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[Leonid A. Ponomarenko](https://orcid.org/0000-0003-1974-0642), Alessandro Principi, Andy D. Niblett, [Wendong Wang](https://orcid.org/0000-0003-1045-7170), [Roman V. Gorbachev](https://orcid.org/0000-0003-3604-5617), [Piranavan Kumaravadivel](https://orcid.org/0000-0002-9817-1697), Alexey I. Berdyugin, [Alexey V. Ermakov](https://orcid.org/0000-0001-9920-9549), [Sergey Slizovskiy](https://orcid.org/0000-0003-0131-0775), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Qi Ge, [Vladimir I. Fal’ko](https://orcid.org/0000-0003-0828-0310), [Laurence Eaves](https://orcid.org/0000-0002-5334-0987), [Mark T. Greenaway](https://orcid.org/0000-0003-3243-3794), [Andre K. Geim](https://orcid.org/0000-0003-2861-8331)

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[Extreme electron–hole drag and negative mobility in the Dirac plasma of graphene](https://mdr.nims.go.jp/datasets/780b1938-1284-4d01-83d7-becfadf845ac)

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Extreme electron–hole drag and negative mobility in the Dirac plasma of grapheneArticle https://doi.org/10.1038/s41467-024-54198-xExtreme electron–hole drag and negativemobility in the Dirac plasma of grapheneLeonid A. Ponomarenko 1,2 , Alessandro Principi2 , Andy D. Niblett1,Wendong Wang 3, Roman V. Gorbachev 3, Piranavan Kumaravadivel 3,Alexey I. Berdyugin2, Alexey V. Ermakov 2, Sergey Slizovskiy 2,3,Kenji Watanabe 4, Takashi Taniguchi 4, Qi Ge5, Vladimir I. Fal’ko 2,3,Laurence Eaves 6, Mark T. Greenaway 6,7 & Andre K. Geim 2,3Coulomb drag between adjacent electron and hole gases has attracted con-siderable attention, being studied in various two-dimensional systems,including semiconductor and graphene heterostructures. Here we reportmeasurements of electron–hole drag in the Planckian plasma that develops inmonolayer graphene in the vicinity of its Dirac point above liquid-nitrogentemperatures. The frequent electron–hole scattering forcesminority carriers tomove against the applied electric field due to the drag induced by majoritycarriers. This unidirectional transportof electrons andholes results innominallynegative mobility for the minority carriers. The electron–hole drag is found tobe strongest near room temperature, despite being notably affected byphononscattering. Our findings provide better understanding of the transport prop-erties of charge-neutral graphene, reveal limits on its hydrodynamic descrip-tion, and also offer insight into quantum-critical systems in general.If electron- and hole-doped two-dimensional (2D) conductors areplaced in close proximity to each other, Coulomb interactionsbetween charge carriers in adjacent layers lead to electron–hole drag(for review, see refs. 1,2). The drag was extensively studied usingvarious electronic systems based on GaAs heterostructures and,more recently, graphene1–12. The strength of Coulomb interactionrapidly increases with decreasing the distance between 2D systems,and the ultimately strong drag is expected if electrons and holescoexist within the same atomic plane. Graphene near its Dirac orneutrality point (NP) provides the realization of such an electronicsystem. Indeed, close to the NP, a finite temperature T leads tothermal excitations of electrons and holes, whereas their relativeconcentrations can be controlled by gate voltage. The resultingelectron–hole plasma is strongly interacting and represents a quan-tum critical system where particle–particle collisions are governedby Planckian dissipation13–23. The system is also often referred to asDirac fluid, assuming inter–carrier scattering dominates other scat-tering mechanisms. Because the Dirac plasma in graphene is a rela-tively simple and tunable electronic system, its behavior can beinsightful for understanding of electron transport in more complexPlanckian systems including “strange metals” and high-temperaturesuperconductors in the normal state24,25. There is also an interestingconceptual overlap with relativistic electron–positron plasmas