Electronic transport in graphene-based structures: An effective cross-section approach

We show that transport in low-dimensional carbon structures with finite concentrations of scatterers can be modeled by utilizing scaling theory and effective cross sections. Our results are based on large-scale numerical simulations of carbon nanotubes and graphene nanoribbons, using a tight-binding model with parameters obtained from first-principles electronic structure calculations. As shown by a comprehensive statistical analysis, the scattering cross sections can be used to estimate the conductance of a quasi-one-dimensional system both in the Ohmic and localized regimes. They can be computed with good accuracy from the transmission functions of single defects, greatly reducing the computational cost and paving the way toward using first-principles methods to evaluate the conductance of mesoscopic systems, consisting of millions of atoms.

Figure 1

Scattering cross section $\sigma \left(E\right)$ of a monovacancy in a (40,0)-CNT, computed from the transmission function of a single defect [shown in the inset together with the transmission function of the corresponding defect-free conductor $T_0\left(E\right)$] using Eq. (1). The arrows show the locations of the energy values used in the statistical analysis of Fig. 2. The energy is given with respect to the top of the valence band $E_v$.

Figure 2

(a) and (b): Length dependence of the estimated scattering cross sections for a monovacancy in (40,0)-CNTs at the two energies shown in Fig. 1. (c) and (d): Sample skewnesses of the distributions of $T$ and $\ln \left(T\right)$. The defect density is $9.6\times {10}^{-3}$ nm${}^{-2}$, and the vertical bars indicate the localization lengths based on the scattering cross section shown in Fig. 1. (e) Distribution of $T\left(E_1\right)$ at $L\approx 300$ nm, marked by the arrow in (c). (f) Distribution of $\ln \left[T\left(E_2\right)\right]$ at $L\approx 1000$ nm, marked by the arrow in (d).

Figure 3

Predicted (solid lines) and calculated (markers) localization lengths $\xi \left(E\right)$ for (40,0)-CNTs with monovacancies at three defect densities.

Figure 4

(a) Comparison between DFT and TB based conductances for a 16-AGNR with a single Stone-Wales defect. (b) Scattering cross sections for Stone-Wales defects, 555777 defects, and monovacancies.

Figure 5

Predicted and calculated conductances of a realistically sized AGNR with 100 each of Stone-Wales defects, 555777 defects, and monovacancies, as well as the conductance of a pristine conductor $T_0\left(E\right)$. In the Ohmic (localized) regimes, $T_{\mathrm{pred}}\left(E\right)$ refers to the estimate given by Eq. (5) [Eq. (6)] and $T_{\mathrm{calc}}\left(E\right)$ to the mean (typical) transmission.