Asymmetry gap in the electronic band structure of bilayer graphene

A tight-binding model is used to calculate the band structure of bilayer graphene in the presence of a potential difference between the layers that opens a gap $\Delta$ between the conduction and valence bands. In particular, a self-consistent Hartree approximation is used to describe imperfect screening of an external gate, employed primarily to control the density $n$ of electrons on the bilayer, resulting in a potential difference between the layers and a density dependent gap $\Delta \left(n\right)$. We discuss the influence of a finite asymmetry gap $\Delta \left(0\right)$ at zero excess density, caused by the screening of an additional transverse electric field, on observations of the quantum Hall effect.

Figure 1

(a) Schematic of the electronic bands near the $K$ point in the presence of finite layer asymmetry $\Delta$ (for illustrative purposes a very large asymmetry $\Delta ={\gamma }_1$ is used); (b) numerically calculated dependence $\tilde{\Delta }\left(n\right)$ (solid line) compared with analytic expression (dashed line) using Eq. (1). For these values of density $\tilde{\Delta }\approx \Delta$ (Ref. 10).

Figure 2

Numerical evaluation of (a) the bilayer asymmetry $\Delta \left(n\right)$ for different values of the bare asymmetry ${\Delta }_0=0$ (solid line), ${\Delta }_0=0.1{\gamma }_1=39\mathrm{meV}$ (dashed line), and ${\Delta }_0=0.2{\gamma }_1=78\mathrm{meV}$ (dotted line), using typical parameter values (Ref. 11); (b) layer densities $n_2$ (solid) and $n_1$ (dashed) as functions of $n$ for ${\Delta }_0=0.2{\gamma }_1=78\mathrm{meV}$; (c) $\Delta \left(0\right)$ as a function of ${\Delta }_0$ for ${\epsilon }_r=1$ (solid line) and $2$ (dashed); and (d) the cyclotron mass $m_c$ in units of the bare electronic mass $m_e$ for different values of ${\Delta }_0$ as in (a).

Figure 3

Landau levels for zero (left) and finite asymmetry $\Delta$ (right). Brackets $(N,\xi )$ indicate LL number $N$ and valley index $\xi =\pm 1$. In the center the predicted Hall conductivity ${\sigma }_{xy}$ as a function of carrier density for bare asymmetry ${\Delta }_0=0$ (dashed line) is compared to that for ${\Delta }_0\ne 0$ (solid line).