Alice Marcotte

Mathieu Lizee

Baptiste Coquinot

Nikita Kavokine

Karen Sobnath

Clément Barraud

Ankit Bhardwaj

Boya Radha

Antoine Niguès

Lydéric Bocquet

Alessandro Siria

Strong electronic winds blowing under liquid flows on carbon surfaces

The interface between a liquid and a solid is the location of plethora of intrincate mechanisms at the nanoscale, at the root of their specific emerging properties in natural processes or technological applications [1,2]. However, while the structural properties and chemistry of interfaces have been intensively explored, the effect of the solid-state electronic transport at the fluid interface has been broadly overlooked up to now. It has been reported that water flowing against carbon-based nanomaterials, such as carbon nanotubes [3][4][5][6][7][8] or graphene sheets [9][START_REF] Yin | Harvesting energy from water flow over graphene?[END_REF][START_REF] Lee | Flow-induced voltage generation in nonionic liquids over monolayer graphene[END_REF][START_REF] Kuriya | Output density quantification of electriciy generation by flowing deionized water on graphene[END_REF][START_REF] Yin | Generating electricity by moving a droplet of ionic liquid along graphene[END_REF][START_REF] Park | Identification of dropletflow-induced electric energy on electrolyteinsulator-semiconductor structure[END_REF], does induce electronic currents, but the mechanism at stake remains controversial [9,[START_REF] Yin | Harvesting energy from water flow over graphene?[END_REF][START_REF] Pei | Liquid flow-induced electricity in carbon nanomaterials[END_REF]. Here, we unveil the molecular mechanisms underlying the hydro-electronic couplings by investigating the electronic conversion under flow at the nanoscale. We use a tuning fork-Atomic Force Microscope (AFM) to deposit and displace a micrometric droplet of both ionic and nonionic liquids on a multilayer graphene sample, while recording the electrical current across the carbon flake. We report measurements of an oscillation-induced current which is several orders of magnitude larger than previously reported for water on carbon [START_REF] Yin | Generating electricity by moving a droplet of ionic liquid along graphene[END_REF], and further boosted by the presence of surface wrinkles on the carbon layer. Our results point to a peculiar momentum transfer mechanism between fluid molecules and charge carriers in the carbon walls mediated by phonon excitations in the solid [3,[START_REF] Lee | Flow-induced voltage generation in nonionic liquids over monolayer graphene[END_REF]. Our findings pave the way for active control of fluid transfer at the nanoscale by harnessing the complex interplay between collective excitations in the solid and the molecules in the fluid.

Transport of fluid across or at the interface of carbon unveiled unexpected properties in term of very fast permeation, non-linear ionic transport and sieving and even energy storage and harnessing [2]. These results highlight the subtle interactions of water and charges with the carbon surface. Classically, the interaction of a fluid with a solid wall is based on picturing the solid as a static periodic potential that acts on the fluid molecules: interactions result from collisions on the surface roughness, as flows induced on the roughness scale dissipate mechanical energy. However, recent results for fluid and ion transport in carbon nanotubes and nanochannels made of graphitic materials have pointed out the limits of such standard description for fluid-solid interactions and existing couplings of liquid transport with electronic degrees of freedom inside the confining walls. This takes the form of electronic currents induced in the solid surface under flowing † These authors contributed equally to this work liquid [4,5,[7][8][9][START_REF] Yin | Harvesting energy from water flow over graphene?[END_REF][START_REF] Lee | Flow-induced voltage generation in nonionic liquids over monolayer graphene[END_REF][START_REF] Kuriya | Output density quantification of electriciy generation by flowing deionized water on graphene[END_REF][START_REF] Yin | Generating electricity by moving a droplet of ionic liquid along graphene[END_REF][START_REF] Park | Identification of dropletflow-induced electric energy on electrolyteinsulator-semiconductor structure[END_REF], the modification of wetting by electronic screening [START_REF] Comtet | Nanoscale capillary freezing of ionic liquids confined between metallic interfaces and the role of electronic screening[END_REF][START_REF] Schlaich | Electronic screening using a virtual Thomas?Fermi fluid for predicting wetting and phase transitions of ionic liquids at metal surfaces[END_REF], or anomalies in the water-carbon friction [START_REF] Kavokine | Fluctuation-induced quantum friction in nanoscale water flows[END_REF]. The interaction mechanisms at stake remain largely unexplained and to be properly understood.

In this article, we report the generation of electric current in a few-layer graphene sample by the displacement of a sub-micrometer scale liquid droplet along its surface. We deposited a thin graphite flake (1 to 70 nm in thickness) onto a Si/SiO 2 substrate; two gold electrodes, separated by a distance of ∼ 10µm and connected to a low noise current amplifier, allowed us to measure the electric current passing through the flake (see Figure S1). The sample was placed in a tuning fork atomic force microscope (AFM), that was specifically designed for controlled liquid deposition, see Figure 1a. The liquid was introduced into a quartz capillary, which was then glued to the quartz tuning fork (see Figure S2). A piezo-dither induced a nanometric oscillation of the tuning fork at its resonant frequency (f 0 ≈ 32 kHz), a typical resonance curve is shown in the inset of Figure 1a. By monitoring precisely the tuning The HOPG crystals are purchased from HQ Graphene. Each bulk crystal is mechanically exfoliated and flakes are first randomly deposited on separate 280 nm-thick Si/SiO2 substrates. The thickness of the SiO2 layer is chosen to maximize the optical contrast [2].

