Manifestation of nonlocal electron-electron interaction in graphene

Graphene is an ideal platform to study many-body effects due to its semimetallic character and the possibility to dope it over a wide range. Here we study the width of graphene's occupied $\pi$ band as a function of doping using angle-resolved photoemission. Upon increasing electron doping, we observe the expected shift of the band to higher binding energies. However, this shift is not rigid and the bottom of the band moves less than the Dirac point. We show that the observed shift cannot be accounted for by single-particle effects and local self-energies alone, but that nonlocal many-body effects, in particular exchange interactions, must be taken into account.

Figure 1

Angle-resolved photoemission data used to determine the doping and $\pi$-band width for five different epitaxial graphene systems: oxygen-intercalated graphene on Ir(111) (GR/O/Ir) [23], hydrogen-intercalated graphene on SiC (GR/H/SiC) [24, 25, 26], graphene on Ir(111) (GR/Ir) [23, 28], graphene on SiC (GR/SiC) [17], and rubidium-doped, oxygen-intercalated graphene on Ir(111) (GR/Rb/O/Ir) [32]. (a)–(e) Dispersion near $\overline{K}$, showing how the Dirac point energy $E_D$ is obtained by extrapolation of the $\pi$ band (red dashed lines) [34]. (f)–(j) Corresponding dispersion near the bottom of the band at $\overline{\mathrm{\Gamma }}$ where the position is obtained by a simple peak fitting [34]. The inset in (a) shows the color scale used and the insets in (b) and (g) give the scan directions used for the dispersion near $E_D$ and near $E_{\pi }$, respectively.

Figure 2

(a) Schematic band structure for the $\pi$ band of graphene defining the Dirac point energy/doping level $E_D$, the band bottom energy/occupied bandwidth $E_{\pi }$, and the total $\pi$-band width $\mathrm{\Delta }\pi$. (b) Occupied bandwidth $E_{\pi }$ as a function of doping level $E_D$. The solid line is a linear fit to the data points. The dashed line is a fit with the constraint $\alpha =1$, where $\alpha$ is the slope of the line. (c) Total bandwidth $\mathrm{\Delta }\pi$ as a function of $E_D$. The red square in (c) is the result of a $GW$ calculation for charge-neutral graphene [36].

Figure 3

(a) DFT band structure of graphene at different doping levels indicated by the Dirac point energies $E_D$ relative to the Fermi level. The band structures are aligned such that $E_D$ is at $E_{\mathrm{bin}}=0$ in all cases. (b) Bandwidth as a function of the Dirac point energy considering the Coulomb interaction in the Hartree-Fock approximation (dashed line) compared to the experimental data and $\mathit{ab}$ initio calculations using the GGA (squares), the HSE06 (triangles), and the PBE0 (diamonds) functionals. All results are shifted to the same bandwidth at the charge neutrality level. Lines connecting the $\mathit{ab}$ initio results are guides to the eye.