1(c)], thereby relaxing the criteria forhydrodynamic behavior (lee < ldiff ,w) for properly designedchannels. The value of ldiff is computed considering a Drudemodel,ldiff = σm∗v fne2, (1)where σ is the channel conductivity, m∗ the cyclotron mass[34], and v f = 106 m s−1, the Fermi velocity of graphene. Weobserve a reduction in the ldiff with increasing temperature (es-pecially at T = 337 K), which can be attributed to increasedmomentum relaxing scattering events [16]. Temperature de-pendence of ldiff is provided in Fig. S1 [33].023075-2ELECTRONIC POISEUILLE FLOW IN HEXAGONAL BORON … PHYSICAL REVIEW RESEARCH 5, 023075 (2023)(a)(c)(d) (e)(b)FIG. 2. Current bias measurements in GrFETs. (a) Schematic depiction of a typical GrFET device with circuit connections for current biasmeasurements. IAC and IDC represent the AC and DC bias current. Vg is the gate bias. (b) Optical image of the device used for current biasmeasurements. Red and yellow dot lines highlight the bottom and top h-BN flakes, respectively. The scale bar is 5 μm. (c) Color plots ofdV /dI as a function of IDC and n at T = 9 (left) and 150 K (right). (d) Plots depicting the IDC dependence of differential resistance at T =9 K for three different n values indicated by white dashed lines in (c). The number density ranges for the different dV /dI scaling behaviorsare indicated at the top of the color plot (left panel in c) by solid lines of the same color as the corresponding scatter plot in (d). (e) Evolutionof dV /dI curves at n = 1.6 × 1011 cm−2 (white dashed line for T = 9 and 150 K data in (c) for increasing values T from 9 (top) to 300 K(bottom). The Knudsen, Poiseuille, and diffusive transport regimes are indicated by color bars in the top panel.III. CURRENT BIAS MEASUREMENTSWith the high quality of our GrFETs established, we turnto current bias measurements which consist of measuring thedevice conductance for increasing DC bias current. The roleof the DC bias (|IDC| � 300 μA) is to increase the electronictemperature (Te) of the devices, thereby providing a knobto increase the interparticles interactions and reduce lee. Atthe same time, the weak electron-phonon coupling in SLGensures the lattice temperature does not significantly increase,023075-3WENHAO HUANG et al. PHYSICAL REVIEW RESEARCH 5, 023075 (2023)hBNhBNGraphenethBN45ow14540 100 200 3000.050.10.150.2T (K)n = 7 x 1011cm-20 100 200 300100150200250300T (K)n = 7 x 1011cm-20 100 200 3000.40.60.811.21.4T (K)n = 7 x 1011cm-2w1l(a) (b)(c) (d) (e)0 2 4 64681012w ( m)T = 200 Kn = 7 x 1011cm-2IACVV +V -VgV -FIG. 3. Width-dependent measurements in GrFETs. (a) Left: schematic overview of GrFET design and circuit connection used for width-dependent measurements. Regions of different widths (w) are fabricated on the same heterostructure using EBL and RIE (see Methods).Top right: false colored scanning electron microscopy (SEM) image, used for measuring the channel dimensions. Scale bar is 1 μm. TheSEM image is from a region depicted by a red dotted line in the left panel. Bottom right: schematic of the device cross-section depicting thecorrection introduced in the calculation of w due to the etching angle. (b) Scaling of σ as a function of w at T = 200 K. Solid line is a fit to themeasured values (closed circles) following Eq. (5). (c), (d), and (e) represent ldiff , Dν , and lv values, respectively, as a function of T , extractedfrom the fits in (b). Red line (both solid and dotted) and shaded region in e. represent theoretically predicted scaling of lν , following Eq. (7)and the temperature range for Poiseuille efffects in GrFETs from width-dependent measurements, respectively. The red colored bar above (e)represents the temperature range for Poiseuille effects observed from the current bias measurements (Fig. 2 and Sec. III). Error bars in (c), (d),and (e) are calculated from the fitting error in (b).which would lead to an increase in diffusive scattering pro-cesses, and hence a decrease in ldiff [13–15]. The decouplingof the two length scales therefore creates an ideal scenario forhydrodynamic charge flow [9]. We discuss the validity of thisassumption below.In the current work, we use IDC to induce hot electrons inthe GrFET. The dissipated power heats up the charge car-riers and follows a I2DC scaling. Following previous reports[35,36], we can calculate a value of �Te ∼ Te − T ∼ 200 K(at T = 9 K), in the measured device. This is similar tovalues reported for graphene field effect transistors [6,36].However, the lattice remains largely unaffected by IDC. Thiswas experimentally verified by measuring the shift in the 2Dpeak of graphene as a function of input bias voltage [36]. Theauthors of the aforementioned work report a 30 K/V increasein lattice temperature due to bias induced phonon heating. Forthe devices in the current work, this would correspond to alattice temperature change of �T ∼ 10.5 K at IDC = 300 μA(at T = 9 K). The value of �T is negligible when comparedto �Te ∼ 200 K . Additionally, the bias induced phonon heat-ing process shows negligible temperature dependence [36],enabling an effective control of Te for a wide range of tem-peratures using IDC.Figure 2(a) presents the device geometry and circuit con-nections used for the current bias measurements and Fig. 2(b)is an optical micrograph of the measured device. The mea-sured device has a channel length and width of 1.1 and 0.55μm, respectively. The transfer characteristics, mobility andldiff of the device are presented in Fig. S2 [33]. Figure 2(c)shows color plots of dV /dI as a function of n and IDC recordedat 9 and 150 K. At T = 9 K, for changing number densities,three types of IDC curves are observed. The number densitiesin which these behaviors are observed are indicated above thecolor plot. For each behavior, a typical IDC curve is plotted inFig. 2(d), with the number densities highlighted by the verticaldashed lines in Fig. 2(c). For low n (scaling behavior 1),the dV /dI values follow a monotonic decreasing trend withincreasing IDC until about 5 × 10−5 A (Jmin ≈ 0.09 mA/μm).For larger values of IDC, the curve is stable with only a minuteincrease in resistance. This behavior results in a central brightspot in the color map. With increasing n, we observe a non-monotonicity in the measured dV /dI , as apparent from the023075-4ELECTRONIC POISEUILLE FLOW IN HEXAGONAL BORON … PHYSICAL REVIEW RESEARCH 5, 023075 (2023)central bright spot that splits into two ridges and a valleyappearing in the center (scaling behavior 2).For scaling behavior 2, the IDC curves exhibit three dis-tinct regions. First, for low DC currents, the dV /dI valuesincrease with current. dV /dI then reaches a peak and inthe second region, decreases for increasing current (reachinga minimum of dV /dI), while in the third region, dV /dIsteadily increases with DC current. The experimentally ob-served nonmonotonicity in the differential resistance hintstowards the presence of viscous effects [n = 1.6 × 1011 cm−2in Fig. 2(d)], as we will discuss later on. Finally, for the largestn (scaling behavior 3), the negative slope region vanishes andthe dV /dI is constant for small IDC with a monotonic increasefor higher current values.The three different scaling behaviors with changing n canbe rationalized as follows. For scaling behavior 1, electricaltransport is dominated by the formation of charge puddles,leading to an insulating behavior at low DC currents, witha transition to metallic behavior for increasing DC current[31,32,37,38]. The initial drop in resistance is associatedwith electric field assisted interband carrier generation [37].We can calucate the charge inhomogeneity (�n) near theDirac point from the minima in the dV /dI . This gives�n = Jmin/ev f ≈ 5 × 1010 cm−2, which is similar to thosereported in encapsulated graphene [38]. The metallic behaviorat higher IDC has been attributed to a Schwinger productionof electron hole plasma in graphene [38]. For scaling be-havior 2, increasing the DC current leads to three distinctregions in the IDC curve in which the charge transport isquasiballistic/Knudsen, hydrodynamic/Poiseuille and diffu-sive, respectively. For low DC currents, the charge carriersflow nearly collisionless, leading to a quasiballistic transport,also known as the electronic Knudsen regime. The trans-fer characteristics also indicate the presence of quasiballistictransport (Fig. S2 [33]) [39,40]. Upon increasing IDC, theelectron temperature increases and consequently the interpar-ticle collisions are increased. Even though such collisions aremomentum conserving and lead to no resistance, they make itmore probable for the carriers to reach and diffusively interactwith the sample boundaries [9], leading to an increase inthe resistance [d (dV /dI)/dIDC > 0]. A further increase inIDC until lee < w brings about a transition to the Poiseuilleregime. Here, the interparticle collisions actively prevent alarge fraction of the charge carriers (ones near the centerof the channel) from seeing the boundaries, resulting in aslope reversal, d (dV /dI)/dIDC < 0, and a minimum in thedifferential resistance. This negative slope for intermediateIDC and minima of dV /dI in clean conductors are signaturesof the electronic Gurzhi effect [41]. This has been experimen-tally observed previously and is indicative of the presence ofviscous/Poiseuille flow [6,9]. This regime lasts until diffu-sive scattering processes within the sample gain prominence,restoring the positive slope of dV /dI as expected in standardmetallic behavior or the diffusive regime (d(dV /dI)/dIDC >0). A similar behavior is observed in four different samples(see Methods for details). Finally, for higher number densities(scaling behavior 3), the interparticle interactions are screened(larger lee) [42,43] while the diffusive scattering remainslargely unaffected [constant ldiff in Fig. S2 [33] and Fig. 1(c)][6]. A combination of these effects leads to the vanishing ofthe Poiseuille flow (negative slope of dV /dI). The dV /dIvalues are nearly constant for low IDC (quasiballistic regime)and show an increasing trend once diffusive scattering startsdominating the transport around DC currents of 3 × 10−5 A(0.05 mA/μm). Similar observations were also made for holetransport in GrFETs (negative n), and are presented in Fig. S3[33].We also investigate the effect of the lattice temperature (T )on the Gurzhi effect by recording the conductance maps atdifferent temperatures. The color plots at all temperatures arepresented in Fig. S4 [33]. First, we compare the conductancemap at 150 K [Fig. 2(c) (right panel)] to the one obtainedat 9 K [Fig. 2(c) (left panel)]. Two major differences areobserved. First, the central bright spot has a reduced inten-sity and, second, the Knudsen flow is absent for higher Tvalues. Color plots at intermediate temperatures are presentedin the Fig. S4 [33]. The reduction in intensity of the centralbright spot is indicative of a reduced Dirac point resistanceat higher values of T in GrFETs. This is an effect of the en-hanced particle-hole excitation, similar to our observations inFig. 1(b). The disappearance of the Knudsen regime is bettervisualized when the IDC curves for an intermediate numberdensity of n = 1.6 × 1011 cm−2 are investigated for all tem-peratures [Fig. 2(e)]. At low T , all three regimes, Knudsen,Poiseuille and diffusive are visible. However, with increasingT , the Knudsen regime first reduces in size, until it disappearsat T = 150 K. Thereafter, at T = 300 K, the Poiseuille regimealso vanishes and the GrFETs only demonstrate signatures ofdiffusive transport.This T dependence can be understood by considering thevariation of lee and ldiff with lattice temperature. At T = 9 K,lee, ldiff � w, which results in a quasiballistic or Knudsen flowpersisting upto large values of IDC (Fig. S2 [33]). A transi-tion from Knudsen to Poiseuille flow occurs when lee < w.With increasing T , the Poiseuille flow condition is satisfiedat smaller values of IDC, leading to a diminishing Knudsenflow regime. This trend persists until the lattice temperature ishigh enough for the sample to enter the Poiseuille flow regimeat IDC = 0. This is indicated by the absence of the Knudsenregime and is observed at T = 150 K (Fig. S2 [33]). Thesample remains in the Poiseuille regime until diffusive scatter-ing processes gain prominence (ldiff < lee) and the differentialresistance shows a monotonic positive slope [300 K data inFig. 