Strong electronic winds blowing under liquid flows on carbon surfaces

Alice Marcotte†,1 Mathieu Lizee†,1 Baptiste Coquinot,1, 2 NikitaKavokine,2 Karen Sobnath,3 Clément Barraud,3 Ankit Bhardwaj,4, 5 Boya

Radha,4, 5 Antoine Niguès,1 Lydéric Bocquet,1 and Alessandro Siria11Laboratoire de Physique de l’École Normale Supérieure,ENS, Université PSL, CNRS, 3 Sorbonne Université,

Université Paris Cité, 75005 Paris, France2Center for Computational Quantum Physics, Flatiron Institute, New York, USA

3Université Paris Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris, France4National Graphene Institute, The University of Manchester, UK

5Department of Physics and Astronomy, The University of Manchester, UK

The interface between a liquid and a solid is the location of plethora of intrincatemechanisms at the nanoscale, at the root of their specific emerging properties in naturalprocesses or technological applications [1, 2]. However, while the structural propertiesand chemistry of interfaces have been intensively explored, the effect of the solid-stateelectronic transport at the fluid interface has been broadly overlooked up to now.It has been reported that water flowing against carbon-based nanomaterials, such ascarbon nanotubes [3–8] or graphene sheets [9–14], does induce electronic currents,but the mechanism at stake remains controversial [9, 10, 15]. Here, we unveil themolecular mechanisms underlying the hydro-electronic couplings by investigating theelectronic conversion under flow at the nanoscale. We use a tuning fork-Atomic ForceMicroscope (AFM) to deposit and displace a micrometric droplet of both ionic and non-ionic liquids on a multilayer graphene sample, while recording the electrical currentacross the carbon flake. We report measurements of an oscillation-induced currentwhich is several orders of magnitude larger than previously reported for water oncarbon [13], and further boosted by the presence of surface wrinkles on the carbonlayer. Our results point to a peculiar momentum transfer mechanism between fluidmolecules and charge carriers in the carbon walls mediated by phonon excitations inthe solid [3, 11]. Our findings pave the way for active control of fluid transfer at thenanoscale by harnessing the complex interplay between collective excitations in thesolid and the molecules in the fluid.

Transport of fluid across or at the interface ofcarbon unveiled unexpected properties in termof very fast permeation, non-linear ionic trans-port and sieving and even energy storage andharnessing [2]. These results highlight the sub-tle interactions of water and charges with thecarbon surface. Classically, the interaction ofa fluid with a solid wall is based on pictur-ing the solid as a static periodic potential thatacts on the fluid molecules: interactions re-sult from collisions on the surface roughness,as flows induced on the roughness scale dissi-pate mechanical energy. However, recent re-sults for fluid and ion transport in carbon nan-otubes and nanochannels made of graphitic ma-terials have pointed out the limits of such stan-dard description for fluid-solid interactions andexisting couplings of liquid transport with elec-tronic degrees of freedom inside the confiningwalls. This takes the form of electronic cur-rents induced in the solid surface under flowing

† These authors contributed equally to this work

liquid [4, 5, 7–14], the modification of wettingby electronic screening [16, 17], or anomaliesin the water-carbon friction [18]. The interac-tion mechanisms at stake remain largely unex-plained and to be properly understood.

In this article, we report the generation ofelectric current in a few-layer graphene sam-ple by the displacement of a sub-micrometerscale liquid droplet along its surface. We de-posited a thin graphite flake (1 to 70 nm inthickness) onto a Si/SiO2 substrate; two goldelectrodes, separated by a distance of ∼ 10µmand connected to a low noise current amplifier,allowed us to measure the electric current pass-ing through the flake (see Figure S1). The sam-ple was placed in a tuning fork atomic force mi-croscope (AFM), that was specifically designedfor controlled liquid deposition, see Figure 1a.The liquid was introduced into a quartz capil-lary, which was then glued to the quartz tuningfork (see Figure S2). A piezo-dither induceda nanometric oscillation of the tuning fork atits resonant frequency (f0 ≈ 32 kHz), a typ-ical resonance curve is shown in the inset ofFigure 1a. By monitoring precisely the tuning

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3

I. GRAPHENE-BASED DEVICE’S MICRO-FABRICATION

A. Transferred samples

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The HOPG crystals are purchased from HQ Graphene. Each bulk crystal is mechani-

cally 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] in a 100 x 100 µm2 area in the middle of a 2x2 mm2 pre-

patterned 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).

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).

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Figure 1: Experimental set-up. a, Schematic of the experiment. A quartz capillary of tipdiameter 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 atits extremity. The scanner allows for droplet relative motion on the graphite flake in theXY-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 thetuning 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 thestiffness 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, thesample is put into horizontal motion with respect to the droplet while a PID controller keeps aconstant 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 (1.5nm-thick). e, Schematic of droplet motion. f, Current (nA) as a function of time (s) resultingfrom 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, Thecorresponding 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 thepipette tip approach the sample, we achieveda fully controlled contact between the carbonsurface and the pipette’s aperture, leading tothe formation of a liquid capillary bridge. Thecapillary bridge then evolved into a droplet, ofradius fixed by the outer diameter of the tip(500 nm) [19, 20]. This novel experimental set-

up allows to deposit very viscous liquid at nanoand micro scale over a broad range of substrates[21]. We used a room temperature ionic liquid(BMIM-PF6) and a neutral and apolar siliconoil (Polyphenyl-methylsiloxane), both very vis-cous with a viscosity of 0.3 Pa·s and 0.1 Pa·s, re-spectively, and with very small vapor pressures,allowing us to safely neglect evaporation. The

3

continuous monitoring of the contact with boththe in-situ optical microscope and the AFM fre-quency shift signal allows us to ensure that thedroplet remained hooked to the tip, sliding overthe carbon surface. Moreover, motion does pre-serve the roughly spherical shape of the droplet(see Figure S2d).

