Edge of Brillouin zone,1D crystal, graphene, standing wave, band gap

[Spinnor]

Consider the standing wave of probability at the edge of the first Brillouin zone in a 1D crystal for an electron, first figure b below,

which gives rise to a band gap,

It is clear from the first figure b where we get the gap in energy, for an electron more likely near the ion cores, lower energy and for an electron more likely between the ion cores, higher energy.

Now consider the case of graphene at the edge of the first Brillouin zone at a Dirac point. Should we be able to come up with a graph like the first figure b above that makes clear why there is no energy gap for the standing wave associated with the Dirac points? Can the standing probability wave be shifted forward or backward in the direction of the momentum vectors associated with a Dirac points and the energy of an electron does not change? Does that seem correct (Maybe we would need to consider circular boundary conditions?)? If so I think it would be instructional. I hope I was clear enough, thanks.

Images copied from https://www.physicsforums.com/threads/energy-gap-between-energy-bands-in-solid-state-physics.953750/ which are from Kittel I think.

 

[anuttarasammyak]

Graphene is not a single-atom lattice. It has two sublattices (A and B).
At a Dirac point (K point), the wavefunction is not a simple scalar — it has two components. That makes difference. Please check this layman’s conjecture by yourself.

 

[Spinnor]

That is one of the first things you learn when you study the physics of graphene. A standing wave in graphene is a bit more complicated. I am not sure if you answered my question though,

"Can the standing probability wave be shifted forward or backward in the direction of the momentum vectors associated with a Dirac points and the energy of an electron does not change?"

There is a lot of good information on the web about graphene including the paper by Wallace, May 1st 1947, "The Band Theory of Graphite", and many good videos including,



It is slow going for me but I think I am making some progress.

Thank you.