Electron delocalization in bilayer graphene induced by an electric field

Electronic localization is numerically studied in disordered bilayer graphene with an electric-field-induced energy gap. Bilayer graphene is a zero-gap semiconductor, in which an energy gap can be opened and controlled by an external electric field perpendicular to the layer plane. We found that, in the smooth disorder potential not mixing the states in different valleys ($K$ and $K^{\prime }$ points), the gap opening causes a phase transition at which the electronic localization length diverges. We show that this can be interpreted as the integer quantum Hall transition at each single valley, even though the magnetic field is absent.

Figure 1

Atomic structure of the bilayer graphene. $+\Delta$ and $-\Delta$ represent the potential of the top and bottom layers, respectively.

Figure 2

Energy spectrum Eq. (6) plotted against $\Delta$ at $K$ valley in the bilayer graphene with a uniform magnetic field.

Figure 3

(a) Density of states as a function of energy in bilayer graphene with several $\Delta$’s at fixed $\Gamma$. (b) Thouless number $g$ at zero energy plotted against $\Delta$.

Figure 4

Inverse of the localization as a function of energy in bilayer graphene with several $\Delta$’s.

Figure 5

Hall conductivity of a single-valley $K$ Hamiltonian of bilayer graphene with several $\Delta$’s plotted against the energy.

Figure 6

Hall plateau diagram against $\Delta$ and energy $E$. Dots represent the critical energies estimated from the size dependence of ${\sigma }_{xy}^K$. Dashed curves are guides for the eyes for the phase boundary.