Figure 1

Theoretical calculations for the correlation function $C\left(r\right)$ using the Thomas-Fermi approximation. This autocorrelation function depends separately on the typical distance $d$ of long-ranged impurities from the graphene sheet, and $q_{\mathrm{TF}}$ the inverse effective screening length, allowing them to be determined independently. Symbols show the correlation length $\xi$, defined as the HWHM length.Reuse & Permissions

Figure 2

Effective carrier density as a function of impurity density assuming $d=1\mathrm{nm}$, $r_s=0.8$, and $n_0=2.3\times {10}^{12}{\mathrm{cm}}^{-2}$. For bilayer graphene, the blue circles show the Thomas-Fermi approximation and the red squares are RPA results. The empirical relation $n_{\mathrm{eff}}=\sqrt{n_{\mathrm{imp}}{n}_1}$ adequately captures the RPA results, with $n_1=6.8\times {10}^{11}{\mathrm{cm}}^{-2}$.Reuse & Permissions

Figure 3

The random phase approximation polarizability function $\Pi \left(q\right)$ normalized by the density of states for monolayer graphene and for bilayer graphene with a hyperbolic dispersion. Also shown is the parabolic approximation for the bilayer, which can be obtained from the hyperbolic dispersion when $\eta \ll 1$. The Thomas-Fermi approximation discussed in the text corresponds to the assumption that the normalized $\Pi \left(q\right)=1$ for all $q$.Reuse & Permissions

Figure 4

Potential autocorrelation function $C\left[0\right]$ for monolayer and bilayer graphene. At the Dirac point, $k_F$ is the Fermi wave vector arising from the effective carrier density, i.e., $k_F=\sqrt{\pi n_{\mathrm{eff}}}$. For large density $4k_F{r}_{s}d\gg 1$, both the monolayer and bilayer results approach the “complete screening” limit, defined here as $C\left(0\right)={\left(4k_F{r}_{s}d\right)}^{-2}$. Notice that the Thomas-Fermi approximation shown as dashed lines captures the correct qualitative behavior, but can give significantly larger values for $C\left[0\right]$, and is therefore unsuitable for quantitative comparisons.Reuse & Permissions

Figure 5

Theoretical results for the puddle correlation length at the Dirac point as a function of impurity concentration. While the puddle size in bilayer graphene ($\xi \approx 3.5\mathrm{nm}$) is relatively insensitive to the disorder concentration, the size of the puddles in monolayer graphene varies from $3\mathrm{nm}$ in dirty samples to more than $35\mathrm{nm}$ in clean samples.Reuse & Permissions

Figure 6

Comparison of theoretical results with experimental data. Top panel shows the normalized correlation function $A\left(r\right)=C\left(r\right)/C\left(0\right)$ for bilayer graphene. The circles are from the experimental data and the solid curve is the theory for bilayer graphene with $d=1\mathrm{nm}$. The theory curve is insensitive to impurity concentration and doping away from the Dirac point. The error bars indicate single standard deviation uncertainties (Ref.21). The small oscillation in the data over the monotonic decrease is a result of the finite size of the experimental image. Bottom panel is the normalized puddle correlation function in monolayer graphene at the Dirac point. Note the change in $x$-axis scale from bilayer graphene in top panel. The solid curve is obtained from the self-consistent screening theory. The black squares are the results of a numerical mesoscopic density functional theory calculation for the ground-state properties of monolayer graphene (Ref.22), while the circles are experimental data taken from Deshpande et al. (Ref.12). Transport measurements on that same device set $n_{\mathrm{imp}}={10}^{11}{\mathrm{cm}}^{-2}$, which is the value used for the theory curves. The theory also uses $d=1\mathrm{nm}$, which is the typical distance of the impurities from the graphene sheet extracted from transport measurements of graphene on SiO${}_2$ (Ref.23).Reuse & Permissions