First-Principles Study of Electron Linewidths in Graphene

We present first-principles calculations of the linewidths of low-energy quasiparticles in $n$-doped graphene arising from both the electron-electron and the electron-phonon interactions. The contribution to the electron linewidth arising from the electron-electron interactions varies significantly with wave vector at fixed energy; in contrast, the electron-phonon contribution is virtually wave vector independent. These two contributions are comparable in magnitude at a binding energy of $\sim 0.2\mathrm{eV}$, corresponding to the optical phonon energy. The calculated linewidths, with both electron-electron and electron-phonon interactions included, explain to a large extent the linewidths seen in recent photoemission experiments.

Figure 1

(a)–(d) Calculated imaginary part of the electron self-energy arising from the $e\mathrm{\text{−}}e$ interaction, $\mathrm{Im}{\Sigma }_{n\mathbf{k}}^{e\mathrm{\text{−}}e}({\epsilon }_{n\mathbf{k}})$, versus the LDA energy ${\epsilon }_{n\mathbf{k}}$ (solid black lines) in $n$-doped graphene. The Dirac point energy $E_D$ is 1.0 eV below the Fermi level. The contributions to $\mathrm{Im}{\Sigma }_{n\mathbf{k}}^{e\mathrm{\text{−}}e}({\epsilon }_{n\mathbf{k}})$ coming from electronic transitions to the upper linear bands and to the lower linear bands are shown as dashed red lines and dash-dotted blue lines, respectively. The self-energy is evaluated along the reciprocal space segments shown in the insets. (a) and (c) are results for suspended graphene with a background dielectric constant of ${\epsilon }_b=1.0$, whereas (b) and (d) are results for graphene with a background dielectric constant of ${\epsilon }_b=(1+{\epsilon }_{\mathrm{SiC}})/2=3.8$. The Fermi level and $E_D$ are indicated by vertical lines. (e) Calculated plasmon energy dispersion relation ${\omega }_{00}^{\mathrm{pl}}(\mathbf{q})$, given by ${ϵ}_{\mathbf{G}=0,{\mathbf{G}}^{\prime }=0}[\mathbf{q},{\omega }_{00}^{\mathrm{pl}}(\mathbf{q})]=0$, versus $\hslash v_0|\mathbf{q}|$ along the $\Gamma M$ direction. The solid lines are guides to the eye and the dashed line corresponds to $\omega (q)=\hslash v_{0}q$.

Figure 2

Calculated Im ${\Sigma }_{n\mathbf{k}}({\epsilon }_{n\mathbf{k}})$ versus the LDA energy eigenvalue ${\epsilon }_{n\mathbf{k}}$ in $n$-doped graphene ($E_D=-1.0\mathrm{eV}$) on a model substrate (${\epsilon }_b=3.8$). The total self-energy, the self-energy arising from the $e\mathrm{\text{−}}e$ interaction, and that arising from the $e\mathrm{\text{−}}\mathrm{ph}$ interaction are shown in solid black, dashed red, and dash-dotted blue lines, respectively. The self-energy is evaluated along the reciprocal space segments shown in the insets.

Figure 3

MDC width versus binding energy in $n$-doped graphene ($E_D=-1.0\mathrm{eV}$). Calculated quantities for suspended graphene (${\epsilon }_b=1.0$) and for graphene on a model substrate (${\epsilon }_b=3.8$) are shown in dash-dotted blue and dashed red lines, respectively. The experimental result measured for sample corresponding to the highest level of doping in Fig. 3 of Ref. 19 are shown as the solid black line [54]. Both the experimental and the calculated results are along the KM and the $K\Gamma$ direction of the Brillouin zone when the electron energy is above and below $E_D$, respectively.