Optical conductivity of disordered graphene beyond the Dirac cone approximation

In this paper we systemically study the optical conductivity and density of states of disordered graphene beyond the Dirac cone approximation. The optical conductivity of graphene is computed by using the Kubo formula, within the framework of a full $\pi$-band tight-binding model. Different types of noncorrelated and correlated disorder are considered, such as random or Gaussian potentials, random or Gaussian nearest-neighbor hopping parameters, randomly distributed vacancies or their clusters, and randomly adsorbed hydrogen atoms or their clusters. For a large enough concentration of resonant impurities, an additional peak in the optical conductivity is found, associated with transitions between the midgap states and the Van Hove singularities of the main $\pi$ band. We further discuss the effect of doping on the spectrum, and find that small amounts of resonant impurities are enough to obtain a background contribution to the conductivity in the infrared part of the spectrum, in agreement with recent experiments.

Figure 1

Numerical results for the density of states (left panels) and optical conductivity (right panels) of undoped graphene with different kinds of noncorrelated disorder: (a),(b) random on-site potentials, (c),(d) random hopping parameters, (e),(f) randomly distributed vacancies, and (g),(h) randomly distributed hydrogen adatoms. Size of the samples is $4096\times 4096$ for the DOS and $8192\times 8192$ for the optical conductivity. In the right column, the insets show a zoom of the optical conductivity in the infrared region of the spectrum.

Figure 2

Numerical results for the DOS (left panels) and optical conductivity (right panels) of undoped graphene with different kinds of correlated disorder: (a),(b) Gaussian potentials, (c),(d) Gaussian hoppings, (e),(f) vacancy clusters, and (g),(h) hydrogen clusters. The distributions of the clusters of impurities used for the results (e)–(h) are sketched in Fig. 3.

Figure 3

Sketch of a graphene sheet with vacancies (left panels) or hydrogen adatoms (right panels). The vacancies are presented as missing carbon atoms, whereas the hydrogen adatoms are highlighted (in red). From top to bottom, resonant impurites are distributed as formation I ($R=0)$, II ($0⩽R⩽3a$), and III ($R=3a$) as described in the text. For illustrative purposes, the size of the sample shown in this sketch is $60\times 40$, and the concentration of impurities is approximately equal to $2\%$.

Figure 4

Contour plot of the amplitudes of quasieigenstates at energy $E=0$ or $E=0.1094t$. The radius of the resonant clusters is fixed at $R_c=5a$.

Figure 5

Simulation results for the optical conductivity of doped graphene with diffenrent kinds of noncorrelated disorders. The chemical potential is $\mu =0.1t$ in (a) and $0.2t$ in (b).

Figure 6

Optical conductivity of doped graphene with resonant scatterers. Upper panels: Fixed concentration of impurities; $\sigma \left(\omega \right)$ for different values of $\mu$. Lower panels: Fixed chemical potential $\mu$; $\sigma \left(\omega \right)$ for different concentrations of impurities.

Figure 7

As Fig. 6 but for doped graphene with on-site potential disorder distributed in Gaussian clusters.