Optical conductivity of bilayer graphene with and without an asymmetry gap

When a bilayer of graphene is placed in a suitably configured field effect device, an asymmetry gap can be generated and the carrier concentration made different in each layer. This provides a tunable semiconducting gap, and the valence and conductance bands no longer meet at the two Dirac points of the graphene Brillouin zone. We calculate the optical conductivity of such a semiconductor with particular emphasis on the optical spectral weight redistribution brought about by changes in gap and chemical potential due to charging. We derive an algebraic formula for an arbitrary value of the chemical potential for the case of the bilayer conductivity without a gap.

Figure 1

(Color online) Upper frame: Frequency-dependent conductivity $\sigma \left(\Omega \right)$ of the bilayer normalized to twice that of a single graphene sheet ${\sigma }_0$ versus $\Omega ∕\gamma$ for three values of the chemical potential, as indicated, and $\Delta =0$. Middle frame: Dispersion of ${ϵ}_1$ (solid line) and ${ϵ}_2$ (dashed) of the bilayer near the $K$ point with the bands split by $\gamma$, showing the four types of transitions possible when $\mu =0$. Bottom frames: Transitions possible in the band structure for finite chemical potentials of $\mu =0.2\gamma$ (left hand side) and $1.2\gamma$ (right hand side). See text for discussion.

Figure 2

(Color online) The evolution of the positive frequency spectral weight $W$, found under each of the delta functions in Eq. (19), as function of chemical potential. The case of an uncoupled graphene bilayer is given for comparison, and the quantity $c\left(\mu \right)$ represents the spectral weight missing in the conductivity for $0<\Omega <2\mu$ relative to the $\mu =0$ case.

Figure 3

(Color online) Comparison between the full numerical evaluation of the conductivity [Eq. (16)] and the analytical formula of Eq. (19). For Eq. (19), the delta functions are broadened into Lorentzians with a scattering rate to match the effective scattering rate in the numerical work. Two different regimes of $\mu$ are shown: $\mu =0.2\gamma$ (upper frame) and $1.2\gamma$ (lower frame).

Figure 4

(Color online) Partial optical sum $I\left(\Omega \right)$ in units of $\gamma$ versus $\Omega ∕\gamma$ for various values of chemical potential, as indicated in the figure.

Figure 5

(Color online) Top frame: Band structure around the Fermi level in the presence of the asymmetry gap $\Delta$ for realistic values of $\Delta =0.6\mathrm{eV}$ and $\gamma =0.4\mathrm{eV}$. A finite chemical potential of $\mu =0.25\mathrm{eV}$ is shown. Various important energies are indicated in the figure and displayed in Table . Bottom frame: The density of states $N\left(ϵ\right)$, in units of $\gamma ∕{\hslash }^2{v}_F^2$, for the bilayer for several values of $\Delta$, as indicated in the figure, in comparison with the case of $\gamma =\Delta =0$ corresponding to the uncoupled graphene bilayer. The inset shows the partial density of states $N_1\left(ϵ\right)$ and $N_2\left(ϵ\right)$ for the two separate bands ${ϵ}_1$ and ${ϵ}_2$, respectively, for the case of $\Delta ∕\gamma =1.5$.

Figure 6

(Color online) Optical conductivity in the presence of an asymmetry gap $\Delta$. The upper frame shows the results for varying $\mu$ with fixed $\Delta =0.2\gamma$. The bottom frame shows the result for realistic values of the parameters, i.e., $\Delta =0.6\mathrm{eV}$ and $\gamma =0.4\mathrm{eV}$. Here, two curves are shown, one for $\mu =0.25\mathrm{eV}$, where the chemical potential lies above the gap in the band structure but below the hat maximum, and $\mu =0.35\mathrm{eV}$, where it lies above the hat maximum.

Figure 7

(Color online) The partial optical sum $I\left(\Omega \right)$ in units of $\gamma$ versus $\Omega ∕\gamma$ for various values of $\Delta$ with $\mu =0$. Inset: $I\left(\Omega \right)$ for the more realistic parameters used for $\sigma \left(\Omega \right)$ in Fig. 6b, i.e., $\gamma =0.4\mathrm{eV}$, $\Delta =0.6\mathrm{eV}$, and $\mu =0.25\mathrm{eV}$ (dashed red curve) and $0.35\mathrm{eV}$ (solid blue curve).