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Daniel Vaquero, Vito Clericò, Michael Schmitz, Juan Antonio Delgado-Notario, Adrian Martín-Ramos, Juan Salvador-Sánchez, Claudius S. A. Müller, Km Rubi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Bernd Beschoten, Christoph Stampfer, Enrique Diez, Mikhail I. Katsnelson, Uli Zeitler, Steffen Wiedmann, Sergio Pezzini

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[Phonon-mediated room-temperature quantum Hall transport in graphene](https://mdr.nims.go.jp/datasets/94f40125-1c0d-4bb8-b2f6-7641a2a6633a)

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Phonon-mediated room-temperature quantum Hall transport in grapheneArticle https://doi.org/10.1038/s41467-023-35986-3Phonon-mediated room-temperaturequantum Hall transport in grapheneDaniel Vaquero 1,10, Vito Clericò 1,10, Michael Schmitz2,3,Juan Antonio Delgado-Notario 1,4, Adrian Martín-Ramos1,Juan Salvador-Sánchez 1, Claudius S. A. Müller 5,6, Km Rubi5,6,Kenji Watanabe 7, Takashi Taniguchi 8, Bernd Beschoten 2,Christoph Stampfer 2,3, Enrique Diez 1, Mikhail I. Katsnelson 6,Uli Zeitler 5,6, Steffen Wiedmann 5,6 & Sergio Pezzini 9The quantum Hall (QH) effect in two-dimensional electron systems (2DESs) isconventionally observed at liquid-helium temperatures, where lattice vibra-tions are strongly suppressed and bulk carrier scattering is dominated bydisorder. However, due to large Landau level (LL) separation (~2000 K atB = 30T), graphene can support the QH effect up to room temperature (RT),concomitant with a non-negligible population of acoustic phonons with awave-vector commensurate to the inverse electronic magnetic length. Here,we demonstrate that graphene encapsulated in hexagonal boron nitride (hBN)realizes a novel transport regime, where dissipation in the QH phase is gov-erned predominantly by electron-phonon scattering. Investigating thermally-activated transport at filling factor 2 up to RT in an ensemble of back-gateddevices, we show that the high B-field behaviour correlates with their zero B-field transport mobility. By this means, we extend the well-accepted notion ofphonon-limited resistivity in ultra-clean graphene to a hitherto unexploredhigh-field realm.Van der Waals heterostructures of graphene and hBN have recentlygranted experimental access to novel phenomena in condensedmatter1. The use of hBN as atomically-flat encapsulating dielectric, inparticular, permits a drastic reduction of extrinsic disorder in gra-phene devices2, leading to the observation of zero-field transportregimes dominated by either electron-electron3, electron-hole4 orelectron-phonon (e-ph) interaction5, which manifest over differentcarrier density and temperature ranges. Toward RT (T ~ 300K), thescattering of electrons with acoustic phonons was theoreticallyidentified as the main intrinsic contribution to the electrical resistivityin graphene6–8, implying a carrier mobility exceeding 105 cm2V−1s−1 atlow carrier concentration (n < 1012 cm−2). While such figures couldalready be inferred from early data on disordered SiO2-supportedgraphene (~104 cm2V−1s−1 mobility)9,10, at present, the reach of the zero-field acoustic-phonon-limit is firmly established as a generic propertyof high-quality graphene devices5, also when encapsulated in hBNcrystals from different sources11 or engineered to high doping levels(n > 1013cm−2)12. Notable exceptions to the cleanness-implies-high-RT-Received: 26 November 2021Accepted: 10 January 2023Check for updates1Nanotechnology Group, USAL–Nanolab, Universidad de Salamanca, E-37008 Salamanca, Spain. 2JARA-FIT and 2nd Institute of Physics, RWTH AachenUniversity, 52074 Aachen, Germany. 3Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany. 4CENTERA Laboratories, Instituteof High Pressure Physics, Polish Academy of Sciences, 29/37 Sokołowska Str, Warsaw, Poland. 5High Field Magnet Laboratory (HFML-EMFL), RadboudUniversity, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands. 6Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJNijmegen, The Netherlands. 7Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan.8International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan. 9NEST, IstitutoNanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa, Italy. 10These authors contributed equally: Daniel Vaquero, Vito Clericò.e-mail: sergio.pezzini@nano.cnr.itNature Communications |          (2023) 14:318 11234567890():,;1234567890():,;http://orcid.org/0000-0001-7025-125Xhttp://orcid.org/0000-0001-7025-125Xhttp://orcid.org/0000-0001-7025-125Xhttp://orcid.org/0000-0001-7025-125Xhttp://orcid.org/0000-0001-7025-125Xhttp://orcid.org/0000-0001-6646-8309http://orcid.org/0000-0001-6646-8309http://orcid.org/0000-0001-6646-8309http://orcid.org/0000-0001-6646-8309http://orcid.org/0000-0001-6646-8309http://orcid.org/0000-0001-9714-8180http://orcid.org/0000-0001-9714-8180http://orcid.org/0000-0001-9714-8180http://orcid.org/0000-0001-9714-8180http://orcid.org/0000-0001-9714-8180http://orcid.org/0000-0002-3043-6417http://orcid.org/0000-0002-3043-6417http://orcid.org/0000-0002-3043-6417http://orcid.org/0000-0002-3043-6417http://orcid.org/0000-0002-3043-6417http://orcid.org/0000-0001-7369-8807http://orcid.org/0000-0001-7369-8807http://orcid.org/0000-0001-7369-8807http://orcid.org/0000-0001-7369-8807http://orcid.org/0000-0001-7369-8807http://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/0000-0003-2359-2718http://orcid.org/0000-0003-2359-2718http://orcid.org/0000-0003-2359-2718http://orcid.org/0000-0003-2359-2718http://orcid.org/0000-0003-2359-2718http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0001-7964-4148http://orcid.org/0000-0001-7964-4148http://orcid.org/0000-0001-7964-4148http://orcid.org/0000-0001-7964-4148http://orcid.org/0000-0001-7964-4148http://orcid.org/0000-0001-5165-7553http://orcid.org/0000-0001-5165-7553http://orcid.org/0000-0001-5165-7553http://orcid.org/0000-0001-5165-7553http://orcid.org/0000-0001-5165-7553http://orcid.org/0000-0002-5293-2673http://orcid.org/0000-0002-5293-2673http://orcid.org/0000-0002-5293-2673http://orcid.org/0000-0002-5293-2673http://orcid.org/0000-0002-5293-2673http://orcid.org/0000-0002-9122-7117http://orcid.org/0000-0002-9122-7117http://orcid.org/0000-0002-9122-7117http://orcid.org/0000-0002-9122-7117http://orcid.org/0000-0002-9122-7117http://orcid.org/0000-0003-4289-907Xhttp://orcid.org/0000-0003-4289-907Xhttp://orcid.org/0000-0003-4289-907Xhttp://orcid.org/0000-0003-4289-907Xhttp://orcid.org/0000-0003-4289-907Xhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-35986-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-35986-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-35986-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-35986-3&domain=pdfmailto:sergio.pezzini@nano.cnr.itmobility scenario are suspended graphene samples, where flexuralphonons dramatically contribute to carrier scattering leading to a T2behaviour of the resistivity13, and rotationally faulted graphene bilay-ers close to magic-angle, showing strong phonon-driven T-linearresistivity14. The difference between freely suspended graphene andgraphene encapsulated in hBN is due to the fact that in the latter casevan der Waals interaction between graphene and substrate makesflexural phonons harder, suppressing an intrinsic rippling instability15.In this work, we address the fundamental question whether thee-ph mechanism in clean graphene could also govern the electricaltransport in the QH regime16 at temperatures close to RT. In this sense,we note that previous literature on the RT-QH effect in graphene17–20exclusively includes experiments on SiO2-supported devices, pre-cluding such investigation.ResultsThe QH effect in 2DESs manifests when the Fermi level (EF) lies on thelocalised states between two LLs, formed in a perpendicular magneticfield and separated by an energy gap ΔLL. The interplay between thisenergy scale and the thermal energy kT governs the basic phenom-enology of the electrical transport in theQH regime.When kT≪ΔLL, noconduction takes place in the 2D bulk, while 