Stéphane Berciaud

Marek Potemski

Clément Faugeras

Many-body effects in suspended graphene probed through magneto-phonon resonances

We make use of micro-magneto Raman scattering spectroscopy to probe magneto-phonon resonances (MPR) in suspended mono-to penta-layer graphene. MPR correspond to avoided crossings between zone-center optical phonons (G-mode) and optically-active inter Landau level (LL) transitions and provide a tool to perform LL spectroscopy at a fixed energy (≈ 197 meV) set by the G-mode phonon. Using a single-particle effective bilayer model, we readily extract the velocity parameter associated with each MPR. A single velocity parameter slightly above the bulk graphite value suffices to fit all MPR for N ≥ 2 layer systems. In contrast, in monolayer graphene, we find that the velocity parameter increases significantly from ≈ (1.23 ± 0.01) m.s -1 up to ≈ (1.45 ± 0.02) m.s -1 as the first to third optically-active inter LL transition couple to the G-mode phonon. This result is understood as a signature of enhanced many-body effects in unscreened graphene.

Pristine suspended monolayer graphene is a welldefined, unscreened two-dimensional electronic system, in which a wealth of intriguing electronic [START_REF] Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF][START_REF] Du | Approaching ballistic transport in suspended graphene[END_REF][START_REF] Bolotin | Observation of the fractional quantum hall effect in graphene[END_REF][START_REF] Elias | Dirac cones reshaped by interaction effects in suspended graphene[END_REF][START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF], optical [START_REF] Mak | Optical spectroscopy of graphene: From the far infrared to the ultraviolet[END_REF] and mechanical [START_REF] Castellanos-Gomez | Mechanics of freely-suspended ultrathin layered materials[END_REF] properties have been uncovered. In particular, spectacular deviations from a simple oneelectron picture of graphene's band structure (i.e., the Dirac cones) emerge at low carrier densities (below a few 10 11 cm -2 ). Due to electron-electron interactions, the velocity parameter diverges logarithmically as the Fermi energy approaches the Dirac point [START_REF] Elias | Dirac cones reshaped by interaction effects in suspended graphene[END_REF], reaching values, well above the Fermi velocities of bulk graphite and supported graphene (∼ 1 × 10 6 m.s -1 ) [START_REF] Basov | Colloquium : Graphene spectroscopy[END_REF].

In the presence of a transverse magnetic field B, the electronic states of graphene merge into discrete, highly degenerate Landau Levels (LL) [START_REF] Orlita | Dirac electronic states in graphene systems: Optical spectroscopy studies[END_REF]. The energy and lifetime of LL can be probed using magneto-transport measurements [START_REF] Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF][START_REF] Elias | Dirac cones reshaped by interaction effects in suspended graphene[END_REF], scanning tunnelling spectroscopy [START_REF] Li | Scanning tunneling spectroscopy of graphene on graphite[END_REF] and magneto-optical spectroscopies [START_REF] Orlita | Dirac electronic states in graphene systems: Optical spectroscopy studies[END_REF][START_REF] Chen | Observation of an intrinsic bandgap and landau level renormalization in graphene/boronnitride heterostructures[END_REF]. Recently, micro-magneto-Raman spectroscopy (MMRS) [START_REF] Faugeras | Tuning the electron-phonon coupling in multilayer graphene with magnetic fields[END_REF][START_REF] Faugeras | Magneto-Raman scattering of graphene on graphite: Electronic and phonon excitations[END_REF][START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF][START_REF] Kim | Measurement of filling-factor-dependent magnetophonon resonances in graphene using raman spectroscopy[END_REF][START_REF] Neumann | Low b field magneto-phonon resonances in single-layer and bilayer graphene[END_REF][START_REF] Kazimierczuk | Time-resolved magneto-raman study of carrier dynamics in low landau levels of graphene[END_REF] has been employed to demonstrate that electronic excitations between LL may be strongly affected by manybody effects, i.e., electron-electron interactions [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF][START_REF] Neumann | Low b field magneto-phonon resonances in single-layer and bilayer graphene[END_REF][START_REF] Sonntag | Impact of many-body effects on landau levels in graphene[END_REF] and significant Coulomb binding within a magnetoexciton [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF]. As a result, the velocity parameter associated with each LL (hereafter denoted v F for simplicity) increases logarithmically with decreasing B. Interestingly, at a given B, magneto-Coulomb binding also depends on the LL index n, leading to n-dependent v F [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF]. Thus far, Raman signatures of many body effects have predominantly appeared on the electronic Raman scattering response of graphene [START_REF] Riccardi | Gate-dependent electronic raman scattering in graphene[END_REF] under a transverse magnetic field [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF][START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF]. Recently, many-body effects have also been unveiled by monitoring magneto-phonon resonances [START_REF] Faugeras | Tuning the electron-phonon coupling in multilayer graphene with magnetic fields[END_REF] (MPR) between optically-active inter-LL transitions [START_REF] Russell | Many-particle effects in the cyclotron resonance of encapsulated monolayer graphene[END_REF] and zone-center optical phonons (i.e., Raman G-mode phonons [START_REF] Ferrari | Raman spectroscopy as a versatile tool for studying the properties of graphene[END_REF]) in graphene encapsulated in hexagonal boron nitride (BN) films [START_REF] Neumann | Low b field magneto-phonon resonances in single-layer and bilayer graphene[END_REF][START_REF] Sonntag | Impact of many-body effects on landau levels in graphene[END_REF]. More prominent effects are expected in suspended graphene, where electron-electron interactions are minimally screened.