2(e)]. Our differential resistance measurements, thus, hinttowards the presence of a viscous Poiseuille flow in GrFETsat temperatures between 150 and 300 K.IV. WIDTH-DEPENDENT MEASUREMENTSIn addition to differential resistance measurements, we alsoverify the presence of Poisuille flow using geometric effectsin the channel conductivity (σ ). For diffusive transport, σis a material parameter and therefore the current density (J)is uniform in the central part of the channel, vanishing atthe boundaries, as shown in Ref. [23,26,44]. However, in thePoiseuille regime, the parabolic profile of J (or carrier veloc-ity) [Fig. 1(a)] manifests itself as a strong width-dependencein σ . For a 2D Fermi liquid in the Poiseuille regime, thetransport equations are given by [20]∇ · J(r) = 0, (2)023075-5WENHAO HUANG et al. PHYSICAL REVIEW RESEARCH 5, 023075 (2023)σ0e∇φ(r) + D2ν∇2J(r) = J(r), (3)where Eqs. (2) and (3) are the electrical analogues of thecontinuity and Navier-Stokes equations, respectively. σ0 isthe Drude conductivity, φ(r) the electric potential and e theelectronic charge. Dν , the Gurzhi length, is a charcteristiclength scale determining the strength of viscous effects. Itsfunctional dependence is given byDν =√νldiffv f, (4)where ν is the kinematic viscosity. The second term of Eq. (3)is the viscous term and is responsible for the geometry de-pendence of σ . An analytical relation for σ is obtained bysolving Eqs. (2) and (3) with a diffusive/no-slip boundarycondition. Our choice of no-slip boundary condition followsfrom current profile imaging experiments in similar deviceswhich demonstrate a decrease of the flow velocity near thechannel boundaries [23,26,44]. The resulting functional de-pendence isσ = σ0[1 − 2Dνwtanh(w2Dν)]. (5)From Eq. (5), we can define an asymptotic limit of strongPoiseuille flow using the condition Dν � w. In this case, weobtain, σ = σ0w2/12Dν . This w2 scaling of σ has been re-ported previously in the Fermi liquid of Weyl semimetals [8].However, comparing the values of w used in the current work(0.6 < w < 5 μm) and previously reported values of Dν ≈0.2–0.4 μm in graphene [6], we expect that the measuredGrFETs do not satisfy the conditions for strong Poiseuilleflow. Hence, unlike previous reports [8], we find it suitable tofit the experimental σ with the nonasymptotic relation givenby Eq. (5).Figure 3(a) shows a schematic of the GrFETs used forour width-dependent measurements. To fabricate these de-vices, the h-BN encapsulated graphene heterostructure wasshaped into channels of different widths using reactive ionetching (RIE) (see Methods). Channel widths (w1) and lengths(l) were confirmed by scanning electron micrographs [topright panel of Fig. 3(a)]. The actual channel width (w) wasestimated from the observed width (w1), plus taking intoconsideration the height of the top h-BN and the 45◦ anglegenerated by the RIE process [bottom right panel of Fig. 3(a)][30]. The width-dependent fitting allows us to extract severalflow parameters, providing further insight into the viscouscharge transport in GrFETs. A detailed discussion of the pro-cess is presented below.The transfer characteristic and ldiff of the width-dependentdevice is presented in Figs. 1(b) and 1(c), respectively. Fig-ure 3(b) shows the fitting of σ for lattice temperature T =200 K and n = 7 × 1011 cm−2. This particular choice of T andn is guided by previous reports of viscous effects in grapheneat similar values of these parameters [6]. The closed circlesare measured values of σ , while the solid line is a fit followingEq. (5), with σ0 and Dν as fitting parameters. From the fit, onecan directly extract the values of σ0 and Dν . The extractedvalue of σ0 enables us to compute the diffusive mean free pathldiff using Eq. (1). We find ldiff ∼ 1 μm, which is a commonoccurrence in high quality graphene devices [Fig. 3(c)] [6,30].The value of Dν ∼ 0.2 μm [Fig. 3(d)], obtained from the fit isalso in close agreement with previous reports [6]. Substitutingldiff and Dν in Eq. (4) gives us an estimate of the viscosity(ν) of the charge carriers. We obtain a high kinematic vis-cosity, ν ≈ 0.04 m2s−1 (T = 200 K), making the flow moreviscous than honey, a phenomenon which has been reportedpreviously (Fig. S5 [33]) [6]. This high viscosity is also oneof the assumptions for using the linearized Navier-Stokesequation in Eq. (3). Additionally, the flow viscosity, whicharises due to the interparticle scattering, can be directly relatedto the viscous mean free path, lν , a parameter which is closelyrelated to lee [6,19]. The exact relationship at low excitationfrequencies ( f << 1012 Hz) is given bylν = 4m∗νh̄k f. (6)The extracted lν ≈ 0.14 μm at T = 200 K, is the small-est length scale in the system (lν << w, ldiff ), indicative ofthe presence of Poiseuille flow in GrFETs at T = 200 K[Fig. 3(e)]. To explore further, we repeat the width-dependentmeasurements at different lattice temperatures and explore thetemperature