1D chiral edge states carrythe electrical current ballistically, leading to zero longitudinal resis-tivity (ρxx) when measured in four-probe configuration (Fig. 1a, upperpanel). As the temperature increases and kT ∼ΔLL, thermal excitationof extended bulk states (close to the LLs centre) exponentially restoresbulk conduction and carrier scattering (Fig. 1a, lower panel), resultingin a finite value of the longitudinal resistivity minimum according toρxx =ρ0exp �ΔLL=2kT� �. This relation is vastly employed to estimatethe inter-LL separation via T-dependent measurements of the localresistivity minimum (under the precaution that the activationenergy underestimates ΔLL due to disorder-broadening of the LLs21).The pre-factor to the exponential term, ρ0, which is often not con-sidered explicitly, determines the magnitude of the T-activated resis-tivity (shaded yellow area Fig. 1a, lower) and contains informationregarding the disorder potential22,23. In perpendicular magnetic fields,e-ph scattering requires lattice vibrations with a wave-vector in theorder of the inverse of the magnetic length (lB ∼ 25nm=ffiffiffiffiffiffiffiffiffiB½T�p)24,which defines a third energy scale relevant to our problem Eph = _vs=lB(where vs is the sound velocity in the material). In conventional 2DESs,the small ΔLL leads to a complete suppression of the QH effect within afew K25, where the Eph-controlled phonon population can be con-sidered negligible. Although the low electronic mass in 2DESs such asInSb26 and HgTe27–29 enables the observation of the QHE up to liquid-nitrogen temperature, this is insufficient to ensure kT≫Eph andtherefore insufficient to realize a predominance of e-ph interaction.This condition, as sketched in Fig. 1b, is instead fulfilled by graphene inthe RT-QH regime (the field dependence of Eph and the correspondingT-dependent excitation probability for acoustic phonons in grapheneat B = 30T are shown in Fig. S1). Under this circumstance, the T-acti-vated resistivity (shadeddarkcyanarea in Fig. 1b) shoulddirectly relateto e-ph scattering24.Figure 1c shows a representative measurement of the RT-QHeffect, acquired at B = 30 T in a hBN/graphene/hBNback-gatedHall bar(sample D2). The Hall conductivity (σxy) presents weak slope changesaround filling factors ν = ±2 (Vg ~±20 V), while the shelves-like featuresat low carrier concentration originate from the onset of electron-holecoexistence in the highly-degenerate N =0 LL30. ρxx, in addition to thepronounced maximum around the charge neutrality point (CNP),shows two sizable minima (Fig. 1c, inset), indicative of T-activated QHstates. Notably, the overall robustness of the RT-QH signatures dra-matically differs in high-mobility graphene with respect to SiO2-sup-ported samples;17 we thoroughly address this striking observation in aseparate work, where we study the suppression of the σxy plateaus inFig. 1 | Dissipation regimes in the quantum Hall phase: high-quality grapheneatRT. a Schematics of temperature-dependent transport in conventional quantumHall systems, such as 2DESs in semiconductors. At low T (relative to the LLseparation, upper part), the electrical current is carried by chiral edge states,leading to zero longitudinal resistance. At higher T (lower part), thermally-excitedbulk states give a finite resistivity due to disorder scattering (yellow shading), withnegligible contribution from lattice vibrations.bAtRT, graphene supports both theQH effect (due to large inter-LL spacing) and predominant e-ph scattering in high-mobility samples, enabling the realization of a different transport regime, withphonon-mediated dissipation at high magnetic fields (dark cyan shading). c ρxx(black) and σxy (red) as a function of the back-gate voltage (corrected by a 5.2 Voffset from the CNP), measured in hBN-encapsulated sample D2 at B = 30T andT = 295 K. Inset: zoom-in of ρxx in the vicinity of filling factor ν = 2 (the dark cyanshading indicates the finite value of the resistivity minimum).Article https://doi.org/10.1038/s41467-023-35986-3Nature