Quantum Hall effect in bilayer and multilayer graphene with finite Fermi energy

We discuss the quantum Hall effect of bilayer graphene with finite gate voltage where the Fermi energy exceeds the interlayer hopping energy. We calculated magnetic susceptibility, and diagonal and off-diagonal conductivities in finite-magnetic-field formalism. We also observed crossover of integer quantum Hall effect from two independent monolayer type systems to a strongly coupled bilayer system by changing the ratio of interlayer hopping energy and the Fermi energy. We also discuss the case of multilayer systems with Bernal stacking.

Figure 1

(Color online) Schematic of the anomalous integer quantum Hall effect (per valley and per spin) of (a) monolayer and (b) bilayer graphenes. The twice larger step at $1/B=0$ in (b) is due to the twofold degeneracy of the Landau level at zero energy. The insets show dispersion relations of both systems.

Figure 2

(Color online) Quantized Hall conductivity for $\Gamma =0$ [Eq. (15)] versus inverse magnetic field $1/B$ for (a) bilayer systems, and (b) $N$-layer systems $\left(N=1–5\right)$ with Bernal stacking and $t/\mu =0.5$.

Figure 3

(Color online) Crossover of quantum Hall effect from two independent monolayerlike regimes to strongly coupled bilayer regime, observed as changing the ratio of interlayer hopping energy $t$ and the gate voltage (Fermi energy $\mu$). The red (solid) and green (dashed) lines denote off-diagonal $\left({\sigma }_{xy}\right)$ and diagonal $\left({\sigma }_{xx}\right)$ conductivities with $\Gamma /\mu =0.01$, respectively. The blue (dotted) lines indicate ${\sigma }_{xy}$ with $\Gamma =0$.