Dirac and Normal Fermions in Graphite and Graphene: Implications of the Quantum Hall Effect

Spectral analysis of the Shubnikov–de Haas magnetoresistance oscillations and the quantum Hall effect (QHE) measured in quasi-2D highly oriented pyrolytic graphite (HOPG) [Phys. Rev. Lett. 90, 156402 (2003)] reveals two types of carriers: normal (massive) electrons with Berry phase 0 and Dirac-like (massless) holes with Berry phase $\pi$. We demonstrate that recently reported integer- and semi-integer QHEs for bilayer and single-layer graphenes take place simultaneously in HOPG samples.

Figure 1

Field dependence of basal-plane magnetoresistance $R_{xx}(B)$ and Hall resistance $R_{xy}(B)$ measured for the HOPG-3 sample with magnetic field $B$ applied along the sample $c$ axis [1].

Figure 2

Spectral intensity of SdH oscillations of magnetoresistance $|R_{xx}(B_0)|$ and Hall conductance $|G_{xy}(B_0)|$ in the HOPG-3 sample in comparison with dHvA oscillations of susceptibility $|\chi (B_0)|$ in the HOPG-UC sample [3]. Peaks $e_1$ and $h_1$ correspond to the normal electrons and Dirac holes, respectively.

Figure 3

Hall conductance $G_{xy}$ (solid line a) and $\Delta {\sigma }_{xx}\sim -\Delta R_{xx}$ (solid line b) for the HOPG-3 sample as a function of the normalized “filling factor” $\nu =B_0/B$ (where $B_0=4.68\mathrm{T}$ is the SdH oscillation frequency). Normal electrons give the dominant contribution in the quantum oscillations. The profile of $G_{xy}(\nu )$ is fitted by the integer QHE staircase; the minima of $\Delta {\sigma }_{xx}$ correspond to the QHE plateaus. Hall conductance is normalized on $G_{0xy}=28{\Omega }^{-1}$ equal to the step between QHE plateaus. The results are compared with SdH oscillations in thick film of graphite (dashed line c) [13] (having approximately the same $B_0=4.65\mathrm{T}$) and with the QHE in bilayer graphene [11] obtained as a function of both magnetic field (dashed line d) ($\nu =B_0/B$, $B_0=27.8\mathrm{T}$) and concentration (dotted line e) ($\nu =n/n_0$, with $n_0=1.7\times {10}^{12}{\mathrm{cm}}^{-2}$ at $B=20\mathrm{T}$).

Figure 4

Solid lines a and b: the same as in Fig. 3 (with $B_0=6.41\mathrm{T}$, $G_{0xy}=19{\Omega }^{-1}$) but after filtering elimination of the normal electrons contribution. The $G_{xy}(\nu )$ profile and positions of the minima in $\Delta {\sigma }_{xx}\sim -\Delta R_{xx}$ are coherent with the Dirac QHE staircase. The results are compared with the Dirac QHE in graphene monolayer obtained both as a function of magnetic field ($\nu =B_0/B$ with $B_0=6.67\mathrm{T}$) (dashed line c) [10] and carrier concentration ($\nu =n/n_0$ with $n_0=1.5\times {10}^{12}{\mathrm{cm}}^{-2}$ at $B=14\mathrm{T}$) (dotted line d) [9].

Figure 5

Values of $B^{-1}$ for the normal (N) and Dirac-like (D) charge carriers at which the $n$th Landau level crosses the Fermi level. These values are found as maxima (○) in the SdH oscillations of magnetoconductivity $\Delta {\sigma }_{xx}(B)$. The minima positions (×) in $\Delta {\sigma }_{xx}(B)$ are shifted by $1/2$ to the left. Linear extrapolation of the data to $B^{-1}=0$ (solid lines) allows for the phase factor [$-\gamma +\delta$, Eq. (2)] definition: $\gamma =1/2$, $\delta \simeq -1/8$ for normal electrons and $\gamma =0$, $\delta =0$ for Dirac-like holes.