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Aaron H. Barajas-Aguilar, Jasen Zion, Ian Sequeira, Andrew Z. Barabas, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Eric B. Barrett, Thomas Scaffidi, Javier D. Sanchez-Yamagishi

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[Electrically driven amplification of terahertz acoustic waves in graphene](https://mdr.nims.go.jp/datasets/31ce91eb-7fa4-4b37-8e76-e2e3b5ce9d67)

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Electrically driven amplification of terahertz acoustic waves in grapheneArticle https://doi.org/10.1038/s41467-024-46819-2Electrically driven amplification of terahertzacoustic waves in grapheneAaron H. Barajas-Aguilar 1, Jasen Zion2, Ian Sequeira1, Andrew Z. Barabas1,Takashi Taniguchi 3, Kenji Watanabe 3, Eric B. Barrett1, Thomas Scaffidi1 &Javier D. Sanchez-Yamagishi 1In graphenedevices, the electronic drift velocity can easily exceed the speedofsound in the material at moderate current biases. Under these conditions, theelectronic system can efficiently amplify acoustic phonons, leading to anexponential growth of sound waves in the direction of the carrier flow. Here,we show that such phonon amplification can significantlymodify the electricalproperties of graphene devices. We observe a superlinear growth of theresistivity in the direction of the carrier flow when the drift velocity exceedsthe speed of sound— resulting in a sevenfold increase over a distance of 8 µm.The resistivity growth is observed at carrier densities away from the Diracpoint and is enhanced at cryogenic temperatures. We develop a theoreticalmodel for the resistivity growth due to the electrical amplification of acousticphonons — reaching frequencies up to 2.2 THz — where the wavelength iscontrolled by gate-tunable transitions across the Fermi surface. These findingsprovide a route to on-chip high-frequency sound generation and detection inthe THz frequency range.Sound waves are important as high-frequency signal carriers and as ameans to distort crystal lattices in space and time. Due to the slow speedof sound compared to light, ultrashort sound wavelengths in the nan-ometer scale are attainable in the terahertz (THz) domain, the highestsound frequencies in solids. The control and generation of THz soundwaves offers a route to nanoscale ultrasonic imaging, the generation ofTHz electromagnetic radiation1–3, and the dynamic modulation of elec-tronic behaviors4–7. However, an electrical on-chip source of THzacoustic waves remains elusive. Coherent THz sound waves have onlybeen achieved via ultrafast optical pumping8,9, while conventional pie-zoelectric transducers produce maximum frequencies of ~10 GHz10.Accelerated electrons can generate and amplify traveling waves.Famous examples include the traveling wave amplifier11 and Cher-enkov radiation. The common gain condition is that the electronvelocity exceeds thewave phase velocity. In solids, an analogous effectoccurs when the carrier drift velocity (vD) exceeds the speed of sound(vS), resulting in acoustic wave amplification12,13. Unlike the freeelectron case, the amplification has a characteristic wavelength givenby transitions across the Fermi surface (Fig. 1a). Such acoustic ampli-fication has been studied in bulk semiconductors14,15 and has recentlybeen used to make nonreciprocal acoustic amplifiers operating in thegigahertz frequency range16.Two-dimensional van der Waals (vdW) materials present manyadvantages for acoustoelectric devices because phonons are naturallyconfined to atomic layers, leading to long lifetimes17 andmore efficientcoupling to electrons18. Acoustic waves offer a way to dynamicallymodulate lattice strain in both space and time, which can couple todiverse vdW heterostructure phenomena19, as well as to quantumdefects20. Despite this interest, acoustic studies in vdWmaterials havebeen limited by the challenge of generating propagating sound wavesat high frequencies, as the confinednatureof the two-dimensional (2D)phonons makes it difficult to excite with external transducers.Graphene is an attractive host for the electrical amplification ofacoustic phonons. Its high Fermi velocity and electron mobility (μ)Received: 16 October 2023Accepted: 12 March 2024Check for updates1Department of Physics and Astronomy, University of California, Irvine, Irvine, CA, USA. 2T. J. Watson Laboratory of Applied Physics, California Institute ofTechnology, Pasadena, CA, USA. 