WOLFRAM|DEMONSTRATIONS PROJECT

Solving the Secular Equation for Zigzag and Bearded Graphene Nanoribbons

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q = 2 cos(k/2)
1.385
carbon atom pairs, w
1
2
3
4
5
6
7
8
9
10
edge type
zigzag
bearded
sin((w + 1)θ)
q sin(wθ)
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
sin(wθ)-qsin((w+1)θ)=0
, while for a ribbon with bearded edges it is
sin((w+1)θ)-qsin(wθ)=0
. For ribbons containing
w
pairs of carbon atoms in their unit cells, these equations quantize the transverse momentum of the electron,
θ
, and couple it to the longitudinal momentum,
k=2arccos(q/2)
. The equation for a ribbon with zigzag edges is the same as equation (19) in [1]. The equation for a ribbon with bearded edges,
sin((w+1)θ)-qsin(wθ)=0
, is different from the equation
sin((w+1)θ)+qsin(wθ)=0
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].