High-energy plasmon spectroscopy of freestanding multilayer graphene

V. Borka Jovanović, I. Radović, D. Borka, and Z. L. Mišković

Phys. Rev. B 84, 155416 – Published 13 October 2011

Abstract

We present several applications of the layered electron gas model to electron energy loss spectroscopy of freestanding films consisting of $N$ graphene layers in a scanning transmission electron microscope. Using a two-fluid model for the single-layer polarizability, we discuss the evolution of high-energy plasmon spectra with $N$ and find good agreement with the recent experimental data for both multilayer graphene with $N < 10$ and thick slabs of graphite. Such applications of these analytical models help shed light on several features observed in the plasmon spectra of those structures, including the role of plasmon dispersion, dynamic interference in excitations of various plasmon eigenmodes, as well as the relevance of the bulk plasmon bands, rather than surface plasmons, in classifying the plasmon peaks.

Received 2 January 2011

DOI:https://doi.org/10.1103/PhysRevB.84.155416

Authors & Affiliations

V. Borka Jovanović, I. Radović, and D. Borka

VINČA Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia

Z. L. Mišković^*

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

^*zmiskovi@math.uwaterloo.ca

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Issue

Vol. 84, Iss. 15 — 15 October 2011

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Images

Figure 1
Probability density $P_{N} (ω)$ (in 1/eV) versus energy loss $ω$ (in eV), evaluated from Eq. (10) for $N$ = 1, 2, 5, and 13 graphene layers [smooth solid (yellow, green, blue, and pink) curves], along with the corresponding experimental EEL spectra from Ref. 2 [noisy (gray) curves].Reuse & Permissions
Figure 2
Probability density $P_{N} (ω)$ (in 1/eV) versus energy loss $ω$ (in eV), evaluated from Eq. (10) for $N$ = 1 [lower (red) curves] and 2 [upper (blue) curves] graphene layers with nonzero damping (smooth thin solid lines) and vanishing damping (dashed lines), along with the corresponding experimental data [noisy (gray) curves] and the ab initio curves (smooth thick solid lines) from Ref. 2.Reuse & Permissions
Figure 3
Dispersion curves for the $π$ (lower group) and $σ + π$ (upper group) plasmons in MLG, obtained from the LEG model with $N = 1$ [dotted (red) lines] and $N = 2$ [dashed (blue) lines], as well as $N = 5$ [only the upper group is shown by (black) solid lines]. The upper and the lower edges are shown for both the $π$ and $σ + π$ plasmon bands in HOPG [thick (orange) solid lines]. The (dark cyan) diamonds show the experimental plasmon peak positions in the EEL spectra of HOPG from Ref. 50, whereas the (black) squares show the peak positions obtained by ab initio calculations for graphite in Ref. 18.Reuse & Permissions
Figure 4
Probability density $P_{N} (ω)$ (in 1/eV) versus energy loss $ω$ (in eV) evaluated from Eq. (10) for $N$ in the range from 1 to 15 with unit steps, using the same parameters as in Fig. 1 [solid (black) curves]. Also shown are the results of ab initio calculations from Ref. 2 for graphene with $N = 1, 2$ , and 3 layers [dashed (red, blue, and green, respectively) curves].Reuse & Permissions
Figure 5
Dynamic structure factor [solid (blue) lines] is evaluated from Eq. (20) and appropriately scaled to compare it with the corresponding experimental IXS data (points) from Ref. 5 with a fixed in-plane wave number of $q =$ 0.37 Å $^{- 1}$ for the indicated values of the perpendicular wave number $k$ . Also shown are the results of ab initio calculations from Ref. 5 [dashed (red) lines].Reuse & Permissions
Figure 6
The EEL spectra versus energy loss (in eV), evaluated from $P_{\infty} (ω)$ in Eq. (13) [thick (black) solid line], along with the corresponding experimental spectrum of HOPG from Ref. 3 [the (violet) chain curve], which is theoretically decomposed into a laser-generated peak [thin (black) solid line], $π$ plasmon peak [thin (black) dash-dot line], a “surface” $σ + π$ plasmon peak [thin (black) dashed line], and a “bulk” $σ + π$ plasmon peak [thin (black) dotted line].Reuse & Permissions

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