Electronic Band Structure of Armchair and Zigzag Graphene Nanoribbons

Graphene nanoribbons (GNRs) are nanometer-wide stripes of carbon atoms arranged in a honeycomb lattice. According to the geometry of their edge, in their simplest forms, they can be either armchair (AGNRs) or zigzag (ZGNRs), although more complicated GNRs with irregular edges are also possible. The width of a GNR is given by the number of dimer lines , whereas the number of carbon atoms inside a GNR unit cell is . For instance, an armchair GNR with 8 dimer lines can be denoted concisely as 8-AGNR.
This Demonstration shows the electronic structure of both armchair and zigzag graphene nanoribbons obtained by diagonalization of the tight-binding (TB) Hamiltonian matrix in the -sampled 1D Brillouin zone. The TB Hamiltonian matrix depends on the value of the nearest-neighbor hopping parameter for electrons, which is about 2.6–2.8 eV for graphene-based materials. Since AGNRs can be obtained by unzipping zigzag single-walled carbon nanotubes (ZSWCNTs), their electronic structure is very reminiscent of that of ZSWCNTs, whereas ZGNRs can be related to armchair single-walled carbon nanotubes (ASWCNTs).
In tight-binding approximation, AGNRs can be either metallic () or semiconducting ( or ), where is an integer. For instance, 5-AGNR is metallic and 7-AGNR is semiconducting. AGNRs are thus said to have an "oscillating gap". On the other hand, ZGNRs are always metallic since they have a zero-energy localized state at the Fermi level with . This Demonstration lets you explore this behavior by choosing the number of dimer lines and the appropriate geometry of the graphene nanoribbons.
However, according to the most recent literature results on the computation of electronic properties of AGNRs based on the solution of the more refined ab initio and Hubbard-model-based Hamiltonians, it has to be noted that the metallic state of AGNRs is unstable against bond deformation at the edges, electron-electron interactions, or longer-range hoppings; thus, these systems always display semiconducting behavior (see Details section for further reading).

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Snapshot 1: tight-binding electronic band structure of a metallic armchair graphene nanoribbon (8-AGNR)
Snapshot 2: tight-binding electronic band structure of a zigzag graphene nanoribbon (6-ZGNR) with points sampled in the Brillouin zone
Snapshot 3: tight-binding electronic band structure of a semiconducting armchair graphene nanoribbon (7-AGNR)
The matrix expressions are taken from [1]. For further developments beyond tight-binding electronic structure methods for GNRs, a comprehensive review can be found in [2].
References
[1] A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, and S. Roche, "Charge Transport in Disordered Graphene-Based Low Dimensional Materials," Nano Research, 1, 2008 pp. 361–394. doi:10.1007/s12274-008-8043-2.
[2] J. Alfonsi and M. Meneghetti, "Excitonic Properties of Armchair Graphene Nanoribbons from Exact Diagonalization of the Hubbard Model," New Journal of Physics, 14, 2012 053047. doi:10.1088/1367-2630/14/5/053047.
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