$D$ band Raman intensity calculation in armchair edged graphene nanoribbons

The $D$ band Raman intensity is calculated for armchair edged graphene nanoribbons using an extended tight-binding method in which the effect of interactions up to the seventh nearest neighbor is taken into account. The possibility of a double resonance Raman process with multiple scattering events is considered by calculating a $T$ matrix through a direct diagonalization of the nanoribbon Hamiltonian. We show that long-range interactions play an important role in the evaluation of both the $D$ band intensity and that the main effect of multiple scattering events on the calculated $D$ band is an overall increase in intensity by a factor of 4. The $D$ band intensity is shown to be independent of the nanoribbon widths for widths larger than 17 nm, leading to the well-known linear dependence of the $I_D/I_G$ ratio on the inverse of the crystalline size. The $D$ band intensity was shown to be nearly independent of the laser excitation energy and to have a maximum value for incident and scattering photons polarized along the direction of the edge.

Figure 1

(a) Schematics of a graphene ribbon unit cell for $N_y=5$, corresponding to $L~1.1$ nm. The filled symbols represent the atoms in the ribbon unit cell, while the open symbols represent adjacent unit cells in the $x$ direction. The box shows a 4-atom unit cell, which is used to generate the graphene ribbon. (b) and (c) show, respectively, the real- and reciprocal-space structures of graphene, highlighting the 4-atom unit cell and its corresponding Brilluoin zone.

Figure 2

Dependence of the scattering amplitude $T_{{\mathit{kk}}^{\prime }}$ for $k=0.86\pi /a$ and $k^{\prime }=-0.86\pi /a$ on the broadening factor $\epsilon$. Squares (circles) are for scattering within the valence (conduction) band. In both cases, the $T$ matrix converges for $\epsilon >0.4$ eV.

Figure 3

(a) Comparison between the $T$ matrix calculated for different nanoribbon sizes (number of atoms varying between 32 and 320) for an initial state $k=(0,-5.5/a)$ and final states in the $K\Gamma$ direction (left), and for an initial state $k=(0,4.6/a)$ and final states in the $\mathit{KM}$ direction (right). (b) Dependence of the normalized single scattering (square) and multiple (triangles) matrix elements on the nanoribbon width $L$ ($N_y$).

Figure 4

(a) Laser excitation energy dependence of the $D$ band Raman spectra for the ETB-$m$ model. Dependence of (b) the $D$ band integrated intensity and (c) $I_D/I_G$ ratio on the laser excitation energy (log-log scale). The lines are a guide to the eyes showing the expected $E_{\mathrm{laser}}^{-4}$ dependence.

Figure 5

Dependence of the calculated $D$ band intensity on the scattered light polarization (${\Theta }_e$) for (a) ${\Theta }_a=0^{°}$ and (b) ${\Theta }_a={90}^{°}$. The polarization scheme is shown for each case. The green vertical arrow in (a) (horizontal arrow in (b)) represents the direction of ${\Theta }_a$ relative to the graphene nanoribbon edge.

Figure 6

Dependence of (a) the $D$ band intensity on the nanoribbon width $N_y$ and of (b) the $I_D/I_G$ ratio on the inverse nanoribbon width ($1/N_y$) for the ETB-$s$ (squares) and ETB-$m$ (triangle) models for $E_{\mathrm{laser}}=2.0$ eV. The solid lines are a guide to the eye showing the expected $1/N_y$ dependence of the $I_D/I_G$ ratio.