Graphene flakes are then carefully selected for their thickness (ranging from 1 nm to 70 nm), homogeneity and size (> 15 µm). The graphene flakes are transferred thanks to the "hot pick-up" technique [3,4]

B. Un-transferred samples

As discussed in the text, wrinkles have a strong influence on viscous flow driven current generation at the carbon surface. To assess the role of wrinkles in current generation, we use a transfer-free fabrication technique to prepare flat graphite samples. We evaporate gold electrodes on the SiO2 substrate where the graphite was exfoliated in the first place and thus obtain wrinkle-free samples which can be compared to high wrinkle-density samples of the same thickness obtained when transferring the flakes before evaporating the contacts (see Figure S1 c). A quartz capillary of tip diameter of 500 nm is filled with liquid and glued on the lower prong of a millimetric quartz tuning fork. Once the tip is in contact with the graphite substrate, a liquid droplet is formed at its extremity. The scanner allows for droplet relative motion on the graphite flake in the XY-plane, while current is measured between both gold electrodes. An additional electric potential drop ∆V can be imposed between the electrodes. Inset: The resonance curve of the tuning fork whose frequency shift is used to deposit the droplet and maintain a gentle mechanical contact. b, Frequency shift response of the AFM during a typical experiment, indicating the stiffness of the pipette-sample contact. First, the pipette is brought into contact with the sample (1). After the contact, a slow relaxation (2) indicates the formation of the droplet. Finally, the sample is put into horizontal motion with respect to the droplet while a PID controller keeps a constant frequency shift (3). c, AFM image of a multilayer graphene sample (3.6 nm thick) exhibiting a high density of wrinkles. d, AFM image of a flat multilayer graphene sample (1.5 nm-thick). e, Schematic of droplet motion. f, Current (nA) as a function of time (s) resulting from droplet oscillations of 2 µm amplitude and 0.5 Hz frequency, on the wrinkled sample displayed on c (black dots) and on the flat graphite sample displayed on d (red dots). g, The corresponding current spectra as a function of frequency. For the wrinkled sample (black curve), the spectrum displays a peak at the droplet oscillation frequency (0.5 Hz). fork oscillation's phase and amplitude as the pipette tip approach the sample, we achieved a fully controlled contact between the carbon surface and the pipette's aperture, leading to the formation of a liquid capillary bridge. The capillary bridge then evolved into a droplet, of radius fixed by the outer diameter of the tip (500 nm) [START_REF] Fang | Control of droplet size in liquid nanodispensing[END_REF][START_REF] Fabié | Writing with liquid using a nanodispenser: Spreading dynam-ics at the sub-micron scale[END_REF]. This novel experimental set-up allows to deposit very viscous liquid at nano and micro scale over a broad range of substrates [21]. We used a room temperature ionic liquid (BMIM-PF6) and a neutral and apolar silicon oil (Polyphenyl-methylsiloxane), both very viscous with a viscosity of 0.3 Pa•s and 0.1 Pa•s, respectively, and with very small vapor pressures, allowing us to safely neglect evaporation. The continuous monitoring of the contact with both the in-situ optical microscope and the AFM frequency shift signal allows us to ensure that the droplet remained hooked to the tip, sliding over the carbon surface. Moreover, motion does preserve the roughly spherical shape of the droplet (see Figure S2d). In a typical experiment, the droplet is put into oscillatory motion along the carbon surface by a piezo-scanner with an amplitude of a few microns at a frequency of the order of 1 Hz. An electric current through the carbon surface is then measured at the droplet oscillation frequency (Figure 1e and Figure S3). As shown on Figure 1a, our set-up enables to further impose a constant potential drop ∆V (typically between -50 mV and 50 mV) between the electrodes. We report electric currents in the tens of nano-Ampere range induced by a droplet moving at a few micrometers per second. To allow for a comparison with other current generation experiments, we define the current density as the AC-current amplitude divided by the diameter of the droplet (of order 1µm) : j = I/(2R drop ) typically in the tens of nA/µm range.

Firstly, several benchmark measurements allowed us to ensure that the current generation is due to the specific interaction of the liquid droplet with the few-layer graphene. We checked that no alternating current is recorded when the deposited droplet oscillates over the SiO 2 substrate between the electrodes but without the multilayer graphene substrate. We further controlled that no current is generated when the pipette is not in contact with the graphite but slightly above or even when a tungsten tip is used instead of the liquid filled pipette, see Figure S4. Altogether, we can safely conclude that the measured effect is due to the displacement of the liquid droplet on the carbon surface.

On Figure 2a, we report results of several samples (with thickness varying between 1.7 and 8.2 nm), where an electro-fluidic current is measured. We observe that for a fixed oscillation frequency f and bias voltage ∆V , the amplitude of the AC generated current increases linearly with the droplet oscillation amplitude a, or equivalently, with the droplet peak velocity (v = 2πf a). Taking advantage of the demonstrated linear dependency of the current density with respect to the droplet peak velocity, we define the electro-fluidic conductivity of our samples as

σ ef = j v ,
quantifying the cross-coupling between electronic current j and liquid flow v. Among the five samples presented here, we show an electrofluidic conductivity ranging from 2.8 to 21.6 nA.s/µm 2 for a droplet size in the micrometer range.

