Two-dimensional and $\pi$ plasmon spectra in pristine and doped graphene

The spectra of electronic excitations in graphene are calculated using first principles time-dependent density functional theory formalism, and used to obtain $\pi$ and $\pi +\sigma$ plasmon dispersion curves. The spectra and dispersion are in excellent agreement with recent experimental results, and they are used to investigate the anisotropy and splitting of a $\pi$ plasmon, which has also been experimentally verified. The high accuracy of this calculation enabled the discovery of some different features in the spectra, especially the $\Gamma M$-$\Gamma K$ anisotropy of the two-dimensional (2D) plasmon dispersion curve, which qualitatively agrees with recent experimental results. Our ab initio 2D plasmon dispersion curves are compared with the ones obtained in some recently proposed 2D models. They show strong disagreement with the dispersion curve obtained using a simple one-band 2D theory, as well as some discrepancies with respect to the commonly used Das Sarma et al.'s dispersion curve, even in the isotropic region. Excellent agreement of the calculated spectrum in pristine graphene with the electron energy loss spectroscopy spectrum measured for lower momentum transfers is demonstrated.

Figure 1

Schematic representation of the graphene monolayer. The unit cell parameter in the parallel direction is $a=4.651$ a.u., in the perpendicular direction it is $L=5a$, and the thickness of the electronic density is taken to be $2a$.

Figure 2

Graphene Brillouin and irreducible Brillouin zone. Blue squares: $12\times 12\times 1$ Monkhorst-Pack sampling used in the LDA ground state calculations; black dots: $101\times 101\times 1$ Monkhorst-Pack sampling used in the RPA calculation. The thick lines represent the path corresponding to momentum transfers $Q$ for which the loss intensities are calculated and shown in Figs. 5 and 8.

Figure 3

Graphene band structure. The blue line shows the occupied $\pi$ band and the red line shows the unoccupied ${\pi }^{*}$ band.

Figure 4

LDA graphene electronic pseudodensity averaged over the $xy$ plane. $L$ is the thickness of one supercell. The vertical dotted lines represent positions of the image planes in the pristine graphene calculated in Ref. 32. The scale on the abscissa is the unit cell parameter in the $xy$ plane, $a=4.651\text{a.u.}$

Figure 5

Intensities of the electronic excitations in pristine graphene.

Figure 6

Spectra of electronic excitations in pristine graphene for particular wave vectors $\mathbf{Q}$ in the (a) $\Gamma K$ and (b) $\Gamma M$ directions. The minimum absolute value of the wave vectors $\mathbf{Q}$ are for (a) $\Delta Q_{\Gamma M}=0.0154\text{a.u.}$ and for (b) $\Delta Q_{\Gamma K}=0.02675\text{a.u.}$. For each higher graph the wave vector increases with the corresponding step $\Delta Q$.

Figure 7

Comparison between the spectrum of electronic excitations in pristine graphene calculated from (19) for $\mathbf{Q}=0.046\text{a.u.}$ in the $\Gamma M$ direction and the experimental results taken from Ref. 14, where $\mathbf{Q}=0.053\text{a.u.}$, also in the $\Gamma M$ direction.

Figure 8

Intensities of electronic excitations in doped graphene, $E_F=1\text{eV}$.

Figure 9

Spectra of electronic excitations in doped graphene ($E_F=1\text{eV}$) for particular wave vectors $\mathbf{Q}$ in the (a) $\Gamma K$ and (b) $\Gamma M$ directions. The corresponding absolute values of the wave vectors are the same as in Fig. 6.

Figure 10

Dispersion curve of a 2D plasmon in doped graphene. Red squares: Frequencies of the low-energy peaks in Fig. 9a; black dots: frequencies of the low-energy peaks in Fig. 9b; blue dotted-dashed line: one-band model dispersion curve (27); green dashed line: dispersion curve obtained by using a free electron response function taken from Ref. 17. The solid lines denote the borders of the intraband electron-hole region and the dashed line with a negative slope represents the lower interband electron-hole edge. The inset shows the frequencies of 2D plasmons calculated for very low momentum transfers, $Q_{\Gamma M}=0.0078\text{a.u.}$ (brown dot) and $Q_{\Gamma K}=0.0134\text{a.u.}$ (yellow square).