Ab initio study of energy loss and wake potential in the vicinity of a graphene monolayer

A propagator of the dynamically screened Coulomb interaction in the vicinity of a graphene monolayer is calculated using ground-state Kohn-Sham orbitals, and the imaginary part of this propagator is used to calculate the energy-loss rate of a static blinking point charge due to excitation of electronic modes in graphene. Energy loss calculated for all $(Q,\omega )$ modes gives intensities of electronic excitations, including plasmon dispersions in graphene, with low-energy two-dimensional (2D) and high-energy ${\pi }_1$, ${\pi }_2$, and $\pi +\sigma$ plasmons. Plasmon energies are in good agreement with experimental results. This spectral analysis also enables us to study the contribution of each plasmon mode to the stopping power and potential induced by a point charge moving parallel to the graphene. We find the bow waves that in pristine graphene appear for higher velocities ($v\ge 2v_F$) and predominantly originate from excitation of $\pi$ plasmons. Doping induces extra features which appear for lower $v\approx v_F$ velocities and predominantly originate from the excitation of 2D or Drude plasmons.

Figure 1

Schematic representation of the graphene monolayer. Unit cell parameter in the parallel direction is $a=4.651$ a.u., in the perpendicular direction is $L=5a$, and the thickness of the electronic density is $2a$.

Figure 2

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

Figure 3

Schematic representation of independent electron response function or LDA ground-state electronic density in periodically repeated graphene layers.

Figure 4

Static blinking point charge placed at position ${\mathbf{r}}_0=(\mathbf{R},z_0)$ close to the graphene monolayer.

Figure 5

Spectrum of electronic excitations in pristine graphene.

Figure 6

Spectrum of electronic excitations in doped graphene, $E_F=1$ eV. The white line marks the frequencies and wave vectors of the modes that can be excited by a moving charge at velocity $\mathbf{v}=v\widehat{x}$.

Figure 7

Plasmon dispersion curves in a doped graphene monolayer with $E_F=1$ eV, obtained by tracking the intensity maxima in Fig. 6, represent ${\pi }_1$ plasmon (blue squares), ${\pi }_2$ plasmon (red dots), and 2D plasmon (black diamonds). Solid lines represent the corresponding linear [Eq. (24)] and square root [Eq. (25)] fits. Parameters are listed in Table . Dotted lines represent $v{Q}_x$ for $v=1,2,3,4,5v_F$. The black dashed line represents the upper edge of ${\pi }^{*}\to {\pi }^{*}$ intraband $e$-$h$ transitions.

Figure 8

Point charge moving parallel to the graphene monolayer. Red dot shows the current position of moving charge, and blue dot shows the point at which the potential is calculated.

Figure 9

Stopping power $S(v,z_0)$ for particle in parallel motion as a function of its velocity $v$ for pristine (red dots) and doped (blue squares) graphene. Doping parameter is $E_F=1$ eV and $z_0=10$ a.u. The vertical dashed line denotes velocity which corresponds to a graphene cone constant $\gamma =0.63v_F$.

Figure 10

Intensities of the potential induced by point charge moving at distance $z_0=10$ a.u. from the center of the pristine graphene monolayer ($E_F=0$) as functions of coordinates $(x-vt,y$).

Figure 11

Intensities of the potential induced by point charge moving at distance $z_0=10\mathrm{a}.\mathrm{u}.$ from the center of doped graphene monolayer ($E_F=1\mathrm{eV}$) as a function of coordinates $(x-vt,y)$.