Quantum Hall effects of graphene with multiorbitals: Topological numbers, Boltzmann conductance, and semiclassical quantization

Hall conductance ${\sigma }_{xy}$ as the Chern numbers of the Berry connection in the magnetic Brillouin zone is calculated for a realistic multiband tight-binding model of graphene with a nonorthogonal basis. It is confirmed that the envelope of ${\sigma }_{xy}$ coincides with a semiclassical result when the magnetic field is sufficiently small. The Hall resistivity ${\rho }_{xy}$ from the weak-field Boltzmann theory also explains the overall behavior of the ${\sigma }_{xy}$ if the Fermi surface is composed of a single energy band. The plateaux of ${\sigma }_{xy}$ are explained from semiclassical quantization and necessary modification is proposed for the Dirac fermion regimes.

Figure 1

(Color online) The lower panel shows the calculated Chern number $\left({\sigma }_{xy}\right)$ for $\varphi =1/53$ as a function of the chemical potential. The upper and middle panels present energy bands and total density of states when magnetic field is absent. Vertical dashed lines indicate positions of van Hove singularities.

Figure 2

(Color online) Comparison of the Hall conductance obtained from Chern number with semiclassical formulations. The solid line shows the ${\sigma }_{xy}$ obtained from Chern number and open circles indicate that from semiclassical theory with clean limit. The dashed line is $-{\rho }_{xy}^{-1}$ obtained from weak-field limit of Boltzmann theory.

Figure 3

(Color online) Chern numbers for $\varphi =1/200$ at several energy regions. The upper panels show the corresponding Fermi surface calculated at the energies indicated by vertical lines. The shaded regions are filled by electrons. The dashed lines indicate the semiclassical estimation $c_{\text{scq}}$ defined by Eq. (8). For (b) and (c), the estimations by Eq. (9) are also plotted but invisible because they trace almost same points with solid lines.