# Solving the Secular Equation for Zigzag and Bearded Graphene Nanoribbons

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 [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].

sin(wθ)-qsin((w+1)θ)=0

sin((w+1)θ)-qsin(wθ)=0

w

θ

k=2arccos(q/2)

sin((w+1)θ)-qsin(wθ)=0

sin((w+1)θ)+qsin(wθ)=0