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Article https://doi.org/10.1038/s41467-023-35986-3 Fig. 2 | Phonon-limited transport and residual disorder at zero magnetic field. 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- 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 ( μ = 1=ðneρxx Þ), 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 (cal- culated 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 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). 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. 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 pic- ture 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 con- ductivity 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 phe- nomenology 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 Nature Communications | (2023)14:318 3PDF Image | Phonon-mediated quantum Hall transport in graphene
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