Communications |          (2023) 14:318 2ultra-high-quality devices at temperatures significantly lower than RT.In the following, we will focus on the magnitude of ρxx in the RT-QHregime and identify the underlyingmechanism employing a collectionof dry-assembled hBN/graphene/hBN heterostructures.In Fig. 2 we present the main transport characteristics of ourdevices (details on the fabrication are given in Methods), measured atzero magnetic field and at elevated temperatures. Figure 2a shows theRT mobility of three hBN-encapsulated devices, calculated accordingto the Drudemodel (μ= 1=ðneρxxÞ), as a function of the carrier densityn. All the mobility curves are well above the typical values for SiO2-supported graphene (grey shaded area) over the whole n range.Importantly, sample D3 shows a μ(n) dependence comparable to thedata of ref. 5 (dash-dotted line), demonstrating the standardfingerprintof phonon-limited RT mobility in zero magnetic field11,12 (as confirmedby temperature-dependent resistivity data shown in Fig. S2). We notethat, althoughWang et al. employed a 15μm-wide vander Pauwdevice,e-ph scattering imposes a ~1μm upper bound to the electronic meanfree path at B =0 and RT5. Therefore, the zero-field e-ph limit can alsobe realized using narrow Hall bars, provided that their channel widthexceeds 1μm(1.5μmto 2.3μm in our devices). The overall high qualityof the samples is also supported by the observation of fractional QHstates at liquid-helium temperature (see data for sample D2 in Fig. S3,and ref. 31 for sample D4, fabricated using CVD-grown graphene). InFig. 2b we explore the correlation between the carrier mobility (cal-culated using the field-effect formula32) and the charge inhomogeneityin the CNP region, estimated as the usual n* parameter33 (see Fig. 2binset for an example of the extraction). We consider data at T = 220K,where clear thermal activation is observed in the RT-QH regime. n*values above the intrinsic CNP thermal broadening (~2.6 × 1010 cm−2 at220 K, beginning of the x-axis in Fig. 2b) quantify the residual disorder,which, in our devices, remains well below the typical observations forgraphene on SiO2 (n* in the few-1011 cm−2 range). In addition, as forrefs. 33,34, the linear μ−1(n*) dependence (see shaded area in Fig. 2b)indicates scattering from long-range potentials, attributed to randomstrain variations generic to graphene on substrates35. We can thereforeconclude that the devices at disposal (i) span a low-disorder rangeunexplored in previous RT-QH experiments, and (ii) present a well-defined disorder type, with increasing impact along the D4-to-D1sequence.We then employ the sample temperature as an experimental knobto control the excitation of both phonons (see Fig. S1) and bulk-extended electronic states in strong magnetic fields. In Fig. 3a wesketch the effect of increasing T on the Landau-quantized electrons ingraphene at B = 30 T. Toward RT, the broadening of the Fermi-Diracdistribution around EF (experimentally set by Vg) ensures excitedcharge carriers from both the N =0 and N = 1 LLs, across the giant gapΔLL. Accordingly, the local resistivity minimum at filling factor ν = 2leaves zero and displays increasing finite values, as shown in theexperimental curves of Fig. 3b. In Fig. 3c, we present a complete pic-ture of the T-dependence of ρxx (ν = 2) for samples D1-4, at selectedmagneticfields (30T and 25 T in themain panel and inset, respectively;data at ν = −2 are shown in Fig. S4). In addition to our data, we showreferencepoints from ref. 20 (blackdiamonds,ρxx (ν = 2) in grapheneonSiO2), and two theoretical calculations defining different dissipationlimits (continuous lines). In both cases we take an activation energyequal to ΔLL/2: this was shown to be accurate for high B-fields in ref. 20and should hold true for clean graphene with reduced LL broadening.The upper line (yellow) assumes the universal conductivity pre-factordue to long-range disorder (2e2/h)23, multiplied by a factor 4 to takeinto account the LLdegeneracyof