To understand the mechanism driving this electro-fluidic current and having in mind the strongly energy-dependent electronic density of graphene close to the Fermi level [22], we tune the bias voltage ∆V between the electrodes and measure its influence on the electro-fluidic current. We show that the alternating current induced by the droplet oscillation strongly depends on ∆V , with a linear dependency. This is highlighted in Figures 2b-c, where the electrofluidic conductivity is shown to scale roughly linearly with the potential drop ∆V , with a small or negligeable shift at zero potential drop and a sign reversal of the current for negative versus positive ∆V . Finally, the slope of this linear dependency σ ef /∆V is in the range of 0.05 to 0.2 nA.s/µm 2 /mV, see Figure 2c, and slightly varies across samples. Now, an unexpected feature of the experimental results is the sharp distinction among carbon samples regarding the amplitude of the the electro-fluidic conductivity. Over all few-layer-graphene measured, with thicknesses ranging from 1 nm up to 70 nm, we observe that the thinner ones tends to exhibit the largest electro-fluidic conductivity, see Figure S7a. However, investigating the surface topography of the carbon samples using AFM imaging, such as on Figure 1c-d, we rather reveal a correlation between the amplitude of the electro-fluidic conductivity and the existence of surface corrugation in the forms of folds and wrinkles. Notably, thinner samples exhibit a large number of folds and wrinkles whereas thicker samples present a flat surface without folds or defects (see Figure S1). This difference is due to the fabrication process involving the transfer of the exfoliated graphene flake onto the oxidized silicon wafer (See Sec. S-I) : the thinner flakes are softer and more flexible than the thicker ones, and more likely to exhibit wrinkles [START_REF] Liu | The origin of wrinkles on transferred graphene[END_REF][START_REF] Deng | Wrinkled, rippled and crumpled graphene: An overview of formation mechanism, electronic properties and applications[END_REF], see Figure 1a and Figure S7. In order to disentangle the role of wrinkles from that of sample thickness, we implemented an alternative fabrication method in which all transfer steps are avoided and few-layer graphene flakes were directly exfoliated on the Si/SiO 2 wafer. This allowed us to obtain sub-10 nm, un-transferred and smooth graphene samples with a uniform wrinkle-free surface (see ness showing a much larger surface corrugation (see Figure S9 for a Raman analysis of wrinkles). As shown on Figure 2d, droplet motion on smooth samples results in no detectable current. A contrario, crumpled flakes of the same thickness have a high wrinkle density and show the strongest electro-fluidic conductivity, in the nA.s/µm 2 range. Our results reveal that wrinkles are instrumental to the current generation mechanism, while there is no evidence of an effect of graphene thickness. Further evidence of the crucial role of wrinkles is provided by the dependence of the generated current on the droplet oscillation direction, see Figure S8. As expected, the current amplitude is 180 • periodic with respect to the angle between the droplet trajectory and the electrodes. However, the maximal amplitude is not necessarily reached when the droplet oscillation is perpendicular to the electrodes, and the direction yielding the maximum current strongly depends on the position on the sample's surface. We argue that this spatial and directional variability is related to the random distribution in height, position and direction of the wrinkles on the carbon surface. This further highlights the role of wrinkles as current amplifiers. Let us note for completeness that the data displayed on Figure 2 are obtained along the direction with maximal amplitude.

Summarizing our experimental results, we have shown that the motion of a micrometric droplet on a crumpled, few-layer graphene surface generates a strong electro-fluidic current in the flake, which is proportional to the velocity of the droplet, as well as to the bias voltage applied between the electrodes. A remarkable feature is the strong dependence of the magnitude of the generated current on the surface corrugation of the flake.

While our results echo previous reports of electric potential differences induced by water flow in carbon-based nanomaterials [3][4][5][6][7][8][9][START_REF] Yin | Harvesting energy from water flow over graphene?[END_REF][START_REF] Lee | Flow-induced voltage generation in nonionic liquids over monolayer graphene[END_REF][START_REF] Kuriya | Output density quantification of electriciy generation by flowing deionized water on graphene[END_REF][START_REF] Yin | Generating electricity by moving a droplet of ionic liquid along graphene[END_REF][START_REF] Park | Identification of dropletflow-induced electric energy on electrolyteinsulator-semiconductor structure[END_REF], our nanoscale investigation puts us in position to disentangle various effects at the root of electronic current generation under fluid flow, and get much insights into the molecular mechanism underlying the phenomenon. Previously, sev-eral mechanisms have been invoked to rationalize such conversion, including streaming potential effects [5], adsorbed/desorbed ion hopping [6,9], Coulomb drag [4,8] or charging/discharging of a pseudo-capacitance [START_REF] Yin | Generating electricity by moving a droplet of ionic liquid along graphene[END_REF][START_REF] Park | Identification of dropletflow-induced electric energy on electrolyteinsulator-semiconductor structure[END_REF]. These mechanisms involve Coulomb interactions between the polar liquid -exhibiting molecular partial charges -and the carbon surface as the main pathway for momentum transfer from the liquid to the electrons. As we now show, these previous explanations fail to account for the phenomenon at stake. Indeed, the strong effect of surface corrugation reported here suggests that current generation proceeds through a 'mechanical' transfer of momentum of the fluid towards the electronic degrees of freedom. In order to confirm this hypothesis and rule out any effect caused by the charges in the ionic liquid, we first repeated the experiment, now with an apolar silicone oil replacing the ionic liquid (and a wrinkled few-layer graphene as a substrate). We do observe qualitatively and quantitatively similar current generation for the silicone oil and the ionic liquid, see Figure S5. This discards coulombic couplings between the liquid molecules and the electrons and all corresponding mechanisms based on charge-charge interactions across the interface. These considerations naturally lead us to reconsider phonon-drag mechanisms, as first proposed theoretically in 2001 [3] as the driving mechanism for the measured electro-fluidic current. The phonon drag relies on hydrodynamic (viscous) friction at the solid-liquid interface, which transfers momentum to the solid's phonons, thereby exciting a 'phonon wind'. Through the electron-phonon interaction, the phonon wind 'blows' on the Fermi sea, so that electrons near the Fermi surface are dragged along the direction of the liquid flow, hence producing an electric current (see Figure 3a). A complete theory of phonon drag at the quantum level, within the nonequilibrium Keldysh framework is developed in [25]. Here, we give only a semi quantitative description that allows us to capture the main physical ingredients.

First, assuming a Couette flow within the liquid droplet, the hydrodynamic friction force is F 0 ∼ ηA drop v drop /h drop , v drop is the drop velocity, η its viscosity, A drop its surface and h drop its height (here typically, η = 0.3 Pa • s and h drop ≈ 1 µm). In line with observations, wrinkles are expected to enhance this hydrodynamics friction force. In a simple picture, wrinkles are modelled as protusions on which the flowing liquid exerts a Stokes force scaling (per wrinkle) as F W ∝ ηh W v drop , where h W is the typical height of the wrinkles (h W ∼ 10 -30 nm).

