Importance of $\sigma$ Bonding Electrons for the Accurate Description of Electron Correlation in Graphene

Electron correlation in graphene is unique because of the interplay between the Dirac cone dispersion of $\pi$ electrons and long-range Coulomb interaction. Because of the zero density of states at Fermi level, the random phase approximation predicts no metallic screening at long distance and low energy, so one might expect that graphene should be a poorly screened system. However, empirically graphene is a weakly interacting semimetal, which leads to the question of how electron correlations take place in graphene at different length scales. We address this question by computing the equal time and dynamic structure factor $S(\mathit{q})$ and $S(\mathit{q},\omega )$ of freestanding graphene using ab initio fixed-node diffusion Monte Carlo simulations and the random phase approximation. We find that the $\sigma$ electrons contribute strongly to $S(\mathit{q},\omega )$ for relevant experimental values of $\omega$ even at distances up to around 80 Å. These findings illustrate how the emergent physics from underlying Coulomb interactions results in the observed weakly correlated semimetal.

Figure 1

Band structure of graphene, hydrogen, and the tight-binding (TB) model (with nearest-neighbor hopping $t=2.7\mathrm{eV}$). Graphene and hydrogen band structures are from DFT calculation with PBE functional.

Figure 2

$S(q)$ of different systems obtained through different methods. G, graphene; ${\mathrm{G}}_{\pi }$, $\pi$ electrons only graphene; H, hydrogen; TB, $\pi$-band tight-binding model with $1/r$ interactions; Hubbard, the Hubbard model with $U/t=1.6$.

Figure 3

Effect of $\sigma$ electrons. [(a)–(c)] Imaginary part of the response function $\chi (q,\omega )$ of G, the H system, and the TB model from either $x$-ray experiment or RPA calculations at $q=0.21,1.25$, and $2.80{\AA }^{-1}$. (d) $S(q)$ of graphene from integration of $S(q,\omega )$ with different upper frequency limit ${\omega }_c$: G(IXS)—2,000 eV; G(IXS12)—12 eV. (e) $\pi \to {\pi }^{*}$ plasmon dispersion. (f) Effective screening function from $\sigma$ electrons.

Figure 4

Joint density of states of G, H, and the TB model computed from PBE band structure and tight-binding band structure.