Theory of optical transitions in graphene nanoribbons

Matrix elements of electron-light interactions for armchair and zigzag graphene nanoribbons are constructed analytically using a tight-binding model. The changes in wave number ($\Delta n$) and pseudospin are the necessary elements if we are to understand the optical selection rule. It is shown that incident light with a specific polarization and energy induces an indirect transition ($\Delta n=\pm 1$), which results in a characteristic peak in the absorption spectra. Such a peak provides evidence that the electron standing wave is formed by multiple reflections at both edges of a ribbon. It is also suggested that the absorption of low-energy light is sensitive to the position of the Fermi energy, direction of light polarization, and irregularities in the edge. The effect of depolarization on the absorption peak is briefly discussed.

Figure 1

(a) The structure of an $N$ armchair nanoribbon. The unit length along the edge is denoted by $b$ ($\equiv \sqrt{3}a$). Carbon atoms are divided into $A$ ($•$) and $B$ ($\circ$) atoms. (b) The BZ of armchair nanoribbons: $\theta \in (0,\pi )$ and $kb\in [-\pi ,\pi )$. The circle stands for the Dirac point. (c) The definition of the polar angle $\Theta (k,\theta )$. The circle represents an energy contour (1 eV). (d) Schematic diagrams for intra- and interband transitions from the initial state with ${\Theta }_n=0$ (on the $\theta$ axis).

Figure 2

(a) The calculated $\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ for an $N=15$ (semiconducting) armchair nanoribbon. The interband transition processes relevant to the absorption peaks of $\mathrm{Re}\left({\sigma }_{yy}\right)$ and $\mathrm{Re}\left({\sigma }_{xx}\right)$ are shown in (b) and (c), respectively. (d) The optically allowed indirect interband transition (from ${\phi }_n^v$ to ${\phi }_{n-1}^c$ or from ${\phi }_{n-1}^v$ to ${\phi }_n^c$). Bottom: The pseudospin (denoted by arrows) is useful for showing that only forward scattering is allowed in the interband optical transitions.

Figure 3

$\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ of an $N=55$ armchair nanoribbon with an energy gap about 0.2 eV, calculated for (a) ${\varepsilon }_{\mathrm{F}}=0$ eV and (b) ${\varepsilon }_{\mathrm{F}}=0.3$ eV.

Figure 4

(a) The structure of an $N$ zigzag nanoribbon. (b) The BZ of a zigzag nanoribbon: $\theta \in (0,\pi )$ and $ka\in [0,2\pi )$. The empty circles show the positions of the $K$ and $K$${}^{\prime }$ points. The dashed semicircles around the $K$ and $K$${}^{\prime }$ points represent an energy contour (1 eV). (c) The plot of ${\theta }_n\left(k\right)$ for an $N$ zigzag nanoribbon.

Figure 5

(a) Calculated $\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ for an $N=11$ zigzag nanoribbon. The optical transition processes relevant to (b) $\mathrm{Re}\left({\sigma }_{xx}\right)$ and (c) $\mathrm{Re}\left({\sigma }_{yy}\right)$. The flat band at the band center represents the energy dispersion of the edge states.

Figure 6

The calculated $\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ of an $N=50$ zigzag nanoribbon for different Fermi energies: (a) ${\varepsilon }_{\mathrm{F}}=0$ eV and (b) ${\varepsilon }_{\mathrm{F}}=0.3$ eV.

Figure 7

The calculated $\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ in the presence (thick lines) or absence (thin lines) of irregularities for (a) $N=15$ armchair and (c) $N=11$ zigzag nanoribbons. (b) compares $\mathrm{Re}\left({\sigma }_{\alpha \alpha }\right)$ in the presence of an irregularity consisting of two sublattices (thin lines) with that consisting of one sublattice (thick lines). In the simulation, we assume that the length of the nanoribbon is 40 nm, $\delta =50$ meV, and ${\varepsilon }_{\mathrm{F}}=0$ eV.