One then obtains a total friction force F W ∼ F 0 (1+W ), where we introduced the dimensionless wrinkling number, W = 3πn W h W h drop , with n W the wrinkle density to account for the increase in viscous friction due to surface corrugation.

Our model is sketched on Figure 3a-d : the hydrodynamic friction force transfers momentum from the liquid (Figure 3a) to the solid exciting acoustic phonons with dispersion relation q = qc where c is the phonon (sound) velocity (c ≈ 2 × 10 4 m • s -1 [START_REF] Nika | Phonon thermal conduction in graphene: Role of Umklapp and edge roughness scattering[END_REF][START_REF] Ochoa | Temperaturedependent resistivity in bilayer graphene due to flexural phonons[END_REF][START_REF] Cong | Probing the Acoustic Phonon Dispersion and Sound Velocity of Graphene by Raman Spectroscopy[END_REF]). The phonons are mainly relaxed by transferring their momentum to the crystal (Figure 3d) through Umklapp processes [START_REF] Nika | Phonon thermal conduction in graphene: Role of Umklapp and edge roughness scattering[END_REF], on a typical timescale τ um ∼ 10 ps [START_REF] Klemens | Thermal conductivity of graphite in the basal plane[END_REF]. As sketched on Figure 3b, the constant influx of momentum modifies the phononic distribution by shifting the energetic cost of a phonon of wavevector q according to q → qq • v eff , which is lower along the direction of the liquid flow. Here v eff is the effective velocity of the phonon wind: it is determined by balancing the momentum fluxes in and out of the phonon system, according to τ um F W /A drop = ∆P ph /A drop = dq 2π q n B ( qcq • v eff ), with n B the Bose-Einstein distribution and q is the phonon wavevector, that we assume two-dimensional. This yields

v eff 2π 3ζ(3) 2 c 4 (k B T ) 3 ×(1+W )η v drop h drop ×τ um (1)
Then, v eff ≈ 9 • 10 4 × (1 + W ) × v drop . As the electrons scatter on the phonons, they acquire an average velocity equal to the phonon wind velocity; our complete theory shows that this is true whatever the form of the electron-phonon interaction [25]. As sketched on Figure 3c, in a 2d model of the solid, this amounts to shifting the band energies according to q → qq • v eff . Assuming for concreteness that the liquid flows to the right, this roughly means that in an energy window of width k F v eff around the Fermi level (k F is the Fermi wavevector), the electrons move to the right at the Fermi velocity v F . The corresponding current density reads

j = ev F × N ( F ) × [ k F v eff ], (2)
where N ( F ) is the density of states at the Fermi level F . We obtain a current density proportional to the droplet velocity, to the liquid viscosity and to the wrinkling number W in excellent agreement with the experiment. Moreover, the current is independent of the chemical nature of the liquid as observed in experiment (ionic liquid and silicone oil generate similar Finally this asymmetry propagates to the electronic distribution c by electron-phonon interaction. This asymmetry in the electronic distribution translates into the generation of a net electric current, whose sign depends on the nature of charge carriers at the Fermi level (electrons or holes). We stress that this is an out of equilibrium phenomenon in which most of the momentum is relaxed to the crystal d by the Umklapp process with a rate 1/τ um . Finally, on e, we compare the theoretical derivation of the electro-fluidic conductivity σ ef versus the bias voltage ∆V . We stress that the experimental data is the same as in Figure 2c and that the solid lines are plotted using Eq. ( 4) assuming the bulk graphite effective mass m = 0.1m e and W the wrinkling number as the only fitting parameter chosen between 5 and 20. This wrinkling number characterizes the enhancement of Stokes friction due to wrinkles on the graphene surface. current). According to the complete derivation presented in [25] (see Sec. S-III for the main ingredients), we add in the following a correction factor of order 1/2 to the current density. Electronic bands in graphene have a linear dispersion, q = v F q ; however, in our multilayer samples of few-nanometer thickness (see Figure S7a) one expects the electronic properties to be better described by a parabolic band with effective mass m [30-32]: q = 2 q 2 /2m. The density of states is then independent of energy: N ( ) = N ( F ) = m/ 2 . Altogether, we obtain the electro-fluidic cross conductivity

σ ef = e m 2 F × v eff v drop . (3)
The prediction in Eq. ( 3) can now be compared with experimental results. Experimentally, the applied bias voltage, sets the Fermi level F ≈ e∆V /2, so that

σ ef ≈ 0.1 × m/m e × (1 + W ) × ∆V [nA.s/µm 2 ]
(4) with ∆V in mV. This estimate quantitatively reproduces the conductivity measured experimentally of a few nA.s/µm 2 with the correct linear scalings of the electro-fluidic conductivity with the droplet velocity and the bias voltage. In addition, this current is proportional to the wrinkling number, in line with the observed impact of the presence of surface wrinkles as current amplifiers. On Figure 3e, we compare this theoretical prediction with the experimental measurement of σ ef as a function of the bias voltage. For the electronic effective mass, we consider graphite in a parabolic approximation, m ∼ 0.1m e , where m e is the electron mass [START_REF] Gruneis | Tightbinding description of the quasiparticle dispersion of graphite and few-layer graphene[END_REF]. The theoretical estimate involves no fitting parameter other than the wrinkling number W chosen between 5 and 20 to reproduce the experimental data.) Note that although the above discussion focussed on electrons as charge carriers, one expects that the current will change sign if the charge carriers are holes. This explains why the generated current flips sign along with ∆V : for positive bias, the charge carriers are electrons in the conduction band, whereas for negative bias, the charge carriers are holes in the valence band (a seamless transition is ensured by the fact that the bands are touching). Altogether, the proposed modeling strongly supports the hypothesis that the experimentally observed electric current is due to charge carriers being dragged by a phonon wind. Our model describes the momentum transfer from the liquid to the phonon distribution where an imbalance in the distribution develops. Scattering on the electron bath then results in an electronic current.