3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan.e-mail: javier.sanchezyamagishi@uci.eduNature Communications |         (2024) 15:2550 11234567890():,;1234567890():,;http://orcid.org/0000-0002-0432-5924http://orcid.org/0000-0002-0432-5924http://orcid.org/0000-0002-0432-5924http://orcid.org/0000-0002-0432-5924http://orcid.org/0000-0002-0432-5924http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0001-9703-6525http://orcid.org/0000-0001-9703-6525http://orcid.org/0000-0001-9703-6525http://orcid.org/0000-0001-9703-6525http://orcid.org/0000-0001-9703-6525http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46819-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46819-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46819-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46819-2&domain=pdfmailto:javier.sanchezyamagishi@uci.edumeans that large drift velocities (vD = μE) can be achieved at relativelysmall electric biases. Under a current bias, the drifting Fermi dis-tribution results in an effective population inversion with an energydifference ΔE = ℏvD2kF between forward and backward propagatingcarriers (at zero temperature), where kF is the magnitude of the Fermiwavevector.When vD > vS (vS-TA = 13 km/s and vS-LA = 21 km/s for TA andLA phonons, respectively21), electrons can relax energy and momen-tum by emitting and amplifying waves via inelastic backscattering(Fig. 1a). Importantly, over a large range of parameters, the only exci-tations that graphene can generate are acoustic waves22,23, which are inthe THz frequency range (Supplementary Fig. 6). Evidence for acousticphonon amplification has come fromnoisemeasurements in graphenedevices24. However, the direct effects of THz acoustic waves on theelectronic properties of materials are still unexplored.Here, we study the transport behavior of clean graphene devicesas a function of voltage bias at cryogenic temperatures. In contrast toprevious bias studies25–28, we measure the position-dependent resis-tance, focus on gate voltages away from the Dirac point so interbandtransitions are suppressed, and apply moderate source-drain voltagebiases to avoid optical phonon generation. Our primary findings arethat graphene resistivity grows strongly in the direction of carrier flowwhen the drift velocity exceeds the speedof sound.Our results arewellexplained by the electrically-induced amplification of terahertzacoustic waves and their associated strong scattering of grapheneelectrons.ResultsTo study the spatial dependence of the graphene resistance, we fab-ricated long graphene devices encapsulated in hexagonal boronnitride (hBN) with equally spaced voltage probes along the channellength (inset Fig. 1c). A DC source-drain voltage bias is applied to thedevice, and by measuring the voltage difference between adjacentelectrodes we probe the spatial distribution of the voltage drop acrossthe channel as a function of the current flowing between the sourceand drain electrodes. The resulting V–I curves are shown in Fig. 1c (toppanel), measured at an electron carrier density n = 1.4 × 1012 cm−2 andcryostat temperature of 1.5 K. At low currents, all curves show linearOhmic behavior corresponding to an average resistivity of 19.7Ω/square with less than 6% deviations across the sample. The averagecarrier mobility is 2.3 × 105 cm2/V*s. At higher current magnitudes, allthe curves deviate strongly from Ohmic behavior with a differentialresistance (dV/dI) that grows superlinearly with the magnitude of thesource-drain current (Fig. 1c, bottom panel).Strikingly, the nonlinear resistance is highly asymmetric with thecurrent direction and strongly position-dependent. The largest asym-metry and nonlinearity are found for the measurements closest to thesource and drain electrodes (blue and orange lines Fig. 1c). Forexample, for contacts 1–2, the differential resistivity at −1mA is 44Ω,but at +1mA it rises to 612Ω, a factor of 14× difference for oppositecurrent directions. Measurements on the opposite side of the device(5–6) produce similar nonlinear curves but with opposite dependenceon the current direction.The position dependence of the graphene resistivity showssuperlinear growth in the direction of carrier flow (Fig. 1d): For elec-tron-doping, the resistivity growth is opposite to the current; for hole-doping, growth is in the direction of the current. The growth is sub-stantial at moderate current densities j = 0.34mA/μm, leading to a 7×increase in resistivity and 12× increase in differential resistivity overFig. 