graphene. The lower line (dark cyan)is based on the work by Alexeev et al.24, who calculated the con-ductivity mediated by two-phonon scattering for graphene in the RT-QH regime. The relevant e-ph process conserves the LL number, butmodifies the in-plane electronic momentum. We note that this phe-nomenology is fundamentally different from that of magnetophononoscillations, recently discovered in extra-wide graphene devices36,which rely on resonant inter-LL scattering at T < 200K. Here, two-phonon scattering within each LL contributes with a conductivity pre-factor σ0 = σN(T/300K)(B/10 T)1/2, which depends both on temperatureand magnetic field (in contrast to the constant pre-factor commonlyFig. 2 | Phonon-limited transport and residual disorder at zero magnetic field.aRT carriermobility (calculated according to theDrudemodel) as a function of thecarrier concentration, for three hBN-encapsulated devices. The reference dash-dotted line are data from ref. 5, indicating a carrier mobility limited by electron-acoustic phonon scattering. The grey-shaded area shows the typical mobility forSiO2-supported graphene devices, 1–2 × 104 cm2V–1s–1. b Inverse of the high-temperature (220 K) field-effectmobility as a function of charge inhomogeneity n*,for hBN/graphene/hBNdevices D1-4. The shaded area covers a linear fit to the data,as in ref. 33, ± one standard error on the best-fit intercept and slope. Inset: Log-Logplot of the longitudinal conductivity of sample D1 as a function of the carrierdensity, exemplifying the extraction of n* (black arrow).Article https://doi.org/10.1038/s41467-023-35986-3Nature Communications |          (2023) 14:318 3assumed in QH studies). In the ν = 2 state, the predominant contribu-tion to the σN terms comes from the N =0 LL (0.65 e2/h, one order ofmagnitude larger with respect to N = 1, 0.06 e2/h)24. Strikingly, theresulting activated behaviour, not including any free parameter, is wellapproximated by our devices, while the reference data from grapheneon SiO2 follow the long-rangedisorder limit. The qualitative agreementbetween theoretical calculations and experimental data, together withthe contrasting behaviour with respect to previous reports20, indicatethat graphene/hBN heterostructures support an e-ph-dominatedtransport in the RT-QH regime. Arrhenius-type fits to theconductivity37, shown in Fig.S5, confirm the contrasting magnitude ofthe pre-factor for the two generations of graphene devices (as well asthe correctness of the assumed gap size).Despite the presence of long-range potentials (Fig. 2b), our dataclearly indicate that the e-ph pre-factor does not simply add up to thestandard long-range disorder term. To elucidate this point, we quan-titatively analyse the deviation from the phonon-mediated limit in thedifferent devices. We proceed by fitting the data from samples D1-3(only two high T curves are acquired for D4 due to experimental lim-itations) with a generalized relation (Fig. 4, inset), which adds to thetheoretical e-ph dependence from ref. 24 an activation part with aconstant pre-factor (ρD). This term is intended to account for the effectof residual disorder, and it is the only free parameter in the fits. In Fig. 4we plot the extracted ρD for the three samples at different magneticfields, as a function of the n* parameter (averaged between the elec-tron andhole-side). The linearρD(n*) behaviour observed here (shadedarea in Fig. 4) indicates that the random strain variations inducing theCNPbroadening are also responsible for ρxx exceeding the e-ph limit inthe RT-QH regime. Notably, the only device to display an exact e-ph-type dependence (D3, ρD ~ 0), is also the one to show a Drudemobilitycomparable to the zero-field e-ph limit5. Taking into account thesample-dependent correction due to residual disorder, in SI (Fig. S6)we proceed to a quantitative investigationof the field and temperaturedependence of the conductivity pre-factor in our samples, revealingthe expected B1/2 behaviour of the e-ph term. However, we note thatthe simplified pre-factor proposed in ref. 24 is the result of severalapproximations and, more importantly, it neglects the effect of dis-order. To better understand the interplay between the differentFig. 