Effective screening and the plasmaron bands in graphene

Electron-plasmon coupling in graphene has been shown recently to give rise to a “plasmaron” quasiparticle excitation. The strength of this coupling has been predicted to depend on the effective screening, which in turn is expected to depend on the dielectric environment of the graphene sheet. Here we compare the strength of environmental screening for graphene on four different substrates by evaluating the separation of the plasmaron bands from the hole bands using angle-resolved photoemission spectroscopy. Comparison with $G_{0}W$-random phase approximation predictions are used to determine the effective dielectric constant of the underlying substrate layer. We also show that plasmaron and electronic properties of graphene can be independently manipulated, an important aspect of a possible use in “plasmaronic” devices.

Figure 1

The linear band dispersion in (a) the noninteracting single-particle (bare band) description and (b) the case of interactions between elementary charges and plasmons. In (c) a linear cut of the Dirac region in (b) ($k_{\mathrm{y}}$ = 0) is shown. Hole bands (upper) are shown in red while plasmaron bands (lower) are shown in black. The Dirac energy crossing $E_{\mathrm{D}}$ in (a) is split into three crossings by the interactions with the plasmons: $E_0$, the hole band crossing, $E_2$, the plasmaron band crossing, and $E_1$, the ringlike crossing of the hole and plasmaron bands. The scaled momentum and energy widths of the diamond region in (c) are given by $\delta k=|k_{+}-k_{-}|/|k_{\mathrm{F}}|$ and $\delta E=|E_2-E_0|/|E_0-E_{\mathrm{F}}|$, respectively. In (d) a plot of the graphene Brillouin zone, showing the principle points and principle directions referenced in the paper, is presented.

Figure 2

Experimental spectral function of graphene on $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC (i), graphene on Au-SiC (ii), graphene on H-SiC (iii), and graphene on F-SiC (iv) in the $K$-$K$ direction (a) and the Fermi surface (b). The graphene on the $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC sample is $n$ doped by the substrate ($n_e\approx 1.2\times {10}^{13}$cm${}^{-2}$) while the graphene on Au-SiC ($n_h\approx 7\times {10}^{11}$cm${}^{-2}$), graphene on H-SiC ($n_h\approx 6\times {10}^{12}$cm${}^{-2}$), and graphene on F-SiC ($n_h\approx 4.5\times {10}^{13}$ cm${}^{-2}$) are all $p$ doped by the substrate.

Figure 3

Experimental spectral function of graphene on Au-SiC (i), graphene on $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC (ii), graphene on F-SiC (iii), and graphene on H-SiC (iv) in the $K$-$K$ direction (a) and the $K$-$\Gamma$ direction (b). As opposed to Fig. 2, the order is now in terms of increasing ${\alpha }_{\mathrm{G}}$. All samples are $n$ doped to $n_e\approx 6\times {10}^{13}$ cm${}^{-2}$ by the addition of K atoms to the surface. All of the plots are presented with the energy scaled to the Dirac energy ($E_{\mathrm{scaled}}=E_{\mathrm{B}}/E_0$) and the momentum scaled to the Fermi vector ($k_{\mathrm{scaled}}=k/\left|k_{\mathrm{F}}\right|$). The results of the energy line profile peak fitting to the electron hole bands and the plasmaron bands are shown in red (upper) and black (lower), respectively, while the $k/|k_{\mathrm{F}}|=0$ line profile and fitted peaks are shown in (c). The separation of the hole and plasmaron bands increases from the left-hand side to the right-hand side, and in all cases two bands (hole and plasmaron) are observed with the the hole (plasmaron) band showing greater intensity above (below) the Dirac crossings.

Figure 4

Constant energy ($E_1$) (i) and constant momentum ($k=0$) (ii) intensity profiles as a function of K doping for graphene on Au-SiC (a), graphene on $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC (b), graphene on F-SiC (c), and graphene on H-SiC (d) in the $K$-$\Gamma$ direction. All of the plots are presented with the energy scaled to the Dirac energy ($E_{\mathrm{scaled}}=E_{\mathrm{B}}/E_0$) and the momentum scaled to the Fermi vector ($k_{\mathrm{scaled}}=k/\left|k_{\mathrm{F}}\right|$). The momentum ($\delta k$) and energy ($\delta E$) separations in (i) and (ii) are constant except close to the zero doping level, where the vanishing density of states act to converge the hole and plasmaron bands.

Figure 5

Peak amplitudes and Gaussian widths from the peaks shown in Fig. 3 for graphene on Au-SiC (a), graphene on $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC (b), graphene on F-SiC (c), and graphene on H-SiC (d) in the $K$-$\Gamma$ direction. In all cases we observe a similar increase (reduction) in the amplitude (Gaussian line width) of the hole band peaks, shown in red. The reverse is true for the plasmaron band peaks, shown in black. The fitted positions from Fig. 3 clearly show that for $k_{\mathrm{y}}$ / $k_{\mathrm{F}}$ $<0$ ($k_{\mathrm{y}}$ / $k_{\mathrm{F}}$ $>0$) the positions are below (above) the Dirac crossing.

Figure 6

The energy separations $\delta E$ (black crosses) and momentum separations $\delta k$ (red plus signs) of the diamond region as a function of the graphene effective dielectric constant ${\alpha }_{\mathrm{G}}$ as determined from one-particle Green's function calculations within the random phase approximation ($G_{0}W$-RPA) (Ref. 7). The values for ${\alpha }_{\mathrm{G}}$ that correspond to the experimental widths shown in Table are indicated by dashed lines.

Figure 7

One-particle Green's function calculations within the Random Phase Approximation ($G_{0}W$-RPA ) (Ref. 7) of the spectral function of graphene in the $K$-$K$ (a) and $K$-$\Gamma$ (b) directions for the effective coupling constants (${\alpha }_{\mathrm{G}}=e^2/\epsilon \hslash v_{\mathrm{F}}\sim 2.2/\epsilon$) found experimentally. Overlayed are the experimental fitted positions from Fig. 3 for graphene on Au-SiC (i), graphene on $(6\sqrt{3}\times 6\sqrt{3})R{30}^{\circ }$ C-SiC (ii), graphene on F-SiC (iii), and graphene on H-SiC (iv). Above the lower Dirac crossing $E_2$, good agreement between the experiment and theory is observed, however, below $E_2$ the hole (upper, red) and plasmaron (lower, black) bands quickly hybridize, deviating from the experimental prediction. The intensity at high binding energy is also overestimated by the theory (see Fig. 3).