1 | Acoustic phonon amplification and observation of resistance growth inthe direction of carrier flow. a Top: Schematic of the graphene electronic dis-tribution with a drift velocity (vD). The blue shaded region shows the occupiedstates. The Fermi surface is tilted such that the energy difference between right andleft moving carriers is ħvD2kF. When vD > vS, the carriers can backscatter and emit aphonon of energy ħvS2kF (transition is indicated by the orange arrow). Bottom:device under the phonon amplification conditions, the phonon population growsexponentially with distance in the direction of the carrier flow. b Resistivity versuselectron carrier density for device A, the absence of satellite peaks indicates thatthe graphene and hBN layers of the device are unaligned. c top: Voltage difference(ΔV) between pairs of consecutive contacts vs. source-drain current for device A.The voltage differences ΔV1-2 (blue) and ΔV5-6 (orange) exhibit the largest non-Ohmic behavior (n = 1.4 × 1012 cm−2), bottom: differential resistivity vs. source-draincurrent for the outermost pairs of contacts (ΔV1-2 and ΔV5-6) showing asymmetricnon-Ohmic behavior. Inset: the optical image of device A with a 13μm length, 3μmwidth, and center-to-center distance between voltage probe contacts of 2 μm. Thecolored bars label the pairs of contacts used to measure the voltage differencesplotted in the top and bottom panels. d Position dependence of the resistivity atdifferent drift velocities for n = +1.4 × 1012 cm−2 (top panels) and n = −1.4 × 1012 cm−2(bottom panels). Lines are a visual aid connecting data points. The device cartoonsshow the carrier flow direction and carrier type for each case, the source contact ison the left, as in (c). The maximum drift velocity in the top panels is 154 km/s(j =0.34mA/μm), and 104 km/s in the bottom panels. Note that the maximum vDachieved for holes is lower than for electrons due to larger source-drain contactresistance. All measurements are performed at T = 1.5 K.Article https://doi.org/10.1038/s41467-024-46819-2Nature Communications |         (2024) 15:2550 28μm (Supplementary Fig. 1). As a result of the highly nonuniformresistivity at this bias, 81% of the graphene resistance occurs in the last46% of the channel length (Supplementary Fig. 2).The nonlinear resistance growth occurs for a wide range of carrierdensities. This is seen in Fig. 2a, where, for contacts 1–2, we observestrongly asymmetric dV/dI versus current curves for carrier densitiesranging from 0.6 × 1012 to 3.5 × 1012 cm−2. For all densities, the resis-tance is larger in the direction where the carriers travel longer dis-tances in the device (downstream, Fig. 2a).When plotting dV/dI versus drift velocity (vD = j/ne) (Fig. 2b), thecurves with carrier densities above 1.1 × 1012 cm−2 collapse together,suggesting that the physics of this phenomenon is dictated by adrifting electronic carrier distribution. On a logarithmic scale plot(Fig. 2b inset), the normalized differential resistivity versus vD shows asharp transition from constant to a growing non-Ohmic behavior. Thethreshold drift velocities, which we define as a 1.5× increase in differ-ential resistivity, vary with carrier density from 21.8 to 83.3 km/s.Notably, non-Ohmic behavior is only observed above the lowest gra-phene sound velocity (vS-TA = 13 km/s). The sharp transition betweenthe Ohmic and non-Ohmic regimes can be observed for any pair ofcontacts in the device (Supplementary Fig. 3).When the voltage drop ismeasured closest to the carrier injectionpoint (upstream, Fig. 2c), the resistance versus carrier density curvesfollow the typical graphene Dirac peak response with a weak depen-dence on drift velocity, indicating mostly Ohmic behavior. When thecarrier flow is reversed, the carriers travel 10.5 µm before reaching themeasuring contacts (downstream, Fig. 2d). Here, for drift velocitieslower than vS, the line traces show a typical graphene response—decreasing in resistance with increasing carrier density magnitude(Fig. 1b). But, when vD is larger than the speedof sound (light blue line),the differential resistivity instead grows rapidly with carrier density forn larger than 0.4 × 1012 cm−2, surpassing the value of the resistance atthe Dirac peak. This effect can be seen symmetrically for both electronand hole carriers. As the carrier density increases, we observe a peak inthe dV/dI