Introduction

Van der Waals heterostructures of graphene and hBN have recently granted experimental access to novel phenomena in condensed matter1. The use of hBN as atomically-flat encapsulating dielectric, in particular, permits a drastic reduction of extrinsic disorder in graphene devices2, leading to the observation of zero-field transport regimes dominated by either electron-electron3, electron-hole4 or electron-phonon (e-ph) interaction5, which manifest over different carrier density and temperature ranges. Toward RT (T ~ 300 K), the scattering of electrons with acoustic phonons was theoretically identified as the main intrinsic contribution to the electrical resistivity in graphene6,7,8, implying a carrier mobility exceeding 105 cm2V−1s−1 at low carrier concentration (n < 1012 cm−2). While such figures could already be inferred from early data on disordered SiO2-supported graphene (~104 cm2V−1s−1 mobility)9,10, at present, the reach of the zero-field acoustic-phonon-limit is firmly established as a generic property of high-quality graphene devices5, also when encapsulated in hBN crystals from different sources11 or engineered to high doping levels (n > 1013 cm−2)12. Notable exceptions to the cleanness-implies-high-RT-mobility scenario are suspended graphene samples, where flexural phonons dramatically contribute to carrier scattering leading to a T2 behaviour of the resistivity13, and rotationally faulted graphene bilayers close to magic-angle, showing strong phonon-driven T-linear resistivity14. The difference between freely suspended graphene and graphene encapsulated in hBN is due to the fact that in the latter case van der Waals interaction between graphene and substrate makes flexural phonons harder, suppressing an intrinsic rippling instability15.

In this work, we address the fundamental question whether the e-ph mechanism in clean graphene could also govern the electrical transport in the QH regime16 at temperatures close to RT. In this sense, we note that previous literature on the RT-QH effect in graphene17,18,19,20 exclusively includes experiments on SiO2-supported devices, precluding such investigation.

Results

The QH effect in 2DESs manifests when the Fermi level (EF) lies on the localised states between two LLs, formed in a perpendicular magnetic field and separated by an energy gap ΔLL. The interplay between this energy scale and the thermal energy kT governs the basic phenomenology of the electrical transport in the QH regime. When \({kT}\ll {\varDelta }_{{LL}}\), no conduction takes place in the 2D bulk, while 1D chiral edge states carry the electrical current ballistically, leading to zero longitudinal resistivity (ρxx) when measured in four-probe configuration (Fig. 1a, upper panel). As the temperature increases and \({kT} \sim {\varDelta }_{{LL}}\), thermal excitation of extended bulk states (close to the LLs centre) exponentially restores bulk conduction and carrier scattering (Fig. 1a, lower panel), resulting in a finite value of the longitudinal resistivity minimum according to \({\rho }_{xx}={\rho }_{0}{\exp }\left(-{\varDelta }_{{LL}}/2{kT}\right)\). This relation is vastly employed to estimate the inter-LL separation via T-dependent measurements of the local resistivity minimum (under the precaution that the activation energy underestimates ΔLL due to disorder-broadening of the LLs21). The pre-factor to the exponential term, ρ0, which is often not considered explicitly, determines the magnitude of the T-activated resistivity (shaded yellow area Fig. 1a, lower) and contains information regarding the disorder potential22,23. In perpendicular magnetic fields, e-ph scattering requires lattice vibrations with a wave-vector in the order of the inverse of the magnetic length (\({l}_{B} \sim 25\,{{{{{\rm{nm}}}}}}/\sqrt{B[{{{{{\rm{T}}}}}}]}\))24, which defines a third energy scale relevant to our problem \({E}_{{ph}}={{\hslash }}{{{{{{\rm{v}}}}}}}_{s}/{l}_{B}\) (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 a few K25, where the Eph-controlled phonon population can be considered negligible. Although the low electronic mass in 2DESs such as InSb26 and HgTe27,28,29 enables the observation of the QHE up to liquid-nitrogen temperature, this is insufficient to ensure \({kT}\gg {E}_{{ph}}\) and therefore insufficient to realize a predominance of e-ph interaction. This condition, as sketched in Fig. 1b, is instead fulfilled by graphene in the RT-QH regime (the field dependence of Eph and the corresponding T-dependent excitation probability for acoustic phonons in graphene at B = 30 T are shown in Fig. S1). Under this circumstance, the T-activated resistivity (shaded dark cyan area in Fig. 1b) should directly relate to e-ph scattering24.