gen-erated in cosmic events, which are difficult to recreate in laboratoryexperiments26. Previous experimental studies of the Dirac plasmareported its hydrodynamic flow20, the violation of theWiedemann–Franz law17, giant linear magnetoresistance23, and otheranomalies indicative of the quantum-critical regime18–23. So far, thepossibility of probing mutual drag between electron and hole sub-systems within the Dirac plasma has escaped attention.Received: 23 August 2024Accepted: 31 October 2024Check for updatesDepartment of Physics, University of Lancaster, Lancaster, UK. Department of Physics and Astronomy, University of Manchester, Manchester, UK. NationalGraphene Institute, University of Manchester, Manchester, UK. Research Center for Electronic and Optical Materials, National Institute for Materials Science,Tsukuba, Japan. Institute for Functional Intelligent Materials, National University of Singapore, Singapore, Singapore. School of Physics and Astronomy,University of Nottingham,Nottingham, UK. Department of Physics, LoughboroughUniversity, Loughborough, UK. e-mail: l.ponomarenko@lancaster.ac.uk;alessandro.principi@manchester.ac.uk; m.t.greenaway@lboro.ac.uk; geim@manchester.ac.ukNature Communications |         (2024) 15:9869 11234567890():,;1234567890():,;http://orcid.org/0000-0003-1974-0642http://orcid.org/0000-0003-1974-0642http://orcid.org/0000-0003-1974-0642http://orcid.org/0000-0003-1974-0642http://orcid.org/0000-0003-1974-0642http://orcid.org/0000-0003-1045-7170http://orcid.org/0000-0003-1045-7170http://orcid.org/0000-0003-1045-7170http://orcid.org/0000-0003-1045-7170http://orcid.org/0000-0003-1045-7170http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0002-9817-1697http://orcid.org/0000-0002-9817-1697http://orcid.org/0000-0002-9817-1697http://orcid.org/0000-0002-9817-1697http://orcid.org/0000-0002-9817-1697http://orcid.org/0000-0001-9920-9549http://orcid.org/0000-0001-9920-9549http://orcid.org/0000-0001-9920-9549http://orcid.org/0000-0001-9920-9549http://orcid.org/0000-0001-9920-9549http://orcid.org/0000-0003-0131-0775http://orcid.org/0000-0003-0131-0775http://orcid.org/0000-0003-0131-0775http://orcid.org/0000-0003-0131-0775http://orcid.org/0000-0003-0131-0775http://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/0000-0002-1467-3105http://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/0000-0003-0828-0310http://orcid.org/0000-0003-0828-0310http://orcid.org/0000-0003-0828-0310http://orcid.org/0000-0003-0828-0310http://orcid.org/0000-0003-0828-0310http://orcid.org/0000-0002-5334-0987http://orcid.org/0000-0002-5334-0987http://orcid.org/0000-0002-5334-0987http://orcid.org/0000-0002-5334-0987http://orcid.org/0000-0002-5334-0987http://orcid.org/0000-0003-3243-3794http://orcid.org/0000-0003-3243-3794http://orcid.org/0000-0003-3243-3794http://orcid.org/0000-0003-3243-3794http://orcid.org/0000-0003-3243-3794http://orcid.org/0000-0003-2861-8331http://orcid.org/0000-0003-2861-8331http://orcid.org/0000-0003-2861-8331http://orcid.org/0000-0003-2861-8331http://orcid.org/0000-0003-2861-8331http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54198-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54198-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54198-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54198-x&domain=pdfmailto:l.ponomarenko@lancaster.ac.ukmailto:alessandro.principi@manchester.ac.ukmailto:m.t.greenaway@lboro.ac.ukmailto:geim@manchester.ac.ukwww.nature.com/naturecommunicationsResultsLongitudinal and Hall resistivity of the Dirac plasmaOur devices were multi-terminal Hall bars made from encapsulatedmonolayer graphene. It was essential to make them larger than 10μmin width to avoid an obscuring contribution from edge scattering andcharge accumulation at boundaries12,27. The devices exhibited highcarrier mobilities (∼106 cm²V−1 s−1) and little inhomogeneity (Meth-ods). We studied several such devices, 3 of which were chosen fordetailed analysis of their longitudinal and Hall resistivities near the NP(ρ and RH, respectively). All