File:Optical selection rules of zigzag graphene nanoribbons.gif

English: This gif animation demonstrates wave function symmetry and electric dipole transition selection rules in zigzag graphene nanoribbons. It also demonstrates wave function localization at the edges of the ribbon as electron wave vector ${\displaystyle k}$ passes transition point ${\displaystyle k_{t}}$, where bulk states meet the edge states. The upper left plot shows the wave functions ${\displaystyle \Psi _{J(s)}}$ as functions of normalized transverse coordinate of the ribbon ${\displaystyle x_{i}/W}$ for the initial state and final electron energy states depicted in red and blue, respectively. The states are shown in the upper right band structure plot with the same colors. In the band structure dipole transition is shown by vertical arrow, which is crossed in the case of the transition forbidden by selection rules. The lower left plot shows the wave function product (overlapping). The lower right plot presents the optical transition matrix element ${\displaystyle M_{J_{2}(s_{2})\rightarrow J_{1}(s_{1})}}$ for longitudinal polarization of the incident light as a function of the electron wave vector for the transition depicted in the band structure plot.