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Article https://doi.org/10.1038/s41467-023-35986-3 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 bilay- ers 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 graphene17–20 exclusively includes experiments on SiO2-supported devices, pre- cluding 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 phenom- enology of the electrical transport in the QH regime. When kT≪ΔLL, no conduction takes place in the 2D bulk, while 1D chiral edge states carry the electrical current ballistically, leading to zero longitudinal resis- tivity (ρxx) when measured in four-probe configuration (Fig. 1a, upper panel). As the temperature increases and kT ∼ ΔLL, 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 ρxx =ρ0exp ΔLL=2kT . 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 con- sidered explicitly, determines the magnitude of the T-activated resis- tivity (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 pffiffiffiffiffiffiffiffiffi 24 order of the inverse of the magnetic length (lB ∼ 25 nm= B1⁄2T) , 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 a few K25, where the Eph-controlled phonon population can be con- sidered negligible. Although the low electronic mass in 2DESs such as InSb26 and HgTe27–29 enables the observation of the QHE up to liquid- nitrogen temperature, this is insufficient to ensure kT≫Eph 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-acti- vated resistivity (shaded dark cyan area in Fig. 1b) should directly relate to e-ph scattering24. 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 dra- matically differs in high-mobility graphene with respect to SiO2-sup- ported samples;17 we thoroughly address this striking observation in a separate work, where we study the suppression of the σxy plateaus in Fig. 1 | Dissipation regimes in the quantum Hall phase: high-quality graphene at RT. 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). Nature Communications | (2023)14:318 2PDF Image | Phonon-mediated quantum Hall transport in graphene
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