Effect of the band structure topology on the minimal conductivity for bilayer graphene with symmetry breaking

Using the Kubo formula we develop a general and simple expression for the minimal conductivity in systems described by a $2\times 2$ Hamiltonian. As an application we derive an analytical expression for the minimal conductivity tensor of bilayer graphene as a function of a complex parameter $w$ related to recently proposed symmetry breaking mechanisms resulting from electron-electron interaction or strain applied to the sample. The number of Dirac points changes with varying parameter $w$, and this directly affects the minimal conductivity. Our analytic expression is confirmed using an independent calculation based on the Landauer approach, and we find remarkably good agreement between the two methods. We demonstrate that the minimal conductivity is very sensitive to the change of the parameter $w$ and the orientation of the electrodes with respect to the sample. Our results show that the minimal conductivity is closely related to the topology of the low-energy band structure.

Figure 1

(a) The topologically distinct regions of the complex $w$ plane. Inside (outside) of the triangular curve ABC given by $w_{▵}$ (red/gray solid line) there are four (two) separate Dirac cones. The dashed and solid lines in regions I and II correspond to constant parameters $\alpha$ and $\beta$, respectively. The components of the conductivity tensor (in units of ${\sigma }_0$) in the complex $w$ plane: (b) ${\sigma }_{xx}\left(w\right)$, (c) ${\sigma }_{yy}\left(w\right)$, and (d) ${\sigma }_{xy}\left(w\right)={\sigma }_{yx}\left(w\right)$.

Figure 2

The conductivity ${\sigma }_{\mathrm{Kubo}}\left(w_0\right)$ (black solid line) and by numerical evaluation of the Landauer formula (red dashed line) as functions of the parameter $w_0$, where $w=w_0{e}^{i\omega }$, and (a) $\omega ={27}^{\circ }$, $\theta =-{42}^{\circ }$ and (b) $\omega ={57}^{\circ }$, $\theta ={42}^{\circ }$. The units are the same as in Fig. 1. The dotted vertical lines indicate the values of the parameter $w_0$ where the number of Dirac points changes.