dependence of ldiff , Dν , lν , and ν in Figs. 3(c)–3(e), and Fig. S5 [33], respectively. The flow parametersdemonstrate a T dependence only in an intermediate temper-ature range of 178 < T < 300 K and saturate (except for ldiff )for T < 178 K and T > 300 K.To understand the physical relevance of these values, wecompare the extracted lν with its predicted T dependence forthe Dirac fermionic liquid in a doped graphene sheet [19]. Theanalytical relation for lν is given bylν = 45v f (1 + Nf αee)2h̄ε f64π√2Nf α2ee(kBT )2, (7)where Nf = 4 is the number of Fermionic flavours ingraphene, αee = 7.3 the fine structure constant, and kB theBoltzmann constant. αee is computed from the relation [19,45]αee = e2ε h̄v f, (8)where ε is the electrical permittivity. For the current work,we have used the permittivity of h-BN considering a dielec-tric constant of 4 [46]. Eq. (7) [solid line in Fig. 3(e)] hasno fitting parameters and matches the extracted values of lν[closed circles in Fig. 3(e)] for the intermediate temperaturerange, 178 < T < 300 K [shaded region in Fig. 3(e)]. Theextracted flow parameters are thus physically relevant onlyfor these intermediate temperatures. Additionally, we note thatthe assumptions of viscous flow made in the formulation of thetransport equations [Eqs. (2) and (3)] limit their validity, andas a result, the validity of the geometry-dependent analysis forvalues of T where Poiseuille flow is present. In combination,the physical relevance of the extracted flow parameters andthe validity of the viscous flow equations confirm the presenceof Poiseuille flow for 178 < T < 300 K in GrFETs. Finally,this observation is strengthened by the fact that similar Tvalues are obtained for both the current bias measurements(Sec. III) and width-dependent measurements [Figs. 2(e) and023075-6ELECTRONIC POISEUILLE FLOW IN HEXAGONAL BORON … PHYSICAL REVIEW RESEARCH 5, 023075 (2023)40 K178 K(b)(c)250 K337 K(d)(e)(a)(f)FIG. 4. Finite element modeling of graphene channel. (a) Plots of Jnorm as a function of w and Posit ion (norm.) at T = 178 K depicting aparabolic current flow profile for small values of w and a uniform Jnorm for the largest simulated widths. (b) Flow curvature (solid line) as afunction of w computed from flow profiles in (a) at Posit ion (norm.) = 0. Filled circles indicate the curvatures for measured values of w in thecurrent work. The dashed line represents curvature for an ideal parabolic flow. Shaded region indicates w values for strong Poiseuille flow (frominset of (d). [(c), (d), (e), and (f)] Experimental (filled circles) and simulated (solid line) values of σ as a function of w for T = 40, 178, 250,and 337 K, respectively. Dashed line in (d) and (e) indicates the w2 scaling of σ expected for strong Poiseuille effects. Inset of (d) comparesthe w2 scaling and simulated σ for w � 0.5 μm. Shaded region represents the predicted w values for strong Poiseuille flow. The finite elementmodeling is performed at n = 7 × 1011 cm−2.3(e)]. We also look at the number density dependence ofthe flow parameters in Fig. S6 [33]. The geometry-dependentmeasurements indicate the presence of hydrodynamic effectsfor number densities >1011 cm−2. Further discussions areprovided in Fig. S6 [33].V. FINITE ELEMENT METHODSFinally, we turn to finite element calculations (imple-mented in COMSOL MULTIPHYSICS 5.6) to model the Poiseuilleflow in GrFETs. This process involves numerically solvingEqs. (2) and (3) with an uniform electric field (∇φ) andno-slip boundary conditions for different values of w. Theinput parameters, Dν and σ0, are calculated from the theoret-ical predictions of lν [Eq. (7)] and the field effect mobility(μ). Our channels are oriented along the x axis while theirwidth follows the y axis. An example of a simulated flowprofile is depicted in Fig. 4(a) for T = 178 K. We observestrong Poiseuille effects for small channel widths, indicatedby the parabolic dependence of the normalized current den-sity [Jnorm = Jx(y)/Jx(y = 0)] on the normalized channelposition [Posit ion (norm.) = y/(w/2)], and a weak Poiseuilleflow, indicated by a constant Jnorm, for larger values of w. Thecurvature of the flow profile computed at the center of thechannel [Posit ion (norm.) = 0] also exhibits a width depen-dence [solid line in Fig. 4(b)]. It saturates to a value of 2 forthe smallest widths