# Tight binding in graphene (complex energy?)

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1. May 27, 2013

### Soob

Hi, I'm a 4th-year physics undergrad and I have a question about calculating the band structure of graphene using tight binding. Following the calculation here https://wiki.physics.udel.edu/phys8...rgy_quasiparticles,_Berry_phase,_and_all_that , the E(k) is a complex number, and the modulus must be taken to get its expectation value. I asked my tutor about this, since energy is a physical observable, how can it be a complex number? And he said that it's because here the nearest neighbours to any atom are not equivalent to that atom (they are on different sub-lattices) and to do the calculation properly, you'd need to treat the basis like a C-C dimer and consider the transfer integral between nearest-neighbour dimers. He also said there may be some subtle reason why it's acceptable to do the calculation like in the link above, and take the modulus of the complex energy, but he doesn't know what that reason might be.

Can anyone point me in the right direction?

I'm not sure if I should be putting this in the homework help forum, but it's not that I can't do the calculation, just that I can't understand it - apologies if this post shouldn't be here!

2. May 27, 2013

### DrDu

The energies are real, what is complex are the off-diagonal elements of the hamiltonian matrix.
However, as this matrix is hermitian, the eigenvalues are real, automatically.

3. May 27, 2013

### Soob

It seems that both my tutor and I got bogged down in irrelevant details and missed the obvious - thank you so much, Dr Du!