Transport in bilayer graphene: Calculations within a self-consistent Born approximation

The transport properties of a bilayer graphene are studied theoretically within a self-consistent Born approximation. The electronic spectrum is composed of $k$-linear dispersion in the low-energy region and $k$-square dispersion as in an ordinary two-dimensional metal at high energy, leading to a crossover between different behaviors in the conductivity on changing the Fermi energy or disorder strengths. We find that the conductivity approaches $2e^2∕{\pi }^2\hslash$ per spin in the strong-disorder regime, independently of the short- or long-range disorder.

Figure 1

(Left) Top view of the atomic structure in a bilayer graphene. Solid and dashed lines represent the top ($A$ and $B$ sites) and bottom layers ($A^{\prime }$ and $B^{\prime }$ sites), respectively. (Right) Definition of the hopping parameters in a tight-binding model, ${\gamma }_0$ between nearest-neighbor sites in each layer, ${\gamma }_1$ between $A$ and $B^{\prime }$, and ${\gamma }_3$ between $B$ and $A^{\prime }$.

Figure 2

Equienergy lines (top) and the three-dimensional (3D) plot (bottom) of the energy dispersion of the bilayer graphene around the $K$ point. In the latter only the lower half $(\epsilon <0)$ is shown.

Figure 3

Density of states per spin calculated in the SCBA. Plots become identical for the short- and long-range scatterers.

Figure 4

Calculated SCBA conductivity (solid) and Boltzmann conductivity (dotted) per spin in the case of short-range disorder. Upper and lower panels show smaller and larger $W$’s, respectively. For the Boltzmann conductivity in the lower panel we used the expression (37) which is valid for $\epsilon >{\epsilon }_0$. In the lower panel the approximate result given by Eq. (42) is also shown.

Figure 5

Calculated SCBA conductivity (solid) and Boltzmann conductivity (dotted) per spin in the case of long-range disorder, with upper and lower panels showing smaller and larger $W$’s, respectively. For the Boltzmann conductivity in the lower panel we used the expression in (52) which is valid for $\epsilon >{\epsilon }_0$. In the lower panel the approximate result given by Eq. (53) is also shown.