Fig. 1: Dissipation regimes in the quantum Hall phase: high-quality graphene at RT.
figure 1

a Schematics of temperature-dependent transport in conventional quantum Hall systems, such as 2DESs in semiconductors. At low T (relative to the LL separation, upper part), the electrical current is carried by chiral edge states, leading to zero longitudinal resistance. At higher T (lower part), thermally-excited bulk states give a finite resistivity due to disorder scattering (yellow shading), with negligible contribution from lattice vibrations. b At RT, graphene supports both the QH effect (due to large inter-LL spacing) and predominant e-ph scattering in high-mobility samples, enabling the realization of a different transport regime, with phonon-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 V offset from the CNP), measured in hBN-encapsulated sample D2 at B = 30 T and T = 295 K. Inset: zoom-in of ρxx in the vicinity of filling factor ν = 2 (the dark cyan shading indicates the finite value of the resistivity minimum).

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Figure 1c shows a representative measurement of the RT-QH effect, acquired at B = 30 T in a hBN/graphene/hBN back-gated Hall bar (sample D2). The Hall conductivity (σxy) presents weak slope changes around filling factors ν = ±2 (Vg ~±20 V), while the shelves-like features at low carrier concentration originate from the onset of electron-hole coexistence in the highly-degenerate N = 0 LL30. ρxx, in addition to the pronounced maximum around the charge neutrality point (CNP), shows two sizable minima (Fig. 1c, inset), indicative of T-activated QH states. Notably, the overall robustness of the RT-QH signatures dramatically differs in high-mobility graphene with respect to SiO2-supported samples;17 we thoroughly address this striking observation in a separate work, where we study the suppression of the σxy plateaus in ultra-high-quality devices at temperatures significantly lower than RT. In the following, we will focus on the magnitude of ρxx in the RT-QH regime and identify the underlying mechanism employing a collection of dry-assembled hBN/graphene/hBN heterostructures.

In Fig. 2 we present the main transport characteristics of our devices (details on the fabrication are given in Methods), measured at zero magnetic field and at elevated temperatures. Figure 2a shows the RT mobility of three hBN-encapsulated devices, calculated according to the Drude model (\(\,\mu=1/({ne}{\rho }_{{xx}})\)), as a function of the carrier density n. 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 the data of ref. 5 (dash-dotted line), demonstrating the standard fingerprint of phonon-limited RT mobility in zero magnetic field11,12 (as confirmed by temperature-dependent resistivity data shown in Fig. S2). We note that, although Wang et al. employed a 15 μm-wide van der Pauw device, e-ph scattering imposes a ~1 μm upper bound to the electronic mean free path at B = 0 and RT5. Therefore, the zero-field e-ph limit can also be realized using narrow Hall bars, provided that their channel width exceeds 1 μm (1.5 μm to 2.3 μm in our devices). The overall high quality of the samples is also supported by the observation of fractional QH states at liquid-helium temperature (see data for sample D2 in Fig. S3, and ref. 31 for sample D4, fabricated using CVD-grown graphene). In Fig. 2b we explore the correlation between the carrier mobility (calculated using the field-effect formula32) and the charge inhomogeneity in the CNP region, estimated as the usual n* parameter33 (see Fig. 2b inset for an example of the extraction). We consider data at T = 220 K, where clear thermal activation is observed in the RT-QH regime. n* values above the intrinsic CNP thermal broadening (~2.6 × 1010 cm−2 at 220 K, beginning of the x-axis in Fig. 2b) quantify the residual disorder, which, in our devices, remains well below the typical observations for graphene on SiO2 (n* in the few-1011 cm−2 range). In addition, as for refs. 33,34, the linear μ−1(n*) dependence (see shaded area in Fig. 2b) indicates scattering from long-range potentials, attributed to random strain variations generic to graphene on substrates35. We can therefore conclude that the devices at disposal (i) span a low-disorder range unexplored in previous RT-QH experiments, and (ii) present a well-defined disorder type, with increasing impact along the D4-to-D1 sequence.

Fig. 2: Phonon-limited transport and residual disorder at zero magnetic field.
figure 2

a RT carrier mobility (calculated according to the Drude model) as a function of the carrier 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 for SiO2-supported graphene devices, 1–2 × 104 cm2V–1s–1. b Inverse of the high-temperature (220 K) field-effect mobility as a function of charge inhomogeneity n*, for hBN/graphene/hBN devices 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-Log plot of the longitudinal conductivity of sample D1 as a function of the carrier density, exemplifying the extraction of n* (black arrow).