at n ~ 2 × 1012 cm−2 and an eventual downturn at higher elec-tron doping. Similar behavior is observed for all the other pairs ofcontacts along the device (Supplementary Fig. 4).The resistance growth is most prominent at cryogenic tempera-tures. Figure 3a displays the differential resistivity versus drift velocitymeasured for contacts 1–2 from T = 1.5 to 280K. At T= 1.5 K, a highlyasymmetric curve is obtained with a large nonlinear resistivity growthwhen measuring far from the carrier injection point (downstream,−2 −1 0 1 2Current (mA)0200400600800−300 −150 0 150 300− (km/s)vD100 101 102| |vD100101R'/R' 0−3 −2 −1 0 1 2 3Carrier density (10 cm )12 −20100200300400500600700DifferentialResistivity(Ω)−3 −2 −1 0 1 2 3n (1012 cm-2)0.59 3.49 vSvS322 200 |vD|(km/s)adcbh+h+e- e-e- e-e- e-Fig. 2 | The resistance growth is sensitive to carrier density and only occurswhen vD > vS.Differential resistivity as a functionof current (a) anddrift velocity (b)at different carrier densities. The shadowed regions indicate drift velocitiesbetween 13 and 21 km/s, corresponding to the speed of sound for TA and LAphonons, respectively. Inset: Logarithmic plot of differential resistivity normalizedto the value at vD = 0. c, d Same differential resistivity data plotted vs. carrierdensity for constant drift velocities (positive and negative for (c) and (d),respectively). The device cartoons indicate the carrier type and flow direction, aswell as the contacts being measured in each case (contacts 1–2 for all the panels).The highlighted blue traces correspond to a drift velocity of 32 km/s, which is thelowest drift velocity above vS-LA = 21 km/s shown in this plot. All measurements areperformed at T = 1.5 K. Note that data were taken within a maximum source-drainvoltage of ±0.6 V. Hence the curves appear with different ranges when plottedversus current, vD, or for constant values of vD.Article https://doi.org/10.1038/s41467-024-46819-2Nature Communications |         (2024) 15:2550 3negative vD). As the temperature increases, the nonlinearity and asym-metry are steadily reduced but are still evident even at T=280K. Cor-respondingly, the resistance dependence on temperature shows anopposite trend for different current bias directions (Fig. 3b). In anupstream configuration (top panel Fig. 3b), a steady increase of theresistivity with temperature is observed as is typical for graphene at lowbiases, consistent with increased scattering from thermally-occupiedacoustic phonons29,30. For the downstreammeasurement (bottom panelFig. 3b), a similar behavior is observed for small drift velocities. But, as vDincreases beyond 37 km/s (blue trace), the opposite dependence isobserved, with resistivity decreasing as the temperature rises above40K. At 1.5 K, the effect of drift velocity on resistivity is 3.5 times greaterthan the effect of heating from 1.5K to 280K.To summarize, we observe the graphene resistivity to growsuperlinearly in the direction of carrier flow when the drift velocityexceeds the speed of sound. The resistance growth is suppressed atlow carrier densities, and it is strongest at cryogenic temperatures.These observations were made in two graphene devices where thegraphene is misaligned with its encapsulating hBN layers (Supple-mentary Figs. 8–10). The directional resistance growth is notobserved in devices where the graphene and hBN form a moirésuperlattice (Supplementary Fig. 11) or where there is too muchdisorder (Supplementary Fig. 12).DiscussionIn low-disorder conductors, acoustic phonons are the simplest low-energy excitation that can relax the momentum of electrons acceler-ated by electric fields and create resistance. At low temperatures, thisform of dissipation is only unlocked when the drift velocity exceedsthe speed of sound due to energy-momentum conservation. This canbe understood from the form of the tilted Fermi distribution in adrifting rest frame fD(ϵ) = 1/(1 + exp(ϵ − µ − ħvD·k)/kBT), where theeffective population inversion between forward and backwardmovingcarriers enables electrons to backscatter by emitting acoustic phononsin the direction of the carrier flow, with characteristic wavevectorskphonon ~ 2kF and energy ħvSkphonon (Fig. 1a).In this work, the range of drift velocities where the resistivitygrowth is observed points to an acoustic-phonon mechanism. Thethreshold behavior for the drift velocity is particularly sharp, with littlechange to the local resistivity (<15%) and highly symmetric