Enhanced Screening in Chemically Functionalized Graphene

Resonant scatterers such as hydrogen adatoms can strongly enhance the low-energy density of states in graphene. Here, we study the impact of these impurities on electronic screening. We find a two-faced behavior: Kubo formula calculations reveal an increased dielectric function $\epsilon$ upon creation of midgap states but no metallic divergence of the static $\epsilon$ at small momentum transfer $q\to 0$. This bad metal behavior manifests also in the dynamic polarization function and can be directly measured by means of electron energy loss spectroscopy. A new length scale $l_c$ beyond which screening is suppressed emerges, which we identify with the Anderson localization length.

Figure 1

Static dielectric function of graphene with different concentrations of impurities $n_i$ as function of wave vector $q$ at fixed chemical potential $\mu =0$. The maxima in $\epsilon (q,\omega =0)$ indicate bad metal behavior. The wave vector $\mathbf{q}$ is in the $\Gamma \mathrm{\text{−}}K$ direction. Periodic boundary conditions are used in a sample containing $4096\times 4096$ carbon atoms.

Figure 2

(a,b) Static polarization function of graphene with different concentrations of impurities $n_i$ as a function of wave vector $q$. The chemical potential is fixed as $\mu =0$. (c,d) $q$-dependent static polarization function of graphene at different chemical potentials $\mu =0$, $0.1t$, and $0.2t$. Results for graphene without impurities (solid lines) are compared to those with an impurity concentration of $n_i=0.5\%$ (dashed lines). (a,c) Results from the numerically exact Kubo formula calculations. (b,d) Results obtained by neglecting vertex corrections [Eq. (2)].

Figure 3

(a,b) Quasieigenstates at energies $E=-0.031t$ (a) and $+0.031t$ (b) for graphene with hydrogen adatoms at concentration $n_i=0.5\%$. (c) Energy dependent localization lengths in graphene with different concentrations $n_i$ of hydrogen adatoms. (d) Critical screening length $l_c=1/q_c$, Anderson localization length $\xi$ and mean free path $L_E$ as functions of average interimpurity distance $1/\sqrt{n_i}$. $l_c$ is evaluated for $\mu =0$, whereas $\xi$ and $L_E$ are evaluated for $\mu$ inside the tails of the impurity band.

Figure 4

Dynamical polarization function of graphene at chemical potential ($\mu =0$) with (a) different concentrations of impurities and the same wave vector $q=0.15a^{-1}$ as well as (b) with different wave vectors for the same concentration of impurities. (c) Electron energy loss function $-\mathrm{Im}(1/\epsilon )$ of graphene with adsorbed hydrogen impurities at concentrations between $n_i=0$ and $n_i=2\%$ with fixed $q=0.15a^{-1}$. There is an impurity-induced EELS intensity at energies $\omega <v_{F}q$, which is energetically forbidden in pristine graphene. (d) $-\mathrm{Im}(1/\epsilon )$ in the low-energy region for different momentum transfers $q$ at fixed impurity concentration $n_i=0.5\%$.