In a typical experiment, the droplet is putinto oscillatory motion along the carbon sur-face by a piezo-scanner with an amplitude ofa few microns at a frequency of the order of1 Hz. An electric current through the carbonsurface is then measured at the droplet oscilla-tion frequency (Figure 1e and Figure S3). Asshown on Figure 1a, our set-up enables to fur-ther impose a constant potential drop ∆V (typ-ically between -50 mV and 50 mV) betweenthe electrodes. We report electric currents inthe tens of nano-Ampere range induced by adroplet moving at a few micrometers per sec-ond. To allow for a comparison with other cur-rent generation experiments, we define the cur-rent density as the AC-current amplitude di-vided by the diameter of the droplet (of order1µm) : j = I/(2Rdrop) typically in the tens ofnA/µm range.

Firstly, several benchmark measurements al-lowed us to ensure that the current generationis due to the specific interaction of the liq-uid droplet with the few-layer graphene. Wechecked that no alternating current is recordedwhen the deposited droplet oscillates over theSiO2 substrate between the electrodes but with-out the multilayer graphene substrate. We fur-ther controlled that no current is generatedwhen the pipette is not in contact with thegraphite but slightly above or even when atungsten tip is used instead of the liquid filledpipette, see Figure S4. Altogether, we cansafely conclude that the measured effect is dueto the displacement of the liquid droplet on thecarbon surface.

On Figure 2a, we report results of severalsamples (with thickness varying between 1.7and 8.2 nm), where an electro-fluidic current ismeasured. We observe that for a fixed oscilla-tion frequency f and bias voltage ∆V , the am-plitude of the AC generated current increaseslinearly with the droplet oscillation amplitudea, or equivalently, with the droplet peak ve-locity (v = 2πfa). Taking advantage of thedemonstrated linear dependency of the currentdensity with respect to the droplet peak veloc-ity, we define the electro-fluidic conductivity ofour samples as

σef =j

v,

quantifying the cross-coupling between elec-tronic current j and liquid flow v. Among thefive samples presented here, we show an electro-fluidic conductivity ranging from 2.8 to 21.6nA.s/µm2 for a droplet size in the micrometerrange.

To understand the mechanism driving thiselectro-fluidic current and having in mind thestrongly energy-dependent electronic density ofgraphene close to the Fermi level [22], we tunethe bias voltage ∆V between the electrodes andmeasure its influence on the electro-fluidic cur-rent. We show that the alternating currentinduced by the droplet oscillation strongly de-pends on ∆V , with a linear dependency. This ishighlighted in Figures 2b-c, where the electro-fluidic conductivity is shown to scale roughlylinearly with the potential drop ∆V , with asmall or negligeable shift at zero potential dropand a sign reversal of the current for negativeversus positive ∆V . Finally, the slope of thislinear dependency σef/∆V is in the range of0.05 to 0.2 nA.s/µm2/mV, see Figure 2c, andslightly varies across samples.

Now, an unexpected feature of the experi-mental results is the sharp distinction amongcarbon samples regarding the amplitude ofthe the electro-fluidic conductivity. Over allfew-layer-graphene measured, with thicknessesranging from 1 nm up to 70 nm, we observethat the thinner ones tends to exhibit thelargest electro-fluidic conductivity, see FigureS7a. However, investigating the surface to-pography of the carbon samples using AFMimaging, such as on Figure 1c-d, we ratherreveal a correlation between the amplitude ofthe electro-fluidic conductivity and the exis-tence of surface corrugation in the forms of foldsand wrinkles. Notably, thinner samples exhibita large number of folds and wrinkles whereasthicker samples present a flat surface withoutfolds or defects (see Figure S1). This differ-ence is due to the fabrication process involv-ing the transfer of the exfoliated graphene flakeonto the oxidized silicon wafer (See Sec. S-I) :the thinner flakes are softer and more flexiblethan the thicker ones, and more likely to exhibitwrinkles [23, 24], see Figure 1a and Figure S7.In order to disentangle the role of wrinkles fromthat of sample thickness, we implemented an al-ternative fabrication method in which all trans-fer steps are avoided and few-layer grapheneflakes were directly exfoliated on the Si/SiO2

wafer. This allowed us to obtain sub-10 nm,un-transferred and smooth graphene sampleswith a uniform wrinkle-free surface (see Fig-ure 1b and Figure S1) to compare with thetransferred and crumpled flakes of similar thick-

4

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Figure 2: Generation of a droplet motion-induced electronic current. a, Current density(nA/µm) as a function of fluid velocity v = 2πfa for 4 different samples. Each curve is measured

at a bias ∆V indicated in the corresponding plot. The multi-layer graphene subtrates havethickness 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 dropletoscillation for various voltage bias ∆V . c, Effect of the voltage bias ∆V on the electro-fluidicconductivity σef = j/v, showing a linear dependency. The dotted lines are guides to the eye. d,Electro-fluidic conductivity σef (at ∆V = 20 mV) versus the substrate corrugation, measured

using AFM images. The black diamonds correspond to current measurements below the detectionlevel.

ness showing a much larger surface corrugation(see Figure S9 for a Raman analysis of wrin-kles). As shown on Figure 2d, droplet mo-tion on smooth samples results in no detectablecurrent. A contrario, crumpled flakes of thesame thickness have a high wrinkle density andshow the strongest electro-fluidic conductivity,in the nA.s/µm2 range. Our results reveal thatwrinkles are instrumental to the current gen-eration mechanism, while there is no evidenceof an effect of graphene thickness. Further evi-dence of the crucial role of wrinkles is providedby the dependence of the generated current onthe droplet oscillation direction, see Figure S8.As expected, the current amplitude is 180◦ pe-riodic with respect to the angle between thedroplet trajectory and the electrodes. How-ever, the maximal amplitude is not necessar-ily reached when the droplet oscillation is per-pendicular to the electrodes, and the directionyielding the maximum current strongly dependson the position on the sample’s surface. We ar-gue that this spatial and directional variabilityis related to the random distribution in height,

position and direction of the wrinkles on thecarbon surface. This further highlights the roleof wrinkles as current amplifiers. Let us note forcompleteness that the data displayed on Figure2 are obtained along the direction with maxi-mal amplitude.