of them showed practically identicalcharacteristics so that, for brevity and consistency, we illustrate theobserved behavior using the data obtained fromone of the devices. Itsoptical micrograph is shown in the inset of Fig. 1a.Near the NP, where both electrons and holes are present, the totalcharge density in graphene is given by en = e(ne − nh) where ne and nhare the sheet densities of electrons and holes, respectively, and e is theelectron charge. The charge density en can be controlled capacitivelyby gate voltage (Supplementary Note 1). The device’s resistivity ρ as afunction of n is shown in Fig. 1a (positive and negative n correspond toelectrons and holes, respectively). At low T, ρ(n) exhibits a sharp peakat the NP (red curve). It is instructive18,23 to replot ρ(n) in a logarithmicscale (right inset) which reveals that ρ is weakly density-dependent forn≲ 1010 cm−2. The point at which ρ becomes notably dependent on n islabeled as δn (arrow in the inset of Fig. 1a). The value of δn at low Tprovides a measure of residual charge inhomogeneity (“electron–holepuddles”)18,23. Despite its extra-large size (15 × 30μm2), the deviceexhibited δn of only ∼5 × 109 cm−2 at low T. As T increased, the peak inρ(n) became wider and smaller because of thermally excited electronsand holes (black curves in Fig. 1a, b). At room T, themeasured value ofδn increased by an order of magnitude with respect to that atliquid–helium T (inset of Fig. 1b).The corresponding behavior of Hall resistivity RH near the NP isshown in Fig. 1c. A small magnetic field Bwas applied perpendicular tographene, and its value (4 mT) was carefully chosen to keep electrontransport deep in the weak–field limit where RH remained linear in B(nonlinearities started emerging typically above 10mT) and, at thesame time, to ensure a large enough Hall response to record RH withhigh accuracy. Both conditions were essential for our analysis descri-bed below. Away from the NP, at densities |n | > 1011 cm−2, RH evolved asB/ne, as expected for transport dominated by one type of chargecarriers (Fig. 1c). Near the NP, RH(n) departed from this dependencedue to the presence of both electrons andholes and changed its sign atthe NP, indicating a switch from majority hole to majority electrontransport. The range of n over which both electrons and holes con-tributed to the Hall effect can be characterized by δnH, the distancebetween the maximum and minimum in RH (see the inset of Fig. 1c).δnH did not depend on B (in the discussed limit of weak B) andincreased with T as the density of thermally excited charge carriersincreased. This is illustrated in Fig. 2a which shows RH(n) at three dif-ferent T. As the temperature increased, the extrema in RH were broa-dened andmoved further apart. Figure 2b shows δnHmeasured over awide range of T and compares the behavior with δn(T) determinedfrom the broadening of the peak in ρ(n). Both δnH and δn exhibitsimilar values and a roughly parabolic T dependence. At low T, theytend to have a constant value due to residual charge inhomogeneity.The transport behavior described above and illustrated byFigs. 1 and 2 is archetypical of high-quality graphene. It was previouslyobserved in numerous experiments but not subjected to in-depthanalysis. Most often, ρ(n) curves have been used only to evaluate thecharge inhomogeneity of a device (as described above) and extract thefield-effect mobility defined as μ(n) = 1/neρ(n). The latter expression isvalid only in the caseof one typeof carrier so that, unsurprisingly,μhasbeen found to diverge near the NP because n goes through zero (bluecurve in Fig. 3a). As for the behavior ofRH(n), the region close to theNPhas usually been ignored with reference to the presence ofelectron–hole puddles. This is justified at liquid–He temperatures but,as thermal excitations overpower the effects of charge inhomogeneitywith increasing T, electron transport at the NP becomes intrinsic. Thishigh-T regime was overlooked previously and merits a better under-standing, which is provided