with a parabolic flow profile [w = 0.1 μmin Fig. 4(a)] and reduces to 0 for larger widths with a constantJnorm [w = 5 μm in Fig. 4(a)]. The filled circles in Fig. 4(b)correspond to the measured values of w in the current work.Further, we extract the channel conductivity (σ ) from thesimulated flow profile. In Figs. 4(c) to 4(f), the simulated(solid red line) and experimental (filled circles) values of σ arecompared for four different values of T . The simulated resultsmatch experimental data for temperatures in the Poiseuilleflow regime (T = 178 and 250 K), with mismatches at lower(T = 40 K) and higher (T = 337 K) temperatures, where ourexperiments indicate the absence of hydrodynamic effects.In what follows, we quantify our observations and predictthe values of w necessary for a strong Poiseuille flow. Asdiscussed before, the strong Poiseuille regime is characterizedby a w2 scaling of σ . This is indicated by the dashed bluelines in Figs. 4(d) and 4(e), computed with the same param-eter values as in the simulation (solid red line). Indeed, theexperimentally measured widths in our GrFETs devices aretoo large for observing this effect as the flow is in the weakPoiseuille regime. The strong Poiseuille regime is determinedby computing the relative difference between the simulated023075-7WENHAO HUANG et al. PHYSICAL REVIEW RESEARCH 5, 023075 (2023)(solid red line) and strong Poiseuille (dashed blue line) valuesof σ . Values of w for which the relative difference remainsbelow 10% are considered to be in the strong Poiseuilleregime. The criteria is satisfied for w < 0.15 μm, indicated bythe shaded region in the inset of Fig. 4(d). Another indicatorof the strong Poiseuille regime is the saturation of the flowcurvature to a value of 2 [shaded region in Fig. 4(b)]. Inconclusion, our finite element model confirms our GrFETs arein the weak Poiseuille regime and predicts strong Poiseuilleflow for w < 0.15 μm in GrFETs. Comparing the measuredvalues of lν [Fig. 3(e)] with the computed length scale forstrong hydrodynamics, we would expect this behavior to bepresent at a slightly higher temperature than the range in-dicated in the current work in order to satisfy the criterialee < w.VI. DISCUSSIONS AND CONCLUSIONSFabrication related constraints, specifically, the availabilityof sizes of exfoliated graphene and h-BN flakes, governed thedesign of the width-dependent device measured in the currentwork. The design, presented in Fig. S10 [33], was adoptedto increase the number of widths in a single graphene flake.This led to a variation in the aspect ratio of the channels inthe regions of different widths. Notably, for the two largestmeasured widths, the separation between the voltage probes(∼1.5 μm) is smaller than w ∼ 2.5 and 5 μm, making thechannel length, the smallest geometric length scale. To inves-tigate the consequences of such a design choice, we simulatethe flow profile in the measured device and compare it withthat of a model geometry (Fig. S10 [33]). The geometry of themeasured device is obtained from the gds file used to shape thedevice during fabrication. The model geometry is obtained bymodifying the measured device and increasing the length ofthe segments with the largest widths (i.e., w= 2.5 and 5 μm).Comparing the flow profiles in the two geometries we findno difference for the smallest measured widths (w = 0.6, 0.8and 1 μm), and a slight variation for the two largest widths(Fig. S10(c) [33]). The variation in the flow profile leads to avariation in the channel conductivity �σ ∼ 7 − 15% for w =2.5 and 5 μm when compared to the model geometry. Whilethese values are not negligible, the extracted flow parametersfrom the model geometry fall within the variation introducedby the width-dependent fitting (demonstrated by error barsin Figs. 3(c)–3(e) and hence do not modify the conclusionsdrawn in the current work.Additionally, we also investigate the injection effects thatarise due to the connection of channels of different widths.This is done by extracting the flow profile at various regionsin the channel (details in Fig. S11 [33]). We find negligibleinjection effects in the smallest measured widths (w = 0.6,0.8, and 1 μm) (Fig. S11(b) [33]). We observe some injectioneffects in the largest measured widths, but these flow insta-bilities die down within 300 nm of the voltage probes