Tight binding in graphene (complex energy?)

In summary, the question is about calculating the band structure of graphene using tight binding, and how the energy can be a complex number. The tutor explains that this is due to the non-equivalent nature of nearest neighbors and suggests treating the basis as a C-C dimer. However, it is pointed out that the energies are actually real due to the Hermitian nature of the hamiltonian matrix. The original question is then solved with this understanding.
  • #1
Soob
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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!
 
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  • #2
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
It seems that both my tutor and I got bogged down in irrelevant details and missed the obvious - thank you so much, Dr Du!
 

1. What is tight binding in graphene?

Tight binding is a method used to describe the electronic structure of a solid material, such as graphene. It takes into account the interactions between atoms and their surrounding electrons, and is based on the principle that the wavefunctions of electrons in a solid can be approximated by a combination of atomic wavefunctions.

2. How does tight binding explain the complex energy of graphene?

Tight binding calculations can accurately predict the electronic band structure of graphene, including the complex energy states. This is because tight binding takes into account the unique hexagonal lattice structure of graphene, as well as the interactions between the π orbitals of its carbon atoms.

3. What are the advantages of using tight binding to study graphene?

Tight binding is a relatively simple and computationally efficient method for studying the electronic properties of graphene. It allows researchers to gain insight into the energy levels and band structure of this material, which is important for understanding its unique properties and potential applications in electronics and other fields.

4. Are there any limitations to the tight binding approach in graphene research?

While tight binding is a useful tool for studying graphene, it does have some limitations. For example, it may not accurately capture the effects of disorder or impurities in the material. In addition, it does not take into account the effects of electron-electron interactions, which can be important in certain situations.

5. How does tight binding compare to other methods for studying graphene?

There are several other methods that can be used to study the electronic properties of graphene, including first-principles calculations and experimental techniques. Each method has its own strengths and limitations, and the choice of method will depend on the specific research question being addressed. Tight binding is often used as a starting point for understanding graphene, and can provide a useful framework for interpreting experimental results.

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