3 | Temperature-activated resistivity and phonon-mediated dissipation inthe quantum Hall effect. a Density of states (DOS) of graphene as a function ofenergy, at B = 30T (with a realistic value of LL broadening of 15 K). On top of theDOS we show the Fermi-Dirac distribution, with EF positioned in the middle of theN =0 and N = 1 LL, at two different temperatures, representative of the experi-mental range considered. b Temperature-activated longitudinal resistivity in thevicinity of ν = 2, measured in sample D1. c Minimum of ρxx at ν = 2 as a function oftemperature, for the hBN-encapsulated devices. The reference data (black dia-monds) are from ref. 20. The yellow and dark cyan continuous line are theoreticalcalculations based on refs. 23,24, respectively (the shading covers resistivity valueswithin the two theoretical calculations). Themagneticfield is 30T (25 T) in themainpanel (inset).4 6 80.00.20.4300 250 200 1500.00.10.20.3D(h/e2 )n* (1010 cm-2)D1D2D3D1D2D3D1D220 T25 T30 Txx(=2)(kΩ)T (K)B = 25 TFig. 4 | Sample-dependent disorder contribution to the activated resistivity.Correlation between the T-independent pre-factor to the activated resistivity andn*(220K) for devices D1-3. The shaded area is defined as in Fig. 2b. The error barscorrespond to ± one standard error from the fits shown in the inset. Inset:fits to theminimum resistivity as a function of temperature (continuous lines), using thegeneralized formula including both e-ph and disorder contributions, at B = 25T.Article https://doi.org/10.1038/s41467-023-35986-3Nature Communications |          (2023) 14:318 4scattering mechanisms underlying the activated resistivity, in SI(Figs. S7 and S8) we discuss additional data at lower temperature(down to 50K) and magnetic field (down to 1 T). We find that ρDdrastically increases toward low T, with the activated resistivityexceeding the e-ph limit by more than one order of magnitude in aclean sample. However, as the temperature and magnetic field areincreased, ρD progressively drops (i.e., the activated resistivity tendstoward the e-ph limit), suggesting a temperature-driven crossoverbetween regimes dominatedby either disorder or e-ph interaction (thelatter being realized only close to RT).While it is not surprising that thee-ph limit works as a lower bound to the activated resistivity of realsamples, the non-universality (i.e., the sample and temperaturedependence) of the disorder contribution deserves particular atten-tion in future theoretical treatments of the RT-QH in graphene.DiscussionThe physics of graphene is essentially determined by its deviationsfrom flatness (that is, ripples), due to either thermal fluctuationsassociated to flexural phonons for freely suspended samples or toroughness of substrate like for graphene on SiO215. In both cases, rip-ples induce inhomogeneity of electron density with electron and holepuddles in the vicinity of the CNP38,39. In particular, for the case ofgraphene on SiO2 the amplitude of induced inhomogeneity of charge-carrier density is estimated as 3 × 1011cm–239, in agreement with theabove cited experimental values of n*. This makes the system stronglydisordered, and any intrinsic scattering mechanisms become irrele-vant. Oppositely, the hBN substrate is atomically flat1 and at the sametime suppresses intrinsic ripples which increases the RT carriermobility by an order of magnitude and makes intrinsic scatteringmechanisms dominant15. Indeed, experimentally measured n* for oursamples is an order-of-magnitude smaller than what is supposed to beinduced by ripples at RT. This results in an essentially different pictureof QH physics at high enough temperatures.In conclusion, we showed experimental evidence of predominante-ph scattering in the QH regime. This is realized by uniquely com-bining strong magnetic fields, high temperatures and