Summarizing our experimental results, wehave shown that the motion of a micrometricdroplet on a crumpled, few-layer graphene sur-face generates a strong electro-fluidic current inthe flake, which is proportional to the velocityof the droplet, as well as to the bias voltageapplied between the electrodes. A remarkablefeature is the strong dependence of the mag-nitude of the generated current on the surfacecorrugation of the flake.

While our results echo previous reports ofelectric potential differences induced by waterflow in carbon-based nanomaterials [3–14], ournanoscale investigation puts us in position todisentangle various effects at the root of elec-tronic current generation under fluid flow, andget much insights into the molecular mechanismunderlying the phenomenon. Previously, sev-

5

eral mechanisms have been invoked to rational-ize such conversion, including streaming poten-tial effects [5], adsorbed/desorbed ion hopping[6, 9], Coulomb drag [4, 8] or charging/discharg-ing of a pseudo-capacitance [13, 14]. Thesemechanisms involve Coulomb interactions be-tween the polar liquid – exhibiting molecularpartial charges – and the carbon surface as themain pathway for momentum transfer from theliquid to the electrons. As we now show, theseprevious explanations fail to account for thephenomenon at stake. Indeed, the strong effectof surface corrugation reported here suggeststhat current generation proceeds through a ’me-chanical’ transfer of momentum of the fluid to-wards the electronic degrees of freedom. In or-der to confirm this hypothesis and rule out anyeffect caused by the charges in the ionic liquid,we first repeated the experiment, now with anapolar silicone oil replacing the ionic liquid (anda wrinkled few-layer graphene as a substrate).We do observe qualitatively and quantitativelysimilar current generation for the silicone oiland the ionic liquid, see Figure S5. This dis-cards coulombic couplings between the liquidmolecules and the electrons and all correspond-ing mechanisms based on charge-charge interac-tions across the interface. These considerationsnaturally lead us to reconsider phonon-dragmechanisms, as first proposed theoretically in2001 [3] as the driving mechanism for the mea-sured electro-fluidic current. The phonon dragrelies on hydrodynamic (viscous) friction at thesolid-liquid interface, which transfers momen-tum to the solid’s phonons, thereby exciting a’phonon wind’. Through the electron-phononinteraction, the phonon wind ’blows’ on theFermi sea, so that electrons near the Fermi sur-face are dragged along the direction of the liq-uid flow, hence producing an electric current(see Figure 3a). A complete theory of phonondrag at the quantum level, within the non-equilibrium Keldysh framework is developed in[25]. Here, we give only a semi quantitativedescription that allows us to capture the mainphysical ingredients.

First, assuming a Couette flow within the liq-uid droplet, the hydrodynamic friction force isF0 ∼ ηAdropvdrop/hdrop, vdrop is the drop ve-locity, η its viscosity, Adrop its surface and hdrop

its height (here typically, η = 0.3Pa · s andhdrop ≈ 1µm). In line with observations, wrin-kles are expected to enhance this hydrodynam-ics friction force. In a simple picture, wrinklesare modelled as protusions on which the flowingliquid exerts a Stokes force scaling (per wrin-kle) as FW ∝ ηhWvdrop, where hW is the typi-cal height of the wrinkles (hW ∼ 10 − 30 nm).

One then obtains a total friction force FW ∼F0(1+W ), where we introduced the dimension-less wrinkling number, W = 3πnWhWhdrop,with nW the wrinkle density to account for theincrease in viscous friction due to surface cor-rugation.

Our model is sketched on Figure 3a-d : thehydrodynamic friction force transfers momen-tum from the liquid (Figure 3a) to the solidexciting acoustic phonons with dispersion rela-tion εq = ~qc where c is the phonon (sound)velocity (c ≈ 2 × 104 m · s−1 [26, 28, 29]).The phonons are mainly relaxed by transfer-ring their momentum to the crystal (Figure 3d)through Umklapp processes [26], on a typicaltimescale τum ∼ 10 ps [27]. As sketched onFigure 3b, the constant influx of momentummodifies the phononic distribution by shiftingthe energetic cost of a phonon of wavevectorq according to εq 7→ εq − ~q · veff , which islower along the direction of the liquid flow.Here veff is the effective velocity of the phononwind: it is determined by balancing the momen-tum fluxes in and out of the phonon system,according to τumFW/Adrop = ∆Pph/Adrop =∫

dq2π ~qnB(~qc − ~q · veff), with nB the Bose-

Einstein distribution and q is the phononwavevector, that we assume two-dimensional.This yields

veff '2π

3ζ(3)

~2c4

(kBT )3×(1+W )η

vdrop

hdrop×τum (1)