below.Two-fluid model for the Dirac plasmaFor two types of charge carriers present in graphene near its NP, it issensible to try to describe the transport characteristics using thestandard two-carrier Drude model28:ρðnÞ= 1eðμene +μhnhÞð1ÞRHðnÞ=Beneμ2e � nhμ2hðnhμh +neμeÞ2ð2Þwhere μe and μh are the mobilities of electrons and holes, respec-tively. Their densities are given by ne,h =Rf ± εk,Θ� �DoS εk� �dεkwhere f ± εk,Θ� �is the Fermi–Dirac distribution for electrons (+) and-20 -10 0 10 200123(k�)n (1011 cm-2)300 K5 K0.01 0.1 10.11(k�)n (1011 cm-2)�na-3 -2 -1 0 1 2 3-30-1501530RH(�)n (1011 cm-2)300 KexperimentDrudesingle carrier4.0 mTcRHn2�nH-3 -2 -1 0 1 2 300.40.81.2(k�)n (1011 cm-2)300 K0.01 0.1 10.11(k�)n (1011 cm-2)�nbFig. 1 | Transport characteristics of monolayer graphene near theneutrality point. a Resistivity at room and low T in zero magnetic field. Left inset:opticalmicrographof oneof the studieddevices. Scale bar, 10 µm.Right inset:ρ (n)measured at 5 K is replotted on a log–log scale. The arrow marks δn at whichpoint the resistivity starts responding to gate voltage. b Zooming in on the beha-vior of ρ in the vicinity of the NP at 300K (black symbols, same curve as in a). Inset:same as the inset in (a) but at room T. c Room-T Hall resistivity in small B (opensymbols). The green curve plots RH expected from the standard Drude modelassuming electron-hole symmetry μe(n) = μh(n) (the curve does not depend on themobilities’ absolute values). Blue curves: RH = B/ne as expected for a single carriertype. The inset explains how we define δnH that is analogous to δn in panels a, b.Article https://doi.org/10.1038/s41467-024-54198-xNature Communications |         (2024) 15:9869 2www.nature.com/naturecommunicationsholes (−), and DoSðεkÞ is the density of states. For a given n, theelectrochemical potential Θ can be found by solving the integralequation n = ne − nh (Supplementary Note 2), which in turn allows usto find ne,h as a function of n. At the NP (n and Θ = 0),ne = nh ≡ nth = (2π3/3)(kBT/hvF)2 where kB and h are the Boltzmannand Planck constants, respectively, and vF is graphene’s Fermivelocity. At room T, the observed intrinsic broadening δnH ≈ δnwas approximately twice smaller than nth (Fig. 2b). The room-Tresistivity of ∼0.9 kOhm at the NP (Fig. 1) corresponds toμe = μh ≈ 47,000 cm2 V−1 s−1 and yields the scattering rate of∼0.3 ps, inagreement with the Planckian frequency τP−1 ≈ CkBT/h where C is theconstant of about unity13–23.If the electron and hole subsystems were to respond indepen-dently to the electricfield E, as the standardDrudemodel assumes, theelectron–hole symmetry of graphene’s spectrum would imply equaldrift velocities and, therefore, μe = μh (although the mobilities’ valuemaydependon n). Then, Eq. 2 simplifies toRH(n) = nB/e(ne + nh)2 whichis independent of scattering times. This dependence is shown in Fig. 1cby the green curve that has no adjustable parameters. This curve isprofoundly different from those observed experimentally. Theextrema of the Drude curve are much shallower and occur furtheraway from the NP than in the experiment. It also yields δnH ≈ 2.07nth(dashed curve in Fig. 2b), which is ∼4 times larger than δnH measuredat room T. If a finite charge inhomogeneity is included within theDrude model (solid blue curve in Fig. 2b and Supplementary Note 4),we achieve a better match between experimental and theoreticalcurves for δnH at low T but obviously, this cannot resolve the dis-crepancy at high T. The failure to explain the sharp transition in RH(n)near the NP shows that the standard Drude model, assuming non-interacting fluids, is inadequate to describe the Dirac plasma’s trans-port properties.Next, we relax assumptions and, empirically, allow electron andhole mobilities to be unequal and even negative (the latter contradictsthe Drude model’s assumptions). Equations 1 and 2 contain twounknown functions μe(n) and μh(n) and, for each n, their values canuniquely be evaluated from the two measured variables, ρ and RH.0 100 200 30001234�n,�n H(1010cm-2)T (K)Experiment:from RHfromDrude model:no puddlespuddlesb-1 0 1-90-60-300306090RH(�)n (1011 cm-2)300 K200 K150 K4.0 mTaFig. 