In this letter, we report the results of MMRS measurements performed on suspended mono-to pentalayer graphene. For each number of layers N , we resolve a set of well-defined MPR. Using a single-particle effective bilayer model, we readily extract v F associated with each MPR. While a single parameter slightly above the bulk graphite value ( 1.05 m.s -1 ) suffices to fit all MPR for N ≥ 2 layer systems, We find that v F increases significantly up to ≈ 1.45 m.s -1 in monolayer graphene. This result is understood as a signature of enhanced manybody effects in unscreened graphene.

The low-energy electronic electronic bands and magneto-optical response of mono-and N -layer graphene have been extensively discussed using an effective bilayer model [START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF][START_REF] Latil | Charge carriers in few-layer graphene films[END_REF][START_REF] Koshino | Orbital diamagnetism in multilayer graphenes: Systematic study with the effective mass approximation[END_REF][24][START_REF] Orlita | Graphite from the viewpoint of landau level spectroscopy: An effective graphene bilayer and monolayer[END_REF][START_REF] Mak | The evolution of electronic structure in few-layer graphene revealed by optical spectroscopy[END_REF]. To lowest order, this model uses only two parameters, namely the singleparticle Fermi velocity v F = 3/2aγ 0 / , where γ 0 is the nearest neighbor hopping parameter, a = 0.142 nm is the c-c bond length, and the nearest neighbor interlayer coupling constant γ 1 . The band structure of Bernal-or equivalently AB-stacked N ≥ 2 layer systems exhibits 2N bands, including one pair of linear (monolayer-like) bands only present for odd N and a set of N/2 effective bilayer bands (here, denotes the integer part) with a rescaled γ1 = 2γ 1 cos θ [START_REF] Koshino | Orbital diamagnetism in multilayer graphenes: Systematic study with the effective mass approximation[END_REF][24][START_REF] Orlita | Graphite from the viewpoint of landau level spectroscopy: An effective graphene bilayer and monolayer[END_REF][START_REF] Mak | The evolution of electronic structure in few-layer graphene revealed by optical spectroscopy[END_REF], which corresponds to half the energy gap between their split-off bands. The angle θ represent the quantized transverse momenta θ = k z c/2 at which two-dimensional cuts are made in the three-dimensional electronic dispersion of graphite, with c/2 = 0.34 nm the interlayer separation in bulk graphite. The band structure of mono-to trilayer graphene and corresponding values of γ1 and θ are shown in Fig. 1a-c.