Then, veff ≈ 9 · 104 × (1 + W ) × vdrop. As theelectrons scatter on the phonons, they acquirean average velocity equal to the phonon windvelocity; our complete theory shows that this istrue whatever the form of the electron-phononinteraction [25]. As sketched on Figure 3c, in a2d model of the solid, this amounts to shiftingthe band energies according to εq 7→ εq − ~q ·veff . Assuming for concreteness that the liquidflows to the right, this roughly means that inan energy window of width ~kFveff around theFermi level (kF is the Fermi wavevector), theelectrons move to the right at the Fermi velocityvF. The corresponding current density reads

j = evF ×N(εF)× [~kFveff ], (2)

where N(εF) is the density of states at theFermi level εF. We obtain a current density pro-portional to the droplet velocity, to the liquidviscosity and to the wrinkling number W in ex-cellent agreement with the experiment. More-over, the current is independent of the chemicalnature of the liquid as observed in experiment(ionic liquid and silicone oil generate similar

6

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m2 ) Sample 1

Sample 2Sample 3W = 20W = 12W = 5

Figure 3: Phonon drag mechanism. The flowing liquid a exerts a friction force FW on thegraphite 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 netelectric 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 themomentum is relaxed to the crystal d by the Umklapp process with a rate 1/τum. Finally, on e,

we compare the theoretical derivation of the electro-fluidic conductivity σef versus the biasvoltage ∆V . We stress that the experimental data is the same as in Figure 2c and that the solidlines are plotted using Eq. (4) assuming the bulk graphite effective mass m = 0.1me and W thewrinkling 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 derivationpresented in [25] (see Sec. S-III for the main in-gredients), we add in the following a correctionfactor of order 1/2 to the current density.

Electronic bands in graphene have a lineardispersion, εq = ~vFq ; however, in our multi-layer samples of few-nanometer thickness (seeFigure S7a) one expects the electronic proper-ties to be better described by a parabolic bandwith effective mass m [30–32]: εq = ~2q2/2m.The density of states is then independent of en-ergy: N(ε) = N(εF) = m/~2. Altogether, weobtain the electro-fluidic cross conductivity

σef = em

~2εF ×

veffvdrop

. (3)

The prediction in Eq. (3) can now be com-pared with experimental results. Experimen-tally, the applied bias voltage, sets the Fermilevel εF ≈ e∆V/2, so that

σef ≈ 0.1×m/me × (1 +W )×∆V [nA.s/µm2](4)

with ∆V in mV. This estimate quantitativelyreproduces the conductivity measured experi-mentally of a few nA.s/µm2 with the correctlinear scalings of the electro-fluidic conductiv-ity with the droplet velocity and the bias volt-age. In addition, this current is proportional tothe wrinkling number, in line with the observedimpact of the presence of surface wrinkles ascurrent amplifiers. On Figure 3e, we compare

7

this theoretical prediction with the experimen-tal measurement of σef as a function of the biasvoltage. For the electronic effective mass, weconsider graphite in a parabolic approximation,m ∼ 0.1me, where me is the electron mass [30].The theoretical estimate involves no fitting pa-rameter other than the wrinkling number Wchosen between 5 and 20 to reproduce the ex-perimental data.)Note that although the above discussion fo-cussed on electrons as charge carriers, one ex-pects that the current will change sign if thecharge carriers are holes. This explains whythe generated current flips sign along with ∆V :for positive bias, the charge carriers are elec-trons in the conduction band, whereas for neg-ative bias, the charge carriers are holes in thevalence band (a seamless transition is ensuredby the fact that the bands are touching). Al-together, the proposed modeling strongly sup-ports the hypothesis that the experimentallyobserved electric current is due to charge car-riers being dragged by a phonon wind. Ourmodel describes the momentum transfer fromthe liquid to the phonon distribution where animbalance in the distribution develops. Scat-tering on the electron bath then results in anelectronic current.

In conclusion we report the generation of avery large electronic current induced by thecontrolled motion of a liquid droplet at the sur-face of carbon nanomaterials. Our results sug-gest that the current generation is mediated byhydrodynamic momentum transfer from the liq-uid molecules to the phonons in the solid. Ourspecifically tailored system is particularly wellsuited to evidence the phonon-drag inducedelectronic current. Indeed, the combinationof a local micrometric flow on strongly wrin-kled graphene samples along with the choiceof a very viscous and neutral liquid allows thephonon-drag as a direct momentum transfer to

dominate any other current generation mecha-nism. The versatility of our setup allows to tunethe fluid velocity as well as the solid’s electronicproperties using the bias between the electrodesto fully characterize the phenomenon. Finally,all experimental results were rationalized withour new dedicated model of the phonon-drag.Our findings pave the way for the active controlof fluid transport at the nanoscale by harness-ing the interplay between fluid molecules andthe collective excitation in the solid suggest-ing that novel nanofluidic functionalities can beproposed where a flow of fluid at the interfacewith tunable solid can be generated by exter-nally excitable phonons or structural deforma-tion waves. Further, the coupling of electronicand micro-fluidic currents bears the promiseof ubiquitous and environmentally friendly en-ergy production as demonstrated recently intitanium carbide MXene based transpiration-driven electrokinetic power generators [33].

ACKNOWLEDGEMENTS

L.B. acknowledges funding from the EUH2020 Framework Programme/ERC AdvancedGrant agreement number 785911-Shadoks.A.S. acknowledges funding from FET-openProID. The Flatiron Institute is a division ofthe Simons Foundation. B.C. acknowledgesfunding from a J.-P. Aguilar grant of the CFMFoundation, and thanks Olivier Parcollet forhosting him at the Flatiron Institute. B.R.acknowledges the Royal Society University Re-search Fellowship (URF/R1/180127) and Euro-pean Research Council Starting Grant (agree-ment number 852674 - AngstroCAP). A.B.thanks Dr Ashok Keerthi for help in fabrica-tion.