In conclusion we report the generation of a very large electronic current induced by the controlled motion of a liquid droplet at the surface of carbon nanomaterials. Our results suggest that the current generation is mediated by hydrodynamic momentum transfer from the liquid molecules to the phonons in the solid. Our specifically tailored system is particularly well suited to evidence the phonon-drag induced electronic current. Indeed, the combination of a local micrometric flow on strongly wrinkled graphene samples along with the choice of a very viscous and neutral liquid allows the phonon-drag as a direct momentum transfer to dominate any other current generation mechanism. The versatility of our setup allows to tune the fluid velocity as well as the solid's electronic properties using the bias between the electrodes to fully characterize the phenomenon. Finally, all experimental results were rationalized with our new dedicated model of the phonon-drag. Our findings pave the way for the active control of fluid transport at the nanoscale by harnessing the interplay between fluid molecules and the collective excitation in the solid suggesting that novel nanofluidic functionalities can be proposed where a flow of fluid at the interface with tunable solid can be generated by externally excitable phonons or structural deformation waves. Further, the coupling of electronic and micro-fluidic currents bears the promise of ubiquitous and environmentally friendly energy production as demonstrated recently in titanium carbide MXene based transpirationdriven electrokinetic power generators [START_REF] Bae | Il-Doo Kim Towards Watt-Scale Hydroelectric Energy Harvesting by Ti3C2Tx-Based Transpiration-Driven Electrokinetic Power Generators[END_REF].

The HOPG crystals are purchased from HQ Graphene. Each bulk crystal is mechanically exfoliated and flakes are first randomly deposited on separate 280 nm-thick Si/SiO2 substrates. The thickness of the SiO2 layer is chosen to maximize the optical contrast [2]. Graphene flakes are then carefully selected for their thickness (ranging from 1 nm to 70 nm), homogeneity and size (> 15 µm). h-BN flakes of thicknesses between 7 and 15 nm are chosen. In the case of graphite/SiO2 devices, the graphene flakes are transferred thanks to the † These authors contributed equally to this work "hot pick-up" technique [3,4] in a 100 x 100 µm 2 area in the middle of a 2x2 mm 2 prepatterned substrate with Au electrodes. In the case of h-BN supported graphene devices, h-BN and graphene flakes are first stacked and then transferred on the pre-patterned devices using the same technique. The devices are then fabricated by defining two parallel electrical contacts over the graphene flake by e-beam lithography using a double layered PMMA process: a 50 K lower layer of PMMA and a 950 K upper layer of PMMA both spin-coated at 4000 RPM. The spacing between the two contacts is set to at least 10 µm for the AFM experiment. The metallic contacts are subsequently deposited under high vacuum (a few 10-7 mBar) by e-beam evaporation. 5 nm of Ti covered by 45 nm (or 95 nm) of Au are deposited at very low evaporation rates (0.1 nm/s). The last fabrication step is a lift-off process in which the remaining resist is dissolved in an aceton bath for 20 min. The samples are finally rinsed into isopropanol for 2 min. This fabrication method involves a transfer of the graphite flake and thus yields a high density of wrinkles on thinner flakes as measured by AFM (see Figure S1 a,b). As the viscous-flow induced current did not show any clear difference between SiO2 and hBN supported devices, we disregard this detail in the following.

B. Un-transferred samples

As discussed in the text, wrinkles have a strong influence on viscous flow driven current generation at the carbon surface. To assess the role of wrinkles in current generation, we used a transfer-free fabrication technique to prepare flat graphite samples. We evaporated gold electrodes on the SiO2 substrate where the graphite was exfoliated in the first place and thus ob-

II. DROPLET MOTION-INDUCED CURRENT GENERATION

a. Micro-pipette AFM preparation The capillary tip is obtained by locally heating and simultaneously pulling a 10 cm long quartz capillary of inner diameter 0.5 mm and outer diameter 1 mm (World Precision Instruments), with a Sutter Instruments P-2000 pipette puller. The capillary tube is separated into two pipettes. Each of them has one cylindrical end, and one conical tapper on the other end with a tip outer diameter of 500 nm, that can be varied by tuning the pulling parameters.

The pipette is then filled with ionic liquid (Bmim-PF6 of high purity from Merck) or with neutral silicon oil (Polyphenyl-methylsiloxane, 100 mPa.s, from Sigma Aldrich). This step is greatly facilitated by the inner filament in the pipette, which enhances the capillary filling of the tip. Once the pipette is filled, its very extremity is glued with epoxy glue on the lateral side of a millimetric quartz tuning fork prong. With a diamond tip, the largest part of the capillary is then cut free from its extremity and from the tuning fork. These steps are summa-rized by Figure S2a.

Finally, the tuning fork and its probe are examined under Scanning Electron Microscopy, to measure the outer diameter of the pipette extremity and to make sure the previous fabrication steps did not damage the tip, see Figure S2b.

b. Liquid droplet creation and control All our experiments are conducted under ambient conditions. To deposit a very small and controlled amount of liquid on the graphite, and to put the deposited droplet in motion over the sample while keeping constant the degree of squeezing of the droplet, we use the tuning fork as an autosensitive Atomic Force Microscope (AFM) mechanical oscillator. The tuning fork is excited via a piezo-dither at its resonance frequency f 0 in the normal mode (f 0 ≈ 32 kHz), while the amplitude and phase shift of the tuning fork oscillations with regards to the excitation AC voltage are obtained from the piezoelectric current flowing through the tuning fork electrodes. Our home-made AFM is used in frequency-modulation mode : a Phase Lock Loop (PLL) ensures that the tuning fork is systematically excited at its resonant frequency. Furthermore, a PID servo loop maintains a constant oscillation amplitude at resonance by tuning the excitation voltage. Interactions with the substrate are detected by the change in the resonance frequency and in the excitation amplitude. As shown on Figure S2c, we use a camera to perform the coarse approach of the AFM Figure S2: a Steps for the preparation of the tuning fork and its probe: Firstly, the pipette is filled with liquid. Then, it is glued with epoxy on the side of one tuning fork prong. Finally, the cylindrical part of the capillary is cut free from its extremity and from the tuning fork with a diamond tip. b Scanning Electron Microscopy images of the lower prong of a millimetric tuning-fork, and of the extremity of the pipette glued on it (inset). c In-situ microscope view of the graphite sample connected by two Ti/Au electrodes, while the capillary tip is moving an ionic liquid droplet on the substrate. d Post-experiment Scanning Electron Microscopy image of a graphite sample. Two roughly spherical ionic liquid droplets remain on the flake.

