Solving the Secular Equation for Zigzag and Bearded Graphene Nanoribbons
This Demonstration explores the solutions of the secular equations for graphene nanoribbons with zigzag and bearded edges. For graphene nanoribbons with zigzag edges, the secular equation is , while for a ribbon with bearded edges it is . For ribbons containing pairs of carbon atoms in their unit cells, these equations quantize the transverse momentum of the electron, , and couple it to the longitudinal momentum, . The equation for a ribbon with zigzag edges is the same as equation (19) in . The equation for a ribbon with bearded edges, , is different from the equation given in reference . The sign in front of the second term is of no physical importance, but the minus is more convenient for mathematical treatment, as has been discussed for zigzag ribbons .
The lowest solutions of the secular equations disappear for and for in the case of a zigzag  and bearded  graphene nanoribbon, respectively.
Snapshot 1: solutions in light blue regions for a zigzag graphene nanoribbon
Snapshot 2: solutions in the light blue regions and one missing solution in the light green region for a zigzag graphene nanoribbon
Snapshot 3: solutions in the light blue regions for a bearded graphene nanoribbon
Snapshot 4: solutions in the light blue regions and one missing solution in the light green region for a bearded graphene nanoribbon
The missing solution restored by the analytical continuation gives rise to so-called edge states [2, 3]. The localization of the solutions within the light blue regions defines the parity of the electron wavefunction and, therefore, optical selection rules .
 V. A. Saroka, M. V. Shuba and M. E. Portnoi, "Optical Selection Rules of Zigzag Graphene Nanoribbons," Physical Review B, 95(15), 2017 155438. doi:10.1103/PhysRevB.95.155438.
 D. J. Klein, "Graphitic Polymer Strips with Edge States," Chemical Physics Letters, 217(3), 1994 pp. 261–265. doi:10.1016/0009-2614(93)E1378-T.
 M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, "Peculiar Localized State at Zigzag Graphite Edge," Journal of the Physical Society of Japan, 65(7), 1996 pp. 1920–1923. doi:10.1143/JPSJ.65.1920.