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

Alice Marcotte†,1 Mathieu Lizee†,1 Baptiste Coquinot,1, 2 NikitaKavokine,2 Karen Sobnath,3 Clément Barraud,3 Ankit Bhardwaj,4, 5 Boya

Radha,4, 5 Antoine Niguès,1 Lydéric Bocquet,1 and Alessandro Siria11Laboratoire de Physique de l’École Normale Supérieure,ENS, Université PSL, CNRS, 3 Sorbonne Université,

Université Paris Cité, 75005 Paris, France2Center for Computational Quantum Physics, Flatiron Institute, New York, USA

3Université Paris Cité, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris, France4National Graphene Institute, The University of Manchester, UK

5Department of Physics and Astronomy, The University of Manchester, UK

CONTENTS

I. Graphene-based device’smicro-fabrication 1A. Transferred samples 1B. Un-transferred samples 1

II. Droplet motion-induced currentgeneration 2A. Controls 4B. Wrinkles are current amplifiers 4

III. Theoretical model 6A. Electron drag: Phonon drag vs.

Coulomb drag 6B. Electronic state and electric current 8C. Model for the solid 9

References 9

I. GRAPHENE-BASED DEVICE’SMICRO-FABRICATION

A. Transferred samples

The HOPG crystals are purchased from HQGraphene. Each bulk crystal is mechanicallyexfoliated and flakes are first randomly de-posited on separate 280 nm-thick Si/SiO2 sub-strates. The thickness of the SiO2 layer ischosen to maximize the optical contrast [2].Graphene flakes are then carefully selected fortheir thickness (ranging from 1 nm to 70 nm),homogeneity and size (> 15 µm). h-BN flakesof thicknesses between 7 and 15 nm are cho-sen. In the case of graphite/SiO2 devices, thegraphene flakes are transferred thanks to the

† These authors contributed equally to this work

“hot pick-up” technique [3, 4] in a 100 x 100µm2 area in the middle of a 2x2 mm2 pre-patterned substrate with Au electrodes. In thecase of h-BN supported graphene devices, h-BNand graphene flakes are first stacked and thentransferred on the pre-patterned devices usingthe same technique. The devices are then fab-ricated by defining two parallel electrical con-tacts over the graphene flake by e-beam lithog-raphy using a double layered PMMA process:a 50 K lower layer of PMMA and a 950 K up-per layer of PMMA both spin-coated at 4000RPM. The spacing between the two contactsis set to at least 10 µm for the AFM experi-ment. The metallic contacts are subsequentlydeposited under high vacuum (a few 10-7 mBar)by e-beam evaporation. 5 nm of Ti covered by45 nm (or 95 nm) of Au are deposited at verylow evaporation rates (0.1 nm/s). The last fab-rication step is a lift-off process in which theremaining resist is dissolved in an aceton bathfor 20 min. The samples are finally rinsed intoisopropanol for 2 min. This fabrication methodinvolves a transfer of the graphite flake andthus yields a high density of wrinkles on thin-ner flakes as measured by AFM (see Figure S1a,b). As the viscous-flow induced current didnot show any clear difference between SiO2 andhBN supported devices, we disregard this detailin the following.

B. Un-transferred samples

As discussed in the text, wrinkles have astrong influence on viscous flow driven currentgeneration at the carbon surface. To assess therole of wrinkles in current generation, we useda transfer-free fabrication technique to prepareflat graphite samples. We evaporated gold elec-trodes on the SiO2 substrate where the graphitewas exfoliated in the first place and thus ob-

arX

iv:2

205.

0503

7v1

[co

nd-m

at.m

es-h

all]

10

May

202

2

2

Wrinkled 3.6 nm-thick samplea Flat 68 nm-thick sample

Transferred flakes

Flat 1.5 nm-thick sample

25 nm

05

10

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20 um 20 um

5 um75 nm

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10152025303540455055606570

5 um

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30 um

cb

Figure S1: Samples prepared by the two alternative techniques. On a and b, we show opticaland AFM images of transferred samples. The few nanometers thick samples like a show a highwrinkle 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).

tained wrinkle-free samples which can be com-pared to high wrinkle-density samples of thesame thickness obtained when transferring theflakes before evaporating the contacts (see Fig-ure S1 c).

II. DROPLET MOTION-INDUCEDCURRENT GENERATION

a. Micro-pipette AFM preparation Thecapillary tip is obtained by locally heating andsimultaneously pulling a 10 cm long quartz cap-illary of inner diameter 0.5 mm and outer di-ameter 1 mm (World Precision Instruments),with a Sutter Instruments P-2000 pipettepuller. The capillary tube is separated into twopipettes. Each of them has one cylindrical end,and one conical tapper on the other end witha tip outer diameter of 500 nm, that can bevaried by tuning the pulling parameters.

The pipette is then filled with ionic liquid(Bmim-PF6 of high purity from Merck) or withneutral silicon oil (Polyphenyl-methylsiloxane,100 mPa.s, from Sigma Aldrich). This step isgreatly facilitated by the inner filament in thepipette, which enhances the capillary filling ofthe tip. Once the pipette is filled, its very ex-tremity is glued with epoxy glue on the lateralside of a millimetric quartz tuning fork prong.With a diamond tip, the largest part of the cap-illary is then cut free from its extremity andfrom the tuning fork. These steps are summa-

rized by Figure S2a.Finally, the tuning fork and its probe are ex-

amined under Scanning Electron Microscopy, tomeasure the outer diameter of the pipette ex-tremity and to make sure the previous fabrica-tion steps did not damage the tip, see FigureS2b.