tip and we monitor the gentle approach via the PLL to detect contact with the substrate and the formation of the capillary bridge, and to ensure a constant contact stiffness (frequencyshift) during the droplet motion on the sample.

c. Current measurement The sample's motion is controlled by a XYZ-scanner. Once the droplet is formed on the graphite sample, the X and Y position of the scanner are modulated by a Lock-In at the chosen amplitude (between 100 nm and a few microns) and at the chosen frequency (between 100 mHz and a few Hertz) while the electric current flowing through the electrodes on the graphite sample is demodulated at the droplet oscillation frequency. The direction of the droplet oscillation can be chosen, this allows to vary the angle between the droplet's motion axis and the inter-electrodes axis. Prior to any measure-ment of the generated current dependency on the modulation amplitude, we search for the scan angle leading to an alternating current with maximal amplitude, for a given frequency and amplitude of oscillations, around a given central position. Then we measure around the same central position and along the optimal axis, how the current amplitude varies with the oscillation amplitude at a given frequency, and how it depends on the bias voltage ∆V .

d. Shape-preserving motion of the droplet

To make sure the reported phenomena are not related to the deformation of the interaction surface between the liquid droplet and graphite, we imaged the samples after the current measurements under Scanning Electron Microscopy. As shown on Figure S2d, the remaining droplets display a roughly spherical shape, which confirms that scanning of the sam-ple with the capillary tip does not result in the deposition of a liquid line, but instead, in the sliding of a hemispherical droplet over graphite.

e. Data processing To extract the dropletmotion induced current, we record the generated current as a function of time for at least 50 periods of droplet oscillations. The generated current amplitude is defined as the Fourier component of current corresponding to the droplet oscillation frequency (see Figure S3).

A. Controls

To make sure that the measured current is indeed related to the viscous flow of liquid on graphite, we performed extensive controls which we detail here. We measure the current in various experimental configurations and show that the motion of a viscous droplet on top of the graphite flake is necessary to measure a non-zero current. Finally, we demonstrate the predominance of viscosity over charge or polarity of the liquid to convince the reader that the viscous friction of liquid on graphite is at the origin of the motion induced current. a. Pipette above the droplet We first checked that no current can be generated if the pipette is not in contact with the liquid droplet but is slightly above instead (see Figure ??). In this configuration, a liquid droplet is deposited on graphite and the pipette is subsequently retracted and brought a few µm above the liquid. Thus, during the oscillation, no motion is induced between the droplet and the sample. As shown on Figure S4, we do not measure any motion induced current in this situation.

b. On SiO 2 We also checked that no current can be generated if the droplet moves over the SiO 2 surface comprised between the electrodes, see Figure S4b. Finally, these controls show unambiguously that current generation requires the droplet to slide on the graphite surface.

c. With a neutral and apolar silicon oil droplet Finally, we performed an additional control to confirm a mechanism relying on momentum transfer-mediated current generation by using neutral and apolar silicon oil (Polyphenyl-methylsiloxane) instead of ionic liquid, with similar viscosity (0.1 Pa.s instead of 0.3 Pa.s for the ionic liquid). As evidenced by Figure S5, the current generation is also possible in this configuration with induced currents of the same order of magnitude than with ionic liquid. This confirms that the motion-induced current is caused by a momentum transfer which dominates any electrostatic contribution.

d. Frequency dependency of generated current

The current density displays a puzzling dependency on the oscillation frequency: While at a given frequency, the current amplitude scales linearly with the oscillation amplitude/velocity, when fixing the oscillation amplitude, the current amplitude decreases with frequency. Thus, the moving droplet-graphite system behaves as a low-pass filter with a cutoff frequency in the order of 1 Hz, see Figure S6. This dependency is rather unexpected since such long time-scales cannot come from the solid state where electrons and phonons relax much faster. Thus, we attribute this behavior to a thin wetting film spread on the droplet trajectory, which would reduce the effective friction (and therefore the generated current) and could have such slow dynamics. In this scenario, increasing the frequency is not equivalent to increasing the oscillation amplitude.

B. Wrinkles are current amplifiers

As discussed previously, our sample fabrication techniques allow us to produce graphite flakes of desired thickness with either a high wrinkle density (transferred flakes) or a very flat surface showing negligible roughness (untransferred flakes). We were therefore able to study the dependency of generated current on both thickness and wrinkle density independently. On Figure S7 a, we illustrate the thickness-dependency of the current induced by droplet oscillations of 1 µm.s -1 for both transferred (diamonds) and un-transferred (circles) graphite flakes. Our experiments reveal that for transferred samples, the thinner the graphite flake, the more efficient the current generation. Primarily, we observe a transition for the current generation around 10 nm in thickness: for flakes thicker than 10 nm, no current generation is possible, while current can be induced at the surface of flakes thinner than 10 nm. On the other hand, we report no generated current for flat un-transferred samples down to single nanometer thickness.