V-I curvesuntil the sound velocity is exceeded (Fig. 2b).When vD = vS-LA, the tilt ofthe Fermi distribution isħvD2kF = 5.8meV atn = 1.4 × 1012 cm−2. At theselow energy scales, acoustic phonons are the only excitation availablefor inelastic energy transfer. At the highest drift velocities that weprobe at this carrier density (200 km/s), the energy tilt is 55meV and isnot sufficient to directly excite the lowest energy optical phonons ofthe device (102meV for hexagonal boron nitride31).Typically, electron-phonon scattering is considered solely asource of dissipation, leading to local heating of the electron andphonon distributions. However such Joule heating effects would notproduce the strongly asymmetric resistance profile that we observe(see Methods section on heating effects). Instead, if the acousticphonons are long-lived, theywill propagate downstreamand stimulatethe emission of additional phonons, producing an exponential growthin the direction of carrier flow. The only condition for growth is thatthe net rate of phonon emission exceeds the phonon decay rate, thelatter being small at cryogenic temperatures32. The phononpopulationgrowth will be mirrored in the resistivity, as each phonon emissionoccurs with an electron backscattering. Thus, acoustic-phononamplification results in an exponentially growing resistance in thedirection of carrier flow when vD > vS.We calculate the phonon amplification rates and effects on thegraphene resistance using a model of the driven electron–phonondynamics across the channel length (seeMethods and SI Section 5).Wefind a cone of phonon modes in the direction of the drift velocity thatwill be amplified, resulting in exponential growth in thedirectionof thecarrier flow as exp(Γk*(x/vs)), where Γk is the amplification rate formode k, and x is the position. Γk reaches levels of 1–40GHz for vD−160 −80 0 80 160V (km/s)D0100200300400500600700DifferentialResistivity(Ω)10305070Resistivity(Ω)0 40 80 120 160 200 240 280T (K)1030507090110130150|vD|3 km/s 140 km/s|vD|3 km/s 140 km/se-e- e-e-a b80 K T (K)1.551020304080100120160200240280Fig. 3 | Resistance due to amplified acoustic phonons is largest at low tem-peratures. a Differential resistivity vs. drift velocity wasmeasured for contacts 1–2at different temperatures from 1.5 to 280K at n = 1.4 × 1012 cm−2. b Resistivity forcontacts 1–2 as a function of temperature for constant drift velocities. The bluearrows indicate the direction of growth of the vD magnitude. The blue tracescorrespond to vD = 37 km/s, from which the temperature dependence of theresistivity inverts when the carriers move downstream. The excess resistivityinduced by high vD at 1.5 K is 130Ω. The excess resistivity induced by heating from1.5 K to 280K for vD = 3 km/s is 37Ω.Article https://doi.org/10.1038/s41467-024-46819-2Nature Communications |         (2024) 15:2550 4values from2 to 10*vS-LA, with a broadmaximumnear k = kF along the xdirection (Supplementary Fig. 13). Extending beyond previous works,we calculate the spatial growth rate of the resistivity, and find it to bewell approximated by the peak value of Γk/vS with values of0.3–2.5μm−1 (Supplementary Fig. 14). Such micron-scale growthlengths are comparable with our experimental observations. For thedata measured at vD = 7.33*vS-LA (154 km/s), n = 1.4 × 1012 cm−2, theobserved trend iswellfit by anexponentialwith a characteristicgrowthrate ~0.32μm−1 (Supplementary Fig. 2). This value is 5× less than thetheoretical calculation, which is not surprising given that the modelneglects phonon lossmechanisms such as anharmonic decay and edgescattering.The nonmonotonic dependence of the resistivity growth on car-rier density can also be understood within the phonon amplificationmodel. Near the Dirac point, we expect phonon amplification to besuppressed due to competing pathways for energy relaxation viainterband excitations, such as electron–hole generation, when theFermi tilt is comparable to the Fermi energy27. Moreover, theelectron–phonon coupling and amplification rates are reduced withsmaller kphonon ~ kF = √πn (Supplementary Fig. 13). This explains why atlower carrier densities, higher drift velocities are required to observethe phonon amplification effect (Fig. 2b inset). Conversely, as thecarrier density increases, the larger kphonon will have stronger electron-phonon couplings and subsequent higher rates of amplification andresistive scattering with electrons24,29. The larger