b. Liquid droplet creation and control Allour experiments are conducted under ambientconditions. To deposit a very small and con-trolled amount of liquid on the graphite, andto put the deposited droplet in motion overthe sample while keeping constant the degreeof squeezing of the droplet, we use the tuningfork as an autosensitive Atomic Force Micro-scope (AFM) mechanical oscillator. The tuningfork is excited via a piezo-dither at its resonancefrequency f0 in the normal mode (f0 ≈ 32kHz), while the amplitude and phase shift ofthe tuning fork oscillations with regards to theexcitation AC voltage are obtained from thepiezoelectric current flowing through the tun-ing fork electrodes. Our home-made AFM isused in frequency-modulation mode : a PhaseLock Loop (PLL) ensures that the tuning fork issystematically excited at its resonant frequency.Furthermore, a PID servo loop maintains a con-stant oscillation amplitude at resonance by tun-ing the excitation voltage. Interactions with thesubstrate are detected by the change in the res-onance frequency and in the excitation ampli-tude. As shown on Figure S2c, we use a cam-era to perform the coarse approach of the AFM

3

capillary tip

tip reflection

Ti/Au electrode

graphite sample

epoxy

capillary

ionic liquid

diamond tip

tuning fork1

2

3

500 um500 um

5 um

IL droplets

Ti/Au electrode

graphite

a

c

b

d

10 um

Figure S2: a Steps for the preparation of the tuning fork and its probe: Firstly, the pipette isfilled with liquid. Then, it is glued with epoxy on the side of one tuning fork prong. Finally, thecylindrical 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 millimetrictuning-fork, and of the extremity of the pipette glued on it (inset). c In-situ microscope view ofthe 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 agraphite sample. Two roughly spherical ionic liquid droplets remain on the flake.

tip and we monitor the gentle approach via thePLL to detect contact with the substrate andthe formation of the capillary bridge, and toensure a constant contact stiffness (frequency-shift) during the droplet motion on the sample.

c. Current measurement The sample’smotion is controlled by a XYZ-scanner. Oncethe droplet is formed on the graphite sample,the X and Y position of the scanner are mod-ulated by a Lock-In at the chosen amplitude(between 100 nm and a few microns) and atthe chosen frequency (between 100 mHz anda few Hertz) while the electric current flowingthrough the electrodes on the graphite sampleis demodulated at the droplet oscillation fre-quency. The direction of the droplet oscillationcan be chosen, this allows to vary the anglebetween the droplet’s motion axis and theinter-electrodes axis. Prior to any measure-

ment of the generated current dependency onthe modulation amplitude, we search for thescan angle leading to an alternating currentwith maximal amplitude, for a given frequencyand amplitude of oscillations, around a givencentral position. Then we measure around thesame central position and along the optimalaxis, how the current amplitude varies withthe oscillation amplitude at a given frequency,and how it depends on the bias voltage ∆V .

d. Shape-preserving motion of the dropletTo make sure the reported phenomena arenot related to the deformation of the inter-action surface between the liquid droplet andgraphite, we imaged the samples after the cur-rent measurements under Scanning ElectronMicroscopy. As shown on Figure S2d, the re-maining droplets display a roughly sphericalshape, which confirms that scanning of the sam-

4

ple with the capillary tip does not result in thedeposition of a liquid line, but instead, in thesliding of a hemispherical droplet over graphite.

e. Data processing To extract the droplet-motion induced current, we record the gener-ated current as a function of time for at least 50periods of droplet oscillations. The generatedcurrent amplitude is defined as the Fourier com-ponent of current corresponding to the dropletoscillation frequency (see Figure S3).

A. Controls

To make sure that the measured current isindeed related to the viscous flow of liquidon graphite, we performed extensive controlswhich we detail here. We measure the cur-rent in various experimental configurations andshow that the motion of a viscous droplet on topof the graphite flake is necessary to measure anon-zero current. Finally, we demonstrate thepredominance of viscosity over charge or polar-ity of the liquid to convince the reader that theviscous friction of liquid on graphite is at theorigin of the motion induced current.

a. Pipette above the droplet We firstchecked that no current can be generated if thepipette is not in contact with the liquid dropletbut is slightly above instead (see Figure ??). Inthis configuration, a liquid droplet is depositedon graphite and the pipette is subsequently re-tracted and brought a few µm above the liquid.Thus, during the oscillation, no motion is in-duced between the droplet and the sample. Asshown on Figure S4, we do not measure anymotion induced current in this situation.

b. On SiO2 We also checked that no cur-rent can be generated if the droplet moves overthe SiO2 surface comprised between the elec-trodes, see Figure S4b. Finally, these controlsshow unambiguously that current generationrequires the droplet to slide on the graphite sur-face.

c. With a neutral and apolar silicon oildroplet Finally, we performed an additionalcontrol to confirm a mechanism relying onmomentum transfer-mediated current genera-tion by using neutral and apolar silicon oil(Polyphenyl-methylsiloxane) instead of ionicliquid, with similar viscosity (0.1 Pa.s instead of0.3 Pa.s for the ionic liquid). As evidenced byFigure S5, the current generation is also possi-ble in this configuration with induced currentsof the same order of magnitude than with ionicliquid. This confirms that the motion-inducedcurrent is caused by a momentum transferwhich dominates any electrostatic contribution.

d. Frequency dependency of generated cur-rent The current density displays a puz-zling dependency on the oscillation frequency:While at a given frequency, the current ampli-tude scales linearly with the oscillation ampli-tude/velocity, when fixing the oscillation am-plitude, the current amplitude decreases withfrequency. Thus, the moving droplet-graphitesystem behaves as a low-pass filter with a cut-off frequency in the order of 1 Hz, see FigureS6. This dependency is rather unexpected sincesuch long time-scales cannot come from thesolid state where electrons and phonons relaxmuch faster. Thus, we attribute this behaviorto a thin wetting film spread on the droplet tra-jectory, which would reduce the effective fric-tion (and therefore the generated current) andcould have such slow dynamics. In this sce-nario, increasing the frequency is not equivalentto increasing the oscillation amplitude.