2 | Hall resistivity in lowmagneticfields and thermal broadening at the NP.a Examples of RH(n) at different T (color coded). b Characteristic width of theregion where both electrons and holes are present. Symbols: δnH and δn extractedfrom the Hall and longitudinal resistivities, respectively. Error bars: standarddeviations. Blue curves: δnH(T) expected from the standard Drude model thatignores electron–hole drag.100 200 30010100mobility(m2 V-1s-1 )T (K)T -2NP2�1011 cm-2c-4 -2 0 2 4-4004080e,h(m2 V-1s-1 )n (1011 cm-2)300 K150 K200 K250 Kb-4 -2 0 2 4-1001020e,h(m2 V-1s-1 )n (1011 cm-2)300 KaFig. 3 | Trade-off between electron and hole mobilities near the Dirac point.a Room–Tmobilities of electrons (solid symbols) and holes (open) extracted fromHall and longitudinal resistivities at low B using the modified Drude model. As thedensity of majority carriers increases away from the NP, their mobility alsoincreases but the mobility of minority carriers rapidly becomes negative. Bluecurve: field-effect mobility extracted under the assumption of one type of chargecarrier. The error bars arise from noise of ∼0.2Ohm in the measured Hall resis-tance. b Same as in (a) but at a different T (color-coded). cMobility at the NP (blacksymbols) and at the density of 2 × 1011 cm−2 (red) as a function of T. For self-consistency check, the green curve shows themobility calculated directly from theminimum conductivity rather than using Eq. 3. Blue curve, T −2 dependence.Article https://doi.org/10.1038/s41467-024-54198-xNature Communications |         (2024) 15:9869 3www.nature.com/naturecommunicationsCombining Eqs. 1 and 2, we obtain the following expression:μe, h nð Þ= ±1neρ1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinh, ene, h1� eRHBn� �s" #ð3ÞThe electron and holemobilities extracted using Eq. 3 are plottedin Fig. 3a, b where we limit our analysis to T ≥ 150 K so that the densityof thermally excited carriers dominates over the residual chargeinhomogeneity (Fig. 2b).At the NP, the electron and hole mobilities are found to be equalas required by symmetry. Away from the NP, as the carrier density ofeither electrons or holes is increased, their mobility also increases,until it saturates at n of several nth, where the charge density isdominated by one type of carrier (see Supplementary Fig. 2). In con-trast, themobility of theminority carriers rapidly decreases away fromthe NP and becomes negative at |n | ≳ nth. Near room T, the mobility ofminority carriers saturates to an absolute value comparable to that ofmajority carriers (Fig. 3a). This means that the minority carriers aredragged by majority carriers in the direction opposite to their expec-ted drift direction and, if one type of carrier dominates, the other oneis forced to drift along with a similar velocity. We observe this reversalin the drift direction for minority carriers over our entire temperaturerange (Fig. 3b) and for all devices. The behavior is attributed to strongCoulomb interaction between electrons and holes. For completeness,Fig. 3c shows the T dependence of the extracted mobilities at the NPwhere μe ≡ μh and at a finite density where one carrier type remainspresent. For charge–neutral graphene, the mobilities evolve approxi-mately as ∝1/T2 and start saturating below 150K (Fig. 3c) whereelectron–hole puddles can no longer be neglected. Square depen-dence is expected because the Planckian scattering time τP and theeffectivemass of theDirac fermions are both linearly dependent on 1/T(ref. 29).DiscussionBoltzmann model for the Dirac plasmaEquations 1–3 are inadequate to accurately describe an interactingplasma. The fundamental reason for this is that electron–hole scat-tering leads tomomentum relaxation in the electric field direction butnot in the perpendicular Hall field direction23. Because of this relaxa-tion anisotropy, electron–hole scattering (described by time τeh)contributes to the transport