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We then employ the sample temperature as an experimental knob to control the excitation of both phonons (see Fig. S1) and bulk-extended electronic states in strong magnetic fields. In Fig. 3a we sketch the effect of increasing T on the Landau-quantized electrons in graphene at B = 30 T. Toward RT, the broadening of the Fermi-Dirac distribution around EF (experimentally set by Vg) ensures excited charge carriers from both the N = 0 and N = 1 LLs, across the giant gap ΔLL. Accordingly, the local resistivity minimum at filling factor ν = 2 leaves zero and displays increasing finite values, as shown in the experimental curves of Fig. 3b. In Fig. 3c, we present a complete picture of the T-dependence of ρxx (ν = 2) for samples D1-4, at selected magnetic fields (30 T and 25 T in the main panel and inset, respectively; data at ν = −2 are shown in Fig. S4). In addition to our data, we show reference points from ref. 20 (black diamonds, ρxx (ν = 2) in graphene on SiO2), and two theoretical calculations defining different dissipation limits (continuous lines). In both cases we take an activation energy equal to ΔLL/2: this was shown to be accurate for high B-fields in ref. 20 and should hold true for clean graphene with reduced LL broadening. The upper line (yellow) assumes the universal conductivity pre-factor due to long-range disorder (2e2/h)23, multiplied by a factor 4 to take into account the LL degeneracy of graphene. The lower line (dark cyan) is based on the work by Alexeev et al.24, who calculated the conductivity mediated by two-phonon scattering for graphene in the RT-QH regime. The relevant e-ph process conserves the LL number, but modifies the in-plane electronic momentum. We note that this phenomenology is fundamentally different from that of magnetophonon oscillations, recently discovered in extra-wide graphene devices36, which rely on resonant inter-LL scattering at T < 200 K. Here, two-phonon scattering within each LL contributes with a conductivity pre-factor σ0 = σN(T/300 K)(B/10 T)1/2, which depends both on temperature and magnetic field (in contrast to the constant pre-factor commonly assumed in QH studies). In the ν = 2 state, the predominant contribution to the σN terms comes from the N = 0 LL (0.65 e2/h, one order of magnitude larger with respect to N = 1, 0.06 e2/h)24. Strikingly, the resulting activated behaviour, not including any free parameter, is well approximated by our devices, while the reference data from graphene on SiO2 follow the long-range disorder limit. The qualitative agreement between theoretical calculations and experimental data, together with the contrasting behaviour with respect to previous reports20, indicate that graphene/hBN heterostructures support an e-ph-dominated transport in the RT-QH regime. Arrhenius-type fits to the conductivity37, shown in Fig.S5, confirm the contrasting magnitude of the pre-factor for the two generations of graphene devices (as well as the correctness of the assumed gap size).

Fig. 3: Temperature-activated resistivity and phonon-mediated dissipation in the quantum Hall effect.
figure 3

a Density of states (DOS) of graphene as a function of energy, at B = 30 T (with a realistic value of LL broadening of 15 K). On top of the DOS we show the Fermi-Dirac distribution, with EF positioned in the middle of the N = 0 and N = 1 LL, at two different temperatures, representative of the experimental range considered. b Temperature-activated longitudinal resistivity in the vicinity of ν = 2, measured in sample D1. c Minimum of ρxx at ν = 2 as a function of temperature, for the hBN-encapsulated devices. The reference data (black diamonds) are from ref. 20. The yellow and dark cyan continuous line are theoretical calculations based on refs. 23,24, respectively (the shading covers resistivity values within the two theoretical calculations). The magnetic field is 30 T (25 T) in the main panel (inset).