Fermi surface alsoamplifies a larger range of modes, further increasing the resistivitygrowth. Anopposing effect is the increaseof the phonondecay rates atlarger phonon wavectors33, either due to anharmonic decay or short-range disorder, which will lower the amplification rates at higher car-rier densities. Our data is in agreement with these aspects of thephonon amplificationmodel, where we observe a sharp increase in theresistivity growth aswedope away from theDirac point, amaximumatn ~ 2 × 1012 cm−2, and a downturn for higher n values (Fig. 2d).The requirement that the phonon emission rate exceeds thedecay rate for net amplificationmakes the process also sensitive to theoverall lattice temperature. The phonon decay rate will increase withthermal mode occupation due to anharmonic processes, which willsuppress the amplification process. Indeed, the resistance growth isstrongest at T = 1.5 K and is only suppressed when the temperatureexceeds the energy scale of the amplified modes (ħvS2kF/kB = 40K forn = 1.5 × 1012 cm−2) (Fig. 3b). The unique end result is a metallic con-ductorwith a larger resistance at cryogenic temperatures than at roomtemperature when biased.Other processes that scatter acoustic phonons will also suppressphonon amplification. This explains why we do not observe the resis-tance growth in disordered devices (i.e., with low electronic mobility,Supplementary Fig. 12) or with a moiré superlattice potential (Sup-plementary Fig. 11). In the case of aligned graphene-hBN devices, themoiré superlattice can be as large as 14 nm, which is comparable to thewavelengths of the emitted phonons (9–40 nm). As such, the super-lattice will induce phonon–phonon Umklapp scattering, which willreduce the phonon lifetime and the amplification effect34.In summary, we demonstrate an unprecedented directional growthof the graphene resistance induced by electrically amplified acousticphonons. Our results have important implications for graphene appli-cations, especially at high current densities or over long distances wherephonon amplification is likely to be a limiting factor. At the same time,these observations show the unique potential of high-frequencyacoustic waves to remotely modulate the electrical properties of amaterial. The strong modulation of the graphene resistance is onlypossible due to the large wavevectors of the generated phonons, whichcan backscatter electrons across the Fermi surface. Low wavevectorphonons with kphonon≪ 2kF, as would be predominantly generated by athermal pulse, can only induce small-angle scattering and hence weaklyaffect the resistivity. For similar reasons, traditional acoustic-electronicstudies, which use surface acoustic waves with wavelengths ≫ 100nm,can only act as slow and long length-scale perturbations for carrierdensities above 1 × 1011 cm−2 (λF < ~100nm). In our devices, we estimatethat we are predominantly amplifying acoustic phonons with wave-lengths of 40nm down to 9nm. Interestingly, such length scales arecomparable to the moiré superlattices found in twisted van der Waalsheterostructures35,36, motivating future work studying the dynamicspatiotemporal strain effects of acoustic waves.The characteristic frequencies of the amplified acoustic phononsare in the terahertz range (0.3–2.2 THz) and are tunable with the gra-phene Fermi energy Ephonon ~ (vS/vF)EF (Supplementary Fig. 6). Cur-rently, there are no alternative demonstrations of electrical generationof terahertz acoustic phonons. Transducing the mechanical motion ofthe acoustic wave to an electric field would offer a route toward aterahertz electromagnetic source. Acoustic amplification should beobservable in other high-mobility vdW materials, such as transitionmetal dichalcogenides37, where intrinsic piezoelectricity can convertthe sound waves into electric waves. Lastly, Cherenkov phononamplification in graphene offers a unique route to the electrical gen-eration of other high-frequency and largewavevector excitations in 2Dheterostructures, such as acoustic plasmons38 and magnons39, whichare otherwise challenging to source and probe.MethodsFabricationAll graphene and hBN layers were exfoliated from bulk crystals. Stackswere fabricated by the dry transfer method40 using stamps made of apolycarbonate (PC) film on top of a polydimethylsiloxane (PDMS)square on a glass slide. All the lithographic processes were made byelectron beam lithography (EBL) using a layer of poly(methyl