Our results demonstrate that wrinkles are a prerequisite to the viscous flow induced current generation. Another experimental evidence of the predominant role played by the wrinkles resides in the strong spatial anisotropy of the generated current (current depends on oscillation direction). As shown in Figure S8, the directional dependency depends strongly on the position on the sample surface. This variability is linked to the random distribution in height, position and direction of the wrinkles over the a. Raman spectroscopy To strengthen our claim that wrinkles are folded graphene structures that increase liquid friction and thus motion-induced current by a mere geometrical effect, we perform Raman spectroscopy on one transferred graphite sample and demonstrate that wrinkles show a ratio of D-peak intensity over G-peak intensity of the same order as flatter regions and that they show the usual spectroscopic properties of graphite samples. The measurements are presented on Figure S9. On a, we show the AFM scan of a transferred sample. On this sample, the transfer process has torn appart a top layer of graphite which is therefore showing wrinkles on top of the un-derlying graphite crystal (dark region). We use this particular sample to study the structural properties of wrinkles compared to a flat crystal. On b, we display a typical Raman spectrum measured with a 532 nm laser on this sample. Furthemore, on c, we display the G peak intensity map, revealing the in-homogeneity of thickness in the folded regions (higher G peak means thicker region). Finally, on d, we show the ratio of D-peak over G-peak intensity. This shows that wrinkles and folded regions are by no means defects. This analysis strengthen our theoretical treatment of wrinkles and folded regions as graphite or graphene with an enhanced corrugation. The electrons are dragged both by phonon drag and Coulomb drag. These two phenomena have recently been studied formally in the Keldysh framework by Coquinot et al. [5]. According to the authors, they can be treated similarly as bosons, either phonons or liquid fluctuations of the drop (called hydrons), interacting with electrons. The liquid fluctuations drag the electrons at the interfacial liquid velocity v and v v drop as the slip length is small compared with the drop height. On the other side, the phonons drag the electrons at the effective velocity v eff calculated in the main text: v eff ≈ 9•10 4 ×(1+W )×v drop . In the stationary regime, the electrons reach an electronic velocity

v e (q) = τ q τ e/ph q v eff + τ q τ e/h q v ≈ τ q τ e/ph q v eff (1)
Here τ e/b is the electron-boson interaction time and τ is the total electronic scattering time (1/τ = 1/τ e/ph + 1/τ e/h ). Therefore, the electronic velocity is the convex combination of the phonon effective velocity and the interfacial liquid velocity. In practice, in our model, τ q /τ e/ph q ≈ 1/2. In Eq. ( 1) the first term describes the effect of the phonon drag while the second term describes the effect of the Coulomb drag. Let us notice that the Coulomb drag effect reduces the electronic velocity: it then contributes negatively to the electric current generation.

B. Electronic state and electric current

In stationary regime, the electronic distribution of band dispersion q is shifted according to q → qq • v e . Therefore, the distribution will be filled above the doping level F in the velocity direction, and below in the opposition direction. Since the solid is 2D, the angle integration will provide a trigonometric factor 1/2. The distribution is not at zero tempera-ture, thus the energies close to the doping level up to the thermal energy matter. Moreover, the interaction broadens the electronic distribution around the band structure. According to [5], in stationary regime, the electronic distribution reads n e (q, ω) = 2τ / × n F (ω -q • v e (q))

1 + τ 2 ( ω + q -F ) 2 (2)
where n F is the Fermi-Dirac distribution and τ is the electronic scattering time. Therefore, taking into account the spin degeneracy, the authors of [5] predict an electric current j = 2e d dω (∇ q q ) N ( ) q v e 4k B T ch 2 ω+ -F 2kBT τ /π 1 + τ 2 ω 2

(3) where describes the band energy levels. Here, (∇ q q ) is the band velocity and q the momentum both at the band energy level . Going to the limit of small interaction and low temperature τ -1 k B T F , we recover Eq. ( 2) of the main text.

C. Model for the solid

Using a linear dispersion relation to describe graphene yields an energy-dependent density of states. When the doping vanishes we expect no current because of the electron-hole symmetry. For large dopings F k B T , only one band matters and the electric current writes j ≈ ev e 2 F /( v F ) 2 , where v F = 10 6 m/s is the Fermi velocity of graphene. The current is then quadratic in doping, which is consistent with the total average charge of the charge carriers. At smaller doping, temperature effects appear and a linear regime exists when F k B T . Experimentally, most measurements are in this regime but one may expect nonlinear effects for the higher bias voltage applied. As no such effect is measured in practice the relevance of this model may be questioned. Moreover, numerical applications lead to an electric current 1 order of magnitude smaller than experimentally measured.

To describe more precisely our samples, we used a model for multi-layer graphene made of two independent parabolic bands with effective mass m [6-8]: q = 2 q 2 /2m. In particular, the density of states does not depend on the doping, which is key to get a linear dependence for the current. The predicted current is then j ≈ 2ev e m F / 2 from which is deduced Eq. ( 4) of the main text.

3 I

 3 . GRAPHENE-BASED DEVICE'S MICRO-FABRICATION A. Transferred samples a cos(!t) !0

  in a 100 x 100 µm 2 area in the middle of a 2x2 mm 2 prepatterned substrate with Au electrodes. The devices are then fabricated by defining two parallel electrical contacts over the graphene flake by e-beam lithography using a double layered PMMA process: a 50 K lower layer of PMMA and a 950 K upper layer of PMMA both spin-coated at 4000 RPM. The spacing between the two contacts is set to at least 10 µm for the AFM experiment. The metallic contacts are subsequently deposited under high vacuum (a few 10-7 mBar) by e-beam evaporation. 5 nm of Ti covered by 45 nm (or 95 nm) of Au are deposited at very low evaporation rates (0.1 nm/s). The last fabrication step is a lift-off process in which the remaining resist is dissolved in an aceton bath for 20 min. The samples are finally rinsed into isopropanol for 2 min. This fabrication method involves a transfer of the graphite flake and thus yields a high density of wrinkles on thinner flakes as measured by AFM (see Figure S1 a,b).

