Solving the Secular Equation for Zigzag and Bearded Graphene Nanoribbons

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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 [1]. The equation for a ribbon with bearded edges, , is different from the equation given in reference [2]. 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 [1].

Contributed by: Vasil Saroka (June 2017)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The lowest solutions of the secular equations disappear for and for in the case of a zigzag [1] and bearded [2] 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 [1].

References

[1] 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.

[2] 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.

[3] 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.



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