metha-crylate) (PMMA) resist. To write the patterns for the one-dimensional(1D) edge contacts, PMMA 950 A5 was spun for 2min at 2000 rpmproducing a ~500 nm thick layer. The EBL patterns were written at1.6 nA with 30 kV excitation and then developed for 2–3min in a coldmixture of 3:1 isopropyl alcohol (IPA)/water. After writing and devel-oping the patterns, reactive ion etching (RIE) was used to expose thegraphene with the following parameters: a flow of 10 standard cubiccentimeters per minute (SCCM) of SF6, 2 SCCM of O2, and 30W ofradio frequency power, at 100mTorr for 30 s41. Then, 3 nm of Cr and100nm of Au were deposited in an electron beam evaporator systemat 1 Å/s. Liftoff was performed by soaking the sample in acetone for1–2 h and agitating with a pipette. To define the geometry of thedevice, amaskwas written with EBL, and finally, a two-step RIE processwasmade,first anSF6 etchingusing the sameparameters described forthe 1D edge contacts and then anO2 etching with a flow of 20 SCCMofO2, 30W of radiofrequency power at 70mTorr for 15 s.Device measurements detailsThe devices weremeasured in a variable temperature cryostat. For thetransport measurements, a source-drain DC voltage bias was appliedto the devices while measuring the sourced current using a Keithley2400 SMU. The voltage drop between consecutive pairs of contactswas measured using digital multimeters. A gate voltage (VG) wasapplied to the silicon back gate to control the carrier density. Tocalibrate the gate capacitancevalue fordeviceA in themain text,whichdetermines the calculated values of the drift velocity, the Landau fanfeatures up to B = 3 T were measured and fitted. Using the measuredcapacitance and thicknesses of the dielectric layers, we extract a valuefor the out-of-plane dielectric constant of hBN ε॥ = 3.44, which agreeswith the reported value42. This dielectric constant value for hBN wasused in the data analysis for the rest of the devices.Data analysisFrom themeasurements, we obtain a dataset ofV-I curves for eachpairof consecutive contacts over a range of applied gate voltages (−50 toArticle https://doi.org/10.1038/s41467-024-46819-2Nature Communications |         (2024) 15:2550 550 V). From thesedata,we calculate the local differential resistivity andlocal resistivity between adjacent contacts, which we use to plot thespatial resistivity profile of the device (Fig. 1c bottom and Fig. 1d). Wealso calculate the carrier density n =CVG/e (where C is the gate capa-citance per area and e is the electron charge) and the drift velocityvD = j/ne for each data point, which is used to create Fig. 2. To producecurves of constant vD values, we use 1D interpolation to determine thedifferential resistance for equally-spaced vD values (Fig. 2c, d). Thisprocedure is well-behaved, as the differential resistance is a smoothfunction of vD (Fig. 2b).Uncertainty analysisThe maximum uncertainties for measurements of the current andvoltages are 0.1% and 0.2%, respectively. This corresponds to a max-imumuncertainty of 0.2% for the resistance anddifferential resistance.Considering a 100 nm accuracy in the dimensions of the device, theuncertainty for the resistivity and differential resistivity is 6%. Theuncertainty in the carrier density is determined by the 5% uncertaintyin the dielectric constants of the hBN and SiO2. This corresponds to amaximum uncertainty of 6% for the carrier density and 7% for the driftvelocity. In summary, none of these uncertainties affect the conclu-sions reached in this work.Theory calculationsTo calculate the resistance in a long graphene channel under a currentbias, we assume a drifting Fermi–Dirac distribution with drift velocityvD for the electrons in order to calculate the phonon amplification rateΓqamp due to electron-phonon coupling via the deformation potential.We then use it to calculate the position-dependent out-of-equilibriumphonon distribution in the sample. Finally, we find the position-dependent electric field needed to sustain the assumed vD based onthe electronic Boltzmann equation in which the phonon distributionenters through the electron-phonon scattering integral.Our primary assumption is that of a drifting Fermi–Dirac dis-tribution for the electrons, with a nominal temperature that is vD-independent. In the actual system, we would expect that Joule heatingwould lead to increased electronic temperatures at high vD. We alsoneglect phonon loss mechanisms such as disorder scattering oranharmonic