In the presence of a transverse magnetic field, the ef- fective mono-and bilayers give rise to independent sets of Landau fans. The energy L θ n L θ -n = -L θ n of the n th electron (n ≥ 0) or hole (n ≤ 0) LL arising from the gapless bands in an effective bilayer writes [START_REF] Orlita | Dirac electronic states in graphene systems: Optical spectroscopy studies[END_REF][START_REF] Mccann | Landau-level degeneracy and quantum Hall effect in a graphite bilayer[END_REF]:

L θ |n| = γ2 1 2 + |n| + 1 2 E 2 1 - γ4 1 4 + |n| + 1 2 E 2 1 γ2 1 + E 4 1 4 , (1)
where

E 1 = v F √ 2e B. At θ = π/2, one obtains the well-known LL fan of monolayer graphene L π/2 |n| = √ nE 1 . For θ = π/2, nearly linear scalings of L θ n (B)
, characteristic of effective bilayers are observed. The corresponding LL fans are shown in 1d-f for mono to trilayer graphene, respectively. As previously established [START_REF] Kashuba | Signature of electronic excitations in the Raman spectrum of graphene[END_REF][START_REF] Mucha-Kruczyński | Spectral features due to interlandau-level transitions in the raman spectrum of bilayer graphene[END_REF][START_REF] Kossacki | Electronic excitations and electronphonon coupling in bulk graphite through raman scattering in high magnetic fields[END_REF], optically-allowed transitions, such that δ |n| = ±1 couple to the graphene G-mode phonons (with energy ω G ≈ 1585 cm -1 or equivalently ≈ 197 meV) and give rise to series of MPR [START_REF] Faugeras | Tuning the electron-phonon coupling in multilayer graphene with magnetic fields[END_REF][START_REF] Ando | Magnetic oscillation of optical phonon in graphene[END_REF][START_REF] Goerbig | Filling-factor-dependent magnetophonon resonance in graphene[END_REF], as indicated by the vertical arrows in Fig. 1d-f. In contrast, symmetric inter LL transitions (δ |n| = 0, denoted L n , with n ≥ 1) are Raman-allowed [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF][START_REF] Faugeras | Magneto-Raman scattering of graphene on graphite: Electronic and phonon excitations[END_REF][START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF] but are not expected to couple to optical phonons [START_REF] Kashuba | Signature of electronic excitations in the Raman spectrum of graphene[END_REF]. In the following, the energy of the optically active inter LL transitions associated with a given value of θ will be simply denoted

T n = L θ ±n+1 -L θ ∓(n) , n ≥ 0 (2)
and the (n + 1) th MPR (n ≥ 0) occurs when

T n = ω G .
The suspended graphene samples investigated here were prepared by mechanical cleavage of Bernal stacked natural graphite and dry transfer onto 8 µm-diameter pits etched in Si/SiO 2 substrates as described in details elsewhere [START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF][START_REF] Berciaud | Probing the intrinsic properties of exfoliated graphene: Raman spectroscopy of free-standing monolayers[END_REF][START_REF] Metten | All-optical blister test of suspended graphene using micro-raman spectroscopy[END_REF]. The samples were held in a homebuilt magneto-optical cryostat where MMRS measure- ments were performed in a back-scattering geometry under transverse magnetic fields up to 29 T. A cw laser beam at 514 nm was focused onto a ∼ 2 µm-diameter spot. A cross-circular polarization configuration was implemented, where incoming and Raman scattered photons have opposite helicity. In these conditions, the magneto-Raman spectra are dominated by MPR [START_REF] Kashuba | Signature of electronic excitations in the Raman spectrum of graphene[END_REF][START_REF] Kühne | Polarization-resolved magneto-raman scattering of graphenelike domains on natural graphite[END_REF]. We have employed laser powers ∼ 1 µW at our samples to minimise laser-induced heating effects while still having a sufficiently large MMRS signal. All measurements were performed at a base temperature of 4 K. Raman G-mode spectra were fit to Lorentzian functions and the peak frequencies and full-width at half maximum are determined with an experimental error that is typically smaller than the symbol size.