andthe flow in the majority of the measured channel is stable(although with a slight variation from the model geometryas mentioned previously) (Fig. S11(c) [33]). This makes usbelieve that the charge injection effects do not play a majorrole in the measured device. All the simulations are performedat T = 178 K and n = 7 × 11 cm−2, following similar consid-erations as present in the discussion of Sec. V in the main text.Strikingly, we observe hydrodynamic effects at differentcharge number densities in the geometry dependence study(n > 1 × 1011 cm−2) (Fig. S6 [33]) and the current bias mea-surements (1.6 × 1011 � n � 2.4 × 1011 cm−2), for which wehave no clear, unique explanation. Similar behavior has beenseen in previous reports, with current bias measurementsshowing viscous effects for lower number density ranges com-pared to negative vicinity resistance [6]. This could indicateeither a number density selectivity in the transport probes ofcharge hydrodynamics or a number density variability of thehydrodynamic regime in graphene.In conclusion, we demonstrate the electrical signaturesof Poiseuille flow in charge transport through graphenechannels using the Gurzhi effect and geometry-dependentmeasurements. We observe a negative slope in the differentialresistance measurements and a strong geometry dependenceof the channel conductivity, both indicative of the presenceof viscous effects. The two different measurements types, inconjunction, enable us to define a temperature range for theviscous effects to occur, which reaches as high as 300 K.Moreover, we provide an electrical transport-based character-ization framework for detecting Poiseuille flow in conductorsthat can easily be extended to other conductors. Our ex-perimental observations are corroborated by finite-elementcalculations that shed light on the channel geometries requiredfor a strong Poiseuille flow. Our findings offer promis-ing prospects for functional hydrodynamic graphene devicesin the near future, such as geometric rectifiers like Teslavalves and charge amplifiers based on the electronic Venturieffects [47–49].VII. METHODSA. Device fabricationGrFETs measured in the current work are fabricated usingthe following process: the heterostructure assembly beginswith the mechanical exfoliation of individual layers frombulk crystals (the graphite is obtained commercially fromNGS Trading & Consulting GmbH and the h-BN is from ourcoauthors in National Institute for Materials Science, Japan).Exfoliated flakes are optically screened using an optical mi-croscope (Zeiss Axio Imager M2m). The layer number andquality of grphene flakes are determined using Raman spec-troscopy (see Fig. S7 [33]) [50]. We use a Raman systemwith a backscattering geometry confocal Raman microscope(WITec, Alpha 300 R). An excitation laser with a wave-length of 532 nm from a diode laser is used for all Ramanmeasurements. The thickness of h-BN layers is characterizedusing atomic force microscopy in the tapping mode (BrukerIcon3 AFM). Selected flakes are aligned and stacked layerby layer using commercially available micromanipulator (HQGraphene 2D Heterostructure Transfer System). The pick-upprocess is performed using PDMS backed polycarbonate (PC)stamps. PC shows good adherence to h-BN for 330 K � T �350 K. After successful pick-up of the top h-BN layer, themiddle graphene layer and bottom h-BN layer are pickedup using the van der Waals interaction between the differ-ent two-dimensional materials. Finally, the heterostructure is023075-8ELECTRONIC POISEUILLE FLOW IN HEXAGONAL BORON … PHYSICAL REVIEW RESEARCH 5, 023075 (2023)deposited on a pre-patterned Si++/SiO2 (285 nm) wafer atT = 450 K. The PC is dissolved in dchlorometane (DCM).The transfer process followed is similar to that in Ref. [29].Following this, a thermal annealing step is performed at 650 Kwith H2/Ar 35/200 sccm for 3 hours to improve the quality ofthe heterostructures by reducing interfacial bubbles, contami-nants etc. (See Figs. S8 and S9 [33].) The small electrodes aredefined using EBL and RIE (CHF3 = 40 sccm, O2 = 4 sccm,P = 60 mTorr, Power = 60 W). Next, we deposit 5/25 nmCr/Au directly using ebeam evaporator for edge-contactedelectrodes and the pattern is lifted off using acetone for45 min [30]. After that, the second EBL and metals deposition(5/65 nm Cr/Au) are performed for contact