Resonant Scattering by Realistic Impurities in Graphene

We develop a first-principles theory of resonant impurities in graphene and show that a broad range of typical realistic impurities leads to the characteristic sublinear dependence of the conductivity on the carrier concentration. By means of density functional calculations various organic groups as well as adatoms such as H absorbed to graphene are shown to create midgap states within $\pm 0.03\mathrm{eV}$ around the neutrality point. A low energy tight-binding description is mapped out. Boltzmann transport theory as well as a numerically exact Kubo formula approach yield the conductivity of graphene contaminated with these realistic impurities in accordance with recent experiments.

Figure 1

(a) Band structures of $4\times 4$ graphene supercells with ${\mathrm{CH}}_3$, ${\mathrm{C}}_2{\mathrm{H}}_5$, ${\mathrm{CH}}_2\mathrm{OH}$, H, and OH adsorbates and the respective adsorption geometries of the ${\mathrm{CH}}_3$, ${\mathrm{C}}_2{\mathrm{H}}_5$, ${\mathrm{CH}}_2\mathrm{OH}$ (c)–(e) groups. (b) Comparison of the supercell band structure of graphene with ${\mathrm{CH}}_3$ as obtained from DFT to the TB models with $V=2t$ and on-site energies ${ϵ}_d=-0.16\mathrm{eV}$ and ${ϵ}_d=-0.65\mathrm{eV}$.

Figure 2

Conductivity $\sigma$ in the Boltzmann approach as function of charge carrier concentration $n_e$ (in units of electrons per atom) for different impurities: Impurities with hybridization $V=2t=5.2\mathrm{eV}$ and on-site energies ${ϵ}_d=-0.26$, 0.26, and 1.7 eV in concentration $n_i=0.1\%$. (Curves are labeled by the corresponding ${ϵ}_d$.) Fits to the $V\to \infty$ limit of Eq. (3) with $n_i=0.06\%$ (dashed) as well as Eq. (4) with $q_0=0.02{\AA }^{-1}$ (dash dotted) are shown. (Here, $n_e=E_F^2/D^2$ corresponds to the clean graphene DOS.)

Figure 3

Conductivity $\sigma$ as a function of charge carrier concentration $n_e$ (in units of electrons per atom) for different resonant impurity (${\epsilon }_d=-t/16$, $V=2t$) or vacancy concentrations ($n_x$): (a) $n_i=n_x=0.1\%$, (b) 0.2%, (c) 1%, (d) 5%. Periodic boundary conditions are used with a sample containing (a) $8192\times 8192$ and (b)–(d) $4096\times 4096$ carbon atoms. The carrier concentrations $n_e$ are obtained from the integral of the corresponding DOS depicted in Fig. 4.

Figure 4

Density of states as a function of energy $E$ for different resonant impurity (${\epsilon }_d=-t/16$, $V=2t$) or vacancy concentrations: $n_i(n_x)=0.1\%$, 0.2%, 1%, 5%.