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Despite the presence of long-range potentials (Fig. 2b), our data clearly indicate that the e-ph pre-factor does not simply add up to the standard long-range disorder term. To elucidate this point, we quantitatively analyse the deviation from the phonon-mediated limit in the different devices. We proceed by fitting the data from samples D1-3 (only two high T curves are acquired for D4 due to experimental limitations) with a generalized relation (Fig. 4, inset), which adds to the theoretical e-ph dependence from ref. 24 an activation part with a constant pre-factor (ρD). This term is intended to account for the effect of residual disorder, and it is the only free parameter in the fits. In Fig. 4 we plot the extracted ρD for the three samples at different magnetic fields, as a function of the n* parameter (averaged between the electron and hole-side). The linear ρD(n*) behaviour observed here (shaded area in Fig. 4) indicates that the random strain variations inducing the CNP broadening are also responsible for ρxx exceeding the e-ph limit in the 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 Drude mobility comparable to the zero-field e-ph limit5. Taking into account the sample-dependent correction due to residual disorder, in SI (Fig. S6) we proceed to a quantitative investigation of the field and temperature dependence of the conductivity pre-factor in our samples, revealing the expected B1/2 behaviour of the e-ph term. However, we note that the simplified pre-factor proposed in ref. 24 is the result of several approximations and, more importantly, it neglects the effect of disorder. To better understand the interplay between the different scattering mechanisms underlying the activated resistivity, in SI (Figs. S7 and S8) we discuss additional data at lower temperature (down to 50 K) and magnetic field (down to 1 T). We find that ρD drastically increases toward low T, with the activated resistivity exceeding the e-ph limit by more than one order of magnitude in a clean sample. However, as the temperature and magnetic field are increased, ρD progressively drops (i.e., the activated resistivity tends toward the e-ph limit), suggesting a temperature-driven crossover between regimes dominated by either disorder or e-ph interaction (the latter being realized only close to RT). While it is not surprising that the e-ph limit works as a lower bound to the activated resistivity of real samples, the non-universality (i.e., the sample and temperature dependence) of the disorder contribution deserves particular attention in future theoretical treatments of the RT-QH in graphene.

Fig. 4: Sample-dependent disorder contribution to the activated resistivity.
figure 4

Correlation between the T-independent pre-factor to the activated resistivity and n*(220 K) for devices D1-3. The shaded area is defined as in Fig. 2b. The error bars correspond to ± one standard error from the fits shown in the inset. Inset: fits to the minimum resistivity as a function of temperature (continuous lines), using the generalized formula including both e-ph and disorder contributions, at B = 25 T.

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Discussion

The physics of graphene is essentially determined by its deviations from flatness (that is, ripples), due to either thermal fluctuations associated to flexural phonons for freely suspended samples or to roughness of substrate like for graphene on SiO215. In both cases, ripples induce inhomogeneity of electron density with electron and hole puddles in the vicinity of the CNP38,39. In particular, for the case of graphene on SiO2 the amplitude of induced inhomogeneity of charge-carrier density is estimated as 3 × 1011 cm–239, in agreement with the above cited experimental values of n*. This makes the system strongly disordered, and any intrinsic scattering mechanisms become irrelevant. Oppositely, the hBN substrate is atomically flat1 and at the same time suppresses intrinsic ripples which increases the RT carrier mobility by an order of magnitude and makes intrinsic scattering mechanisms dominant15. Indeed, experimentally measured n* for our samples is an order-of-magnitude smaller than what is supposed to be induced by ripples at RT. This results in an essentially different picture of QH physics at high enough temperatures.

In conclusion, we showed experimental evidence of predominant e-ph scattering in the QH regime. This is realized by uniquely combining strong magnetic fields, high temperatures and hBN-encapsulation of graphene. Although the RT-QH in graphene has long been known, we showed that mitigation of disorder via van der Waals engineering provides novel insights on the transport mechanisms in this phenomenon.

Methods

Graphene-hBN van der Waals assembly and device fabrication

hBN/graphene/hBN samples D1-3 are assembled using the standard van der Waals dry pick-up5, starting from micromechanically exfoliated graphene flakes previously identified by optical and Raman microscopy. Sample D4 is obtained by CVD growth on Cu foil and direct hBN-mediated pick-up after controlled decoupling via Cu surface oxidation31. All the devices are fabricated making use of electron beam lithography, reactive ion etching and e-beam evaporation of Cr/Au 1D edge contacts5.

Magnetotransport measurements

We use standard lock-in acquisition at low frequency (13 Hz), with simultaneous ρxx and ρxy measurements in four-probe configuration, either under a constant current excitation (12.5 nA, sample D1-D3) or a constant voltage bias (300 µV, sample D4). The devices are mounted in a VTI system with low-pressure 4He serving as exchange gas, coupling the samples to a liquid-N2 reservoir. The cryogenic system is accommodated in the access bore of a resistive Bitter magnet at HFML-EMFL, with a maximum field of 33 T.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.