pads. Finally, weperform the third ebeam processing, followed by the secondRIE for defining the channel shape. The morphology of thedevice is recorded by the same optical microscope mentionedabove. Channel widths of devices are acquired from SEM(Hitachi S-4800). For the width-dependent device the chan-nel width (w) was varied while keeping the channel lengthconstant.For the current bias measurements, four devices on fourdifferent chips were fabricated. All samples demonstrate asimilar scaling of dV /dI at base temperature. We observe thecurrent biasing measurements have a tendency to generate ir-reversible hysteresis and electron doping (shifting of the Diracpoint to negative gate voltages) in devices. Due to this wecould perform the temperature characterization in one deviceand partial temperature characterization in a second device.For the width-dependent measurements, the largest fourvalues of w are from a single large graphene channel whichwas shaped into channels of different widths, while the small-est w value is obtained from a second device on a differentchip made following similar fabrication protocols.B. Measurement DetailsThe four-probe resistance of GrFETs was measured usinga lock-in technique, with an AC source-drain current(of IAC =100 nA (231.45 Hz); IDC = 0) while measuring the resultingvoltage drop (Lockins used are SRS 830 or EG&G 7265). TheSRS CS580 voltage controlled current source is used to supplyIAC to the sample. The gate voltage is supplied by an AdWinGold 2 connected to a voltage amplifier (Physics Basel SP908). From the applied gate voltage, the number density iscomputed using n = Cox(Vg − Vd )/e [51], where, Cox is theoxide capacitance (series combination of SiO2 (285 nm) andbottom h-BN), e is the electron charge, Vg and Vd are the gateand the Dirac point voltage, respectively.For the current bias measurements, the DC bias is suppliedby an AdWin Gold 2 and the AC bias by a lockin. The twobias voltages are added using a summing amplifier SIM980.The current source CS580 converts the summed voltage to thesupplied current bias. An IAC = 100 nA (231.5 Hz) is usedfor the measurements and the output voltage from the samplemeasured using the lockin.The width-dependent measurements follow similar mea-surement technique as the transfer characteristic measure-ments.The current bias measurements are carried out on a closedcycle manual probe station (Lakeshore). For width depen-dence measurements, the samples are wire bonded to a chipcarrier and measured in a commercially available closed cyclecryostat from Advanced Research Systems (ARS).The data that support the findings of the study are availablefrom the corresponding author upon reasonable request.ACKNOWLEDGMENTSW.H., T.P., and M.C. acknowledge funding from the SwissNational Science Foundation under the Sinergia Grant no.189924 (Hydronics). K.W. and T.T. acknowledge supportfrom the JSPS KAKENHI (Grant Numbers 19H05790 and20H00354). M.L.P. acknowledges funding from the SwissNational Science Foundation under Spark grant No. 196795.and the Eccellenza Professorial Fellowship No. PCEFP2-203663, as well as supported by the Swiss State Secretariatfor Education, Research and Innovation (SERI) under contractnumber MB22.00076. The authors acknowledge support fromthe Multiphysics Hub @ Empa for the COMSOL Multiphyicscalculations. We thank the Cleanroom Operations Team of theBinnig and Rohrer Nanotechnology Center (BRNC) for theirhelp and support. We also thank our project partners BerndGotsmann, Ilaria Zardo, Nicola Marzari, Ivan Shorubalko fordiscussions. W.H. would like to thank Ilaria Zardo for hermentorship.W.H., T.P., M.L.P., and M.C. conceived and designedthe experiments. W.H. and T.P. prepared the devices andperformed the electrical measurements. W.H. and T.P. per-formed the annealing, Raman, SEM and AFM measurements.K.W. and T. T. synthesized the hBN flakes. W.H., T.P.,M.L.P., and M.C. did the data analysis. M.L.P. performedthe finite element calculations, with the help of T.P., W.H.,T.P., M.L.P., and M.C. discussed the figures and wrote themanuscript. M.L.P. and M.C. supervised the study. All authorsdiscussed the results and implications and commented on themanuscript.The authors declare that there are no competing interests.[1] N. W. Ashcroft and N. D. 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