Fingerprints of disorder source in graphene

We present a systematic study of the electronic, transport, and optical properties of disordered graphene, including the next-nearest-neighbor hopping. We show that this hopping has a nonnegligible effect on resonant scattering but is of minor importance for long-range disorder such as charged impurities, random potentials, or hoppings induced by strain fluctuations. Different types of disorders can be recognized by their fingerprints appearing in the profiles of dc conductivity, carrier mobility, optical spectroscopy, and Landau level spectrum. The minimum conductivity $4e^2/h$ found in the experiments is dominated by long-range disorder and the value of $4e^2/\pi h$ is due to resonant scatterers only.

Figure 1

The dc conductivity as a function of carrier density $n_e$ for disordered graphene. Left panels show the results without the NNN hopping $t^{\prime }$, and right panels with $t^{\prime }=0.1t$. For CI, we use $\kappa =6$ of hexagonal-boron nitride as a typical value of dielectric constant for graphene on a substrate. The use of other $\kappa$ for different substrate such as ${\mathrm{SiO}}_2$ does not change the results quantitatively. Here $0.01\%$ disorder corresponds to a concentration of $3.82\times {10}^{11}{\text{cm}}^{-2}$.

Figure 2

The carrier mobility as a function of carrier density $n_e$ for disorder graphene with $t^{\prime }$.

Figure 3

The optical conductivity as a function of energy for disordered graphene with ${\mu }_F=0.1t$ and $t^{\prime }=0.1t$. Here, ${\sigma }_0=\pi e^2/\left(2h\right)$ is the universal optical conductivity of graphene. All along the work the temperature of optical calculation is $T=45$ K, the same as in the experiment of Ref. [14].

Figure 4

The optical conductivity as a function of energy for disordered graphene (a) LRDP, (b) CI, and (c) RS with $t^{\prime }$. The disorder concentrations are determined via the best fit to the experimental results of optical spectroscopy [14, 15]. The chemical potential ${\mu }_F$ changes from $0.05t$ to $0.1t$. The inner panels show the corresponding carrier mobility for the same concentration of disorder. The dashed line in (c) is a guide to the eye separating two regions in which the spectrum changes differently in the presence of RS.

Figure 5

Density of states as a function of energy for disordered graphene in the presence of a uniform perpendicular magnetic field ($B=50\mathrm{T}$).

Figure 6

The reciprocal of DOS as a function of carry density $n_e$ for disordered graphene in the presence of a uniform perpendicular magnetic field ($B=50\mathrm{T}$).

Figure 7

The reciprocal of DOS as a function of carry density $n_e$ for disordered graphene in the presence of a uniform perpendicular magnetic field ($B=50\mathrm{T}$).