v F = 1.23 v 0 F w G (cm -1 ) v F = 1.35 v 0 F v F = 1.45 v 0 F (c) (d) N = 1 q = p/2
Figure 2 shows MMRS data for a suspended graphene monolayer with minimal native doping well below 10 12 cm -2 . MPR appear as anticrossings involving significant frequency shifts and line broadening of the G-mode feature. Due to electronic broadening in graphene [START_REF] Bostwick | Quasiparticle dynamics in graphene[END_REF][START_REF] Neumann | Probing electronic lifetimes and phonon anharmonicities in high-quality chemical vapor deposited graphene by magneto-raman spectroscopy[END_REF], anti-crossings are not fully resolved, particularly so at low field. The third and second MPR, involving tran-sitions T 2 and T 1 appear at B 2 = 1.45 ± 0.1 T and B 1 = 2.75 ± 0.1 T, respectively. Remarkably, for the first time in suspended graphene, we report the T 0 MPR at B 0 = 19.4 ± 0.1 T. The prominence of the T 0 MPR originates from the fact that the coupling strength in MPR theory [START_REF] Goerbig | Filling-factor-dependent magnetophonon resonance in graphene[END_REF] is directly proportional to the LL degeneracy, which grows proportionally to B. As a result, MPR at low B involving LL with higher n are fainter.

The resonant fields observed here are considerably lower than in substrate-supported systems [START_REF] Faugeras | Tuning the electron-phonon coupling in multilayer graphene with magnetic fields[END_REF][START_REF] Kim | Measurement of filling-factor-dependent magnetophonon resonances in graphene using raman spectroscopy[END_REF][START_REF] Neumann | Low b field magneto-phonon resonances in single-layer and bilayer graphene[END_REF] as discussed in more details later. As shown in Fig. 2b, near B 0 , the G-mode exhibit a double-Lorentzian lineshape, with a broad, prominent feature that shows an avoided crossing with the T 0 transition and a sharp, fainter feature near 1588 cm -1 (not displayed in Fig. 2c,d) that is virtually independent on B. Such an uncoupled Gmode subfeature appears more prominently in substratesupported graphene and is tentatively assigned to residual inhomegeneous doping and strain [START_REF] Kim | Measurement of filling-factor-dependent magnetophonon resonances in graphene using raman spectroscopy[END_REF]. Here, the comparably lower spectral weight of the uncoupled feature illustrates the improved homogeneity of suspended graphene. In Fig. 2c, one can clearly see an increase of the phonon linewidth, a signature of a variation of the phonon lifetime induced by an interaction, at B 1 = (4.7 ± 0.1) T. This resonance implies the G-mode phonon and the symmetric inter LL excitation L 1 . Although symmetric electronic excitations should not couple to the optical phonons [START_REF] Kashuba | Signature of electronic excitations in the Raman spectrum of graphene[END_REF], a similar signature has been observed in graphene on graphite [START_REF] Faugeras | Magneto-Raman scattering of graphene on graphite: Electronic and phonon excitations[END_REF] and on graphene encapsulated in hexagonal Boron Nitride (BN) [START_REF] Neumann | Low b field magneto-phonon resonances in single-layer and bilayer graphene[END_REF]. In the present case, we investigate suspended graphene and substrateinduced effects can be ruled out to rationalize this forbidden coupling. These results point towards an intrinsic effect, possibly originating from landau level mixing.

We now consider how MPR evolve for thicker graphene stacks, as the electronic structure gets increasingly complex. Figure 3 shows MPR data for suspended bilayer and trilayer graphene. Data for tetra-and penta-layer graphene are reported in Fig. 4. The latter data were acquired using a lower spectral resolution that for N = 1, 2, 3. As a result, we have not attempted to resolve signatures of pronounced anticrossings at high fields for N = 4 and N = 5.

