Theory of 2D Transport in Graphene for Correlated Disorder

We theoretically revisit graphene transport properties as a function of carrier density, taking into account possible correlations in the spatial distribution of the Coulomb impurity disorder in the environment. We find that the charged impurity correlations give rise to a density-dependent graphene conductivity, which agrees well qualitatively with the existing experimental data. We also find, quite unexpectedly, that the conductivity could increase with increasing impurity density if there is sufficient interimpurity correlation present in the system. In particular, the linearity (sublinearity) of graphene conductivity at lower (higher) gate voltage is naturally explained as arising solely from impurity correlation effects in the Coulomb disorder.

Figure 1

(a) Density plot of structure factor $S(\mathbf{q})$ obtained from Monte Carlo simulations for $n_i=0.95\times {10}^{12}{\mathrm{cm}}^{-2}$, $a_0=4.92\AA$, and $r_0=5a_0$. (b) Structure factor $S(\mathbf{q})$ using Eq. (2) (solid line) and Monte Carlo simulations. Dot-dashed and dashed lines show the Monte Carlo results for two different directions of $\mathbf{q}$ from the $x$ axis: $\theta =0$ and $\theta =30°$, respectively.

Figure 2

The carrier density for a single disorder realization obtained from the Thomas-Fermi-Dirac theory (a) for the uncorrelated case and (b) $r_0=10a_0$ with $n_i=0.95\times {10}^{12}{\mathrm{cm}}^{-2}$. Carrier probability distribution function $P(n)$ is shown in (c), (d), and (e) for $⟨n⟩=0$, 1.78, and $7.7\times {10}^{12}{\mathrm{cm}}^{-2}$, respectively. In (f) the ratio $n_{\mathrm{rms}}/n_i$ is shown as a function of $r_0/r_i$ for $n_i=0.95\times {10}^{12}{\mathrm{cm}}^{-2}$ (solid lines) and $n_i=4.8\times {10}^{12}{\mathrm{cm}}^{-2}$ (dashed lines). We use $⟨n⟩=7.7$, 3.14, 0.94, and $0\times {10}^{12}{\mathrm{cm}}^{-2}$ for the solid lines (from top to bottom) and $⟨n⟩=8.34$, 4.10, 1.7, and $0\times {10}^{12}{\mathrm{cm}}^{-2}$ for the dashed lines.

Figure 3

Calculated $\sigma (n)$ with $S(\mathbf{q})$ obtained from the Monte Carlo simulations (symbols) and $S(\mathbf{q})$ given by Eq. (2) (solid lines) for (a) $n_i=0.95\times {10}^{12}{\mathrm{cm}}^{-2}$ and (b) $n_i=4.8\times {10}^{12}{\mathrm{cm}}^{-2}$. (c) and (d) show the results for $\sigma (⟨n⟩)$ obtained from the effective medium theory. The value used for $n_i$ in (c) and (d) is the same as in (a) and (b), respectively. The insets in (c) and (d) show the value of ${\sigma }_{\mathrm{mim}}$ as a function of $r_0/r_i$. In (e), the resistivity $\rho$ is shown as a function of impurity density $n_i$ for different carrier densities with $r_0=5a_0$. (f) The relationship between $r_i/r_0$ and $\sqrt{n}{r}_0$ where the conductivity is minimum. The dashed line is obtained by using Eq. (5).