This paper investigates the electronic band structure of bilayer graphene in the presence of a potential difference between the layers, which opens a gap $\Delta$ between the conduction and valence bands. The study uses a tight binding model and a self-consistent Hartree approximation to describe the imperfect screening of an external gate, which controls the electron density $n$. The authors discuss how a finite asymmetry gap $\Delta(0)$ at zero excess density, caused by screening of an additional transverse electric field, affects the quantum Hall effect. They find that the addition of density $n \sim 10^{12} \text{cm}^{-2}$ yields a gap $\Delta \sim 10 \text{meV}$. The paper also provides an analytic expression for the gap $\Delta(n)$ in the limit where the interlayer coupling $\gamma_1$ and the Fermi energy $\epsilon_F$ are much larger than the asymmetry $\Delta$. The results suggest that the finite bilayer asymmetry gap $\Delta(0)$ could introduce a plateau at zero density in the Hall conductivity, which could be used to extract the value of $\Delta(0)$ from temperature dependence measurements.This paper investigates the electronic band structure of bilayer graphene in the presence of a potential difference between the layers, which opens a gap $\Delta$ between the conduction and valence bands. The study uses a tight binding model and a self-consistent Hartree approximation to describe the imperfect screening of an external gate, which controls the electron density $n$. The authors discuss how a finite asymmetry gap $\Delta(0)$ at zero excess density, caused by screening of an additional transverse electric field, affects the quantum Hall effect. They find that the addition of density $n \sim 10^{12} \text{cm}^{-2}$ yields a gap $\Delta \sim 10 \text{meV}$. The paper also provides an analytic expression for the gap $\Delta(n)$ in the limit where the interlayer coupling $\gamma_1$ and the Fermi energy $\epsilon_F$ are much larger than the asymmetry $\Delta$. The results suggest that the finite bilayer asymmetry gap $\Delta(0)$ could introduce a plateau at zero density in the Hall conductivity, which could be used to extract the value of $\Delta(0)$ from temperature dependence measurements.