Figure 1

Ball and stick model of the hydrogen passivated acGNR8. The unit cell for this structure is highlighted in gray, which consists of 16 carbon and 4 hydrogen atoms. x- and y- are the longitudinal and transverse directions, respectively. Atomic visualization is done by Hückel-NV.[13]Reuse & Permissions

Figure 2

Band structure for acGNRs calculated using the pzTB model. (a) $N=2$, (b) $N=5$, (c) $N=8$, (d) $N=11$.Reuse & Permissions

Figure 3

Overlap integral $|\langle k^{\prime };1|k;\overline{1}\rangle |$ for the lowest conduction band (1) and the highest valence band ($\overline{1}$) as a function of $k$ for acGNR2 (overlaps for the linear bands in other semimetallic acGNR exhibit similar behavior). In the top panel, we present a qualitative diagram of the nature of the overlap integral. In this panel, lighter shades indicate larger overlap and darker shades indicate smaller overlap. The shading of the upper left and lower right parts of the diagram (regions where the overlap integral $\approx 1$) has been adjusted to clearly show the small variation of the overlap from 0.97–1. The shading of the upper-right and lower-left parts of the diagram (regions where the overlap integral $\ll 1$) has been adjusted to clearly show the variation of the overlap from 0–0.15. In the bottom panel, a quantitative comparison of the overlap integral $|\langle k^{\prime }=3\pi /4d|k\rangle |$ for the pzTB (solid curve) and continuum (dotted curve) models is shown. The pzTB curve depicts a section through the top panel at $k^{\prime }d/\pi =0.75$. It should be noted that the pzTB overlap is zero only for $k=k^{\prime }$, in contrast to the result for the continuum model where the overlap is zero for all $k$, $k^{\prime }$ of the same sign. Thus, in contrast to results predicted from the continuum model, for nonzero values of $q$ backscattering will be allowed. As a result, in the pzTB approximation there is a decay path for plasmons into free carriers for all $q\ne 0$.Reuse & Permissions

Figure 4

$v_{\overline{1},1,\overline{1},1}\left(q\right)$ for acGNR$N$ calculated using Eq. (11) with $w=1.08984\AA$ and $0.90\AA$. Ribbon atomic widths for each panel are: (a) $N=2$, (b) $N=5$, (c) $N=8$, and (d) $N=11$. For comparison purposes, $v_{\overline{1},1,\overline{1},1}\left(q\right)$ from Eq. (9) of Ref. 19 is shown as well for the two nanoribbons of even atomic width.Reuse & Permissions

Figure 5

Various off-diagonal $v\left(q\right)$ matrix elements for acGNR5 calculated using Eq. (11) with $w=1.08984\AA$. The magnitude of the matrix elements is shown in the top panel, and the phase of the matrix elements is shown in the bottom panel.Reuse & Permissions

Figure 6

Plasmon dispersion curves for acGNR$N$ calculated using Eq. (12) with $w=1.08984\AA$ and $0.90\AA$. Ribbon atomic widths for each panel are: (a) $N=2$, (b) $N=5$, (c) $N=8$, and (d) $N=11$. For comparison purposes, the two-band plasmon dispersion calculated at $T=0$, using Eqs. (14) and (15) of Ref. 19 is shown as well for the two nanoribbons of even atomic width. The free carrier dispersion is shown as a dotted line.Reuse & Permissions

Figure 7

Plot of the plasmon wavenumber $q$ normalized by the free-space wavenumber $q_0$ as a function of free space wavelength ${\lambda }_0$ for $p_z$ wavefunction localization parameter $w=1.08984\AA$.Reuse & Permissions

Figure 8

The group velocity $v_g$ of acGNR$N$ ($N=2,5,8,11$) with $q\to 0$ is plotted for values of the $p_z$ wavefunction localization parameter near $w=1\AA$.Reuse & Permissions

Figure 9

Schematic illustration of the difference between the intrinsic and extrinsic plasmon excitations. The linear conduction and valence bands are shown as the diagonal bold lines. Intrinsic transitions are interband transitions that occur across the intrinsic chemical potential $\mu =0$ at the Dirac point from a filled (empty) state in the valence (conduction) band to an empty (filled) state in the conduction (valence) band. Intrinsic transitions to a higher (lower) energy state contribute to the ${\Pi }_{\overline{1}1}$ (${\Pi }_{1\overline{1}}$) polarizability term. Extrinsic transitions are intraband transitions that occur across the extrinsic chemical potential $\mu ={\mu }_{\mathrm{ext}}$ from a filled (empty) state in the conduction band to an empty (filled) state in the conduction band. All extrinsic transitions contribute to the ${\Pi }_{11}$ polarizability term. Negligible differences in the magnitude of the wavefunction overlaps (the overlaps are nearly 1 in both cases) between the interband and intraband transitions account for the small group velocity differences between plasmons. Otherwise, the plasmon dispersion is identical between the two cases.Reuse & Permissions