decay, which will limit the phonon amplification processat large temperatures and large phonon population levels. As such, themodel is best at capturing the qualitative aspects of phonon amplifi-cation, especially at lower drift velocities. For a detailed explanation ofthe theoretical calculations, see Supplementary Note 5.Relation to previous noise studies of acoustic-phonon amplifi-cation in grapheneA previous work by Andersen et al.24 provided evidence for acoustic-phonon amplification in graphene via noise measurements and byextracting phonon transit timescales from AC transport measure-ments. Our study is differentiated by showing how acoustic-phononamplification can directly modify the graphene DC resistance, leadingto dramatic and easily measurable effects that are spatially varying.Other mechanisms for spatially-dependent resistance athigh biasLarge source-drain voltage biases can lead to a spatially dependentcarrier density across the channel26. This can lead to a spatially varyingdevice resistance which is significant if the voltage bias is comparableto the gate voltage. In ourmeasurements, themaximumapplied biasesare 0.6 V, leading to a Δn ~ 4 × 1010 cm−2 change in the carrier densityacross the device channel. This is small (1%) compared to the muchlarger gate voltages and carrier densities, which are the focus of thepresented data (VG = ± 50V and n ~ ± 3.3 × 1012 cm−2), and thus cannotexplain the observed resistance growth (Supplementary Fig. 7).Joule heating due to resistive dissipation can lead to a spatially-dependent temperature profile with corresponding spatially-dependent resistances and thermovoltages. In a diffusive system,Joule heating depends only on themagnitude of the current, leading toa temperature profile that is symmetric with the current. The longdevices that we measure are in the diffusive regime, as the electronmean free path is substantially smaller than the channel length due toedge scattering (sample width = 3 µm, length = 13 µm) and will getsmaller as the electron-acoustic phonon scattering increases whenvD > vS. 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Express 9,065901 (2022).AcknowledgementsThe authors acknowledge the use of facilities and instrumentation at theIntegrated Nanosystems Research Facility (INRF), in the Samueli Schoolof Engineering at the University of California, Irvine, and at the UC IrvineMaterials Research Institute (IMRI), which is supported in part by theNSFMRSEC through the UC Irvine Center for Complex and Active Materials.The authors also acknowledge the use of the UCI Laser SpectroscopyLab. The authors thank L. Jauregui, V. Fatemi, and E. Pop for productivediscussions, as well as the technical assistance of Q. Lin, R. Chang, M.Kebali, J. Hes, and D. Fishman. This work was partially supported by theNational Science Foundation Career Award DMR-2046849. A.H.B.A.acknowledges the University of California Institute for Mexico and theUnited States (UC MEXUS) for partial financial support. I.S. acknowl-edges fellowship support from the UCI Eddleman Quantum Institute.Author contributionsA.H.B.A., J.Z., I.S., andA.Z.B. prepared the samples. A.H.B.A., J.Z., and I.S.performed the device measurements. A.H.B.A., J.Z., and J.D.S.Y. ana-lyzed the experimental data. E.B.B. and T.S. developed the theoreticalmodel and performed the theoretical calculations. T.T. and K.W. grewand provided the hBN crystals. A.H.B.A., J.Z., I.S., A.Z.B., E.B.B., T.S., andJ.D.S.Y. discussed the findings and wrote the paper. J.D.S.Y supervisedthe project.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-46819-2.Correspondence and requests for materials should be addressed toJavier D. Sanchez-Yamagishi.Peer review information Nature Communications thanks Yang Liu, andthe other, anonymous, reviewer(s) for their contribution to the peerreview of this work. A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-46819-2Nature Communications |         (2024) 15:2550 7https://doi.org/10.1038/s41467-024-46819-2http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Electrically driven amplification of terahertz acoustic waves in graphene Results Discussion Methods Fabrication Device measurements details Data analysis Uncertainty analysis Theory calculations Relation to previous noise studies of acoustic-phonon amplification in graphene Other mechanisms for spatially-dependent resistance at high�bias Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information