# What is the boundary condition of graphene flake with zigzag edges?

It is a question about free carrier behavior in graphene flakes. (or may be called charge confinement)

Say if we have a perfect hexagonal free standing graphene flake terminated with zigzag edges. Since there are two subatoms (let's call them A,B) in every unit cell. Edges will be terminated, let say, ABABAB (as shown in the figure).

My question is how to calculate (understand) electron wavefunction in this system (only considering relative low energy case, charge carrier can be seen as massless fermions). Another way to ask it what type of boundary condition one shall use for those two types of edges? According to this paper(pdf), scattering at the zigzag boundary dosen't couple the two valleys. I think we can only consider one valley picture(is this true?), which means the most simple Dirac equation $-i\hbar \nu_F\nabla \Psi_i=E\Psi_i$, where $i=A,B$ (or Klein-Gordon eq). For simplicity, spin effect here can also be ignored(no magnetic field). I am not only asking about the edge states, also for higher energy states. What confusing me is how to distinguish this from the particle in a box case (relativistic particle, here I mean).

Intuitively, I think electron will behave as billiard in side of the flake. But there could be two cases, either electrons bouncing between A,B edges, or they do it only in one type of edge, such as only bouncing at edges terminated with subatom A. Can it be understood this way? If yes, which of the two picture is the real one?

Hope it is clear. Thanks.

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