Figure 1 :

 1 Figure 1: Experimental set-up. a, Schematic of the experiment. A quartz capillary of tip diameter of 500 nm is filled with liquid and glued on the lower prong of a millimetric quartz tuning fork. Once the tip is in contact with the graphite substrate, a liquid droplet is formed at its extremity. The scanner allows for droplet relative motion on the graphite flake in the XY-plane, while current is measured between both gold electrodes. An additional electric potential drop ∆V can be imposed between the electrodes. Inset: The resonance curve of the tuning fork whose frequency shift is used to deposit the droplet and maintain a gentle mechanical contact. b, Frequency shift response of the AFM during a typical experiment, indicating the stiffness of the pipette-sample contact. First, the pipette is brought into contact with the sample (1). After the contact, a slow relaxation (2) indicates the formation of the droplet. Finally, the sample is put into horizontal motion with respect to the droplet while a PID controller keeps a constant frequency shift (3). c, AFM image of a multilayer graphene sample (3.6 nm thick) exhibiting a high density of wrinkles. d, AFM image of a flat multilayer graphene sample (1.5 nm-thick). e, Schematic of droplet motion. f, Current (nA) as a function of time (s) resulting from droplet oscillations of 2 µm amplitude and 0.5 Hz frequency, on the wrinkled sample displayed on c (black dots) and on the flat graphite sample displayed on d (red dots). g, The corresponding current spectra as a function of frequency. For the wrinkled sample (black curve), the spectrum displays a peak at the droplet oscillation frequency (0.5 Hz).

8 Figure 2 :

 82 Figure 2: Generation of a droplet motion-induced electronic current. a, Current density (nA/µm) as a function of fluid velocity v = 2πf a for 4 different samples. Each curve is measured at a bias ∆V indicated in the corresponding plot. The multi-layer graphene subtrates have thickness 3.3, 1.7, 3.6 and 8.2nm, respectively and their corresponding surface roughness measured using AFM are [55,50,40,20] nm respectively. b, Time-dependent current under droplet oscillation for various voltage bias ∆V . c, Effect of the voltage bias ∆V on the electro-fluidic conductivity σ ef = j/v, showing a linear dependency. The dotted lines are guides to the eye. d, Electro-fluidic conductivity σ ef (at ∆V = 20 mV) versus the substrate corrugation, measured using AFM images. The black diamonds correspond to current measurements below the detection level.

5 Figure 3 :

 53 Figure3: Phonon drag mechanism. The flowing liquid a exerts a friction force F W on the graphite sample. This force is related to a momentum transfer to the phonons, that generates an asymmetry in the phonon distribution b which is biased in the direction of the liquid flow. Finally this asymmetry propagates to the electronic distribution c by electron-phonon interaction. This asymmetry in the electronic distribution translates into the generation of a net electric current, whose sign depends on the nature of charge carriers at the Fermi level (electrons or holes). We stress that this is an out of equilibrium phenomenon in which most of the momentum is relaxed to the crystal d by the Umklapp process with a rate 1/τ um . Finally, on e, we compare the theoretical derivation of the electro-fluidic conductivity σ ef versus the bias voltage ∆V . We stress that the experimental data is the same as in Figure2cand that the solid lines are plotted using Eq. (4) assuming the bulk graphite effective mass m = 0.1m e and W the wrinkling number as the only fitting parameter chosen between 5 and 20. This wrinkling number characterizes the enhancement of Stokes friction due to wrinkles on the graphene surface.

Figure S1 :

 S1 Figure S1: Samples prepared by the two alternative techniques. On a and b, we show optical and AFM images of transferred samples. The few nanometers thick samples like a show a high wrinkle density while the thicker ones (see b) are very flat. On c, we show an un-transferred sample which displays negligible wrinkle density even for very low thickness (1.5 nm).

FigureFigure

  FigureS3: a, Current as a function of time, for droplet oscillations of 2 µm amplitude and 0.5 Hz frequency. b, Current spectrum as a function of frequency. The spectrum displays a peak at the droplet oscillation frequency (0.5 Hz). The peak maximal value is defined as the generated current amplitude.

Figure

  FigureS5: a Current spectrum (nA) as a function of the frequency (Hz) for a micrometric silicone oil droplet oscillating at 0.5 Hz and 3 µm in amplitude on the graphite sample. b Generated current amplitude as a function of the silicon oil droplet oscillation amplitude, at 0.5 Hz. The circles are the experimental data, while the dotted line is the best linear fit. All these measurements are taken at a bias voltage ∆V = 0mV .

Figure

  Figure S7: a Semilog plot of the normalized generated current given in nA/µm.s -1 , as a function of the flake thickness. b AFM image of a flat 1.5 nm-thick sample (un-transferred). c AFM image of a current generating wrinkled 3.6 nm-thick sample (transferred). d AFM image of a flat 68 nm-thick sample (transferred). No wrinkles are induced and thus no current is measured. On samples thicker than 10 nm, no wrinkles are observed and the droplet oscillations do not result in the generation of a measurable current. Similarly, on the thinner samples, current generation requires the wrinkles.

Figure

  Photon counts

ACKNOWLEDGEMENTS L.B. acknowledges funding from the EU H2020 Framework Programme/ERC Advanced Grant agreement number 785911-Shadoks. A.S. acknowledges funding from FET-open ProID. The Flatiron Institute is a division of the Simons Foundation. B.C. acknowledges funding from a J.-P. Aguilar grant of the CFM Foundation, and thanks Olivier Parcollet for hosting him at the Flatiron Institute. B.R. acknowledges the Royal Society University Research Fellowship (URF/R1/180127) and European Research Council Starting Grant (agreement number 852674 -AngstroCAP). A.B. thanks Dr Ashok Keerthi for help in fabrication.

Supplementary Material for: Strong electronic winds blowing under liquid flows on carbon surfaces

Alice Marcotte † , 1 Mathieu Lizee † , 1 Baptiste Coquinot, 1, 2 Nikita Kavokine, 2 Karen Sobnath, 3 Clément Barraud, 3 Ankit Bhardwaj, 4, 5 Boya Radha, 4, 5 Antoine Niguès, 1 Lydéric Bocquet, 1 and Alessandro Siria