Kohn anomalies in graphene nanoribbons

The quantum corrections to the energies of the $\Gamma$ point optical phonon modes (Kohn anomalies) in graphene nanoribbons (NRs) are investigated. We show theoretically that the longitudinal optical (LO) modes undergo a Kohn anomaly effect, while the transverse optical (TO) modes do not. In relation to Raman spectroscopy, we show that the longitudinal optical modes are not Raman active near the zigzag edge, while the transverse optical modes are not Raman active near the armchair edge. These results are useful for identifying the orientation of the edge of graphene nanoribbons by G band Raman spectroscopy, as is demonstrated experimentally. The differences in the Kohn anomalies for nanoribbons and for metallic single-wall nanotubes are pointed out, and our results are explained in terms of pseudospin effects.

Figure 1

(a) The lattice structure of a Z-NR. The solid (empty) circles denote the $A$ $\left(B\right)$ sublattice. We use integers $I∊\left[0,M\right]$ and $J∊\left[0,N\right]$ for the axes. The width (length) $W$ $\left(L\right)$ of a Z-NR is given by $N\ell$ ($aM$ where $a\equiv \sqrt{3}{a}_{\text{cc}}$). (b) An $\left(n,n\right)$ armchair SWNT is cut along its axis and flattened out to make a $N=2n-1$ Z-NR. There is a metallic energy band in both structures [see Figs. 2a, 2b]. (c) and (d) The phonon eigenvectors of the $\Gamma$ point LO and TO modes are illustrated. The LO mode satisfies $\delta {\gamma }_{0,1}=0$ and $\delta {\gamma }_{0,2}=-\delta {\gamma }_{0,3}$. The TO mode is characterized by $\delta {\gamma }_{0,1}=-2\delta {\gamma }_{0,2}$ and $\delta {\gamma }_{0,2}=\delta {\gamma }_{0,3}$, and $\delta {\gamma }_{0,1}$, $\delta {\gamma }_{0,2}$, and $\delta {\gamma }_{0,3}$ are defined in the inset.

Figure 2

The energy band structure of (a) a $N=9$ Z-NR and (b) a (5,5) armchair SWNT. (c) $\left|V_{\text{L}}/\delta \gamma \right|$ as a function of $ka$. The matrix elements of the vertical transitions, denoted by the arrows (1,2) in (a) and (b), are shown as the solid and dashed curves, respectively.

Figure 3

(Color online) (a) The $E_{\text{F}}$ dependence of the renormalized energies of the LO and TO modes in a $N=9$ Z-NR and of the LO and TO modes in a (5,5) SWNT. (b) The $E_{\text{F}}$ dependence of the renormalized frequencies of the LO and TO modes in a $N=9$ A-NRand of the LO and TO modes in a (9,0) SWNT.

Figure 4

(a) The lattice structure of an A-NR. (b) and (c) The eigenvectors of the $\Gamma$ point LO and TO phonon modes are illustrated. The LO mode satisfies $\delta {\gamma }_{0,1}=-2\delta {\gamma }_{0,2}$ and $\delta {\gamma }_{0,2}=\delta {\gamma }_{0,3}$. The TO mode is characterized by $\delta {\gamma }_{0,1}=0$ and $\delta {\gamma }_{0,2}=-\delta {\gamma }_{0,3}$.

Figure 5

(Color online) (a) An optical image of a graphene sample. The sample is characterized as a multilayer graphene ($\sim 5$ layers) sample. (b) The position dependence of the G band frequency. From the theoretical results obtained in this paper, we conclude that the G band appearing near the upper (lower) side of the edge consists only of the TO (LO) mode, and that the upper (lower) edge is dominated by the zigzag (armchair) edge.

Figure 6

In the $k$ space, we consider a state near the $K$ point (solid circle) and states, which are scattered by the zigzag and armchair edges (empty circles). $k_x$ $\left(k_y\right)$ is taken as parallel to the zigzag (armchair) edge. The arrows in the insets indicate the direction of the pseudospins. The $⟨{\sigma }_y⟩$ component of the pseudospin is reversed at the zigzag edge, while the $⟨{\sigma }_x⟩$ component is preserved at the armchair edge.