Quantum Hall effects of graphene with multi orbitals: Topological numbers, Boltzmann conductance and Semi-classical quantization

Hall conductance $σ_{xy}$ as the Chern numbers of the Berry connection in the magnetic Brillouin zone is calculated for a realistic multi band tight-band model of graphene with non-orthogonal basis. It is confirmed that the envelope of $σ_{xy}$ coincides with a semi-classical result when magnetic field is sufficiently small. The Hall resistivity $ρ_{xy}$ from the weak-field Boltzmann theory also explains the overall behaviour of the $σ_{xy}$ if the Fermi surface is composed of a single energy band. The plateaux of $σ_{xy}$ are explained from semi-classical quantization and necessary modification is proposed for the Dirac fermion regimes.

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