coefficients in a different way comparedto scattering by phonons and impurities which can be parametrized byanother time τ. Accordingly, to describe electron transport in theDiracplasma at finite B, we have used the linearized Boltzmannmodel30 thatis presented in Supplementary Note 3. In brief, the Boltzmann modelyields the following coupled Drude-like equations:±eEme, h+ue, hτ±ρh, eρe + ρhue � uhτeh� �=0 ð4Þwhere ue, h are the drift velocities of electrons and holes,me, h are theirenergy–dependent effective masses, and ρe, h are the mass densities(Supplementary Note 3). Both ρe, h and me, h are positive and dependon n and T (Supplementary Fig. 3). The first two terms of Eq. 4 haveexactly the same form as the standard Drude equation describing theforce acting on a charge carrier due to the electric field and anopposing “frictional” force proportional to 1=τ. The third term corre-sponds to an additional frictional force caused by electron–holescattering. This term can attain a value opposite to the electric fieldterm and dominate over it. In the latter case, charge carriers would bedragged in the direction opposite to E. Solving Eq. 4, we determine ρand RH as a function of n and the two scattering times (these bulky butanalytical expressions are provided in Supplementary Note 3). Foreach n, we again have only two unknowns (τeh and τ) that fully define ρand RH whereas all the other relevant parameters are determined bygraphene’s electronic spectrum. The resulting coupled nonlinearequations can be solved numerically, which has allowed us to obtainboth scattering times as shown in Fig. 4a, b.Near room T, the extracted τ is practically independent of n. Withdecreasing T, τ starts exhibiting a dependence close to √n which isexpected for charged impurities and other mechanisms sensitive toscreening by charge carriers. This square-root dependence yieldsmobility independent of carrier density29, typical of graphene at low T.At the NP, τ depends relatively weakly on T over our entire tempera-ture range. Nonetheless, note that τ(n =0) first increases withincreasing T, presumably due to stronger screening by the increasinglydense Dirac plasma. Then, above 200K, τ decreases because of pho-non scattering (Fig. 4a). As for τeh, it exhibits relatively weak depen-dence on doping (note that the prefactor in the third term of Eq. 4accounts for the n dependence of electron–hole friction caused by thevarying mass densities). Some electron–hole asymmetry observedbelow 200K (Fig. 4b) originates from subtly asymmetric ρ(n) andRH(n) found in the experiment, probably because of remnant doping.With increasing T, τeh evolves as Planckian scattering, that is, τeh≈h/CkBT where C ≈0.6, in good agreement with the coefficient, reportedpreviously23. Furthermore, the inset of Fig. 4b plots the T dependence-4 -2 0 2 4-1001020e,h(m2 V-1s-1 )n (1011 cm-2)300 Kc-3 -2 -1 0 1 2 300.511.52eh(ps)n (1011 cm-2)300 K250 K200 K150 Kb150 300024T (K)/ eh-3 -2 -1 0 1 2 3012345(ps)n (1011 cm-2)300 K250 K200 K150 KaFig. 4 | Scattering times in graphene’s Dirac plasma and comparison betweentheBoltzmann andmodifiedDiracmodels. a Extracted τ(n) caused by impuritiesand phonons at different T (color coded). b Similarly for electron–hole scattering.Experimental errors rapidly increase away from the NP because a contribution ofτeh towards the transport coefficients is exponentially small beyond a few nth. Theinset shows the temperature dependence of τ=τeh at the NP. c Comparison of themobilities found using the modified Drudemodel (black curves, same as in Fig. 3a)and calculated from the Boltzmann model (red) using the scattering times frompanels (a, b).Article https://doi.org/10.1038/s41467-024-54198-xNature Communications |         (2024) 15:9869 4www.nature.com/naturecommunicationsof τ=τeh the NP. It exhibits relatively little scatter thanks to the fact thatRH(n) depends only on the ratio τ=τeh rather than the individual timesand is very sensitive to its absolute value,whichminimizes errors in ournumerical analysis (Supplementary Note 3).We emphasize that the ratio τ=τeh does not exceed 4 at any T,meaning that phonon and impurity scattering significantly affectelectron–hole drag in the Dirac plasma, especially below 200K. Thisbears ramifications for a hydrodynamic description of the Diracplasma. Indeed, to observe a viscous flow, it is imperative to haveparticle–particle scattering more frequent than momentum-relaxingscattering. The particle–particle scattering time τv that defines theelectron viscosity of the Dirac plasma is generally expected to becomparable to τeh. This means that, even under the most favorableconditions (close to room T), the ratio τ=τv near the neutrality point ismodest (a factor of several atmost), suppressing viscous effects, whichagrees with recent observations31,32. At lower Twhere values of τeh andτ becomeclose, it wouldbedifficult, if not impossible, to observe evenremnants of electron hydrodynamics.Justifying the modified Drude modelIt is instructive to calculate electron and hole mobilities from thescattering times found using the Boltzmann model (SupplementaryNote 3). The results are shown in Fig. 4c for the case of room T whereour accuracy was highest because of the largest τ=τeh ≈ 4. As expected,the Boltzmann analysis also yields negative mobilities for minoritycarriers at |n | > nth and saturating drift velocities in the same directionfor both electrons and holes, if doping is larger than a few nth. In thelimit τ → ∞, both electrons and holes are expected to drift with thesame velocity (Supplementary Note 3). The finite τ=τeh reduces thedrift velocity of minority carriers and, at room T, it is approximatelytwice as small as the velocity of majority carriers. Although the mod-ified Drude model does not distinguish between electron–hole, andelectron–phonon scattering, it agrees surprisingly well with theBoltzmann analysis. Notable deviations occur only for minority car-riers and do not exceed∼20% (Fig. 4c). The agreementwas found to besimilar for all the studied devices at T above 150K. This shows that,however empirical, the Drude model with different and sign-varyingμe(n) and μh(n) can be used for a semi-quantitative description of theDirac plasma in weak fields (RH should remain linear in B; see Supple-mentary Note 3). Furthermore, both Boltzmann and modified–Drudemodels describe equally well themeasured dependence δnH(T) shownin Fig. 2b (Supplementary Figs. 4 and 5).SummaryGraphene’s transport characteristics near the NP cannot possibly beunderstood without considering the strong interaction between theelectron and hole subsystems within the Dirac plasma because min-ority carriers are dragged in the same direction as majority carriers.The observed behavior of both the longitudinal and Hall resistivities isaccurately described by our Boltzmann analysis, which allows quanti-tative evaluation of the scattering rates. Inevitable scattering by pho-nons and impurities reduces the achievable value of the ratio τ=τeh sothat the minority carriers in the Dirac plasma always lag behind themajority ones. For high–quality encapsulated graphene,mutual drag isstrongest near room T where the minority carriers drift at approxi-mately half the velocity of the majority carriers. This shows thatimpurity and phonon scattering significantly affect the transportproperties of graphene’s Dirac plasma and, in particular, suppresses itsviscous (hydrodynamic) behavior.While the paper was under review, two preprints appeared onarXiv33,34, reporting magneto and magneto-thermal transport in gra-phene and interpreting deviations from conventional behavior interms of electronic viscosity. In all cases, the reported anomalies stemfrom frequent momentum–conserving electron scattering. However,unlike our study, the experiments33,34 required the two-probeCorbino–disk geometry and finite doping and involved significantcurrent-flow distortions caused by viscosity. We used the standardfour-probe Hall bar geometrywith no detectable distortions in currentflow, andwe could probeboth charge-neutral