B. Wrinkles are current amplifiers

As discussed previously, our sample fabrica-tion techniques allow us to produce graphiteflakes of desired thickness with either a highwrinkle density (transferred flakes) or a veryflat surface showing negligible roughness (un-transferred flakes). We were therefore ableto study the dependency of generated currenton both thickness and wrinkle density inde-pendently. On Figure S7 a, we illustrate thethickness-dependency of the current induced bydroplet oscillations of 1 µm.s−1 for both trans-ferred (diamonds) and un-transferred (circles)graphite flakes. Our experiments reveal that fortransferred samples, the thinner the graphiteflake, the more efficient the current generation.Primarily, we observe a transition for the cur-rent generation around 10 nm in thickness: forflakes thicker than 10 nm, no current genera-tion is possible, while current can be inducedat the surface of flakes thinner than 10 nm. Onthe other hand, we report no generated currentfor flat un-transferred samples down to singlenanometer thickness.

Our results demonstrate that wrinkles are aprerequisite to the viscous flow induced currentgeneration. Another experimental evidence ofthe predominant role played by the wrinkles re-sides in the strong spatial anisotropy of the gen-erated current (current depends on oscillationdirection). As shown in Figure S8, the direc-tional dependency depends strongly on the po-sition on the sample surface. This variabilityis linked to the random distribution in height,position and direction of the wrinkles over the

5

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Figure S3: a, Current as a function of time, for droplet oscillations of 2 µm amplitude and 0.5Hz frequency. b, Current spectrum as a function of frequency. The spectrum displays a peak atthe droplet oscillation frequency (0.5 Hz). The peak maximal value is defined as the generated

current amplitude.

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tungsten tipIL droplet

SiO2Graphite

Figure S4: a Current spectrum (nA) as a function of the frequency (Hz) while the capillary tipmoves at 5 µm and 0.5 Hz in amplitude and frequency, 0.5 µm (clear blue curve) and 1.4 µm

(dark blue curve) above the graphite, and in contact with the graphite via the ionic liquid droplet(red curve). b Current spectrum (nA) as a function of the frequency (Hz) for a droplet motion of

2 µm amplitude and 0.5 Hz frequency, on graphite (red curve) and on SiO2 (blue curve).

graphite surface.

a. Raman spectroscopy To strengthen ourclaim that wrinkles are folded graphene struc-tures that increase liquid friction and thusmotion-induced current by a mere geometricaleffect, we perform Raman spectroscopy on onetransferred graphite sample and demonstratethat wrinkles show a ratio of D-peak intensityover G-peak intensity of the same order as flat-ter regions and that they show the usual spec-troscopic properties of graphite samples. Themeasurements are presented on Figure S9. Ona, we show the AFM scan of a transferred sam-ple. On this sample, the transfer process hastorn appart a top layer of graphite which istherefore showing wrinkles on top of the un-

derlying graphite crystal (dark region). We usethis particular sample to study the structuralproperties of wrinkles compared to a flat crys-tal. On b, we display a typical Raman spec-trum measured with a 532 nm laser on this sam-ple. Furthemore, on c, we display the G peakintensity map, revealing the in-homogeneity ofthickness in the folded regions (higher G peakmeans thicker region). Finally, on d, we showthe ratio of D-peak over G-peak intensity. Thisshows that wrinkles and folded regions are byno means defects. This analysis strengthen ourtheoretical treatment of wrinkles and folded re-gions as graphite or graphene with an enhancedcorrugation.

6

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Figure S5: a Current spectrum (nA) as a function of the frequency (Hz) for a micrometricsilicone 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 0.5Hz. 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 .

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35

40

Sample 3, a = 2µm, ∆V = 25mV

0.5 1.0 1.5 2.00

2

4

6

8

10

12

Sample 4, a = 2µm, ∆V = 20mV

Frequency (Hz)

Cur

rent

dens

ity

(nA

/µ

m)

Figure S6: Generated current amplitude as a function of the oscillation frequency. We show thedependency measured on three different samples (corresponding to those described in the maintext) namely samples 1, 3 and 4. The oscillation amplitude and bias are given above each plot.

III. THEORETICAL MODEL

A. Electron drag: Phonon drag vs.Coulomb drag

The electrons are dragged both by phonondrag and Coulomb drag. These two phenom-ena have recently been studied formally in theKeldysh framework by Coquinot et al. [5]. Ac-cording to the authors, they can be treated sim-ilarly as bosons, either phonons or liquid fluc-tuations of the drop (called hydrons), interact-ing with electrons. The liquid fluctuations dragthe electrons at the interfacial liquid velocity

v` and v` � vdrop as the slip length is smallcompared with the drop height. On the otherside, the phonons drag the electrons at the ef-fective velocity veff calculated in the main text:veff ≈ 9 ·104×(1+W )×vdrop. In the stationaryregime, the electrons reach an electronic veloc-ity

ve(q) =τq

τe/phq

veff +τq

τe/hq

v` ≈τq

τe/phq

veff (1)

Here τ e/b is the electron-boson interaction timeand τ is the total electronic scattering time(1/τ = 1/τ e/ph + 1/τ e/h). Therefore, the elec-tronic velocity is the convex combination of

7

100 101 102

Thickness (nm)

10°1

100

101

Con

duct

ivit

ys e

f(n

A.s

/µ

m2 )

Un-transferred samplesThick transferred samplesSample 1Sample 2Sample 3Sample 4

5 um

Un-

tran

sfer

red

flake 25 nm

05

10

15

20

25 nm

20 nm

15 nm

10 nm

5 nm

0

5 um

Flat 1.5 nm-thick sample

75 nm

0

10152025303540455055606570

5 um

Wrinkled 3.6 nm-thick sample

a

c

b

d

Sam

ple

3 (tr

ansf

erre

d fla

ke)

Thic

k tr

ansf

erre

d fla

ke

75 nm

0

20

30

40

50

60

5 um

Flat 68 nm-thick sample

Detection level

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

DV = 10mV

°30

°20

°10

0

10

20

30Thin transferred flakes

Figure S7: a Semilog plot of the normalized generated current given in nA/µm.s−1, as a functionof 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 flat68 nm-thick sample (transferred). No wrinkles are induced and thus no current is measured. Onsamples thicker than 10 nm, no wrinkles are observed and the droplet oscillations do not result inthe generation of a measurable current. Similarly, on the thinner samples, current generation

requires the wrinkles.

