Impact of Many-Body Effects on Landau Levels in Graphene

We present magneto-Raman spectroscopy measurements on suspended graphene to investigate the charge carrier density-dependent electron-electron interaction in the presence of Landau levels. Utilizing gate-tunable magnetophonon resonances, we extract the charge carrier density dependence of the Landau level transition energies and the associated effective Fermi velocity $v_F$. In contrast to the logarithmic divergence of $v_F$ at zero magnetic field, we find a piecewise linear scaling of $v_F$ as a function of the charge carrier density, due to a magnetic-field-induced suppression of the long-range Coulomb interaction. We quantitatively confirm our experimental findings by performing tight-binding calculations on the level of the Hartree-Fock approximation, which also allow us to estimate an excitonic binding energy of $\approx 6\mathrm{meV}$ contained in the experimentally extracted Landau level transitions energies.

Figure 1

(a), (b) Position and width of the Raman $G$ peak as a function of the $B$ field and gate voltage (graphs offset for clarity; compare horizontal dashed lines). Black and orange arrows highlight the $T_1$-MPR positions for $V_g=0$ and 14.5 V, respectively. Dashed black curves represent our theoretical calculation [35], and full black lines depict Lorentzian fits to ${\mathrm{\Gamma }}_G$ near the $T_1$-MPR. Inset: False-color scanning electron micrograph of the measured device. The white dot represents the laser spot, and the scale bar is $2\mu \mathrm{m}$. (c) Illustration of the LL transition energies and density of states near $E_F$. (d) Full lines: LL transition energies for $v_F=1.35\times {10}^6\mathrm{m}/\mathrm{s}$. Dashed line: Phonon energy at $B=0\mathrm{T}$.

Figure 2

(a) $G$ peak width at the $T_1$-MPR and (b) position of the $T_1$-MPR as a function of $n_{\mathrm{el}}$. Blue line: Theoretical calculation [35] with the average parameters from all measurements. Upper axis: Filling factor $\nu =n_{\mathrm{el}}h/(eB)$ at $B=3\mathrm{T}$.

Figure 3

Effective Fermi velocity as a function of $n_{\mathrm{el}}$. Solid (dashed) line: Calculation without (with) considering the excitonic binding energy. Gray-shaded area: Uncertainty due to $B$-field-dependent renormalization [19, 42]. Upper axis: Filling factor $\nu$ at $B=3\mathrm{T}$. Inset: $v_F$ in the presence of LLs (black dots and lines as in main panel) and at a low $B$ field (blue dots from Ref. [13], purple squares from SdHO measurements on our device). Blue line: Theoretical expectation at low $B$ field. Illustrations: Renormalization of the band structure with $n_{\mathrm{el}}$ at a low (left) and high (right) $B$ field.