For N = 2, three MPR are resolved at B 3 = 7.9±0.1 T, B 2 = 10.8 ± 0.1 T, B 1 = 16.6 ± 0.1 T and are assigned to the T 3 , T 2 and T 1 MPR of bilayer graphene (θ = π/3), respectively. In trilayer graphene (Fig. 3c,d), we are able to resolve the first three monolayer-like MPR (T 0 , T 1 , T 2 for θ = π/2) and the T 1 , T 2 , T 3 MPR associated with the effective bilayer (θ = π/4). At high fields B > 14 T, the G-mode lineshape becomes more complex due to the overlap between the second bilayer-like MPR (T 1 , θ = π/4) and first monolayer-like MPR (T 0 , θ = π/2). These spectra are well fit to a bilayer Lorentzian form for for 13 T < B < 21 T and 26 T < B < 29 T and to a triple Lorentzian (red, yellow and dark yellow) for 21 T < B < 26 T. The narrow, central feature (red symbols in Fig. 3c) is assigned to the second bilayer-like MPR (T 1 , θ = π/4), while the broader and shifted features (orange symbols for double-Lorentzian fits then dark yellow and yellow symbols for triple-Lorentzian fits in Fig. 3c) are assigned to the first monolayer-like MPR (T 0 , θ = π/2).

In tetralayer graphene (Fig. 4a,b), under transverse magnetic fields up to B = 29 T, we observe six MPR associated with gapless effective bilayer bands at θ = π/5 and θ = 2π/5, respectively [START_REF] Faugeras | Probing the band structure of quadri-layer graphene with magneto-phonon resonance[END_REF]. Up to nine MPR are observed in the same window for pentalayer graphene (Fig. 4c,d). These MPR are associated with the effective monolayer (θ = pi/2), with two other subsets of MPR stemming from the gapless effective bilayer bands at θ = π/3 and θ = π/6, respectively. Let us note that as the number of electronic subbands increase, MPR involving distinct values of θ and n may be nearly degenerate. As a result, we are not able to clearly resolve T 1 , θ = 2π/5 and T 3 , θ = π/5 near 12 T for N = 4 (Fig. 4c,d). Similarly, for N = 5 (Fig. 4c,d), the MPR associated with T 2 , θ = π/3 and T 4 , θ = π/6 near 11 T as well as with T 0 , θ = π/2 and T 1 , θ = π/6 near 27-28 T appear as single slightly broadened MPR features. Thus, as N increases, the growing complexity of the LL spectrum prevents detailed studies of MPR for N > 5, until the limit of the LL fan diagram of bulk graphite is reached.

N = 4 v F = 1 . 0 6 v 0 F B 4 B 3 B 2 B 1 B 3 B 2 B 1 0 1 0 2 0 3 0 2 0 2 5 M a g n e t i c F i e l d ( T ) ( d ) B 3 B 2 B 1 B 4 B 3 B 4 B 3 B 2 B 1 B 2 B 0 B 1 θ=π/ 6 θ=π/ 3 θ=π/ 2 T 3 T 2 T 1 T 0 T 4 T 3 T 2 T 1 T 4 T 3 T 2 T 1 N = 5 v F = 1 . 0 5 v 0 F ω G ( c m -1 ) T 3 T 2 T 1 T 3 T 2 T 1 θ=π/ 5 θ=2 π/ 5 ( c ) ( b ) ( a )
From the data in Fig. 234and using the effective bilayer model (Eq. ( 2)), we can now determine the value of v F associated with each MPR (i.e., evaluated at ω G ), as shown in Fig. 5a. For simplicity, we have considered a constant value of γ 1 = 400 meV for all N and have attempted a global fit of all the resonant magnetic fields for a given N . For N = 2 to N = 5, This procedure yields a very good agreement between Eq. ( 2) and our experiments with a constant v F that decreases smoothly from 1.08 ± 0.01 × 10 6 m.s -1 for N = 2 down to 1.05 ± 0.01 × 10 6 m.s -1 for N = 5. These values converge towards the bulk graphite value of 1.02 ± 0.01 × 10 6 m.s -1 [START_REF] Orlita | Graphite from the viewpoint of landau level spectroscopy: An effective graphene bilayer and monolayer[END_REF] and are in excellent agreement with previous electronic Raman measurements on the same samples [START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF].

