Time-dependent density functional theory of coupled electronic lattice motion in quasi-two-dimensional crystals

Electron-holes, phonons, and plasmons come in close proximity to each other in the low-energy range of the excitation spectrum of two-dimensional (2D) crystals, breaking the validity of the weakly interacting-quasiparticles picture. By including the lattice oscillations into the scheme of time-dependent density-functional theory, we open a pathway to the ab initio treatment of the coupled low-energy excitations in 2D crystals. With the use of graphene as an important test system, we find the strong coupling of the elementary excitations, giving rise to new hybrid collective modes. The total (including both the electronic and ionic response) dielectric function ${\varepsilon }_{\text{tot}}\left(\omega \right)$ is constructed and the picture of the low-energy excitation spectrum of graphene is redrawn.

Figure 1

Left: Energy-loss function of the monolayer graphene doped with $1.2\times {10}^{14}$ cm${}^{-2}$ electrons corresponding to the Fermi energy shift up by 1 eV. Plasmon peaks with linear (acoustic) dispersion are dominant in the low-frequency range of the spectra. The variation of $\mathbf{q}$ is along the $\Gamma K$ direction. Right: Dispersion of phonons (black solid lines). From bottom to top: three acoustic phonons, $z$-polarized optical phonon, two $xy$-polarized optical phonons, and acoustic plasmon (red symbols) dispersion. The red dashed line is the linear best fit to the acoustic plasmon.

Figure 2

Real part of the total (including both the electronic and ionic response) dielectric function (red solid line) and its frozen-lattice counterpart (black dashed line). The calculation was carried out for monolayer graphene doped with $1.2\times {10}^{14}$ cm${}^{-2}$ electrons corresponding to the Fermi energy shift up by 1 eV. Vertical black arrows show positions of uncoupled phonons.

Figure 3

Same as Fig. 2, but for the imaginary part of the dielectric function.

Figure 4

Same as Fig. 2, but for the 2D energy-loss function. The strong feature that does not fit into the plot is the 2D (sheet) plasmon which practically does not couple to phonons.

Figure 5

Energy transfer per unit cell per unit time from the ionic to the electronic subsystem.

Figure 6

Ions' displacement amplitudes squared versus the frequency.

Figure 7

2D energy-loss function within a higher energy range than in Fig. 4. The 2D (sheet) plasmon dominates this part of the spectra. The dotted black line represents the frozen-lattice calculation.

Figure 8

Evolution of the acoustic plasmon in graphene with the increase of the wave vector. Top: The energy-loss function $-\text{Im}1/{\varepsilon }_{2\mathrm{D}}(q,\omega )$. Bottom: The dielectric function ${\varepsilon }_{2\mathrm{D}}(q,\omega )$. Lines labeled $1,2,3,$ and 4 correspond to $q=0.077,0.092,0.105,$ and $0.120$ a.u., respectively.

Figure 9

Experimental loss spectrum for 1 ML graphene grown on H-etched SiC(0001) for $q=0.026$ a.u. [35] (black solid line) and our theory with account of the screening by the substrate (red dashed line).

Figure 10

Total dielectric function of graphene with the full dynamic coupling of the electronic and phononic degrees of freedom (red solid line, labeled “Coupled”) and that obtained without the last term in Eq. (13) (black dashed line, labeled “Uncoupled”).