anddopedDirac plasma,but not the highly degenerate electron liquid as in refs. 33,34. Ouranalysis allowed direct extraction of mobilities and scattering times(τeh and τ) rather than the electron viscosity. A more subtle differenceis that the electron-hole drag we observed is caused by electron-holescattering whereas allmomentum-conserving events (described by τv)contribute to viscosity in refs. 33,34.MethodsThe studied devices were made from monolayer graphene encapsu-lated between two crystals of hexagonal boron nitride. Relatively thickgraphite placed under the trilayer heterostructures served as a gateelectrode. This allowed charge carrier mobilities to reach∼106 cm² V−1 s−1 at low T (measured at finite carrier densities of a few1011 cm−2). The remnant doping was low, typically ∼2 × 1010 cm−2. Theelectrical measurements were carried out using the standard low-frequency lock-in technique. Further details are provided in Supple-mentary Information.Data availabilityThe authors declare that the data presented in this study are availableon request from LAP.References1. Rojo, A. G. Electron-drag effects in coupled electron systems. J.Phys. Cond. Matter 11, R31 (1999).2. Narozhny, B. N. & Levchenko, A. Coulombdrag. Rev.Mod. Phys.88,025003 (2016).3. Sivan, U., Solomon, P. M. & Shtrikman, H. Coupled electron–holetransport. Phys. Rev. Lett. 68, 1196–1199 (1992).4. Croxall, A. F. et al. Anomalous Coulomb drag in electron–holebilayers. Phys. Rev. Lett. 101, 246801 (2008).5. Seamons, J. A., Morath, C. P., Reno, J. L. & Lilly, M. P. Coulomb dragin the exciton regime in electron–hole bilayers. Phys. Rev. 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Preprint at https://doi.org/10.48550/arXiv.2407.05026 (2024).AcknowledgementsWeacknowledgefinancial support from the European ResearchCouncil(grant VANDER) and the Lloyd’s Register Foundation (grant DesignerNanomaterials) (A.K.G. and V.I.F.). A.P. was supported by the EuropeanCommission under the EU Horizon 2020 MSCA-RISE-2019 program(project 873028 HYDROTRONICS) and the Leverhulme Trust (grant RPG−2023-253). M.T.G. acknowledges support from the Engineering andPhysical Sciences Research Council (grant EP/V008110/1). K.W. and T.T.acknowledge support from the Elemental Strategy Initiative of Japan(grant JPMXP0112101001) and JSPS KAKENHI (19H05790, 20H00354,and 21H05233).Author contributionsL.A.P. initiated the project, carried out electrical measurements (withhelp from A.I.B. and A.D.N.), and suggested interpretation using themodified Drude model (with help from A.K.G.). A.P., M.T.G., and A.K.G.provided the analysis using the Boltzmann model. A.K.G., A.P., andM.T.G. wrote the paper with contributions from L.A.P. and L.E. W.W.,R.V.G., P.K., K.W., and T.T. provided structures and devices. V.I.F., A.V.E.,S.S., A.I.B., andQ.G. provided theoretical support and discussed results.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-54198-x.Correspondence and requests for materials should be addressed toLeonid A. Ponomarenko, Alessandro Principi, Mark T. Greenaway orAndre K. Geim.Peer review information Nature Communications thanks Kayoung Lee,and the other, anonymous, reviewer(s) for their contribution to the peerreview of this work. 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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/s41467-024-54198-xNature Communications |         (2024) 15:9869 6https://doi.org/10.48550/arXiv.2406.13799https://doi.org/10.48550/arXiv.2406.13799https://doi.org/10.48550/arXiv.2407.05026https://doi.org/10.48550/arXiv.2407.05026https://doi.org/10.1038/s41467-024-54198-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Extreme electron–hole drag and negative mobility in the Dirac plasma of graphene Results Longitudinal and Hall resistivity of the Dirac plasma Two-fluid model for the Dirac plasma Discussion Boltzmann model for the Dirac plasma Justifying the modified Drude model Summary Methods Data availability References Acknowledgements Author contributions Competing interests Additional information