In contrast, a single velocity parameter would grossly fail to determine the three observed MPR in monolayer graphene. Instead, using Eq. ( 2), we determine that v F increases significantly from ≈ (1.23 ± 0.01) m.s -1 up to ≈ (1.45±0.02) m.s -1 as the transverse magnetic field decreases from 19.4 T down to 1.40 T. As shown in Fig. 5b, this behavior is consistent with recent observations of a logarithmic divergence of v F , due to electron-electron interactions, as the carrier energy approaches the Dirac point [START_REF] Elias | Dirac cones reshaped by interaction effects in suspended graphene[END_REF][START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF]. Here, we should mention that MPR probe charge carriers at a fixed energy set by ω G . However, each MPR occurs at a resonant field B n that, in a oneelectron picture (Eq. ( 2)) scales as 1/ √ n. Our previous experimental and theoretical studies of symmetric inter LL excitations (L n , see Fig. 5b) indicate that at a given carrier energy, the velocity parameter augments as the LL index n increases [START_REF] Faugeras | Landau level spectroscopy of electronelectron interactions in graphene[END_REF]. Our data are in line with these results, since the velocity parameter associated with the T 2 MPR lines up on the logarithmic divergence observed for the L 2 electronic transition, whereas the velocity parameter associated with the T 1 and T 1 MPR are much closer to the trend observed for the L 1 transition. Remarkably, in keeping with our previous measurements of L n transitions [START_REF] Berciaud | Probing electronic excitations in mono-to pentalayer graphene by micro magneto-raman spectroscopy[END_REF], many-body effects are essentially observed in monolayer graphene and nearly vanish for N ≥ 2. Such observations are consistent with recent works showing that the reduction of the exciton binding energy in an atomically thin semiconductor coupled to graphene layers is readily maximized with one graphene monolayer and does not increase further if a bilayer graphene is used instead [START_REF] Lorchat | Filtering the photoluminescence spectra of atomically thin semiconductors with graphene[END_REF]. In addition, the parabolic dispersion of massive fermions in effective bilayers (Fig. 1b,c) creates a finite density of states at the Dirac point, which may be sufficient to quench electronelectron interactions and the subsequent renormalization of electronic bands. Further experimental and theoretical investigations of many-body magneto-excitonic effects are required to account for the abrupt transition between the mono and bilayer cases [START_REF] Russell | Many-particle effects in the cyclotron resonance of encapsulated monolayer graphene[END_REF][START_REF] Shizuya | Renormalization and cyclotron resonance in bilayer graphene with weak electron-hole asymmetry[END_REF][START_REF] Sokolik | Many-body filling factor dependent renormalization of fermi velocity in graphene in strong magnetic field[END_REF][START_REF] Shizuya | Many-body effects on landau-level spectra and cyclotron resonance in graphene[END_REF].

In conclusion, we have demonstrated that magnetophonon resonances provide invaluable fingerprints of the low-energy electronic dispersion of suspended graphene layers, which are in excellent quantitative agreement with a one-electron effective bilayer model for N ≥ 2. In the monolayer limit, electrons undergo minimal screening, leading to pronounced many-body effects which, to experimental accuracy, do not yield sizeable experimental signatures for N ≥ 2. Many body effects in graphene and Dirac fermions in general continue to attract considerable interest and future directions include the interplay between of many-body effects and electronic lifetime in graphene and related systems [START_REF] Ho-Kin | The role of electron-electron interactions in two-dimensional dirac fermions[END_REF].

FIG. 1 .

 1 FIG.1. Single-particle low-energy electronic structure and Landau levels of mono-, bi-and trilayer graphene. The electronic dispersions obtained from the effective bilayer model described in the text are shown in (a-c), respectively. The corresponding dispersion of the Landau levels arising from the gapless bands are shown in (d,e,f) for mono-, bi-and trilayer graphene, respectively. Red lines in (d) and (f) correspond to monolayer-like Landau levels (θ = π/2), while dark yellow lines in (e,f) correspond to Landau levels arising from effective bilayers obtained at quantized values of θ = π/2. The vertical arrows indicate the optically allowed inter LL transitions Tn that give rise to the magneto-phonon resonances. The calculations are performed with vF = 1.05 × 10 6 m/s and γ1 = 400 meV .

