Splitting of the quantum Hall transition in disordered graphenes

Integer quantum Hall effect is studied for a noninteracting electron in a monolayer graphene. We calculate the Hall conductivity within a single Landau level in the presence of randomness in the bond couplings and in the on-site potential, and estimate the critical energies for the quantum Hall transition. We show that valley-degenerated ($K$ and $K^{\prime }$ points) Landau levels contain two displaced critical energies indicating that an extra Hall plateau appears inside, while the localization is found to be much stronger in the Landau level $N=0$ than in $N=1$.

Figure 1

(a) Hall conductivity per spin, (b) its difference measured from the smallest sample, and (c) the inverse of the localization length in the Landau level $N=0$ in graphenes with ${\Gamma }_s={\Gamma }_h=\Gamma ∕\sqrt{2}$ and several sizes $L$. Vertical dashed lines represent energies for ${\sigma }_{xy}∕(-e^2∕h)=\pm 1∕2$.

Figure 2

Critical energy $E_c$ plotted as a function of ${\Gamma }_h$ for $N=0$ and $N=1$. (inset) Ratio of $E_c$ to ${\Gamma }_h$ against ${\Gamma }_h$ for $N=0$.

Figure 3

Log-Log plots of the localization length against the Fermi energy measured from the critical energy, for (a) $N=0$ with ${\Gamma }_s∕{\Gamma }_h=1$ (Fig. 1) and (b) $N=0$ with ${\Gamma }_s∕{\Gamma }_h=0$ (Fig. 4). Fitting with $\nu =2.34$ is shown as straight lines. Vertical dashed lines show the energy region used to estimate the critical exponent.

Figure 4

Plots similar to Fig. 1 for the Landau level $N=1$ with ${\Gamma }_s∕{\Gamma }_h=0$.