0 50 100 150

Scan angle (°)

0

5

10

15

20

Curr

ent (n

A)

a bPosition 1Position 2Position 3

Figure S8: a Scanning electron microscopy image of a graphite sample. Current is generated bydroplet motion of constant amplitude and frequency, along axis forming tilted with respect to theelectrode axis by the ’scan angle’. We stress that a is a schematic and that our in-situ microscopedoes not allow such a fine positioning. Here, as detailed on b, we merely show that the angular

dependency of the generated current also depends of the region of the sample that we arescanning. b Generated current (nA) as a function of the scan angle (◦) of droplet motion (at 0.5Hz frequency and 1 µm amplitude), around various central positions. As expected, the current is

a 180◦-periodic function of the oscillations angle, but interestingly the phase depends on thecentral positions of the oscillations. For each location, the experimental points are pictured by

circles, while the solid line is the best sinusoidal fit.

8

5µm

99 nm

820

30

40

50

60

70

80

90

a

Tran

sfer

red

flake

AFM Scan

c

1600

1700

1800

1900

2000

2100

2200

2300

1600

1700

1800

1900

2000

2100

2200

2300

3 um3 um

G peak intensity d D/G peak intensity ratio

0.010

0.012

0.014

0.016

0.018

0.020

0.010

0.012

0.014

0.016

0.018

0.020

3 um

Raman spectrum (532 nm laser)b

1400 1600 1800 2000 2200 2400 2600 2800

Raman shift (cm�1)

0

500

1000

1500

2000

Phot

onco

unts

G

D

2D

Figure S9: A transferred graphite sample shows wrinkles as discussed above and shown by AFMimaging in a. On b, a typical Raman spectrum is shown. The Raman G-peak intensity is

displayed on c and is stronger in the thicker regions. On d, the intensity ratio between the D andG peak is displayed. The low value (1%) of this ratio indicate a very low level of defects.

Moreover, wrinkles, as opposed to flake edges do not display any local D-peak enhancement. Thisindicates that wrinkles are indeed folded graphene structures and neither pollution nor defects.

the phonon effective velocity and the interfa-cial liquid velocity. In practice, in our model,τq/τ

e/phq ≈ 1/2. In Eq. (1) the first term de-

scribes the effect of the phonon drag while thesecond term describes the effect of the Coulombdrag. Let us notice that the Coulomb drag ef-fect reduces the electronic velocity: it then con-tributes negatively to the electric current gen-eration.

B. Electronic state and electric current

In stationary regime, the electronic distribu-tion of band dispersion εq is shifted accordingto εq 7→ εq − ~q · ve. Therefore, the distribu-tion will be filled above the doping level εF inthe velocity direction, and below in the opposi-tion direction. Since the solid is 2D, the angleintegration will provide a trigonometric factor1/2. The distribution is not at zero tempera-

ture, thus the energies close to the doping levelup to the thermal energy matter. Moreover, theinteraction broadens the electronic distributionaround the band structure. According to [5],in stationary regime, the electronic distributionreads

ne(q, ω) =2τ/~× nF(ω − q · ve(q))

1 +(τ~)2

(~ω + εq − εF)2(2)

where nF is the Fermi-Dirac distribution andτ is the electronic scattering time. Therefore,taking into account the spin degeneracy, the au-thors of [5] predict an electric current

〈j〉 = 2e

∫dεdω

(∇qεq)εN(ε)~qεve

4kBT ch2[~ω+ε−εF

2kBT

] τε/π

1 + τ2ε ω

2

(3)where ε describes the band energy levels. Here,(∇qεq)ε is the band velocity and qε the momen-tum both at the band energy level ε. Going to

9

the limit of small interaction and low tempera-ture ~τ−1 � kBT � εF, we recover Eq. (2) ofthe main text.

C. Model for the solid

Using a linear dispersion relation to describegraphene yields an energy-dependent density ofstates. When the doping vanishes we expectno current because of the electron-hole symme-try. For large dopings εF � kBT , only oneband matters and the electric current writes〈j〉 ≈ eveε

2F/(~vF)2, where vF = 106 m/s is the

Fermi velocity of graphene. The current is thenquadratic in doping, which is consistent withthe total average charge of the charge carriers.At smaller doping, temperature effects appear

and a linear regime exists when εF � kBT .Experimentally, most measurements are in thisregime but one may expect nonlinear effects forthe higher bias voltage applied. As no such ef-fect is measured in practice the relevance of thismodel may be questioned. Moreover, numericalapplications lead to an electric current 1 orderof magnitude smaller than experimentally mea-sured.

To describe more precisely our samples, weused a model for multi-layer graphene made oftwo independent parabolic bands with effectivemass m [6–8]: εq = ~2q2/2m. In particular,the density of states does not depend on thedoping, which is key to get a linear dependencefor the current. The predicted current is then〈j〉 ≈ 2evemεF/~2 from which is deduced Eq.(4) of the main text.

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