  FIG. 2. Magneto-phonon resonances in suspended monolayer graphene. (a,b) Raman G-mode spectra at selected values of the transverse magnetic field B. The spectra are acquired for the same duration and incident laser power and offset for clarity. The dark and light gray lines are Lorentzian fits to the data. (c) G-mode frequency ωG (red circles), extracted from Lorentzian fits of the G-mode spectra, overlaid on a false-color plot of the Raman spectra as a function of B. The dashed red lines are calculated dispersions of the Tn transitions and of the faint L1 symmetric electronic excitation (in grey). The velocity parameters (estimated at ωG) associated with the three observed MPR are indicated in units of v 0 F = 1.00 × 10 6 m.s -1 . (d) Lorentzian full-width at half maximum ΓG as a function of the magnetic field. The resonant fields Bn are indicated. The insets in (c) and (d) show close-ups on the regions boxed with thin dotted lines.

FIG. 3 .

 3 FIG. 3. Magneto-phonon resonances in suspended bi-(a,b) and trilayer (c,d) graphene. (a,c) G-mode frequency ωG (red circles), extracted from a Lorentzian fit of the G-mode spectra, overlaid on a false-color plot of the Raman spectra as a function of the transverse magnetic field B. The red and dark yellow dashed lines are calculated dispersions of the Tn transitions for monolayer-like and effective bilayer bands. The single vF extracted from all MPR is indicated in both cases in units of v 0 F = 1.00 × 10 6 m.s -1 . (b,d) Lorentzian full width at half maximum ΓG as a function of the magnetic field. The resonant fields Bn are indicated with color-coded values of θ. In the trilayer case (c, d), the G-mode spectra are fit to a double Lorentzian (red and orange symbols) for 13 T < B < 21 T and 26 T < B < 29 T and to a triple Lorentzian (red, yellow and dark yellow) for 21 T < B < 26 T near the first monolayer-like and second bilayer-like MPR. The insets in (a,b,d) show close-ups on the regions boxed with red dotted lines in (a,d) and on the most prominent G-mode subfeature in (c).

  n e t i c F i e l d ( T )

T 4 FIG. 4 .

 44 FIG. 4. Magneto-phonon resonances in suspended tetra-(a,b) and pentalayer (c,d) graphene. (a,c) G-mode frequency ωG (red circles), extracted from a Lorentzian fit of the G-mode spectra as a function of the magnetic field B. The red, dark yellow and orange dashed lines are calculated dispersions of the Tn transitions for monolayer-like and effective bilayer bands with color-coded values of θ. The single vF extracted from all MPR is indicated in both cases in units of v 0 F = 1.00 × 10 6 m.s -1 . (b,d) Lorentzian full width at half maximum ΓG as a function of the magnetic field. The resonant fields Bn are indicated with color-coded values of θ.

FIG. 5 .

 5 FIG. 5. (a) Velocity parameters vF extracted from the magneto-phonon resonances in Fig. 2-4 as a function of N . (c) Velocity parameters in suspended monolayer graphene (filled red symbols) as a function of ln B/ B, with B = 1 T. These velocity parameters are compared with those obtained from electronic Raman scattering measurements reproduced from Ref. 5), with the symmetric L1 and L2 transitions in the same sample (open red symbols) and in a graphene sample encapsulated in hexagonal boron nitride (BN, open blue symbols). The solid and dashed lines are theoretical calculations considering electron-electron interactions and magneto-excitonic effects. For comparison, we show (blue symbols) the values of vF measured in Ref. 16 in BN-capped graphene.

Acknowledgements

We are grateful to D.M. Basko for fruitful discussions. We thank I. Breslavetz, the StNano clean room staff, M. Romeo, F. Chevrier, A. Boulard and the IPCMS workshop for technical support. We acknowledge financial support from the Agence Nationale de Recherche (ANR) under grants H2DH ANR-15-CE24-0016, ANR-17-CE24-0030, 2D-POEM ANR-18-ERC1-0009, the Labex NIE project ANR-11-LABX-0058-NIE and the USIAS GOLEM project, within the Investissement d'Avenir program ANR-10-IDEX-0002-02. M.P. and C.F. acknowledge support from the EC Graphene Flagship project (no. 604391). Part of this work was performed at the LNCMI-Grenoble, a member of the European Magnetic Field Laboratory (EMFL).