hBN-encapsulation of graphene. Although the RT-QH in graphene haslong been known, we showed that mitigation of disorder via van derWaals engineering provides novel insights on the transport mechan-isms in this phenomenon.MethodsGraphene-hBN van der Waals assembly and device fabricationhBN/graphene/hBN samples D1-3 are assembled using the standardvanderWaals dry pick-up5, starting frommicromechanically exfoliatedgraphene flakes previously identified by optical and Raman micro-scopy. SampleD4 isobtainedbyCVDgrowthonCu foil anddirect hBN-mediated pick-up after controlled decoupling via Cu surfaceoxidation31. All the devices are fabricatedmaking use of electron beamlithography, reactive ion etching and e-beam evaporation of Cr/Au 1Dedge contacts5.Magnetotransport measurementsWe use standard lock-in acquisition at low frequency (13Hz), withsimultaneous ρxx and ρxy measurements in four-probe configuration,either under a constant current excitation (12.5 nA, sample D1-D3) or aconstant voltage bias (300 µV, sample D4). The devices aremounted ina VTI system with low-pressure 4He serving as exchange gas, couplingthe samples to a liquid-N2 reservoir. The cryogenic system is accom-modated in the access bore of a resistive Bittermagnet at HFML-EMFL,with a maximum field of 33 T.Reporting summaryFurther information on research design is available in the NaturePortfolio Reporting Summary linked to this article.Data availabilityThe data presented in this study are available at https://doi.org/10.5281/zenodo.7352031.References1. Yankowitz,M.,Ma,Q., Jarillo-Herrero, P. & LeRoy, B. J. van derWaalsheterostructures combining graphene and hexagonal boronnitride. Nat. Rev. Phys. 1, 112–125 (2019).2. Rhodes, D., Chae, S. H., Ribeiro-Palau, R. & Hone, J. Disorder in vander Waals heterostructures of 2D materials. Nat. Mater. 18,541–549 (2019).3. Bandurin, D. A. et al. Negative local resistance caused byviscous electron backflow in graphene. Science 351, 1055–1058(2016).4. Crossno, J. et al. 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J.A.D-N thanks the support from theUniversidad de Salamanca for the María Zambrano postdoctoral grantfunded by the Next Generation EU Funding for the Requalification of theSpanish University System 2021–23, Spanish Ministry of Universities.K.W. and T.T. acknowledge support from the Elemental Strategy Initia-tive conducted by the MEXT, Japan (Grant Number JPMXP0112101001)and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and21H05233).Author contributionsU.Z., S.W. and S.P. conceived the experiment and coordinated the col-laboration. D.V., V.C. and M.S. fabricated the graphene devices andperformed the transport measurements. J.A.D.-N., A.M.-R. and J.S.-S.provided technical assistance in the cleanroom processing. C.S.A.M.and K.R. provided technical assistance during the high-field experi-ments. K.W. and T.T. provided single crystals of hBN. B.B., C.S. and E.D.supervised the experimental work. D.V., V.C., M.S., and S.P. performedthe data analysis. M.I.K. provided theoretical input for the interpretationof the results. S.P. wrote the manuscript with input from all the co-authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-35986-3.Correspondence and requests for materials should be addressed toSergio Pezzini.Peer review information NatureCommunications thanksMikhail Portnoiand the anonymous reviewers for their contribution to the peer reviewofthis work.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 license, and indicate ifchanges were made. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-35986-3Nature Communications |          (2023) 14:318 6https://doi.org/10.1038/s41467-023-35986-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Phonon-mediated room-temperature quantum�Hall transport in graphene Results Discussion Methods Graphene-hBN van der Waals assembly and device fabrication Magnetotransport measurements Reporting summary Data availability References Acknowledgements Author contributions Competing interests Additional information